Lagrangian quantum field theory in momentum picture. IV. Commutation
  relations for free fields
Lagrangian quantum field theory
in momentum picture
IV. Commutation relations for free fields
Bozhidar Z. Iliev
∗ † ‡
Short title: QFT in momentum picture: IV
Produced: → November 4, 2018
http://www.arXiv.org e-Print archive No. : arXiv:0704.0066[hep-th]
R© TM
Subject Classes:
Quantum field theory
2000 MSC numbers:
81Q99, 81S05, 81T99
2003 PACS numbers:
03.70.+k, 11.10.Ef, 11.10.-z,
11.90.+t, 12.90.+b
Key-Words:
Commutation relations, Anticommutation relations, Free quantum fields
Paracommutation relations, Parafermi and parabose commutation relations
Heisenberg relations (equations), Euler-Lagrange equations, Equations of motion
Lagrangians for free fields, Momentum operator, Angular momentum operator
Spin and orbital angular momentum operators, Normal ordering
∗Laboratory of Mathematical Modeling in Physics, Institute for Nuclear Research and Nuclear
Energy, Bulgarian Academy of Sciences, Boul. Tzarigradsko chaussée 72, 1784 Sofia, Bulgaria
†E-mail address: bozho@inrne.bas.bg
‡URL: http://theo.inrne.bas.bg/∼bozho/
http://arxiv.org/abs/0704.0066v1
http://www.arXiv.org
http://arxiv.org/abs/0704.0066
http://theo.inrne.bas.bg/~bozho/
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations i
Contents
1 Introduction 1
2 The momentum picture 3
3 Lagrangians, Euler-Lagrange equations and dynamical variables 5
4 On the uniqueness of the dynamical variables 10
5 Heisenberg relations 14
6 Types of possible commutation relations 20
6.1 Restrictions related to the momentum operator . . . . . . . . . . . . . . . . . 21
6.2 Restrictions related to the charge operator . . . . . . . . . . . . . . . . . . . . 25
6.3 Restrictions related to the angular momentum operator(s) . . . . . . . . . . . 27
7 Inferences 31
8 State vectors, vacuum and mean values 37
9 Commutation relations for several coexisting different free fields 43
9.1 Commutation relations connected with the momentum operator. Problems
and their possible solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
9.2 Commutation relations connected with the charge and angular momentum
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
9.3 Commutation relations between the dynamical variables . . . . . . . . . . . . 50
9.4 Commutation relations under the uniqueness conditions . . . . . . . . . . . . 52
10 Conclusion 54
References 55
This article ends at page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Abstract
Possible (algebraic) commutation relations in the Lagrangian quantum theory of free
(scalar, spinor and vector) fields are considered from mathematical view-point. As sources of
these relations are employed the Heisenberg equations/relations for the dynamical variables
and a specific condition for uniqueness of the operators of the dynamical variables (with
respect to some class of Lagrangians). The paracommutation relations or some their gen-
eralizations are pointed as the most general ones that entail the validity of all Heisenberg
equations. The simultaneous fulfillment of the Heisenberg equations and the uniqueness re-
quirement turn to be impossible. This problem is solved via a redefinition of the dynamical
variables, similar to the normal ordering procedure and containing it as a special case. That
implies corresponding changes in the admissible commutation relations. The introduction of
the concept of the vacuum makes narrow the class of the possible commutation relations;
in particular, the mentioned redefinition of the dynamical variables is reduced to normal
ordering. As a last restriction on that class is imposed the requirement for existing of an
effective procedure for calculating vacuum mean values. The standard bilinear commutation
relations are pointed as the only known ones that satisfy all of the mentioned conditions and
do not contradict to the existing data.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 1
1. Introduction
The main subject of this paper is an analysis of possible (algebraic) commutation relations
in the Lagrangian quantum theory1 of free fields. These relations are considered only from
mathematical view-point and physical consequence of them, like the statistics of many-par-
ticle systems, are not investigated.
The canonical quantization method finds its origin in the classical Hamiltonian mechan-
ics [9, 10] and naturally leads to the canonical (anti)commutation relations [3, 11,12]. These
relations can be obtained from different assumptions (see, e.g., [1,13–15]) and are one of the
basic corner stones of the present-day quantum field theory.
Theoretically there are possible also non-canonical commutation relations. The best
known example of them being the so-called paracommutation relations [16–18]. But, however,
it seems no one of the presently known particles/fields obeys them.
In the present work is shown how different classes of commutation relations, understood
in a broad sense as algebraic connections between creation and/or annihilation operators,
arise from the Lagrangian formalism, when applied to three types of Lagrangians describing
free scalar, spinor and vector fields. Their origin is twofold. One one hand, a requirement for
uniqueness of the dynamical variables (that can be calculated from Lagrangians leading to
identical Euler-Lagrange equation) entails a number of specific commutation relations. On
another hand, any one of the so-called Heisenberg relations/equations [3, 11], implies cor-
responding commutation relations; for example, the paracommutation relations arise from
the Heisenberg equations regarding the momentum operator, when ‘charge symmetric’ La-
grangian is employed.2 The combination of the both methods leads to strong, generally
incompatible, restrictions on the admissible types of commutation relations.
The introduction of the concept of vacuum, combined with the mentioned uniqueness of
the operators of the dynamical variables, changes the situation and requires a redefinition
of these operators in a way similar to the one known as the normal ordering [1, 3, 11, 12],
which is its special case. Some natural assumptions reduce the former to the letter one; in
particular, in that way are excluded the paracommutation relations. However, this does not
reduce the possible commutation relations to the canonical ones. Further, the requirement
to be available an effective procedure for calculating vacuum mean (expectation) values, to
which reduce all predictable results in the theory, puts new restriction, whose only realistic
solution at the time being seems to be the standard canonical (anti)commutation relations.
The layout of the work is as follows.
Sect. 2 gives an idea of the momentum picture of motion and discusses the relations
between the creation and annihilation operators in it and in Heisenberg picture. In Sect. 3
are reviewed some basic results from [13–15], part of which can be found also in papers like [1,
3, 11, 12]. In particular, the explicit expression of the dynamical variables via the creation
1 In this paper we considered only the Lagrangian (canonical) quantum field theory in which the quantum
fields are represented as operators, called field operators, acting on some Hilbert space, which in general
is unknown if interacting fields are studied. These operators are supposed to satisfy some equations of
motion, from them are constructed conserved quantities satisfying conservation laws, etc. From the view-point
of present-day quantum field theory, this approach is only a preliminary stage for more or less rigorous
formulation of the theory in which the fields are represented via operator-valued distributions, a fact required
even for description of free fields. Moreover, in non-perturbative directions, like constructive and conformal
field theories, the main objects are the vacuum mean (expectation) values of the fields and from these are
reconstructed the Hilbert space of states and the acting on it fields. Regardless of these facts, the Lagrangian
(canonical) quantum field theory is an inherent component of the most of the ways of presentation of quantum
field theory adopted explicitly or implicitly in books like [1–8]. Besides, the Lagrangian approach is a source
of many ideas for other directions of research, like the axiomatic quantum field theory [3,7,8].
2 Ordinary [3,11], the commutation relations are postulated and the validity of the Heisenberg relations is
then verified. We follow the opposite method by postulating the Heisenberg equations and, then, looking for
commutation relations that are compatible with them.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 2
and annihilation operators are presented (without assuming some commutation relations or
normal ordering) and it is pointed to the existence of a family of such variables for a given
system of Euler-Lagrange equations for free fields. The last fact is analyzed in Sect. 4,
where a number of its consequences, having a sense of commutation relations, are drawn.
The Heisenberg relations and the commutation relations between the dynamical variables
are reviewed and analyzed in Sect. 5. It is pointed that the letter should be consequences
from the former ones. Arguments are presented that the Heisenberg equation concerning
the angular momentum operator should be split into two independent ones, representing its
‘orbital’ and ‘spin’ parts, respectively.
Sect. 6 contains a method for assigning commutation relations to the Heisenberg equa-
tions. It is shown that the Heisenberg equation involving the ‘orbital’ part of the angular
momentum gives rise to a differential, not algebraic, commutation relation and the one con-
cerning the ‘spin’ part of the angular momentum implies a complicated integro-differential
connections between the creation and annihilation operators. Special attention is paid to
the paracommutation relations, whose particular kind are the ordinary ones, which ensure
the validity of the Heisenberg equations concerning the momentum operator. Partially is
analyzed the problem for compatibility of the different types of commutation relations de-
rived. It is proved that some generalization of the paracommutation relations ensures the
fulfillment of all of the Heisenberg relations.
Sect. 7 is devoted to consequences from the commutation relations derived in Sect. 6
under the conditions for uniqueness of the dynamical variables presented in Sect. 4. Gen-
erally, these requirements are incompatible with the commutation relations. To overcome
the problem, it is proposed a redefinition of the dynamical variables via a method similar to
(and generalizing) the normal ordering. This, of course, entails changes in the commutation
relations, the new versions of which happen to be compatible with the uniqueness conditions
and ensure the validity of the Heisenberg relations.
The concept of the vacuum is introduced in Sect. 8. It reduces (practically) the redefini-
tion of the operators of the dynamical variables to the one obtained via the normal ordering
procedure in the ordinary quantum field theory, but, without additional suppositions, does
not reduce the commutation relations to the standard bilinear ones. As a last step in specify-
ing the commutation relations as much as possible, we introduce the requirement the theory
to supply an effective way for calculating vacuum mean values of (anti-normally ordered)
products of creation and annihilation operators to which are reduced all predictable results,
in particular the mean values of the dynamical variables. The standard bilinear commutation
relation seems to be the only ones know at present that survive that last condition, however
their uniqueness in this respect is not investigated.
Sect. 9 deals with the same problems as described above but for systems containing at
least two different quantum fields. The main obstacle is the establishment of commutation
relations between creation/annihilation operators concerning different fields. Argument is
presented that they should contain commutators or anticommutators of these operators.
The major of corresponding commutation relations are explicitly written and the results
obtained turn to be similar to the ones just described, only in ‘multifield’ version.
Section 10 closes the paper by summarizing its main results.
The books [1–3] will be used as standard reference works on quantum field theory. Of
course, this is more or less a random selection between the great number of (text)books and
papers on the theme to which the reader is referred for more details or other points of view.
For this end, e.g., [4, 12,19] or the literature cited in [1–4,12,19] may be helpful.
Throughout this paper ~ denotes the Planck’s constant (divided by 2π), c is the velocity of
light in vacuum, and i stands for the imaginary unit. The superscripts † and ⊤ mean respec-
tively Hermitian conjugation and transposition (of operators or matrices), the superscript ∗
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 3
denotes complex conjugation, and the symbol ◦ denotes compositions of mappings/operators.
By δfg, or δ
f or δ
fg (:= 1 for f = g, := 0 for f = g) is denoted the Kronecker δ-symbol,
depending on arguments f and g, and δn(y), y ∈ Rn, stands for the n-dimensional Dirac
δ-function; δ(y) := δ1(y) for y ∈ R.
The Minkowski spacetime is denoted by M . The Greek indices run from 0 to dimM −1 =
3. All Greek indices will be raised and lowered by means of the standard 4-dimensional
Lorentz metric tensor ηµν and its inverse ηµν with signature (+ − −−). The Latin indices
a, b, . . . run from 1 to dimM − 1 = 3 and, usually, label the spacial components of some
object. The Einstein’s summation convention over indices repeated on different levels is
assumed over the whole range of their values.
At last, we ought to give an explanation why this work appears under the general title
“Lagrangian quantum field theory in momentum picture” when in it all considerations are
done, in fact, in Heisenberg picture with possible, but not necessary, usage of the creation and
annihilation operators in momentum picture. First of all, we essentially employ the obtained
in [13–15] expressions for the dynamical variables in momentum picture for three types of
Lagrangians. The corresponding operators in Heisenberg picture, which in fact is used in this
paper, can be obtained via a direct calculation, as it is partially done in, e.g., [1] for one of
the mentioned types of Lagrangians. The important point here is that in Heisenberg picture
it suffice to be used only the standard Lagrangian formalism, while in momentum picture one
has to suppose the commutativity between the components of the momentum operator and
the validity of the Heisenberg relations for it (see below equations (2.6) and (2.7)). Since for
the analysis of the commutation relations we intend to do the fulfillment of these relations is
not necessary (they are subsidiary restrictions on the Lagrangian formalism), the Heisenberg
picture of motion is the natural one that has to be used. For this reason, the expression for the
dynamical variables obtained in [13–15] will be used simply as their Heisenberg counterparts,
but expressed via the creation and annihilation operators in momentum picture. The only
real advantage one gets in this way is the more natural structure of the orbital angular
momentum operator. As the commutation relations considered below are algebraic ones, it
is inessential in what picture of motion they are written or investigated.
2. The momentum picture
Since the momentum picture of motion will be used only partially in this work, below is
presented only its definition and the connection between the creation/annihilation operators
in it and in Heisenberg picture. Details concerning the momentum picture can be found
in [20,21] and in the corresponding sections devoted to it in [13–15].
Let us consider a system of quantum fields, represented in Heisenberg picture of motion
by field operators ϕ̃i(x) : F → F , i = 1, . . . , n ∈ N, acting on the system’s Hilbert space
F of states and depending on a point x in Minkowski spacetime M . Here and henceforth,
all quantities in Heisenberg picture will be marked by a tilde (wave) “˜” over their kernel
symbols. Let P̃µ denotes the system’s (canonical) momentum vectorial operator, defined via
the energy-momentum tensorial operator T̃ µν of the system, viz.
P̃µ :=
x0=const
T̃0µ(x) d3x. (2.1)
Since this operator is Hermitian, P̃†µ = P̃µ, the operator
U(x, x0) = exp
(xµ − xµ0 ) P̃µ
, (2.2)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 4
where x0 ∈ M is arbitrarily fixed and x ∈ M ,3 is unitary, i.e. U†(x0, x) := (U(x, x0))† =
U−1(x, x0) := (U(x, x0))−1 and, via the formulae
X̃ 7→ X (x) = U(x, x0)( X̃ ) (2.3)
Ã(x) 7→ A(x) = U(x, x0) ◦ ( Ã(x)) ◦ U−1(x, x0), (2.4)
realizes the transition to the momentum picture. Here X̃ is a state vector in system’s Hilbert
space of states F and Ã(x) : F → F is (observable or not) operator-valued function of x ∈ M
which, in particular, can be polynomial or convergent power series in the field operators
ϕ̃i(x); respectively X (x) and A(x) are the corresponding quantities in momentum picture.
In particular, the field operators transform as
ϕ̃i(x) 7→ ϕi(x) = U(x, x0) ◦ ϕ̃i(x) ◦ U−1(x, x0). (2.5)
Notice, in (2.2) the multiplier (xµ − xµ0 ) is regarded as a real parameter (in which P̃µ is
linear). Generally, X (x) and A(x) depend also on the point x0 and, to be quite correct, one
should write X (x, x0) and A(x, x0) for X (x) and A(x), respectively. However, in the most
situations in the present work, this dependence is not essential or, in fact, is not presented
at all. For that reason, we shall not indicate it explicitly.
The momentum picture is most suitable in quantum field theories in which the compo-
nents P̃µ of the momentum operator commute between themselves and satisfy the Heisenberg
relations/equations with the field operators, i.e. when P̃µ and ϕ̃i(x) satisfy the relations:
[ P̃µ, P̃ν ] = 0 (2.6)
[ ϕ̃i(x), P̃µ] = i~∂µ ϕ̃i(x). (2.7)
Here [A,B]± := A ◦ B ± B ◦ A, ◦ being the composition of mappings sign, is the commuta-
tor/anticommutator of operators (or matrices) A and B.
However, the fulfillment of the relations (2.6) and (2.7) will not be supposed in this paper
until Sect. 6 (see also Sect. 5).
Let a±s (k) and a
s (k) be the creation/annihilation operators of some free particular field
(see Sect. 3 below for a detailed explanation of the notation). We have the connections
ã±s (k) = e
xµkµ U−1(x, x0) ◦ a±s (k) ◦ U(x, x0)
ã†±s (k) = e
xµkµ U−1(x, x0) ◦ a†±s (k) ◦ U(x, x0)
m2c2 + k2 (2.8)
whose explicit form is
ã±s (k) = e
kµa±s (k)
ã†±s (k) = e
kµa†±s (k)
m2c2 + k2. (2.9)
Further it will be assumed ã±s (k) and ã
s (k) to be defined in Heisenberg picture, indepen-
dently of a±s (k) and a
s (k), by means of the standard Lagrangian formalism. What concerns
the operators a±s (k) and a
s (k), we shall regard them as defined via (2.9); this makes them
independent from the momentum picture of motion. The fact that the so-defined operators
a±s (k) and a
s (k) coincide with the creation/annihilation operators in momentum picture
(under the conditions (2.6) and (2.7)) will be inessential in the almost whole text.
3 The notation x0, for a fixed point in M , should not be confused with the zeroth covariant coordinate
µ of x which, following the convention xν := ηνµx
µ, is denoted by the same symbol x0. From the context,
it will always be clear whether x0 refers to a point in M or to the zeroth covariant coordinate of a point
x ∈ M .
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 5
3. Lagrangians, Euler-Lagrange equations
and dynamical variables
In [13–15] we have investigated the Lagrangian quantum field theory of respectively scalar,
spin 1
and vector free fields. The main Lagrangians from which it was derived are respectively
(see loc. cit. or, e.g. [1, 3, 11,12]):
L̃′sc = L̃′sc( ϕ̃, ϕ̃†) =−
1 + τ( ϕ̃)
m2c4 ϕ̃(x) ◦ ϕ̃†(x) + 1
1 + τ( ϕ̃)
c2~2(∂µ ϕ̃(x)) ◦ (∂µ ϕ̃†(x))
(3.1a)
L̃′sp = L̃′sp( ψ̃,
ψ) =− 1
i~c{ ˜̆ψ
(x)C−1γµ ◦ (∂µ ψ̃(x))
− (∂µ ˜̆ψ
(x))C−1γµ ◦ ψ̃(x)}+mc2 ˜̆ψ
(x)C−1 ◦ ψ̃(x)
(3.1b)
L̃′v = L̃′v( Ũ , Ũ†) =
1 + τ( Ũ)
Ũ†µ ◦ Ũ
1 + τ( Ũ)
−(∂µ Ũ
ν) ◦ (∂
µ Ũν) + (∂µ Ũ
) ◦ (∂ν Ũ
(3.1c)
Here it is used the following notation: ϕ̃(x) is a scalar field, a tilde (wave) over a symbol
means that it is in Heisenberg picture, the dagger † denotes Hermitian conjugation, ψ̃ :=
( ψ̃0, ψ̃1, ψ̃2, ψ̃3) is a 4-spinor field,
ψ := C ψ̃
:= C( ψ̃†γ0) is its charge conjugate with γµ
being the Dirac gamma matrices and the matrix C satisfies the equations C−1γµC = −γµ
and C⊤ = −C, Uµ is a vector field, m is the field’s mass (parameter) and the function
τ(A) :=
1 for A† = A (Hermitian operator)
0 for A† 6= A (non-Hermitian operator)
, (3.2)
with A : F → F being an operator on the systems Hilbert space F of states, takes care of
is the field charged (non-Hermitian) or neutral (Hermitian, uncharged). Since a spinor field
is a charged one, we have τ( ψ̃) = 0; sometimes below the number 0 = τ( ψ̃) will be written
explicitly for unification of the notation.
We have explored also the consequences from the ‘charge conjugate’ Lagrangians
L̃′′sc = L̃′′sc( ϕ̃, ϕ̃†) := L̃′sc( ϕ̃†, ϕ̃) (3.3a)
L̃′′sp = L̃′′sp( ψ̃,
ψ) := L̃′sp(
ψ, ψ̃) (3.3b)
L̃′′v = L̃′′v( Ũ , Ũ†) := L̃′v( Ũ†, Ũ), (3.3c)
as well as from the ‘charge symmetric’ Lagrangians
L̃′′′sc = L̃′′′sc( ϕ̃, ϕ̃†) :=
L̃′sc + L̃′′sc
L̃′sc( ϕ̃, ϕ̃†) + L̃′sc( ϕ̃†, ϕ̃)
(3.4a)
L̃′′′sp = L̃′′′sp( ψ̃,
ψ) :=
L̃′sp + L̃′′sp
L̃′sp( ψ̃,
ψ) + L̃′sp(
ψ, ψ̃)
(3.4b)
L̃′′′v = L̃′′′v ( Ũ , Ũ†) :=
L̃′v + L̃′′v
L̃′v( Ũ , Ũ†) + L̃′v( Ũ†, Ũ)
. (3.4c)
It is essential to be noted, for a massless, m = 0, vector field to the Lagrangian formalism
are added as subsidiary conditions the Lorenz conditions
∂µ Ũµ = 0 ∂
µ Ũ†µ = 0 (3.5)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 6
on the solutions of the corresponding Euler-Lagrange equations. Besides, if the opposite is
not stated explicitly, no other restrictions, like the (anti)commutation relations, are supposed
to be imposed on the above Lagrangians. And a technical remark, for convenience, the fields
ϕ̃, ψ̃ and Ũ and their charge conjugate ϕ̃†, ˜̆ψ and Ũ†, respectively, are considered as
independent field variables.
Let L̃′ denotes any one of the Lagrangians (3.1) and L̃′′ (resp. L̃′′′) the corresponding
to it Lagrangian given via (3.3) (resp. (3.4)). Physically the difference between L̃′ and L̃′′ is
that the particles for L̃′ are antiparticles for L̃′′ and vice versa. Both of the Lagrangians L̃′
and L̃′′ are not charge symmetric, i.e. the arising from them theories are not invariant under
the change particle↔antiparticle (or, in mathematical terms, under some of the changes
ϕ̃ ↔ ϕ̃†, ψ̃ ↔ ˜̆ψ, Ũ ↔ Ũ†) unless some additional hypotheses are made. Contrary to
this, the Lagrangian L̃′′′ is charge symmetric and, consequently, the formalism on its base
is invariant under the change particle↔antiparticle.4
The Euler-Lagrange equations for the Lagrangians L̃′, L̃′′ and L̃′′′ happen to coin-
cide [13–15]:5
∂ L̃′
( ∂ L̃′
∂(∂µχ)
≡ ∂ L̃
( ∂ L̃′′
∂(∂µχ)
≡ ∂ L̃
( ∂ L̃′′′
∂(∂µχ)
= 0, (3.6)
where χ = ϕ̃, ϕ̃†, ψ̃,
ψ, Ũ , Ũ† for respectively scalar, spinor and vector field.
Since the creation and annihilation operators are defined only on the base of Euler-La-
grange equations [1, 3, 11–15], we can assert that these operators are identical for the La-
grangians L̃′, L̃′′ and L̃′′′. We shall denote these operators by a±s (k) and a
s (k) with the
convention that a+s (k) (resp. a
s (k)) creates a particle (resp. antiparticle) with 4-momen-
tum (
m2c2 + k2,k), polarization s (see below) and charge (−q) (resp. (+q))6 and a†−s (k)
(resp. a−s (k)) annihilates/destroys such a particle (resp. antiparticle). Here and henceforth
k ∈ R3 is interpreted as (anti)particle’s 3-momentum and the values of the polarization index
s depend on the field considered: s = 1 for a scalar field, s = 1 or s = 1, 2 for respectively
massless (m = 0) or massive (m 6= 0) spinor field, and s = 1, 2, 3 for a vector field.7 Since
massless vector field’s modes with s = 3 may enter only in the spin and orbital angular mo-
menta operators [15], we, for convenience, shall assume that the polarization indices s, t, . . .
take the values from 1 to 2j+1− δ0m(1− δ0j), where j = 0, 12 , 1 is the spin for scalar, spinor
and vector field, respectively, and δ0m := 1 for m = 0 and δ0m := 0 for m 6= 0;8 if the value
s = 3 is important when j = 1 and m = 0, it will be commented/considered separately. Of
course, the creation and annihilation operators are different for different fields; one should
write, e.g., ja
(k) for a±s (k), but we shall not use such a complicated notation and will
assume the dependence on j to be an implicit one.
4 Besides, under the same assumptions, the Lagrangian L̃′′′ does not admit quantization via anticommu-
tators (commutators) for integer (half-integer) spin field, while L̃′ and L̃′′ do not make difference between
integer and half-integer spin fields.
5 Rigorously speaking, the Euler-Lagrange equations for the Lagrangian (3.4b) are identities like 0 = 0 —
see [22]. However, bellow we shall handle this exceptional case as pointed in [14].
6 For a neutral field, we put q = 0.
7 For convenience, in [14], we have set s = 0 if m = 0 and s = 1, 2 if m 6= 0 for a spinor field. For a massless
vector field, one may set s = 1, 2, thus eliminating the ‘unphysical’ value s = 3 for m = 0 — see [1, 11, 15].
In [13], for a scalar field, the notation ϕ±
(k) and ϕ
(k) is used for a±
(k) and a
(k), respectively.
8 In this way the case (j, s,m) = (1, 3, 0) is excluded from further considerations; if (j,m) = (1, 0) and
q = 0, the case considered further in this work corresponds to an electromagnetic field in Coulomb gauge,
as the modes with s = 3 are excluded [15]. However, if the case (j, s,m) = (1, 3, 0) is important for some
reasons, the reader can easily obtain the corresponding results by applying the ones from [15].
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 7
The following settings will be frequently used throughout this chapter:
0 for scalar field
for spinor field
1 for vector field
1 for q = 0 (neutral (Hermitian) field)
0 for q 6= 0 (charged (non-Hermitian) field)
ε := (−1)2j =
+1 for integer j (bose fields)
−1 for half-integer j (fermi fields)
(3.7)
[A,B]± := [A,B]±1 := A ◦B ±B ◦A, (3.8)
where A and B are operators on the system’s Hilbert space F of states.
The dynamical variables corresponding to L̃′, L̃′′ and L̃′′′ are, however, completely dif-
ferent, unless some additional conditions are imposed on the Lagrangian formalism [13–15].
In particular, the momentum operators P̃ωµ , charge operators Q̃ω, spin operators S̃ωµν and
orbital operators L̃ωµν , where ω = ′, ′′, ′′′, for these Lagrangians are [13–15]:
P̃ ′µ =
1 + τ
2j+1−δ0m(1−δ0j )
d3kkµ|
m2c2+k2
{a†+s (k) ◦ a−s (k) + εa†−s (k) ◦ a+s (k)}
(3.9a)
P̃ ′′µ =
1 + τ
2j+1−δ0m(1−δ0j )
d3kkµ|
m2c2+k2
{a+s (k) ◦ a†−s (k) + εa−s (k) ◦ a†+s (k)}
(3.9b)
P̃ ′′′µ =
2(1 + τ)
2j+1−δ0m(1−δ0j )
d3kkµ|
m2c2+k2
{[a†+s (k), a−s (k)]ε + [a+s (k), a†−s (k)]ε}
(3.9c)
Q̃′ = +q
2j+1−δ0m(1−δ0j)
d3k{a†+s (k) ◦ a−s (k)− εa†−s (k) ◦ a+s (k)} (3.10a)
Q̃′′ = −q
2j+1−δ0m(1−δ0j)
d3k{a+s (k) ◦ a†−s (k)− εa−s (k) ◦ a†+s (k)} (3.10b)
Q̃′′′ = 1
2j+1−δ0m(1−δ0j )
d3k{[a†+s (k), a−s (k)]ε − [a+s (k), a†−s (k)]ε} (3.10c)
S̃ ′µν =
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s (k) ◦ a−s′(k)
+ σss
µν (k)a
s (k) ◦ a+s′(k)
(3.11a)
S̃ ′′µν = ε
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s′(k) ◦ a
s (k)
+ σss
µν (k)a
(k) ◦ a†+s (k)
(3.11b)
S̃ ′′′µν =
(−1)j−1/2j~
2(1 + τ)
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)[a
s (k), a
(k)]ε
+ σss
µν (k)[a
s (k), a
s′(k)]ε
(3.11c)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 8
L̃′µν =x0µ P̃ ′ν − x0 ν P̃ ′µ
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s (k) ◦ a−s′(k)
+ lss
µν (k)a
s (k) ◦ a+s′(k)
2(1 + τ)
2j+1−δ0m(1−δ0j )
a†+s (k)
←−−−−−→
←−−−−−→
◦ a−s (k)
− εa†−s (k)
←−−−−−→
←−−−−−→
◦ a+s (k)
m2c2+k2
(3.12a)
L̃′′µν =x0µ P̃ ′′ν − x0 ν P̃ ′′µ
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s′(k) ◦ a
s (k)
+ lss
µν (k)a
(k) ◦ a†+s (k)
2(1 + τ)
2j+1−δ0m(1−δ0j )
a+s (k)
←−−−−−→
←−−−−−→
◦ a†−s (k)
− εa−s (k)
←−−−−−→
←−−−−−→
◦ a†+s (k)
m2c2+k2
(3.12b)
L̃′′′µν =x0µ P̃ ′′′ν − x0 ν P̃ ′′′µ
(−1)j−1/2j~
2(1 + τ)
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)[a
s (k), a
(k)]ε
+ lss
µν (k)[a
s (k), a
(k)]ε
4(1 + τ)
2j+1−δ0m(1−δ0j )
a†+s (k)
←−−−−−→
←−−−−−→
◦ a−s (k)
− εa−s (k)
←−−−−−→
←−−−−−→
◦ a†+s (k) + a+s (k)
←−−−−−→
←−−−−−→
◦ a†−s (k)
− εa†−s (k)
←−−−−−→
←−−−−−→
◦ a+s (k)
m2c2+k2
(3.12c)
Here we have used the following notation: (−1)n+1/2 := (−1)ni for all n ∈ N and i := +
←−−−−−→
◦B(k) := −
∂A(k)
◦B(k) +
A(k) ◦ kµ
∂B(k)
←−−−→
◦B(k)
(3.13)
for operators A(k) and B(k) having C1 dependence on k,9 and σ
ss′,±
µν (k) and l
ss′,±
µν (k) are
9 More generally, if ω : {F → F} → {F → F} is a mapping on the operator space over the system’s Hilbert
space, we put A
ω ◦ B := −ω(A) ◦ B + A ◦ ω(B) for any A,B : F → F . Usually [2, 12], this notation is used
for ω = ∂µ.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 9
some functions of k such that10
µν (k) = −σss
νµ (k) l
ss′,±
µν (k) = −lss
νµ (k)
µν (k) = l
ss′,±
νµ (k) = 0 for j = 0 (scalar field)
µν (k) = −σss
µν (k) =: σ
µν (k) = −σs
µν (k) = −σss
νµ (k) for j = 1 (vector field)
µν (k) = −lss
µν (k) =: l
µν (k) = −ls
µν (k) = −lss
νµ (k) for j = 1 (vector field).
(3.14)
A technical remark must be make at this point. The equations (3.9)–(3.12) were de-
rived in [13–15] under some additional conditions, represented by equations (2.6) and (2.7),
which are considered bellow in Sect. 5 and ensure the effectiveness of the momentum pic-
ture of motion [21] used in [13–15]. However, as it is partially proved, e.g., in [1], when the
quantities (3.9)–(3.12) are expressed via the Heisenberg creation and annihilation operators
(see (2.9)), they remain valid, up to a phase factor, and without making the mentioned
assumptions, i.e. these assumptions are needless when one works entirely in Heisenberg pic-
ture. For this reason, we shall consider (3.9)–(3.12) as pure consequence of the Lagrangian
formalism.
We should emphasize, in (3.11) and (3.12) with S̃ωµν and L̃ωµν , ω = ′, ′′, ′′′, are denoted
the spin and orbital, respectively, operators for L̃ω, which are the spacetime-independent
parts of the spin and orbital, respectively, angular momentum operators [14, 23]; if the last
operators are denoted by S̃ωµν and L̃
µν , the total angular momentum operator of a system
with Lagrangian L̃ω is [23]
M̃ωµν = L̃
µν + S̃
µν = L̃ωµν + S̃ωµν , ω = ′, ′′, ′′′ (3.15)
and S̃ωµν = S̃ωµν (and hence L̃
µν = L̃ωµν) iff S̃
µν is a conserved operator or, equivalently, iff
the system’s canonical energy-momentum tensor is symmetric.11
Going ahead (see Sect. 6), we would like to note that the expressions (3.9c) and, conse-
quently, the Lagrangian L̃′′′ are the base from which the paracommutation relations were
first derived [16].
And a last remark. Above we have expressed the dynamical variables in Heisenberg picture
via the creation and annihilation operators in momentum picture. If one works entirely in
Heisenberg picture, the operators (2.9), representing the creation and annihilation operators
in Heisenberg picture, should be used. Besides, by virtue of the equations
(a±s (k))
† = a†∓s (k) (a
s (k))
† = a∓s (k) (3.16)
ã±s (k)
= ã†∓s (k)
ã†±s (k)
= ã∓s (k), (3.17)
some of the relations concerning a
s (k), e.g. the Euler-Lagrange and Heisenberg equations,
are consequences of the similar ones regarding a±s (k). In view of (2.9), we shall consider (3.9)–
(3.12) as obtained form the corresponding expressions in Heisenberg picture by making the
replacements ã±s (k) 7→ a±s (k) and ã
s (k) 7→ a†±s (k). So, (3.9)–(3.12) will have, up to a
phase factor, a sense of dynamical variables in Heisenberg picture expressed via the cre-
ation/annihilation operators in momentum picture.
10 For the explicit form of these functions, see [13–15]; see also equation (6.57) below.
11 In [14,23] the spin and orbital operators are labeled with an additional left superscript ◦, which, for brevity,
is omitted in the present work as in it only these operators, not S̃
µν and L̃
µν , will be considered. Notice,
the operators S̃
µν and L̃
µν are, generally, time-dependent while the orbital and spin ones are conserved, as
a result of which the total angular momentum is a conserved operator too [14,23].
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 10
4. On the uniqueness of the dynamical variables
Let D = Pµ, Q, Sµν , Lµν denotes some dynamical variable, viz. the momentum, charge,
spin, or orbital operator, of a system with Lagrangian L. Since the Euler-Lagrange equations
for the Lagrangians L′, L′′ and L′′′ coincide (see (3.6)), we can assert that any field satisfying
these equations admits at least three classes of conserved operators, viz. D′, D′′ and D′′′ =
D′+D′′
.Moreover, it can be proved that the Euler-Lagrange equations for the Lagrangian
Lα,β := αL′ + β L′′ α+ β 6= 0 (4.1)
do not depend on α, β ∈ C and coincide with (3.6). Therefore there exists a two parameter
family of conserved dynamical variables for these equations given via
Dα,β := αD′ + βD′′ α+ β 6= 0. (4.2)
Evidently L′′′ = L 1
and D′′′ = D 1
. Since the Euler-Lagrange equations (3.6) are linear
and homogeneous (in the cases considered), we can, without a lost of generality, restrict the
parameters α, β ∈ C to such that
α+ β = 1, (4.3)
which can be achieved by an appropriate renormalization (by a factor (α+β)−1/2) of the field
operators. Thus any field satisfying the Euler-Lagrange equations (3.6) admits the family
Dα,β, α + β = 1, of conserved operators. Obviously, this conclusion is valid if in (4.1) we
replace the particular Lagrangians L′ and L′′ (see (3.1) and (3.3)) with any two Lagrangians
(of one and the same field variables) which lead to identical Euler-Lagrange equations. How-
ever, the essential point in our case is that L′ and L′′ do not differ only by a full divergence,
as a result of which the operators Dα,β are different for different pairs (α, β), α+ β = 1.12
Since one expects a physical system to possess uniquely defined dynamical characteristics,
e.g. energy and total angular momentum, and the Euler-Lagrange equations are considered
(in the framework of Lagrangian formalism) as the ones governing the spacetime evolution of
the system considered, the problem arises when the dynamical operators Dα,β, α+β = 1, are
independent of the particular choice of α and β, i.e. of the initial Lagrangian one starts off.
Simple calculation show that the operators (4.2), under the condition (4.3), are independent
of the particular values of the parameters α and β if and only if
D′ = D′′. (4.4)
Some consequences of the condition(s) (4.4) will be considered below, as well as possible ways
for satisfying these restrictions on the Lagrangian formalism.
Combining (3.9)–(3.12) with (4.4), for respectively D = Pµ, Q, Sµν , Lµν , we see that a
free scalar, spinor or vector field has a uniquely defined dynamical variables if and only if
the following equations are fulfilled:
2j+1−δ0m(1−δ0j )
d3k kµ
m2c2+k2
a†+s (k) ◦ a−s (k)− εa−s (k) ◦ a†+s (k)
− a+s (k) ◦ a†−s (k) + εa†−s (k) ◦ a+s (k)
= 0 (4.5)
12 Note, no commutativity or some commutation relations between the field operators and their charge (or
Hermitian) conjugate are presupposed, i.e., at the moment, we work in a theory without such relations and
normal ordering.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 11
2j+1−δ0m(1−δ0j )
a†+s (k) ◦ a−s (k)− εa−s (k) ◦ a†+s (k)
+ a+s (k) ◦ a†−s (k)− εa†−s (k) ◦ a+s (k)
= 0 (4.6)
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s (k) ◦ a−s′(k)− εσ
ss′,−
µν (k)a
(k) ◦ a†+s (k)
− εσss′,+µν (k)a+s′(k) ◦ a
s (k) + σ
ss′,+
µν (k)a
s (k) ◦ a+s′(k)
= 0 (4.7)
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s (k) ◦ a−s′(k)− εl
ss′,−
µν (k)a
(k) ◦ a†+s (k)
− εlss′,+µν (k)a+s′(k) ◦ a
s (k) + l
ss′,+
µν (k)a
s (k) ◦ a+s′(k)
2j+1−δ0m(1−δ0j )
a†+s (k)
←−−−−−→
←−−−−−→
◦a−s (k)+εa−s (k)
←−−−−−→
←−−−−−→
◦a†+s (k)
−a+s (k)
←−−−−−→
←−−−−−→
◦a†−s (k)−εa†−s (k)
←−−−−−→
←−−−−−→
◦a+s (k)
m2c2+k2
(4.8)
In (4.6) is retained the constant factor q as in the neutral case it is equal to zero and,
consequently, the equation (4.6) reduces to identity.
Since the Euler-Lagrange equations do not impose some restrictions on the creation and
annihilation operators, the equations (4.5)–(4.8) can be regarded as subsidiary conditions
on the Lagrangian formalism and can serve as equations for (partial) determination of the
creation and annihilation operators. The system of integral equations (4.5)–(4.8) is quite
complicated and we are not going to investigate it in the general case. Below we shall
restrict ourselves to analysis of only those solutions of (4.5)–(4.8), if any, for which the
integrands in (4.5)–(4.8) vanish. This means that we shall replace the system of integral
equations (4.5)–(4.8) with respect to creation and annihilation operators with the following
system of algebraic equations (do not sum over s and s′ in (4.12) and (4.13)!):
a†+s (k) ◦ a−s (k) − εa−s (k) ◦ a†+s (k) − a+s (k) ◦ a†−s (k) + εa†−s (k) ◦ a+s (k) = 0 (4.9)
a†+s (k) ◦ a−s (k) − εa−s (k) ◦ a†+s (k) + a+s (k) ◦ a†−s (k) − εa†−s (k) ◦ a+s (k) = 0 if q 6= 0
(4.10)
a†+s (k)
←−−−−−→
←−−−−−→
◦ a−s (k) + εa−s (k)
←−−−−−→
←−−−−−→
◦ a†+s (k)
−a+s (k)
←−−−−−→
←−−−−−→
◦a†−s (k)−εa†−s (k)
←−−−−−→
←−−−−−→
◦a+s (k)
m2c2+k2
(4.11)
µν (k)a
s (k) ◦ a−s′(k)− εσ
ss′,−
µν (k)a
(k) ◦ a†+s (k)
− εσss′,+µν (k)a+s′(k) ◦ a
s (k) + σ
ss′,+
µν (k)a
s (k) ◦ a+s′(k)
= 0 (4.12)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 12
µν (k)a
s (k) ◦ a−s′(k)− εl
ss′,−
µν (k)a
(k) ◦ a†+s (k)
− εlss′,+µν (k)a+s′(k) ◦ a
s (k) + l
ss′,+
µν (k)a
s (k) ◦ a+s′(k)
= 0 (4.13)
Here: s = 1, . . . , 2j + 1 − δ0m(1 − δ0j) in (4.9)–(4.11) and s, s′ = 1, . . . , 2j + 1 − δ0m(1 −
δ1j) in (4.12) and (4.13). (Notice, by virtue of (3.14), the equations (4.12) and (4.13) are
identically valid for j = 0, i.e. for scalar fields.) Since all polarization indices enter in (4.5)
and (4.6) on equal footing, we do not sum over s in (4.9)–(4.11). But in (4.12) and (4.13)
we have retain the summation sign as the modes with definite polarization cannot be singled
out in the general case. One may obtain weaker versions of (4.9)–(4.13) by summing in them
over the polarization indices, but we shall not consider these conditions below regardless of
the fact that they also ensure uniqueness of the dynamical variables.
At first, consider the equations (4.9)–(4.11). Since for a neutral field, q = 0, we have
s (k) = a
s (k), which physically means coincidence of field’s particles and antiparticles,
the equations (4.9)–(4.11) hold identically in this case.
Let consider now the case q 6= 0, i.e. the investigated field to be charged one. Using the
standard notation (cf. (3.8))
[A,B]η := A ◦B + ηB ◦A, (4.14)
for operators A and B and η ∈ C, we rewrite (4.9) and (4.10) as
[a†+s (k), a
s (k)]−ε − [a+s (k), a†−s (k)]−ε = 0 (4.9′)
[a†+s (k), a
s (k)]−ε + [a
s (k), a
s (k)]−ε = 0 if q 6= 0, (4.10′)
which are equivalent to
[a†±s (k), a
s (k)]−ε = 0 if q 6= 0. (4.15)
Differentiating (4.15) and inserting the result into (4.11), one can verify that (4.11) is
tantamount to
a†+s (k),
◦ a−s (k)
a+s (k),
◦ a†−s (k)
m2c2+k2
= 0 if q 6= 0, (4.16)
Consider now (4.12) and (4.13). By means of the shorthand (4.14), they read
µν (k)[a
s (k), a
(k)]−ε + σ
ss′,+
µν (k)[a
s (k), a
(k)]−ε
= 0 (4.17)
µν (k)[a
s (k), a
(k)]−ε + l
ss′,+
µν (k)[a
s (k), a
(k)]−ε
= 0. (4.18)
For a scalar field, j = 0, these conditions hold identically, due to (3.14). But for j 6= 0 they
impose new restrictions on the formalism. In particular, for vector fields, j = 1 and ε = +1
they are satisfied iff (see (3.14))
[a†+s (k), a
(k)]−ε − [a†−s (k), a+s′(k)]−ε − [a
(k), a−s (k)]−ε + [a
(k), a+s (k)]−ε = 0. (4.19)
One can satisfy (4.17) and (4.18) if the following generalization of (4.15) holds
[a†±s (k), a
(k)]−ε = 0. (4.20)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 13
For spin j = 1
(and hence ε = −1 – see (3.7)), the conditions (4.12) and (4.13) cannot
be simplified much, but, if one requires the vanishment of the operator coefficients after
ss′,±
µν (k) and l
ss′,±
µν (k), one gets
a†±s (k) ◦ a∓s′(k) = 0 j =
ε = −1. (4.21)
Excluding some special cases, e.g. neutral scalar field (q = 0 and j = 0), the equa-
tions (4.15) and (4.21) are unacceptable from many viewpoints. The main of them is that they
are incompatible with the ordinary (anti)commutation relations (see, e.g., e.g. [1, 11, 12, 18]
or Sect. 6, in particular, equations (6.13) bellow); for example, (4.21) means that the acts of
creation and annihilation of (anti)particles with identical characteristics should be mutually
independent, which contradicts to the existing theory and experimental data.
Now we shall try another way for achieving uniqueness of the dynamical variables for
free fields. Since in (4.9)–(4.13) naturally appear (anti)commutators between creation and
annihilation operators and these (anti)commutators vanish under the standard normal or-
dering [1,11,12,18], one may suppose that the normally ordered expressions of the dynamical
variables may coincide. Let us analyze this method.
Recall [1, 3, 11, 12], the normal ordering operator N (for free field theory) is a linear
operator on the operator space of the system considered such that to a product (composition)
c1 ◦ · · · ◦ cn of n ∈ N creation and/or annihilation operators c1, . . . cn it assigns the operator
(−1)f cα1 ◦ · · · cαn . Here (α1, . . . , αn) is a permutation of (1, . . . , n), all creation operators
stand to the left of all annihilation ones, the relative order between the creation/annihilation
operators is preserved, and f is equal to the number of transpositions among the fermion
operators (j = 1
) needed to be achieved the just-described order (“normal order”) of the
operators c1 ◦ · · · ◦ cn in cα1 ◦ · · · cαn .13 In particular this means that
a+s (k) ◦ a
t (p)
= a+s (k) ◦ a
t (p) N
a†+s (k) ◦ a−t (p)
= a†+s (k) ◦ a−t (p)
a−s (k) ◦ a
t (p)
t (p) ◦ a−s (k) N
a†−s (k) ◦ a+t (p)
= εa+t (p) ◦ a†−s (k) (4.22)
and, consequently, we have
[a†±s (k), a
t (p)]−ε
= 0 N
[a±s (k), a
t (p)]−ε
= 0, (4.23)
due to ε := (−1)2j = ±1 (see (3.7)). (In fact, below only the equalities (4.22) and (4.23),
not the general definition of a normal product, will be applied.)
Applying the normal ordering operator to (4.9′), (4.10′), (4.17) and (4.18), we, in view
of (4.23), get the identity 0 = 0, which means that the conditions (4.9), (4.10), (4.12)
and (4.13) are identically satisfied after normal ordering. This is confirmed by the application
of N to (3.9) and (3.10), which results respectively in (see (4.22))
N ( P̃ ′µ) = N ( P̃ ′′µ)
1 + τ
2j+1−δ0m(1−δ0j )
d3kkµ|
m2c2+k2
{a†+s (k) ◦ a−s (k) + a+s (k) ◦ a†−s (k)} (4.24)
13 We have slightly modified the definition given in [1,3,11,12] because no (anti)commutation relations are
presented in our exposition till the moment. In this paper we do not concern the problem for elimination of
the ‘unphysical’ operators a±
(k) and a
(k) from the spin and orbital momentum operators when j = 1; for
details, see [15], where it is proved that, for an electromagnetic field, j = 1 and q = 0, one way to achieve this
is by adding to the number f above the number of transpositions between a±s (k), s = 1, 2, and a
(k) needed
for getting normal order.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 14
N ( Q̃′) = N ( Q̃′′) = 1
1 + τ
2j+1−δ0m(1−δ0j )
d3k{a†+s (k) ◦ a−s (k)− a+s (k) ◦ a†−s (k)}.
(4.25)
Therefore the normal ordering ensures the uniqueness of the momentum and charge operators,
if we redefine them respectively as
P̃µ := N ( P̃ ′µ) Q̃ := N ( Q̃′). (4.26)
Putting ωµν := kµ
− kν ∂∂kµ and using (4.22), one can verify that
a+s (k)
←−−−→
ωµν ◦ a†−s (k)
= a+s (k)
←−−−→
ωµν ◦ a†−s (k)
a†+s (k)
←−−−→
ωµν ◦ a−s (k)
= a†+s (k)
←−−−→
ωµν ◦ a−s (k)
a−s (k)
←−−−→
ωµν ◦ a†+s (k)
= −εa†+s (k)
←−−−→
ωµν ◦ a−s (k)
a†−s (k)
←−−−→
ωµν ◦ a+s (k)
= −εa+s (k)
←−−−→
ωµν ◦ a†−s (k).
(4.27)
As a consequence of these equalities, the action of N on the l.h.s. of (4.11) vanishes. Com-
bining this result with the mentioned fact that the normal ordering converts (4.12) and (4.13)
into identities, we see that the normal ordering procedure ensures also uniqueness of the spin
and orbital operators if we redefine them respectively as:
S̃µν := N ( S̃ ′µν) := N ( S̃ ′′µν) =
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s (k) ◦ a−s′(k) + εσ
ss′,+
µν (k)a
s′(k) ◦ a
s (k)
(4.28)
L̃µν := N ( L̃′µν) := N ( L̃′′µν) = x0µ P̃ν − x0 ν P̃µ +
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s (k) ◦ a−s′(k) + εl
ss′,+
µν (k)a
(k) ◦ a†−s (k)
2(1 + τ)
2j+1−δ0m(1−δ0j )
a†+s (k)
←−−−−−→
←−−−−−→
◦ a−s (k)
+ a+s (k)
←−−−−−→
←−−−−−→
◦ a†−s (k)
m2c2+k2
(4.29)
where (3.14) was applied.
5. Heisenberg relations
The conserved operators, like momentum and charge operators, are often identified with the
generators of the corresponding transformations under which the action operator is invari-
ant [1, 3, 11, 12]. This leads to a number of commutation relations between the components
of these operators and between them and the field operators. The relations of the letter
set are known/referred as the Heisenberg relations or equations. Both kinds of commuta-
tion relations are from pure geometric origin and, consequently, are completely external to
the Lagrangian formalism; one of the reasons being that the mentioned identification is, in
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 15
general, unacceptable and may be carried out only on some subset of the system’s Hilbert
space of states [23, 24]. Therefore their validity in a pure Lagrangian theory is questionable
and should be verified [11]. However, the considered relations are weaker conditions than
the identification of the corresponding operators and there are strong evidences that these
relations should be valid in a realistic quantum field theory [1,11]; e.g., the commutativity be-
tween the momentum and charge operators (see below (5.18)) expresses the experimental fact
that the 4-momentum and charge of any system are simultaneously measurable quantities.
It is known [1,11], in a pure Lagrangian approach, the field equations, which are usually
identified with the Euler-Lagrange, 14 are the only restrictions on the field operators. Besides,
these equations do not determine uniquely the field operators and the letter can be expressed
through the creation and annihilation operators. Since the last operators are left completely
arbitrary by a pure Lagrangian formalism, one is free to impose on them any system of
compatible restrictions. The best known examples of this kind are the famous canonical
(anti)commutation relations and their generalization, the so-called paracommutation rela-
tions [16,18]. In general, the problem for compatibility of such subsidiary to the Lagrangian
formalism system of restrictions with, for instance, the Heisenberg relations is open and
requires particular investigation [11]. For example, even the canonical (anti)commutation
relations for electromagnetic field in Coulomb gauge are incompatible with the Heisenberg
equation involving the (total) angular momentum operator unless the gauge symmetry of this
field is taken into account [11, § 84]. However, the (para)commutation relations are, by con-
struction, compatible with the Heisenberg relations regarding momentum operator (see [16]
or below Subsect. 6.1). The ordinary approach is to be imposed a system of equations on
the creation and annihilation operators and, then, to be checked its compatibility with, e.g.,
the Heisenberg relations. In the next sections we shall investigate the opposite situation:
assuming the validity of (some of) the Heisenberg equations, the possible restrictions on
the creation and annihilation operators will be explored. For this purpose, below we briefly
review the Heisenberg relations and other ones related to them.
Consider a system of quantum fields ϕ̃i(x), i = 1, . . . , N ∈ N, where ϕ̃i(x) denote the
components of all fields (and their Hermitian conjugates), and P̃µ, Q̃ and M̃µν be its
momentum, charge and (total) angular momentum operators, respectively. The Heisenberg
relations/equations for these operators are [1, 3, 11,12]
[ ϕ̃i(x), P̃µ] = i~
∂ ϕ̃i(x)
(5.1)
[ ϕ̃i(x), Q̃] = e( ϕ̃i)q ϕ̃i(x) (5.2)
[ ϕ̃i(x), M̃µν ] = i~{xµ∂ν ϕ̃i(x)− xν∂µ ϕ̃i(x)}+ i~
ϕ̃i′(x). (5.3)
Here: q = const is the fields’ charge, e( ϕ̃i) = 0 if ϕ̃
i = ϕ̃i, e( ϕ̃i) = ±1 if ϕ̃
i 6= ϕ̃i with
e( ϕ̃i)+e( ϕ̃
i ) = 0, and the constants I
iµν = −Ii
iνµ characterize the transformation properties
of the field operators under 4-rotations. (If ε( ϕ̃i) 6= 0, it is a convention whether to put
ε( ϕ̃i) = +1 or ε( ϕ̃i) = −1 for a fixed i.)
We would like to make some comments on (5.3). Since its r.h.s. is a sum of two operators,
the first (second) characterizing the pure orbital (spin) angular momentum properties of
the system considered, the idea arises to split (5.3) into two independent equations, one
involving the orbital angular momentum operator and another concerning the spin angular
momentum operator. This is supported by the observation that, it seems, no process is known
for transforming orbital angular momentum into spin one and v.v. (without destroying the
14 Recall, there are Lagrangians whose classical Euler-Lagrange equations are identities. However, their
correct and rigorous treatment [22] reveals that they entail field equations which are mathematically correct
and physically sensible.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 16
system). So one may suppose the existence of operators M̃orµν and M̃
µν such that
[ ϕ̃i(x), M̃orµν ] = i~{xµ∂ν ϕ̃i(x)− xν∂µ ϕ̃i(x)} (5.4)
[ ϕ̃i(x), M̃spµν ] = i~
iµν ϕ̃i′(x) (5.5)
M̃µν = M̃orµν + M̃spµν . (5.6)
However, as particular calculations demonstrate [5,14,15], neither the spin (resp. orbital)
nor the spin (resp. orbital) angular momentum operator is a suitable candidate for M̃spµν
(resp. M̃orµν). If we assume the validity of (5.1), then equations (5.4) and (5.5) can be
satisfied if we choose
M̃orµν(x) = L̃extµν := xµ P̃ν − xν P̃µ (5.7)
M̃spµν(x) = M̃(0)µν (x) := M̃µν − L̃extµν = S̃µν + L̃µν − {xµ P̃ν − xν P̃µ} (5.8)
with M̃µν satisfying (5.3). These operators are not conserved ones. Such a representation
is in agreement with the equations (3.12), according to which the operator (5.7) enters addi-
tively in the expressions for the orbital operator.15 The physical sense of the operator (5.7)
is that it represents the orbital angular momentum of the system due to its movement as a
whole. Respectively, the operator (5.8) describes the system’s angular momentum as a result
of its internal movement and/or structure.
Since the spin (orbital) angular momentum is associated with the structure (movement)
of a system, in the operator (5.8) are mixed the spin and orbital angular momenta. These
quantities can be separated completely via the following representations of the operators
Morµν and M
µν in momentum picture (when (5.1) holds)
Morµν = xµ Pν − xµPµ + Lintµν (5.9)
Mspµν = Mµν − (xµ Pν − xµ Pµ)− Lintµν , (5.10)
where Lintµν describes the ‘internal’ orbital angular momentum of the system considered and
depends on the Lagrangian we have started off. Generally said, Lintµν is the part of the
orbital angular momentum operator containing derivatives of the creation and annihilation
operators. In particular, for the Lagrangians L′, L′′ and L′′′ (see Sect. 3), the explicit forms
of the operators (5.9) and (5.10) respectively are:
M′ orµν =xµP ′ν − xν P ′µ
2(1 + τ)
2j+1−δ0m(1−δ0j)
a†+s (k)
←−−−−−→
←−−−−−→
◦ a−s (k)
− εa†−s (k)
←−−−−−→
←−−−−−→
◦ a+s (k)
m2c2+k2
(5.11a)
M′′ orµν =xµP ′′ν − xν P ′′µ
2(1 + τ)
2j+1−δ0m(1−δ0j)
a+s (k)
←−−−−−→
←−−−−−→
◦ a†−s (k)
− εa−s (k)
←−−−−−→
←−−−−−→
◦ a†+s (k)
m2c2+k2
(5.11b)
15 This is evident in the momentum picture of motion, in which xµ stands for x0µ in (3.12) — see [13–15].
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 17
M′′′ orµν =xµP ′′′ν − xν P ′′′µ
4(1 + τ)
2j+1−δ0m(1−δ0j)
a†+s (k)
←−−−−−→
←−−−−−→
◦ a−s (k)
− εa−s (k)
←−−−−−→
←−−−−−→
◦ a†+s (k) + a+s (k)
←−−−−−→
←−−−−−→
◦ a†−s (k)
− εa†−s (k)
←−−−−−→
←−−−−−→
◦ a+s (k)
m2c2+k2
(5.11c)
M′ spµν =
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k) + l
ss′,−
µν (k))a
s (k) ◦ a−s′(k)
+ (σss
µν (k) + l
ss′,+
µν (k))a
s (k) ◦ a+s′(k)
(5.12a)
M′′ spµν = ε
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k) + l
ss′,+
µν (k))a
(k) ◦ a†−s (k)
+ (σss
µν (k) + σ
ss′,−
µν (k))a
(k) ◦ a†+s (k)
(5.12b)
M′′′ spµν =
(−1)j−1/2j~
2(1 + τ)
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k) + l
ss′,−
µν (k))[a
s (k), a
s′(k)]ε
+ (σss
µν (k) + l
ss′,+
µν (k))[a
s (k), a
(k)]ε
(5.12c)
Obviously (see Sect. 2), the equations (5.12) have the same form in Heisenberg picture in
terms of the operators (2.9) (only tildes over M and a must be added), but the equa-
tions (5.11) change substantially due to the existence of derivatives of the creation and
annihilation operators in them [13–15]:
M̃′ orµν =
2(1 + τ)
2j+1−δ0m(1−δ0j )
ã†+s (k)
←−−−−−→
←−−−−−→
◦ ã−s (k)
− εã†−s (k)
←−−−−−→
←−−−−−→
◦ ã+s (k)
m2c2+k2
(5.13a)
M̃′′ orµν =
2(1 + τ)
2j+1−δ0m(1−δ0j )
ã+s (k)
←−−−−−→
←−−−−−→
◦ ã†−s (k)
− εã−s (k)
←−−−−−→
←−−−−−→
◦ ã†+s (k)
m2c2+k2
(5.13b)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 18
M̃′′′ orµν =
4(1 + τ)
2j+1−δ0m(1−δ0j )
ã†+s (k)
←−−−−−→
←−−−−−→
◦ ã−s (k)
− εã−s (k)
←−−−−−→
←−−−−−→
◦ ã†+s (k) + ã+s (k)
←−−−−−→
←−−−−−→
◦ ã†−s (k)
− εã†−s (k)
←−−−−−→
←−−−−−→
◦ ã+s (k)
m2c2+k2
(5.13c)
From (5.13) and (5.12) is clear that the operators M̃orµν and M̃
µν so defined are conserved
(contrary to (5.7) and (5.8)) and do not depend on the validity of the Heisenberg rela-
tions (5.1) (contrary to expressions (5.11) in momentum picture).
The problem for whether the operators (5.12) and (5.13) satisfy the equations (5.4)
and (5.5), respectively, will be considered in Sect. 6.
There is an essential difference between (5.4) and (5.5): the equation (5.5) depends on
the particular properties of the operators ϕ̃i(x) under 4-rotations via the coefficients I
(see (5.25) below), while (5.4) does not depend on them. This is explicitly reflected in (5.11)
and (5.12): the former set of equations is valid independently of the geometrical nature of the
fields considered, while the latter one depends on it via the ‘spin’ (‘polarization’) functions
ss′,±
µν (k) and l
ss′,±
µν (k). Similar remark concerns (5.3), on one hand, and (5.1) and (5.2), on
another hand: the particular form of (5.3) essentially depends on the geometric properties
of ϕ̃i(x) under 4-rotations, the other equations being independent of them.
It should also be noted, the relation (5.3) does not hold for a canonically quantized
electromagnetic field in Coulomb gauge unless some additional terms it its r.h.s., reflecting
the gauge symmetry of the field, are taken into account [11, § 84].
As it was said above, the relations (5.1)–(5.3) are from pure geometrical origin. However,
the last discussion, concerning (5.4)–(5.8), reveals that the terms in braces in (5.3) should be
connected with the momentum operator in the (pure) Lagrangian approach. More precisely,
on the background of equations (3.11a)–(3.12c), the Heisenberg relation (5.3) should be
replaced with
[ ϕ̃i(x), M̃µν ] = xµ[ ϕ̃i(x), P̃ν ] − xν [ ϕ̃i(x), P̃µ] + i~
iµν ϕ̃i′(x), (5.14)
which is equivalent to (5.3) if (5.1) is true. An advantage of the last equation is that it is valid
in any picture of motion (in the same form) while (5.3) holds only in Heisenberg picture.16
Obviously, (5.14) is equivalent to (5.5) with M̃spµν defined by (5.8).
The other kind of geometric relations mentioned at the beginning of this section are
connected with the basic relations defining the Lie algebra of the Poincaré group [7, pp. 143–
147], [8, sect. 7.1]. They require the fulfillment of the following equations between the com-
ponents P̃µ of the momentum and M̃µν of the angular momentum operators [3, 5, 7, 8]:
[ P̃µ, P̃ν ] = 0 (5.15)
[M̃µν , P̃λ] = −i~(ηλµ P̃ν − ηλν P̃µ). (5.16)
[M̃κλ, M̃µν ] = −i~
ηκµ M̃λν − ηλµ M̃κν − ηκν M̃λµ + ηλν M̃κµ
. (5.17)
We would like to pay attention to the minus sign in the multiplier (−i~) in (5.16) and (5.17)
with respect to the above references, where i~ stands instead of −i~ in these equations. When
16 In other pictures of motion, generally, additional terms in the r.h.s. of (5.3) will appear, i.e. the functional
form of the r.h.s. of (5.3) is not invariant under changes of the picture of motion, contrary to (5.14).
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 19
(a representation of) the Lie algebra of the Poincaré group is considered, this difference in the
sign is insignificant as it can be absorbed into the definition of M̃µν . However, the change
of the sign of the angular momentum operator, M̃µν 7→ −M̃µν , will result in the change
i~ 7→ −i~ in the r.h.s. of (5.3). This means that equations (5.15), (5.16) and (5.3), when
considered together, require a suitable choice of the signs of the multiplier i~ in their right
hand sides as these signs change simultaneously when M̃µν is replaced with −M̃µν . Since
equations (5.3), (5.16) and (5.17) hold, when M̃µν is defined according to the Noether’s
theorem and the ordinary (anti)commutation relations are valid [13–15], we accept these
equations in the way they are written above.
To the relations (5.15)–(5.17) should be added the equations [3, p. 78]
[ Q̃, P̃µ] = 0 (5.18)
[ Q̃, M̃µν ] = 0, (5.19)
which complete the algebra of observables and express, respectively, the translational and
rotational invariance of the charge operator Q̃; physically they mean that the charge and
momentum or the charge and angular momentum are simultaneously measurable quantities.
Since the spin properties of a system are generally independent of its charge or momentum,
one may also expect the validity of the relations17
[ S̃µν , P̃µ] = 0 (5.20)
[ S̃µν , Q̃] = 0. (5.21)
But, as the spin describes, in a sense, some of the rotational properties of the system,
equality like [ S̃µν , L̃κλ] = 0 is not likely to hold. Indeed, the considerations in [13–15] reveal
that (5.20) and (5.21), but not the last equation, are true in the framework of the Lagrangian
formalism with added to it standard (anti)commutation relations. Notice, if (5.20) and (5.21)
hold, then, respectively, (5.16) and (5.19) are equivalent to
[ L̃µν , P̃λ] = −i~(ηλµ P̃ν − ηλν P̃µ). (5.22)
[ Q̃, L̃µν ] = 0. (5.23)
It is intuitively clear, not all of the commutation relations (5.1)–(5.3) and (5.15)–(5.21)
are independent: if D̃ denotes some of the operators P̃µ, Q̃, M̃µν , S̃µν or L̃µν and the
commutators [ ϕ̃i(x), D̃] , i = 1, . . . , N , are known, then, in principle, one can calculate the
commutators [Γ( ϕ̃1(x), . . . , ϕ̃N (x)), D̃] , where Γ( ϕ̃1(x), . . . , ϕ̃N (x)) is, for example, any
function/functional bilinear in ϕ̃1(x), . . . , ϕ̃N (x); to prove this fact, one should apply the
identity [A,B ◦ C] = [A,B] ◦ C + B ◦ [A,C] a suitable number of times. In particular, if
D̃1 and D̃2 denote any two (distinct) operators of the dynamical variables, and [ ϕ̃i(x), D̃1]
is known, then the commutator [ D̃1, D̃2] can be calculated explicitly. For this reason, we
can expect that:
(i) Equation (5.1) implies (5.15), (5.16), (5.18), (5.20) and (5.22).
(ii) Equation (5.2) implies (5.18), (5.19), (5.21), and (5.23).
(iii) Equation (5.3) implies (5.16), (5.17), and (5.19).
Besides, (5.3) may, possibly, entail equations like (5.17) with S or L forM , with an exception
of M̃µν in the l.h.s., i.e.
[ S̃κλ, M̃µν ] = −i~
ηκµ S̃λν − ηλµ S̃κν − ηκν S̃λµ + ηλν S̃κµ
[ L̃κλ, M̃µν ] = −i~
ηκµ L̃λν − ηλµ L̃κν − ηκν L̃λµ + ηλν L̃κµ
} (5.24)
17 Recall, S̃µν (resp. L̃µν) is the conserved spin (resp. orbital) operator, not the generally non-conserved
spin (resp. orbital) angular momentum operator [23].
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 20
The validity of assertions (i)–(iii) above for free scalar, spinor and vector fields, when respec-
tively
ϕ̃i(x) 7→ ϕ̃(x), ϕ̃†(x) Ii
iµν 7→ Iµν = 0 e( ϕ̃) = −e( ϕ̃†) = +1 (5.25a)
ϕ̃i(x) 7→ ψ̃(x), ˜̆ψ(x) Ii
iµν 7→ Iψµν = Iψ̆µν = −
σµν e( ψ̃) = −e( ˜̆ψ) = +1 (5.25b)
ϕ̃i(x) 7→ Ũµ(x), Ũ†µ(x) Ii
iµν 7→ Iσρµν = I†σρµν = δσµηνρ − δσν ηµρ e( Ũµ) = −e( Ũ†µ) = +1,
(5.25c)
where σµν := i
[γµ, γν ] with γµ being the Dirac γ-matrices [1, 25], is proved in [13–15],
respectively. Besides, in loc. cit. is proved that equations (5.24) hold for scalar and vector
fields, but not for a spinor field.18
Thus, we see that the Heisenberg relations (5.1)–(5.3) are stronger than the commutation
relations (5.15)–(5.23), when imposed on the Lagrangian formalism as subsidiary restrictions.
6. Types of possible commutation relations
In a broad sense, by a commutation relation we shall understand any algebraic relation
between the creation and annihilation operators imposed as subsidiary restriction on the
Lagrangian formalism. In a narrow sense, the commutation relations are the equations (6.13),
with ε = −1, written below and satisfied by the bose creation and annihilation operators. As
anticommutation relations are known the equations (6.13), with ε = +1, written below and
satisfied by the fermi creation and annihilation operators. The last two types of relations
are often referred as the bilinear commutation relations [18]. Theoretically are possible also
trilinear commutation relations, an example being the paracommutation relations [16, 18]
represented below by equations (6.18) (or (6.20)).
Generally said, the commutation relations should be postulated. Alternatively, they
could be derived from (equivalent to them) different assumptions added to the Lagrangian
formalism. The purpose of this section is to be explored possible classes of commutation
relations, which follow from some natural restrictions on the Lagrangian formalism that are
consequences from the considerations in the previous sections. Special attention will be paid
on some consequences of the charge symmetric Lagrangians as the free fields possess such a
symmetry [1, 3, 11,12].
As pointed in Sect 3, the Euler-Lagrange equations for the Lagrangians L̃′, L̃′′ and L̃′′′
coincide and, in quantum field theory, the role of these equations is to be singled out the
independent degrees of freedom of the fields in the form of creation and annihilation operators
a±s (k) and a
s (k) (which are identical for L̃′, L̃′′ and L̃′′′). Further specialization of these
operators is provided by the commutation relations (in broad sense) which play a role of field
equations in this situation (with respect to the mentioned operators).
Before proceeding on, we would like to simplify our notation. As a spin variable, s say, is
always coupled with a 3-momentum one, k say, we shall use the letters l, m and n to denote
pairs like l = (s,k), m = (t,p) and n = (r, q). Equipped with this convention, we shall write,
e.g., a±l for a
s (k) and a
l for a
s (k). We set δlm := δstδ
3(k−p) and a summation sign like
l should be understood as
d3k, where the range of the polarization variable s will
be clear from the context (see, e.g., (3.9)–(3.12)).
18 The problem for the validity of assertions (i)–(iii) or equations (5.24) in the general case of arbitrary
fields (Lagrangians) is not a subject of the present work.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 21
6.1. Restrictions related to the momentum operator
First of all, let us examine the consequences of the Heisenberg relation (5.1) involving the
momentum operator. Since in terms of creation and annihilation operators it reads [1,13–15]
[a±s (k), Pµ] = ∓kµa±s (k) [a†±s (k), Pµ] = ∓kµa†±s (k) k0 =
m2c2 + k2, (6.1)
the field equations in terms of creation and annihilation operators for the Lagrangians (3.1),
(3.3) and (3.4) respectively are (see [13–15] or (6.1) and (3.9)):
2j+1−δ0m(1−δ0j )
m2c2+q2
a±s (k), a
t (q) ◦ a−t (q) + εa
t (q) ◦ a+t (q)
± (1 + τ)a±s (k)δstδ3(k − q)
d3q = 0
(6.2a)
2j+1−δ0m(1−δ0j )
m2c2+q2
a†±s (k), a
t (q) ◦ a−t (q) + εa
t (q) ◦ a+t (q)
± (1 + τ)a†±s (k)δstδ3(k − q)
d3q = 0
(6.2b)
2j+1−δ0m(1−δ0j )
m2c2+q2
a±s (k), a
t (q) ◦ a
t (q) + εa
t (q) ◦ a
t (q)
± (1 + τ)a±s (k)δstδ3(k − q)
d3q = 0
(6.3a)
2j+1−δ0m(1−δ0j )
m2c2+q2
a†±s (k), a
t (q) ◦ a
t (q) + εa
t (q) ◦ a
t (q)
± (1 + τ)a†±s (k)δstδ3(k − q)
d3q = 0
(6.3b)
2j+1−δ0m(1−δ0j )
m2c2+q2
a±s (k), [a
t (q), a
t (q)]ε + [a
t (q), a
t (q)]ε
± (1 + τ)a±s (k)δstδ3(k − q)
d3q = 0
(6.4a)
2j+1−δ0m(1−δ0j)
m2c2+q2
a†±s (k), [a
t (q), a
t (q)]ε + [a
t (q), a
t (q)]ε
± (1 + τ)a†±s (k)δstδ3(k − q)
d3q = 0,
(6.4b)
where j and ε are given via (3.7), the generalized commutation function [·, ·]ε is defined
by (4.14), and the polarization indices take the values
s, t = 1, . . . , 2j + 1− δ0m(1− δ0j) =
1 for j = 0 or for j = 1
and m = 0
1, 2 for j = 1
and m 6= 0 or for j = 1 and m = 0
1, 2, 3 for j = 1 and m 6= 0
(6.5)
The “b” versions of the equations (6.2)–(6.4) are consequences of the “a” versions and the
equalities
(a±l )
† = a
l ) = a
l (6.6)
[A,B]η
= η[A†, B†]η for [A,B]η = η[B,A]η η = ±1. (6.7)
Applying (6.2)–(6.4) and the identity
[A,B ◦ C] = [A,B]η ◦ C − ηB ◦ [A,C]η for η = ±1 (6.8)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 22
for the choice η = −1, one can prove by a direct calculation that
[ P̃µ, P̃ν ] = 0 [ Q̃, P̃µ] = 0 [ S̃µν , P̃λ] = 0
[ L̃µν , P̃λ] = −i~{ηλµ P̃ν − ηλν P̃µ} [M̃µν , P̃λ] = −i~{ηλµ P̃ν − ηλν P̃µ},
(6.9)
where the operators P̃µ, Q̃, S̃µν , L̃µν , and M̃µν denote the momentum, charge, spin,
orbital and total angular momentum operators, respectively, of the system considered and
are calculated from one and the same initial Lagrangian. This result confirms the supposition,
made in Sect. 5, that the assertion (i) before (5.24) holds for the fields investigated here.
Below we shall study only those solutions of (6.2)–(6.4) for which the integrands in them
vanish, i.e. we shall replace the systems of integral equations (6.2)–(6.4) with the following
systems of algebraic equations (see the above convention on the indices l and m and do not
sum over indices repeated on one and the same level):
a±l , a
m ◦ a−m + εa†−m ◦ a+m
± (1 + τ)δlma±l = 0 (6.10a)
l , a
m ◦ a−m + εa†−m ◦ a+m
± (1 + τ)δlma†±l = 0 (6.10b)
a±l , a
m ◦ a†−m + εa−m ◦ a†+m
± (1 + τ)δlma±l = 0 (6.11a)
l , a
m ◦ a†−m + εa−m ◦ a†+m
± (1 + τ)δlma†±l = 0 (6.11b)
a±l , [a
m , a
m]ε + [a
± 2(1 + τ)δlma±l = 0 (6.12a)
, [a†+m , a
m]ε + [a
± 2(1 + τ)δlma†±l = 0. (6.12b)
It seems, these are the most general and sensible trilinear commutation relations one may
impose on the creation and annihilation operators.
First of all, we should mentioned that the standard bilinear commutation relations, viz. [1,
3, 11–15]
[a±l , a
m]−ε = 0 [a
l , a
m ]−ε = 0
[a∓l , a
m]−ε = (±1)2j+1τδlm idF [a
l , a
m ]−ε = (±1)2j+1τδlm idF
[a±l , a
m ]−ε = 0 [a
l , a
m]−ε = 0
, a†±m ]−ε = (±1)2j+1δlm idF [a
, a±m]−ε = (±1)2j+1δlm idF , (6.13)
provide a solution of any one of the equations (6.10)–(6.12) in a sense that, due to (3.7)
and (6.8), with η = −ε any set of operators satisfying (6.13) converts (6.10)–(6.12) into
identities.
Besides, this conclusion remains valid also if the normal ordering is taken into account,
i.e. if, in this particular case, the changes a
m ◦ a+m 7→ εa+m ◦ a
m and a
m ◦ a
m 7→ εa†+m ◦ a−m
are made in (6.10)–(6.12).
Now we shall demonstrate how the trilinear relations (6.12) lead to the paracommuta-
tion relations. Equations (6.12) can be ‘split’ into different kinds of trilinear commutation
relations into infinitely many ways. For example, the system of equations
a±l , [a
± (1 + τ)δlma±l = 0 (6.14a)
a±l , [a
m , a
± (1 + τ)δlma±l = 0 (6.14b)
, [a+m, a
± (1 + τ)δlma†±l = 0 (6.14c)
l , [a
m , a
± (1 + τ)δlma†±l = 0 (6.14d)
provides an evident solution of (6.12). However, it is a simple algebra to be seen that
these relations are incompatible with the standard (anti)commutation relations (6.13) and,
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 23
in this sense, are not suitable as subsidiary restrictions on the Lagrangian formalism. For
our purpose, the equations
a+l , [a
+ 2δlma
l = 0 (6.15a)
a+l , [a
m , a
+ 2τδlma
l = 0 (6.15b)
a−l , [a
− 2τδlma−l = 0 (6.15c)
a−l , [a
m , a
− 2δlma−l = 0 (6.15d)
and their Hermitian conjugate provide a solution of (6.12), which is compatible with (6.13),
i.e. if (6.13) hold, the equations (6.15) are converted into identities.
The idea of the paraquantization is in the following generalization of (6.15)
a+l , [a
+ 2δlna
m = 0 (6.16a)
a+l , [a
m , a
+ 2τδlna
m = 0 (6.16b)
a−l , [a
− 2τδlma−n = 0 (6.16c)
a−l , [a
m , a
− 2δlma−n = 0 (6.16d)
which reduces to (6.15) for n = m and is a generalization of (6.13) in a sense that any set
of operators satisfying (6.13) converts (6.16) into identities, the opposite being generally not
valid.19
Suppose that the field considered consists of a single sort of particles, e.g. electrons or
photons, created by b
and annihilated by bl := a
. Then the equation Hermitian
conjugated to (6.15a) reads
[bl, [b
m, bm]ε] = 2δlmbm. (6.17)
This is the main relation from which the paper [16] starts. The basic paracommutation
relations are [16–18,26]:
[bl, [b
m, bn]ε] = 2δlmbn (6.18a)
[bl, [bm, bn]ε] = 0. (6.18b)
The first of them is a generalization (stronger version) of (6.17) by replacing the second index
m with an arbitrary one, say n, and the second one is added (by ”hands”) in the theory as
an additional assumption. Obviously, (6.18) are a solution of (6.15) and therefore of (6.12)
in the considered case of a field consisting of only one sort of particles.
The equations (6.15) contain also the relativistic version of the paracommutation rela-
tions, when the existence of antiparticles must be respected [18, sec. 18.1]. Indeed, noticing
that the field’s particles (resp. antiparticles) are created by b
:= a+
(resp. c
) and
annihilated by bl := a
(resp. cl := a
), from (6.15) and the Hermitian conjugate to them
equations, we get
[bl, [b
m, bm]ε] = 2δlmbm [cl, [c
m, cm]ε] = 2δlmcm (6.19a)
, [c†m, cm]ε] = −2τδlmb†m [c
, [b†m, bm]ε] = −2τδlmc†m. (6.19b)
Generalizing these equations in a way similar to the transition from (6.17) to (6.18), we
obtain the relativistic paracommutation relations as (cf. (6.16))
[bl, [b
m, bn]ε] = 2δlmbn [bl, [bm, bn]ε] = 0 (6.20a)
[cl, [c
m, cn]ε] = 2δlmcn [cl, [cm, cn]ε] = 0 (6.20b)
l , [c
m, cn]ε] = −2τδlnb†m [c
l , [b
m, bn]ε] = −2τδlnc†m. (6.20c)
19 Other generalizations of (6.15) are also possible, but they do not agree with (6.13). Moreover, it is easy
to be proved, any other (non-trivial) arrangement of the indices in (6.16) is incompatible with (6.13).
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 24
The equations (6.20a) (resp. (6.20b)) represent the paracommutation relations for the field’s
particles (resp. antiparticles) as independent objects, while (6.20c) describe a pure relativistic
effect of some “interaction” (or its absents) between field’s particles and antiparticles and
fixes the paracommutation relations involving the bl’s and cl’s, as pointed in [18, p. 207]
(where bl is denoted by al and cl by bl). The relations (6.17) and (6.20) for ε = +1 (resp.
ε = −1) are referred as the parabose (resp. parafermi) commutation relations [18]. This
terminology is a natural one also with respect to the commutation relations (6.16), which
will be referred as the paracommutation relations too.
As first noted in [16], the equations (6.13) provide a solution of (6.20) (or (6.18) in the
nonrelativistic case) but the latter equations admit also an infinite number of other solutions.
Besides, by taking Hermitian conjugations of (some of) the equations (6.18) or (6.20) and
applying generalized Jacobi identities, like
α[[A,B]ξ , C]η + ξη[[A,C]−α/ξ , B]−α/η − α2[[B,C]ξη/α, A]1/α = 0 αξη 6= 0
β[A, [B,C]α, ]−βγ + γ[B, [C,A]β , ]−γα + α[C, [A,B]γ , ]−αβ = 0 α, β, γ = ±1
[[A,B]η, C]− + [[B,C]η, A]− + [[C,A]η , B]− = 0 η = ±1
[[A,B]ξ, [C,D]η ]− = [[A,B]ξ , C]−,D]η + η[[A,B]ξ ,D]−, C]1/η η 6= 0,
(6.21)
one can obtain a number of other (para)commutation relations for which the reader is referred
to [16,18,26].
Of course, the paracommutation relations (6.16), in particular (6.18) and (6.20) as their
stronger versions, do not give the general solution of the trilinear relations (6.12). For
instance, one may replace (6.12) with the equations
a+l , [a
m , a
n ]ε + [a
+ 2(1 + τ)δlna
m = 0 (6.22a)
a−l , [a
m , a
n ]ε + [a
− 2(1 + τ)δlma−n = 0. (6.22b)
and their Hermitian conjugate, which in terms of the operators bl and cl introduced above
[bl, [b
m, bn]ε + [c
m, cm]ε] = 2(1 + τ)δlmbn (6.23a)
[cl, [b
m, bn]ε + [c
m, cm]ε] = 2(1 + τ)δlmcn, (6.23b)
and supplement these relations with equations like (6.18b). Obviously, equations (6.16) con-
vert (6.22) into identities and, consequently, the (standard) paracommutation relations (6.20)
provide a solution of (6.23). On the base of (6.23) or other similar equations that can be
obtained by generalizing the ones in (6.10)–(6.12), further research on particular classes of
trilinear commutation relations can be done, but, however, this is not a subject of the present
work.
Let us now pay attention to the fact that equations (6.10), (6.11) and (6.12) are generally
different (regardless of existence of some connections between their solutions). The cause for
this being that the momentum operators for the Lagrangians L′, L′′ and L′′′ are generally
different unless some additional restrictions are added to the Lagrangian formalism (see
Sect. 4). A necessary and sufficient condition for (6.10)–(6.12) to be identical is
[a±l , [a
m , a
m]−ε − [a+m, a†−m ]−ε] = 0, (6.24)
which certainly is valid if the condition (4.9′), viz.
[a†+m , a
m]−ε − [a+m, a†−m ]−ε = 0, (6.25)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 25
ensuring the uniqueness of the momentum operator are, holds. If one adopts the standard
bilinear commutation relations (6.13), then (6.25), and hence (6.24), is identically valid, but
in the framework of, e.g., the paracommutation relations (6.16) (or (6.20) in other form) the
equations (6.25) should be postulated to ensure uniqueness of the momentum operator and
therefore of the field equations.
On the base of (6.10) or (6.11) one may invent other types of commutation relations,
which will not be investigated in this paper because we shall be interested mainly in the
case when (6.10), (6.11) and (6.12) are identical (see (6.24)) or, more generally, when the
dynamical variables are unique in the sense pointed in Sect. 4.
6.2. Restrictions related to the charge operator
The consequences of the Heisenberg relations (5.2), involving the charge operator for a charged
field, q 6= 0 (and hence τ = 0 – see (3.7)), will be examined in this subsection. In terms of
creation and annihilation operators it is equivalent to [1, 13–15]
[a±s (k), Q] = qa±s (k) [a†±s (k), Q] = −qa†±s (k), (6.26)
the values of the polarization indices being specified by (6.5). Substituting here (3.10), we
see that, for a charged field, the field equations for the Lagrangians L′, L′′ and L′′′ (see
Sect. 3) respectively are:
2j+1−δ0m(1−δ0j )
d3p{[a±s (k), a
t (p) ◦ a−t (p)− εa
t (p) ◦ a+t (p)] − a±s (k)δstδ3(k − p)} = 0
(6.27a)
2j+1−δ0m(1−δ0j )
d3p{[a†±s (k), a
t (p) ◦ a−t (p)− εa
t (p) ◦ a+t (p)] + a†±s (k)δstδ3(k − p)} = 0
(6.27b)
2j+1−δ0m(1−δ0j )
d3p{[a±s (k), a+t (p) ◦ a
t (p)− εa−t (p) ◦ a
t (p)] + a
s (k)δstδ
3(k − p)} = 0
(6.28a)
2j+1−δ0m(1−δ0j )
d3p{[a†±s (k), a+t (p) ◦ a
t (p)− εa−t (p) ◦ a
t (p)] − a†±s (k)δstδ3(k − p)} = 0
(6.28b)
2j+1−δ0m(1−δ0j )
d3p{[a±s (k), [a
t (p), a
t (p)]ε − [a+t (p), a
t (p)ε] − 2a±s (k)δstδ3(k − p)} = 0
(6.29a)
2j+1−δ0m(1−δ0j )
d3p{[a†±s (k), [a
t (p), a
t (p)]ε − [a+t (p), a
t (p)ε] + 2a
s (k)δstδ
3(k − p)}=0.
(6.29b)
Using (6.27)–(6.29) and (6.8), with η = ε = −1, or simply (6.26), one can easily verify
the validity of the equations
[ P̃µ, Q̃] = 0 [ L̃µν , Q̃] = 0
[ S̃µν , Q̃] = 0 [M̃µν , Q̃] = 0,
(6.30)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 26
where the operators P̃µ, Q̃, S̃µν , L̃µν and M̃µν are calculated from one and the same
initial Lagrangian according to (3.9)–(3.12). This result confirms the validity of assertion (ii)
before (5.24) for the fields considered.
Following the above considerations, concerning the momentum operator, we shall now
replace the systems of integral equations (6.27)–(6.29) with respectively the following stronger
systems of algebraic equations (by equating to zero the integrands in (6.27)–(6.29)):
a±l , a
m ◦ a−m − εa†−m ◦ a+m
− δlma±l = 0 (6.31a)
, a†+m ◦ a−m − εa†−m ◦ a+m
+ δlma
= 0 (6.31b)
a±l , a
m ◦ a†−m − εa−m ◦ a†+m
+ δlma
l = 0 (6.32a)
, a+m ◦ a†−m − εa−m ◦ a†+m
− δlma†±l = 0 (6.32b)
a±l , [a
m , a
m]ε − [a+m, a†−m ]ε
− 2δlma±l = 0 (6.33a)
, [a†+m , a
m]ε − [a+m, a†−m ]ε
+ 2δlma
= 0. (6.33b)
These trilinear commutation relations are similar to (6.10)–(6.12) and, consequently, can be
treated in analogous way.
By invoking (6.8), it is a simple algebra to be proved that the standard bilinear commu-
tation relations (6.13) convert (6.31)–(6.33) into identities. Thus (6.13) are stronger version
of (6.31)–(6.33) and, in this sense, any type of commutation relations, which provide a
solution of (6.31)–(6.33) and is compatible with (6.13), is a suitable candidate for general-
izing (6.13). To illustrate that idea, we shall proceed with (6.33) in a way similar to the
‘derivation’ of the paracommutation relations from (6.12).
Obviously, the equations (cf. (6.14) with τ = 0, as now q 6= 0)
, [a+m, a
m ]ε] + δlma
m = 0 (6.34a)
, [a†+m , a
m]ε] − δlma±m = 0 (6.34b)
and their Hermitian conjugate provide a solution of (6.33), but, as a direct calculations shows,
they do not agree with the standard (anti)commutation relations (6.13). A solution of (6.33)
compatible with (6.13) is given by the equations (6.15), with τ = 0 as the field considered is
charged one — see (3.7). Therefore equations (6.16), with τ = 0, also provide a compatible
with (6.13) solution of (6.33), from where immediately follows that the paracommutation
relations (6.20), with τ = 0, convert (6.33) into identities. To conclude, we can say that the
paracommutation relations (6.20), in particular their special case (6.13), ensure the simul-
taneous validity of the Heisenberg relations (5.1) and (5.2) for free scalar, spinor and vector
fields.
Similarly to (6.22), one may generalize (6.33) to
a+l , [a
m , a
n ]ε − [a+m, a†−n ]ε
− 2δlna+m = 0 (6.35a)
a−l , [a
m , a
n ]ε − [a+m, a†−n ]ε
− 2δlma−n = 0. (6.35b)
which equations agree with (6.13), (6.15), (6.16) and (6.20), but generally do not agree
with (6.22), with τ = 0, unless the equations (6.16), with τ = 0, hold.
More generally, we can assert that (6.33) and (6.12), with τ = 0, hold simultaneously if
and only if (6.15), with τ = 0, is fulfilled. From here, again, it follows that the paracommu-
tation relations ensure the simultaneous validity of (5.1) and (5.2).
Let us say now some words on the uniqueness problem for the Heisenberg equations
involving the charge operator. The systems of equations (6.31)–(6.33) are identical iff
a±l , [a
m , a
m]−ε + [a
m ]−ε
= 0, (6.36)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 27
which, in particular, is satisfied if the condition
[a†+m , a
m]−ε + [a
m ]−ε = 0, (6.37)
ensuring the uniqueness of the charge operator (see (4.10′)), is valid. Evidently, equa-
tions (6.36) and (6.24) are compatible iff
a+l , [a
m , a
a−l , [a
m , a
= 0 (6.38)
which is a weaker form of (4.15) ensuring simultaneous uniqueness of the momentum and
charge operator.
6.3. Restrictions related to the angular momentum operator(s)
It is now turn to be investigated the restrictions on the creation and annihilation operators
that follow from the Heisenberg relations (5.3) concerning the angular momentum operator.
They can be obtained by inserting the equations (3.11) and (3.12) into (5.3). As pointed
in Sect. 5, the resulting equalities, however, depend not only on the particular Lagrangian
employed, but also on the geometric nature of the field considered; the last dependence
being explicitly given via (5.25) and the polarization functions σss
µν (k) and l
ss′m±
µν (k) (see
also (3.14)).
Consider the terms containing derivatives in (5.3),
L̃orµν := i~
ϕ̃i(x). (6.39)
If ϕ̃
(k) denotes the Fourier image of ϕ̃i(x), i.e.
ϕ̃i(x) = Λ
d4ke−
kµxµ ϕ̃
(k), (6.40)
with Λ being a normalization constant, then the Fourier image of (6.39) is
(k). (6.41)
Comparing this expression with equations (3.12), we see that the terms containing derivatives
in (3.12) should be responsible for the term (6.39) in (5.3).20 For this reason, we shall suppose
that the momentum operator M̃µν admits a representation
M̃µν = M̃orµν + M̃spµν (6.42)
such that the operators M̃orµν and M̃
µν satisfy the relations (5.4) and (5.5), respectively.
Thus we shall replace (5.3) with the stronger system of equations (5.4)–(5.5). Besides,
we shall admit that the explicit form of the operatorsM̃orµν and M̃
µν are given via (5.13)
and (5.12) for the fields investigated in the present work.
Let us consider at first the ‘orbital’ Heisenberg relations (5.4), which is independent
of the particular geometrical nature of the fields studied. Substituting (5.13) and (6.40)
into (5.4), using that ϕ̃
(±k), with k2 = m2c2, is a linear combination of ã±s (k) with classical,
not operator-valued, functions of k as coefficients [1, 13–15] and introducing for brevity the
operator
ωµν(k) := kµ
, (6.43)
20 The terms proportional to the momentum operator in (3.12) disappear if the creation and annihilation
operators (2.9) in Heisenberg picture are employed (see also [13–15]).
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 28
we arrive to the following integro-differential systems of equations:
2j+1−δ0m(1−δ0j )
(−ωµν(p) + ωµν(q))([ã±s (k), ã
t (p) ◦ ã−t (q)
− εã†−t (p) ◦ ã+t (q)] )
m2c2+p2
= 2(1 + τ)ωµν(k)(ã
s (k)) (6.44a)
2j+1−δ0m(1−δ0j )
(−ωµν(p) + ωµν(q))([ã† ±s (k), ã
t (p) ◦ ã−t (q)
− εã†−t (p) ◦ ã+t (q)] )
m2c2+p2
= 2(1 + τ)ωµν(k)(ã
s (k)) (6.44b)
2j+1−δ0m(1−δ0j )
(−ωµν(p) + ωµν(q))([ã±s (k), ã+t (p) ◦ ã
t (q)
− εã−t (p) ◦ ã
t (q)] )
m2c2+p2
= 2(1 + τ)ωµν(k)(ã
s (k)) (6.45a)
2j+1−δ0m(1−δ0j )
(−ωµν(p) + ωµν(q))([ã† ±s (k), ã+t (p) ◦ ã
t (q)
− εã−t (p) ◦ ã
t (q)] )
m2c2+p2
= 2(1 + τ)ωµν(k)(ã
s (k)) (6.45b)
2j+1−δ0m(1−δ0j )
(−ωµν(p) + ωµν(q))([ã±s (k), [ã
t (p), ã
t (q)]ε
+ [ã+t (p), ã
t (q)]ε] )
m2c2+p2
= 4(1 + τ)ωµν(k)(ã
s (k)) (6.46a)
2j+1−δ0m(1−δ0j )
(−ωµν(p) + ωµν(q))([ã† ±s (k), [ã
t (p), ã
t (q)]ε
+ [ã+t (p), ã
t (q)]ε] )
m2c2+p2
= 4(1 + τ)ωµν(k)(ã
s (k)), (6.46b)
where k0 =
m2c2 + k2 is set after the differentiations are performed (see (6.43)). Follow-
ing the procedure of the previous considerations, we replace the integro-differential equa-
tions (6.44)–(6.46) with the following differential ones:
(−ω◦µν(m) + ω◦µν(n))([ã±l , ã
m ◦ ã−n − εã†−m ◦ ã+n ] )
= 2(1 + τ)δlmω
µν(l)(ã
l ) (6.47a)
(−ω◦µν(m)+ω◦µν(n))([ã
l , ã
m ◦ ã−n − εã†−m ◦ ã+n ] )
= 2(1+ τ)δlmω
µν(l)(ã
l ) (6.47b)
(−ω◦µν(m) + ω◦µν(n))([ã±l , ã
m ◦ ã†−n − εã−m ◦ ã†+n ] )
= 2(1 + τ)δlmω
µν(l)(ã
) (6.48a)
(−ω◦µν(m)+ω◦µν(n))([ã
l , ã
m ◦ ã†−n − εã−m ◦ ã†+n ] )
= 2(1+ τ)δlmω
µν(l)(ã
l ) (6.48b)
(−ω◦µν(m) + ω◦µν(n))([ã±l , [ã
m , ã
n ]ε + [ã
m, ã
n ]ε] )
= 4(1 + τ)δlmω
µν(l)(ã
l ) (6.49a)
(−ω◦µν(m) + ω◦µν(n))([ã
, [ã†+m , ã
n ]ε + [ã
m, ã
n ]ε] )
= 4(1 + τ)δlmω
µν(l)(ã
(6.49b)
where we have set (cf. (6.43))
ω◦µν(l) := ωµν(k) = kµ
if l = (s,k) (6.50)
and k0 =
m2c2 + k2 is set after the differentiations are performed.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 29
Remark. Instead of (6.47)–(6.49) one can write similar equations in which the operator
−ω◦µν(m) or +ω◦µν(n) is deleted and the factor +12 or −
, respectively, is added on their right
hand sides. These manipulations correspond to an integration by parts of some of the terms
in (6.44)–(6.46).
The main difference of the obtained trilinear relations with respect to the previous ones
considered above is that they are partial differential equations of first order.
The relations (6.49) agree with the equations (6.16) in a sense that if (6.16) hold,
then (6.49) become identically valid. Indeed, since
(−ω◦µν(m) + ω◦µν(n))(ã±mδln)
= −2δlmω◦µν(m)(ã±m)
(−ω◦µν(m) + ω◦µν(n))(ã±n δlm)
= +2δlmω
µν(m)(ã
(6.51)
due to (6.50), (6.43) and the equality
dδ(x)
f(x) = −δ(x)df(x)
for a C1 function f , the
application of the operator (−ω◦µν(m) + ω◦µν(n)) to (6.16) and subsequent setting n = m
entails (6.49). In particular, this means that the paracommutation relations (6.20) and,
moreover, the standard (anti)commutation relations (6.13) convert (6.49) into identities.
Therefore the ‘orbital’ Heisenberg relations (5.4) hold for scalar, spinor and vector fields
satisfying the bilinear or para commutation relations.
It should be noted, the paracommutation relations are not the only trilinear commutation
relations that are solutions of (6.49). As an example, we shall present the trilinear relations
a+l , [a
a+l , [a
m , a
= −(1 + τ)δlna+m (6.52a)
a−l , [a
a−l , [a
m , a
= +(1 + τ)δlma
n , (6.52b)
which reduce to (6.14) for n = m, do not agree with (6.13), but convert (6.49) into identities
(see (6.51)). Other example is provided by the equations (6.22), which are compatible with
the paracommutation relations and, as a result of (6.51), convert (6.49) into identities. Prima
facie one may suppose that any solution of (6.12) provides a solution of (6.49), but this is
not the general case. A counterexample is provided by the commutation relations
a±l , [a
m , a
n ]ε + [a
± 2(1 + τ)δlna±m = 0, (6.53)
which reduce to (6.12) for n = m, satisfy (6.49) with ã+l for ã
l , and do not satisfy (6.49)
with ã−l for ã
l (see (6.51) and cf. (6.22)).
From (5.13) follows that the operator M̃orµν is independent of the Lagrangian L′, L′′ or
L′′′ one starts off if and only if (see (4.11))
(−ω◦µν(m) + ω◦µν(n))
[ã†+m , ã
n ]−ε − [ã+m, ã†−n ]−ε
= 0. (6.54)
This condition ensures the coincidence of the systems of equations (6.47), (6.48) and (6.49)
too. However, the following necessary and sufficient condition for the coincidence of these
systems is expressed by the weaker equations
(−ω◦µν(m) + ω◦µν(n))
ã±l , [ã
m , ã
n ]−ε − [ã+m, ã†−n ]−ε
= 0. (6.55)
It is now turn to be considered the ‘spin’ Heisenberg relations (5.5).
Recall, the field operators ϕi for the fields considered here admit a representation [13–15]
ϕi = Λ
i (p)a
t (p) + v
i (p)a
t (p)
, (6.56)
where Λ is a normalization constant and v
i (p) are classical, not operator-valued, complex
or real functions which are linearly independent. The particular definition of v
i (p) depends
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 30
on the geometrical nature of ϕi and can be found in [13–15] (see also [1]), where the reader
can find also a number of relations satisfied by v
i (p). Here we shall mention only that
i (p) = 1 for a scalar field and v
i (p) = v
i (p) =: v
i(p) = (v
i(p))
∗ for a vector field.
The explicit form of the polarization functions σ
ss′,±
µν (k) and l
ss′,±
µν (k) (see Sect. 3, in
particular (3.14)) through v
i (k) are [13–15]:
µν (k) =
(−1)j
j + δj0
i (k))
∗Iii′µνv
µν (k) =
(−1)j
2j + δj0
i (k))
←−−−−−→
←−−−−−→
i (k),
(6.57)
with an exception that σ
ss′,±
0a (k) = σ
ss′,±
a0 (k) = 0, a = 1, 2, 3, for a spinor field, j =
, [14].
Evidently, the equations (3.14) follow from the mentioned facts (see also (5.25)).
Substituting (6.56) and (5.12) into (5.5), we obtain the following systems of integral
equations (corresponding respectively to the Lagrangians L′, L′′ and L′′′):
(−1)j+1j
1 + τ
s,s′,t
i (p)
µν (k) + l
ss′,−
µν (k))[a
t (p), a
s (k) ◦ a−s′(k)]
+ (σss
µν (k) + l
ss′,+
µν (k))[a
t (p), a
s (k) ◦ a+s′(k)]
d3pIi
(p)a±t (p) (6.58)
(−1)j+1j
1 + τ
s,s′,t
i (p)
µν (k) + l
ss′,+
µν (k))[a
t (p), a
(k) ◦ a†−s (k)]
+ (σss
µν (k) + l
ss′,−
µν (k))[a
t (p), a
(k) ◦ a†+s (k)]
d3pIi
(p)a±t (p) (6.59)
(−1)j+1j
2(1 + τ)
s,s′,t
i (p)
µν (k) + l
ss′,−
µν (k))
a±t (p), [a
s (k), a
(k)]ε
+ (σss
µν (k) + l
ss′,+
µν (k))
a±t (p), [a
s (k), a
(k)]ε
d3pIi
(p)a±t (p).
(6.60)
For the difference of all previously considered systems of integral equations, like (6.2)–
(6.4), (6.27)–(6.29) and (6.44)–(6.46), the systems (6.58)–(6.60) cannot be replaced by ones
consisting of algebraic (or differential) equations. The cause for this state of affairs is that
in (6.58)–(6.60) enter polarization modes with arbitrary s and s′ and, generally, one cannot
‘diagonalize’ the integrand(s) with respect to s and s′; moreover, for a vector field, the modes
with s = s′ are not presented at all (see (3.14)). That is why no commutation relations can
be extracted from (6.58)–(6.60) unless further assumptions are made. Without going into
details, below we shall sketch the proof of the assertion that the commutation relations (6.16)
convert (6.60) into identities for massive spinor and vector fields.21 In particular, this entails
that the paracommutation and the bilinear commutation relations provide solutions of (6.60).
Let (6.16) holds. Combining it with (6.60), we see that the latter splits into the equations
21 The equations (6.58)–(6.60) are identities for scalar fields as for them Iµν = 0 and v
i (k) = 1, which
reflects the absents of spin for these fields.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 31
(−1)jj
1 + τ
i (p)
τ(σst,−µν (p) + l
µν (p)) + ε(σ
µν (p) + l
µν (p))
a+s (p),
(p)a+s (p) (6.61a)
(−1)j+1j
1 + τ
i (p)
(σts,−µν (p) + l
µν (p)) + ετ(σ
µν (p) + l
µν (p))
a−s (p),
i′ (p)a
s (p). (6.61b)
Inserting here (6.57), we see that one needs the explicit definition of v
i (k) and formulae for
sums like ρii′(k) :=
i (k)(v
(k))∗, which are specific for any particular field and can
be found in [13–15]. In this way, applying (5.25), (3.7) and the mentioned results from [13–15],
one can check the validity of (6.61) for massive fields in a way similar to the proof of (5.3)
in [13–15] for scalar, spinor and vector fields, respectively.
We shall end the present subsection with the remark that the equations (4.17) and (4.18),
which together with (4.15) ensure the uniqueness of the spin and orbital operators, are
sufficient conditions for the coincidence of the equations (6.58), (6.59) and (6.60).
7. Inferences
To begin with, let us summarize the major conclusions from Sect. 6. Each of the Heisenberg
equations (5.1)–(5.3), the equations (5.3) being split into (5.4) and (5.5), induces in a natural
way some relations that the creation and annihilation operators should satisfy. These rela-
tions can be chosen as algebraic trilinear ones in a case of (5.1) and (5.2) (see (6.10)–(6.12)
and (6.31)–(6.33), respectively). But for (5.4) and (5.5) they need not to be algebraic and
are differential ones in the case of (5.4) (see (6.47)–(6.49)) and integral equations in the case
of (5.5) (see (6.58)–(6.60)). It was pointed that the cited relations depend on the initial
Lagrangian from which the theory is derived, unless some explicitly written conditions hold
(see (6.24), (6.37) and (6.55)); in particular, these conditions are true if the equations (4.9)–
(4.13), ensuring the uniqueness of the corresponding dynamical operators, are valid. Since
the ‘charge symmetric’ Lagrangians (3.4) seem to be the ones that best describe free fields,
the arising from them (commutation) relations (6.12), (6.33), (6.49) and (6.60) were stud-
ied in more details. It was proved that the trilinear commutation relations (6.16) convert
them into identities, as a result of which the same property possess the paracommutation
relations (6.20) and, in particular, the bilinear commutation relations (6.13). Examples of tri-
linear commutation relations, which are neither ordinary nor para ones, were presented; some
of them, like (6.14), (6.34) and (6.52), do not agree with (6.13) and other ones, like (6.16),
(6.22) and (6.35), generalize (6.20) and hence are compatible with (6.13). At last, it was
demonstrated that the commutators between the dynamical variables (see (5.15)–(5.23)) are
uniquely defined if a Heisenberg relation for one of the operators entering in it is postulated.
The chief aim of the present section is to be explored the problem whether all of the
reasonable conditions, mentioned in the previous sections and that can be imposed on the
creation and annihilation operators, can hold or not hold simultaneously. This problem is
suggested by the strong evidences that the relations (5.1)–(5.3) and (5.15)–(5.23), with a
possible exception of (5.3) (more precisely, of (5.5)) in the massless case, should be valid in
a realistic quantum field theory [1, 3, 7, 8, 11, 12]. Besides, to the arguments in loc. cit., we
shall add the requirement for uniqueness of the dynamical variables (see Sect. 4).
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 32
As it was shown in Sect. 6, the relations (5.1), (5.2), (5.4) and (5.5) are compatible if
one starts from a charge symmetric Lagrangian (see (3.4)), which best describes a free field
theory; in particular, the commutation relations (6.16) (and hence (6.20) and (6.13)) ensure
their simultaneous validity.22 For that reason, we shall investigate below only commutation
relations for which (5.1), (5.2), (5.4) and (5.5) hold. It will be assumed that they should be
such that the equations (6.10)–(6.12), (6.31)–(6.33), (6.47)–(6.49) and (6.58)–(6.60), respec-
tively, hold.
Consider now the problem for the uniqueness of the dynamical variables and its consis-
tency with the commutation relations just mentioned for a charged field. It will be assumed
that this uniqueness is ensured via the equations (4.9)–(4.11).
The equation (4.15), viz.
[a†±m , a
m]−ε = 0, (7.1)
is a necessary and sufficient conditions for the uniqueness of the momentum and charge
operators (see Sect. 4 and the notation introduced at the beginning of Sect. 6). Before
commenting on this relation, we would like to derive some consequences of it. Applying
consequently (6.8) for η = −ε, (7.1) and the identity
[A,B ◦ C]+ = [A,B]η ◦ C − ηB ◦ [A,C]−η η = ±1 (7.2)
for η = +ε,−ε, we, in view of (7.1), obtain
[a+m, [a
m ]ε] = [a
m , [a
m]−ε]+ = (1− ε)[a†−m , a+m]ε ◦ a+m
[a−m, [a
m , a
m]ε] = ε[a
m , [a
m]−ε]+ = ε(1 − ε)[a†+m , a−m]ε ◦ a−m.
(7.3)
Forming the sum and difference of (6.12a), for τ = 0, and (6.33a), we see that the system
of equations they form is equivalent to
[a+l , [a
m , a
m]ε] = 0 [a
l , [a
m ]ε] = 0 (7.4a)
, [a+m, a
m ]ε] + 2δlma
= 0 [a−
, [a†+m , a
m]ε] − 2δlma−l = 0. (7.4b)
Combining (7.4b), for l = m, with (7.3), we get
(1− ε)[a†−m , a+m]ε ◦ a+m + 2a+m = 0 ε(1− ε)[a†+m , a−m]ε ◦ a−m − 2a−m = 0. (7.5)
Obviously, these equations reduce to
a±m = 0 (7.6)
for bose fields as for them ε = +1 (see (3.7)). Since the operators (7.6) describe a completely
unobservable field, or, more precisely, an absence of a field at all, the obtained result means
that the theory considered cannot describe any really existing physical field with spin j = 0, 1.
Such a conclusion should be regarded as a contradiction in the theory. For fermi fields, j = 1
and ε = −1, the equations (7.5) have solutions different from (7.6) iff a±m are degenerate
operators, i.e. with no inverse ones, in which case (7.4a) is a consequence of (7.5) and (7.1)
(see (6.8) and (7.3) too).
The source of the above contradiction is in the equation (7.1), which does not agree with
the bilinear commutation relations (6.13) and contradicts to the existing correlation between
creation and annihilation of particles with identical characteristics (m = (t,p) in our case)
as (7.1) can be interpreted physically as mutual independence of the acts of creation and
annihilation of such particles [1, § 10.1].
At this point, there are two ways for ‘repairing’ of the theory. On one hand, one can
forget about the uniqueness of the dynamical variables (in a sense of Sect. 4), after which
22 The special case(s) when (5.5) may not hold for a massless field will not be considered below.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 33
the formalism can be developed by choosing, e.g., the charge symmetric Lagrangians (3.4)
and following the usual Lagrangian formalism; in fact, this is the way the parafield theory is
build [16,18]. On another hand, one may try to change something at the ground of the theory
in such a way that the uniqueness of the dynamical variables to be ensured automatically.
We shall follow the second method. As a guiding idea, we shall have in mind that the
bilinear commutation relations (6.13) and the related to them normal ordering procedure
provide a base for the present-day quantum field theory, which describes sufficiently well
the discovered elementary particles/fields. On this background, an extensive exploration of
commutation relations which are incompatible with (6.13) is justified only if there appear
some evidences for fields/particles that can be described via them. In that connection it
should be recalled [17, 18], it seems that all known particles/fields are described via (6.13)
and no one of them is a para particle/field.
Using the notation introduced at the beginning of Sect. 4, we shall look for a linear
mapping (operator) E on the operator space over the system’s Hilbert space F of states such
E(D′) = E(D′′). (7.7)
As it was shown in Sect. 4, an example of an operator E is provided by the normal ordering
operator N . Therefore an operator satisfying (7.7) always exists. To any such operator E
there corresponds a set of dynamical variables defined via
D = E(D′). (7.8)
Let us examine the properties of the mapping E that it should possess due to the re-
quirement (7.7).
First of all, as the operators of the dynamical variables should be Hermitian, we shall
require
= E(B†) (7.9)
for any operator B, which entails
D† = D, (7.10)
due to (3.9)–(3.12) and (7.8).
As in Sect. 4, we shall replace the so-arising integral equations with corresponding alge-
braic ones. Thus the equations (4.5)–(4.20) remain valid if the operator E is applied to their
left hand sides.
Consider the general case of a charged field, q 6= 0. So, the analogue of (4.15) reads
[a†±m , a
= 0, (7.11)
which equation ensures the uniqueness of the momentum and charge operators. Respectively,
the condition (4.11) transforms into
(−ω◦µν(m) + ω◦µν(n))
E([a†+m , a−n ]−ε)− E([a+m, a†−n ]−ε)
= 0, (7.12)
which, by means of (7.11) can be rewritten as (cf. (4.16))
ω◦µν(n)
E([a†+m , a−n ]−ε)− E([a+m, a†−n ]−ε)
= 0. (7.13)
At the end, equations (4.17) and (4.18) now should be written as
µν (k) E
[a†+s (k), a
(k)]−ε
+ σss
µν (k) E
[a†−s (k), a
(k)]−ε
= 0 (7.14)
µν (k) E
[a†+s (k), a
(k)]−ε
+ lss
µν (k) E
[a†−s (k), a
(k)]−ε
= 0. (7.15)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 34
These equations can be satisfied if we generalize (7.11) to (cf. (4.20))
[a†±s (k), a
(k)]−ε
= 0 (7.16)
for any s and s′. At last, the following stronger version of (7.16)
[a†±m , a
n ]−ε
= 0, (7.17)
for any m = (t,p) and n = (r, q), ensures the validity of (7.14) and (7.15) and thus of the
uniqueness of all dynamical variables.
It is time now to call attention to the possible commutation relations. The replacement
D′, D′′, D′′′ 7→ D := E(D′) = E(D′′) = E(D′′′) results in corresponding changes in the
whole of the material of Sect. 6. In particular, the systems of commutation relations (6.10)–
(6.12), (6.31)–(6.33), (6.47)–(6.49) and (6.58)–(6.60) should be replaced respectively with:23
a±l , E(a
m ◦ a−m) + ε E(a†−m ◦ a+m)
± (1 + τ)δlma±l = 0 (7.18)
a±l , E(a
m ◦ a−m)− ε E(a†−m ◦ a+m)
− δlma±l = 0 (7.19)
(−ω◦µν(m) + ω◦µν(n))([ã±l , E(ã
m ◦ ã−n )− ε E(ã†−m ◦ ã+n )] )
= 2(1 + τ)δlmω
µν(l)(ã
(7.20)
(−1)j+1j
1 + τ
s,s′,t
i (p)
µν (k) + l
ss′,−
µν (k))[a
t (p), E(a†+s (k) ◦ a−s′(k))]
+ (σss
µν (k) + l
ss′,+
µν (k))[a
t (p), E(a†−s (k) ◦ a+s′(k))]
d3pIi
i′ (p)a
t (p).
(7.21)
Due to the uniqueness conditions (7.11)–(7.14), one can rewrite the terms E(a†±m ◦ a∓m)
in (7.18)–(7.21) in a number of equivalent ways; e.g. (see (7.11))
E(a†±m ◦ a∓m) = ε E(a∓m ◦ a†±m ) =
E([a†±m , a∓m]ε). (7.22)
Consider the general case of a charged field, q 6= 0 (and hence τ = 0). The system of
equations (7.18)–(7.19) is then equivalent to
, E(a†±m ◦ a∓m)
= 0 (7.23a)
, E(a†−m ◦ a+m)
+ εδlma
= 0 (7.23b)
a−l , E(a
m ◦ a−m)
− δlma−l = 0. (7.23c)
These (commutation) relations ensure the simultaneous fulfillment of the Heisenberg rela-
tions (5.1) and (5.2) involving the momentum and charge operators, respectively. To ensure
also the validity of (7.20), with τ = 0, and, consequently, of (5.4), we generalize (7.23) to
a±l , E(a
m ◦ a∓n )
= 0 (7.24a)
, E(a†−m ◦ a+n )
+ εδlma
n = 0 (7.24b)
, E(a†+m ◦ a−n )
− δlma−n = 0, (7.24c)
for any l = (s,k), m = (t,p) and n = (t, q) (see also (6.51)). In the way pointed in
Sect. 6, one can verify that (7.24) for any l = (s,k), m = (t,p) and n = (r,p) entails (7.21)
and hence (5.5). At last, to ensure the validity of all of the mentioned conditions and a
23 To save some space, we do not write the Hermitian conjugate of the below-written equations.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 35
suitable transition to a case of Hermitian field, for which q = 0 and τ = 1 (see (3.7)), we
generalize (7.24) to
a+l , E(a
m ◦ a−n )
+ τδlna
m = 0 (7.25a)
a−l , E(a
m ◦ a+n )
− ετδlna−m = 0 (7.25b)
, E(a†−m ◦ a+n )
+ εδlma
n = 0, (7.25c)
, E(a†+m ◦ a−n )
− δlma−n = 0 (7.25d)
where l, m and n are arbitrary. As a result of (7.17), which we assume to hold, and τa
τa±l (see (3.7)), the equations (7.25a) and (7.25c) (resp. (7.25b) and (7.25d)) become identical
when τ = 1 (and hence a
l = a
l ); for τ = 0 the system (7.25) reduces to (7.24). Recalling
that ε = (−1)2j (see (3.7)), we can rewrite (7.25) in a more compact form as
a±l , E(a
m ◦ a∓n )
+ (±1)2j+1τδlna±m = 0 (7.26a)
a±l , E(a
m ◦ a±n )
− (∓1)2j+1τδlma±n = 0. (7.26b)
Since the last equation is equivalent to (see (7.17)) and use that ε = (−1)2j)
, E(a±m ◦ a†∓n )
+ (±1)2j+1δlna±m = 0, (7.26b′)
it is evident that the equations (7.26a) and (7.26b) coincide for a neutral field.
Let us draw the main moral from the above considerations: the equations (7.17) are
sufficient conditions for the uniqueness of the dynamical variables, while (7.26) are such
conditions for the validity of the Heisenberg relations (5.1)–(5.5), in which the dynamical
variables are redefined according to (7.8). So, any set of operators a±
and E , which are
simultaneous solutions of (7.17) and (7.26), ensure uniqueness of the dynamical variables
and at the same time the validity of the Heisenberg relations.
Consider the uniqueness problem for the solutions of the system of equations consisting
of (7.17)and (7.26). Writing (7.17) as
E(a†±m ◦ a∓n ) = ε E(a∓n ◦ a†±m ) =
E([a†±m , a∓n ]ε), (7.27)
which reduces to (7.22) for n = m, and using ε = (−1)2j (see (3.7)), one can verify that (7.26)
is equivalent to
a+l , E([a
n ]ε)
+ 2δlna
m = 0 (7.28a)
a+l , E([a
m , a
n ]ε)
+ 2τδlna
m = 0 (7.28b)
a−l , E([a
n ]ε)
− 2τδlma−n = 0 (7.28c)
a−l , E([a
m , a
n ]ε)
− 2δlma−n = 0. (7.28d)
The similarity between this system of equations and (6.16) is more than evident: (7.28) can
be obtained from (6.16) by replacing [·, ·]ε with E([·, ·]ε).
As it was said earlier, the bilinear commutation relations (6.13) and the identification of
E with the normal ordering operator N ,
E = N , (7.29)
convert (7.27)–(7.28) into identities; by invoking (6.8), for η = −ε, the reader can check
this via a direct calculation (see also (4.23)). However, this is not the only possible solution
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 36
of (7.27)–(7.28). For example, if, in the particular case, one defines an ‘anti-normal’ ordering
operator A as a linear mapping such that
A(a+m ◦ a†−n ) := εa†−n ◦ a+m A(a†+m ◦ a−n ) := εa−n ◦ a†+m
A(a−m ◦ a†+n ) := a−m ◦ a†+n A(a†−m ◦ a+n ) := a†−m ◦ a+n ,
(7.30)
then the bilinear commutation relations (6.13) and the setting E = A provide a solution
of (7.27)–(7.28); to prove this, apply (6.8) for η = −ε. Evidently, a linear combination of
N and A, together with (6.13), also provides a solution of (7.27)–(7.28).24 Other solution
of the same system of equations is given by E = id and operators a±
satisfying (6.16), in
particular the paracommutation relations (6.20), and a
m ◦ a,∓n = εa,∓n ◦ a†±m . The problem
for the general solution of (7.27)–(7.28) with respect to E and a±l is open at present.
Let us introduce the particle and antiparticle number operators respectively by (see (7.27),
(7.9) and (3.16))
Nl :=
= E(a+
◦ a†−
) = (Nl)† =: N †l
†Nl :=
l , a
= E(a†+l ◦ a
l ) = (
†Nl)† =: †Nl†.
(7.31)
As a result of the commutation relations (7.28), with n = m, they satisfy the equations25
[Nl, a+m]− = δlma+l (7.32a)
[ †Nl, a+m]− = τδlma+l (7.32b)
[Nl, a†+m ]− = τδlma
l (7.32c)
[ †Nl, a†+m ]− = δlma
l . (7.32d)
Combining (3.9)–(3.12) and (5.11)–(5.13) with (7.8), (7.27) and (7.31), we get the following
expressions for the operators of the (redefined) dynamical variables:
P̃µ =
1 + τ
m2c2+k2
(Nl + †Nl) l = (s,k) (7.33)
Q̃ = q
(−Nl + †Nl) (7.34)
S̃µν =
(−1)j−1/2j~
1 + τ
{εσmn,+µν Nnm + σmn,−µν †Nmn)}
m=(s,k)
n=(s′,k)
(7.35)
L̃µν = x0µ P̃ν − x0 ν P̃µ +
(−1)j−1/2j~
1 + τ
{εlmn,+µν Nnm + lmn,−µν †Nmn)}
m=(s,k)
n=(s′,k)
2(1 + τ)
−ω◦µν(l) + ω◦µν(m)
(Nl + †Nl)
m=l=(s,k)
(7.36)
M̃spµν =
(−1)j−1/2j~
1 + τ
{ε(σmn,+µν + lmn,+µν )Nnm + (σmn,−µν + lmn,−µν ) †Nmn)}
m=(s,k)
n=(s′,k)
(7.37)
M̃orµν =
2(1 + τ)
−ω◦µν(l) + ω◦µν(m)
(Nl + †Nl)
m=l=(s,k)
. (7.38)
24 If we admit a±
to satisfy the ‘anomalous” bilinear commutation relations (8.27) (see below), i.e. (6.13)
with ε for −ε and (±1)2j for (±1)2j+1, then E = N , A also provides a solution of (7.27)–(7.28). However,
as it was demonstrated in [13–15], the anomalous commutation relations are rejected if one works with the
charge symmetric Lagrangians (3.4).
25 The equations (7.32a) and (7.32b) correspond to (7.28a) and (7.28b), respectively, and (7.32c) and (7.32d)
correspond to the Hermitian conjugate to (7.28c) and (7.28d), respectively.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 37
Here ω◦µν(l) is defined via (6.50), we have set
σmn,±µν := σ
ss′,±
µν (k) l
µν := l
ss′,±
µν (k) for m = (s,k) and n = (s
′,k), (7.39)
and (see (7.27))
Nlm :=
[a+l , a
= E(a+l ◦ a
m ) = (Nml)† =: N
†Nlm :=
l , a
= E(a†+l ◦ a
m) = (
†Nml)† =: †Nml†
(7.40)
are respectively the particle and antiparticle transition operators (cf. [26, sec. 1] in a case of
parafields). Obviously, we have
Nl = Nll †Nl = †Nll. (7.41)
The choice (7.29), evidently, reduces (7.33)–(7.36) to (4.24), (4.25), (4.28) and (4.29), respec-
tively.
In terms of the operators (7.38), the commutation relations (7.28) can equivalently be
rewritten as (see also (7.9))
[Nlm, a+n ]− = δmna+l (7.42a)
[ †Nlm, a+n ]− = τδmna+l (7.42b)
[Nlm, a†+n ]− = τδmna
(7.42c)
[ †Nlm, a†+n ]− = δmna
l . (7.42d)
If m = l, these relations reduce to (7.32), due to (7.39).
We shall end this section with the remark that the conditions for the uniqueness of
the dynamical variables and the validity of the Heisenberg relations are quite general and
are not enough for fixing some commutation relations regardless of a number of additional
assumptions made to reduce these conditions to the system of equations (7.27)–(7.28).
8. State vectors, vacuum and mean values
Until now we have looked on the commutation relations only from pure mathematical view-
point. In this way, making a number of assumptions, we arrived to the system (7.27)–(7.28) of
commutation relations. Further specialization of this system is, however, almost impossible
without making contact with physics. For the purpose, we have to recall [1, 3, 11, 12] that
the physically measurable quantities are the mean (expectation) values of the dynamical
variables (in some state) and the transition amplitudes between different states. To make
some conclusions from these basic assumption of the quantum theory, we must rigorously
said how the states are described as vectors in system’s Hilbert space F of states, on which
all operators considered act.
For the purpose, we shall need the notion of the vacuum or, more precisely, the assumption
of the existence of unique vacuum state (vector) (known also as the no-particle condition).
Before defining rigorously this state, which will be denoted by X0, we shall heuristically
analyze the properties it should possess.
First of all, the vacuum state vector X0 should represent a state of the field without any
particles. From here two conclusions may be drawn: (i) as a field is thought as a collection
of particles and a ‘missing’ particle should have vanishing dynamical variables, those of the
vacuum should vanish too (or, more generally, to be finite constants, which can be set equal
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 38
to zero by rescaling some theory’s parameters) and (ii) since the operators a−l and a
l are
interpreted as ones that annihilate a particle characterize by l = (s,k) and charge −q or +q,
respectively, and one cannot destroy an ‘absent’ particle, these operators should transform
the vacuum into the zero vector, which may be interpreted as a complete absents of the field.
Thus, we can expect that
D(X0) = 0 (8.1a)
a−l (X0) = 0 a
l (X0) = 0. (8.1b)
Further, as the operators a+l and a
l are interpreted as ones creating a particle charac-
terize by l = (s,k) and charge −q or +q, respectively, state vectors like a+l (X0) and a
l (X0)
should correspond to 1-particle states. Of course, a necessary condition for this is
X0 6= 0, (8.2)
due to which the vacuum can be normalize to unit,
〈 X0| X0〉 = 1, (8.3)
where 〈·|·〉 : F × F → C is the Hermitian scalar (inner) product of F . More generally, if
, . . .) is a monomial only in i ∈ N creation operators, the vector
ψl1l2... := M(a+l1 , a
, . . .)(X0) (8.4)
may be expected to describe an i-particle state (with i1 particles and i2 antiparticles, i1+i2 =
i, where i1 and i2 are the number of operators a
l and a
l , respectively, in M(a
, . . .)).
Moreover, as a free field is intuitively thought as a collection of particles and antiparticles,
it is natural to suppose that the vectors (8.4) form a basis in the Hilbert space F . But the
validity of this assumption depends on the accepted commutation relations; for its proof,
when the paracommutation relations are adopted, see the proof of [18, p. 26, theorem I-1].
Accepting the last assumption and recalling that the transition amplitude between two
states is represented via the scalar product of the corresponding to them state vectors, it
is clear that for the calculation of such an amplitude is needed an effective procedure for
calculation of scalar products of the form
〈ψl1l2...|ϕm1m2...〉 := 〈 X0|(M(a+l1 , a
, . . .))† ◦ M′(a+m1 , a
, . . .)X0〉, (8.5)
with M and M′ being monomials only in the creation operators. Similarly, for computation
of the mean value of some dynamical operator D in a certain state, one should be equipped
with a method for calculation of scalar products like
〈ψl1l2...| Dϕm1m2...〉 := 〈X0|(M(a+l1 , a
, . . .))† ◦ D ◦ M′(a+m1 , a
, . . .)X0〉. (8.6)
Supposing, for the moment, the vacuum to be defined via (8.1), let us analyze (8.1)–(8.6).
Besides, the validity of (7.27)–(7.28) will be assumed.
From the expressions (7.8) and (3.9)–(3.12) for the dynamical variables, it is clear that
the condition (8.1a) can be satisfied if
E(a†±m ◦ a∓n )(X0) = 0, (8.7)
which, in view of (7.27), is equivalent to any one of the equations
E(a±m ◦ a†∓n )(X0) = 0 (8.8a)
E([a±m, a†∓n ]ε)(X0) = 0. (8.8b)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 39
Equation (8.7) is quite natural as it expresses the vanishment of all modes of the vacuum
corresponding to different polarizations, 4-momentum and charge. It will be accepted here-
after.
By means of (8.8) and the commutation relations (7.28) in the form (7.42), in particu-
lar (7.32), one can explicitly calculate the action of any one of the operators (7.33)–(7.38)
on the vectors (8.4): for the purpose one should simply to commute the operators Nlm (or
Nl = Nll) with the creation operators in (8.4) according to (7.42) (resp. (7.32)) until they
act on the vacuum and, hence, giving zero, as a result of (8.8) and (7.42) (resp. (7.32)). In
particular, we have the equations (k0 =
m2c2 + k2):
a+l (X0)
= kµa
l (X0) P̃µ
l (X0)
= kµa
l (X0) l = (s,k) (8.9)
= −qa+
(X0) Q̃
= +qa
(X0) (8.10)
l=(s,k)
(−1)j−1/2j~
1 + τ
{εσlm,+µν + τσml,−µν }
m=(t,k)
a+m|m=(t,k)(X0)
l=(s,k)
(−1)j−1/2j~
1 + τ
{ετσlm,+µν + σml,−µν }
m=(t,k)
a†+m |m=(t,k)(X0)
(8.11)
l=(s,k)
= (x0 µkν − x0 νkµ)(a+l )(X0)− i~
ω◦µν(l)(a
(−1)j−1/2j~
1 + τ
{εllm,+µν + τ lml,−µν }
m=(t,k)
a+m|m=(t,k)(X0)
l=(s,k)
= (x0 µkν − x0 νkµ)(a†+l )(X0)− i~
ω◦µν(l)(a
(−1)j−1/2j~
1 + τ
{ετ llm,+µν + lml,−µν }
m=(t,k)
a†+m |m=(t,k)(X0)
(8.12)
M̃spµν
l=(s,k)
(−1)j−1/2j~
1 + τ
{ε(σlm,+µν + llm,+µν )
+ τ(σml,−µν + l
µν )}
m=(t,k)
a+m|m=(t,k)(X0)
M̃spµν
l=(s,k)
(−1)j−1/2j~
1 + τ
{ετ(σlm,+µν + llm,+µν )
+ (σml,−µν + l
µν )}
m=(t,k)
a†+m |m=(t,k)(X0)
(8.13)
M̃orµν
ã+l (X0)
= −i~
ω◦µν(l)(ã
(X0) M̃orµν
l (X0)
= −i~
ω◦µν(l)(ã
(X0). (8.14)
These equations and similar, but more complicated, ones with an arbitrary monomial in the
creation operators for a+
are the base for the particle interpretation of the quantum
theory of free fields. For instance, in view of (8.9) and (8.10), the state vectors a+l (X0) and
l (X0) are interpreted as ones representing particles with 4-momentum (
m2c2 + k2,k)
and charges −q and +q, respectively; similar multiparticle interpretation can be given to the
general vectors (8.4) too.
The equations (8.9)–(8.12) completely agree with similar ones obtained in [13–15] on the
base of the bilinear commutation relations (6.13).
By means of (8.7), the expression (8.6) can be represented as a linear combination of
terms like (8.5). Indeed, as D is a linear combinations of terms like E(a†±m ◦a∓n ), by means of
the relations (7.28) we can commute each of these terms with the creation (resp. annihilation)
operators in the monomial M′(a+m1 , a
m2 , . . .) (resp. (M(a+l1 , a
, . . .))† = M′′(a†−
, . . .))
and thus moving them to the right (resp. left) until they act on the vacuum X0, giving
the zero vector — see (8.7). In this way the matrix elements of the dynamical variables,
in particular their mean values, can be expressed as linear combinations of scalar products
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 40
of the form (8.5). Therefore the supposition (8.7) reduces the computation of mean values
of dynamical variables to the one of the vacuum mean value of a product (composition)
of creation and annihilation operators in which the former operators stand to the right of
the latter ones. (Such a product of creation and annihilation operators can be called their
‘antinormal’ product; cf. the properties (7.30) of the antinormal ordering operator A.)
The calculation of such mean values, like (8.5) for states ψ,ϕ 6= X0, however, cannot
be done (on the base of (7.27)–(7.28), (8.7) and (8.1a)) unless additional assumption are
made. For the purpose one needs some kind of commutation relations by means of which
the creation (resp. annihilation) operators on the r.h.s. of (8.5) to be moved to the left (resp.
right) until they act on the left (resp. right) vacuum vector X0; as a result of this operation,
the expressions between the two vacuum vectors in (8.5) should transform into a linear
combination of constant terms and such with no contribution in (8.5). (Examples of the last
type of terms are E(a†±m ◦ a∓) and normally ordered products of creation and annihilation
operators.) An alternative procedure may consists in defining axiomatically the values of all
or some of the mean values (8.5) or, more stronger, the explicit action of all or some of the
operators, entering in the r.h.s. of (8.5), on the vacuum.26 It is clear, both proposed schemes
should be consistent with the relations (7.27)–(7.28), (8.1b) and (8.7)–(8.8).
Let us summarize the problem before us: the operator E in (7.27)–(7.28) has to be fixed
and a method for computation of scalar products like (8.5) should be given provided the
vacuum vector X0 satisfies (8.1b), (8.2), (8.3) and (8.7). Two possible ways for exploration
of this problem were indicated above.
Consider the operator E . Supposing E(a†±m ◦ a∓n ) to be a function only of a
m and a
we, in view of (8.1b), can write E(a†±m ◦ a∓n ) = f±(a
m ◦ a∓n ) ◦ b with b = a−n (upper sign) or
b = a
m (lower sign) and some functions f
±. Applying (7.27), we obtain (do not sum over l)
E(a†+m ◦ a−l ) = f
+(a†+m , a
l ) ◦ a
l E(a
m ◦ a
l ) = f
−(a+m, a
l ) ◦ a
◦ a†+m ) = εf+(a†+m , a−l ) ◦ a
E(a†−
◦ a+m) = εf−(a+m, a
) ◦ a†−
Since E is a linear operator, the expression E(a†±m ◦a∓n ) turns to be a linear and homogeneous
function of a
m and a
n , which immediately implies f
±(A,B) = λ±A for operators A and
B and some constants λ± ∈ C. For future convenience, we assume λ± = 1, which can be
achieved via a suitable renormalization of the creation and annihilation operators.27 Thus,
the last equations reduce to
E(a†+m ◦ a−l ) = a
m ◦ a−l E(a
m ◦ a
) = a+m ◦ a
(8.15a)
E(a−l ◦ a
m ) = εa
m ◦ a−l E(a
l ◦ a
m) = εa
m ◦ a
l . (8.15b)
Evidently, these equations convert (7.27), (8.7) and (8.8) into identities. Comparing (8.15)
and (4.22), we see that the identification
E = N (8.16)
of the operator E with the normal ordering operator N is quite natural. However, for our
purposes, this identification is not necessary as only the equations (8.15), not the general
definition of N , will be employed.
26 Such an approach resembles the axiomatic description of the scattering matrix [1,7,8].
27 Since λ+ = 0 or/and λ− = 0 implies D = 0, due to (7.8), these values are excluded for evident reasons.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 41
As a result of (8.15), the commutation relations (7.28) now read:
[a+l , a
m ◦ a†−n ] + δlna+m = 0 (8.17a)
, a†+m ◦ a−n ] + τδlna+m = 0 (8.17b)
, a+m ◦ a†−n ] − τδlma−n = 0 (8.17c)
[a−l , a
m ◦ a−n ] − δlma−n = 0. (8.17d)
(In a sense, these relations are ‘one half’ of the (para)commutation relations (6.16): the
latter are a sum of the former and the ones obtained from (8.17) via the changes a+m ◦a
n ◦ a+m and a
m ◦ a−n 7→ εa−n ◦ a
m ; the last relations correspond to (7.28) with E = A, A
being the antinormal ordering operator — see (7.30). Said differently, up to the replacement
a±i 7→
for all l, the relations (8.17) are identical with (6.16) for ε = 0; as noted in [26, the
remarks following theorem 2 in sec. 1], this is a quite exceptional case from the view-point
of parastatistics theory.) By means of (6.8) for η = −ε, one can verify that equations (8.17)
agree with the bilinear commutation relations (6.13), i.e. (6.13) convert (8.17) into identities.
The equations (8.15) imply the following explicit forms of the number operators (7.31)
and the transition operators (7.40):
Nl = a+l ◦ a
†Nl = a†+l ◦ a
l (8.18)
Nlm = a+l ◦ a
†Nlm = a†+l ◦ a
m. (8.19)
As a result of them, the equations (7.33)–(7.36) are simply a different form of writing of (4.24),
(4.25), (4.28) and (4.29), respectively.
Let us return to the problem of calculation of vacuum mean values of antinormal ordered
products like (8.5). In view of (8.1b) and (8.3), the simplest of them are
〈 X0|λ idF (X0)〉 = λ 〈X0|M±(X0)〉 = 0 (8.20)
where λ ∈ C and M+ (resp. M−) is any monomial of degree not less than 1 only in the
creation (resp. annihilation) operators; e.g. M± = a±l , a
l , a
◦a±l2 , a
◦a†±l2 . These equations,
with λ = 1, are another form of what is called the stability of the vacuum: if Xi denotes an
i-particle state, i ∈ N∪{0}, then, by virtue of (8.20) and the particle interpretation of (8.4),
we have
〈 Xi| X0〉 = δi0, (8.21)
i.e. the only non-forbidden transition into (from) the vacuum is from (into) the vacuum.
More generally, if Xi′,0 and X0,j′′ denote respectively i′-particle and j′′-antiparticle states,
with X0,0 := X0, then
〈Xi′,0| X0,j′′〉 = δi′0δ0j′′ , (8.22)
i.e. transitions between two states consisting entirely of particles and antiparticles, respec-
tively, are forbidden unless both states coincide with the vacuum. Since we are dealing with
free fields, one can expect that the amplitude of a transitions from an (i′-particle + j′-an-
tiparticle) state Xi′,j′ into an (i′′-particle + j′′-antiparticle) state Xi′′,j′′ is
〈 Xi′,j′| Xi′′,j′′〉 = δi′i′′δj′j′′ , (8.23)
but, however, the proof of this hypothesis requires new assumptions (vide infra).
Let us try to employ (8.17) for calculation of expressions like (8.5). Acting with (8.17)
and their Hermitian conjugate on the vacuum, in view of (8.1b), we get
a+m ◦ (−a†−n ◦ a+l + δln idF)(X0) = 0 a
n ◦ (a−m ◦ a
− δlm idF )(X0) = 0
a†+m ◦ (−a−n ◦ a+l + τδln idF )(X0) = 0 a
n ◦ (a†−m ◦ a
l − τδlm idF )(X0) = 0.
(8.24)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 42
These equalities, as well as (8.17), cannot help directly to compute vacuum mean values
of antinormally ordered products of creation and annihilation operators. But the equa-
tions (8.24) suggest the restrictions28
◦ a+m(X0) = δlm X0 a−l ◦ a
m (X0) = δlm X0
a−l ◦ a
m(X0) = τδlm X0 a
l ◦ a
m (X0) = τδlm X0
(8.25)
to be added to the definition of the vacuum. These conditions convert (8.24) into identities
and, in this sense agree with (8.17) and, consequently, with the bilinear commutation rela-
tions (6.13). Recall [16, 18], the relations (8.25) are similar to ones accepted in the parafield
theory and coincide with that for parastatistics of order p = 1; however, here we do not sup-
pose the validity of the paracommutation relations (6.20) (or (6.16)). Equipped with (8.25),
one is able to calculate the r.h.s. of (8.5) for any monomial M (resp. M′) and monomials
M′ (resp. M) of degree 1, degM′ = 1 (resp. degM = 1).29 Indeed, (8.25), (8.1b) and (8.3)
entail:
〈 X0|a†−l ◦ a
m(X0)〉 = 〈X0|a−l ◦ a
m (X0)〉 = δlm
〈 X0|a−l ◦ a
m(X0)〉 = 〈X0|a
◦ a†+m (X0)〉 = τδlm
〈 X0|(M(a+l1 , a
, · · · ))† ◦ a+m(X0)〉 = 〈 X0|(M(a+l1 , a
, · · · ))† ◦ a†+m (X0)〉 = 0 degM≥ 2
〈 X0|a−l ◦ M(a
, a†+m2 , · · · )(X0)〉 = 〈X0|a
l ◦ M(a
, a†+m2 , · · · )(X0)〉 = 0 degM≥ 2.
(8.26)
Hereof the equation (8.23) for i′ + j′ = 1 (resp. i′′ + j′′ = 1) and arbitrary i′′ and j′′ (resp. i′
and j′) follows.
However, it is not difficult to be realized, the calculation of (8.5) in cases more general
than (8.20) and (8.26) is not possible on the base of the assumptions made until now.30 At
this point, one is free so set in an arbitrary way the r.h.s. of (8.5) in the mentioned general
case or to add to (8.17) (and, possibly, (8.25)) other (commutation) relations by means of
which the r.h.s. of (8.5) to be calculated explicitly; other approaches, e.g. some mixture
of the just pointed ones, for finding the explicit form of (8.5) are evidently also possible.
Since expressions like (8.5) are directly connected with observable experimental results, the
only criterion for solving the problem for calculating the r.h.s. of (8.5) in the general case
can be the agreement with the existing experimental data. As it is known [1, 3, 11, 12], at
present (almost?) all of them are satisfactory described within the framework of the bilinear
commutation relations (6.13). This means that, from physical point of view, the theory
should be considered as realistic one if the r.h.s. of (8.5) is the same as if (6.13) are valid
or is reducible to it for some particular realization of an accepted method of calculation,
e.g. if one accepts some commutation relations, like the paracommutation ones, which are
a generalization of (6.13) and reduce to them as a special case (see, e.g., (6.20)). It should
be noted, the conditions (8.1b)–(8.3) and (8.25) are enough for calculating (8.5) if (6.16),
or its versions (6.17) or (6.20), are accepted (cf. [16]). The causes for that difference are
replacements like [a+m, a
n ] 7→ 2a+m◦a
n , when one passes from (6.16) to (8.17); the existence
of terms like a
n ◦ a+ma+l in (6.16) are responsible for the possibility to calculate (8.5).
28 Since the operators a±
and a
are, generally, degenerate (with no inverse ones), we cannot say that (8.24)
implies (8.25).
29 For degM′ = 0 (resp. degM′ = 0) — see (8.20).
30 It should be noted, the conditions (8.1b)–(8.3) and (8.25) are enough for calculating (8.5) if the rela-
tions (6.16), or their version (6.20), are accepted (cf. [16]). The cause for that difference is in replacements
like [a+m, a
n ] 7→ 2a
m ◦ a
n , when one passes from (6.16) to (8.17); the existence of terms like a
n ◦ a
m ◦ a
in (6.16) is responsible for the possibility to calculate (8.5), in case (6.16) hold.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 43
If evidences appear for events for which (8.5) takes other values, one should look, e.g.,
for other commutation relations leading to desired mean values. As an example of the last
type can be pointed the following anomalous bilinear commutation relations (cf. (6.13))
, a±m]ε = 0 [a
, a†±m ]ε = 0
[a∓l , a
m]ε = (±1)2jτδlm idF [a
l , a
m ]ε = (±1)2jτδlm idF
[a±l , a
m ]ε = 0 [a
l , a
m]ε = 0
[a∓l , a
m ]ε = (±1)2jδlm idF [a
l , a
m]ε = (±1)2jδlm idF , (8.27)
which should be imposed after expressions like E(a†±m ◦ a∓n ) are explicitly calculated. These
relations convert (8.17) and (8.25) into identities and by their means the r.h.s. of (8.5) can be
calculated explicitly, but, as it is well known [1,3,11,12,27] they lead to deep contradictions
in the theory, due to which should be rejected.31
At present, it seems, the bilinear commutation relations (6.13) are the only known com-
mutation relations which satisfy all of the mentioned conditions and simultaneously provide
an evident procedure for effective calculation of all expressions of the form (8.5). (Besides,
for them and for the paracommutation relations the vectors (8.4) form a base, the Fock
base, for the system’s Hilbert space of states [18].) In this connection, we want to mention
that the paracommutation relations (6.16) (or their conventional version (6.20)), if imposed
as additional restrictions to the theory together with (8.17), reduce in this particular case
to (6.13) as the conditions (8.25) show that we are dealing with a parafield of order p = 1,
i.e. with an ordinary field [17,18].32
Ending this section, let us return to the definition of the vacuum X0. It, generally,
depends on the adopted commutation relations. For instance, in a case of the bilinear com-
mutation relations (6.13) it consists of the equations (8.1a)–(8.3), while in a case of the
paracommutation relations (6.16) (or other ones generalizing (6.13)) it includes (8.1a)–(8.3)
and (8.25).
9. Commutation relations for
several coexisting different free fields
Until now we have considered commutation relations for a single free field, which can be
scalar, or spinor or vector one. The present section is devoted to similar treatment of a
system consisting of several, not less than two, different free fields. In our context, the
fields may differ by their masses and/or charges and/or spins; e.g., the system may consist
of charged scalar field, neutral scalar field, massless spinor field, massive spinor field and
massless neural vector field. It is a priori evident, the commutation relations regarding only
one field of the system should be as discussed in the previous sections. The problem is to be
derived/postulated commutation relations concerning different fields. It will be shown, the
developed Lagrangian formalism provides a natural base for such an investigation and makes
superfluous some of the assumptions made, for example, in [17, p. B 1159, left column] or
in [18, sec. 12.1], where systems of different parafields are explored.
To begin with, let us introduce suitable notation. With the indices α, β, γ = 1, 2, . . . , N
will be distinguished the different fields of the system, with N ∈ N, N ≥ 2, being their
number, and the corresponding to them quantities. Let qα and jα be respectively the charge
31 As it was demonstrated in [13–15], a quantization like (8.27) contradicts to (is rejected by) the charge
symmetric Lagrangians (3.4).
32 Notice, as a result of (8.17), the relations (6.16) correspond to (7.28) for E = A, with A being the
antinormal ordering operator (see (7.30)).
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 44
and spin of the α-th field. Similarly to (3.7), we define
jα :=
0 for scalar α-th field
for spinor α-th field
1 for vector α-th field
τα :=
1 for qα = 0 (neutral (Hermitian) field)
0 for qα 6= 0 (charged (non-Hermitian) field)
εα := (−1)2jα =
+1 for integer jα (bose fields)
−1 for half-integer jα (fermi fields)
(9.1)
Suppose Lα is the Lagrangian of the α-field. For definiteness, we assume Lα for all
α to be given by one and the same set of equations, viz. (3.1), or (3.3) or (3.4). To save
some space, below the case (3.4), corresponding to charge symmetric Lagrangians, will be
considered in more details; the reader can explore other cases as exercises.
Since the Lagrangian of our system of free fields is
Lα, (9.2)
the dynamical variables are
Dα (9.3)
and the corresponding system of Euler-Lagrange equations consists of the independent equa-
tions for each of the fields of the system (see (3.6) with Lα for L). This allows an introduction
of independent creation and annihilation operators for each field. The ones for the α-th field
will be denoted by a±α,sα(k) and a
α,sα(k); notice, the values of the polarization variables
generally depend on the field considered and, therefore, they also are labeled with index
α for the α-th field. For brevity, we shall use the collective indices lα, mα and nα, with
lα := (α, sα,k) etc., in terms of which the last operators are a±
and a
, respectively. The
particular expressions for the dynamical operators Dα are given via (3.9)–(3.12) in which
the following changes should be made:
τ 7→ τα j 7→ jα ε 7→ εα s 7→ sα s′ 7→ s′α
µν (k) 7→ σs
αs′α,±
µν (k) l
ss′,±
µν (k) 7→ ls
αs′α,±
µν (k).
(9.4)
The content of sections 4 and 5 remains valid mutatis mutandis, viz. provided the just
pointed changes (9.4) are made and the (integral) dynamical variables are understood in
conformity with (9.3).
9.1. Commutation relations connected with the momentum operator.
Problems and their possible solutions
In sections 6–8, however, substantial changes occur; for instance, when one passes from (6.12)
or (6.15) to (6.16). We shall consider them briefly in a case when one starts from the charge
symmetric Lagrangians (3.4).
The basic relations (6.12), which arise from the Heisenberg relation (5.1) concerning the
momentum operator, now read (here and below, do not sum over α, and/or β and/or γ if
the opposite is not indicated explicitly!)
a±lα , [a
]εβ + [a
± (1 + τ)δlαmβa±lα = 0 (9.5a)
lα , [a
]εβ + [a
± (1 + τ)δlαmβa
lα = 0. (9.5b)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 45
It is trivial to be seen, the following generalizations of respectively (6.14) and (6.15)
a±lα , [a
± (1 + τβ)δlαmβa±lα = 0 (9.6a)
± (1 + τβ)δlαmβa±lα = 0 (9.6b)
lα , [a
± (1 + τβ)δlαmβa
lα = 0 (9.6c)
± (1 + τβ)δlαmβa
= 0 (9.6d)
a+lα , [a
+ 2δlαmβa
lα = 0 (9.7a)
+ 2τβδlαmβa
= 0 (9.7b)
a−lα , [a
− 2τβδlαmβa−lα = 0 (9.7c)
− 2δlαmβa−lα = 0 (9.7d)
provide a solution of (9.5) in a sense that they convert it into identity. As it was said in
Sect. 6, the equations (9.6) (resp. (9.7)) for a single field, i.e. for β = α, agree (resp. disagree)
with the bilinear commutation relations (6.13).
The only problem arises when one tries to generalize, e.g., the relations (9.7) in a way
similar to the transition from (6.15) to (6.16). Its essence is in the generalization of expres-
sions like [a
]εβ and τ
βδlαmβa
. When passing from (6.15) to (6.16), the indices l and
m are changed so that the obtained equations to be consistent with (6.13); of course, the
numbers ε and τ are preserved because this change does not concern the field regarded. But
the situation with (9.7) is different in two directions:
(i) If we change the pair (mβ,mβ) in [a
]εβ with (m
β, nγ), then with what the num-
ber εβ should be replace? With εβ , or εγ or with something else? Similarly, if the mentioned
changed is performed, with what the multiplier τβ in τβδlαmβa
lα should be replaced? The
problem is that the numbers εβ and τβ are related to terms like a
and a±
◦ a†∓
in the momentum operator, as a whole and we cannot say whether the index β in εβ and τβ
originates from the first of second index mβ in these expressions.
(ii) When writing (mβ , nγ) for (mβ ,mβ) (see (i) above), then shall we replace δlαmβa
with δlαmβa
nγ , or δlαnγa
, or δmβnγa
? For a single field, γ = β = α, this problem is
solved by requiring an agreement of the resulting generalization (of (6.16) in the particular
case) with the bilinear commutation relations (6.13). So, how shall (6.13) be generalized for
several, not less than two, different fields? Obviously, here we meet an obstacle similar to
the one described in (i) above, with the only change that −εβ should stand for εβ .
Let blα and clα denote some creation or annihilation operator of the α-field. Consider the
problem for generalizing the (anti)commutator [blα , clα ]±εα . This means that we are looking
for a replacement
[blα , clα ]±εα 7→ f±(blα , cmβ ;α, β), (9.8)
where the functions f± are such that
f±(blα , cmβ ;α, β)
= [blα , clα ]±εα . (9.9)
Unfortunately, the condition (9.9) is the only restriction on f± that the theory of free fields
can provide. Thus the functions f±, subjected to equation (9.9), become new free parameters
of the quantum theory of different free fields and it is a matter of convention how to choose/fix
them.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 46
It is generally accepted [18, appendix F], the functions f± to have forms ‘maximum’
similar to the (anti)commutators they generalize. More precisely, the functions
f±(blα , cmβ ;α, β) = [blα , cmβ ]±εαβ (9.10)
where εαβ ∈ C are such that
εαα = εα, (9.11)
are usually considered as the only candidates for f±. Notice, in (9.10), εαβ are functions in
α and β, not in lα and/or mβ. Besides, if we assume εαβ to be function only in εα and εβ ,
then the general form of εαβ is
εαβ = uαβεα + (1− uαβ)εβ + vαβ(1− εαεβ) uαβ , vαβ ∈ C, (9.12)
due to (9.1) and (9.11). (In view of (6.13), the value εαβ = +1 (resp. εαβ = −1) corresponds
to quantization via commutators (resp. anticommutators) of the corresponding fields.)
Call attention now on the numbers τα which originate and are associated with each term
[blα , cmα ]±εα . With every change (9.8) one can associate a replacement
τα 7→ g(blα , cmβ ;α, β), (9.13)
where the function g is such that
g(blα , cmβ ;α, β)
= τα. (9.14)
Of course, the last condition does not define g uniquely and, consequently, the function
g, satisfying (9.14), enters in the theory as a new free parameter. Suppose, as a working
hypothesis similar to (9.10)–(9.11), that g is of the form
g(blα , cmβ ;α, β) = τ
αβ , (9.15)
where ταβ are complex numbers that may depend only on α and β and are such that
ταα = τα. (9.16)
Besides, if we suppose ταβ to be functions only in τα and τβ, then
ταβ = xαβτα + yαβτβ + (1− xαβ − yαβ)τατβ xαβ , yαβ ∈ C, (9.17)
as a result of (9.1) and (9.16).
Let us summarize the above discussion. If we suppose a preservation of the algebraic
structure of the bilinear commutation relations (6.13) for a system of different free fields,
then the replacements
[blα , clα ]±εα 7→ [blα , cmβ ]±εαβ εαα = εα (9.18a)
τα 7→ ταα ταα = τα (9.18b)
should be made; accordingly, the relations (6.13) transform into:
]−εαβ = 0 [a
]−εαβ = 0
[a∓lα , a
]−εαβ = τ
αβδlαmβ idF ×
lα , a
]−εαβ = τ
αβδlαmβ idF ×
[a±lα , a
]−εαβ = 0 [a
lα , a
]−εαβ = 0
]−εαβ = δlαmβ idF ×
]−εαβ = δlαmβ idF ×
, (9.19)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 47
where 1 (resp. −εαβ) in
corresponds to the choice of the upper (resp. lower) signs. If
we suppose additionally εαβ (resp. ταβ) to be a function only in εα and εβ (resp. in τα and
τβ), then these numbers are defined up to two sets of complex parameters:
εαβ = uαβεα + (1− uαβ)εβ + vαβ(1− εαεβ) uαβ, vαβ ∈ C (9.20a)
ταβ = xαβτα + yαβτβ + (1− xαβ − yαβ)τατβ xαβ, yαβ ∈ C. (9.20b)
A reasonable further specialization of εαβ and ταβ may be the assumption their ranges
to coincide with those of εα and τα, respectively. As a result of (9.1), this supposition is
equivalent to
vαβ = −uαβ,−uαβ + 1, uαβ − 1, uαβ uαβ ∈ C (9.21a)
(xαβ , yαβ) = (0, 0), (0, 1), (1, 0), (1, 1). (9.21b)
Other admissible restriction on (9.20) may be the requirement εαβ and ταβ to be symmetric,
εαβ(εα, εβ) = εβα(εα, εβ) = εαβ(εβ , εα) (9.22a)
ταβ(τα, τβ) = τβα(τα, τβ) = ταβ(τβ, τα), (9.22b)
which means that the α-th and β-th fields are treated on equal footing and there is no a
priori way to number some of them as the ‘first’ or ‘second’ one.33 In view of (9.20), the
conditions (9.22) are equivalent to
uαβ =
vαβ ∈ C (9.23a)
yαβ = xαβ. (9.23b)
If both of the restrictions (9.21) and (9.23) are imposed on (9.20), then the arbitrariness of
the parameters in (9.20) is reduced to:
(uαβ , uαβ) =
(9.24a)
(xαβ , yαβ) = (0, 0), (1, 1) (9.24b)
and, for any fixed pair (α, β), we are left with the following candidates for respectively εαβ
and ταβ:
(+1 + εα + εβ − εαεβ) (9.25a)
(−1 + εα + εβ + εαεβ) (9.25b)
0 := τ
α + τβ (9.25c)
1 := τ
α + τβ − τατβ. (9.25d)
When free fields are considered, as in our case, no further arguments from mathematical
or physical nature can help for choosing a particular combination (εαβ , ταβ) from the four
possible ones according to (9.25) for a fixed pair (α, β). To end the above considerations of
εαβ and ταβ, we have to say that the choice
(εαβ , ταβ) = (ε
+ , τ
0 ) =
(+1 + εα + εβ − εαεβ), τα + τβ
(9.26)
33 However, nothing can prevent us to make other choices, compatible with (9.18), in the theory of free
fields; for instance, one may set εαβ = εαεβεβα and ταβ = 1
(τα + τβ)τβα.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 48
is known as the normal case [18, appendix F]; in it the relative behavior of bose (resp.
fermi) fields is as in the case of a single field, i.e. they are quantized via commutators (resp.
anticommutators) as (εαβ , ταβ) = (+1, 0) (resp. (εαβ , ταβ) = (−1, 0)), and the one of bose and
fermi field is as in the case of a single fermi field, viz. the quantization is via commutators as
(εαβ , ταβ) = (+1, 0). All combinations between ε
± and τ
0,1 different from (9.26) are referred
as anomalous cases. Above we supposed the pair (α, β) to be fixed. If α and β are arbitrary,
the only essential change this implies is in (9.25), where the choice of the subscripts +, −, 0
and 1 may depend on α and β. In this general situation, the normal case is defined as the one
when (9.26) holds for all α and β. All other combinations are referred as anomalous cases;
such are, for instance, the ones when some fermi and bose operators satisfy anticommutation
relations, e.g. (9.19) with εαβ = −1 for εα + εβ = 0, or some fermi fields are subjected to
commutation relations, like (9.19) with εαβ = +1 for εα = εβ = −1. For some details on this
topic, see, for instance, [18, appendix F], [7, chapter 20] and [27, sect 4-4]. Fields/operators
for which εαβ = +1 (resp. εαβ = −1), with β 6= α, are referred as relative parabose (resp.
parafermi) in the parafield theory [17,18]. One can transfer this terminology in the general
case and call the fields/operators for which εαβ = +1 (resp. εαβ = −1), with β 6= α, relative
bose (resp. fermi) fields/operators.
Further the relations (9.19) will be referred as the multifield bilinear commutation rela-
tions and it will be assumed that they represent the generalization of the bilinear commuta-
tion relations (6.13) when we are dealing with several, not less than two, different quantum
fields. The particular values of εαβ and ταβ in them are insignificant in the following; if one
likes, one can fix them as in the normal case (9.26). Moreover, even the definition (9.19)
of ταβ is completely inessential at all, as ταβ always appears in combinations like ταβδlαmβ
(see (9.19) or similar relations, like (9.27), below), which are non-vanishing if β = α, but
then ταα = τα; so one can freely write τα for ταβ in all such cases.
Equipped with (9.19) and (9.18), we can generalize (9.7) in different ways. For example,
the straightforward generalization of (6.16) is:
, [a+
nγ ]εβγ
+ 2δlαnγa
= 0 (9.27a)
a+lα , [a
, a−nγ ]εβγ
+ 2ταγδlαnγa
= 0 (9.27b)
, [a+
nγ ]εβγ
− 2ταβδlαmβa−nγ = 0 (9.27c)
a−lα , [a
, a−nγ ]εβγ
− 2δlαmβa−nγ = 0. (9.27d)
However, generally, the relations (9.19) do not convert (9.27) into identities. The reason is
that an equality/identity like (cf. (6.8))
[blα , cmβ ◦ dnγ ] = [blα , cmβ ]−εαβ ◦ dnγ + λαβγcmβ ◦ [blα , dnγ ]−εαγ , (9.28)
where blα , cmβ and dnγ are some creation/annihilation operators and λ
αβγ ∈ C, can be valid
only for
λαβγ = εαβ εαγ = 1/εαβ (εαβ 6= 0), (9.29)
which, in particular, is fulfilled if γ = β and εαβ = ±1. So, the agreement between (9.19)
and (9.27) depends on the concrete choice of the numbers εαβ . There exist cases when even
the normal case (9.26) cannot ensure (9.19) to convert (9.27) into identities; e.g. when the
α-th field and β-th fields are fermion ones and the γ-th field is a boson one. Moreover, it can
be proved that (9.19) and (9.27) are compatible in the general case if unacceptable equalities
like a±
◦ a±m = 0 hold.
One may call (9.27) the multifield paracommutation relations as from them a correspond-
ing generalization of (6.18) and/or (6.20) can be derived. For completeness, we shall record
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 49
the multifield version of (6.20):
[blα , [b
, bnγ ]εβγ ] = 2δlαmβbnγ [blα , [bmβ , bnγ ]εβγ ] = 0 (9.30a)
[clα , [c
, cnγ ]εβγ ] = 2δlαmβcnγ [clα , [cmβ , cnγ ]εβγ ] = 0 (9.30b)
lα , [c
, cnγ ]εβγ ] = −2ταγδlαnγb
lα , [b
, bnγ ]εβγ ] = −2ταγδlαnγc
. (9.30c)
For details regarding these multifield paracommutation relations, the reader is referred to [17,
18], where the case τα = τβ = ταβ = 0 is considered.
We leave to the reader as exercise to write down the multifield versions of the commuta-
tion relations (6.22) or (6.23), which provide examples of generalizations of (9.7) and hence
of (9.19) and (9.27).
9.2. Commutation relations connected with the charge and
angular momentum operators
In a case of several, not less than two, different fields, the basic trilinear commutation rela-
tions (6.33), which ensure the validity of the Heisenberg relation (5.2) concerning the charge
operator, read:
a±lα , [a
]εβ − [a+mβ , a
− 2δlαmβa±lα = 0 (9.31a)
lα , [a
]εβ − [a+mβ , a
+ 2δlαmβa
lα = 0. (9.31b)
Of course, these relations hold only for those fields which have non-vanishing charges, i.e.
in (9.31) is supposed (see (9.1))
τα = 0 τβ = 0 (⇐⇒ qαqβ 6= 0). (9.32)
The problem for generalizing (9.31) for these fields is similar to the one for (9.7) in the
case of non-vanishing charges, τβ = 0. Without repeating the discussion of Subsect. 9.1,
we shall adopt the rule (9.18) for generalizing (anti)commutation relations between cre-
ation/annihilation operators of a single field. By its means one can obtain different general-
izations of (9.31). For instance, the commutation relations.
, a−nγ ]εβγ − [a+mβ , a
nγ ]εβγ
− 2δlαnγa+mβ = 0 (9.33a)
a−lα , [a
, a−nγ ]εβγ − [a+mβ , a
nγ ]εβγ
− 2δlαmβa−nγ = 0 (9.33b)
and their Hermitian conjugate contain (9.31) and (6.35) as evident special cases and agree
with (9.19) if γ = β and εαβεβγ = +1. Besides, the multifield paracommutation rela-
tions (9.27) for charged fields, τα = τβ = τγ = 0, convert (9.33) into identities and, in this
sense, (9.33) agree with (contain as special case) (9.27) for charged fields. As an example
of commutation relations that do not agree with (9.27) for charged fields and, consequently,
with (9.33), we shall point the following ones:
a±lα , [a
nγ ]εβγ
+ δlαnγa
= 0 (9.34a)
, a−nγ ]εβγ−
− δlαnγa±mβ = 0, (9.34b)
which are a multifield generalization of (6.34).
The consideration of commutation relations originating from the ‘orbital’ Heisenberg
equation (5.4) is analogous to the one of the same relations regarding the charge operator.
The multifield version of (6.49) is:
(−ω◦µν(mβ) + ω◦µν(nγ))([ã±lα , [ã
, ã−nγ ]εβγ
+ [ã+
nγ ]εβγ ] )
nγ=mβ
= 4(1 + ταβ)δlαmβω
α)(ã±
) (9.35a)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 50
(−ω◦µν(mβ) + ω◦µν(nγ))([ã
lα , [ã
, ã−nγ ]εβγ
+ [ã+
nγ ]εβγ ] )
nγ=mβ
= 4(1 + ταβ)δlαmβω
α)(ã
lα ) (9.35b)
where
ω◦µν(l
α) := ωµν(k) = kµ
if lα = (α, sα,k). (9.36)
Applying (6.51), with mβ for m and nγ for n, one can check that the multifield paracom-
mutation relations (9.27) convert (9.35) into identities and hence provide a solution of (9.35)
and ensure the validity of (5.4), when system of different free fields is considered. An example
of a solution of (9.35) which does not agree with (9.27) is provided by the following multifield
generalization of (6.52):
a+lα , [a
nγ ]εβγ
a+lα , [a
, a−nγ ]εβγ
= −(1 + ταγ)δlαnγa+mβ (9.37a)
a−lα , [a
nγ ]εβγ
a−lα , [a
, a−nγ ]εβγ
= +(1 + ταβ)δlαmβa
nγ , (9.37b)
which provides a solution of (9.5). Notice, the evident multifield version of (6.53) agrees
with (9.5), but disagrees with (9.35) when the lower signs are used.
At last, the multifield exploration of the ‘spin’ Heisenberg relations (5.5) is a mutatis
mutandis (see (9.35)) version of the corresponding considerations in the second part of Sub-
sect. 6.3. The main result here is that the multifield bilinear commutation relations (9.19),
as well as their para counterparts (9.27), ensure the validity of (5.5).
9.3. Commutation relations between the dynamical variables
The aim of this subsection is to be discussed/proved the commutation relations (5.15)–(5.24)
for a system of at least two different quantum fields from the view-point of the commutation
relations considered in subsections 9.1 and 9.2.
To begin with, we rewrite the Heisenberg relations (5.1), (5.2) and (5.4) in terms of
creation and annihilation operators for a multifield system [1,11]:
, Pµ] = ∓kµa±lα [a
, Pµ] = ∓kµa†±lα (9.38)
[a±lα , Q] = qa
lα [a
lα , Q] = −qa
lα (9.39)
,Morµν ] = i~ω◦µν(lα)
,Morµν ] = i~ω◦µν(lα)
, (9.40)
where lα = (α, sα,k), ω◦(lα) is defined by (9.36) and k0 =
m2c2 + k2 is set in (9.38)
and (9.40) (after the differentiations are performed in the last case). The corresponding
version of (5.5) is more complicated and depends on the particular field considered (do not
sum over sα!):
[a±α,sα(k),Mspµν ] = i~gα
αtα,+
µν (k)a
α,tα(k) +
αtα,−
µν (k)a
α,tα(k)
α,sα(k),Mspµν ] = i~hα
αtα,−
µν (k)a
α,tα(k) +
αtα,+
µν (k)a
α,tα(k)
(9.41)
where fsα = −1, 0,+1 (depending on the particular field), gα := −hα := 1jα+δjα0 (−1)
jα+1 and
sαtα,+
µν (k) and
sαtα,−
µν (k) are some functions which strongly depend on the particular field
considered, with ±σ
sαtα,±
µν (k) being related to the spin (polarization) functions σ
sαtα,±
µν (k)
(see (3.14) and (3.11)).34 As a result of (5.6), (9.40) and (9.41), one can easily write the
Heisenberg relations (5.3) in a form similar to (9.38)–(9.41).
34 If φ̃αi (k) are the Fourier images of the α-th field and
i (k) =
i (k)ã
α,sα(k) + v
i (k)ã
α,sα(k)
, (9.42)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 51
The commutation relations involving the momentum operator are:
[Pµ, Pν ] = 0 [Q, Pµ] = 0
[Sµν , Pλ] = [Mspµν , Pλ] = 0
[Lµν , Pλ] = [Morµν , Pλ] = [Mµν , Pλ] = −i~{ηλµ Pν − ηλν Pµ}.
(9.45)
We claim that these equations are consequences from (9.38) and the explicit expressions (3.9)–
(3.12) and (5.11)–(5.13) for the operators of the dynamical variables of the free fields con-
sidered in the present work. In fact, since (9.38) implies
[b±lα ◦ c
, Pµ] = 0 lα = (α, sα,k), mβ = (β, sβ ,k) (9.46a)
[b±lα
←−−−→
ω◦µν (l
α) ◦ c∓
, Pµ] = ±2(kµηνλ − kνηµλ)b±lα ◦ c
, (9.46b)
where b±lα , c
lα = a
lα , a
lα and
←−−−→
ω◦µν (l
α) is defined via (9.36) and (3.13), the verification of (9.45)
reduces to almost trivial algebraic calculations. Further, we assert that any system of commu-
tation relations considered in Subsect. 9.1 entails (9.45): as these relations always imply (9.5)
(or similar multifield versions of (6.10) and (6.11) in the case of the Lagrangians (3.1) or (3.3),
respectively) and, on its turn, (9.5) implies (5.1), the required result follows from the last
assertion and the remark that (5.1) and (9.38) are equivalent. As an additional verification
of the validity of (9.45), the reader can prove them by invoking the identity (6.8) and any
system of commutation relations mentioned in Subsect. 9.1, in particular (9.19) and (9.27).
The commutation relations concerning the charge operator read:
[Pµ, Q] = 0 [Q, Q] = 0
[Lµν , Q] = [Sµν , Q] = 0
[Morµν , Q] = [Mspµν , Q] = [Mµν , Q] = 0.
(9.47)
These equations are trivial corollaries from (3.9)–(3.12) and (5.11)–(5.13) and the observation
that (9.39) implies
lα ◦ a
, Q] = [a±lα ◦ a
, Q] = 0 if qα = qβ, (9.48)
due to (6.8) for η = −1. Since any one of the systems of commutation relations mentioned in
Subsect. 9.2 entails (9.31) (or systems of similar multifield versions of (6.31) and (6.32), if the
Lagrangians (3.1) or (3.3) are employed), which is equivalent to (9.39), the equations (9.47)
hold if some of these systems is valid. Alternatively, one can prove via a direct calculation
that the commutation relations arising from the charge operator entail the validity of (9.47);
where v
i (k) are linearly independent functions normalize via the condition
i (k)
i (k) = δ
, (9.43)
with fs
= 1 for jα = 0, 1
and fs
= 0,−1 for (jα, sα) = (1, 3) or (jα, sα) = (1, 1), (1, 2), respectively, then
µν (k) :=
i (k)
µν (k) :=
i (k)
(9.44)
with Ii
iµν given via (5.25). Besides, σ
µν (k) =
µν (k) with an exception that σ
µν (k) = 0 for
α = 1
and (µ, ν) = (a, 0), (0, a) with a = 1, 2, 3.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 52
for the purpose the identity (6.8) and the explicit expressions for the dynamical variables via
the creation and annihilation operators should be applied.
At last, consider the commutation relations involving the different angular momentum
operators:
[Pλ, Sµν ] = [Pλ,Mspµν ] = 0
[Pλ, Lµν ] = [Pλ,Morµν ] = [Pλ,Mµν ] = +i~{ηλµ Pν − ηλν Pµ}
[Q, Lµν ] = [Q, Sµν ] = [Q,Morµν ] = [Q,Mspµν ] = [Q,Mµν ] = 0
[Sκλ,Mµν ] = −i~
ηκµ Sλν − ηλµ Sκν − ηκν Sλµ + ηλν Sκµ
[Lκλ,Mµν ] = −i~
ηκµ Lλν − ηλµ Lκν − ηκν Lλµ + ηλν Lκµ
[Mκλ,Mµν ] = −i~
ηκµMλν − ηλµMκν − ηκνMλµ + ηλνMκµ
(9.49)
(The other commutators, that can be form from the different angular momentum operators,
are complicated and cannot be expressed in a ‘closed’ form.) The proof of these relations is
based on equations like (see (9.40) and (6.8))
[blα ◦ cmβ ,Morµν ] = i~ω◦µν(lα)
blα ◦ cmβ
lα = (α, sα,k), mβ = (β, sβ ,k), (9.50)
with blα , clα = a
lα , a
lα , a
lα , a
lα , and similar, but more complicated, ones involving the other
angular momentum operators. It, generally, depends on the particular field considered and
will be omitted.
As it was said in Subsect. 6.3, the Heisenberg relations concerning the angular momentum
operator(s) do not give rise to some (algebraic) commutation relations for the creation and
annihilation operators. For this reason, the only problem is which of the commutation
relations discussed in subsections 9.1 and 9.2 imply the validity of the equations (9.49) (or
part of them). The general answer of this problem is not known but, however, a direct
calculation by means of (9.7), if it holds, and (6.8) shows the validity of (9.49). Since (9.19)
and (9.27) imply (9.7), this means that the multifield bilinear and para commutation relations
are sufficient for the fulfillment of (9.49).
To conclude, let us draw the major moral of the above material: the multifield bilinear
commutation relations (9.19) and the multifield paracommutation relations (9.27) ensure
the validity of all ‘standard’ commutation relations (9.45), (9.47) and (9.49) between the
operators of the dynamical variables characterizing free scalar, spinor and vector fields.
9.4. Commutation relations under the uniqueness conditions
As it was said at the end of the introduction to this section, the replacements (9.4) ensure the
validity of the material of Sect. 4 in the multifield case. Correspondingly, the considerations
in Sect. 7 remain valid in this case provided the changes
l 7→ lα m 7→ mβ n 7→ nγ
τδlm 7→ ταβδlαmβ = ταδlαmβ
[bm, bm]ε 7→ [bmβ , bmβ ]εβ [bm, bn]ε 7→ [bmβ , bnγ ]εβγ ,
(9.51)
with bm (or bmβ ) being any creation/annihilation operator, and, in some cases, (9.4) are
made.35 Without going into details, we shall write the final results.
The multifield version of (7.27)–(7.28) is:
E(a†±
◦ a∓nγ ) = εβγ E(a∓nγ ◦ a
E([a†±
, a∓nγ ]εβγ) (9.52)
35 As a result of (7.11), (7.16) and (7.17), in expressions like (7.18)–(7.26) the number ε should be replace
by εαβ, where α and β are the corresponding field indices of the creation/annihilation operators on which the
operator E acts, i.e. ε E(bm ◦ bn) 7→ ε
βγ E(bmβ ◦ bnγ ).
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 53
a+lα , E([a
nγ ]εβγ)
+ 2δlαnγa
= 0 (9.53a)
, E([a†+
, a−nγ ]εβγ )
+ 2ταγδlαnγa
= 0 (9.53b)
a−lα , E([a
nγ ]εβγ)
− 2ταβδlαmβa−nγ = 0 (9.53c)
, E([a†+
, a−nγ ]εβγ )
− 2δlαmβa−nγ = 0 (9.53d)
γ =β. (9.53e)
As one can expect, the relations (9.53a)–(9.53d) can be obtained from the multifield paracom-
mutation relations (9.27) via the replacement [·, ·]ε 7→ E([·, ·]εβγ ). It should be paid special
attention on the equation (9.53e). It is due to the fact that in the expressions for the dynami-
cal variables do not enter ‘cross-field-products’, like a
for β 6= α, and it corresponds to
the condition (ii) in [17, p. B 1159]. The equality (9.53e) is quite important as it selects only
that part of the ‘ E-transformed’ multifield paracommutation relations (9.27) which is com-
patible with the bilinear commutation relations (9.19) (see (9.28) and (9.29)). Besides, (9.53e)
makes (9.53a)–(9.53d) independent of the particular definition of εαβ (see (9.11)).
The equations (9.52) are the only restrictions on the operator E ; examples of this operator
are provided by the normal (resp. antinormal) ordering operator N (resp. A), which has the
properties (cf. (4.22) (resp. (7.30))
◦ a†−nγ
:= a+
◦ a†−nγ N
◦ a−nγ
◦ a−nγ
◦ a†+nγ
:= εβγa
nγ ◦ a−mβ N
◦ a+nγ
:= εβγa+nγ ◦ a
(9.54)
◦ a†−nγ
:= εβγa
nγ ◦ a+mβ A
◦ a−nγ
:= εβγa−nγ ◦ a
◦ a†+nγ
:= a−
◦ a†+nγ A
◦ a+nγ
◦ a+nγ .
(9.55)
The material of Sect. 8 has also a multifield variant that can be obtained via the re-
placements (9.51) and (9.4). Here is a brief summary of the main results found in that
The operator E should possess the properties (9.54) and, in this sense, can be identified
with the normal ordering operator,
E = N . (9.56)
As a result of this fact and εββ = εβ (see (9.11)), the commutation relations (9.53) take the
final form:
a+lα , a
◦ a†−
+ δlαnβa
= 0 (9.57a)
a+lα , a
+ ταβδlαnβa
= 0 (9.57b)
a−lα , a
◦ a†−
− ταβδlαmβa−nβ = 0 (9.57c)
− δlαmβa−nβ = 0 (9.57d)
which is the multifield version of (8.17) and corresponds, up to the replacement a±lα 7→
2a±lα ,
to (9.27) with εβγ = 0.
The vacuum state vector X0 is supposed to be uniquely defined by the following equations
(cf. (8.1b)–(8.3)):
a−lα X0 = 0 a
lα X0 = 0 (9.58a)
X0 6= 0 (9.58b)
〈 X0| X0〉 = 1 (9.58c)
lα ◦ a
(X0) = δlαmβ X0 a−lα ◦ a
(X0) = δlαmβ X0
(X0) = ταβδlαmβ X0 a−lα ◦ a
(X0) = ταβδlαmβ X0.
(9.58d)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 54
The Hilbert space F of state vectors is a direct sum of the Hilbert spaces Fα of the
different fields and it is supposed to be spanned by the vectors
... = M(a
, . . . )(X0) (9.59)
with M(a+
, . . . ) being arbitrary monomial only in the creation operators.
Since (9.58a), (9.56) and (9.54) imply the multifield version of (8.7), the computation of
the mean values of (8.6), with l1 7→ lα11 etc., of the dynamical variables is reduced to the one
of scalar products like (cf. (8.5))
〈ψlα1
...|φmβ1
〉 = 〈 X0|
, . . . )
)† ◦ M′(a+
, . . . )(X0)〉 (9.60)
of basic vectors of the form (9.59). By means of the basic properties (9.58) of the vacuum,
one is able to calculate the simplest forms of the vacuum mean values (9.60), viz. the mul-
tifield versions (see (9.51)) of (8.20) and (8.26). But more general such expression cannot
be calculated by means of (9.57)–(9.58). Prima facie one can suppose that the multifield
commutation relations (9.19), which ensure the vectors (9.59) to form a base of the system’s
Hilbert space of states, can help for the calculation of (9.60) in more complicated cases. In
fact, this is the case which works perfectly well and covers the available experimental data.
In this connection, we must mention that the applicability of (9.19) for calculation of (9.60)
is ensured by the compatibility/agreement between (9.19) and (9.57): by means of (6.8) for
η = −εαβ, one can check that (9.19) converts (9.57) into identities.36
The commutation relations (9.57) admit as a solution also the multifield version of the
anomalous bilinear commutation relations (8.27) but it, as we said earlier, leads to contradic-
tions and must be rejected. The existence of solutions of (9.57) different from it and (9.19)
seems not to be investigated. If there appear date which do not fit into the description by
means of (9.19), one should look for other, if any, solutions of (9.57) or compatible with (9.57)
effective procedures for calculating vacuum mean values like (9.60).
10. Conclusion
In this paper we have investigated two sources of (algebraic) commutation relations in the
Lagrangian quantum theory of free scalar, spinor and vector fields: the uniqueness of the
dynamical variables (momentum, charge and angular momentum) and the Heisenberg rela-
tions/equations for them. If one ignores the former origin, which is the ordinary case, the
paracommutation relations or some their generalizations seems to be the most suitable can-
didates for the most general commutation relations that ensure the validity of all Heisenberg
equations. The simultaneous consideration of the both sources mentioned reveals, however,
their incompatibility in the general case. The outlet of this situation is in the redefinition
of the operators of the dynamical variables, similar to the normal ordering procedure and
containing it as a special case. That operation ensures the uniqueness of the new (redefined)
dynamical variables and changes the possible types of commutation relations. Again, the
commutation relations, connected with the Heisenberg relations concerning the (redefined)
momentum operator, entail the validity of all Heisenberg equations.
36 Recall, equations (9.19) and (9.27), or (9.53a)–(9.53d), for γ 6= β are generally incompatible. For instance,
excluding some special cases, like systems consisting of only fermi (bose) fields or one fermi (bose) field and
arbitrary number of bose (fermi) fields, the only operators satisfying (9.19) and (9.27) for γ 6= β and having
normal spin-statistics connection are such that bmβ ◦ bnγ = 0, with γ 6= β and bmβ and cnγ being any
creation/annihilation operators, which, in particular, means that no states with two particles from different
fields can exist.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 55
Further constraints on the possible commutation relations follow from the definition/in-
troduction of the concept of the vacuum (vacuum state vector). They practically reduce the
redefined dynamical variables to the ones obtained via normal ordering procedure, which
results in the explicit form (8.17) of the admissible commutation relations. In a sense,
they happen to be ‘one half’ of the paracommutation ones. As a last argument in the way
for finding the ‘unique true’ commutation relations, we require the existence of procedure
for calculation of vacuum mean values of anti-normally ordered products of creation and
annihilation operators, to which the mean values of the dynamical variables and the transition
amplitudes between different states are reduced. We have pointed that the standard bilinear
commutation relations are, at present, the only known ones that satisfy all of the conditions
imposed and do not contradict to the existing experimental data.
The consideration of a system of at least two different quantum free fields meets a new
problem: the general relations between creation/annihilation operators belonging to differ-
ent fields turn to be undefined. The cause for this is that the commutation relations for any
fixed field are well defined only on the corresponding to it Hilbert subspace of the system’s
Hilbert space of states and their extension on the whole space, as well as the inclusion in
them of creation/annihilation operators of other fields, is a matter of convention (when free
fields are concerned); formally this is reflected in the structure of the dynamical variables
which are sums of those of the individual fields included in the system under consideration.
We have, however, presented argument by means of which the a priori existing arbitrari-
ness in the commutation relations involving different field operators can be reduced to the
‘standard’ one: these relations should contain either commutators or anticommutators of
the creation/annihilation operators belonging to different fields. A free field theory cannot
make difference between these two possibilities. Accepting these possibilities, the admissible
commutation relations (9.57) for system of several different fields are considered. They turn
to be corresponding multifield versions of the ones regarding a single field. Similarly to the
single field case, the standard multifield bilinear commutation relations seem to be the only
known ones that satisfy all of the imposed restrictions and are in agreement with the existing
data.
Acknowledgments
This research was partially supported by the National Science Fund of Bulgaria under Grant
No. F 1515/2005.
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	Introduction
	The momentum picture
	Lagrangians, Euler-Lagrange equations and dynamical variables
	On the uniqueness of the dynamical variables
	Heisenberg relations
	Types of possible commutation relations
	Restrictions related to the momentum operator
	Restrictions related to the charge operator
	Restrictions related to the angular momentum operator(s)
	Inferences
	State vectors, vacuum and mean values
	Commutation relations for several coexisting different free fields
	Commutation relations connected with the momentum operator. Problems and their possible solutions
	Commutation relations connected with the charge and angular momentum operators
	Commutation relations between the dynamical variables
	Commutation relations under the uniqueness conditions
	Conclusion
	References
	This article ends at page
