Non-Relativistic Propagators via Schwinger's Method
Non-Relativistic Propagators via Schwinger’s Method
A. Aragão,∗ H. Boschi-Filho,† and C. Farina‡
Instituto de F́ısica, Universidade Federal do Rio de Janeiro,
Caixa Postal 68.528, 21941-972 Rio de Janeiro, RJ, Brazil.
F. A. Barone§
Universidade Federal de Itajubá, Av. BPS 1303
Caixa Postal 50 - 37500-903, Itajubá, MG, Brazil.
In order to popularize the so called Schwinger’s method we reconsider the Feynman propagator of
two non-relativistic systems: a charged particle in a uniform magnetic field and a charged harmonic
oscillator in a uniform magnetic field. Instead of solving the Heisenberg equations for the position
and the canonical momentum operators, R and P, we apply this method by solving the Heisenberg
equations for the gauge invariant operators R and π = P − eA, the latter being the mechanical
momentum operator. In our procedure we avoid fixing the gauge from the beginning and the result
thus obtained shows explicitly the gauge dependence of the Feynman propagator.
PACS numbers: 42.50.Dv
Keywords: Schwinger’s method, Feynman Propagator, Magnetic Field, Harmonic Oscillator.
I. INTRODUCTION
In a recent paper published in this journal [1], three
methods were used to compute the Feynman propaga-
tors of a one-dimensional harmonic oscillator, with the
purpose of allowing a student to compare the advan-
tadges and disadvantadges of each method. The above
mentioned methods were the following: the so called
Schwinger’s method (SM), the algebraic method and the
path integral one. Though extremely powerful and el-
egant, Schwinger’s method is by far the less popular
among them. The main purpose of the present paper is
to popularize Schwinger’s method providing the reader
with two examples slightly more difficult than the har-
monic oscillator case and whose solutions may serve as a
preparation for attacking relativistic problems. In some
sense, this paper is complementary to reference [1].
The method we shall be concerned with was introduced
by Schwinger in 1951 [2] in a paper about QED enti-
tled “Gauge invariance and vacuum polarization”. After
introducing the proper time representation for comput-
ing effetive actions in QED, Schwinger was faced with a
kind of non-relativistic propagator in one extra dimen-
sion. The way he solved this problem is what we mean
by Schwinger’s method for computing quantum propa-
gators. For relativistic Green functions of charged parti-
cles under external electromagnetic fields, the main steps
of this method are summarized in Itzykson and Zuber’s
textbook [3] (apart, of course, from Schwinger’s work [2]).
Since then, this method has been used mainly in relativis-
tic quantum theory [4–16].
∗Electronic address: aharagao@if.ufrj.br
†Electronic address: boschi@if.ufrj.br
‡Electronic address: farina@if.ufrj.br
§Electronic address: fbarone@unifei.edu.br
However, as mentioned before, Schwinger’s method is
also well suited for computing non-relativistic propaga-
tors, though it has rarely been used in this context. As
far as we know, this method was used for the first time
in non-relativistic quantum mechanics by Urrutia and
Hernandez [17]. These authors used Schwinger’s action
principle to obtain the Feynman propagator for a damped
harmonic oscillator with a time-dependent frequency un-
der a time-dependent external force. Up to our knowl-
edge, since then only a few papers have been written with
this method, namely: in 1986, Urrutia and Manterola
[18] used it in the problem of an anharmonic charged
oscillator under a magnetic field; in the same year, Hor-
ing, Cui, and Fiorenza [19] applied Schwinger’s method
to obtain the Green function for crossed time-dependent
electric and magnetic fields; the method was later applied
in a rederivation of the Feynman propagator for a har-
monic oscillator with a time-dependent frequency [20];
a connection with the mid-point-rule for path integrals
involving electromagnetic interactions was discussed in
[21]. Finally, pedagogical presentations of this method
can be found in the recent publication [1] as well as in
Schwinger’s original lecture notes recently published [22],
which includes a discussion of the quantum action prin-
ciple and a derivation of the method to calculate propa-
gators with some examples.
It is worth mentioning that this same method was inde-
pendently developed by M. Goldberger and M. GellMann
in the autumn of 1951 in connection with an unpublished
paper about density matrix in statistical mechanics [23].
Our purpose in this paper is to provide the reader with
two other examples of non-relativistic quantum propa-
gators that can be computed in a straightforward way
by Schwinger’s method, namely: the propagator for a
charged particle in a uniformmagnetic field and this same
problem with an additional harmonic oscillator potential.
Though these problems have already been treated in the
context of the quantum action principle [18], we decided
http://arxiv.org/abs/0704.1645v2
mailto:aharagao@if.ufrj.br
mailto:boschi@if.ufrj.br
mailto:farina@if.ufrj.br
mailto:fbarone@unifei.edu.br
to reconsider them for the following reasons: instead of
solving the Heisenberg equations for the position and the
canonical momentum operators, R and P, as is done in
[18], we apply Schwinger’s method by solving the Heisen-
berg equations for the gauge invariant operators R and
π = P − eA, the latter being the mechanical momen-
tum operator. This is precisely the procedure followed
by Schwinger in his seminal paper of gauge invariance
and vacuum polarization [2]. This procedures has some
nice properties. For instance, we are not obligued to
choose a particular gauge at the beginning of calcula-
tions. As a consequence, we end up with an expression
for the propagator written in an arbitrary gauge. As a
bonus, the transformation law for the propagator under
gauge transformations can be readly obtained.
In order to prepare the students to attack more com-
plex problems, we solve the Heisenberg equations in ma-
trix form, which is well suited for generalizations involv-
ing Green functions of relativistic charged particles under
the influence of electromagnetic fields (constant Fµν , a
plane wave field or even combinations of both). For ped-
agogical reasons, at the end of each calculation, we show
how to extract the corresponding energy spectrum from
the Feynman propagator. Although the way Schwniger’s
method must be applied to non-relativistic problems has
already been explained in the literature [1, 18, 22], it is
not of common knowledge so that we start this paper
by summarizing its main steps. The paper is organized
as follows: in the next section we review Schwinger’s
method, in section III we present our examples and sec-
tion IV is left for the final remarks.
II. MAIN STEPS OF SCHWINGER’S METHOD
For simplicity, consider a one-dimensional time-
independent Hamiltonian H and the corresponding non-
relativistic Feynman propagator defined as
K(x, x′; τ) = θ(τ)〈x| exp
[−iHτ
|x′〉, (1)
where θ(τ) is the Heaviside step function and |x〉, |x′〉
are the eingenkets of the position operator X (in the
Schrödinger picture) with eingenvalues x and x′, respec-
tively. The extension for 3D systems is straightforward
and will be done in the next section. For τ > 0 we have,
from equation (1), that
K(x, x′; τ) = 〈x|H exp
[−iHτ
|x′〉. (2)
Inserting the unity 1l = exp [−(i/~)Hτ ] exp [(i/~)Hτ ] in
the r.h.s. of the above expression and using the well
known relation between operators in the Heisenberg and
Schrödinger pictures, we get the equation for the Feyn-
man propagator in the Heisenberg picture,
K(x, x′; τ) = 〈x, τ |H(X(0), P (0))|x′, 0〉, (3)
where |x, τ〉 and |x′, 0〉 are the eingenvectors of opera-
tors X(τ) and X(0), respectively, with the correspond-
ing eingenvalues x and x′: X(τ)|x, τ〉 = x|x, τ〉 and
X(0)|x′, 0〉 = x′|x′, 0〉, with K(x, x′; τ) = 〈x, τ |x′, 0〉. Be-
sides, X(τ) and P (τ) satisfy the Heisenberg equations,
(τ) = [X(τ),H] ; i~
(τ) = [P (τ),H]. (4)
Schwinger’s method consists in the following steps:
(i) we solve the Heisenberg equations forX(τ) and P (τ),
and write the solution for P (0) only in terms of the
operators X(τ) and X(0);
(ii) then, we substitute the results obtained in (i) into
the expression for H(X(0), P (0)) in (3) and using
the commutator [X(0), X(τ)] we rewrite each term
ofH in a time ordered form with all operatorsX(τ)
to the left and all operators X(0) to the right;
(iii) with such an ordered hamiltonian, equation (3) can
be readly cast into the form
K(x, x′; τ) = F (x, x′; τ)K(x, x′; τ), (5)
with F (x, x′; τ) being an ordinary function defined
F (x, x′; τ) =
〈x, τ |Hord(X(τ), X(0))|x
′, 0〉
〈x, τ |x′, 0〉
. (6)
Integrating in τ , the Feynman propagator takes the
K(x, x′; τ) = C(x, x′) exp
F (x, x′; τ ′)dτ ′
, (7)
where C(x, x′) is an integration constant indepen-
dent of τ and
means an indefinite integral;
(iv) last step is concerned with the evaluation of
C(x, x′). This is done after imposing the follow-
ing conditions
〈x, τ |x′, 0〉 = 〈x, τ |P (τ)|x′, 0〉 , (8)
〈x, τ |x′, 0〉 = 〈x, τ |P (0)|x′, 0〉 , (9)
as well as the initial condition
K(x, x′; τ) = δ(x− x′) . (10)
Imposing conditions (8) and (9) means to substitute in
their left hand sides the expression for 〈x, τ |x′, 0〉 given
by (7), while in their right hand sides the operators P (τ)
and P (0), respectively, written in terms of the operators
X(τ) and X(0) with the appropriate time ordering.
III. EXAMPLES
A. Charged particle in an uniform magnetic field
As our first example, we consider the propagator of a
non-relativistic particle with electric charge e and mass
m, submitted to a constant and uniform magnetic fieldB.
Even though this is a genuine three-dimensional problem,
the extension of the results reviewed in the last section
to this case is straightforward. Since there is no electric
field present, the hamiltonian can be written as
(P− eA)
, (11)
where P is the canonical momentum operator, A is the
vector potential and π = P − eA is the gauge invariant
mechanical momentum operation. We choose the axis
such that the magnetic field is given by B = Be3. Hence,
the hamiltonian can be decomposed as
π21 + π
= H⊥ +
, (12)
with an obvious definition for H⊥.
Since the motion along the OX 3 direction is free,
the three-dimensional propagator K(x,x′; τ) can be
written as a product of a two-dimensional propagator,
K⊥(r, r
′; τ), related to the magnetic field and a one-
dimensional free propagator, K
3 (x3, x
3; τ):
K(x,x′; τ) = K⊥(r, r
′; τ)K
3 (x3, x
3; τ), (τ > 0)
where r = x1e1 + x2e2 and K
3 (x3, x
3; τ) is the well
known propagator of the free particle [24],
3 (x3, x
3; τ) =
2πi~τ
(x3 − x
. (14)
In order to use Schwinger’s method to compute the two-
dimensional propagator K⊥(r, r
′; τ) = 〈r, τ |r′, 0〉, we
start by writing the differential equation
〈r, τ |r′, 0〉 = 〈r, τ |H⊥(R⊥(0),π⊥(0))|r
′, 0〉 , (15)
where R⊥(τ) = X1(τ)e1 + X2(τ)e2 and π⊥(τ) =
π1(τ)e1+π2(τ)e2. In (15) |r, τ〉 and |r
′, 0〉 are the eigen-
vectors of position operators R(τ) = X1(τ)e1 +X2(τ)e2
and R(0) = X1(0)e1 + X2(0)e2, respectively. More
especifically, operators X1(0), X1(τ), X2(0) and X2(τ)
have the eigenvalues x′1, x1, x
2 and x2, respectively.
In order to solve the Heisenberg equations for operators
R⊥(τ) and π⊥(τ), we need the commutators
Xi(τ), π
j (τ)
= 2i~πi(τ) ,
πi(τ), π
j (τ)
= 2i~eBǫij3πj(τ), (16)
where ǫij3 is the usual Levi-Civita symbol. Introducing
the matrix notation
R(τ) =
X1(τ)
X2(τ)
; Π(τ) =
π1(τ)
π2(τ)
, (17)
and using the previous commutators the Heisenberg
equations of motion can be cast into the form
dR(τ)
, (18)
dΠ(τ)
= 2ωCΠ(τ) , (19)
where 2ω = eB/m is the cyclotron frequency and we
defined the anti-diagonal matrix
. (20)
Integrating equation (19) we find
Π(τ) = e2ωCτΠ(0) . (21)
Substituting this solution in equation (18) and integrat-
ing once more, we get
R(τ) −R(0) =
sin (ωτ)
eωCτΠ(0) , (22)
where we used the following properties of C matrix:
2 = −1l; C−1 = −C = CT , eαC = cos (α)1l + sin (α)C
with CT being the transpose of C. Combining equations
(22) and (21) we can write Π(0) in terms of the operators
R(τ) and R(0) as
Π(0) =
sin (ωτ)
e−ωCτ
R(τ)−R(0)
. (23)
In order to express H⊥ = (π
2)/2m in terms of R(τ)
and R(0), we use (23). In matrix notation, we have
ΠT (0)Π(0)
2 sin2 (ωτ)
RT (τ)R(τ) +RT (0)R(0) +
−RT (τ)R(0) −RT (0)R(τ)
. (24)
Last term on the r.h.s. of (24) is not ordered appropri-
ately as required in the step (ii). The correct ordering
may be obtained as follows: first, we write
R(0)TR(τ) = R(τ)TR(0) +
[Xi(0), Xi(τ)] . (25)
Using equation (22), the usual commutator
[Xi(0), πj(0)] = i~δij1l and the properties of matrix
C it is easy to show that
[Xi(0), Xi(τ)] =
2i~ sin(ωτ) cos(ωτ)
, (26)
so that hamiltonian H⊥ with the appropriate time order-
ing takes the form
2 sin2 (ωτ)
R2(τ) +R2(0)− 2RT (τ)R(0)
− i~ω cot(ωτ). (27)
Substituting this hamiltonian into equation (15) and in-
tegrating in τ , we obtain
〈r, τ |r′, 0〉 =
C(r, r′)
sin (ωτ)
cot(ωτ)(r− r′)2
, (28)
where C(r, r ′) is an integration constant to be deter-
mined by conditions (8), (9) and (10), which for the case
of hand read
〈r, τ |πj(τ)|r
′, 0〉 =
− eAj(r)
〈r, τ |r′, 0〉 (29)
〈r, τ |πj(0)|r
′, 0〉 =
− eAj(r
〈r, τ |r′, 0〉 , (30)
〈r, τ |r′, 0〉 = δ(2)(r− r′). (31)
In order to compute the matrix element on the l.h.s.
of (29), we need to express Π(τ) in terms of R(τ) and
R(0). From equaitons (21) and (23), we have
Π(τ) =
sin (ωτ)
R(τ)−R(0)
, (32)
which leads to the matrix element
〈r, τ |πj(τ)|r
′, 0〉 = mω[cot(ωτ)
xj − x
+ ǫjk3 (xk − x
k)]〈r, τ |r
′, 0〉 , (33)
where we used the properties of matrix C and Einstein
convention for repeated indices is summed. Analogously,
the l.h.s. of equation (30) can be computed from (23),
〈r, τ |πj(0)|r
′, 0〉 = mω[cot(ωτ)
xj − x
− ǫjk3 (xk − x
k)]〈r, τ |r
′, 0〉 . (34)
Substituting equations (33) and (34) into (29) and
(30), respectively, and using (28), we have
+ eAj(r)+
eFjk(xk−x
C(r, r ′)= 0,(35)
− eAj(r
eFjk(xk−x
C(r, r ′)= 0,(36)
where we defined Fjk = ǫjk3 B.
Our strategy to solve the above system of differential
equations is the following: we first equation (35) assum-
ing in this equation variables r ′ as constants. Then, we
impose that the result thus obtained is a solution of equa-
tion (36). With this goal, we multiply both sides of (35)
by dxj and sum over j, to obtain
Aj(r)+
Fjk (xk − x
dxj . (37)
Integration of the previous equation leads to
C(r, r ′) = C(r ′, r ′) e
[Aj(ξ)+
Fjk(ξk−x
k)] dξj} ,
where the line integral is assumed to be along curve Γ, to
be specified in a moment. As we shall see, this line inte-
gral does not depend on the curve Γ joining r ′ and r, as
expected, since the l.h.s. of (37) is an exact differencial.
In order to determine the differential equation for
C(r ′, r ′) we must substitue expression (38) into equa-
tion (36). Doing that and using carefully the fundamen-
tal theorem of differential calculus, it is straightforward
to show that
(r ′, r ′) = 0 , (39)
which means that C(r ′, r ′) is a constant, C0, indepen-
dent of r ′. Noting that
[B× (ξ − r ′)]j = −Fjk (ξk − x
k) , (40)
equation (38) can be written as
C(r, r ′)= C0 exp
A(ξ)−
B× (ξ − r ′)
Observe, now, that the integrand in the previous equa-
tion has a vanishing curl,
A(ξ)−
B× (ξ − r ′)
= B−B = 0 ,
which means that the line integral in (42) is path in-
dependent. Choosing, for convenience, the straightline
from r ′ to r, it can be readly shown that
[B× (ξ − r ′)] · dξ = 0 ,
where Γsl means a straightline from r
′ to r. With this
simplification, the C(r ′, r) takes the form
C(r, r ′)= C0 exp
~ Γsl
A(ξ) · dξ
. (42)
Substituting last equation into (28) and using the initial
condition (10), we readly obtain C0 =
. Therefore
the complete Feynman propagator for a charged particle
under the influence of a constant and uniform magnetic
field takes the form
K(x,x′; τ)
2πi~ sin (ωτ)
2πi~τ
A(ξ) · dξ
cot(ωτ)(r − r ′)2
(x3 − x
,(43)
where in the above equation we omitted the symbol Γsl
but, of course, it is implicit that the line integral must be
done along a straightline, and we brought back the free
propagation along the OX 3 direction. A few comments
about the above result are in order.
1. Firstly, we should emphasize that the line integral
which appears in the first exponencial on the r.h.s.
of (43) must be evaluated along a straight line be-
tween r′ and r. If for some reason we want to choose
another path, instead of integral
A(ξ) · dξ, we
must evaluate
[A(ξ)− (1/2)B× (ξ − r ′)] · dξ.
2. Since we solved the Heisenberg equations for the
gauge invariant operators R⊥ and π⊥, our final
result is written for a generic gauge. Note that
the gauge-independent and gauge-dependent parts
of the propagator are clearly separated. The gauge
fixing corresponds to choose a particular expression
for A(ξ). Besides, from (43) we imediately obtain
the transformation law for the propagator under a
gauge transformation A → A+∇Λ, namely,
K(r, r ′; τ) 7−→ e
Λ(r)K(r, r ′; τ) e−
Λ(r ′) .
Although this transformation law was obtained in
a particular case, it can be shown that it is quite
general.
3. It is interesting to show how the energy spectrum
(Landau levels), with the corresponding degener-
acy per unit area, can be extracted from prop-
agator (43). With this purpose, we recall that
the partition function can be obtained from the
Feynman propagator by taking τ = −i~β, with
β = 1/(KBT ), and taking the spatial trace,
Z(β) =
dx2 K(r, r;−i~β) .
Substituting (43) into last expression, we get
Z(β) =
2π~ senh(~βω)
where we used the fact that sin(−iθ) = −i sinh θ.
Observe that the above result is divergent, since
the area of the OX 1X2 plane is infinite. This is a
consequence of the fact that each Landau level is
infinitely degenerated, though the degeneracy per
unit area is finite. In order to proceed, let us as-
sume an area as big as we want, but finite. Adopt-
ing this kind or regularization, we write
∫ L/2
∫ L/2
dx2 K (r, r;−i~β) ≈
2π~ senh(~βω)
L2 eB
~βω − e−~βω
L2 eB
1− e−~βωc
L2 eB
−β(n+ 1
)~ωc ,
where we denoted by ωc = eB/2m the ciclotron
frequency. Comparing this result with that of a
partition function whose energy level En has de-
generacy gn, given by
Z(β) =
−βEn ,
we imediately identify the so called Landau leves
and the corresponding degeneracy per unit area,
~ωc ;
(n = 0, 1, ...) .
B. Charged harmonic oscillator in a uniform
magnetic field
In this section we consider a particle with mass m and
charge e in the presence of a constant and uniform mag-
netic field B = Be3 and submitted to a 2-dimensional
isotropic harmonic oscillator potential in the OX 1X2
plane, with natural frequency ω0. Using the same no-
tation as before, we can write the hamiltonian of the
system in the form
H = H⊥ +
, (44)
where
π21 + π
X21 +X
. (45)
As before, the Feynman propagator for this problem
takes the form K(x,x′; τ) = K⊥(r, r
′; τ)K
3 (x3, x
3; τ),
with K
3 (x3, x
3; τ) given by equation (14). The propa-
gator in the OX 1X2-plane satisfies the differential equa-
tion (15) and will be determined by the same used in the
previous example.
Using hamiltonian (45) and the usual commutation re-
lations the Heisenberg equations are given by
dR(τ)
, (46)
dΠ(τ)
= 2ωCΠ(τ) −mω20R(τ) , (47)
where we have used the matrix notation introduced in
(17) and (20). Equation (46) is the same as (18), but
equation (47) contains an extra term when compared to
(19). In order to decouple equations (46) and (47), we
differentiate (46) with respect to τ and then use (47).
This procedure leads to the following uncoupled equation
d2R(τ)
− 2ωC
dR(τ)
+ ω20R(τ) = 0 (48)
After solving this equation, R(τ) and Π(τ) are con-
strained to satisfy equations (46) and (47), respectively.
A straightforward algebra yields the solution
R(τ) = M−R(0) + NΠ(0) (49)
Π(τ) = M+Π(0)−m2ω20NR(0) , (50)
where we defined the matrices
sin (Ωτ)
eωτC (51)
± = eωτC
cos (Ωτ)1l±
sin (Ωτ)C
, (52)
and frequency Ω =
ω2 + ω20 . Using (49) and (50), we
write Π(0) and Π(τ) in terms of R(τ) and R(0),
Π(0) = N−1R(τ)− N−1M−R(0) , (53)
Π(τ)= M+N−1R(τ)−
−+m2ω20N
R(0). (54)
Now, we must order appropriately the hamiltonian
operator H⊥ = Π
T (0)Π(0)/(2m) + mω20R
T (0)R(0)/2,
which, with the aid of equation (53), can be written as
RT (τ)(N−1)T −RT (0)(M−)T (N−1)T
−1R(τ)− N−1M−R(0)
+mω20R
T (0)R(0)
2 sin2 (Ωτ)
RT (τ) −RT (0)(M−)T
R(τ)−M−R(0)
+mω20R
T (0)R(0)
2 sin2 (Ωτ)
RT (τ)R(τ)−RT (τ)M−R(0)
−RT (0)(M−)TRT (τ) +RT (0)(M−)TM−R(0)
+mω20R
T (0)R(0)
2 sin2 (Ωτ)
R2(τ) −RT (τ)M−R(0)
−RT (0)(M−)TR(τ) +R2(0)
, (55)
where superscript T means transpose and we have used
the properties of the matrices N and M− given by (51)
and (52). In order to get the right time ordering, observe
first that
RT (0)(M−)TR(τ) = RT (τ)M−R(0)+
−R(0)
,Xi(τ)
where
−R(0)
,Xi(τ)
= i~Tr
N(M−)T
sin (2Ωτ) .
Using the last two equations into (55) we rewrite the
hamiltonian in the desired ordered form, namely,
2 sin2 (Ωτ)
R2(τ) +R2(0)− 2RT (τ)M−R(0)
sin (2Ωτ)
. (56)
For future convenience, let us define
U(τ) = cos (ωτ) cos (Ωτ) +
sin (ωτ) sin (Ωτ) ,(57)
V (τ) = sin (ωτ) cos (Ωτ) −
cos (ωτ) sin (Ωτ) (58)
and write matrix M−, defined in (52), in the form
− = U(τ)1l + V(τ)C. (59)
Substituting (59) in (56) we have
2 sin2 (Ωτ)
R2(τ) +R2(0)− 2U(τ)RT (τ)R(0)
− 2V (τ)RT (τ)CR(0)−
sin (2Ωτ)
. (60)
The next step is to compute the classical function
F (r, r′; τ). Using the following identities
ΩU(τ)
sin2 (Ωτ)
[cos (ωτ)
sin (Ωτ)
, (61)
ΩV (τ)
sin2 (Ωτ)
[ sin (ωτ)
sin (Ωτ)
, (62)
into (60), we write F (r, r′; τ) in the convenient form
F (r, r′; τ) =
(r2 + r′
)csc(Ωτ)2+mΩr · r′
[cos (ωτ)
sin (Ωτ)
+ mΩr · Cr′
[ sin (ωτ)
sin (Ωτ)
− i~Ω
cos (Ωτ)
sin (Ωτ)
. (63)
Inserting this result into the differential equation
〈r, τ |r′, 0〉 = F (r, r′; τ)〈r, τ |r′, 0〉 ,
and integrating in τ , we obtain
〈r, τ |r′,0〉 =
C(r, r′)
sin (Ωτ)
(r2 + r′
) cot (Ωτ)
r · r′
cos (ωτ)
sin (Ωτ)
+ r · Cr′
sin (ωτ)
sin (Ωτ)
. (64)
where C(r, r ′) is an arbitrary integration constantto be
determined by conditions (29), (30) and (31). Using (54)
we can calculate the l.h.s. of condition (29),
〈r,τ |πj(τ)|r
′, 0〉 =
sin (Ωτ)
cos (Ωτ)xj − cos (ωτ)x
sin (Ωτ)xk − sin (ωτ)x
〈r, τ |r′, 0〉, (65)
and using (53) we get the l.h.s. of condition (30),
〈r,τ |πj(0)|r
′, 0〉 =
sin (Ωτ)
cos (ωτ)xj − cos (Ωτ)x
sin (Ωτ)x′k − sin (ωτ)xk
〈r, τ |r′, 0〉. (66)
With the help of the simple identities
(r2 + r′
) = 2xj ;
(r2 + r′
) = 2x′j
r · r′ = x′j ;
r · r′ = xj
r · Cr′ = ǫjk3x
r · Cr′ = −ǫjk3xk.
and also using equation (64), we are able to compute the
right hand sides of conditions (29) and (30), which are
given, respectively, by
C(r, r′)
∂C(r, r′)
cos (Ωτ)
sin (Ωτ)
xj −mΩ
cos (ωτ)
sin (Ωτ)
sin (ωτ)
sin (Ωτ)
ǫjk3x
k − eAj(r)
〈r, τ |r′, 0〉 (67)
C(r, r′)
∂C(r, r′)
cos (Ωτ)
sin (Ωτ)
x′j +mΩ
cos (ωτ)
sin (Ωτ)
sin (ωτ)
sin (Ωτ)
ǫjk3xk − eAj(r
〈r, τ |r′, 0〉. (68)
Equating (65) and (67), and also (66) and (68)), we get
the system of differential equations for C(r, r′)
∂C(r, r′)
Aj(r) +
C(r, r′) = 0 , (69)
∂C(r, r′)
C(r, r′) = 0. (70)
Proceeding as in the previous example, we first integrate
(69). With this goal, we multiply it by dxj , sum in j and
integrate it to obtain
C(r, r ′) = C(r ′, r ′) exp
Aj(ξ) +
where the path of integration Γ will be specified in a
moment. Inserting expression (71) into the second differ-
ential equation (70), we get
C(r ′, r ′) = 0 =⇒ C(r ′, r ′) = C0 ,
where C0 is a constant independent of r
′, so that equa-
tion (71) can be cast, after some convenient rearrange-
ments, into the form
C(r, r′) = C0 exp
A(ξ)−
Note that the integrand has a vanishing curl so that we
can choose the path of integration Γ at our will. Choos-
ing, as before, the straight line between r′ and r, it can
be shown that
A(ξ)−
·dξ =
A(ξ)·dξ+
Br ·Cr′ , (73)
where, for simplicity of notation, we omitted the symbol
Γsl indicating that the line integral must be done along
a straight line. From equations (71), (72) e (73), we get
C(r, r′) = C0 exp
A(ξ) · dξ
r · Cr′
which substituted back into equation (64) yields
〈r, τ |r′, 0〉 =
sin (Ωτ)
A(ξ) · dξ
{ imΩ
2~ sin (Ωτ)
(r2 + r′
) cos (Ωτ)
−2r · r′ cos (ωτ)− 2
sin (ωτ) −
sinΩτ
The initial condition implies C0 = mΩ/(2πi~). Hence,
the desired Feynman propagator is finally given by
K(x,x′; τ) = K⊥(r, r
′; τ)K
3 (x3, x
3; τ)
2 π i ~ sin (Ωτ)
2πi~τ
A(ξ) · dξ
2~ sin (Ωτ)
cos (Ωτ)(r2 + r′
− 2 cos (ωτ)r · r′ − 2
sin (ωτ)−
sin (Ωτ)
r · Cr′
(x3 − x
, (76)
where we brought back the free part of the propagator
corresponding to the movement along the OX 3 direction.
Of course, for ω0 = 0 we reobtain the propagator found
in our first example and for B = 0 we reobtain the prop-
agator for a bidimensional oscillator in the OX 1X2 plane
multiplied by a free propagator in the OX 3 direction, as
can be easily checked.
Regarding the gauge dependence of the propagator,
the same comments done before are still valid here,
namely, the above expression is written for a generic
gauge, the transformation law for the propagator under a
gauge transformation is the same as before, etc. We fin-
ish this section, extracting from the previous propagator,
the corresponding energy spectrum. With this purpose,
we first compute the trace of the propagator,
dx2 K
⊥(x1, x1, x2, x2; τ) =
2πi~ sin(Ωτ)
dx2 exp
2~ sin(Ωτ)
cos(Ωτ) − cos(ωτ)
(x21 + x
2[cos(Ωτ) − cos(ωτ)]
, (77)
where we used the well known result for the Fresnel in-
tegral. Using now the identity
cos(Ωτ)−cos(ωτ) = −2 sin[(Ω+ω)τ/2] sin[(Ω−ω)τ/2)] ,
we get for the corresponding energy Green function
G (E) =−i
dx2 K
⊥(x1, x1, x2, x2; τ)
sen(Ωτ
τ) sen(Ω−ω
e−(l+
)(Ω+ω)τ
e−i(n+
(Ω−ω)τ
where is tacitly assumed that E → E − iε and we also
used that (with the assumption ν → ν − iǫ)
sen(ν
e−i(n+
)ντ .
Changing the order of integration and summations, and
integrating in τ , we finally obtain
G(E) =
l,n=0
E − Enl
, (78)
where the poles of G(E), which give the desired energy
levels, are identified as
Enl = (l+n+1)~Ω+(l−n)~ω , (l, n = 0, 1, ...) . (79)
The Landau levels can be reobtained from the previous
result by simply taking the limit ω0 → 0:
Enl −→ (2l + 1)~ω = (l +
)~ωc , (80)
with l = 0, 1, ... and ωc = eB/m, in agreement to the
result we had already obtained before.
IV. FINAL REMARKS
In this paper we reconsidered, in the context of
Schwinger’s method, the Feynman propagators of two
well known problems, namely, a charged particle under
the influence of a constant and uniform magnetic field
(Landau problem) and the same problem in which we
added a bidimensional harmonic oscillator potential. Al-
though these problems had already been treated from
the point of view of Schwinger’s action principle, the
novelty of our work relies on the fact that we solved
the Heisenberg equations for gauge invariant operators.
This procedure has some nice properties, as for instance:
(i) the Feynman propagator is obtained in a generic
gauge; (ii) the gauge-dependent and gauge-independent
parts of the propagator appear clearly separated and
(iii) the transformation law for the propagator under
gauge transformation can be readly obtained. Besides,
we adopted a matrix notation which can be straightfor-
wardly generalized to cases of relativistic charged parti-
cles in the presence of constant electromagnetic fields and
a plane wave electromagnetic field, treated by Schwinger
[2]. For completeness, we showed explicitly how one can
obtain the energy spectrum directly from que Feynman
propagator. In the Landau problem, we obtained the (in-
finitely degenerated) Landau levels with the correspond-
ing degeneracy per unit area. For the case where we
included the bidimensional harmonic potential, we ob-
tained the energy spectrum after identifying the poles of
the corresponding energy Green function. We hope that
this pedagogical paper may be useful for undergraduate
as well as graduate students and that these two simple
examples may enlarge the (up to now) small list of non-
relativistic problems that have been treated by such a
powerful and elegant method.
Acknowledgments
F.A. Barone, H. Boschi-Filho and C. Farina would like
to thank Professor Marvin Goldberger for a private com-
munication and for kindly sending his lecture notes on
quantum mechanics where this method was explicitly
used. We would like to thank CNPq and Fapesp (brazil-
ian agencies) for partial financial support.
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