Supersymmetric Field Theory Based on Generalized Uncertainty Principle
RIKEN-TH-96
Supersymmetric Field Theory Based on Generalized Uncertainty
Principle
SHIBUSA Yuuichirou
Theoretical Physics Laboratory,
RIKEN (The Institute of Physical and Chemical Research),
Wako, 351-0198, Japan
Abstract
We construct a quantum theory of free fermion field based on the deformed Heisenberg algebra
[x̂, p̂] = i~(1+βp̂2) where β is a deformation parameter using supersymmetry as a guiding principle.
A supersymmetric field theory with a real scalar field and a Majorana fermion field is given explicitly
and we also find that the supersymmetry algebra is deformed from an usual one.
PACS numbers: 03.65.Ca, 03.70.+k, 11.10.Ef, 11.30.Pb
http://arxiv.org/abs/0704.1545v1
I. INTRODUCTION
Physics in extremely high energy regions is particularly of interest to particle physics. In
particular, when we discuss gravity, it is expected that there is a minimal length in principle.
String theory which has a characteristic scale
α′, is one of the most successful theoretical
frameworks which overcome the difficulty of ultra-violet divergence in quantum theory of
gravity. However, string theory has many difficulties in performing practical computations.
Therefore if we construct a field theory which captures some stringy nature and/or includes
stringy corrections, it would play a pivotal role in investigating physics in high energy regions
even near the Planck scale.
Some of the stringy corrections appear as α′ corrections. In other words, it often takes the
form as higher derivative corrections i.e. higher order polynomial of momentum. One way to
discuss these corrections is deforming the Heisenberg uncertainty principle to a generalized
uncertainty principle (GUP):
∆x̂ ≥ ~
∆p̂, (1.1)
where β is a deforming parameter and corresponds to the square of the minimal length scale.
If GUP is realized in a certain string theory context, β would take a value of order the string
scale (β ∼ α′). This relation comes from various types of studies such as on high energy or
short distance behavior of strings [1], [2], gedanken experiment of black hole [3], de Sitter
space [4], the symmetry of massless particle [5] and wave packets [6].
There are several canonical commutation algebra which lead to the GUP. Among these
algebra we will focus on the algebra;
[x̂, p̂] = i~(1 + βp̂2). (1.2)
This algebra is investigated in [7]-[10] and an attempt to construct a field theory with
minimal length scale is made in [11] by using the Bargmann-Fock representation in 1+1
dimensional spacetime. It has also been used in cosmology, especially in physics at an early
universe (see for example, [12]-[15] and references therein).
In our previous paper [16], we investigated the quantization of fields based on the de-
formed algebra (1.2) in the canonical formalism in 1+1 dimensions and in the path integral
formalism as well. Using the path integral formalism we constructed a quantum theory of
scalar field in arbitrary spacetime dimensions. This theory has a non-locality which stems
from the existence of a minimal length.
In this paper, we construct a quantum theory of free fermion field based on the deformed
Heisenberg algebra. Where, we respect supersymmetry as a guiding principle. This is be-
cause a string theory has this symmetry and we intend to construct a field theory which con-
tains the stringy corrections. Moreover, supersymmetry is also an useful tool to understand
physics in ultra-violet momentum regions. It manages a behavior of system in extremely
high energy regions and eases ultra-violet divergence in quantum theory. Therefore we pro-
pose a quantum field theory of fermion to have a supersymmetry for a scalar system which
was given in [16]. In two and three-dimensional spacetime, we give a system with one real
scalar and one Majorana fermion explicitly. This system has a special symmetry between
a boson and a fermion which corresponds to supersymmetry. Although, this symmetry is
deformed from ordinary supersymmetry. From the fermionic part of this system, we propose
an action of fermionic fields based on GUP in general dimensional spacetime.
II. SCALAR FIELD THEORY
In the paper [16], we proposed a field theory of scalar based on GUP in the path-integral
formalism. We begin with a review of this theory.
Our theory is based on the following algebra [7]:
x̂i, p̂j
= i~(1 + βp̂2)δij . (2.1)
This is an extension to higher dimensional spacetime of deformed Heisenberg algebra (1.2).
Here i, j run from 1 to d which is the number of spatial coordinates and p̂2 ≡
(p̂i)
Hereinafter, we use index i, j for spatial coordinates and a, b for all spacetime coordinates.
Jacobi identity determines the full algebra:
x̂i, x̂j
= −2i~β(1 + βp̂2)L̂ij . (2.2)
p̂i, p̂j
= 0. (2.3)
Here L̂ij are angular momentum like operators L̂ij ≡ 1
2(1+βp̂
(x̂ip̂j − x̂j p̂i + p̂j x̂i − p̂ix̂j).
Because operators p̂i commute with each other, we construct a theory in momentum space
representation. In momentum space representation, momentum operators are diagonalized
simultaneously and we do not distinguish eigenvalues of momentum pi from operators p̂i. In
the following, we set Planck constant ~ to be 1 for simplicity.
Lagrangian in d+ 1 dimensional spacetime [16] is
L = −
ddp(1 + βp2)−1φ(−p, t)
∂2t + p
2 +m2
φ(p, t), (2.4)
where, p2 ≡
The difference from ordinary quantum field theory is a prefactor (1+βp2)−1 in Lagrangian.
Using the Bjorken-Johnson-Low prescription[17], from behavior of T∗-product between
φ(p, t) and φ(p′, t′), we obtain canonical commutation relation:
[φ(p, t), ∂tφ(p
′, t)] = i(1 + βp2)δd(p+ p′). (2.5)
As we can see from this equation, a deforming prefactor (1 + βp̂2) of Heisenberg algebra
in the first quantization (2.1) also appears in canonical commutation relation of the second
quantized field theory.
In a fermion field case, we encounter a difficulty at constructing the second quantized
Hilbert space which does not appear in a scalar system. Note that a system of spin 0
particles contains only spin 0 particle. By contrast, a system of spin 1
particles is not closed
with only fermions in the sense that it contains bosons as bound states. Therefore algebra
of fermion fields must be introduced to be consistent with that of bosons fields. Because the
scalar fields in our theory have a different commutation relation (2.5) from ordinary one,
we must construct fermion fields so that the composite fields which correspond to scalar
particles have the same commutation relations. Or, in two-dimensional ordinary quantum
field theory we could use the concepts of bosonization and fermionization which associate
fermion fields with boson fields. However, it is obscure which of these principles which
relate bosons and fermions remains unchanged in GUP or in extremely high energy regions.
Instead of handling this problem directly, we use supersymmetry to construct quantized field
theory of fermion. This is because string theory accommodates this symmetry and therefore
it is expected that this symmetry is reflected in GUP or in extremely high energy regions.
In the next section we construct a quantum field theory of fermions which is consistent
with the above scalar theory by using supersymmetry.
III. SUPERSYMMETRY IN GUP
In two and three-dimensional spacetime, a system with a real scalar and a Majorana
fermion has a special symmetry between a boson and a fermion, namely supersymmetry.
Thus we construct a quantum field theory of fermion in GUP to have a similar symme-
try between bosons and fermions with an above-reviewed scalar system in two and three-
dimensional spacetime.
Our notation for two and three-dimensional spacetime is as follows: In those dimensional
spacetime (with signature −+ or − + +) the Lorentz group has a real (Majorana) two-
component spinor representation ψα. In the following, we explain the notation of three-
dimensional spacetime. Reduction to two-dimensional spacetime is trivial. We define a
representation of Gamma matrices by Pauli matrices1 as follows:
{Γa,Γb} = 2ηab = 2diag(−++), (3.1)
Γ0 = −iσ2,Γ1 = σ1,Γ2 = −σ3. (3.2)
Spinor indices are lowered and raised by charge conjugation matrix Cαβ ≡ Γ0 and its inverse
matrix C−1:
ψα = ψ
βCβα(= ψ̄α), ψ
α = ψβ(C
−1)βα. (3.3)
Because the algebra of scalar field (2.5) is deformed from usual one, it is natural to
expect that supersymmetry algebra may also be deformed from ordinary one. We generalize
supersymmetry algebra and its actions on a scalar field φ, a Majorana fermion ψ and an
auxiliary field F with parameter ǫαas follows:
ǭ1Q̂, ǭ2Q̂
= 2∆ǭ1Γ
aǫ2P̂a, (3.4)
δφ(p, t) = iǭψ(p, t), (3.5)
δψα(p, t) = A1F (p, t)ǫ
α − A2{(ǭΓ0C−1)α∂t + (ǭΓjC−1)α(ipj)}φ(p, t), (3.6)
δF (p, t) = A3iǭ(Γ
0∂t + Γ
j(ipj))ψ(p, t). (3.7)
Here, we introduce factors ∆, Ai as functions of a deforming parameter β and momentum.
These factors should reduce to 1 in the limit of β → 0 and will be determined later by
consistency conditions.
1 Pauli matrices are σ1 =
, σ2 =
, σ3 =
From the closeness of algebra on each fields, we obtain conditions
A1A3 = A2 = ∆. (3.8)
We also generalize a Lagrangian by introducing factorsBi, which are functions of a deforming
parameter β and momentum and are to be determined as well:
φ(−p, t)(∂2t + p2)φ(p, t)−
ψ̄(−p, t)(Γ0∂t + (ipi)Γi +m)ψ(p, t)
B1B3mφ(−p, t)F (p, t) +
F (−p, t)F (p, t)
. (3.9)
Here d is the number of spatial coordinates (1 or 2). By integrating out the field F , we
obtain Lagrangian with the scalar field and the Majorana field:
φ(−p, t)(∂2t + p2 +m2)φ(p, t)
− iB2
ψ̄(−p, t)(Γ0∂t + (ipi)Γi +m)ψ(p, t)
. (3.10)
Invariance of Lagrangian (3.9) under supersymmetry variations (3.5)-(3.7) leads following
conditions;
A1B2 =
B1B3,
A1B2 = A3B3,
B1 = A1A3B2. (3.11)
From conditions (3.8) and (3.11), only B1 and B2 remain to be determined. (Factor A1 can
be absorbed into normalization of a field F and we set it to be 1 for a field F to be an
auxiliary field.) Noether’s current for supersymmetry can be calculated from Lagrangian
(3.10) and supersymmetry charge is found to be
dtdpdB1{ − ψα(−p, t)∂tφ(p, t) + (ΓiΓ0ψ(−p, t))α(ipi)φ(p, t)
+ m(Γ0ψ(−p, t))αφ(p, t)}. (3.12)
Then, we obtain Hamiltonian of this system from supersymmetry charge and algebra (3.4),
H = P 0 = −1
(CΓ0)αβ{Qα, Qβ}. (3.13)
Using the Bjorken-Johnson-Low prescription, from behaviors of T∗-product between fields,
we obtain canonical commutation relations as follows,
[φ(p, t), ∂tφ(q, t)] =
δ(p+ q), (3.14)
α(p, t), ψβ(q, t)
(Γ0C−1)αβ
δ(p+ q). (3.15)
Thus we can write the Hamiltonian in the following form;
{π(−p, t)π(p, t) + φ(−p, t)(p2 +m2)φ(p, t)}
ψ̄(−p, t)((ipi)Γi +m)ψ(p, t). (3.16)
Here, we use conjugate momentum π(p, t) = ∂tφ(−p, t) and indices i runs from 1 to d.
There is another condition which can be used to determine the factors B1 and B2. It
comes from the free energy of supersymmetric vacuum. From algebra (3.4), supersymmetric
state has zero energy:
TrB ln(B1(E
2 + p2 +m2))− 1
TrF ln(B
2 + p2 +m2)). (3.17)
This fact leads to the condition;
B1 = B
2 . (3.18)
Here TrB and TrF represent trace in bosonic and fermionic Hilbert space respectively.
Lastly, we set B1 = (1+βp
2)−1 as we can see from the scalar action (2.4). This determines
all of the introduced factors as follows;
∆ = A2 = A3 = B2 = (1 + βp
2 , (3.19)
A1 = B3 = 1, (3.20)
B1 = (1 + βp
2)−1. (3.21)
Thus we construct quantized fields of fermion which is consistent with scalar fields (2.5) as
ψα(p, t), ψβ(q, t)
= −(1 + βp2)
2 (Γ0C−1)αβδ(p+ q). (3.22)
Note that a factor ∆ is not equal to 1 no matter how we set A1. Therefore this super-
symmetry algebra is deformed from an usual one as
[ǭ1Q, ǭ2Q] = 2(1 + βp
2 ǭ1Γ
aǫ2Pa. (3.23)
There is no difficulty in generalizing the above quantum fields of fermion to higher d+ 1
dimensions than three dimensions. The action is as follows:
(1 + βp2)
ψ̄(−p, t)(Γ0∂t + (ipi)Γi +m)ψ(p, t)
. (3.24)
There appears a universal prefactor (1+βp2)−
2 comparing with usual fermion action regard-
less as to whether there were supersymmetry or not. This prefactor ensures that fermion
fields are compatible with the scalar fields which had been constructed in our previous paper
[16].
From the actions (2.4) and (3.24), we also have supersymmetric field theory in four
dimensions with an complex scalar and a Majorana (or Weyl) fermion just as a corresponding
ordinary field theory has supersymmetry in four dimensions.
IV. CONCLUSION AND DISCUSSIONS
In summary, we have constructed a quantum theory of free fermion field based on the
deformed Heisenberg algebra. It is consistent with already proposed scalar theory through
supersymmetry. We start with a system with an real scalar and a Majorana fermion in two-
and three-dimensional spacetime and determine supersymmetric action. We found that
supersymmetry algebra is deformed from an usual one. An extension to higher dimensions
are trivial and there is also supersymmetric theory in four-dimensional spacetime.
We conclude with a brief discussion on Lorentz invariant extension of our theory. Lorentz
invariant extension of deformed Heisenberg algebra (2.1) is known as a sort of ‘doubly special
relativity’ or ‘κ-deformation’ (for example, see [18], [19] and references therein),
[x̂a, p̂b] = i~(1 + βp̂
2)δab . (4.1)
Here a, b run from 0 to d and p̂2 ≡ −(p̂0)2 +
i=0(p̂i)
2. Thus we claim that an action where
the factor (1+βp2) is replaced with a new factor (1+βp2) describes quantum field theory of
doubly special relativity. In such case, time slice is not well-defined because of the existence
of minimal time interval. Therefore there is no canonical formalism.
Acknowledgments
The author is grateful to T. Matsuo, K. Oda, T. Tada, and N. Yokoi for valuable dis-
cussions. The author is supported by the Special Postdoctoral Researchers Program at
RIKEN.
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http://arxiv.org/abs/hep-th/0505183
http://arxiv.org/abs/gr-qc/0610056
http://arxiv.org/abs/hep-th/9412167
http://arxiv.org/abs/hep-th/0305262
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http://arxiv.org/abs/gr-qc/0410053
http://arxiv.org/abs/gr-qc/0411056
http://arxiv.org/abs/gr-qc/0504135
http://arxiv.org/abs/hep-th/0511031
http://arxiv.org/abs/hep-th/0203065
http://arxiv.org/abs/gr-qc/0210063
	Introduction
	Scalar Field Theory
	Supersymmetry in GUP
	Conclusion and Discussions
	Acknowledgments
	References
