On Charge Conservation and The Equivalence Principle
in the Noncommutative Spacetime
Youngone Lee
Center for Quantum Spacetime, Sogang University,
Seoul 121-742, Korea.
Abstract
We investigate one of the consequences of the twisted Poincaré symmetry. We derive the
charge conservation law and show that the equivalence principle is satisfied in the canonical non-
commutative spacetime. We applied the twisted Poincaré symmetry to the Weinberg’s analysis
[11]. To this end, we generalize our earlier construction of the twisted S matrix [10], which apply
the noncommutativity to the fourier modes, to the massless fields of integer spins. The transforma-
tion formula for the twisted S matrix for the massless fields of integer spin has been obtained. For
massless fields of spin 1, we obtain the conservation of charge, and the universality of coupling
constant for massless fields of spin 2, which can be interpreted as the equality of gravitational
mass and inertial mass, i.e., the equivalence principle.
PACS numbers: 11.10.Nx, 11.30.-j, 11.30.Cp, 11.55.-m
youngone@yonsei.kr
http://arxiv.org/abs/0704.1805v3
1 Introduction
In effort to construct an effective theory of quantum gravity at the Planck scale, noncommutativity of
the spacetime has been considered. The canonical noncommutative spacetime has the commutation
relations between the coordinates [1],
[xµ, xν ] = iθµν , (1)
where θµν(µ, ν = 0 to 3) is a constant antisymmetric matrix.
Field theories in the canonical noncommutative spacetime can be replaced by field theories in the
commutative spacetime with the Moyal product (The Weyl-Moyal correspondence [2]). One of the
significant problems of those theories is that they violate the Lorentz symmetry. One finds that the
symmetry group is SO(1, 1)×SO(2) instead of the Lorentz group, SO(1, 3). Since there is no spinor
or vector representations in that symmetry group, most of the earlier studies performed by using the
spinor, vector representations of the Lorentz group can not be justfied. Moreover, the factors 1(−1)
are multiplied for a boson(fermion)loop without knowing the spin-statistics relation.
To get around this, Chaichian et.al.∗ have deformed the Poincaré symmetry as well as its module
space to which the symmetry acts [3]. The twisted symmetry group has the same representations as
the original Poincaré group and at the same time they successfully retain the physical information of
the canonical noncommutativity. The main idea was that one can change a classical symmetry group
to a quantum group, ISOθ(1, 3) in this case, and twist-deform the module algebra consistently to
reproduce the noncommutativity. In their approach, the noncommutative parameter θµν transform as
an invariant tensor. This reminds us the situation that Einstein had to change the symmetry group of
the spacetime and its module space(to the Minkowski spacetime) when the speed of light is required
to be constant for any observer in an inertial frame. Similarly, Chaichian et.al. have required the
change of the Hopf algebra with its module algebra so that any observer in an inertial frame feel
the noncommutativity in the same way. For the κ-deformed noncommutativity, Majid and Ruegg
found the κ-deformed spacetime [6] as a module space of the κ-deformed symmetry after Lukierski
et.al. discovered the symmetry [7]. The real benefit of the twist is in the use of the same irreducible
representations of original theories unlike general deformed theories, as in the case of the κ-deformed
theory.
Recently, groups of physicists have constructed the quantum field theory in the noncommutative
spacetime by twisting the quantum space as a module space [8],[9],[10]. Espcially, Bu et.al. have
proposed a twisted S-matrix as well as a twisted Fock space for consistency [10]. There we have
obtained the twisted algebra of the creation and annihilation operators and the spin-statistics relation
by applying the twisted Poincaré symmetry on the quantum space consistently. The analysis of this
paper is mainly based on this work.
These works justify the use of the irreducible representations of the Poincaré group and the sign
factors being used in the earlier studies. Are these works merely a change of viewpoint? Mathemat-
ically they look equivalent and seem to have equal amounts of information. But when the physics
∗ Oeckl [4], Wess [5] have proposed the same deformed Poincaré algebra.
is concerned the action of symmetry becomes more subtle than it seems because it confines possible
configurations of physical systems. In this article, we present an example showing the role of the
twisted symmetry for solving physics problems, especially in the canonical noncommutative space-
time. As an example, we derive the conservation law of charge and show the fact that the equivalence
principle is satisfied even in the noncommutative spacetime. In this derivation we consider spin 1 and
2 massless fields for the photon and the graviton, respectively. For this purpose, we extend our ear-
lier study on the scalar field theory to more general field theories and investigate a noncommutative
version of Weinberg’s analysis [11, 12, 13].
Actually, there are many studies for the relation between noncommutativity and gauge theory [14],
[15], [16], [17] or between the noncommutativity and the gravity [18],[19],[20] and between them
[21]. And there were many argument whether the equivalence principle is satisfied at the quantum
level. Some people argued that the equivalence principle is violated in quantum regime [22],[23],
while there are studies which show non-violation of the equivalence principle [24]. Whether the
principle of equivalence is violated or not is an important issue for quantum gravity because the
principle is the core of the general relativity.
The paper is organized as follows. We extend our previous construction of the S⋆ matrix to the
massless fields of integer spin after giving a brief review on the construction and the properties of the
S⋆ matrix. We give an exact transformation formula for the S⋆ matrix elements in section 2. We give
the consequence of requiring the twist invariance to the S⋆ matrix elements for the scattering process.
These results lead to the charge conservation law for the spin 1 field theory and the universality of
the coupling constant for spin 2 field in noncommutative spacetime in section 3. Finally, we discuss
the implication of the twisted symmetry and its applicability to other issues in section 4. We give
some related calculations of the polarization vector, the noncommutative definition of the invariant
M function, and a twisted transformation formula for the S⋆ matrix in the Appendices.
2 Properties of the general S⋆ matrix
2.1 A short introduction of useful properties of the twist-deformation
An algebra with a product · and a coalgebra with a coproduct ∆ constitute a Hopf algebra if it has an
invertible element S called antipode and with some compatibility relations. For a Lie algebra g, there
is a unique universal enveloping algebra U(g) which preserves the Lie algebra properties in terms of
unital associative algebra. The Hopf algebra of a Lie algebra g is denoted as H ≡ {U(g), ·,∆, ǫ, S},
where U(g) is an universal enveloping algebra of the corresponding algebra g and we denotes the
counit as ǫ. The Sweedler notation is being widely used for a shorthand notation of the coproduct,
Y(1) ⊗ Y(2) [25].
The action of a Hopf algebra H to a module algebra A is defined as
Y ⊲ (a · b) =
(Y(1) ⊲ a) · (Y(2) ⊲ b), (2)
where a, b ∈ A, the symbol · is a multiplication in the module algebra A, and the symbol ⊲ denotes
an action of the Lie generators Y ∈ U(g) on the module algebra A. The product · in H and the
multiplication · in A should be distinguished.
If there is an invertible ’twist element’, F =
F(1) ⊗ F(2) ∈ H ⊗H, which satisfies
(F ⊗ 1) · (∆⊗ id)F = (1⊗F) · (id ⊗∆)F , (3)
(ǫ⊗ id)F = 1 = (id ⊗ ǫ)F , (4)
one can obtain a new Hopf algebra HF ≡ {UF (g), ·,∆F , ǫF , SF} from the original one. The relations
between them are
∆FY = F ·∆Y · F−1 , ǫF (Y ) = ǫ(Y ),
SF (Y ) = u · S(Y ) · u−1 , u =
F(1) · S(F(2)), (5)
with the same product in the algebra sector. The ’covariant’ multiplication of the module algebra AF
for the twisted Hopf algebra HF which maintain the form of Eq.(2) is given as
(a ⋆ b) = ·[F−1 (a⊗ b)]. (6)
From the above relations, one can derive an important property of the twist such that it does not
change the representations of the algebra:
DF(Y )(a ⋆ b) = ⋆ [∆FY (a⊗ b)]
= · [F−1 · F∆0YF−1(a⊗ b)]
= · [∆0Y F−1(a⊗ b)]
= D0(Y )(a ⋆ b), (7)
where representations of the coproduct and the twist element is implied, i.e.,
D[∆Y ] =
D(Y(1))⊗D(Y(2)), D[F ] =
D(F(1))⊗D(F(2)). (8)
The above considerations lead us to the golden rule: The irreducible representations are not changed
by a twist and one can regard the covariant action of a twisted Hopf algebra on a twisted module
algebra as the action of the original algebra on the twisted module algebra.
2.2 The S⋆ matrix and its twist invariance
Recently, a quantum field theory has been constructed in such a way to preserve the twisted Poincaré
symmetry [10]. There we confined the construction for the space-space noncommutativity. It is hard
to know whether one can construct a consistent twist Poincaré invariant field theory satisfying the
causality in the case of space-time noncommutativity. They have tried to apply the twisted symmetry
to quantum spaces consistently, especially to the algebra of the creation and annihilation operators (a†p
and ap). As a result, they obtained the twisted algebra of quantum operators. If we use a shorthand
notation, p ∧ q = pµθµνqν , the twisted algebra of a†p and ap can be denoted as:
cp ⋆ cq = e
ep∧eq cp · cq, (9)
where cp can be ap or a
p, p̃ ≡ −p(p) for cp = ap(a†p) and · denotes the ordinary multiplication of
operators in the commutative theories.
This twisted algebra naturally leads to the twisted form of Fock space, S-matrix and quantities
related to the creation and annihilation operators. Thus, we obtain the twisted basis of Fock space and
S⋆-matrix:
|q1, · · · , qn〉 → |q1, · · · , qn〉⋆ = E(q1, · · · , qn)|q1, · · · , qn〉, (10)
where E(q1, · · · , qn) = exp
i<j qi ∧ qj
is a phase factor which has the interesting properties
[10], and
S → S⋆ =
(−i)k
d4x1 · · · d4xk T {H⋆I(x1) ⋆ · · · ⋆H⋆I(xk)} , (11)
where T denotes the time ordering and H⋆I(x) is an interaction Hamiltonian density in the Dyson
formalism.
The explicit form of the S⋆ matrix elements for the scalar φ
n theoryin the momentum space is:
⋆〈β|S⋆|α〉⋆ = E(−β, α)
(−ig)k
· · ·
···cQk
E(Q̃1) · · · E(Q̃k)〈β|Sk(Q̃1 · · · , Q̃k)|α〉, (12)
where Q̃ is the shorthand notation for (q̃1, . . . , q̃n) [10].
In the above, 〈β|Sk(Q̃1 · · · , Q̃k)|α〉 is a gk order term of the S-matrix element of the commutative
theory where g is the coupling constant of the theory. From the momentum conservation, i.e., delta
functions in the 〈β|Sk(Q̃1 · · · , Q̃k)|α〉, one can show that the S⋆ matrix element, ⋆〈β|S⋆|α〉⋆, can be
represented by Feynman diagrams with extra phase factors E(Q̃) for each vertex. The phase factors
drastically change the predictions of the theory. This result agrees with Filk’s result[26], but we have
overall factors E(−β, α) corresponding to external lines in the Feynman diagram which originated
from the twisted Fock space. From the above considerations, the new modified Feynman diagrams
can be obtained from the untwisted ones by changing the phase factors from 1 to E(Q̃i) at each vertex.
The twist invariance of this prescription of the S⋆ matrix is not manifest because non-locality of
the interactions may violate the twist invariance of the S⋆ matrix, in general, i.e.,
[H⋆I(x),H⋆I(y)]⋆ 6= 0 for spacelike (x− y). (13)
However, we see from the form of S⋆ matrix in Eq.(12), that the proposed S⋆ matrix is clearly twist
invariant since it is constructed from phase factors which are twist invariant, and the Feynman propa-
gators. Twisted product of fields operators satisfy
〈0|ψ(x) ⋆ ψ(y)|0〉 = 〈0|ψ(y) ⋆ ψ(x)|0〉, for spacelike (x− y). (14)
Hence we see that the Feynman propagator, same as the twist Feynman propagator 〈0|T [ψ(x) ⋆
ψ(y)]|0〉, is twist invariant. From this, the invariance of the S⋆ matrix elements follows immediately.
2.3 Generalization to arbitrary fields
We need to get the S⋆ matrix for massless field theories of integer spin for the analysis of this paper.
In the previous work [10], we have constructed the S⋆ matrix for scalar field theory and we have
expected that the same formulation could be possible for general field theories. In this section, we
generalize our argument used in that paper to obtain the form of the S⋆ matrix elements for massless
field theories with spin 1 and 2. In the analysis of this paper, we use the (s/2, s/2) representation for
massless integer spin fields. The reason we use it and the considerations for the other representations
are given in section 4.
We have used the perturbation theory in our formulation of the S⋆ matrix in Eq.(11). Another
assumption was the particle interpretation. That is, the field operators are represented as linear com-
binations of the creation and annihilation operators and the fields of spin s transform(twist) as† :
[Dsθ(Λ
−1)]AB Ψ̂
θ (Λx+ a) = Uθ(Λ, a) Ψ̂
θ (x) U
θ (Λ, a), (15)
where Ds denotes the irreducible representation for spin s. Since translations act homogenously on
the fields, twisted tensor fields can be written as
Ψ̂ Aθ (x) =
[aσ(p) ǫ
θ (p, σ) e
ip·x + b†σ(p) χ
θ (p, σ) e
−ip·x], (16)
where A denotes the tensor index, A ≡ (a1 · · · as) for massless spin s fields and σ for the helicity
indices. Thus, the transformation relation of the fields, Eq.(15), can be reduced to
[Dsθ(Λ
−1)]AB Ψ̂
θ (Λx) = Uθ(Λ) Ψ̂
θ (x) U
θ (Λ). (17)
In order that the field operators transform correctly, the ’polarization’ tensor ǫ Aθ (p, σ) are to be trans-
formed as
ǫ Aθ (p, σ) =
(2π)3
[Dsθ(L(p))]AB ǫ Bθ (k, σ),
χ Aθ (p, σ) =
(2π)3
[Dsθ(L(p))]AB χ Bθ (k, σ′) C−1σ′σ, (18)
where L(p) is the Lorentz transformation‡ which reflects the little group and C is a matrix with the
properties [12],
C∗ C = (−1)2s, C† C = 1. (19)
† Recently, Joung and Mourad [27] have argued that a covariant field linear in creation and annihilation operators does
not exist. But the definition of creation and annihilation operators in that paper is different from ours. We try to follow the
view that we use the same fundamental quantities as of those in the untwisted theories changing the algebras only. The
field operators in linear combinations of the creation and annihilation operators can be justified in this basis, because the
free fields are the same as the commutative case.
‡ L(p) is a Lorentz transformation that takes the standard momentum kµ to pµ ≡ (|p|,p) for massless fields. For
the massless case L(p) satisfies L(p) = R(p̂)B(|p|) where R(p̂) is a rotation which takes the direction of the standard
momentum k to the direction of p and B(|p|) is the boost along the p direction [11].
Since the most important properties of the twist is that the representations are the same as in the
original group, the Poincaré group in this case, the above Dsθ(Λ
−1) and Dsθ(L) can be replaced by
Ds(Λ−1) and Ds(L), respectively. Thus the θ-dependance remains only in the polarization tensors.
Furthermore, one can see that there’s no θ-dependance in ǫ Aθ (σ) and χ
′). We can show this by
considering the case of ǫ Aθ (σ). From the group property of the twisted Lorentz transformations, the
related transformation to the polarization tensor ǫ Aθ (σ) can be written as:
[Dsθ(Λ
−1)]AB [D
θ(L)]CD ǫ Dθ (σ) = [Dsθ(Λ−1 · L)]AB ǫ Bθ (σ). (20)
In order for this transformation to be a twist transformation, it is to be satisfied
[Dsθ(Λ
−1 · L)]AB ǫ Bθ (σ) = [Ds(Λ−1 · L)]AB ǫ Bθ (σ), (21)
i.e., twisted transformation has the same representation as the untwisted one. From the primary
relation between the twist and the module algebra, one can see that the form of ǫ Aθ (σ), twisted version
of the commutative polarization tensor, should be
ǫ a1···anθ (σ) = · [F
n ⊲ ǫ
a1(σ)⊗ · · · ⊗ ǫan(σ)], (22)
where F−1n can be obtain from§ the Fθ, and Fθ is the very twist element corresponding to the canonical
noncommutativity:
Fθ = exp
θαβPα ⊗ Pβ
. (23)
Since Pα ⊲ ǫ
a(σ) = 0, we obtain
ǫ a1a2···asθ (σ) ≡ ǫ
a1a2···as(σ) = ǫ(a1σ ǫ
σ · · · ǫas)σ , (24)
where ǫ(a1a2···as)(σ) is the polarization tensor in the corresponding commutative field theory. This
relation leads to the relation
Ψ̂ Aθ (x) ≡ Ψ̂A(x). (25)
This relation is expected because the twist does not change the representations and we write the field
operators by using an irreducible representations of the symmetry group. Explicit calculations for
massless (s/2, s/2) tensor representation which shows the non-dependance on the θ is given in the
Appendix A.
Therefore, one can safely use the same representations for the field operators in the twisted theory
as those of the untwisted ones. What really be twisted are the multiplications of the creation and an-
nihilation operators only. Since the multiplication of the creation and annihilation operators between
different species of particles act like composite mappings on the Hilbert space, one can construct a
§ The associativity of the twisted product ∗ guaranty that it can be obtained by successive applications of Eq.(6).
module algebra from them. That is, when cp’s and dp’s are the creation and annihilation operators
corresponding to different species, their twisted multiplications are
cp ⋆ dq = e
ep∧eq cp · dq, (26)
where cp(dp) can be ap(bp) or a
p), and p̃ and q̃ are defined as in section 2.2.
Consequently, the scheme for twisting S-matrix used in [10] applies for general field theories.
Twist invariance of the S⋆ matrix for general field theories follows immediately. As in the scalar field
theory, the amplitudes can be obtained by multiplying the phase factor E(q1, · · · , qn) for each vertex
in the Feynman diagram of the untwisted theories.
2.4 Exact transformation formula for the S⋆ matrix element for massless fields
Transformation formula for the S⋆ matrix element corresponding to the process in which a massless
particle is emitted with momentum q and helicity s can be inferred as (Appendix C)
S±s⋆ (q, p) =
exp [±isΘ(q,Λ)]S±s⋆ (Λq,Λp). (27)
The S⋆ matrix can be written as the scalar product of a polarization tensor and theM⋆ function(Appendix
S±s⋆ (q, p) =
± (q̂) · · · ǫ
± (q̂)(M
⋆ )µ1···µs(q, p), (28)
where the M⋆ function twist-transform covariantly as
M±µ1···µs⋆ (q, p) = Λ
· · ·Λ µjνs M
±ν1···νs
⋆ (Λq,Λp). (29)
The form of the S⋆ matrix element in Eq.(28) appears to break the twisted Poincaré symmetry
because the polarization vectors do not satisfy the Lorentz covariance, rather they satisfy
(Λ µν − qµΛ 0ν /|q|)ǫ ν± (Λq) = exp {±iΘ(q,Λ)} ǫ
± (q̂). (30)
Hence requiring the twist invariance of the S⋆ matrix would lead to a constraint relation between the
momentum, the polarization vectors and the M⋆ function. From Eq.(27) and Eq.(30), the S⋆ matrix
element in Eq.(28) can be written as
S±s⋆ (q, p) =
exp [±isΘ(q,Λ)] [ǫ µ1± (Λq)− (Λq)µ1Λ 0ν ǫν±(Λq)/|q|]∗
· · · [ǫ µs± (Λq)− (Λq)µsΛ 0ν ǫν±(Λq)/|q|]∗(M⋆)±µ1···µs(Λq,Λp). (31)
Requiring the twist invariance of the S⋆ matrix element results in:
± (q̂) · · · ǫ
± (q̂)M
⋆ µ1···µs
= 0. (32)
This leads us to the desired identities:
±ρµ2···µs
⋆ (q, p) = 0. (33)
3 Charge conservation and the equivalence principle
Since the analysis of the conservation law is just a noncommutative generalization of Weinberg’s
work [11], the derivation of this section will be fairly straightforward. As we saw in section 2, the
differences between the noncommutative field theory and the commutative one are the phase factors
at each vertex of the Feynman diagrams.
3.1 Dynamical definition of charge and gravitational mass
We define the charge and the gravitational mass dynamically as the coupling of the vertex amplitudes
for the soft(very low energy) photon and graviton, respectively. Consider the vertex amplitude for the
process that a soft¶ massless particle of momentum q and spin s is emitted by a particle of momentum
p and spin J . Since the only tensor which can be used to form the invariantM function is known to be
pµ1 · · ·pµs , the noncommutative M function, M⋆ function, is given by the commutative M function
with phase factor E(q̃, p̃, p̃′) multiplied at each vertex. That is, the vertex amplitude can be written as
E(q̃, p̃, p̃′)
2E(p)
pµ1 · · · pµs ǫ
± (q̂) · · · ǫ
± (q̂). (34)
When the emitting particle has spin J we have to multiply to the vertex amplitude δσσ′ [11]. Thus,
the explicit form of the vertex amplitudes can be written as:
2i(2π)4 · es · δσσ′ [pµ ǫ µ∗± (q̂)]s
(2π)9/2[2E(p)]
· E(q̃, p̃, p̃′), (35)
where es is a coupling constant for emitting a soft massless particle of spin s (e.g., photon and gravi-
ton).
These coupling constants for emitting a soft particle can be interpreted as: e1 ≡ e as the elec-
tric charge, and e2 ≡
8πGg, with g the ratio of the gravitational mass and the inertial mass.
Let us consider a near forward scattering of the two particles A and B with the coupling esA and
B , respectively. From the properties of the phase factors [10], the phase factor for this scattering,
E(q̃, p̃A, p̃A′) · E(−q̃, p̃B, p̃B′), goes to:
E(q̃, p̃A, p̃A′) · E(−q̃, p̃B, p̃B′) = E(p̃A, p̃A′, q̃) · E(−q̃, p̃B, p̃B′)
= E(p̃A, p̃A′) · E(p̃B, p̃B′). (36)
However, in the forward scattering limit, the direction of the particles does not change, i.e. pA ‖ p′A.
For space-space noncommutativity pA ∧ p′A goes to zero, i.e., the phase factor goes to 1.
Thus, when the invariant momentum transfer t = −(p′A − pA)2 goes to zero, using the properties
of the polarization vectors in the Appendix B, the S⋆ matrix element can be shown to approach the
same form as in the corresponding commutative quantity which is easily calculated in a well chosen‖
¶ This is to define the charge and gravitational mass as a monopole, not as a multipole moments.
‖ The coordinate system in which q · pA = q · pB = 0 [11].
coordinate system,
δσAσB′δσBσB′
4π2EAEBt
eAeB(pA · pB) + 8πG gAgB
(pA · pB)2 −
m 2Am
. (37)
This coincidence is quite special in the sense that the S⋆ matrix elements are quite different from the
S matrix elements in the commutative theory when there is a momentum transfer. If particle B is at
rest, this gives
δσAσA′δσBσB′
−eAeB
+G · gA
2EA −
· gBmB
. (38)
Hence, one can interpret the coupling constant eA as the usual charge of the particle A. Moreover one
can identify the effective gravitational mass of A as
(mg)A = gA
2EA −
. (39)
For nonrelativistic limit the gA can be interpreted as a ratio of the gravitational mass and the inertial
mass, i.e.,
(mg)A = gA ·mA, EA ≃ mA. (40)
Consequently, if gA does not depend on the species of A, it suggests that the equivalence principle
holds.
3.2 Conservation law
Consider a S matrix element S(α→ β) for some reaction α→ β, where the states α and β consist of
various species of particles. The same reaction with emitting a soft massless particle of spin s (photon
or graviton for s = 1 or 2), momentum q and helicity ±s can occur. We denote the corresponding
S matrix element as S±s(q, α → β). Each amplitude of this process breaks the Lorentz symmetry
because a massless fields of (s/2, s/2) representation break the symmetry [28]. However, a real
physical reaction, to which a S-matrix element (the sums of each amplitude) correspond, should be
Lorentz invariant. By requiring this condition, Weinberg could obtain the conservation relations. In
this section, we investigate the similar relations by requiring the twist invariance of the S⋆ matrix
elements.
Suppose that a soft particle is emitted by ith particle (i = 1, · · · , n). Then by the polology of
the conventional field theory, the S matrix elements will have poles at |q| = 0 when an extra soft
massless particle is emitted by one of the external lines:
(pi + ηi · q)2 +m 2i
2pi · q
, (41)
where ηi = 1(−1) for the emission of a soft particle from an out(in) particle, respectively. By utilizing
the above relation, one can obtain the S matrix elements for the soft massless particles in the |q| → 0
limit [11]:
S±s(q, α→ β) =
(2π)3/2
ηiesi
[pi · ǫ ∗±(q̂)]s
(pi · q)
S(α→ β). (42)
In the noncommutative case, from the result of the section (2) and by using the above relation,
one can deduce the S⋆ matrix elements for the same process in the noncommutative spacetime:
S ±s⋆ (q, α→ β) =
Ei(q̃, α → β) · S±i (q, α → β)
E(q̃, p̃i, p̃i + q̃) · EI(p̃1, · · · , p̃i + ηiq̃, · · · , p̃n) · S±i (q, α→ β)
E(q, p̃i)
(2π)3/2
ηiesi
[pi · ǫ ∗±(q̂)]s
(pi · q)
× EI(p̃1, · · · , p̃i + ηiq̃, · · · , p̃n) · S±i (α→ β), (43)
where E iI = EI(p̃1, · · · , p̃i + q̃, · · · , p̃n) are the phase factors in the internal process∗∗. Since all the
E iI’s are the same when q → 0 (E iI = EI), one obtains in that limit,
S ±s⋆ (q, α→ β) =
(2π)3/2
ηiesi
[pi · ǫ ∗±(q̂)]s
(pi · q)
S⋆(α → β), (44)
where we used the relation S⋆(α → β) = EI · S(α → β). From the transformation properties of the
S⋆ matrix elements, Eq.(28), we obtain,
S ±s⋆ (q, α→ β) →
± (q̂) · · · ǫ
± (q̂)(M
⋆ )µ1···µs(q, α→ β). (45)
Identifying the S⋆ matrix elements in Eq.(44) and Eq.(45) gives the invariant M⋆ functions for spin 1
and 2:
Mµ⋆ (q, α→ β) =
(2π)3/2
ei · ηip µi
(pi · q)
S⋆(α→ β),
Mµν⋆ (q, α→ β) =
(2π)3/2
gi · ηip µi p νi
(pi · q)
S⋆(α → β). (46)
Requiring the twist invariance to the S⋆ matrix elements, Eq.(33), gives:
0 = qµM
⋆ (q, α→ β) →
ηiei = 0,
0 = qµM
⋆ (q, α→ β) →
gi · ηipνi = 0, (47)
∗∗ The integrals over the internal loop momenta have been suppressed because they do not affect the final result since
the change in the phase factor occurs only in the external lines.
in general. Hence, one obtains the charge conservation law for the spin 1 fields. For s = 2, in order
to satisfy the two relations,
ηipi = 0 (4-momentum conservation) and
i gi · ηipνi = 0, gi should
be constants (i.e. independence of the particle species). The universality of this coupling constant as
a ratio of gravitational mass and inertial mass shows that the equivalence principle is satisfied even in
the noncommutative spacetime.
4 Discussion
We have found that the conservation of charge and the equivalence principle are satisfied even in the
canonical noncommutative spacetime. The derivation was fairly straightforward, once we have con-
structed the S⋆ matrix for general field theories. The assumption was that the quanta of the gravitation
is massless spin 2, massless spin 1 for photon.
We extended the construction of the S⋆ matrix to general fields, especially for the massless fields
of integer spin. The twisted Feynman diagrams can be constructed by the same irreducible represen-
tations as those in the untwisted theories with the same rule except for the different phase factors at
each vertex. Hence, we can say that the same reasoning apply to the massive fields.
We use the (s/2, s/2) representation for the field operators mainly because one can obtain the
condition, Eq.(33), requiring the twist Lorentz invariance. In this representation, since the polarization
vectors are not Lorentz four vectors, each amplitude of emitting a real soft massless particle violate
the symmetry. We have used this property to show the conservation law in this paper. Another
representation for the fields, the (s, 0)⊕ (0, s) representation, which has the parity symmetry, can be
made Lorentz covariant. One realizes that due to these properties one cannot derive the conservation
law by the method used in this paper.
Charge conservation in the noncommutative spacetime is expected from the gauge symmetry and
the noncommutative Noether theorem. However, it is not quite certain whether the equivalence princi-
ple is satisfied in the noncommutative spacetime. Since the noncommutative spacetime is not locally
Minkowski nor the local symmetry group is SO(1, 3), one can not guarantee if the principle is satis-
fied. But in our twisted symmetry context, one can expect that the equivalence principle is satisfied
because the algebra structure is the same as the conventional theory though the coalgebra structure
is different. The applicability of the S matrix theoretic proof of the equivalence principle given in
this paper is restrictive because the analysis is perturbative in nature. We expect that the conclusion
of this paper is to be one of the stepping stone towards the further understanding of the nature of the
principle in quantum gravity.
This paper shows an example for the usefulness of the twisted symmetry to derive the physically
important relations in the noncommutative spacetime. We think that the twist analysis is adequate for
the schematic approaches to the noncommutative physics, while the other approaches are suitable for
the explicit calculations though they are equivalent. The future applications of the twisted symmetry
to the other issues in this direction of approach are expected.
Acknowledgement
I express my deep gratitude to Prof. J. H. Yee for his support. I would like to thank Dr. Jake Lee
for helpful discussions and Dr. A. Tureanu for his comments on the draft. This work was supported
in part by Korea Science and Engineering Foundation Grant No. R01-2004-000-10526-0, and by
the Science Research Center Program of the Korea Science and Engineering Foundation through the
Center for Quantum Spacetime(CQUeST) of Sogang University with grant number R11 - 2005 - 021.
A Exact calculation of the polarization tensor for massless fields
The creation and annihilation operators of the massless field transform under the Lorentz transforma-
tion as [28]
a†(Λp, σ) = e−iσΘ[W (Λ,p)] U(Λ) a†(p, σ) U−1(Λ),
a(Λp, σ) = e+iσΘ[W (Λ,p)] U(Λ) a(p, σ) U−1(Λ), (48)
up to phase factors, where W (Λ, p) = L−1(p)Λ−1L(Λp) denotes the Wigner rotation defined as in
[28] (Appendix C), and the Θ[W (Λ, p)] is the related angle to which the little group corresponds.
Hereafter, we abbreviate Θ[W (Λ, p)] as Θ(Λ, p). It is to be noted that we use U(Λ) instead of Uθ(Λ)
in here. This follows immediately from the properties of the twist in section 2.1.
The field Ψ̂ Aθ transform as
[Ds(Λ−1)]AB Ψ̂
θ (Λx) = U(Λ) Ψ̂
θ (x) U
−1(Λ). (49)
In order to satisfy the two relations (48), (49), the polarization tensor should transform as
ǫ Aθ (p, σ) = e
−iσΘ(Λ,p) [Ds(Λ−1)]AB ǫ
θ (Λp, σ). (50)
When Λ = W and p = k, the above relation goes to
[Ds(W )]AB ǫ
θ (k, σ) = e
−iσΘ(W,p)ǫ Aθ (k, σ), (51)
where k denotes the standard momentum. The Wigner rotation can be written as W (φ, α, β) =
R(φ)T (α, β) because the little group is isomorphic to ISO(2) for massless fields [28].
Suppose that the field transforms as (m,n) representation of spin s (m + n = s). When the φ, α
and β are infinitesimal, Ds(W ) can be written as
Ds(W ) ≃ 1− iφ(M3 +N3) + (α + iβ)(M1 − iM2) + (α− iβ)(N1 + iN2). (52)
For M− ≡M1 − iM2, N+ ≡M1 + iM2, and Θ → φ we have:
(M3 +N3) ǫθ(k, σ) = σ ǫθ(k, σ),
M− ǫθ(k, σ) = 0,
N+ ǫθ(k, σ) = 0, (53)
which gives
M3 ǫθ(k, σ) = −m ǫθ(k, σ),
N3 ǫθ(k, σ) = +n ǫθ(k, σ). (54)
The ǫθ(k, σ) satisfys the same equation as ǫ(k, σ). Since the highest or lowest weight corresponds to
a unique state, one obtains the same polarization tensor as a solution, ǫθ(k, σ) = ǫ(k, σ).
B Properties of the polarization vectors
Here, we summarize the properties of the polarization vector. The properties of the polarization
tensors of other rank, Eq.(24), follows from it.
Solving the (53) for σ = ±1 gives the explicit form of the polarization vector for the standard
momentum as
± (k) ≡
(1,±i, 0, 0), (55)
where we made the conventional choice of the phase. The polarization vector of momentum p is
defined as
± (p) = [L(p)]µν ǫ ν± (k), (56)
where L(p) is the Lorentz transformation which takes k to p, i.e., pµ = [L(p)]µν kν . Then the well
known properties of the polarization vectors can be deduced [11]:
± (p̂) = 0, (57)
ǫ ∗±µ(p̂)ǫ
± (p̂) = 1, ǫ±µ(p̂)ǫ
± (p̂) = 0, (58)
± (p̂) = ǫ
∓ (p̂), ǫ
± (p̂) = 0, (59)∑
± (p̂)ǫ
± (p̂) = η
µν + (p̃µpν + p̃νpµ)/2|p|2 ≡ Πµν , p̃ ≡ (|p|,−p), (60)
± (p̂)ǫ
± (p̂)ǫ
± (p̂)ǫ
± (p̂) =
{Πµ1ν1(p̂)Πµ2ν2(p̂) + Πµ1ν2(p̂)Πµ2ν1(p̂)
−Πµ1µ2(p̂)Πν1ν2(p̂)} . (61)
The polarization ’vectors’ are not the Lorentz four vectors††, rather they transform as
±(p) = e
∓iΘ(Λ,p)[Dsǫ (Λ, p)]
±(Λp),
[Dsǫ (Λ, p)]
ν = (Λ
−1)µν −
(Λ−1)0ν . (62)
For general spin s, the polarization tensor can be written as
ǫAσ (p) = e
∓iΘ(Λ,p)[Dsǫ (Λ, p)]
σ (Λp), (63)
up to a phase factor.
††It comes from the fact that the little group is not semisimple for massless fields. Translation in the little group
isomorphic to ISO(2) generates the gradient term in the transformations.
C Invariant M⋆ function for massless field
Let P denotes a shorthand notation for the external lines, P ≡ (p1, . . . , pn), and K denotes a standard
momenta‡‡ K ≡ (k1, . . . , kn). There exists a unique Lorentz transformation satisfying P ≡ LPK.
Then the relation between the Lorentz group and the little group can be described symbolically as:
W (Λ,P )
// ΛP ,
where W (Λ, P ) is the Wigner transformation to which a Lorentz transformation Λ and the momenta
P correspond. Since the twist do not change the group properties, above relation also holds for the
twisted symmetry group. The S⋆ matrix elements transform as
S⋆[P ] = D
θ[W (Λ, P )] S⋆[ΛP ]
= Dsθ[L
P · Λ
−1 · LΛP ] S⋆[ΛP ] (64)
where the indices for the external lines are suppressed. From the golden rule and (48), the explicit
transformation formular for S⋆[P ] can be obtained:
S⋆[P ] =
N(ΛP )
N(P )
e±isΘ(Λ,P ) S⋆[ΛP ], (65)
where N denotes the corresponding normalization factor.
If one defines M⋆[P ] as S⋆[P ] = D
P ]M⋆[P ], one can show that M⋆[P ] transform as
M⋆[P ] = D
−1]M⋆[ΛP ], (66)
i.e., it is twist invariant. We will call it as an invariant M⋆ function as in [29, 30].
Let us find out the invariant M⋆ function for massless fields. First, we define the quanties for the
standard momenta K. The choice for M⋆[K] is
MA⋆ [K] = N(K) ǫ
A[K] S⋆[K]. (67)
If we define M⋆[P ] as
MA⋆ [P ] = [D
θ(LP )]
⋆ [K], (68)
then one can easily show that MA⋆ [P ] is twist invariant.
By using the explicit form of Dsǫ in (62) and kbjM
b1···bs
⋆ [K] = 0, one obtains the relation
ǫC [P ]∗ (M⋆)C [P ] =
e±isΘ(L
,P )[Dǫθ(L
A[K]∗
[Dθ(LP )]
C (M⋆)B[K]
= e±isΘ(L
,P ) ǫA[K]∗ [Dθ(L
P ) ·D
A (M⋆)B[K]
= e±isΘ(L
,P ) ǫA[P ]∗ (M⋆)A[P ]
= e±isΘ(L
,P )N(K) S⋆[K], (69)
‡‡ see [11],[29].
where we used the relation
[Dθ(L
P ) ·D
A (M⋆)B[K] ≡ [D(L−1P ) ·Dǫ(L
A (M⋆)B[K]
δb1a1 −
(LP )
· · ·
δbsas −
(LP )
× (M⋆)b1···bs [K]
= (M⋆)a1···as [K]. (70)
By setting Λ = L−1P in (65), one can see that (69) is S⋆[P ]. Thus, the desired form of S⋆[P ] is
S⋆[P ] = N(P ) ǫ
A[P ]M
⋆ [P ]. (71)
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	Introduction
	Properties of the general S matrix
	A short introduction of useful properties of the twist-deformation
	The S matrix and its twist invariance
	Generalization to arbitrary fields
	Exact transformation formula for the S matrix element for massless fields
	Charge conservation and the equivalence principle 
	Dynamical definition of charge and gravitational mass
	Conservation law
	Discussion
	Exact calculation of the polarization tensor for massless fields
	Properties of the polarization vectors
	Invariant M function for massless field

  We investigate one of the consequences of the twisted Poincare symmetry. We
derive the charge conservation law and show that the equivalence principle is
satisfied in the canonical noncommutative spacetime. We applied the twisted
Poincare symmetry to the Weinberg's analysis. To this end, we generalize our
earlier construction of the twisted S matrix \cite{Bu}, which apply the
noncommutativity to the fourier modes, to the massless fields of integer spins.
The transformation formula for the twisted S matrix for the massless fields of
integer spin has been obtained. For massless fields of spin 1, we obtain the
conservation of charge, and the universality of coupling constant for massless
fields of spin 2, which can be interpreted as the equality of gravitational
mass and inertial mass, i.e., the equivalence principle.

Introduction
In effort to construct an effective theory of quantum gravity at the Planck scale, noncommutativity of
the spacetime has been considered. The canonical noncommutative spacetime has the commutation
relations between the coordinates [1],
[xµ, xν ] = iθµν , (1)
where θµν(µ, ν = 0 to 3) is a constant antisymmetric matrix.
Field theories in the canonical noncommutative spacetime can be replaced by field theories in the
commutative spacetime with the Moyal product (The Weyl-Moyal correspondence [2]). One of the
significant problems of those theories is that they violate the Lorentz symmetry. One finds that the
symmetry group is SO(1, 1)×SO(2) instead of the Lorentz group, SO(1, 3). Since there is no spinor
or vector representations in that symmetry group, most of the earlier studies performed by using the
spinor, vector representations of the Lorentz group can not be justfied. Moreover, the factors 1(−1)
are multiplied for a boson(fermion)loop without knowing the spin-statistics relation.
To get around this, Chaichian et.al.∗ have deformed the Poincaré symmetry as well as its module
space to which the symmetry acts [3]. The twisted symmetry group has the same representations as
the original Poincaré group and at the same time they successfully retain the physical information of
the canonical noncommutativity. The main idea was that one can change a classical symmetry group
to a quantum group, ISOθ(1, 3) in this case, and twist-deform the module algebra consistently to
reproduce the noncommutativity. In their approach, the noncommutative parameter θµν transform as
an invariant tensor. This reminds us the situation that Einstein had to change the symmetry group of
the spacetime and its module space(to the Minkowski spacetime) when the speed of light is required
to be constant for any observer in an inertial frame. Similarly, Chaichian et.al. have required the
change of the Hopf algebra with its module algebra so that any observer in an inertial frame feel
the noncommutativity in the same way. For the κ-deformed noncommutativity, Majid and Ruegg
found the κ-deformed spacetime [6] as a module space of the κ-deformed symmetry after Lukierski
et.al. discovered the symmetry [7]. The real benefit of the twist is in the use of the same irreducible
representations of original theories unlike general deformed theories, as in the case of the κ-deformed
theory.
Recently, groups of physicists have constructed the quantum field theory in the noncommutative
spacetime by twisting the quantum space as a module space [8],[9],[10]. Espcially, Bu et.al. have
proposed a twisted S-matrix as well as a twisted Fock space for consistency [10]. There we have
obtained the twisted algebra of the creation and annihilation operators and the spin-statistics relation
by applying the twisted Poincaré symmetry on the quantum space consistently. The analysis of this
paper is mainly based on this work.
These works justify the use of the irreducible representations of the Poincaré group and the sign
factors being used in the earlier studies. Are these works merely a change of viewpoint? Mathemat-
ically they look equivalent and seem to have equal amounts of information. But when the physics
∗ Oeckl [4], Wess [5] have proposed the same deformed Poincaré algebra.
is concerned the action of symmetry becomes more subtle than it seems because it confines possible
configurations of physical systems. In this article, we present an example showing the role of the
twisted symmetry for solving physics problems, especially in the canonical noncommutative space-
time. As an example, we derive the conservation law of charge and show the fact that the equivalence
principle is satisfied even in the noncommutative spacetime. In this derivation we consider spin 1 and
2 massless fields for the photon and the graviton, respectively. For this purpose, we extend our ear-
lier study on the scalar field theory to more general field theories and investigate a noncommutative
version of Weinberg’s analysis [11, 12, 13].
Actually, there are many studies for the relation between noncommutativity and gauge theory [14],
[15], [16], [17] or between the noncommutativity and the gravity [18],[19],[20] and between them
[21]. And there were many argument whether the equivalence principle is satisfied at the quantum
level. Some people argued that the equivalence principle is violated in quantum regime [22],[23],
while there are studies which show non-violation of the equivalence principle [24]. Whether the
principle of equivalence is violated or not is an important issue for quantum gravity because the
principle is the core of the general relativity.
The paper is organized as follows. We extend our previous construction of the S⋆ matrix to the
massless fields of integer spin after giving a brief review on the construction and the properties of the
S⋆ matrix. We give an exact transformation formula for the S⋆ matrix elements in section 2. We give
the consequence of requiring the twist invariance to the S⋆ matrix elements for the scattering process.
These results lead to the charge conservation law for the spin 1 field theory and the universality of
the coupling constant for spin 2 field in noncommutative spacetime in section 3. Finally, we discuss
the implication of the twisted symmetry and its applicability to other issues in section 4. We give
some related calculations of the polarization vector, the noncommutative definition of the invariant
M function, and a twisted transformation formula for the S⋆ matrix in the Appendices.
2 Properties of the general S⋆ matrix
2.1 A short introduction of useful properties of the twist-deformation
An algebra with a product · and a coalgebra with a coproduct ∆ constitute a Hopf algebra if it has an
invertible element S called antipode and with some compatibility relations. For a Lie algebra g, there
is a unique universal enveloping algebra U(g) which preserves the Lie algebra properties in terms of
unital associative algebra. The Hopf algebra of a Lie algebra g is denoted as H ≡ {U(g), ·,∆, ǫ, S},
where U(g) is an universal enveloping algebra of the corresponding algebra g and we denotes the
counit as ǫ. The Sweedler notation is being widely used for a shorthand notation of the coproduct,
Y(1) ⊗ Y(2) [25].
The action of a Hopf algebra H to a module algebra A is defined as
Y ⊲ (a · b) =
(Y(1) ⊲ a) · (Y(2) ⊲ b), (2)
where a, b ∈ A, the symbol · is a multiplication in the module algebra A, and the symbol ⊲ denotes
an action of the Lie generators Y ∈ U(g) on the module algebra A. The product · in H and the
multiplication · in A should be distinguished.
If there is an invertible ’twist element’, F =
F(1) ⊗ F(2) ∈ H ⊗H, which satisfies
(F ⊗ 1) · (∆⊗ id)F = (1⊗F) · (id ⊗∆)F , (3)
(ǫ⊗ id)F = 1 = (id ⊗ ǫ)F , (4)
one can obtain a new Hopf algebra HF ≡ {UF (g), ·,∆F , ǫF , SF} from the original one. The relations
between them are
∆FY = F ·∆Y · F−1 , ǫF (Y ) = ǫ(Y ),
SF (Y ) = u · S(Y ) · u−1 , u =
F(1) · S(F(2)), (5)
with the same product in the algebra sector. The ’covariant’ multiplication of the module algebra AF
for the twisted Hopf algebra HF which maintain the form of Eq.(2) is given as
(a ⋆ b) = ·[F−1 (a⊗ b)]. (6)
From the above relations, one can derive an important property of the twist such that it does not
change the representations of the algebra:
DF(Y )(a ⋆ b) = ⋆ [∆FY (a⊗ b)]
= · [F−1 · F∆0YF−1(a⊗ b)]
= · [∆0Y F−1(a⊗ b)]
= D0(Y )(a ⋆ b), (7)
where representations of the coproduct and the twist element is implied, i.e.,
D[∆Y ] =
D(Y(1))⊗D(Y(2)), D[F ] =
D(F(1))⊗D(F(2)). (8)
The above considerations lead us to the golden rule: The irreducible representations are not changed
by a twist and one can regard the covariant action of a twisted Hopf algebra on a twisted module
algebra as the action of the original algebra on the twisted module algebra.
2.2 The S⋆ matrix and its twist invariance
Recently, a quantum field theory has been constructed in such a way to preserve the twisted Poincaré
symmetry [10]. There we confined the construction for the space-space noncommutativity. It is hard
to know whether one can construct a consistent twist Poincaré invariant field theory satisfying the
causality in the case of space-time noncommutativity. They have tried to apply the twisted symmetry
to quantum spaces consistently, especially to the algebra of the creation and annihilation operators (a†p
and ap). As a result, they obtained the twisted algebra of quantum operators. If we use a shorthand
notation, p ∧ q = pµθµνqν , the twisted algebra of a†p and ap can be denoted as:
cp ⋆ cq = e
ep∧eq cp · cq, (9)
where cp can be ap or a
p, p̃ ≡ −p(p) for cp = ap(a†p) and · denotes the ordinary multiplication of
operators in the commutative theories.
This twisted algebra naturally leads to the twisted form of Fock space, S-matrix and quantities
related to the creation and annihilation operators. Thus, we obtain the twisted basis of Fock space and
S⋆-matrix:
|q1, · · · , qn〉 → |q1, · · · , qn〉⋆ = E(q1, · · · , qn)|q1, · · · , qn〉, (10)
where E(q1, · · · , qn) = exp
i<j qi ∧ qj
is a phase factor which has the interesting properties
[10], and
S → S⋆ =
(−i)k
d4x1 · · · d4xk T {H⋆I(x1) ⋆ · · · ⋆H⋆I(xk)} , (11)
where T denotes the time ordering and H⋆I(x) is an interaction Hamiltonian density in the Dyson
formalism.
The explicit form of the S⋆ matrix elements for the scalar φ
n theoryin the momentum space is:
⋆〈β|S⋆|α〉⋆ = E(−β, α)
(−ig)k
· · ·
···cQk
E(Q̃1) · · · E(Q̃k)〈β|Sk(Q̃1 · · · , Q̃k)|α〉, (12)
where Q̃ is the shorthand notation for (q̃1, . . . , q̃n) [10].
In the above, 〈β|Sk(Q̃1 · · · , Q̃k)|α〉 is a gk order term of the S-matrix element of the commutative
theory where g is the coupling constant of the theory. From the momentum conservation, i.e., delta
functions in the 〈β|Sk(Q̃1 · · · , Q̃k)|α〉, one can show that the S⋆ matrix element, ⋆〈β|S⋆|α〉⋆, can be
represented by Feynman diagrams with extra phase factors E(Q̃) for each vertex. The phase factors
drastically change the predictions of the theory. This result agrees with Filk’s result[26], but we have
overall factors E(−β, α) corresponding to external lines in the Feynman diagram which originated
from the twisted Fock space. From the above considerations, the new modified Feynman diagrams
can be obtained from the untwisted ones by changing the phase factors from 1 to E(Q̃i) at each vertex.
The twist invariance of this prescription of the S⋆ matrix is not manifest because non-locality of
the interactions may violate the twist invariance of the S⋆ matrix, in general, i.e.,
[H⋆I(x),H⋆I(y)]⋆ 6= 0 for spacelike (x− y). (13)
However, we see from the form of S⋆ matrix in Eq.(12), that the proposed S⋆ matrix is clearly twist
invariant since it is constructed from phase factors which are twist invariant, and the Feynman propa-
gators. Twisted product of fields operators satisfy
〈0|ψ(x) ⋆ ψ(y)|0〉 = 〈0|ψ(y) ⋆ ψ(x)|0〉, for spacelike (x− y). (14)
Hence we see that the Feynman propagator, same as the twist Feynman propagator 〈0|T [ψ(x) ⋆
ψ(y)]|0〉, is twist invariant. From this, the invariance of the S⋆ matrix elements follows immediately.
2.3 Generalization to arbitrary fields
We need to get the S⋆ matrix for massless field theories of integer spin for the analysis of this paper.
In the previous work [10], we have constructed the S⋆ matrix for scalar field theory and we have
expected that the same formulation could be possible for general field theories. In this section, we
generalize our argument used in that paper to obtain the form of the S⋆ matrix elements for massless
field theories with spin 1 and 2. In the analysis of this paper, we use the (s/2, s/2) representation for
massless integer spin fields. The reason we use it and the considerations for the other representations
are given in section 4.
We have used the perturbation theory in our formulation of the S⋆ matrix in Eq.(11). Another
assumption was the particle interpretation. That is, the field operators are represented as linear com-
binations of the creation and annihilation operators and the fields of spin s transform(twist) as† :
[Dsθ(Λ
−1)]AB Ψ̂
θ (Λx+ a) = Uθ(Λ, a) Ψ̂
θ (x) U
θ (Λ, a), (15)
where Ds denotes the irreducible representation for spin s. Since translations act homogenously on
the fields, twisted tensor fields can be written as
Ψ̂ Aθ (x) =
[aσ(p) ǫ
θ (p, σ) e
ip·x + b†σ(p) χ
θ (p, σ) e
−ip·x], (16)
where A denotes the tensor index, A ≡ (a1 · · · as) for massless spin s fields and σ for the helicity
indices. Thus, the transformation relation of the fields, Eq.(15), can be reduced to
[Dsθ(Λ
−1)]AB Ψ̂
θ (Λx) = Uθ(Λ) Ψ̂
θ (x) U
θ (Λ). (17)
In order that the field operators transform correctly, the ’polarization’ tensor ǫ Aθ (p, σ) are to be trans-
formed as
ǫ Aθ (p, σ) =
(2π)3
[Dsθ(L(p))]AB ǫ Bθ (k, σ),
χ Aθ (p, σ) =
(2π)3
[Dsθ(L(p))]AB χ Bθ (k, σ′) C−1σ′σ, (18)
where L(p) is the Lorentz transformation‡ which reflects the little group and C is a matrix with the
properties [12],
C∗ C = (−1)2s, C† C = 1. (19)
† Recently, Joung and Mourad [27] have argued that a covariant field linear in creation and annihilation operators does
not exist. But the definition of creation and annihilation operators in that paper is different from ours. We try to follow the
view that we use the same fundamental quantities as of those in the untwisted theories changing the algebras only. The
field operators in linear combinations of the creation and annihilation operators can be justified in this basis, because the
free fields are the same as the commutative case.
‡ L(p) is a Lorentz transformation that takes the standard momentum kµ to pµ ≡ (|p|,p) for massless fields. For
the massless case L(p) satisfies L(p) = R(p̂)B(|p|) where R(p̂) is a rotation which takes the direction of the standard
momentum k to the direction of p and B(|p|) is the boost along the p direction [11].
Since the most important properties of the twist is that the representations are the same as in the
original group, the Poincaré group in this case, the above Dsθ(Λ
−1) and Dsθ(L) can be replaced by
Ds(Λ−1) and Ds(L), respectively. Thus the θ-dependance remains only in the polarization tensors.
Furthermore, one can see that there’s no θ-dependance in ǫ Aθ (σ) and χ
′). We can show this by
considering the case of ǫ Aθ (σ). From the group property of the twisted Lorentz transformations, the
related transformation to the polarization tensor ǫ Aθ (σ) can be written as:
[Dsθ(Λ
−1)]AB [D
θ(L)]CD ǫ Dθ (σ) = [Dsθ(Λ−1 · L)]AB ǫ Bθ (σ). (20)
In order for this transformation to be a twist transformation, it is to be satisfied
[Dsθ(Λ
−1 · L)]AB ǫ Bθ (σ) = [Ds(Λ−1 · L)]AB ǫ Bθ (σ), (21)
i.e., twisted transformation has the same representation as the untwisted one. From the primary
relation between the twist and the module algebra, one can see that the form of ǫ Aθ (σ), twisted version
of the commutative polarization tensor, should be
ǫ a1···anθ (σ) = · [F
n ⊲ ǫ
a1(σ)⊗ · · · ⊗ ǫan(σ)], (22)
where F−1n can be obtain from§ the Fθ, and Fθ is the very twist element corresponding to the canonical
noncommutativity:
Fθ = exp
θαβPα ⊗ Pβ
. (23)
Since Pα ⊲ ǫ
a(σ) = 0, we obtain
ǫ a1a2···asθ (σ) ≡ ǫ
a1a2···as(σ) = ǫ(a1σ ǫ
σ · · · ǫas)σ , (24)
where ǫ(a1a2···as)(σ) is the polarization tensor in the corresponding commutative field theory. This
relation leads to the relation
Ψ̂ Aθ (x) ≡ Ψ̂A(x). (25)
This relation is expected because the twist does not change the representations and we write the field
operators by using an irreducible representations of the symmetry group. Explicit calculations for
massless (s/2, s/2) tensor representation which shows the non-dependance on the θ is given in the
Appendix A.
Therefore, one can safely use the same representations for the field operators in the twisted theory
as those of the untwisted ones. What really be twisted are the multiplications of the creation and an-
nihilation operators only. Since the multiplication of the creation and annihilation operators between
different species of particles act like composite mappings on the Hilbert space, one can construct a
§ The associativity of the twisted product ∗ guaranty that it can be obtained by successive applications of Eq.(6).
module algebra from them. That is, when cp’s and dp’s are the creation and annihilation operators
corresponding to different species, their twisted multiplications are
cp ⋆ dq = e
ep∧eq cp · dq, (26)
where cp(dp) can be ap(bp) or a
p), and p̃ and q̃ are defined as in section 2.2.
Consequently, the scheme for twisting S-matrix used in [10] applies for general field theories.
Twist invariance of the S⋆ matrix for general field theories follows immediately. As in the scalar field
theory, the amplitudes can be obtained by multiplying the phase factor E(q1, · · · , qn) for each vertex
in the Feynman diagram of the untwisted theories.
2.4 Exact transformation formula for the S⋆ matrix element for massless fields
Transformation formula for the S⋆ matrix element corresponding to the process in which a massless
particle is emitted with momentum q and helicity s can be inferred as (Appendix C)
S±s⋆ (q, p) =
exp [±isΘ(q,Λ)]S±s⋆ (Λq,Λp). (27)
The S⋆ matrix can be written as the scalar product of a polarization tensor and theM⋆ function(Appendix
S±s⋆ (q, p) =
± (q̂) · · · ǫ
± (q̂)(M
⋆ )µ1···µs(q, p), (28)
where the M⋆ function twist-transform covariantly as
M±µ1···µs⋆ (q, p) = Λ
· · ·Λ µjνs M
±ν1···νs
⋆ (Λq,Λp). (29)
The form of the S⋆ matrix element in Eq.(28) appears to break the twisted Poincaré symmetry
because the polarization vectors do not satisfy the Lorentz covariance, rather they satisfy
(Λ µν − qµΛ 0ν /|q|)ǫ ν± (Λq) = exp {±iΘ(q,Λ)} ǫ
± (q̂). (30)
Hence requiring the twist invariance of the S⋆ matrix would lead to a constraint relation between the
momentum, the polarization vectors and the M⋆ function. From Eq.(27) and Eq.(30), the S⋆ matrix
element in Eq.(28) can be written as
S±s⋆ (q, p) =
exp [±isΘ(q,Λ)] [ǫ µ1± (Λq)− (Λq)µ1Λ 0ν ǫν±(Λq)/|q|]∗
· · · [ǫ µs± (Λq)− (Λq)µsΛ 0ν ǫν±(Λq)/|q|]∗(M⋆)±µ1···µs(Λq,Λp). (31)
Requiring the twist invariance of the S⋆ matrix element results in:
± (q̂) · · · ǫ
± (q̂)M
⋆ µ1···µs
= 0. (32)
This leads us to the desired identities:
±ρµ2···µs
⋆ (q, p) = 0. (33)
3 Charge conservation and the equivalence principle
Since the analysis of the conservation law is just a noncommutative generalization of Weinberg’s
work [11], the derivation of this section will be fairly straightforward. As we saw in section 2, the
differences between the noncommutative field theory and the commutative one are the phase factors
at each vertex of the Feynman diagrams.
3.1 Dynamical definition of charge and gravitational mass
We define the charge and the gravitational mass dynamically as the coupling of the vertex amplitudes
for the soft(very low energy) photon and graviton, respectively. Consider the vertex amplitude for the
process that a soft¶ massless particle of momentum q and spin s is emitted by a particle of momentum
p and spin J . Since the only tensor which can be used to form the invariantM function is known to be
pµ1 · · ·pµs , the noncommutative M function, M⋆ function, is given by the commutative M function
with phase factor E(q̃, p̃, p̃′) multiplied at each vertex. That is, the vertex amplitude can be written as
E(q̃, p̃, p̃′)
2E(p)
pµ1 · · · pµs ǫ
± (q̂) · · · ǫ
± (q̂). (34)
When the emitting particle has spin J we have to multiply to the vertex amplitude δσσ′ [11]. Thus,
the explicit form of the vertex amplitudes can be written as:
2i(2π)4 · es · δσσ′ [pµ ǫ µ∗± (q̂)]s
(2π)9/2[2E(p)]
· E(q̃, p̃, p̃′), (35)
where es is a coupling constant for emitting a soft massless particle of spin s (e.g., photon and gravi-
ton).
These coupling constants for emitting a soft particle can be interpreted as: e1 ≡ e as the elec-
tric charge, and e2 ≡
8πGg, with g the ratio of the gravitational mass and the inertial mass.
Let us consider a near forward scattering of the two particles A and B with the coupling esA and
B , respectively. From the properties of the phase factors [10], the phase factor for this scattering,
E(q̃, p̃A, p̃A′) · E(−q̃, p̃B, p̃B′), goes to:
E(q̃, p̃A, p̃A′) · E(−q̃, p̃B, p̃B′) = E(p̃A, p̃A′, q̃) · E(−q̃, p̃B, p̃B′)
= E(p̃A, p̃A′) · E(p̃B, p̃B′). (36)
However, in the forward scattering limit, the direction of the particles does not change, i.e. pA ‖ p′A.
For space-space noncommutativity pA ∧ p′A goes to zero, i.e., the phase factor goes to 1.
Thus, when the invariant momentum transfer t = −(p′A − pA)2 goes to zero, using the properties
of the polarization vectors in the Appendix B, the S⋆ matrix element can be shown to approach the
same form as in the corresponding commutative quantity which is easily calculated in a well chosen‖
¶ This is to define the charge and gravitational mass as a monopole, not as a multipole moments.
‖ The coordinate system in which q · pA = q · pB = 0 [11].
coordinate system,
δσAσB′δσBσB′
4π2EAEBt
eAeB(pA · pB) + 8πG gAgB
(pA · pB)2 −
m 2Am
. (37)
This coincidence is quite special in the sense that the S⋆ matrix elements are quite different from the
S matrix elements in the commutative theory when there is a momentum transfer. If particle B is at
rest, this gives
δσAσA′δσBσB′
−eAeB
+G · gA
2EA −
· gBmB
. (38)
Hence, one can interpret the coupling constant eA as the usual charge of the particle A. Moreover one
can identify the effective gravitational mass of A as
(mg)A = gA
2EA −
. (39)
For nonrelativistic limit the gA can be interpreted as a ratio of the gravitational mass and the inertial
mass, i.e.,
(mg)A = gA ·mA, EA ≃ mA. (40)
Consequently, if gA does not depend on the species of A, it suggests that the equivalence principle
holds.
3.2 Conservation law
Consider a S matrix element S(α→ β) for some reaction α→ β, where the states α and β consist of
various species of particles. The same reaction with emitting a soft massless particle of spin s (photon
or graviton for s = 1 or 2), momentum q and helicity ±s can occur. We denote the corresponding
S matrix element as S±s(q, α → β). Each amplitude of this process breaks the Lorentz symmetry
because a massless fields of (s/2, s/2) representation break the symmetry [28]. However, a real
physical reaction, to which a S-matrix element (the sums of each amplitude) correspond, should be
Lorentz invariant. By requiring this condition, Weinberg could obtain the conservation relations. In
this section, we investigate the similar relations by requiring the twist invariance of the S⋆ matrix
elements.
Suppose that a soft particle is emitted by ith particle (i = 1, · · · , n). Then by the polology of
the conventional field theory, the S matrix elements will have poles at |q| = 0 when an extra soft
massless particle is emitted by one of the external lines:
(pi + ηi · q)2 +m 2i
2pi · q
, (41)
where ηi = 1(−1) for the emission of a soft particle from an out(in) particle, respectively. By utilizing
the above relation, one can obtain the S matrix elements for the soft massless particles in the |q| → 0
limit [11]:
S±s(q, α→ β) =
(2π)3/2
ηiesi
[pi · ǫ ∗±(q̂)]s
(pi · q)
S(α→ β). (42)
In the noncommutative case, from the result of the section (2) and by using the above relation,
one can deduce the S⋆ matrix elements for the same process in the noncommutative spacetime:
S ±s⋆ (q, α→ β) =
Ei(q̃, α → β) · S±i (q, α → β)
E(q̃, p̃i, p̃i + q̃) · EI(p̃1, · · · , p̃i + ηiq̃, · · · , p̃n) · S±i (q, α→ β)
E(q, p̃i)
(2π)3/2
ηiesi
[pi · ǫ ∗±(q̂)]s
(pi · q)
× EI(p̃1, · · · , p̃i + ηiq̃, · · · , p̃n) · S±i (α→ β), (43)
where E iI = EI(p̃1, · · · , p̃i + q̃, · · · , p̃n) are the phase factors in the internal process∗∗. Since all the
E iI’s are the same when q → 0 (E iI = EI), one obtains in that limit,
S ±s⋆ (q, α→ β) =
(2π)3/2
ηiesi
[pi · ǫ ∗±(q̂)]s
(pi · q)
S⋆(α → β), (44)
where we used the relation S⋆(α → β) = EI · S(α → β). From the transformation properties of the
S⋆ matrix elements, Eq.(28), we obtain,
S ±s⋆ (q, α→ β) →
± (q̂) · · · ǫ
± (q̂)(M
⋆ )µ1···µs(q, α→ β). (45)
Identifying the S⋆ matrix elements in Eq.(44) and Eq.(45) gives the invariant M⋆ functions for spin 1
and 2:
Mµ⋆ (q, α→ β) =
(2π)3/2
ei · ηip µi
(pi · q)
S⋆(α→ β),
Mµν⋆ (q, α→ β) =
(2π)3/2
gi · ηip µi p νi
(pi · q)
S⋆(α → β). (46)
Requiring the twist invariance to the S⋆ matrix elements, Eq.(33), gives:
0 = qµM
⋆ (q, α→ β) →
ηiei = 0,
0 = qµM
⋆ (q, α→ β) →
gi · ηipνi = 0, (47)
∗∗ The integrals over the internal loop momenta have been suppressed because they do not affect the final result since
the change in the phase factor occurs only in the external lines.
in general. Hence, one obtains the charge conservation law for the spin 1 fields. For s = 2, in order
to satisfy the two relations,
ηipi = 0 (4-momentum conservation) and
i gi · ηipνi = 0, gi should
be constants (i.e. independence of the particle species). The universality of this coupling constant as
a ratio of gravitational mass and inertial mass shows that the equivalence principle is satisfied even in
the noncommutative spacetime.
4 Discussion
We have found that the conservation of charge and the equivalence principle are satisfied even in the
canonical noncommutative spacetime. The derivation was fairly straightforward, once we have con-
structed the S⋆ matrix for general field theories. The assumption was that the quanta of the gravitation
is massless spin 2, massless spin 1 for photon.
We extended the construction of the S⋆ matrix to general fields, especially for the massless fields
of integer spin. The twisted Feynman diagrams can be constructed by the same irreducible represen-
tations as those in the untwisted theories with the same rule except for the different phase factors at
each vertex. Hence, we can say that the same reasoning apply to the massive fields.
We use the (s/2, s/2) representation for the field operators mainly because one can obtain the
condition, Eq.(33), requiring the twist Lorentz invariance. In this representation, since the polarization
vectors are not Lorentz four vectors, each amplitude of emitting a real soft massless particle violate
the symmetry. We have used this property to show the conservation law in this paper. Another
representation for the fields, the (s, 0)⊕ (0, s) representation, which has the parity symmetry, can be
made Lorentz covariant. One realizes that due to these properties one cannot derive the conservation
law by the method used in this paper.
Charge conservation in the noncommutative spacetime is expected from the gauge symmetry and
the noncommutative Noether theorem. However, it is not quite certain whether the equivalence princi-
ple is satisfied in the noncommutative spacetime. Since the noncommutative spacetime is not locally
Minkowski nor the local symmetry group is SO(1, 3), one can not guarantee if the principle is satis-
fied. But in our twisted symmetry context, one can expect that the equivalence principle is satisfied
because the algebra structure is the same as the conventional theory though the coalgebra structure
is different. The applicability of the S matrix theoretic proof of the equivalence principle given in
this paper is restrictive because the analysis is perturbative in nature. We expect that the conclusion
of this paper is to be one of the stepping stone towards the further understanding of the nature of the
principle in quantum gravity.
This paper shows an example for the usefulness of the twisted symmetry to derive the physically
important relations in the noncommutative spacetime. We think that the twist analysis is adequate for
the schematic approaches to the noncommutative physics, while the other approaches are suitable for
the explicit calculations though they are equivalent. The future applications of the twisted symmetry
to the other issues in this direction of approach are expected.
Acknowledgement
I express my deep gratitude to Prof. J. H. Yee for his support. I would like to thank Dr. Jake Lee
for helpful discussions and Dr. A. Tureanu for his comments on the draft. This work was supported
in part by Korea Science and Engineering Foundation Grant No. R01-2004-000-10526-0, and by
the Science Research Center Program of the Korea Science and Engineering Foundation through the
Center for Quantum Spacetime(CQUeST) of Sogang University with grant number R11 - 2005 - 021.
A Exact calculation of the polarization tensor for massless fields
The creation and annihilation operators of the massless field transform under the Lorentz transforma-
tion as [28]
a†(Λp, σ) = e−iσΘ[W (Λ,p)] U(Λ) a†(p, σ) U−1(Λ),
a(Λp, σ) = e+iσΘ[W (Λ,p)] U(Λ) a(p, σ) U−1(Λ), (48)
up to phase factors, where W (Λ, p) = L−1(p)Λ−1L(Λp) denotes the Wigner rotation defined as in
[28] (Appendix C), and the Θ[W (Λ, p)] is the related angle to which the little group corresponds.
Hereafter, we abbreviate Θ[W (Λ, p)] as Θ(Λ, p). It is to be noted that we use U(Λ) instead of Uθ(Λ)
in here. This follows immediately from the properties of the twist in section 2.1.
The field Ψ̂ Aθ transform as
[Ds(Λ−1)]AB Ψ̂
θ (Λx) = U(Λ) Ψ̂
θ (x) U
−1(Λ). (49)
In order to satisfy the two relations (48), (49), the polarization tensor should transform as
ǫ Aθ (p, σ) = e
−iσΘ(Λ,p) [Ds(Λ−1)]AB ǫ
θ (Λp, σ). (50)
When Λ = W and p = k, the above relation goes to
[Ds(W )]AB ǫ
θ (k, σ) = e
−iσΘ(W,p)ǫ Aθ (k, σ), (51)
where k denotes the standard momentum. The Wigner rotation can be written as W (φ, α, β) =
R(φ)T (α, β) because the little group is isomorphic to ISO(2) for massless fields [28].
Suppose that the field transforms as (m,n) representation of spin s (m + n = s). When the φ, α
and β are infinitesimal, Ds(W ) can be written as
Ds(W ) ≃ 1− iφ(M3 +N3) + (α + iβ)(M1 − iM2) + (α− iβ)(N1 + iN2). (52)
For M− ≡M1 − iM2, N+ ≡M1 + iM2, and Θ → φ we have:
(M3 +N3) ǫθ(k, σ) = σ ǫθ(k, σ),
M− ǫθ(k, σ) = 0,
N+ ǫθ(k, σ) = 0, (53)
which gives
M3 ǫθ(k, σ) = −m ǫθ(k, σ),
N3 ǫθ(k, σ) = +n ǫθ(k, σ). (54)
The ǫθ(k, σ) satisfys the same equation as ǫ(k, σ). Since the highest or lowest weight corresponds to
a unique state, one obtains the same polarization tensor as a solution, ǫθ(k, σ) = ǫ(k, σ).
B Properties of the polarization vectors
Here, we summarize the properties of the polarization vector. The properties of the polarization
tensors of other rank, Eq.(24), follows from it.
Solving the (53) for σ = ±1 gives the explicit form of the polarization vector for the standard
momentum as
± (k) ≡
(1,±i, 0, 0), (55)
where we made the conventional choice of the phase. The polarization vector of momentum p is
defined as
± (p) = [L(p)]µν ǫ ν± (k), (56)
where L(p) is the Lorentz transformation which takes k to p, i.e., pµ = [L(p)]µν kν . Then the well
known properties of the polarization vectors can be deduced [11]:
± (p̂) = 0, (57)
ǫ ∗±µ(p̂)ǫ
± (p̂) = 1, ǫ±µ(p̂)ǫ
± (p̂) = 0, (58)
± (p̂) = ǫ
∓ (p̂), ǫ
± (p̂) = 0, (59)∑
± (p̂)ǫ
± (p̂) = η
µν + (p̃µpν + p̃νpµ)/2|p|2 ≡ Πµν , p̃ ≡ (|p|,−p), (60)
± (p̂)ǫ
± (p̂)ǫ
± (p̂)ǫ
± (p̂) =
{Πµ1ν1(p̂)Πµ2ν2(p̂) + Πµ1ν2(p̂)Πµ2ν1(p̂)
−Πµ1µ2(p̂)Πν1ν2(p̂)} . (61)
The polarization ’vectors’ are not the Lorentz four vectors††, rather they transform as
±(p) = e
∓iΘ(Λ,p)[Dsǫ (Λ, p)]
±(Λp),
[Dsǫ (Λ, p)]
ν = (Λ
−1)µν −
(Λ−1)0ν . (62)
For general spin s, the polarization tensor can be written as
ǫAσ (p) = e
∓iΘ(Λ,p)[Dsǫ (Λ, p)]
σ (Λp), (63)
up to a phase factor.
††It comes from the fact that the little group is not semisimple for massless fields. Translation in the little group
isomorphic to ISO(2) generates the gradient term in the transformations.
C Invariant M⋆ function for massless field
Let P denotes a shorthand notation for the external lines, P ≡ (p1, . . . , pn), and K denotes a standard
momenta‡‡ K ≡ (k1, . . . , kn). There exists a unique Lorentz transformation satisfying P ≡ LPK.
Then the relation between the Lorentz group and the little group can be described symbolically as:
W (Λ,P )
// ΛP ,
where W (Λ, P ) is the Wigner transformation to which a Lorentz transformation Λ and the momenta
P correspond. Since the twist do not change the group properties, above relation also holds for the
twisted symmetry group. The S⋆ matrix elements transform as
S⋆[P ] = D
θ[W (Λ, P )] S⋆[ΛP ]
= Dsθ[L
P · Λ
−1 · LΛP ] S⋆[ΛP ] (64)
where the indices for the external lines are suppressed. From the golden rule and (48), the explicit
transformation formular for S⋆[P ] can be obtained:
S⋆[P ] =
N(ΛP )
N(P )
e±isΘ(Λ,P ) S⋆[ΛP ], (65)
where N denotes the corresponding normalization factor.
If one defines M⋆[P ] as S⋆[P ] = D
P ]M⋆[P ], one can show that M⋆[P ] transform as
M⋆[P ] = D
−1]M⋆[ΛP ], (66)
i.e., it is twist invariant. We will call it as an invariant M⋆ function as in [29, 30].
Let us find out the invariant M⋆ function for massless fields. First, we define the quanties for the
standard momenta K. The choice for M⋆[K] is
MA⋆ [K] = N(K) ǫ
A[K] S⋆[K]. (67)
If we define M⋆[P ] as
MA⋆ [P ] = [D
θ(LP )]
⋆ [K], (68)
then one can easily show that MA⋆ [P ] is twist invariant.
By using the explicit form of Dsǫ in (62) and kbjM
b1···bs
⋆ [K] = 0, one obtains the relation
ǫC [P ]∗ (M⋆)C [P ] =
e±isΘ(L
,P )[Dǫθ(L
A[K]∗
[Dθ(LP )]
C (M⋆)B[K]
= e±isΘ(L
,P ) ǫA[K]∗ [Dθ(L
P ) ·D
A (M⋆)B[K]
= e±isΘ(L
,P ) ǫA[P ]∗ (M⋆)A[P ]
= e±isΘ(L
,P )N(K) S⋆[K], (69)
‡‡ see [11],[29].
where we used the relation
[Dθ(L
P ) ·D
A (M⋆)B[K] ≡ [D(L−1P ) ·Dǫ(L
A (M⋆)B[K]
δb1a1 −
(LP )
· · ·
δbsas −
(LP )
× (M⋆)b1···bs [K]
= (M⋆)a1···as [K]. (70)
By setting Λ = L−1P in (65), one can see that (69) is S⋆[P ]. Thus, the desired form of S⋆[P ] is
S⋆[P ] = N(P ) ǫ
A[P ]M
⋆ [P ]. (71)
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	Introduction
	Properties of the general S matrix
	A short introduction of useful properties of the twist-deformation
	The S matrix and its twist invariance
	Generalization to arbitrary fields
	Exact transformation formula for the S matrix element for massless fields
	Charge conservation and the equivalence principle 
	Dynamical definition of charge and gravitational mass
	Conservation law
	Discussion
	Exact calculation of the polarization tensor for massless fields
	Properties of the polarization vectors
	Invariant M function for massless field
