CPT violation does not lead to violation of Lorentz invariance and vice versa

Abstract

We present a class of interacting nonlocal quantum field theories, in which the CPT invariance is violated while the Lorentz invariance is present. This result rules out a previous claim in the literature that the CPT violation implies the violation of Lorentz invariance. Furthermore, there exists the reciprocal of this theorem, namely that the violation of Lorentz invariance does not lead to the CPT violation, provided that the residual symmetry of Lorentz invariance admits the proper representation theory for the particles. The latter occurs in the case of quantum field theories on a noncommutative space–time, which in place of the broken Lorentz symmetry possesses the twisted Poincaré invariance. With such a CPT-violating interaction and the addition of a C-violating (e.g., electroweak) interaction, the quantum corrections due to the combined interactions could lead to different properties for the particle and antiparticle, including their masses.

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arXiv:1103.0168v2 [hep-th] 3 Apr 2011
CPT Violation Does Not Lead to Violation of Lorentz
Invariance and Vice Versa
Masud Chaichiana, Alexander D. Dolgovb,c,d, Victor A. Novikovd
and Anca Tureanua
aDepartment of Physics, University of Helsinki, P.O.Box 64, FIN-00014 Helsinki, Finland
bDipartimento di Fisica, Universit` a degli Studi di Ferrara, I-44100 Ferrara, Italy
cInstituto Nazionale di Fisica Nucleare, Sezione di Ferrara, I-44100 Ferrara, Italy
dInstitute of Theoretical and Experimental Physics, 113259 Moscow, Russia
(Phys. Lett. B, in press)
Abstract
We present a class of interacting nonlocal quantum field theories, in which the CPT
invariance is violated while the Lorentz invariance is present. This result rules out a
previous claim in the literature that the CPT violation implies the violation of Lorentz
invariance. Furthermore, there exists the reciprocal of this theorem, namely that the
violation of Lorentz invariance does not lead to the CPT violation, provided that the
residual symmetry of Lorentz invariance admits the proper representation theory for
the particles. The latter occurs in the case of quantum field theories on a noncommuta-
tive space-time, which in place of the broken Lorentz symmetry possesses the twisted
Poincar´ e invariance. With such a CPT-violating interaction and the addition of a
C-violating (e.g., electroweak) interaction, the quantum corrections due to the com-
bined interactions could lead to different properties for the particle and antiparticle,
including their masses.

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1Introduction
Lorentz symmetry and the CPT invariance are two of the most fundamental symmetries of Nature,
whose violation has not yet been observed. While the Lorentz invariance is a continuous symmetry
of space-time, the CPT involves the discrete space- and time-inversions, P, T, and the charge
conjugation operation on the fields, C. Although the individual symmetries, C, P and T have
been observed to be violated in various interactions, their combined product, CPT, remarkably
remains still as an exact symmetry. The first proof of CPT theorem was given by L¨ uders and
Pauli [1,2] based on the Hamiltonian formulation of quantum field theory, which involves locality of
the interaction, Lorentz invariance and Hermiticity of the Hamiltonian. Later on the theorem was
proven by Jost [3] (see also [4–6]) within the axiomatic formulation of quantum field theory without
reference to any specific form of interaction. This proof of CPT theorem relaxes the requirement of
locality or ”local commutativity” condition to the so-called ”weak local commutativity”. Lorentz
symmetry has been an essential ingredient of the proof, both in the Hamiltonian and in the
axiomatic proofs.
A simple phenomenological classification of possible C, P, T, CP, PT, TC and CPT-violating
effects is presented in [7]. For consequences of CPT and their experimental tests, as well as some
theoretical considerations on the possibilities of violation of Lorentz invariance and CPT in the
known interactions, we refer to [8–13] and references therein.
It is important to clarify the relation between the CPT and Lorentz invariance and in particular
to see whether the violation of any of them implies the violation of the other. This issue has recently
become a topical one due to the growing phenomenological importance of CPT violating scenarios,
namely in neutrino physics as well as its cosmological and astrophysical consequences. Indeed, the
relation between the CPT and Lorentz invariance has acquired a prominent place in nowadays
particle physics with the attempts of explaining in a unified manner the contradictory results,
”anomalies”, in the interpretation of various neutrino physics experiments, without enlarging the
neutrino sector. The idea was first suggested by Murayama and Yanagida [14] in the form of
different masses for neutrino and antineutrino, based on phenomenological considerations. This
proposal was formalized as a CPT-violating quantum field theory with a mass difference between
neutrino and antineutrino in [15] (see also [16]). The issue was taken up in relation with the
Lorentz symmetry by Greenberg [17], the conclusion of Greenberg’s analysis being that CPT
violation implies violation of Lorentz invariance. This result was given as a ”theorem”, the dispute
on the validity of which is the subject of this Letter.
We should emphasize that a theorem which states that CPT violation implies violation of
Lorentz invariance has to be explicit, first of all about what is meant by the charge conjugation
in a Lorentz violating theory. Is the violation complete or is any subgroup of Lorentz symmetry
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left, which should have the needed spin-representations to which the particles are assigned? Does
the corresponding theory which violates both CPT and Lorentz invariance contain fields with a
plausible description in terms of equations of motion?
2
CPT-violating free field model
A free field model in which particle and antiparticle have different masses was proposed in [15].
Although the model was hoped to be Lorentz-invariant, a closer examination [17] showed that it
is not – the propagator is not Lorentz covariant, unless the masses of particle and antiparticle
coincide. The model is also nonlocal and acausal: the ∆(x,y)-function, i.e. the commutator of
two fields, does not vanish for space-like separation, unless the two masses are the same, thus
violating the Lorentz invariance. This was considered in [17] as supporting a general ”theorem”
that interacting fields that violate CPT symmetry necessarily violate Lorentz invariance.
We would like to point out that the model taken in [17] is utmost pathological and can not
be considered as a quantum field theory. There, the claim was that the model represents a free
complex scalar field, quantized in such a way that the mass of the antiparticle differs from that
of the particle. However, there is no definite equation of motion that this ”field” satisfies, and
no quantization procedure that would support the claim that the mode expansion with different
masses for ”particle” and ”antiparticle” really represents a free quantized field. Also, two such
”free fields” separated by a space-like distance do not commute, i.e. the theory is acausal at the
free level without invoking interaction.
Moreover, by requiring that the classical symmetries and in particular the global U(1) sym-
metry for a free complex scalar field, i.e. the conservation of electric charge, be preserved at the
quantum level, one can show that using the expansion for a free ”field” as proposed in [17], would
bring it back uniquely to the usual field expansion in terms of creation and annihilation operators
with m = ¯ m – otherwise, the electric charge is not conserved.
Furthermore, in a quantum field theory with acausal free fields, as taken in [17], observables,
which are functions of those fields, do not commute when separated by space-like distances. This,
according to Pauli’s proof of the spin-statistics theorem, implies that there is no spin-statistics
relation already for the free fields. Thus, one has no rule whether to apply commutation or
anticommutation relations in quantizing the fields. But the worst is that in such a model, where
Lorentz invariance is violated by the free fields, there is no concept of spin to start with altogether.
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3
CPT-violating but Lorentz-invariant nonlocal model
Here, as an example, we propose a model which preserves Lorentz invariance while breaking the
CPT symmetry through a (nonlocal) interaction. The latter attitude is taken as responsible for
the violations of a symmetry, based on our experience that all the discrete, C, P and T invariances,
as well as other symmetries, are broken in our description of Nature by means of interaction. We
also know that nonlocal field theories appear, in general, as effective field theories of a larger
theory.
Consider a field theory with the nonlocal interaction Hamiltonian of the type
Hint(x) = λ
?
d4y φ∗(x)φ(x)φ∗(x)θ(x0− y0)θ((x − y)2)φ(y) + h.c.,
(3.1)
where λ is a coupling constant with dimension appropriate for the Hamiltonian density, φ(x) is a
Lorentz-scalar field in the interaction picture and θ is the Heaviside step function, with values 0
or 1, for its negative and positive argument, respectively. The combination θ(x0− y0)θ((x − y)2)
in (3.1) ensures the Lorentz invariance, i.e. invariance under the proper orthochronous Lorentz
transformations, since the order of the times x0and y0remains unchanged for time-like intervals,
while for space-like distances the interaction vanishes. Also, the same combination makes the
nonlocal interaction causal at the tree level, which dictates that there is no interaction when the
fields are separated by space-like distances and thus there is a maximum speed of c = 1 for the
propagation of information.
On the other hand, it is clear that C and P invariance are trivially satisfied in (3.1), while T
invariance is broken due to the presence of θ(x0− y0) in the integrand.
One can always insert into the Hamiltonian (3.1), without changing its symmetry properties,
a weight function or form-factor F((x − y)2), for instance of a Gaussian type:
F = exp
?
−(x − y)2
l2
?
,
(3.2)
with l being a nonlocality length in the considered theory. Such a weight function would smear
out the interaction and would guarantee the desired behaviour of the integrand in (3.1); in the
limit of fundamental length l → 0 in (3.2), the Hamiltonian (3.1) would correspond to a local,
CPT- and Lorentz-invariant theory. A weight function such as (3.2) would make the acausality
of the model (see the next section) restricted only to very small distances, of the order of l. The
latter could be looked upon as being a characteristic parameter relating the effective field theory
to its parent one, for instance the radius of a compactified dimension when the parent theory is
a higher-dimensional one. Furthermore, with such a weight function, the interaction vanishes at
infinite (x − y)2separations and thus one can envisage the existence of in- and out-fields.
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There exists a whole class of such CPT-violating, Lorentz-invariant field theories involving
different, scalar, spinor or higher-spin interacting fields. Typical simplest examples are:
Hint(x) = λ
?
?
?
d4y φ∗
1(x)φ1(x)θ(x0− y0)θ((x − y)2)φ2(y) + h.c.,
(3.3)
Hint(x) = λd4y¯ψ(x)ψ(x)θ(x0− y0)θ((x − y)2)φ(y) + h.c.,
(3.4)
Hint(x) = λd4y φ(x)θ(x0− y0)θ((x − y)2)φ2(y) + h.c.
(3.5)
4Quantum theory of such nonlocal interactions
The S-matrix in the interaction picture is obtained as solution of the Lorentz-covariant Tomonaga-
Schwinger equation [18,19] (see also [20,21]):
i
δ
δσ(x)Ψ[σ] = Hint(x)Ψ[σ],
(4.1)
with σ a space-like hypersurface, and the boundary condition:
Ψ[σ0] = Ψ,
(4.2)
where Hintis for instance the Hamiltonian (3.5) with the fields in the interaction picture. Then
Eq. (4.1) with the boundary condition (4.2) represent a well-posed Cauchy problem.
The existence of a unique solution for the Tomonaga-Schwinger equation is ensured if the
integrability condition
δ2Ψ[σ]
δσ(x)δσ(x′)−
δ2Ψ[σ]
δσ(x′)δσ(x)= 0,
(4.3)
with x and x′on the surface σ, is satisfied. The integrability condition (4.3), inserted into (4.1),
requires that the commutator of the interaction Hamiltonian densities vanishes at space-like sep-
aration:
[Hint(x),Hint(y)] = 0,
for (x − y)2< 0.
(4.4)
Since in the interaction picture the field operators satisfy free-field equations, they automat-
ically satisfy Lorentz-invariant commutation rules. The Lorentz-invariant commutation relations
are such that (4.4) is fulfilled only when x and y are space-like separated, (x−y)2< 0 , i.e. when
σ is a space-like surface. As a result, the integrability condition (4.4) is equivalent to the micro-
causality condition for local relativistic QFT. When the surfaces σ are hyperplanes of constant
time, the Tomonaga-Schwinger equation reduce to the single-time Schr¨ odinger equation.
Inserting, e.g., the expression (3.5) into (4.4), we have:
[Hint(x),Hint(y)] = λ2?d4ad4b θ((x − a)2)θ(x0− a0)θ((y − b)2)θ(y0− b0)
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