3.2 Chern–Simons Modified Gravity
29
The corresponding linear equation of motion is
−
1
2
h
i j
T T +
1
2μ
ilk
∂ k h
j
l,T T = 0.
(3.31)
As a result, one immediately concludes that the dispersion relation is the usual one,
k
2
0 = k
2 , and both polarizations propagate with the speed of light.
The natural question is—what is difference of these polarizations? A more careful
analysis [39] shows that, for plane waves proportional to e
iωt−ikz , one finds that there
are two basic (circular) polarizations T = i S and T = −i S, with their intensities
proportional to (1 +
k
μ
)
−2 and (1 −
k
μ
)
−2 respectively. This difference of intensities
can be treated as a consequence of parity breaking.
It should be noted that if we consider, instead of the CS term, the one-derivative
additive term h μν
λαμρ
θ λ ∂ ρ h
ν
α , with θ
λ being a space-like vector, we will have two
polarizations with physically consistent dispersion relations E = ±θ +
p 2 + θ 2 ,
so, in this case the velocities differ from speed of light [47]. However, this term is not
gauge invariant, which, within the gravity context, means that it breaks the general
covariance.
3.2.3 Perturbative Generation
The special interest is attracted to the gravitational CS term within the context of study
of the Lorentz symmetry breaking. The main reason consists in the fact that, besides
of the CPT symmetry breaking, for a special choice of the CS coefficient ϑ = b μ x
μ ,
where b μ is a constant vector (as we already noted in the previous subsection, this
choice is consistent with the Gödel-type solutions), the CS term displays Lorentz
symmetry breaking, taking the form (3.8), or, for the weak field, the linearized form
(3.9). Therefore the natural idea consists in a generation of this term as a perturbative
correction, similarly to the generation of the CFJ term in the extended QED, see
f.e. [48]. This similarity is supported by a natural analogy between the gravitational
anomalies [40] and the Adler–Bell–Jackiw (ABJ) anomaly [49]. Moreover, it follows
from [50] that this anomaly is deeply related with the ambiguity of results, therefore,
it is natural to expect the ambiguity of the gravitational CS term as well.
So, one can start with the action of spinors coupled to gravity, where the Lorentzbreaking vector b μ is introduced:
S =
d
4 xe ¯
ψ(i∂ / − m − b /γ 5 + ω
/)ψ,
(3.32)
here, b / = b
μ e
a
μ γ a , and ω μ =
1
4
ω μbc σ
bc is a (Riemannian) connection. We note that
the CS term dominates in the limit m → 0 while the one-derivative term discussed
in [47] vanishes in this limit. The corresponding one-loop effective action is given
by the following trace of the logarithm:
29
The corresponding linear equation of motion is
−
1
2
h
i j
T T +
1
2μ
ilk
∂ k h
j
l,T T = 0.
(3.31)
As a result, one immediately concludes that the dispersion relation is the usual one,
k
2
0 = k
2 , and both polarizations propagate with the speed of light.
The natural question is—what is difference of these polarizations? A more careful
analysis [39] shows that, for plane waves proportional to e
iωt−ikz , one finds that there
are two basic (circular) polarizations T = i S and T = −i S, with their intensities
proportional to (1 +
k
μ
)
−2 and (1 −
k
μ
)
−2 respectively. This difference of intensities
can be treated as a consequence of parity breaking.
It should be noted that if we consider, instead of the CS term, the one-derivative
additive term h μν
λαμρ
θ λ ∂ ρ h
ν
α , with θ
λ being a space-like vector, we will have two
polarizations with physically consistent dispersion relations E = ±θ +
p 2 + θ 2 ,
so, in this case the velocities differ from speed of light [47]. However, this term is not
gauge invariant, which, within the gravity context, means that it breaks the general
covariance.
3.2.3 Perturbative Generation
The special interest is attracted to the gravitational CS term within the context of study
of the Lorentz symmetry breaking. The main reason consists in the fact that, besides
of the CPT symmetry breaking, for a special choice of the CS coefficient ϑ = b μ x
μ ,
where b μ is a constant vector (as we already noted in the previous subsection, this
choice is consistent with the Gödel-type solutions), the CS term displays Lorentz
symmetry breaking, taking the form (3.8), or, for the weak field, the linearized form
(3.9). Therefore the natural idea consists in a generation of this term as a perturbative
correction, similarly to the generation of the CFJ term in the extended QED, see
f.e. [48]. This similarity is supported by a natural analogy between the gravitational
anomalies [40] and the Adler–Bell–Jackiw (ABJ) anomaly [49]. Moreover, it follows
from [50] that this anomaly is deeply related with the ambiguity of results, therefore,
it is natural to expect the ambiguity of the gravitational CS term as well.
So, one can start with the action of spinors coupled to gravity, where the Lorentzbreaking vector b μ is introduced:
S =
d
4 xe ¯
ψ(i∂ / − m − b /γ 5 + ω
/)ψ,
(3.32)
here, b / = b
μ e
a
μ γ a , and ω μ =
1
4
ω μbc σ
bc is a (Riemannian) connection. We note that
the CS term dominates in the limit m → 0 while the one-derivative term discussed
in [47] vanishes in this limit. The corresponding one-loop effective action is given
by the following trace of the logarithm: