Advances in perturbative thermal field theory - Ultra-relativistic ...

Thermal field theory 407
In the physics of the very Early Universe, which is filled with a hot plasma of various
elementary particles, the gravitational polarization tensor is also of interest. It describes
the (linear) response to metric perturbations, and its IR behaviour determines the evolution
of large-scale cosmological perturbations, which provide the seeds for structure formation
that are nowadays being studied directly through the anisotropies of the cosmic microwave
background and which are being measured with stunning accuracy [446,447]. The gravitational
polarization tensor is also a central quantity in the theory of stochastic (semiclassical) gravity
[448,449], which aims at a general self-consistent description of quantum statistical fluctuations
of matter in a curved background geometry as a stepping-stone towards a full quantum theory
of gravity.
9.1. HTL gravitational polarization tensor
If Ɣ denotes all contributions to the effective action besides the classical Einstein–Hilbert
action, the energy–momentum tensor is given by the one-point 1PI vertex function
2 δƔ
T µν (x) = √ (9.1)
−g(x) δg µν (x)
and the gravitational polarization tensor by the two-point function
δ 2 Ɣ
− µναβ (x, y) ≡
δg µν (x)δg αβ (y) = 1 δ( √ −g(x)T µν (x))
. (9.2)
2 δg αβ (y)
From the last equality, it is clear that αβµν describes the response of the (thermal) matter
energy–momentum tensor to perturbations in the metric. Equating αβµν to the perturbation
of the Einstein tensor gives self-consistent equations for metric perturbations and, in particular,
cosmological perturbations.
In the ultrarelativistic limit where all bare masses can be neglected and in the limit
of temperature much larger than spatial and temporal variations, the effective (thermal)
action is conformally invariant, that is Ɣ[g] = Ɣ[ 2 g] (the conformal anomaly like other
renormalization issues can be neglected in the high-temperature domain).
This conformal invariance is crucial for the application to cosmological perturbations for
two reasons. First, it allows us to have matter in thermal equilibrium despite a space–time
dependent metric. As long as the latter is conformally flat, ds 2 = σ(τ,x)[dτ 2 −dx 2 ], the local
temperature on the curved background is determined by the scale factor, σ . Second, the thermal
correlation functions are simply given by the conformal transforms of their counterparts on
a flat background, so that ordinary momentum–space techniques can be employed for their
evaluation.
The gravitational polarization tensor in the HTL approximation has been first calculated
fully in [450], after earlier work [451–454] had attempted (in vain) to identify the Jeans mass (a
negative Debye mass squared signalling gravitational instability rather than screening of static
sources) from the static limit of the momentum–space quantity ˜ µναβ (k), k 0 = 0, k → 0in
flat space.
Like the HTL self-energies of QED or QCD, µναβ is an inherently non-local object.
Because it is a tensor of rank 4, and the local plasma rest frame singles out the time direction, it
has a much more complicated structure. From η µν , u µ = δµ 0 and k µ, one can build 14 tensors
to form a basis for ˜ µναβ (k). Its HTL limit (k 0 , |k| ≪T ), however, satisfies the Ward identity,
4k µ ˜ µναβ (k) = k ν T αβ − k σ (T ασ η βν + T βσ η αν ), (9.3)
corresponding to diffeomorphism invariance as well as a further one corresponding to
conformal invariance, the ‘Weyl identity’,
η µν ˜ µναβ (k) =− 1 2 T αβ, (9.4)