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

Thermal field theory 357
It is of course more convenient to stick to the rest frame of the heat bath. Lorentz invariance
appears to be broken then, but it is usually a trivial matter to switch to Lorentz covariant
expressions using u µ .
Information about the dynamics of a thermal system can be obtained by considering
thermal expectation values of the generally time-dependent observables (in the Heisenberg
picture). In linear response theory, the time evolution of small disturbances of an equilibrium
system is determined by the correlation functions of pairs of observables [14].
For both the purpose of calculating the partition function, Z, and correlation functions,
there exist two equivalent but rather different looking formalisms to set up perturbation theory.
These correspond to the two most popular choices of a complex-time path in the path integral
formula:
∫
∫ ∫
〈T c ˆϕ 1 ··· ˆϕ n 〉=N Dϕϕ 1 ···ϕ n exp i dt d 3 xL, (2.12)
C
where T c denotes contour ordering along the complex-time path C from t 0 to t 0 − iβ such that
t i ∈ C, and t 1 ≽ t 2 ≽ ··· ≽ t n with respect to a monotonically increasing contour parameter.
L is the Lagrangian, which in the presence of a chemical potential µ ≠ 0 may be replaced
by L → ¯L = L + µN , provided N does not contain time derivatives. Analyticity requires
that along the complex-time path the imaginary part of t is monotonically decreasing [11]; in
the limiting case of a constant imaginary part of t along (parts of) the contour, distributional
quantities (generalized functions) arise.
Perturbation theory is set up in the usual fashion. Using the interaction-picture
representation, one can derive
〈T c O 1 ···O n 〉= Z 0
Z 〈T cO 1 ···O n e i ∫ C L I
〉 0 , (2.13)
where L I is the interaction part of L, and the correlators on the right-hand side can be evaluated
by a Wick(–Bloch–DeDominicis) theorem:
〈T c e i ∫ C d4 xj ϕ 〉 0 = exp
{
− 1 2
∫
C
∫
C
}
d 4 x d 4 x ′ j(x)D c (x − x ′ )j (x ′ ) , (2.14)
where D c is the two-point function and this is the only building block of Feynman graphs with
an explicit T and µ dependence. It satisfies the Kubo–Martin–Schwinger (KMS) condition
D c (t − iβ) =±e −µβ D c (t), (2.15)
stating that e iµt D(t) is periodic (anti-periodic) for bosons (fermions).
A KMS condition can be formulated for all correlation functions in thermal equilibrium and
can in turn be viewed as a general criterion for equilibrium [15]. There exists also a relativistic
version of the KMS condition [16,17], which encodes the stronger analyticity requirements of
relativistic quantum field theories. The relativistic spectrum condition, H |P|, where P is
the total three-momentum, implies analyticity involving all space–time variables x → z ∈ C 4
in tube domains |Im z| < Im z 0