Exact equilibrium distributions in statistical quantum field theory with rotation and acceleration: scalar field

Abstract

We derive a general exact form of the phase space distribution function and the thermal expectation values of local operators for the free quantum scalar field at equilibrium with rotation and acceleration in flat space-time without solving field equations in curvilinear coordinates. After factorizing the density operator with group theoretical methods, we obtain the exact form of the phase space distribution function as a formal series in thermal vorticity through an iterative method and we calculate thermal expectation values by means of analytic continuation techniques. We separately discuss the cases of pure rotation and pure acceleration and derive analytic results for the stress-energy tensor of the massless field. The expressions found agree with the exact analytic solutions obtained by solving the field equation in suitable curvilinear coordinates for the two cases at stake and already — or implicitly — known in literature. In order to extract finite values for the pure acceleration case we introduce the concept of analytic distillation of a complex function. For the massless field, the obtained expressions of the currents are polynomials in the acceleration/temperature ratios which vanish at 2π, in full accordance with the Unruh effect.

Exact equilibrium distributions in statistical quantum field theory with rotation and acceleration: Dirac field

“…Such expressions are very useful, but since the pseudo-momentum variable k is offshell, as we have seen, it is desirable, in several circumstances, to deal with a distribution function with only on-shell momentum as argument, like in the classical kinetic theory. To define it starting from the Wigner function, one could follow the method used in the scalar field case [1], namely recast the mean current (3.5a) as an integral over on-shell momenta multiplied by a four-vector p µ . This method can be applied to particles and antiparticles separately, by taking advantage of the decomposition (3.3); one can then focus on the particle contribution only, the antiparticle being easily obtained from it.…”

“…It will be obtained with the JHEP10(2021)077 same method used in ref. [1]: a factorization of the density operator and an iterative procedure to calculate the expectation values a † s (p) a t (p ) with imaginary thermal vorticity. The integrals of the Wigner function (3.5) are then analytically continued to real thermal vorticity to obtain the physical values.…”

“…As has been mentioned, the method presented in [1] involves two steps. The first step is the factorization of the density operator (1.1):…”

“…In a previous paper [1], we presented a new method to calculate exact expressions of mean values involving the free scalar field in the most general global thermodynamic equilibrium in Minkowski space-time, including acceleration and rotation. In this paper, we extend that method to a free quantum field of any spin and particularly the free Dirac field of JHEP10(2021)077 spin 1/2 particles.…”

“…In ref. [1] we put forward a method to calculate the exact expression of expectation values with the density operator (1.1) (henceforth denoted as thermal expectation values or TEV), based on the factorization of the exponentials in the density operator (1.1) and their analytical continuation to complex thermal vorticity. We introduced a mathematical prescription, dubbed as analytic distillation, to remove the unphysical, non-analytic terms, which makes it possible to obtain the actual physical TEVs by continuing back to real thermal vorticity.…”