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.
Highlights
T0 being the temperature and μ0 the chemical potential coupled to a conserved charge Q, and Z the partition function
The presented derivation does not make use of the solutions of the Klein-Gordon equations in curvilinear coordinates but it is just based on the plane wave expansion of the field in Minkowski space-time and it is suitable for the general case including both rotation and linear acceleration
We have studied the series in two major cases of non-trivial equilibrium, the pure acceleration and the pure rotation and compared with exact known results obtained solving Klein-Gordon equation in Rindler and rotating coordinates respectively
Summary
The building block to calculate any statistical quantity (mean values, correlations) is the mean value of the combination of one creation and one annihilation operator of four-momentum eigenstates: Tr ρ a†(p) a(p ). Before setting out to do that, it should be first pointed out that any other combination of creation or annihilation operators, such as a†(p) a†(p ) , a(p) a(p ) or combination of creation/annihilation operators of particles and antiparticles in case of a charged scalar field, will vanish. This happens because the density operator (1.2), involving just the generators of Lorentz transformation, translations and charge, does not change the number of particles and antiparticles.
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