Abstract
We derive the general exact forms of the Wigner function, of mean values of conserved currents, of the spin density matrix, of the spin polarization vector and of the distribution function of massless particles for the free Dirac field at global thermodynamic equilibrium with rotation and acceleration, extending our previous results obtained for the scalar field. The solutions are obtained by means of an iterative method and analytic continuation, which lead to formal series in thermal vorticity. In order to obtain finite values, we extend to the fermionic case the method of analytic distillation introduced for bosonic series. The obtained mean values of the stress-energy tensor, vector and axial currents for the massless Dirac field are in agreement with known analytic results in the special cases of pure acceleration and pure rotation. By using this approach, we obtain new expressions of the currents for the more general case of combined rotation and acceleration and, in the pure acceleration case, we demonstrate that they must vanish at the Unruh temperature.
Highlights
We have derived a general exact form of the Wigner function and the thermal expectation values of local operators of the free Dirac field in the most general case of global thermodynamic equilibrium in Minkowski space-time, that is with a Killing fourtemperature vector including rotation and acceleration
For the spin 1/2 particles, we have obtained the general form of the Wigner function of a free Dirac field as a formal series for imaginary thermal vorticity including all quantum corrections to the classical term
The analytic continuation to real thermal vorticity and the extraction of finite results, demands the application of the analytic distillation, an operation on complex functions introduced in ref. [1] and extended here to the alternate fermionic series
Summary
Before we get into the main topic of this work, it is necessary to introduce the basic formalism of the Dirac and spin 1/2 particles theory. The spinor form is dictated by the request for the field Ψ(x) to transform according to the irreducible representation (0, 1/2) ⊕ (1/2, 0) of the orthochronous Lorentz group SO(1, 3)↑ [27]. Where the upper arc stands for the hermitian matrix corresponding to the reflected vector: X= Xμσμ ;. Such definitions are well known in the construction of the SL(2, C)−SO(1, 3)↑ morphism [28]. Pp/γ0U(p), where the bar over a four-vector implies the reflection of its space components, that is Xμ = (X0, −X) With this compact form, it can be readily checked that the spinors fulfill the following relations:.
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