Stefan-Boltzmann law and Casimir effect for the scalar field in phase space at finite temperature

In this paper we use the scalar field constructed in phase space to analyze the analogous Stefan-Boltzman law and Casimir effect both of them at finite temperature. The temperature is introduced by Thermo Field Dynamics (TFD) and the quantities are analyzed once projected in the coordinates space.


I. INTRODUCTION
Eugene Paul Wigner [1,2] introduced in 1932 the first formalism to quantum mechanics in phase space, motived by finding a way to treat transport equations for superfluids. Wigner formalism allows mapping between quantum operators, say A, defined in the Hilbert space, S, with classical functions, say a w (q, p), in phase space Γ, through the * is the star-product or Moyal-product. The opposite problem, i. e., from a classical function finding the correspondent operator, is accomplished by the Weyl mapping. The main motivation to define a given physical theory in phase space is its natural interpretation as demonstrated by classical mechanics.
The star-product has been explored in phase space in different ways. Particularly, it has been used to define operators like a w (q, p) * of interest to study irreducible unitary representations of kinematical groups in phase space [3]. In case of non-relativistic symmetries, this leads to a Schrödinger equation in phase space, where the wave function is directly associated with the Wigner function, so with full physical meaning. In this formalism of quantum mechanic, the observables are represented by operators of type a = a w ⋆, which are used to construct a representation of Galilei symmetries.
The Wigner function is given by f w (q, p) = ψ † ⋆ ψ where ψ = ψ(q, p) are the wave functions, solutions of the Schrödinger equation represented in phase space. Since it is a theory of representation, this formalism has been generalized to the relativistic case, leading to Klein-Gordon and Dirac equations in phase space [4]. This method has been applied successfully, for instance, to the analysis of abelian gauge symmetries [5], describing the dynamics (interaction) in the formulation of quantum theory in phase space.
Although the quantum field theory has successfully been applied to several systems, it doesn't take into account the temperature of such systems. This is a fundamental problem since all macroscopic features of a quantum model is related to temperature. Among all schemes to introduce temperature in a physical theory we cite two ways. The first one is to interpret time as temperature by a Wick rotation [6]. This approach is problematic when one is dealing with a time evolution of a physical state. The second one the so called Thermo Field Dynamics (TFD), it's a natural way to deal with dynamical systems. It preserves the time-evolution once the temperature is identified with a rotation in a duplicated Fock space [7]. In this article we explore how to implement TFD in phase space, particularly we analyze the scalar field in phase space at finite temperature. The Casimir effect for scalar field in phase space at zero and finite temperature is calculated. The Casimir effect [8] is measured when two parallel conducting plates are attracted due to vacuum fluctuations. Although the first application has been developed for the electromagnetic field all quantum field should exhibit this phenomenon. In fact it was demonstrated that nonrelativistic fields such as in Schrödinger equation the Casimir effect is present [9]. Particularlly for non-relativistic fields such a effect is physical only at finite temperature. Sparnaay [10] made the first experimental observation of the Casimir effect. Subsequent experiments have established this effect to a high degree of accuracy [11,12].
The article is divided as follows. In section II, the symplectic Klein-Gordon field is introduced.
In section III, we show the canonical quantization of scalar field. Then in section IV, we recall the ideas of Thermo Field Dynamics. In section V, we calculate the energy-momentum tensor of the symplectic scalar field and in section VI, we derive the Stefan-Boltzmann like law and the Casimir effect at finite temperature. Finally in the last section se present our conclusions.

II. SYMPLECTIC KLEIN-GORDON FIELD AND WIGNER FUNCTION
Let us define the star-operator as hence we can derive the following operators and These operators satisfy the Poincaré algebra and act in Hilbert space associated with phase space H(Γ). From them, we construct a symplectic representation of Poincaré-Lie algebra and as a result we obtain the Klein-Gordon equation in phase space [4] P 2 φ(q, p) = p 2 ⋆ φ(q, p) = m 2 φ(q, p) The functions φ(q, p) are defined in phase space Γ and satisfy the condition Eq.(5) can be derived from lagrangian given by [5] where The association with Wigner formalism is obtained from . To show this, we multiply the right-hand side of Eq.(4) by φ * (q, p), but since φ * (q, p) ⋆ p 2 = m 2 φ * (q, p), we also have Subtracting Eq.(8) of Eq.(9), and using the associativity of star-product, we get where the Moyal-bracket is given by Calculating, we obtain a well known result. Another properties of Wigner function, such as non-positiveness, can be derived analogous a non-relativistic case [3].
If we consider the interaction potential V , the follow density of lagrangian should be used This lagrangian induces the equation where V (ψ) = ∂U (ψψ † ) ∂ψ † . Solutions for Eq.(13) can be obtained from Green function method. For this proposal, take the where G is the Green function. By superposition principle, solution of Eq. (14) is given by where ψ 0 (q µ , p µ ) is the solution of free case.
We can find the solution of Eq.(15) taking its Fourier transform.
where F [g] stands the Fourier transform os function g. In this way, we obtain Then, The solution is Wigner function can be derived from Green function by [13] f W (q µ , p µ ) = lim Eq.(20) provide a method to describe interaction process and scattering theory with physical interpretation in phase space.

III. CANONICAL QUANTIZATION OF SCALAR FIELD IN PHASE SPACE
In this section we construct the formalism of canonical quantization of Klein-Gordon field in phase space. From this formalism we obtain the propagator written in phase space.
From lagrangian given in Eq. (12) we define the conjugate momenta associated to fields φ(q, p) .
After some calculations we have Follow usual procedure of quantization, we impose the commutation relations the other commutation relations are nulls.

A. Annihilation and Creation Operators
The fields φ(q, p) and φ † (q, p) may be expanded as and where ω k = [( 1 2 k − p) 2 + m 2 ] 1/2 and the canonical variable related to position is K, i.e., q → k. The functions form a ortonormal set In this way, the fields φ(q, p) and φ † (q, p) may be written as and Inverting Eq.(23) and Eq.(24) we obtain We can show that and The operators a(k, p), a † (k, p), b(k, p) and b † (k, p) play a crucial role in the particle interpretation of the quantised field theory. First, define the operators and It is simple to show that N (k, p) and N (k ′ , p ′ ) commute In analogous sense, In this case, the eigenstates of these operators may be used to form a basis. Let denote the eigenvalue N (k, p)a(k, p)|n(k, p) = (n(k, p) − 1)a(k, p)|n(k, p) , M (k, p)b(k, p)|m(k, p) = (m(k, p) − 1)b(k, p)|m(k, p) .
Eqs. (27,28) tell us that if the state |k, p has eigenvalue n(k, p),and the states a † (k, p)|n(k, p) and a(k, p)|n(k, p) are eigenstates of N (k, p) with respective eigenvalues n(k, p)+1 and n(k, p)−1. And analogous, we note by Eqs. (29,30) tell us that if the state |k, p has eigenvalue m(k, p), and the states b † (k, p)|m(k, p) and b(k, p)|m(k, p) are eigenstates of M (k, p) with respective eigenvalues m(k, p) + 1 and m(k, p) − 1. So, the operators a † (k, p) and a(k, p) are interpreted as creation and annihilation operators of particles, respectively. Then, analogously, b † (k, p) and b(k, p) can be interpreted as creation and annihilation operators of antiparticles, respectively.
Using the creation and annihilation operators, the Hamiltonian of scalar field in phase space can be written by We can also show that the particles that are the quantum of the Klein-Gordon field obey the Bose-Einstein statistics. For this, note that basta notarmos que The connection between the solution of free Klein-Gordon equation, ϕ(q, p) = ξ(p µ )e −ikµq µ , and the canonical quantization formalism is given by ϕ(q, p) = 0|φ(q, p)|k, p .

IV. THERMO FIELD DYNAMICS -TFD
In this section a brief introduction to TFD formalism is presented [14][15][16][17][18]. In this formalism the thermal average of an observable is given by the vacuum expectation value in an extended The standard doublet notation for an arbitrary bosonic operator X is where a, b = 1, 2. The physical variables are described by nontilde operators.
The Bogoliubov transformation introduces a rotation in the tilde and nontilde variables. Then the thermal effects are introduced by a Bogoliubov transformation, U (α), that is defined as where u 2 (α) − v 2 (α) = 1. These quantities u(α) and v(α) are related to the Bose distribution.
The parameter α is associated with temperature, but, in general, it may be associated with other physical quantities. The Bogoliubov transformations for bosons and fermions are different. For bosons are given as where (a † ,ã † ) are creation operators and (a,ã) are destruction operators, with with ω = ω(k).
For fermions the Bogoliubov transformations are with Let's to consider a free scalar field in Minkowski space with diag(g µν ) = (+1, −1, −1, −1) and then analyze its propagator. Using the Bogoliubov transformation the α-dependent scalar field is given by There is a similar equation for tilde field. The propagator for the scalar field, α-dependent, is where τ is the time ordering operator. Using |0(α) = U (α)|0,0 where G (ab) with Then where is the generalized Bogoliubov transformation [19], with d being the number of compactified dimensions, η = 1(−1) for fermions (bosons) and {σ s } denotes the set of all combinations with s elements.
An important note, in phase space the Green function is dependent of the parameters q and p, i.e., G

V. ENERGY-MOMENTUM TENSOR FOR THE SCALAR FIELD IN PHASE SPACE
The lagrangian that describes the scalar field in phase space is given by In order to calculate the Casimir effect we need the energy-momentum tensor that is defined as To avoid divergences, the energy-momentum tensor is written at different space-time points as where τ is the ordering operator and The vacuum expectation value of the energy-momentum tensor is The scalar field propagator in phase space is defined as Using the identity where κ ν (z) is the Bessel function. Using the doublet notation, the physical energy-momentum tensor in terms of the α-parameter is where T µν(ab) (q; α) = T µν(ab) (q; α) − T µν(ab) (q) and VI. SOME APPLICATIONS In this sections the Stefan-Boltzmann law and the Casimir effect at finite temperature are calculated.
is the Green function with n 3 = (0, 0, 0, 1), being the space-like vector. The energy-momentum tensor for this case becomes T 33(11) (β) = 2 lim It is the Casimir pressure at zero temperature in phase space. In the standard quantum mechanics space the usual result is recovered.

VII. CONCLUSIONS
In this article the introduction of temperature into phase space was explored. We defined the scalar field in phase space by means the use of invariants of the respective relativistic algebra. Then we presented the energy-momentum tensor in such a space. This result was used to implement the prescription of Thermo Field Dynamics which allows to deal with some phenomena at finite temperature such as the analogous Casimir effect and Stefan-Boltzmann law. We point out that we projected the mean energy and pressure in the space of coordinates in order to recover the results of literature. If we project our result in the momenta space we should obtain a fundamental energy associated to the given temperature. Such a result should be better understood since the existence of this thermal energy affects the interpretation of phase space.