Research Article | Open Access
R. G. G. Amorim, S. C. Ulhoa, J. S. da Cruz Filho, A. F. Santos, F. C. Khanna, "Spin-1/2 Particles in Phase Space: Casimir Effect and Stefan-Boltzmann Law at Finite Temperature", Advances in High Energy Physics, vol. 2020, Article ID 7032834, 8 pages, 2020. https://doi.org/10.1155/2020/7032834
Spin-1/2 Particles in Phase Space: Casimir Effect and Stefan-Boltzmann Law at Finite Temperature
The Dirac field, spin 1/2 particles, is investigated in phase space. The Dirac propagator is defined. The Thermo Field Dynamics (TFD) formalism is used to introduce finite temperature. The energy-momentum tensor is calculated at finite temperature. The Stefan-Boltzmann law is established, and the Casimir effect is calculated for the Dirac field in phase space at zero and finite temperature. A comparative analysis with these results in standard quantum mechanics space is realized.
The Wigner function formalism [1, 2] and noncommutative geometry  play a fundamental role in the study of phase space quantum mechanics. The Wigner formalism enables a quantum operator, , defined in the Hilbert space, , to have an equivalent function of the type , in phase space , using the Moyal-product or star-product (å). Such a formalism leads to the classical limit of a quantum theory. In fact, quantum mechanics is a noncommutative theory whose representation in phase space is an object of debate. The opposite question, i.e., for a given classical function, what is its quantum counterpart? It is solved by using the Weyl transformation which is formulated independent of the phase space. In fact, it can be established within the configuration space of the generalized coordinates. The phase space has a well-defined physical meaning. The Hamiltonian function is naturally identified with the energy of the system. Establishing a field theory in phase space sheds light on some obscure points in quantum mechanics. For instance, the quantum symmetries are better understood in the symplectic structure of phase space which is similar to the role of Lorentz transformation in the covariant formulation of special relativity. This theoretical framework has to include a finite temperature in order to be suitable for experiments.
The star product has been employed for different objectives. In particular, it has been used for development of a nonrelativistic quantum mechanics formalism in terms of a phase space using the Galilean symmetry representation . Thus, the Schrödinger equation is obtained. In this case, the wave function is a quasiprobability amplitude defined in phase space and the Wigner function is obtained in an alternative way, i.e., by using . The Dirac equation coupled with the electromagnetic field in phase space  and applications  has been obtained.
Our goal is to explore the quasiprobability amplitude to study the effect of temperature using Thermo Field Dynamics (TFD) formalism [7–13] in a system for spin-1/2 particles. The principles of this theory are the duplication of the Fock space using the Bogoliubov transformations. The TFD formalism is used to study the Casimir effect, at zero and finite temperature. The scalar field in phase space has been studied , and some exclusive effects have been found at finite temperature. In addition, the Stefan-Boltzmann law for spin- particles in phase space is described in details.
In Section 2, the symplectic Dirac field is introduced. In Section 3, the Thermo Field Dynamics formalism is presented. In Section 4, the Stefan-Boltzmann law is established and the Casimir effect for the Dirac field is calculated in phase space at zero and finite temperature. In the last section, some concluding remarks are presented.
2. Spin-1/2 Field in Phase Space
A brief outline for spin-1/2 particles in phase space formalism is described. For this purpose, the following star operators in phase space are defined: which satisfy the Heisenberg commutation relation , with . The Poincaré algebra has the form
The operators in Equations ((1)–(3)) are defined on a Hilbert space, , associated with the phase space . The operators and stand for translations, rotations, and boosts, respectively. Functions defined on the Hilbert space are defined as
The Casimir invariants are and , where are Pauli-Lubansky matrices and is the Levi-Civita symbol.
The Dirac equation in phase space is obtained using the invariant operator . It is defined as where are the Dirac matrices. The Lagrangian density for the Dirac equation is where and is the mass of the particle.
The Wigner function provides the physical interpretation  and is given as where the star product is defined by
Using the Noether theorem in phase space , the energy-momentum tensor for the Dirac field is
Then, the Green function, , is defined as which may be written as where . The propagator of the Dirac field is
Taking , the expression is
This leads to the Green function for the Dirac equation in phase space
is defined as where is the modified Bessel function. It should be noted that due to the dependence on the Dirac matrices, the Green’s function has matrix properties itself.
3. Thermo Field Dynamics Formalism
The Thermo Field Dynamics (TFD) is a thermal quantum field theory at finite temperature [7–13]. It has two basics elements: (i) doubling the degrees of freedom in a Hilbert space and (ii) the Bogoliubov transformation. The doubling of Hilbert space is given by the tilde () conjugate rules where the thermal space is , with being the standard Hilbert space and the tilde (dual) space. There is a mapping between the two spaces; i.e., the map between the tilde and nontilde operators is defined by the following tilde conjugation rules: with for bosons and for fermions.
The Bogoliubov transformation corresponds to a rotation of the tilde and nontilde variables. Using the doublet notation, for fermions leads to where are creation operators, are destruction operators, and is the Bogoliubov transformation given by
Taking ( with being the Boltzmann constant and the temperature), the thermal operators are written explicitly as
These thermal operators satisfy the algebraic rules and other anticommutation relations are null. In addition, the quantities and are related to the Fermi distribution, i.e., such that . The parameter is associated with temperature, but, in general, it may be associated with other physical quantities. In general, a field theory on the topology with , is considered. are the space-time dimensions, and is the number of compactified dimensions. This establishes a formalism such that any set of dimensions of the manifold can be compactified, where the circumference of the th is specified by . The parameter is assumed as the compactification parameter defined by . The effect of temperature is described by the choice and .
Any field in the TFD formalism may be written in terms of the parameter. As an example, the scalar field is considered. Then, the -dependent scalar field becomes where the Bogoliubov transformation is used.
The -dependent propagator for the scalar field is where is the time-ordering operator. Using leads to the Green function where with being the Bogoliubov transformation and where is the scalar field propagator and is the scalar field mass. Here, is the complex conjugate of .
It is important to note that the physical quantities are given by the nontilde variables. Then, the physical Green function is written as where is the generalized Bogoliubov transformation , where is the number of compactified dimensions, for fermions (bosons), denotes the set of all permutations with elements, and is the 4-momentum. In the next section, three different topologies are used : (i) the topology , where . In this case, the time axis is compactified in , with circumference ; (ii) the topology with , where the compactification along the coordinate is considered; and (iii) the topology with is used. In this case, the double compactification consists in time and the coordinate . Then, thermal effects are considered for the Casimir effect and Stefan-Boltzmann law.
4. Stefan-Boltzmann Law and Casimir Effect for the Dirac Field in Phase Space
The Stefan-Boltzmann law is calculated by analyzing the energy-momentum tensor given as where
It should be noted that the field is the Dirac field in phase space as a function of the variables (), i.e., . The vacuum expectation value of the energy-momentum tensor is
The Dirac propagator in phase space is defined in Equation (16) as
Then, the energy-momentum tensor has the form
The vacuum average of the energy-momentum tensor in terms of -dependent fields becomes
In order to obtain measurable physical quantities at finite temperature, a renormalization procedure is carried out. The physical energy-momentum tensor is defined as where
Now, the Stefan-Boltzmann law and the Casimir effect in phase space are calculated at finite temperature.
4.1. Stefan-Boltzmann Law
The study of the Stefan-Boltzmann law in phase space corresponds to a choice of the parameter . It is important to note that the parameter is the compactification parameter that is defined as . The temperature effect is described by the choice
The generalized Bogoliubov transformation, Equation (31), for these parameters is
The Green’s function for the Dirac field in phase space is where is a time-like vector. Then, the physical energy-momentum tensor is
In order to calculate the Stefan-Boltzmann law, taking leads to
This is the Stefan-Boltzmann law for the Dirac field in phase space. It is worth pointing out that the result is recovered by taking the limit of the momentum variable. This result in phase space is necessary to compare with experiments. In this sense, we can integrate over the momenta which explicitly yield and take the limit; then, the only remaining part is the factor ofthat leads to the dependency , once the limit of Bessel function is taken. On the other hand, it is possible to project in the momentum space by integrating over coordinates. This process leads to a divergence which is of the same nature of the coordinate projection in the absence of temperature. Hence, a quantity in the momentum space analogous to the temperature is necessary, that is, the thermal energy. The introduction of TFD formalism introduces the role of temperature, but it can equally do the same for the thermal energy. Using phase space and TFD allows us to deal with systems where microscopic energy is dominant.
4.2. Casimir Effect for the Dirac Field in Phase Space
Here, the choice is , then
The Green function is this case is where is a space-like vector. Then, the energy-momentum tensor is
By taking , the Casimir energy for the Dirac field in phase space at zero temperature is
And for , the Casimir pressure in phase space is
It reproduces the usual result when and integrated over the momenta which means the projection on coordinate space. Then, only the factors of is left; the limit of this part yields the dependency . Here, the dependency on matrices should be viewed as part of the phase space formalism which is by its core matricial. This part does not survive once the projection on coordinates is performed, but it is part of the behavior in phase space. In order to be compared with experimental data, the projection on momentum space requires the introduction the thermal energy.
4.3. Casimir Effect for the Dirac Field in Phase Space at Finite Temperature
The effect of temperature is introduced by taking . Then, the generalized Bogoliubov transformation becomes
The first two terms of these expressions correspond, respectively, to the Stefan-Boltzmann term and the Casimir effect at . The third term is analyzed and it leads to the Green function
Then, the Casimir energy at finite temperature is and the Casimir pressure at finite temperature is
It should be noted that in the limit , both the Casimir energy and pressure are real quantities at zero and finite temperature. In the limit , i.e., , the Casimir energy and pressure become
It is important to note that in this limit, both the Casimir energy and pressure depend only on the distance between the plates. The dependence on gamma matrices is not a problem since the formalism of the phase space is matrix. It leads to the conclusion that neither the energy nor the pressure are scalars but components of a tensor.
The Dirac field in phase space is considered. Using the Dirac equation, the propagator for spin-1/2 particles is calculated. This form of the propagator is similar to that in the usual quantum mechanics. The TFD results are obtained by using the temperature effects in the Dirac propagator. TFD, a real-time finite temperature formalism, is a thermal quantum field theory. Using this formalism, a physical (renormalized) energy-momentum tensor is defined. Then, the Stefan-Boltzmann law in phase space and the Casimir effect are calculated at finite temperature. The results lead to the usual results for the Dirac field when they are projected in the quantum field theory space. The TFD formalism allows studying the finite temperature effects in phase space. On the other hand, such a formalism also may be used to explore the role of a thermal energy which is possibly related to the fermionic feature of the field.
This is a theoretical work, and all previous results are listed in the references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work by A. F. S. is supported by CNPq projects 308611/2017-9 and 430194/2018-8.
- E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Physics Review, vol. 40, no. 5, pp. 749–759, 1932.
- M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Physics Reports, vol. 106, no. 3, pp. 121–167, 1984.
- H. Weyl, “Quantenmechanik und gruppentheorie,” Zeitschrift für Physik, vol. 46, no. 1-2, pp. 1–46, 1927.
- M. D. Oliveira, M. C. B. Fernandes, F. C. Khanna, A. E. Santana, and J. D. M. Vianna, “Symplectic quantum mechanics,” Annals of Physics, vol. 312, no. 2, pp. 492–510, 2004.
- R. G. G. Amorim, M. C. B. Fernandes, F. C. Khanna, A. E. Santana, and J. D. M. Vianna, “Non-commutative geometry and symplectic field theory,” Physics Letters A, vol. 361, no. 6, pp. 464–471, 2007.
- R. G. G. Amorim, S. C. Ulhoa, and E. O. Silva, “On the symplectic Dirac equation,” Brazilian Journal of Physics, vol. 45, no. 6, pp. 664–672, 2015.
- F. C. Khanna, A. P. C. Malbouisson, J. M. C. Malboiusson, and A. E. Santana, Themal Quantum Field Theory: Algebraic Aspects and Applications, World Scientific, Singapore, 2009.
- Y. Takahashi and H. Umezawa, “Higher order calculation in thermo field theory,” Collective Phenomena, vol. 2, p. 55, 1975.
- Y. Takahashi and H. Umezawa, “Thermo field dynamics,” International Journal of Modern Physics B, vol. 10, pp. 1755–1805, 1996.
- H. Umezawa, H. Matsumoto, and M. Tachiki, Thermofield Dynamics and Condensed States, North-Holland Pub. Co., North Holland, Amsterdan, 1982.
- H. Umezawa, Advanced Field Theory: Micro, Macro and Thermal Physics, AIP, New York, 1993.
- A. E. Santana, A. M. Neto, J. D. M. Vianna, and F. C. Khanna, “Symmetry groups, density-matrix equations and covariant Wigner functions,” Physica A, vol. 280, no. 3-4, pp. 405–436, 2000.
- F. C. Khanna, A. P. C. Malbouisson, J. M. C. Malbouisson, and A. E. Santana, “Thermoalgebras and path integral,” Annals of Physics, vol. 324, no. 9, pp. 1931–1952, 2009.
- R. G. G. Amorim, J. S. da Cruz Filho, A. F. Santos, and S. C. Ulhoa, “Stefan-Boltzmann Law and Casimir Effect for the Scalar Field in Phase Space at Finite Temperature,” Advances in High Energy Physics, vol. 2018, Article ID 1928280, 9 pages, 2018.
- F. C. Khanna, A. P. C. Malbouisson, J. M. C. Malbouisson, and A. E. Santana, “Quantum fields in toroidal topology,” Annals of Physics, vol. 326, no. 10, pp. 2634–2657, 2011.
- F. C. Khanna, A. P. C. Malbouisson, J. M. C. Malbouisson, and A. E. Santana, “Quantum field theory on toroidal topology: algebraic structure and applications,” Physics Reports, vol. 539, no. 3, pp. 135–224, 2014.
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