Bose–Einstein condensation and free DKP field

doi:10.1016/S0375-9601(03)01018-1

Physics Letters A

Volume 316, Issues 1–2, 15 September 2003, Pages 33-43

Dedicated to Professor Sı́lvio Roberto de Azevedo Salinas on the occasion of his 60th birthday

https://doi.org/10.1016/S0375-9601(03)01018-1 Get rights and content

Abstract

The thermodynamical partition function of the Duffin–Kemmer–Petiau theory is evaluated using the imaginary-time formalism of quantum field theory at finite temperature and path integral methods. The DKP partition function displays two features: (i) full equivalence with the partition function for charged scalar particles and charged massive spin 1 particles; and (ii) the zero mode sector which is essential to reproduce the well-known relativistic Bose–Einstein condensation for both theories.

Introduction

Bose–Einstein condensation [1], [2] (BEC) is a phenomenon long associated to the liquid helium studies (⁴He and ³He–⁴He mixtures). However, recent research deals with BEC shown that it touches several areas of modern physics [3]: in thermodynamics [4] BEC occurs as a phase transition from gas to a new state of matter, quantum mechanics view BEC as a matter–wave coherence [5] arising from the overlapping de Broglie waves of the atoms and draw an analogy between conventional and “atom lasers” [6], quantum statistics explain BEC as more than one atom sharing a phase space cell, in the quantum theory of atomic traps [7] many atoms condense to the ground state of the trap, in quantum field theory BEC is commonly related to spontaneous symmetry breaking [8], [9], [10].

In quantum field theory BEC the complex scalar field was and is used for studying the thermodynamical properties of physical systems composite of bosonic particles with spin 0. However, at zero temperature, there is an alternative way to study not only the properties for complex scalar field but also the complex vectorial field that it is known as the massive Duffin–Kemmer–Petiau (DKP) theory [11].

At zero temperature, one important question concerning DKP theory is about the equivalence or not between its spin 0 and 1 sectors and the theories based on the second order Klein–Gordon (KG) and Proca equations, respectively. From the beginning of the 50s the belief on this equivalence was perhaps the principal reason for the abandon of DKP equation in favor of KGF and Proca ones. However, in the 70s this supposed equivalence began to be investigated in several situations involving breaking of symmetries and hadronic processes, showing that in some cases DKP and KG theories can give different results (for a historical review of the development of DKP theory until the decade of 70s see Ref. [12]). Moreover, DKP theory follows to be richer than the KG one with respect to the introduction of interactions. In this context, alternative DKP-based models were proposed for the study of meson–nucleus interactions, yielding a better adjustment to the experimental data when compared to the KG-based theory [13]. In the same direction, and guided for the formal analogy with Dirac equation, approximation techniques formerly developed in the context of nucleon–nucleus scattering were generalized, giving a good description for experimental data of meson–nucleus scattering. The deuteron–nucleus scattering was also studied using DKP, motivated by the fact that this theory suggests a spin 1 structure from combining two spin $12$ [14], [15].

Recently there have been a renewed interest in DKP theory. For instance, it has been studied in the context of QCD [16], covariant Hamiltonian dynamics [17], in the causal approach [18], [19], in the context of five-dimensional Galilean covariance [20], in the scattering K⁺-nucleus [21], in curved space–times [22], [23], [24], etc.

These examples, among others in literature, in some cases break the equivalence between the theories based on DKP and KG/Proca equations, such as in [13] or in Riemann–Cartan space–times [23], [24]. Nevertheless, the question about the equivalence or not still lacks a complete answer nowadays.

As above mentioned, all accomplished studies on the DKP theory were made at zero temperature. The aim of this work is to study the thermodynamics and Bose–Einstein condensation for the massive charged particles by using the DKP theory. In Section 2 we present the massive DKP theory in the Minkowski space–time and we make a résumé of the constraint analysis of the model which can be see in [25]. In Section 3 we introduce the partition function and the generating functional of the correlation functions of the theory, and we specialize, separately, the spin 0 and spin 1 sector for the explicit calculation of the respective partition functions. We also analyze explicitly the zero mode contribution in both cases. In Section 4 we give our conclusions and perspectives.

Section snippets

The Duffin–Kemmer–Petiau theory

The Duffin–Kemmer–Petiau equation [11] in Minkowski space–time is given by $iβ^{μ} ∂_{μ} ψ−mψ=0,$ where the matrices β^μ obey the DKP algebra, $β^{α} β^{μ} β^{ν} +β^{ν} β^{μ} β^{α} =β^{α} η^{μν} +β^{ν} η^{μα}$ with η^μν being the metric tensor of Minkowski space–time with signature $(+ − − −)$ . The β^μ are singular matrices which have only three irreducible representations of dimensions 1, 5 and 10. The first one is trivial, having no physical meaning and the other two correspond to fields of spin 0 and 1, respectively.

The DKP equation given in (1) is

The partition function

Now we pass to Euclidean β^μ matrices which are given in , . We must put $ψ ̄ =ψ^{†},$ and use β^μ matrices in all , , , , , , , , , , , , , , , , , , , , , , , , .

Then the partition function in statistic for the massive DKP field is $Z=N(β)∫ D p_{ψ} D p_{ψ̄} ∫ periodic D ψ D ψ ̄ δ(θ)δ(θ ̄)δ(ω)δ(ω ̄) exp ∫ 0 β dτ∫d^{3} x ip_{ψ} ∂_{τ} ψ+i∂_{τ} ψ ̄ p_{ψ̄} − H +μj^{τ},$ where N(β) is an infinite normalizing factor which will be determined later [26]. We integrate over periodic DKP field due to its bosonic character.

The term μj^τ in Eq. (9) appears due to

Conclusions and remarks

The free massive DKP theory at finite temperature (FT) is equivalent to both complex massive scalar field and complex vector field theories at FT. It reproduces the relativistic Bose–Einstein condensation in both sectors, where we show in a clean and elegant way the zero mode existence and its contribution to BEC.

The perspectives to follow are to study the DKP field coupled to the quantized electromagnetic field and the implementation of the renormalization process at zero temperature which

Acknowledgements

We thank Profs. L. Tomio and Victo S. Filho for additional references related to the Bose–Einstein condensation. This work was supported by FAPESP/Brazil (R.C., full support grant 01/12611-7; V.Ya.F., grant 01/12585-6; B.M.P., grant 02/00222-9, J.V. full support grant 00/03812-6), RFFI/Russia (V.Ya.F., grant 02-02-16946) and CNPq/Brazil (B.M.P.).

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¹: Permanent address: P.N. Lebedev Institute of Physics, Moscow, Russia.

View full text

Bose–Einstein condensation and free DKP field

Abstract

Introduction

Section snippets

The Duffin–Kemmer–Petiau theory

The partition function

Conclusions and remarks

Acknowledgements

Phys. Rev. Lett.

Z. Phys.

Sitz. Ber. Kgl. Preuss. Akad. Wiss.

Statistical Mechanics

Nature

Science

Science

Nature

Phys. Rev. Lett.

Science

Phys. Rev. Lett.

Science

Phys. Rev. A

Phys. Rep.

Rev. Mod. Phys.

Phys. Rev. A

Phys. Rev. D

Phys. Rev. Lett.

Phys. Rev. D

Phys. Rev. Lett.

Phys. Rev. D

Acad. Roy. Belg. Cl. Sci. Mém. Collect. 8° (2)

Phys. Rev.

Proc. R. Soc. A

Am. J. Phys.

Phys. Rev. C