Thermal effective potential in two- and three-dimensional non-commutative spaces

Abstract

The issue of thermal correlation functions and the associated effective statistical potential in twodimensional Moyal space, arising in the twisted approach to implement rotational symmetry, has been revisited in an operatorial formulation where no explicit star product is used initially. The corresponding results using Moyal and Voros star products are then easily obtained by taking the corresponding overlap with Moyal and Voros bases. in contrast to the Moyal case where the concept of distance and, in particular, the relative separation between a pair of particles remain ambiguous when the Moyal star product is used, the Voros basis is more physical and the inter-particle distance can be introduced unambiguously. The forms of the correlation function and the effective potential are found to be same as the Moyal case except that the thermal wavelength undergoes a non-commutative deformation, ensuring that it has a lower bound of the order of √ θ. It is shown that in a suitable basis (called here quasi-commutative basis) in the multiparticle sector the thermal correlation function coincides with the commutative result both in the Moyal and Voros cases along with the restoration of the Pauli principle, except that in the Voros case the thermal wavelength, again, gets a non-commutative correction. Finally, we extend our result to three-dimensional non-commutative space and compute the correlation function and effective potential using both twisted and quasi-commutative bases in the Moyal and Voros cases. We find that there is SO(3) → SO(2) symmetry breaking in the effective potential, which also violates the Pauli principle, even for a pair of free particles, despite the fact that a deformed co-product is used to construct twisted symmetric/anti-symmetric basis. However, this SO(3) symmetry, along with Pauli principle, is restored once we use the quasi-commutative bases.

Energy Spectrum and Phase Transition of Superfluid Fermi Gas of Atoms on Noncommutative Space

“…By the way, equation (3.6) is the application to the Moyal case [24] of general formulas [23,24] valid for generic twist-induced deformations. The fact that twisted/deformed (anti)commutation relations do not give rise to any deformation in Fermi-Dirac or Bose-Einstein statistics has also been demonstrated very clearly in [8,16].…”

Quantum mechanics with space-time noncommutativity

“…In this context, let us digress for a little while and discuss the status of generalized Schrödinger uncertainty relation for the phase space operators X, T , Px , Pt in the context of commutative quantum mechanics (θ = 0). We essentially follow the approach of [29].…”

On the role of Schwinger’s SU(2) generators for simple harmonic oscillator in 2D Moyal plane

“…We can now express the position and momentum operators in terms ofB L ,B ‡ L ,B R andB ‡ R . By making use of equation (6), the definitionB [5] and its Hermitian conjugate, and the adjoint action of momenta (6) we get:…”

“…It is therefore desirable to start with the formulation of quantum mechanics itself at completely operatorial level, so that one can formulate second quantization and eventually construct quantum field theory at a completely operatorial level. Indeed, such an attempt was made in [6],albeit in a nonrelativistic framework using the so called Hilbert-Schmidt operator which was initiated in [7,8] , with the operator valued space coordinates were taken to correspond to the Moyal-plane satisfying [x i ,x j ] = iθ ij = iθǫ ij (i, j = 1, 2) (1) and time was taken to be c-number variable. Clearly, the representation of this coordinate algebra (1) is furnished by a Hilbert space isomorphic to the quantum Hilbert space of 1-d harmonic oscillator and we refer to this space as classical Hilbert space(H c ), which is nothing but boson Fock space…”

“…It is therefore desirable to start with the formulation of quantum mechanics itself at completely operatorial level, so that one can formulate second quantization and eventually construct quantum field theory at a completely operatorial level. Indeed, such an attempt was made in [6],albeit in a nonrelativistic framework using the so called Hilbert-Schmidt operator which was initiated in [7,8] , with the operator valued space coordinates were taken to correspond to the Moyal-plane satisfying…”