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Brazilian Journal of Physics

Print version ISSN 0103-9733On-line version ISSN 1678-4448

Braz. J. Phys. vol.38 no.1 São Paulo Mar. 2008 



Noncommutative harmonic oscillator at finite temperature: a path integral approach



A. Jahan

Department of Physics, Amirkabir University of Technology P. O. Box 15875-4413, Tehran, Iran




We use the path integral approach to a two-dimensional noncommutative harmonic oscillator to derive the partition function of the system at finite temperature. It is shown that the result based on the Lagrangian formulation of the problem, coincides with the Hamiltonian derivation of the partition function.

Keywords: Noncommutative space; Partition function




Recently, there has been an immense amount of challenge devoted to the subject of noncommutative space-time. Although, the idea of noncommutating space-time coordinates is an old proposal [1], the recent discoveries in string/M theories were the main source of renewed interests in the subject [2]. In particular, much work is dedicated to study the noncommutative version of the quantum field theory (see for example [3] and references there in). In noncommutative field theories one replaces the space-time coordinates xm with operators µ , which do not commute with each other, i.e.

with qmn as a real anti-symmetric matrix. Thus because of the uncertainty relation induced by the relation (1), the short distance scales in the xm direction correspond to the large distance scales in the xn direction and vice versa. So there will be a mixing between the ultraviolet and infrared behaviors of the field theories in noncommutative space-times. Such a problem, which is called the UV/IR mixing problem, is characteristic of the models defined in noncommutative space [4]. In a noncommutative space-time the usual product between the fields must be replaced by the Weyl-Moyal bracket or star-product defined as

The star-product is the usual starting point in the most studies about the noncommutative field theories since it encodes the noncommutativity of space-time for the product of the several fields defined in the same point in the following sense

On the other hand it seems natural to explore the noncommutative version of quantum mechanics since it stands as the one-particle low energy sector of the quantum field theory [5]. The following set of relations among the coordinates and momenta (setting = 1) 2pt

characterize the noncommutative version of the quantum mechanics (Latin indices stand for the spatial coordinates). An equivalent procedure to implement the star-product is to define a new set of commutating coordinates xi via [5]

(Through the paper summation is implied over the repeated indices). Therefore for the interaction potential (i) defined in noncommutative space one gets the effective potential in usual commutative space as

Path integral approach to the noncommutative quantum mechanics has revealed in series of the recent works [6]. It is demonstrated by the authors that how the noncommutativty of space could be implemented in Lagrangian formulation of the quantum mechanics. Here we aim to incorporate the effect of noncommutativty of space on the thermodynamics of a two dimensional harmonic oscillator by means of the formalism developed in [6]. So after a brief review of the Lagrangian aspect of noncommutative quantum mechanics we derive the partition function of the system by identifying the temperature as imaginary time. It is shown that the result is in accordance with one obtained via the Hamiltonian approach to the partition function.



The Hamiltonian governing the dynamics of a harmonic oscillator in noncommutative space is

On implementing the transformation (5), one finds the effective Hamiltonian in usual commutative space as

with k2 = 1 + m2w2q2/4 and corresponding energy spectrum as

The corresponding Lagrangian can be achieved by means of the Legender transformation Lq(xi,i) = pi i- Hq(xi,pi), upon replacing for the momenta from i = Hq/pi. Hence the Lagrangian associated with the Hamiltonian (8) will be

The above Lagrangian admits the equations of motion as differential equations of rank four. When the solutions are inserted in (10) and integrated out over the time variable t , one finds for the action

with boundary condition x'' = x(t) and x' = x(0). Thus one obtains the semiclassical propagator (transition amplitude) as 2pt



The partition function Z(b) plays a vital role in thermodynamical considerations of the physical systems at finite temperature. It can be defined in terms of the propagator of system as [7]

where the inverse temperature parameter is defined as b = it. The symbol Tr stands for the functional trace which for a bi-local function A(x,x') in D dimensions is defined as

When the time parameter in (12) is replaced with the inverse temperature parameter b = it, the semiclasical propagator (12) modifies to


where we have invoked the well-known trigonometric identities cos ia = i cosh a and sinh a = -i sin ia. Thus from the definition (14) we find for the trace of the term e

Therefore one is left with the partition function as 2pt

The above result for the partition function is derived earlier in the context of Hamiltonian formulation of the problem [8] 2pt

with energy spectrum given by Eq. (9). (Note that the result of above summation derived in [8] is false since the arguments of the sinh functions must be multiplied by the factor 1/2 ). The free energy F(b) of a system at finite temperature is related to the partition function as

which at low temperature limit tends to the ground state energy of the system. In particular for the case of harmonic oscillator we have

which coincides with the ground state energy of the system, e0,0 = w (see Eq. (9)). In the limit 0 , the parameter k tends to unity and one recovers the ground state energy of usual two-dimensional harmonic oscillator.


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Received on 2 November, 2007

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