Voros product and the Pauli principle at low energies
aa r X i v : . [ h e p - t h ] J un Voros product and the Pauli principle at low energies
Sunandan Gangopadhyay a,b, ∗ , Anirban Saha a,c † Frederik. G. Scholtz d, ‡ a Department of Physics, West Bengal State University, Barasat, India b Visiting Associate in S.N. Bose National Centre for Basic Sciences,JD Block, Sector III, Salt Lake, Kolkata-700098, India c Visiting Associate in Inter University Centre for Astronomy and Astrophysics,Ganeshkhind, Pune, India d National Institute of Theoretical Physics, Stellenbosch,Stellenbosch 7600, South Africa
Abstract
Using the Voros star product, we investigate the status of the two particle correlation functionto study the possible extent to which the previously proposed violation of the Pauli principlemay impact at low energies. The results show interesting features which are not present inthe computations made using the Moyal star product.
Keywords : Noncomutative geometry
PACS:
It is likely that at short distances spacetime has to be described by different geometrical struc-tures and that the very concept of point and localizability may no longer be adequate. This isone of the main motivations for the introduction of noncommutative (NC) geometry [1]-[3] inphysics. Attempts have been made to formulate quantum mechanics [4]-[11] and quantum fieldtheories [12]-[15] on NC spacetime. However, for the latter case, the issue of the lack of Lorentzinvariance symmetry has remained a challenge since field theories defined on a NC spacetimewith the commutation relation of the coordinate operators[ˆ x µ , ˆ x ν ] = iθ µν (1)where θ µν is a constant antisymmetric matrix, are obviously not Lorentz invariant.The twist approach to NC quantum field theory has been developed to circumvent thisproblem [16]-[23]. It is triggered by the realization that it is possible to twist the coproduct ofthe universal envelope U ( P ) of the Poincar´e algebra such that it is compatible with the starproduct.An interesting result that follows from the twisted implementation of the Poincar´e group isthat the Bose and Fermi commutation relations get deformed [22]. This is controversial thoughas a different point of view was presented in [24] and the issue is not yet resolved. A furthercontroversy involves the use of the Moyal or Voros twist, related to the use of the Moyal or Vorosstar product [25, 26, 28]. In this article we adopt the point of view of [22] and consider the effect ∗ [email protected], [email protected] † [email protected] ‡ [email protected]
1f implementing the twist through the Voros or Moyal twist within a simple nonrelativisticsetting.A very striking consequence of these deformations is the violation of the Pauli exclusionprinciple. In [27], it has been shown that the two particle correlation function for a free fermiongas in 2 + 1 dimensions (with exclusively spatial noncommutativity, i.e. θ i = 0, i = 1 ,
2) doesnot vanish in the limit r → F ( θ ) M = e − i θ ij ∂ i ⊗ ∂ j (2)where (in two spatial dimensions) θ ij = θǫ ij with ǫ ij being the antisymmetric tensor of ordertwo.Recently, a comparison of NC field theories built on two different star products (twist elements),namely the Moyal and Voros, have been made [28]. It has been found that although the Green’sfunctions are different for the two theories, the S-matrix is the same in both cases and is differentfrom the commutative case. This motivates us to investigate the status of the two particlecorrelation function using deformed commutation relations obtained by incorporating the actionof the Voros twist element [28] F ( θ ) V = e − θ∂ + ⊗ ∂ − (3) ∂ ± = 1 √ (cid:18) ∂∂x ∓ i ∂∂x (cid:19) on the usual pointwise product between two fields.To begin the analysis, we first write down the mode expansion of a free nonrelativisticquantum field ψ of mass m (in units with ¯ h = 1) in the NC plane as ψ ( ~r, t ) = Z d ~k a ( ~k ) e k ( ~r ) (4)where e k ( ~r ) = e − i | ~k | t m e i~k.~r and a ( ~k ) satisfy the usual (anti)commutation relation a ( ~k ) a † ( ~k ′ ) − ηa † ( ~k ′ ) a ( ~k ) = (2 π ) δ ( ~k − ~k ′ ) (5)where η is +1 for bosons and − a ( ~k ) hasalready been derived in [30] for the relativistic case and can be readily shown to hold in thenonrelativistic case as well : a ( ~k ) ⋆ M a ( ~k ′ ) = e − (1 / θ ij k i k ′ j a ( ~k ) a ( ~k ′ ) a ( ~k ) ⋆ M a † ( ~k ′ ) = e (1 / θ ij k i k ′ j a ( ~k ) a † ( ~k ′ ) a † ( ~k ) ⋆ M a ( ~k ′ ) = e − (1 / θ ij k i k ′ j a † ( ~k ) a ( ~k ′ ) (6) a ( ~k ) ⋆ V a ( ~k ′ ) = e − θk − k ′ + a ( ~k ) a ( ~k ′ ) a ( ~k ) ⋆ V a † ( ~k ′ ) = e θk − k ′ + a ( ~k ) a † ( ~k ′ ) a † ( ~k ) ⋆ V a ( ~k ′ ) = e − θk − k ′ + a † ( ~k ) a ( ~k ′ ) (7) The Voros product has also played a key role in the obtention of a θ -deformed mass density of a sphericallysymmetric gravitational source [29]. k ± = √ ( k ± ik ).We now compute the two particle correlation function for a free gas in 2+1 dimensions usingthe canonical ensemble, i.e., we are interested in the matrix elements Z h ~r , ~r | e − βH | ~r , ~r i , ( β =1 / ( k B T )) where Z is the canonical partition function and H is the non-relativistic Hamiltonian.The physical meaning of this function is quite simple; it tells us what the probability is tofind particle two at position ~r , given that particle one is at ~r , i.e., it measures two particlecorrelations [31]. The relevant two particle state is given by | ~r , ~r i = ˆ ψ † ( ~r ) ⋆ V ˆ ψ † ( ~r ) | i = Z d ~q (2 π ) d ~q (2 π ) e ∗ q ( ~r ) e ∗ q ( ~r ) a † ( ~q ) ⋆ V a † ( ~q ) | i . (8)The two particle correlation function can therefore be written as C ( ~r , ~r ) = 1 Z h ~r , ~r | e − βH | ~r , ~r i = 1 Z Z d ~k d ~k e − β m ( ~k + ~k ) |h ~r , ~r | ~k , ~k i| (9)where we have introduced a complete set of momentum eigenstates | ~k , ~k i and Z is the partitionfunction of the system.Using eq(s)(8, 7, 5) and noting that | ~k , ~k i = a † ( ~k ) ⋆ V a † ( ~k ) | i (10)we get |h ~r , ~r | ~k , ~k i| = e − θ ( k x k x + k y k y ) { ηe i ( ~k − ~k ) .~r e iθk ∧ k + c.c } (11)where ~r = ~r − ~r , θk ∧ k = θǫ ij k i k j = θ ( k x k y − k y k x ) and c.c implies complex conjugateof the second term in the above expression. Substituting this in eq(9), we obtain C ( ~r , ~r ) = f ( θ ) Z η (1 − m θ β )(1 − m θ β ) e − m (1+ 2 mθβ ) β (1 − m θ β r (12)where r = | ~r − ~r | and f ( θ ) = mπ/β ) − m θ /β . The partition function of the system can now bereadily computed and reads Z = Z d ~r d ~r h ~r , ~r | e − βH | ~r , ~r i = f ( θ ) A (cid:26) η πβmA (cid:18) − mθβ (cid:19)(cid:27) ≈ f ( θ ) A (13)in the limit of the area of the system A → ∞ . Substituting the above result in eq(12), we finallyobtain C ( ~r , ~r ) = 1 A η (1 − π θ λ )(1 − π θ λ ) e − π (1+ 4 πθλ λ − π θ λ r (14)where λ is the mean thermal wavelength given by λ = (cid:18) πβm (cid:19) / ; β = 1 k B T . (15)3e quote the corresponding result in the Moyal case [27] in order to compare with the Vorosresult obtained above (14): C ( ~r , ~r ) = 1 A η
11 + θ λ e − πλ θ λ r . (16)Eq(14) shows many interesting features. Note that there exists two thermal wavelengths λ =2 √ πθ and λ = q √ πθ where the second term in the above expression vanishes. These wave-lengths correspond to two temperatures T = 1 / (2 k B mθ ) and T = 1 / ( √ k B mθ ) at which thecorrelation function is completely independent of whether the particles are bosons or fermions.For temperatures much less than T the correlations exhibit mild deviations away from that ofcommutative bosons or fermions. However, for temperatures T lying between T and T (i.e. T < T < T ), the bosons start behaving as fermions and vice versa. These temperatures are ofcourse extremely high. Indeed, assuming θ to be of the order of the Planck length squared andrestoring ¯ h one finds them to be of the order of 10 K. For temperatures
T > T , the exponentialin eq(14) becomes positive and correlation grow exponentially with distance. This signals theonset of instability, which can best be seen if one considers the exchange potential defined by V ( r ) = − k B T log C ( r , r ). This becomes unstable or complex above this temperature. Thissuggests the existence of a high energy cut off E = k B T . These features are completely absentwhen the computations are made using the Moyal star product [27] and are summarized inFigures 1a, 1b, 2a and 2b, which shows the exchange potential for the Moyal and Voros twistin the case of bosons and fermions, respectively, for values of the dimensionless variable α = θλ below T ( α = 0 . T and T ( α = 0 . r is measured in units of λ .For reference the commutative results ( α = 0) are also shown in which case there is of course nodistinction between Moyal and Voros. The transgression of statistics in the temperature rangebetween T and T can clearly be seen. Of course, if these temperatures are really as high as ad-vertised here, the nonrelativistic limit would be invalid and the conclusions above will probablybe modified in a proper relativistic description. However, the crucial point here is not so muchthe absolute values of θ and λ , but rather the existence of two length scales ( √ θ and λ ), whichis not the case in commutative theories, and the (not unexpected) non-trivial behaviour thatoccurs when these two length scales become comparable. This may be quantitatively differentin a relativistic treatment, but generically one would again expect non-trivial behaviour whenthe two length scales are comparable. In effective noncommutative systems this may happen atmuch lower temperatures where the nonrelativistic approximation is valid. Acknowledgement
The authors would like to thank the referees for very useful comments.
References [1] A. Connes, Noncommutative Geometry (Academic, New York, 1994).[2] G. Landi, An Introduction to Noncommutative Spaces and Their Geometry (Springer, NewYork, 1997).[3] J. M. Gracia-Bondia, J. C. Varilly, H. Figueroa, Elements Of Noncommutative Geometry(Birkhaeuser, Basel, 2001).[4] L. Mezincescu, Star operation in quantum mechanics, [hep-th/0007046].4 = H Moyal L Α= H Voros L Α= - - - - - - - (cid:144) Λ V (cid:144) k T Α= H Moyal L Α= H Voros L Α= (cid:144) Λ V (cid:144) k T Figure 1: The exchange potential in units of k B T for bosons (a) and fermions (b). The solidcurve represents the commutative result ( α = 0), the short dashed curve is the result for theVoros twist and the long dashed curve the result for the Moyal twist. The value of α = 0 .