Recently Gitman and Vassilevich proposed an interesting model of
noncommutative (NC) scalar field theory [1], with a noncommutativity parameter
assumed to ...
ON NONCOMMUTATIVITY WITH BIFERMIONIC PARAMETER CIPRIAN SORIN ACATRINEI National Institute for Nuclear Physics and Engineering P.O. Box MG-6, 077125 Bucharest, Romania
[email protected] Reveived February 12, 2008
We present an alternative view on the locality of a model recently introduced by Gitman and Vassilevich.
Recently Gitman and Vassilevich proposed an interesting model of noncommutative (NC) scalar field theory [1], with a noncommutativity parameter assumed to be the product of two Grassmann variables. They showed in particular that the model posesses a local energy-momentum tensor. Since such a property is quite unusual for a NC model, we provide here an alternative picture, based on an operatorial formulation of NC field theory [2]. It leads to complete locality of the degrees of freedom of the theory, a property in agreement with the termination of the star-product at the second term in its series. Consider first usual NC field theory. A (2 1) -dimensional scalar )(t xˆ yˆ ), defined over a commuting time t and a pair of NC coordinates which satisfy
[ xˆ yˆ ] iT
(1)
The operators xˆ and yˆ act on a harmonic oscillator Hilbert space
in the
may be given a discrete basis { n } formed by eigenstates of
usual way.
2 xˆ 2 yˆ , or a continuous one { x }, composed of eigenstates of, say, xˆ. To quantize ) [2], start with a usual classical commuting field, expanded into normal modes with coefficients a and a*. Upon usual field quantization, a and a* become operators acting on a standard Fock space . To make the underlying space noncommutative, let us introduce (1) and apply the Weyl (not Weyl-Moyal!) quantization procedure to the exponentials ei ( kx x ky y ) . The result is
)
³³
dk x dk y 2S
G ª ˆ ˆ ˆ kx ky ei ( Zk t kx x ky y ) «a G ¬ 2Zk
º
aˆ †k k ei (Zk t kx xˆ ky yˆ ) » , G
x y
Rom. Journ. Phys., Vol. 53, Nos. 5– 6 , P. 685–687, Bucharest, 2008
¼
(2)
686
Ciprian Sorin Acatrinei
2
which means the following: ) is a ‘doubly’-quantum field operator, acting on a direct product of two Hilbert spaces, )
o
. ) creates (destroys), via i ( Z G t k xˆ k yˆ ) † aˆ k k (aˆ kxky ), an excitation represented by an “operatorial plane wave” e k x y . x y
One could work with ) as an operator ready to act on both and
. It is
however simpler to saturate its action on , working with expectation values x c ) x F o F. . At this point bilocality appears. For, consider the family { x } of eigenstates of xˆ : xˆ x x x , yˆ x iT w x . A simple but key wx equation is x c ei ( kx xˆ ky yˆ ) x
eikx ( x ky T 2) G( x c x k y T) e
ik x x x c 2 G( x c
x k y T)
(3)
This is a bilocal expression, and we already see that its span along the x axis, ( x c x ), is proportional to the momentum along the conjugate y direction, i.e. ( x c x ) Tk y . In general, for n pairs of NC directions, one can keep only one coordinate out of every pair; commutativity is gained on the reduced space, at the expenses of locality. Using Eqs. (2) and (3) one sees that
xc ) x where k y
³
ª dk x i ( ZkG t kx x x c ) i ( ZkG t k x x x c ) º † 2 2 » «a e e ˆ ˆ a k k kx k y » 2 S 2Zkx ky «¬ x y ¼
(4)
( x c x )T. Thus, ) annihilates a linear combination of rods of
(arbitrary) momentum kx and (fixed) length Tky, and creates rods of momentum kx and length –Tky. It is not anymore a local operator, in contrast to usual field theory. Consider now the model of Gitman and Vassilevich, with TPQ having bifermionic character, composed of two fermionic (Grassmann odd) parameters. More precisely, [ x P x Q ] iTPQ (5) with TPQ HP HQ (6) 2 HP being Grassmann odd variables. Consequently TPQ
0 and the star product
terminates after the second term of its expansion, leading to rather mild changes with respect to commutative theories. In particular, a locally conserved energy momentum can be defined. To reinterpret the results of [1] from the point of view of the above operatorial formulation, use (6) in the first equality in Eq. (3). One obtains then instead of (3) the equation
3
On noncommutativity with bifermionic parameter
x c ei ( kx xˆ ky yˆ ) x
eikx (1 ik x k y T)[G( x x c) Tk y w x G( x x c)],
687
(7)
which is not anymore bilocal, but simply local. Upon introducing (7) in the field expansion (2), the first term in (7) leads to a local field, whereas the second term may be seen as an “infinitesimal dipole”, which however ultimately displays local properties. Only a bound state formed by an infinity of such dipoles could be a nonlocal object. The fundamental degrees of freedom are thus local – in total contrast with usual NC field theory [2], and this explains why local quantities could be successfully defined in [1]. The expression resulting instead of (4) is however akward to work with, due to the formal divergences occuring upon integrating over ky. For practical calculations it seems preferable in this case to work with the star-product formulation used in [1]. Acknowledgements. The author was supported by the EU Marie Curie Host Fellowships for Transfer of Knowledge Project COCOS, Correlations in Complex Systems and by NATO Grant PST.EAP.RIG.981202.
REFERENCES 1. D. M. Gitman, D. V. Vassilevich, Space-time noncommutativity with a bifermionic parameter, Publicacao IFUSP-1631/2007, arXiv:hep-th/0701110v3. 2. C. S. Acatrinei, Canonical quantization of noncommutative field theory, Phys. Rev. D67 (2003) 045020.
