Abstract
We study the properties of the energy-momentum tensor of gauge fields coupled to matter in non-commutative (Moyal) space. In general, the non-commutativity affects the usual conservation law of the tensor as well as its transformation properties (gauge covariance instead of gauge invariance). It is known that the conservation of the energy-momentum tensor can be achieved by a redefinition involving another star-product. Furthermore, for a pure gauge theory it is always possible to define a gauge invariant energy-momentum tensor by means of a Wilson line. We show that the latter two procedures are incompatible with each other if couplings of gauge fields to matter fields (scalars or fermions) are considered: The gauge invariant tensor (constructed via Wilson line) does not allow for a redefinition assuring its conservation, and vice-versa the introduction of another star-product does not allow for gauge invariance by means of a Wilson line.
Highlights
In view of the infamous time-ordering problems in quantum field theory on Moyal space [19], we restrict ourselves to the Euclidean version of Moyal space
The lack of gauge invariance and local conservation of the EMT is not surprising since the EMT represents, very much like the Lagrangian density, a non-integrated expression and it is only the integral over Moyal space which ensures the cyclic invariance of factors in star-products, and thereby the vanishing of star-commutator terms
We discussed two possible couplings of scalars and fermions to gauge fields corresponding to neutral and charged matter, respectively: In the first case, the basic EMT transforms covariantly and its gauge invariant counterpart could be constructed by using the non-commutative generalization of a Wilson line
Summary
We consider a U (1) gauge field ( Aμ) coupled to an external current (J μ) in four-dimensional flat Euclidean Moyal space: the action. A similar inconsistency occurs in Yang–Mills theory on ordinary commutative space when coupling the gauge field to an external current [21]. The improved EMT for a free (i.e. not coupling to a current) gauge field in Moyal space was already computed in It is symmetric and traceless, and it transforms covariantly under gauge transformations: δλT μν = −ig[T μν , λ]. [23] (see [24]) it was explained how to construct gauge invariant objects in Moyal space out of gauge covariant ones This task is achieved by folding the quantity in question with a straight Wilson line defined by a length vector (lμ) with lμ = θ μνkν ≡ (θ k)μ. We show that this is not the case
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
