# On the energy-momentum tensor in Moyal space

- 451 Downloads
- 2 Citations

## 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 well 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 last 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.

### Keywords

Gauge Transformation Wilson Line Gauge Field Lagrangian Density Scalar Field Model## 1 Introduction

In general, field theoretic models on such spaces suffer from a new type of divergences which arise due to a phenomenon referred to as UV/IR mixing [7, 8] and which render the models non-renormalizable. This problem can be overcome in the case of some special scalar field models [9, 10, 11], one of them having been shown to be solvable even non-perturbatively [12].

The present work is devoted to a basic aspect of classical field theories on Moyal space, namely the *energy-momentum tensor* (hereafter referred to as *EMT*) and its properties at tree level. In earlier studies [13, 14, 15, 16, 17, 18] some modifications to the conservation law of the EMT due to the non-commutativity parameters \(\theta ^{\mu \nu }\) were found in \(\phi ^{\star 4}\) and in gauge theory (without matter couplings). Here, we wish to investigate more generally complex scalars and fermions coupled to \(U_{\star }(1)\) gauge fields. 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 present work is organized as follows. In Sect. 2, we examine the gauge invariance and conservation properties of the EMT for a gauge field in Moyal space. In Sects. 3 and 4 we then extend the discussion to include various couplings to matter.

## 2 EMT for a gauge field in Moyal space

^{1}Moyal space: the action

*S*[

*A*] is invariant under the infinitesimal gauge transformations

Concerning the transformation laws (5) we emphasize that, by contrast to a *U*(1) gauge theory in ordinary Minkowski space, the field strength \(F_{\mu \nu }\) is not a gauge invariant quantity as in electrodynamics. This non-Abelian nature of the theory in Moyal space is due to the non-commutativity of space-time coordinates, which implies that the field strength “feels” the non-commutativity of the space in which it lives. (This even applies to the simplest case of a constant field strength [22].) The transformation law of \(F_{\mu \nu }\) implies that the Lagrangian density \(\mathcal{L} = \frac{1}{4} \, F_{\mu \nu }\star F^{\mu \nu }\) is *not* invariant under gauge transformations since \(\delta _\lambda \mathcal{L} = -\text {i}g[ \mathcal{L}\mathop {,}\limits ^{\star }\lambda ] \): it is only the integral which plays the role of a trace which ensures cyclic invariance of factors and thereby gauge invariance. Henceforth, the lack of gauge invariance of the EMT will not come as a surprise and contrasts the situation for non-Abelian Yang–Mills fields in Minkowski space.

Let us now come back to the local transformation law (8). In Ref. [23] (see also [24]) it was explained how to construct gauge invariant objects in Moyal space out of gauge covariant ones. In fact, this task is achieved by folding the quantity in question with a straight Wilson line defined by a length vector \((l^\mu )\) with \(l^\mu =\theta ^{\mu \nu }k_\nu \equiv (\theta k)^\mu \). Using this procedure, the authors of Ref. [16] obtained a standard local conservation law for the so constructed EMT. In the following we will also follow this strategy for gauge fields, scalars and fermions, and therefore we briefly outline the procedure here.

*U*(

*x*). Hence, \(\int \! \mathrm{d}^4xW(k,x)\star \exp (\text {i}kx)\) is a gauge invariant object because the length vector of the Wilson line is adjusted to be \(\theta ^{\mu \nu }k_\nu \) and \(\exp (\text {i}kx)\) induces a translation of \(U^\dagger \) by \(-\theta k\), cf. [23, 25]. One may now construct (Fourier transforms of) gauge invariant objects from gauge covariant ones by star-multiplication with

*W*(

*k*,

*x*) and \(\exp (\text {i}kx)\) and integrating over \(\mathrm{d}^4x\). The choice of a straight Wilson line is the most natural one because for such a line it makes no difference if the operator is attached to an endpoint of the Wilson line or somewhere in the middle [23]. Furthermore, in the commutative limit \((\theta \rightarrow 0 )\) the Wilson line’s length goes to zero.

^{2}(which reduces in the commutative limit to the ordinary EMT due to \(\lim \nolimits _{\theta \rightarrow 0} W(k,x)=1\)). However, it is not conserved [16],

## 3 Coupling to neutral matter fields

One of the peculiarities of non-commutative space is that even neutral matter (such as neutrinos) can couple to \(U_\star (1)\) gauge fields (photons) via star-commutators [26, 27, 28], i.e. the matter fields can transform with the adjoint representation [see Eq. (15) below] just like the gauge fields. In the following, we study the EMT of such neutral fields before discussing charged fields in Sect. 4.

### 3.1 Complex scalar field

**Scalar field action:**We consider an external \(U_\star (1)\) gauge field \((A^\mu )\) and a complex scalar field \(\phi \) in the

*adjoint representation,*i.e. the infinitesimal gauge transformations read

The gauge transformation laws (15) imply that the covariant derivatives of \(\phi \) and \(\phi ^*\) also transform covariantly, i.e. \(\delta _\lambda (D_\mu \phi )=-\text {i}g[ D_\mu \phi \mathop {,}\limits ^{\star }\lambda ] \) and analogously for \(D_\mu \phi ^*\). It follows that the Lagrangian density \(\mathcal{L}\) in the action integral (16) transforms as \(\delta _\lambda \mathcal{L} =-\text {i}g[ \mathcal{L}\mathop {,}\limits ^{\star }\lambda ] \) so that the action is gauge invariant. A short calculation using the Jacobi identity shows that the matter current (18) also transforms covariantly, \(\delta _\lambda J_\mu =-\text {i}g[ J_\mu \mathop {,}\limits ^{\star }\lambda ] \).

**Addition of the gauge field action:**If we add to the action (16) a kinetic term for the gauge field so as to obtain the total action

**Restoring gauge invariance:**In the present setting, we may follow the same strategy as in Sect. 2 and define an EMT \(\tilde{T}^{\mu \nu }\) which is gauge invariant in analogy to expression (12):

In Ref. [16] a redefinition of the EMT for a \(\phi ^{\star 4}\) theory which implies its conservation was discussed. However, in the present context the same strategy would destroy gauge covariance of \(T^{\mu \nu }_{\text {tot}}\) making the construction of its gauge invariant counterpart via Wilson line impossible, as we will now show.

Thus, the best one can do is to use the gauge invariant expression (30) above with its modified conservation law (31). On the operator level, this means that only the trace over the divergence of the EMT (which is an operator in quantized space) is conserved, a fact which is obscured by the star-product prescription where the trace becomes an integral over space(-time). Therefore, it is not surprising that we have the local equation \(\partial _\mu \tilde{T}^{\mu \nu }\ne 0\), as was already observed in the case of the \(\phi ^{\star 4}\)-theory in Ref. [15]. We note that \(\int \!\mathrm{d}^4y\partial ^y_\mu \tilde{T}^{\mu \nu }_{\text {tot}}(y)=0=\int \! \mathrm{d}^4x\partial _\mu ^xT^{\mu \nu }_{\text {tot}}(x)\) as can be checked explicitly by using the cyclic properties of the star-product under the integral, as well as \(\int \!\mathrm{d}^4k\exp (\text {i}ky)=(2\pi )^4\delta ^4(k)\) and \(W(0,x)=1\).

In the next section, we show that one finds similar results when coupling to fermions instead of (or in addition to) scalars.

### 3.2 Fermions

*adjoint representation*with the gauge transformation properties

^{3}

^{4}(cf. Appendix A.3 of Ref. [35]).

Once combined with the gauge field EMT as discussed in the previous subsection, one may again define the gauge invariant counterpart of the total EMT via Wilson line as in Eq. (30). Once more \(\partial _\mu \tilde{T}^{\mu \nu }_{\text {tot}}\) involves breaking terms which depend on the non-commutativity parameters \(\theta ^{\mu \nu }\) and which cannot be absorbed into a redefined EMT in a gauge invariant way.

## 4 Coupling to charged matter fields

In this section, we discuss the coupling of gauge fields to charged scalars and fermions. In this case the EMT is found to be gauge invariant in general so that one does not have to resort to Wilson lines. Nonetheless we will not have the standard local conservation law for this EMT and the latter cannot be achieved while maintaining gauge invariance.

### 4.1 Complex scalar field

#### 4.1.1 Fundamental representation

*fundamental representation*, i.e. we have the transformation rules

*A*, the action is now defined by

^{5}(49).

^{6}due to a missing trace/integral (typical of non-commutative space) even in this simpler scalar field model. By adding the gauge field action and integrating with \(\int \mathrm{d}^3x\) in Minkowskian Moyal space with \(\theta ^{0i} =0\), we again find a conserved and gauge invariant four-momentum \((P^\nu )\) by virtue of Eqs. (28) and (52).

#### 4.1.2 Antifundamental representation

*antifundamental representation:*

### 4.2 Fermions

#### 4.2.1 Fundamental representation

*fundamental representation*[36, 37] (rather than the adjoint as in Sect. 3.2), the transformation laws being given by

#### 4.2.2 Antifundamental representation

*antifundamental representation*:

## 5 Conclusion

According to the non-commutative generalization of Noether’s theorem [29], some extra \(\theta \)-dependent terms (“source”/star-commutator terms) generally appear in the local conservation law for the EMT for interacting theories. 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.

In the present paper, we have explicitly shown (for complex scalars as well as for fermions coupled to gauge fields) that the standard local conservation law of the EMT \(T^{\mu \nu }\) is always modified due to non-commutative effects and that \(T^{\mu \nu }\) can always be redefined so as to be conserved, but that the so defined EMT is not gauge invariant. (Yet, for dynamical matter and gauge fields we always have a conserved and gauge invariant four-momentum with components \(P^{\nu } = \int \mathrm{d}^3x \, T^{0\nu }\).)

More specifically, 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. In the second case (for which there exist two variants, namely the fundamental and the antifundamental representations), the freedom in the definition of the EMT allows for the choice of a gauge covariant or a gauge invariant tensor. For all cases we found that the consideration of the \(\star '\)-product allows one to achieve the standard local conservation law for the EMT, but at the expense of losing gauge invariance (and symmetry). We note that the tools employed here are also those which are generally considered for the quantization; e.g. see Refs. [23, 30, 38].

Our systematic study is tantamount to a proof that it is not possible to construct a conserved and gauge invariant (and symmetric) EMT for spin 0 and spin 1 / 2 matter fields coupled to a \(U_{\star }(1)\) gauge field in Moyal space.^{7} Yet, in all cases the total energy-momentum \(P^\nu \equiv \int \mathrm{d}^3x \, T^{0\nu }_{\mathrm{tot}}\) of the system represents a conserved and gauge invariant quantity. In practice, the formulation of classical as well as quantum field theories in flat space primarily relies on the conserved charges \(P^\nu \), so that the problematic properties of the EMT that we discussed can somehow be circumvented. However, the situation is quite different in curved space, where one has to couple the EMT to a metric field while taking into account the related non-commutativities.

## Footnotes

- 1.
We recall that certain signs (e.g. some global signs in the actions) change upon passage from Minkowskian to Euclidean signature. The coupling constant is denoted by

*g*and there should be no risk of confusion with the determinant of the metric tensor \((g_{\mu \nu })\) considered for defining the Einstein–Hilbert EMT. - 2.
When considering \(U_\star (N)\) gauge fields rather than \(U_\star (1)\) fields as we do here, an additional trace appears in the product, i.e. \(W\star T\) becomes \(\text {tr}\, (W\star T)\).

- 3.
Its free counterpart (with coupling \(g=0\)) was previously constructed in Ref. [32] where the Sugawara form of the EMT for a free fermion in Moyal space was also established.

- 4.
- 5.
The EMT \(T'^{\mu \nu }\) corresponding to the Lagrangian \(\mathcal{L}' \equiv ( {\bar{D}}^\mu \phi )\star ({\bar{D}}_\mu ^*\phi ^*)\), i.e. \(T'^{\mu \nu }=( {\bar{D}}^\mu \phi ) \star ({\bar{D}}^{*\,\nu }\phi ^* )+ ( \mu \leftrightarrow \nu ) - \delta ^{\mu \nu } ( {\bar{D}}^\rho \phi ) \star ( {\bar{D}}^*_\rho \phi ^* )\) transforms covariantly under a gauge transformation (as does the one corresponding to the Lagrangian given by an anticommutator). Furthermore, its covariant derivative produces additional commutator terms similar to the ones present on the r.h.s. of Eq. (54) below.

- 6.
Note that \(F^{\mu \nu }\star J_\mu = \frac{1}{2} \, \{ F^{\mu \nu }\mathop {,}\limits ^{\star }J_\mu \} + \frac{1}{2} \, [ F^{\mu \nu }\mathop {,}\limits ^{\star }J_\mu ] \) where the first term is the opposite of the covariant divergence of the gauge field EMT and where the second term is a star-commutator.

- 7.
An indirect cure of the problems for the case of neutral scalars appears to be the passage to the matrix model framework since these scalars appear naturally as extra dimensions in this framework and the extra terms we found in the conservation law of the EMT are not present there due to internal symmetries [33, 34].

## Notes

### Acknowledgments

D. B. wishes to thank H. Steinacker for pointing out to us Ref. [35]. Furthermore, we wish to thank the anonymous referees for their valuable comments which helped us to clarify several points.

### References

- 1.H.J. Groenewold, On the principles of elementary quantum mechanics. Physica
**12**, 405–460 (1946)MathSciNetADSCrossRefMATHGoogle Scholar - 2.J.E. Moyal, Quantum mechanics as a statistical theory. Proc. Camb. Philos. Soc.
**45**, 99–124 (1949)MathSciNetADSCrossRefMATHGoogle Scholar - 3.R.J. Szabo, Quantum field theory on noncommutative spaces. Phys. Rep.
**378**, 207–299 (2003). arXiv:hep-th/0109162 - 4.V. Rivasseau, Non-commutative renormalization, in
*Séminaire Poincaré X (2007)—Espaces Quantiques*, ed. by B. Duplantier, V. Rivasseau (Birkhäuser, Boston, 2007). arXiv:0705.0705 [hep-th] - 5.D.N. Blaschke, E. Kronberger, R.I.P. Sedmik, M. Wohlgenannt, Gauge theories on deformed spaces. SIGMA
**6**, 062 (2010). arXiv:1004.2127 [hep-th] - 6.H. Grosse, G. Lechner, T. Ludwig, R. Verch, Wick rotation for quantum field theories on degenerate Moyal space(–time). J. Math. Phys.
**54**, 022307 (2013). arXiv:1111.6856 [hep-th] - 7.S. Minwalla, M. Van Raamsdonk, N. Seiberg, Noncommutative perturbative dynamics. JHEP
**02**, 020 (2000). arXiv:hep-th/9912072 - 8.A. Matusis, L. Susskind, N. Toumbas, The IR/UV connection in the non-commutative gauge theories. JHEP
**12**, 002 (2000). arXiv:hep-th/0002075 - 9.H. Grosse, R. Wulkenhaar, Renormalisation of \(\phi ^4\) theory on noncommutative \(\mathbb{R}^2\) in the matrix base. JHEP
**12**, 019 (2003). arXiv:hep-th/0307017 - 10.H. Grosse, R. Wulkenhaar, Renormalisation of \(\phi ^4\) theory on noncommutative \(\mathbb{R}^4\) in the matrix base. Commun. Math. Phys.
**256**, 305–374 (2005). arXiv:hep-th/0401128 - 11.R. Gurau, J. Magnen, V. Rivasseau, A. Tanasa, A translation-invariant renormalizable non-commutative scalar model. Commun. Math. Phys.
**287**, 275–290 (2009). arXiv:0802.0791 [math-ph] - 12.H. Grosse, R. Wulkenhaar, Self-dual noncommutative \(\phi ^4\) -theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory. Commun. Math. Phys.
**329**, 1069–1130 (2014). arXiv:1205.0465 [math-ph] - 13.A. Micu, M.M. Sheikh Jabbari, Noncommutative \(\Phi ^4\) theory at two loops. JHEP
**01**, 025 (2001). arXiv:hep-th/0008057 - 14.T. Pengpan, X. Xiong, A note on the noncommutative Wess–Zumino model. Phys. Rev. D
**63**, 085012 (2001). arXiv:hep-th/0009070 - 15.A. Gerhold, J. Grimstrup, H. Grosse, L. Popp, M. Schweda, R. Wulkenhaar, The energy-momentum tensor on noncommutative spaces: some pedagogical comments. arXiv:hep-th/0012112
- 16.M. Abou-Zeid, H. Dorn, Comments on the energy momentum tensor in noncommutative field theories. Phys. Lett. B
**514**, 183–188 (2001). arXiv:hep-th/0104244 - 17.J.M. Grimstrup, B. Kloibock, L. Popp, V. Putz, M. Schweda, M. Wickenhauser, The energy momentum tensor in noncommutative gauge field models. Int. J. Mod. Phys. A
**19**, 5615–5624 (2004). arXiv:hep-th/0210288 - 18.A.K. Das, J. Frenkel, On the energy momentum tensor in noncommutative gauge theories. Phys. Rev. D
**67**, 067701 (2003). arXiv:hep-th/0212122 - 19.D. Bahns, S. Doplicher, K. Fredenhagen, G. Piacitelli, On the unitarity problem in space/time noncommutative theories. Phys. Lett. B
**533**, 178–181 (2002). arXiv:hep-th/0201222 - 20.T.C. Adorno, D.M. Gitman, A.E. Shabad, D.V. Vassilevich, Classical noncommutative electrodynamics with external source. Phys. Rev. D
**84**, 065003 (2011). arXiv:1106.0639 [hep-th] - 21.P. Sikivie, N. Weiss, Classical Yang–Mills theory in the presence of external sources. Phys. Rev. D
**18**, 3809 (1978)ADSCrossRefGoogle Scholar - 22.H. Balasin, D.N. Blaschke, F. Gieres, M. Schweda, Wong’s equations and charged relativistic particles in non-commutative space. SIGMA
**10**, 099 (2014). arXiv:1403.0255 [hep-th] - 23.D.J. Gross, A. Hashimoto, N. Itzhaki, Observables of noncommutative gauge theories. Adv. Theor. Math. Phys.
**4**, 893–928 (2000). arXiv:hep-th/0008075 - 24.D. Berenstein, R.G. Leigh, Observations on noncommutative field theories in coordinate space. arXiv:hep-th/0102158
- 25.N. Ishibashi, S. Iso, H. Kawai, Y. Kitazawa, Wilson loops in noncommutative Yang–Mills. Nucl. Phys. B
**573**, 573–593 (2000). arXiv:hep-th/9910004 - 26.M. Hayakawa, Perturbative analysis on infrared aspects of noncommutative QED on \(\mathbb{R}^4\). Phys. Lett. B
**478**, 394–400 (2000). arXiv:hep-th/9912094 - 27.M. Chaichian, P. Prešnajder, M.M. Sheikh-Jabbari, A. Tureanu, Noncommutative gauge field theories: a no-go theorem. Phys. Lett. B
**526**, 132–136 (2002). arXiv:hep-th/0107037 - 28.P. Schupp, J. Trampetic, J. Wess, G. Raffelt, The photon neutrino interaction in non-commutative gauge field theory and astrophysical bounds. Eur. Phys. J. C
**36**, 405–410 (2004). arXiv:hep-ph/0212292 - 29.J. Zahn, Wirkungs- und Lokalitätsprinzip für nichtkommutative skalare Feldtheorien. Master’s thesis, Universität Hamburg (2003)Google Scholar
- 30.H. Liu, J. Michelson, \(*\)-Trek: the one loop \(N=4\) noncommutative SYM action. Nucl. Phys. B
**614**, 279–304 (2001). arXiv:hep-th/0008205 - 31.T. Mehen, M.B. Wise, Generalized \(\star \)-products, Wilson lines and the solution of the Seiberg–Witten equations. JHEP
**12**, 008 (2000). arXiv:hep-th/0010204 - 32.M. Ghasemkhani, Noncommutative Sugawara construction. arXiv:1407.3831 [hep-th]
- 33.Y. Okawa, H. Ooguri, Energy momentum tensors in matrix theory and in noncommutative gauge theories. arXiv:hep-th/0103124
- 34.H. Steinacker, Emergent gravity and noncommutative branes from Yang–Mills matrix models. Nucl. Phys. B
**810**, 1–39 (2009). arXiv:0806.2032 [hep-th] - 35.A. Polychronakos, H. Steinacker, J. Zahn, Brane compactifications and 4-dimensional geometry in the IKKT model. Nucl. Phys. B
**875**, 566–598 (2013). arXiv:1302.3707 [hep-th] - 36.J.M. Gracia-Bondia, C.P. Martín, Chiral gauge anomalies on noncommutative \(\mathbb{R}^4\). Phys. Lett. B
**479**, 321–328 (2000). arXiv:hep-th/0002171 - 37.E.F. Moreno, F.A. Schaposnik, Wess–Zumino–Witten and fermion models in noncommutative space. Nucl. Phys. B
**596**, 439–458 (2001). arXiv:hep-th/0008118 - 38.S. Bellucci, I. Buchbinder, V. Krykhtin, Renormalization of the energy momentum tensor in noncommutative scalar field theory. Nucl. Phys. B
**665**, 402–424 (2003). arXiv:hep-th/0303186

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (
http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}.