On a Lorentz-Invariant Interpretation

of Noncommutative Space-Time

and Its Implications on Noncommutative QFT

M. Chaichian, P. P. Kulish, K. Nishijima and A. Tureanu

High Energy Physics Division, Department of Physical Sciences, University of Helsinki

and

Helsinki Institute of Physics, P.O. Box 64, FIN-00014 Helsinki, Finland

St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia

Nishina Memorial Foundation, 2-28-45 Honkomagome, Bunkyo-ku, Tokyo 113-8941, Japan

Abstract By invoking the concept of twisted Poincaré symmetry of the algebra of functions on a Minkowski space-time, we demonstrate that the noncommutative space-time with the commutation relations , where is a constant real antisymmetric matrix, can be interpreted in a Lorentz-invariant way. The implications of the twisted Poincaré symmetry on QFT on such a space-time is briefly discussed. The presence of the twisted symmetry gives justification to all the previous treatments within NC QFT using Lorentz invariant quantities and the representations of the usual Poincaré symmetry.

PACS: 11.10.Nx

## 1 Introduction

Quantum field theories on noncommutative space-time have been lately thoroughly investigated, especially after it has been shown [1] that they can be obtained as low-energy limits of open string theory in an antisymmetric constant background field (for reviews, see [2], [3]). However, the issue of the lack of Lorentz symmetry has remained a challenge to this moment, since the field theories defined on a space-time with the commutation relation of the coordinate operators

(1.1) |

where is a constant antisymmetric matrix, are obviously not Lorentz-invariant.

In spite of this well-recognized problem, all fundamental issues, like the unitarity [4], causality [5], UV/IR divergences [6], have been discussed in a formally Lorentz invariant approach, using the representations of the usual Poincaré algebra. These results have been achieved using the Weyl-Moyal correspondence, which assigns to every field operator its Weyl symbol defined on the commutative counterpart of the noncommutative space-time. At the same time, this correspondence requires that products of operators are replaced by Moyal -products of their Weyl symbols:

(1.2) |

where the Moyal -product is defined as

(1.3) |

Consequently, the commutators of operators are replaced by Moyal brackets and the equivalent of (1.1) is

(1.4) |

In fact, admitting that noncommutativity should be relevant only at very short distances, the noncommutativity has been often treated as a perturbation and only the corrections to first order in were computed. As a result, the NC QFT was practically considered Lorentz invariant in zeroth order in , with the first order corrections coming only from the -product.

Later the fact that QFT on 4-dimensional NC space-time is invariant under the subgroup of the Lorentz group was used [7] (for several applications, see [8], [9], [10], [11]). However, a serious problem arises from the fact that the representation content of the subgroup is very different from the representation content of the Lorentz group: both and being abelian groups, they have only one-dimensional unitary irreducible representations and thus no spinor, vector etc. representations. In this respect, one encounters a contradiction with previous calculations, in which the representation content for the NC QFT was assumed to be the one of the Poincaré group.

In this letter we shall show that indeed the transformation properties of the NC space-time coordinates can still be regarded as the transformations under the usual Poincaré algebra, with their representation content identical to the one of the commutative case. At the same time, the commutation relation (1.4) appears as the consequence of the noncommutativity of the coproduct (called noncocommutativity) of the twist-deformed (Hopf) Poincaré algebra when acting on the products of the space-time coordinates . As a consequence, the QFT constructed with -product on such a NC space-time, though it explicitly violates the Lorentz invariance, possesses the symmetry under the proper twist-Poincaré algebra.

## 2 Twist deformation of the Poincaré algebra

The usual Poincaré algebra with the generators and
has abelian subalgebra of infinitesimal translations.
Using this subalgebra it is easy to construct a twist element of
the quantum group theory [12] (for detailed explanations, see
the monographs [13], [14]), which permits to deform
the universal enveloping of the Poincaré algebra ^{*}^{*}*For a deformed Poincaré group with twisted classical
algebra, see [15]..

This twist element does not touch the multiplication in , i.e. preserves the corresponding commutation relations among and ,

(2.1) | |||||

(2.2) | |||||

(2.3) |

with the essential physical implication that the representations of the algebra are the same. However, the action of in the tensor product of representations is defined by the coproduct given, in the standard case, by the symmetric map (primitive coproduct)

(2.4) | |||

(2.5) |

for all generators . The twist element changes the coproduct of [12]

(2.6) |

This similarity transformation is consistent with all the
properties of as a Hopf algebra if
satisfies the following twist equation^{†}^{†}†See more detailed
explanations in monographs on quantum groups (e.g. [13],
[14]).:

(2.7) |

Taking the twist element in the form of an abelian twist [16],

(2.8) |

one can check that the twist equation (2.7) is valid.

Since the generators of translations are commutative, their coproduct is not deformed ( is primitive)

(2.9) |

However, the coproduct of the Lorentz algebra generators is changed:

(2.10) |

Using the operator formula and (last
line of (2.1)), we obtain the explicit form of the
coproduct^{‡}^{‡}‡After the submission of the present work to the
hep-th Archive, we were informed that the result (2.11)
appears also in [17], which is an extended version of the
talk given by Julius Wess in the ”Balkan Workshop 2003”.
: and the
commutation relation between

(2.11) | |||||

(2.12) | |||||

(2.13) |

It is known (cf. [13], [18]) that having a representation of a Hopf algebra in an associative algebra consistent with the coproduct of (a Leibniz rule)

(2.14) |

the multiplication in has to be changed after twisting . The new product of consistent with the twisted coproduct is defined as follows: let , then

(2.15) |

where denotes the representation of in , and the action of elements on elements is the same as without twisting.

Let us now consider the commutative algebra of functions, , ,…, depending on coordinates , , in the Minkowski space . In we have the representation of generated by the standard representation of the Poincaré algebra:

(2.16) |

acting on coordinates as follows:

(2.17) |

The Poincaré algebra acts on the Minkowski space , with commutative multiplication:

(2.18) |

When twisting , one has to redefine the multiplication according to (2.15), while retaining the action of the generators of the Poincaré algebra on the coordinates as in (2.17):

(2.19) | |||||

(2.20) |

Specifically, one can now easily compute the commutator of coordinates:

(2.21) | |||||

(2.22) | |||||

(2.23) | |||||

(2.24) |

Hence,

(2.25) |

which is indeed the Moyal bracket (1.4).

## 3 QFT on space-time with twisted Poincaré symmetry

Comparing (1.3) and (2.19) (or equivalently (1.4) and (2.25)), it is obvious that building up the noncommutative quantum field theory through Weyl-Moyal correspondence is equivalent to the procedure of redefining the multiplication of functions, so that it is consistent with the twisted coproduct of the Poincaré generators (2.9), (2.11). The QFT so obtained is invariant under the twisted Poincaré algebra. The benefit of reconsidering NC QFT in the latter approach is that it makes transparent the invariance under the twist-deformed Poincaré algebra, while the first approach highlights the violation of the Lorentz group.

To show this invariance, let us take, as an instructive example, the product . In the standard non-twisted case, the action of the Lorentz generators on this product reads as:

(3.26) |

expressing the fact that is a rank-two Lorentz
tensor. In the twisted case, should be replaced,
according to (2.19), by the symmetrized
expression^{§}^{§}§We use the symmetrization because, due to the
commutation relation (where
is twisted-Poincaré invariant, as shown also in the
consistency check performed below), every tensorial object of the
form can be written as
a sum of symmetric tensors of lower or equal ranks, so that the
basis of the representation algebra is symmetric.
This statement is valid in general in the case of the universal
enveloping algebras of Lie algebras.
, and correspondingly the
action of the Lorentz generator should be applied through the
twisted coproduct:

(3.27) |

In the above equation, denotes the usual Lorentz generator, but with the action of a twisted coproduct. A straightforward calculation gives:

(3.28) |

which is analogous to (3.26), confirming the (expected) covariance under the twisted Poincaré algebra. This argument extends to any symmetrized tensor formed from the -products of ’s. For example, the invariance of Minkowski length is obvious: multiplying (3.28) by , one obtains .

As a consistency check, we shall calculate the action of on the antisymmetric combination :

(3.29) | |||||

(3.30) |

Thus, we have , since , i.e. the antisymmetric tensor is twisted-Poincaré invariant.

Therefore, the Lagrangian obtained by replacing all the usual products of fields in the corresponding commutative theory with -products, though it breaks the Lorentz invariance in the usual sense, it is, however, invariant under the twist-deformed Poincaré algebra.

Another important feature of the QFT with twist-deformed Poincaré symmetry deserves a special highlighting: the representation content of the NC QFT is exactly the same as for its commutative correspondent. It is easy to see that the action of the Pauli-Ljubanski operator, is not changed by the twist (due to the commutativity of the translation generators) and and retain their role of Casimir operators. Consequently, the representations of the twisted Poincaré algebra will be, just as in the commutative case, classified according to the eigenvalues of these invariant operators, and , respectively. Besides justifying the validity of the results obtained so far in NC QFT using the representations of the Poincaré algebra, this aspect will cast a new light on other closely-related fundamental issues, such as the CPT and the spin-statistics theorems in NC QFT [9, 10, 19].

## 4 Conclusions

In this letter we have shown that the quantum field theory on NC space-time possesses symmetry under a twist-deformed Poincaré algebra. The twisted Poincaré symmetry exists provided that: (i) we consider -products among functions instead of the usual one and (ii) we take the proper action of generators specified by the twisted coproduct. As a byproduct with major physical implications, the representation content of NC QFT, invariant under the twist-deformed Poincaré algebra, is identical to the one of the corresponding commutative theory with usual Poincaré symmetry. Some of the applications of the present treatment of the symmetry properties of NC QFT will be considered in a forthcoming communication [20].

Acknowledgements

We are indebted to Peter Prešnajder for many illuminating discussions, remarks and useful suggestions.

The financial support of the Academy of Finland under the Projects No. 54023 and 104368 is greatly acknowledged. The work of PPK was partly supported by the RFBR grant 03-01-00593.

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