On a Lorentz-Invariant Interpretation of Noncommutative Space-Time and Its Implications on Noncommutative QFT

By invoking the concept of twisted Poincar\' e symmetry of the algebra of functions on a Minkowski space-time, we demonstrate that the noncommutative space-time with the commutation relations $[x_\mu,x_\nu]=i\theta_{\mu\nu}$, where $\theta_{\mu\nu}$ is a {\it constant} real antisymmetric matrix, can be interpreted in a Lorentz-invariant way. The implications of the twisted Poincar\'e 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 Poincar\'e symmetry.


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 where θ µν 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 φ(x) its Weyl symbol φ(x) defined on the commutative counterpart of the noncommutative spacetime. At the same time, this correspondence requires that products of operators are replaced by Moyal ⋆-products of their Weyl symbols: where the Moyal ⋆-product is defined as Consequently, the commutators of operators are replaced by Moyal brackets and the equiv- 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 θ were computed. As a result, the NC QFT was practically considered Lorentz invariant in zeroth order in θ µν , 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 SO(1, 1)× SO(2) 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 2 Twist deformation of the Poincaré algebra The usual Poincaré algebra P with the generators M µν and P α 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 U(P) * .
This twist element F ∈ U(P) ⊗ U(P) does not touch the multiplication in U(P), i.e.
preserves the corresponding commutation relations among M µν and P α , with the essential physical implication that the representations of the algebra U(P) are the same. However, the action of U(P) in the tensor product of representations is defined by the coproduct given, in the standard case, by the symmetric map (primitive coproduct) for all generators Y ∈ P. The twist element F changes the coproduct of U(P) [12] ∆ 3) * For a deformed Poincaré group with twisted classical algebra, see [15].

4
This similarity transformation is consistent with all the properties of U(P) as a Hopf algebra if F satisfies the following twist equation † : Taking the twist element in the form of an abelian twist [16], Since the generators of translations P α are commutative, their coproduct is not deformed However, the coproduct of the Lorentz algebra generators is changed: (AdB) n n! C and the commutation relation between M µν and P α (last line of (2.1)), we obtain the explicit form of the coproduct ‡ ∆ t (M µν ): appears also in [17], which is an extended version of the talk given by Julius Wess in the "Balkan Workshop (2.8) It is known (cf. [13], [18]) that having a representation of a Hopf algebra H in an associative algebra A consistent with the coproduct ∆ of H (a Leibniz rule) the multiplication in A has to be changed after twisting H. The new product of A consistent with the twisted coproduct ∆ t is defined as follows: let F = f 1 ⊗ f 2 , then acting on coordinates as follows: The Poincaré algebra acts on the Minkowski space x µ , µ = 0, 1, 2, 3 with commutative multiplication: When twisting U(P), one has to redefine the multiplication according to (2.10), while retaining the action of the generators of the Poincaré algebra on the coordinates as in (2.12): ) . (2.14) Specifically, one can now easily compute the commutator of coordinates: To show this invariance, let us take, as an instructive example, the product f ρσ (x) = x ρ x σ .
In the standard non-twisted case, the action of the Lorentz generators on this product reads as: expressing the fact that f ρσ is a rank-two Lorentz tensor. In the twisted case, f ρσ should be replaced, according to (2.14), by the symmetrized expression , and correspondingly the action of the Lorentz generator should be applied through the twisted coproduct: In the above equation, M t µν denotes the usual Lorentz generator, but with the action of a twisted coproduct. A straightforward calculation gives: which is analogous to (3.17), confirming the (expected) covariance under the twisted Poincaré algebra. This argument extends to any symmetrized tensor formed from the ⋆-products of x's. For example, the invariance of Minkowski length s 2 t = x µ ⋆ x µ = x µ x µ is obvious: multiplying (3.19) by η ρσ , one obtains M t µν s 2 t = 0. § We use the symmetrization because, due to the commutation relation [x µ , x ν ] ⋆ = iθ µν (where θ µν is twisted-Poincaré invariant, as shown also in the consistency check performed below), every tensorial object of the form x µ ⋆ x ν ⋆ · · · ⋆ x σ can be written as a sum of symmetric tensors of lower or equal ranks, so that the basis of the representation algebra A t is symmetric. This statement is valid in general in the case of the universal enveloping algebras of Lie algebras.
As a consistency check, we shall calculate the action of M t µν on the antisymmetric com- Thus, we have M t µν θ ρσ = 0, since θ ρσ = −i[x ρ , x σ ] ⋆ , i.e. the antisymmetric tensor θ ρσ 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, W α = − 1 2 ǫ αβγδ M βγ P δ is not changed by the twist (due to the commutativity of the translation generators) and P 2 and W 2 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, m 2 and m 2 s(s + 1), 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 closelyrelated fundamental issues, such as the CPT and the spin-statistics theorems in NC QFT [9,10,19]. 9

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].