Abstract
We discuss the obstruction to the construction of a multiparticle field theory on a $\kappa$-Minkowski noncommutative spacetime: the existence of multilocal functions which respect the deformed symmetries of the problem. This construction is only possible for a light-like version of the commutation relations, if one requires invariance of the tensor product algebra under the coaction of the $\kappa$-Poincar\'e group. This necessitates a braided tensor product. We study the representations of this product, and prove that $\kappa$-Poincar\'e-invariant N-point functions belong to an Abelian subalgebra, and are therefore commutative. We use this construction to define the 2-point Whightman and Pauli--Jordan functions, which turn out to be identical to the undeformed ones. We finally outline how to construct a free scalar $\kappa$-Poincar\'e-invariant quantum field theory, and identify some open problems.
Highlights
The κ-Minkowski spacetime [1,2,3] is a deformation of the algebra of complex-valued functions on Minkowski spacetime, C1⁄2R3;1 into the noncommutative *-algebra A, generated by the coordinate functions1 i1⁄2xμ ;xν 1⁄4 ðvμ xν − vν xμ Þ;κ μ 1⁄4 0;...;3; ðxμ Þ† 1⁄4 xμ ; ð1:1Þ μ μ where v are four arbitrary real numbers, and the x operators generalize the Cartesian coordinate functions.The constant κ has the dimensions of an inverse length, supposedly identified with the Planck energy
N-point functions that do not address the issue of multilocal functions, which we describe in the following
Our result is coherent with the one found by Jurić, Meljanac and Pikutić [59] using a Drinfeld twist. They too obtained that a covariant deformation of the tensor product can only be obtained for the lightlike case. This particular choice for the vector v is remarkable for several other reasons, and the result we just derived makes it the only viable algebra, in the κ-Minkowski family, to construct a well-defined quantum field theory
Summary
The κ-Minkowski spacetime [1,2,3] is a deformation of the algebra of complex-valued functions on Minkowski spacetime, C1⁄2R3;1 into the noncommutative *-algebra A, generated by the coordinate functions i. They too obtained that a covariant deformation of the tensor product can only be obtained for the lightlike case This particular choice for the vector v is remarkable for several other reasons, and the result we just derived makes it the only viable algebra, in the κ-Minkowski family, to construct a well-defined quantum field theory. A QFT on a commutative spacetime (in particular Minkowski) is entirely defined in terms of all N-point functions [61] We can import this definition into our noncommutative setting, the. We already found the algebra that these coordinates close, Eq (3.7), and the main feature we would like to highlight is that the algebra realizes an action of the xμcm generators on the yμa ones, because the xμcm generators close a subalgebra, and their commutators with yνb gives a linear combination of yνb Eq (4.4) in the following form: d4 k1 ...d4 kN fðkaμ Þeiqμ y1 eikμ xcm eiqμ y2 eikμ xcm ...eiqμ yN eikμ xcm d4 k1 ...d4 kN fðkaμ Þeiqμ y1 eiðk ⊳q
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