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Register a new userkappa-Poincare as a symmetry of flat quantum spacetime
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Authors
J. Kowalski-Glikman
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In this short note, based on the talk given at the 3rd Conference of the Polish Society on Relativity, I present the basic points of our recent paper Symmetries of quantum spacetime in three dimensions, stressing their physical meaning, and avoiding technical details.
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We propose that finite cutoff regions of holographic spacetimes represent quantum circuits that map between boundary states at different times and Wilsonian cutoffs, and that the complexity of those quantum circuits is given by the gravitational action. The optimal circuit minimizes the gravitational action. This is a generalization of both the complexity equals volume conjecture to unoptimized circuits, and path integral optimization to finite cutoffs. Using tools from holographic $Tbar T$, we find that surfaces of constant scalar curvature play a special role in optimizing quantum circuits. We also find an interesting connection of our proposal to kinematic space, and discuss possible circuit representations and gate counting interpretations of the gravitational action.
There is a growing number of physical models, like point particle(s) in 2+1 gravity or Doubly Special Relativity, in which the space of momenta is curved, de Sitter space. We show that for such models the algebra of space-time symmetries possesses a natural Hopf algebra structure. It turns out that this algebra is just the quantum $kappa$-Poincare algebra.
The classical $r$-matrix for $N=1$ superPoincar{e} algebra, given by Lukierski, Nowicki and Sobczyk is used to describe the graded Poisson structure on the $N=1$ Poincar{e} supergroup. The standard correspondence principle between the even (odd) Poisson brackets and (anti)commutators leads to the consistent quantum deformation of the superPoincar{e} group with the deformation parameter $q$ described by fundamental mass parameter $kappa quad (kappa^{-1}=ln{q})$. The $kappa$-deformation of $N=1$ superspace as dual to the $kappa$-deformed supersymmetry algebra is discussed.
Using the methods of ordinary quantum mechanics we study $kappa$-Minkowski space as a quantum space described by noncommuting self-adjoint operators, following and enlarging arXiv:1811.08409. We see how the role of Fourier transforms is played in this case by Mellin transforms. We briefly discuss the role of transformations and observers.
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 $kappa$-Poincare group. This necessitates a braided tensor product. We study the representations of this product, and prove that $kappa$-Poincare-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 $kappa$-Poincare-invariant quantum field theory, and identify some open problems.
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