Generalizing the Noether Theorem for Hopf-Algebra Spacetime Symmetries

Abstract

Over these past few years several quantum-gravity research groups have been exploring the possibility that in some Planck-scale nonclassical descriptions of spacetime one or another form of nonclassical spacetime symmetries might arise. One of the most studied scenarios is based on the use of Hopf algebras, but previous attempts were not successful in deriving constructively the properties of the conserved charges one would like to obtain from the Hopf structure, and this in turn did not allow a crisp physical characterization of the new concept of spacetime symmetry. Working within the example of κ-Minkowski noncommutative spacetime, known to be particularly troublesome from this perspective, we observe that these past failures in the search of the charges originated from not recognizing the crucial role that the noncommutative differential calculus plays in the symmetry analysis. We show that, if the properties of the κ-Minkowski differential calculus are correctly taken into account, one can easily perform all the steps of the Noether analysis and obtain an explicit formula relating fields and energy-momentum charges. Our derivation also exposes the fact that an apparent source of physical ambiguity in the description of the Hopf-algebra rules of action, which was much emphasized in the literature, actually only amounts to a choice of conventions and in particular does not affect the formulas for the charges.1 The space indices j, l take values in {1, 2, 3} while 0 is the time index. We shall later also use the spacetime indices µ, ν, α, which take values in {0, 1, 2, 3}.2 Rather than our length scale λ a majority of authors use the energy scale κ, which is the inverse of λ (λ → 1/κ). 3 One notices that κ-Minkowski and the κ-Poincaré Hopf algebra form a "Heisenberg double" [8,9], i.e. κ-Minkowski and κ-Poincaré are linked, as algebras, in a way that is rather similar to the relationship between classical Minkowski spacetime and the classical Poincaré Lie algebra.

Twisted statistics inκ-Minkowski spacetime

“…For λ = 1 2 , the same factor appears on the left and right hand side symmetrically [28]. In this sense it can be named as symmetric ordering, but it should be kept in mind that it is completely different from the totally symmetric Weyl ordering.…”

Covariant realizations of kappa-deformed space

“…Although a single space may be realized in many different ways, physical results do not depend on concrete realizations, i.e. orderings (see also [24]). …”

Hidden quantum groups symmetry of super-renormalizable gravity

“…It is well known in literature that the algebra of symmetries for the non-commutative space-time in (24) is a twisted Hopf algebra P θ called θ-Poincaré Hopf algebra, and that a theory with a covariant action gives rise in this framework to conserved Noether charges (see e.g. [37], which extends the work done in [38] for the case of the κ-Poincaré Hopf algebra). Therefore θ-Poincaré truly represents an external symmetry (Hopf) algebra, and the same holds for its Lorentz sub-algebra.…”

“…Indeed, one might argue that the θ ∼ 1/Λ 2 modifications to the co-algebraic sector induces a θ modification to the many-particle states at the quantum level, and establishing firmly this point would open the path to the study of entanglement effects, as studied in the last one of Refs. [38] for the case of the κ-Poincaré algebra. Finally, a clear understanding of the non-commutative space-time reformulation of the models studied in may be derived through a comparison of the interacting terms contained in the SRQG theory at the perturbative level with an appropriate expansion of a selected non-commutative theory of gravity, which may be described by [39] or some other model.…”

“…This latter reflects Leibnitz rule), leaving unchanged the algebraic structure of the Hopf-algebra 4 . In other words, the enlargement of the symmetry structure we are dealing with, will be affect a modification of the Fock space of the theory (see for instance the last reference of [38]), without chancing the Hilbert space of the theory. This is also reflected in the fact that the quantum groups structure emerges at tree-level, though the evaluation of the graviton propagator.…”