Dirac spinors for doubly special relativity and  -Minkowski noncommutative spacetime

Abstract

We construct a Dirac equation that is consistent with one of the recently-proposed schemes for a "doubly-special relativity", a relativity with both an observer-independent velocity scale (still naturally identified with the speed-of-light constant) and an observer-independent length/momentum scale (possibly given by the Planck length/momentum). We find that the introduction of the second observer-independent scale only induces a mild deformation of the structure of Dirac spinors. We also show that our modified Dirac equation naturally arises in constructing a Dirac equation in the κ-Minkowski noncommutative spacetime. Previous, more heuristic, studies had already argued for a possible role of doubly-special relativity in κ-Minkowski, but remained vague on the nature of the consistency requirements that should be implemented in order to assure the observer-independence of the two scales. We find that a key role is played by the choice of a differential calculus in κ-Minkowski. A much-studied choice of the differential calculus does lead to our doublyspecial relativity Dirac equation, but a different scenario is encountered for another popular choice of differential calculus.

A freely falling frame at the interface of gravitational and quantum realms

“…by Agostini, Amelino-Camelia, and Arzano in [39] but my interpretation differs dramatically. In what follows, I also incorporate an important phase factor, not appreciated in [39].…”

“…In what follows, I also incorporate an important phase factor, not appreciated in [39]. The neglect of that phase amounts to projecting out antiparticles from NSR-2s as I have already noted in [42].…”

“…For this reason, the O(ℓ P ) departures given in [39] are artifacts of the used variables. The same holds true for the O(ℓ P ) corrections at the level of vector potential given in [40].…”

Topics in Noncommutative Geometry Inspired Physics

“…Furthermore the NC quantum theories are qualitatively different with an inherent nonlocality that obviously can not be captured in the canonical set up which, however, provides a very convenient framework to construct the quantum theory. An explicit example is given [177] where we construct the DSR generalization of Dirac fermions in a very simple way as compared to the original derivation [178].…”

Some Encouraging and Some Cautionary Remarks on Doubly Special Relativity in Quantum Gravity

“…For example in the description of fields in κ-Minkowski one must of course introduce a differential calculus, for which various alternatives have been proposed in the κ-Minkowski literature. In particular there has been interest in a four-dimensional [22] and in a fivedimensional [23] differential calculus, and both of these differential calculi deserve equal consideration at the mathematics level, but if one insists on compatibility with the DSR principles only the five-dimensional differential calculus turns out [24] to be acceptable.…”