Generalized Quantum Relativistic Kinematics: a Stability Point of View

Abstract

We apply Lie algebra deformation theory to the problem of identifying the stable form of the quantum relativistic kinematical algebra. As a warm up, given Galileo's conception of spacetime as input, some modest computer code we wrote zeroes in on the Poincare-plus-Heisenberg algebra in about a minute. Further ahead, along the same path, lies a three dimensional deformation space, with an instability double cone through its origin. We give physical as well as geometrical arguments supporting our view that moment, rather than position operators, should enter as generators in the Lie algebra. With this identification, the deformation parameters give rise to invariant length and mass scales. Moreover, standard quantum relativistic kinematics of massive, spinless particles corresponds to non-commuting moment operators, a purely quantum effect that bears no relation to spacetime non-commutativity, in sharp contrast to earlier interpretations.

26 pages, 2 figures

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• Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
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• Geometry, stochastic calculus and quantum fields in a non-commutative space-time
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• Lie Superalgebra Stability and Branes

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