Born reciprocity and the granularity of space-time

Abstract

The Schrödinger-Robertson inequality for relativistic position and momentum operators X^μ, P_ν, μ, ν= 0,1,2,3, is interpreted in terms of Born reciprocity and `non-commutative' relativistic phase space geometry. For states which saturate the Schrödinger-Robertson inequality, a typology of semiclassical limits is pointed out, characterised by the orbit structure within its unitary irreducible representations, of the full invariance group of Born reciprocity, the so-called `quaplectic' group U(3,1)xH(3,1) (the semi-direct product of the unitary relativistic dyamical symmetry U(3,1) with the Weyl-Heisenberg group H(3,1)). The example of the `scalar' case, namely the relativistic oscillator, and associated multimode squeezed states, is treated in detail. In this case,it is suggested that the semiclassical limit corresponds to the separate emergence of space-time and matter, in the form of the stress-energy tensor, and the quadrupole tensor, which are in general reciprocally equivalent.

arXiv comment12 pages, LaTeX. Minor wording changes and references updated

References in corpus (5)

• Robertson Intelligent States
• Representations of the Canonical group, (the semi-direct product of the Unitary and Weyl-Heisenberg groups), acting as a dynamical group on noncommuting extended phase space
• Reciprocal relativity of noninertial frames and the quaplectic group
• Mixed symmetry gauge fields in a flat background
• Poincare and Heisenberg quantum dynamical symmetry: Casimir invariant field equations of the quaplectic group

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• Reciprocal relativity of noninertial frames: quantum mechanics
• Gouy Phase for Relativistic Quantum Particles
• Inverse momentum expectation values for hydrogenic systems
• Born reciprocity and the 1/r potential
• Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics
• Constraint quantisation of a worldline system invariant under reciprocal relativity. II
• Relativity implications of the quantum phase
• A Robertson-type Uncertainty Principle and Quantum Fisher Information