Born Reciprocity and the Granularity of Spacetime

Abstract

The Schrödinger-Robertson inequality for relativistic position and momentum operators X µ , Pν , µ, ν = 0, 1, 2, 3 , is interpreted in terms of Born reciprocity and 'non-commutative' relativistic positionmomentum 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) ⊗s H(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.

Born Reciprocity and the 1/r Potential

“…It is explicitly non-relativistic and should at least be made to conform with special relativity, To carry out that program sensibly one would need to study the quaplectic group [11][12][13] and augment the electromagnetic or gravitational field (photon/graviton exchange) by their reciprocity analogues or some other contributions. This signifies overhauling the whole of standard field theory and the task seems rather difficult, if not vague, at this stage.…”

A Dynamical Uncertainty Principle in von Neumann Algebras by Operator Monotone Functions

“…The relevance of the above inequality in quantum physics is well-known and we refer the reader to [3,4,13,[27][28][29]. This means that the standard uncertainty principle says nothing "quantum" for an odd number of observables.…”

World-line quantization of a reciprocally invariant system

“…The idea of reciprocity has found resonance with various attempts to generalize the framework for the fundamental interactions-for example, in the context of string and M-theory [5,6], in the guise of bi-crossproduct algebras and physics at the Planck scale [7], in 'two-time' formulations [8], or in ad hoc 'noncommutative geometry' extensions of perturbative field theory [9]. Born-Green reciprocity can be viewed [10][11][12][13] as an alternative paradigm for generalized wave equations, which specify unitary irreducible representations of the full symmetry group, in the same way that relativistic wave equations establish unitary irreducible representations of the Poincaré group in four dimensions. It can be argued [10][11][12][13] that the appropriate invariance group is the so-called quaplectic group Q(3, 1) ∼ = U(3, 1) H (4), or more generally in D spacetime dimensions, the group Q(D − 1, 1) ∼ = U(D − 1, 1) H (D) of reciprocal relativity, the semi-direct product of the pseudo-unitary group of linear transformations between x µ and p µ which preserve both the extended metric d 2 and the symplectic form, with the WeylHeisenberg group.…”