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# A modern approach to born reciprocity

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posted on 2023-05-26, 06:02 authored by Morgan, SOIn the early twentieth century, Max Born attempted to develop a theory he called the principle of reciprocity‚ÄövÑvp. He observed that many formulae of physics remain unchanged under the following transformation of space-time coordinates and momentum-energy variables: x\\(^˜í¬¿\\) ‚Äövúv¿ p\\(^˜í¬¿\\); p\\(^˜í¬¿\\) ‚Äövúv¿ ‚ÄövÑvÆx\\(^˜í¬¿\\). Examples include Hamilton's equations and Heisenberg's commutation relations. Born's attempts to expand this observation to a general theory were largely unsuccessful. More recently, Stephen Low has made use of the group theoretical methods of Eugene Wigner, Valentine Bargmann and George Mackey to study a group which possesses Born reciprocity as an intrinsic symmetry, called the \quaplectic group\" \\(Q\\)(1 3). This involves the postulation of a new fundamental constant: the maximum rate of change of momentum (or maximum force) - denoted by \\(b\\). It also involves a new space-time-momentum-energy line element: ds\\(^2\\) = ‚Äöv†v¿dt\\(^2\\) + 1/c\\(^2\\)dx\\(^2\\) + 1/b\\(^2\\)(dp\\(^2\\) ‚Äöv†v¿ 1/c\\(^2\\)de\\(^2\\)) which remains invariant under quaplectic transformations. In this work we consider the different contraction limits of the quaplectic group in analogy to the contraction of the Poincare group to the Galilei group in the limit c ‚Äövúv¿ ‚Äöv†vª . In particular the quaplectic group contracts to the Poincare group in the limit b ‚Äövúv¿ ‚Äöv†vª (under the constraint that the reference frames must be inertial) and to the Hamilton group - the group of non-inertial classical mechanics - in the limit b c ‚Äövúv¿ ‚Äöv†vª. For the compact group \\(Q\\)(2) acting on two dimensional Euclidean space we consider the branching rules and use the \\(P^2\\) Casimir operator of the Euclidean subgroup to label states. We consider the implications of Born reciprocity to the Schr‚àö‚àÇdinger-Robertson inequality concluding that the covariance matrix ‚Äöv†v´ is quaplectic invariant and that physically distinct semi-classical limits of two different but unitarily-equivalent minimal uncertainty states must be related by a unitary transformation which does not belong to the quaplectic group. Finally we explore the worldline quantisation of a system invariant under reciprocal relativity finding that the square of the energy-momentum tensor is continuous over the entire real line. The resulting states therefore include tachyonic and null states as well as massless states of continuous spin which cannot be projected out in the current formulation. These states are discussed along with the massive states."

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