Summary
The linear canonical transformations leave the symplectic metric on time-position-energy-momentum space invariant. The Lorentz metric defines a non-compact orthogonal metric on time-position space and separately on energy-momentum space. A physical constantb extends the orthogonal metric to time-position-energy-momentum space as defined by Born and Caianiello. The constantb bounds the rate of change of momentum and hence acceleration. TheU(3,1) group of transformations leaving both of these metrics invariant is studied and it is shown to reduce to the expected form in the Newtonian limit.
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References
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Low, S.G. U(3,1) Transformations with invariant symplectic and orthogonal metrics. Il Nuovo Cimento B 108, 841–851 (1993). https://doi.org/10.1007/BF02828732
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DOI: https://doi.org/10.1007/BF02828732