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Space-time operators and the incompatibility of quantum mechanics with both finite-dimensional spinor fields and Lagrangian dynamics in the context of special relativity

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Abstract

It is shown that the present procedure used to quantize relativistic systems is inconsistent. Three mutually supporting arguments are given to sustain this conclusion. First, it is noted that the use of wave functions which transform according to any representation of 0(1, 3) whether finite or infinite dimensional is inappropriate because such a description allows for too many degrees of freedom. The phenomenon of Thomas precession indicates that internal structure such as spin and multipole moments must be described by mass shell (rest system) three-tensors rather than by unconstrained four-tensors. Second, even if representations of 0(1, 3) are employed, the momentum space construction for position-time operators, which is quite general and is applicable in any Euclidean or pseudo-Euclidean space, requires that the infinite-dimensional UIRs of 0(1, 3) be used rather than the finite-dimensional, nonunitary spinor representations. Third, various anomalous features of the customary kinematic formalism can be readily understood provided that this formalism is viewed as anad hoc blend of two other formalisms which, while self-consistent, are incompatible except for the trivial case of free one-particle states. These criticisms focus attention on a number of specific weaknesses of the kinematic foundations of relativistic quantum mechanics and relativistic quantum field theory. These weaknesses are sufficiently serious to require a radical revision of the current theory even at the kinematic level.

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Work supported in part by the National Research Council of Canada.

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Coleman, R.A. Space-time operators and the incompatibility of quantum mechanics with both finite-dimensional spinor fields and Lagrangian dynamics in the context of special relativity. Int J Theor Phys 17, 83–141 (1978). https://doi.org/10.1007/BF00686954

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Keywords

  • Spinor Representation
  • Mass Shell
  • Spinor Field
  • Multipole Moment
  • Relativistic Quantum Mechanic