Group Theoretical Derivation of Consistent Free Particle Theories
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Group Theoretical Derivation of Consistent Free Particle Theories Giuseppe Nisticò1,2 Received: 5 February 2020 / Accepted: 20 July 2020 / Published online: 14 August 2020 © The Author(s) 2020
Abstract The difficulties of relativistic particle theories formulated by means of canonical quantization, such as those of Klein–Gordon and Dirac, ultimately led theoretical physicists to turn to quantum field theory to model elementary particle physics. In order to overcome these difficulties, the theories of the present approach are developed deductively from the physical principles that specify the system, without making use of canonical quantization. For a free particle these starting assumptions are invariance of the theory and covariance of position with respect to Poincaré transformations. In pursuing the approach, the effectiveness of group theoretical methods is exploited. The coherent development of our program has shown that robust classes of representations of the Poincaré group, discarded by the known particle theories, can in fact be taken as bases for perfectly consistent theories. For massive spin zero particles, six inequivalent theories have been determined, two of which do not correspond to any of the current ones; all of these theories overcome the difficulties of Klein–Gordon one. The present lack of the explicit transformation properties of position with respect to boosts prevents the complete determination of non zero spin particle theories. In the past a particular form of these transformation properties was adopted by Jordan and Mukunda. We check its consistency within the present approach and find that for spin 21 particles there is only one consistent theory, which is unitarily related to Dirac’s; yet, once again, it requires classes of irreducible representations previously discarded.
1 Motivations and Overview Canonical quantization was the primary method for formulating specific relativistic particle theories [1–3]; unlike the non-relativistic case, the results were affected by problems. The first problem was the order of the wave equation for a spin zero free
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Giuseppe Nisticò [email protected]
1
Dipartimento di Matematica e Informatica, Università della Calabria, Rende, Italy
2
INFN – gruppo collegato di Cosenza, Via Bucci 30B, 87036 Rende, Italy
123
978
Foundations of Physics (2020) 50:977–1007
particle, i.e. the Klein–Gordon equation, that turned out to be second order with respect to time, while according to the general laws of quantum theory it should have been first order. A related problem concerns the probability density ρˆ of the position of the particle, that the theory allowed to be negative [4]. The proposals [5] to solve this paradoxical situation, by a posteriori reinterpreting ρˆ in terms of charge density, violate the peculiar relativistic invariance required by the theory [6]. The difficulties of relativistic particle theory, effectively shown by Weinberg in full detail [7] also for Dirac equation [8–10], ultimately led theoretical physicists to turn to qua
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