Lie-isotopic lifting of unitary symmetries and of Wigner’s theorem for extended, deformable particles

Abstract

As is well known, the notions of symmetries and Wigner's theorem constitute some of the ultimate foundations of quantum mechanics. Nevertheless, the theory is crucially dependent on the simplest possible realization of Lie's theory, that via unitary linear or antilinear operators, which is characterized by enveloping associative algebras ℰ of operatorsA, B … with trivial associative productAB. A series of recent, mathematical and physical studies have established the existence of the Lie-isotopic reformulation of Lie's theory, which is based on enveloping algebrasE that are still associative, yet are realized via the less trivial associative-isotopic productA*B=AgB, whereg is a suitable, fixed, operator. Furthermore, it has been proved that the Lieisotopic theory can be consistently formulated on a Hilbert space, by providing realistic possibilities of achieving a generalization of quantum mechanics known under the name of «hadronic mechanics». In this paper, we present the notion of Lie-isotopic lifting of unitary linear and antilinear symmetries and of Wigner's theorem within the context of (the closed-exterior branch of) hadronic mechanics. The results are applied to the isotopic lifting of the operator formulation of the rotational symmetry. It is shown that the generalized symmetry can provide the invariance of all possible ellipsoidical deformations of spherical particles. This confirms the general lines of hadronic mechanics conjectured earlier, that space-time (and other) symmetries can be exact for extended particles, provided that they are expressed in a structurally more general way (isotopic-unitary) while the same symmetries can be violated when expressed via the simplest possible (unitary) realizations for pointlike approximations.