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arXiv:hep-th/0209001

Quantum Theory over a Galois Field and Spin-statistics theorem

Felix M. Lev

Published Aug 30, 2002• 0 citations
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Abstract

In our previous papers it has been shown that quantum theory based on a Galois field (GFQT) possesses a new symmetry between particles and antiparticles, and for massless particles this symmetry (called the AB one) is compatible with all representation operators of the symmetry algebra. In the present paper, it is shown that the AB symmetry is compatible with all representation operators of the symmetry algebra in the general case. As a consequence of simple arithmetical considerations, this symmetry is compatible with the vacuum condition, only for particles with half-integer spin (in usual units). If the AB symmetry is combined with the spin-statistics theorem, one arrives at the following conclusions: in quantum theory based on a Galois field i) any neutral particle can be only composite but not elementary (this property has been proved for the massless case in hep-th/0207192); ii) any interaction can involve only an even number of creation/annihilation operators.

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In our previous papers it has been shown that quantum theory based on a Galois field (GFQT) possesses a new symmetry between particles and antiparticles, and for massless particles this symmetry (called the AB one) is compatible with all representation operators of the symmetry algebra. In the present paper, it is shown that the AB symmetry is compatible with all representation operators of the symmetry algebra in the general case. As a consequence of simple arithmetical considerations, this symmetry is compatible with the vacuum condition, only for particles with half-integer spin (in usual units). If the AB symmetry is combined with the spin-statistics theorem, one arrives at the following conclusions: in quantum theory based on a Galois field i) any neutral particle can be only composite but not elementary (this property has been proved for the massless case in hep-th/0207192); ii) any interaction can involve only an even number of creation/annihilation operators.
We consider in detail an approach (proposed by the author earlier) where quantum states are described by elements of a linear space over a Galois field, and operators of physical quantities - by linear operators in this space. The notion of Galois fields (which is extremely simple and elegant) is discussed in detail and we also discuss the conditions when our description gives the same predictions as the conventional one. In quantum theory based on a Galois field, all operators are well defined and divergencies cannot not exist in principle. A particle and its antiparticle are described by the same modular irreducible representation of the symmetry algebra. This automatically explains the existence of antiparticles and shows that a particle and its antiparticle are the different states of the same object. As a consequence, a new symmetry arises, and the structure of the theory is considerably simplified. In particular, one can work with only creation operators or only annihilation ones since they are no more independent. In our approach the problem arises whether the existence of neutral elementary particles (e.g. the photon) is compatible with the usual relation between spin and statistics, or in other words whether neutral particles can be elementary or only composite.
As shown in our previous papers (hep-th/0209001 and reference therein), quantum theory based on a Galois field (GFQT) possesses a new symmetry between particles and antiparticles, which has no analog in the standard approach. In the present paper it is shown that this symmetry (called the AB one) is also compatible with supersymmetry. We believe this is a strong argument in favor of our assumption that the AB symmetry is a fundamental symmetry in the GFQT (and in nature if it is described by quantum theory over a Galois field). We also consider operatorial formulations of space inversion and X inversion in the GFQT. It is shown in particular that the well known fact, that the parity of bosons is real and the parity of fermions is imaginary, is a simple consequence of the AB symmetry.
A nonrelativistic proof of the spin-statistics theorem is given in terms of the field operators satisfying commutation and anticommutation relations, which are introduced here in the coordinate space as a means to build the permutation symmetry into the brackets of identical particles. An eigenvalue problem of a πππ-rotation for a product of two annihilation operators is combined with an analysis on its rotational property to prove the connection that the field operators for integral-spin and half-integral-spin particles obey the commutation and anticommutation relations, respectively.
We consider a quantum theory based on a Galois field. In this approach infinities cannot exist, the cosmological constant problem does not arise, and one irreducible representation (IR) of the symmetry algebra splits into independent IRs describing a particle an its antiparticle only in the approximation when de Sitter energies are much less than the characteristic of the field. As a consequence, the very notions of particles and antiparticles are only approximate and such additive quantum numbers as the electric, baryon and lepton charges are conserved only in this approximation. There can be no neutral elementary particles and the spin-statistics theorem can be treated simply as a requirement that standard quantum theory should be based on complex numbers.
We define statistics of quantum mechanical particles, obtaining the Bose–Einstein and Fermi–Dirac varieties of indistinguishable particles. After finding the rotation and Lorentz matrices in different Lorentz representations, we describe the spin-statistics theorem, relating fermions with half-integer spin and bosons with integer spin. We explain two simple proofs and say some words on two others. We end by discussing symmetries in more generality, and we discuss the fact that internal symmetries must commute with spacetime ones, due to the Coleman–Mandula theorem.
Conventional proofs of the spin–statistics theorem either use relativistic quantum field theory or concern field theory solitons. We here construct new proofs of this theorem for spinning particles. They are extensions of previous work on spinless particles and require neither relativity nor field theory. They do, however, assume certain continuity conditions involving the existence of antiparticles. Non-Abelian statistics, such as parastatistics and braid statistics of order 2 or more, are excluded in our framework.