Paraparticles of order p = 2 must be pair produced, so the least massive are absolutely stable. Consequently, paraparticles are excellent candidates to be associated with dark matter and/or dark energy. For a fixed number of paraparticles, in a "generalized ray space" there are simple orthonormal bracket "ray representatives" constructed from paraparticle creation operators. Each such ray representative is associated with a Young Diagram with a unique "P-bar-sum" eigenvalue. "P-bar-sum" is the sum of the particle-exchange operators which exchange two identical particles. In this paper, by using totally symmetric and totally antisymmetric brackets, generalized ray spaces for order p=2 are constructed for parabosons (pB), for parafermions (pB), and for possible para-Families (pFam). Para-Families occur in the case of non-commuting pB and pF creation operators. Explicit arbitrary order p ray representatives for a few paraparticles may be useful to establish the order p if, indeed, dark matter and/or dark energy is discovered to exhibit parastatistic properties. Up to p identical pB's (pF's) can occupy a totally antisymmetric (symmetric) state, unlike for normal boson (fermion) statistics.
We show that it is possible to formulate a consistent first-quantized theory of paraparticles, i.e., particles which are neither bosons nor fermions. We examine a number of properties of the theory and show that the formulation of Messiah and Greenberg in terms of generalized rays can be replaced by an equivalent formulation in which states are represented by rays in the usual way. We use this alternative formulation to establish some results of ordinary quantum mechanics. We examine in detail the consistency of the theory with the cluster law and show that paraparticles must have states associated with whole families of different permutation symmetries, according to the following rule: If a given particle has
N-particle states associated with a given Young diagram, then it must have ($N\ensuremath{-}1$)-, ($N\ensuremath{-}2$)-,..., two-particle states associated with all Young diagrams which can be obtained from the first by successively removing squares. This gives rise to infinitely many different kinds of paraparticle, all with rather complicated properties.
We show that with one exception every possible statistical type of particle consistent with the cluster properties of quantum mechanics can be uniquely identified by a single pair of integers (
p,q). A particle of type (
p,q) has states corresponding to all Young diagrams whose first
p columns (and only these) have arbitrary depth and whose first
q rows (and only these) have arbitrary length. Parabosons and parafermions of order
p are included in this scheme as types (0,
p) and (
p, 0), respectively. If
p and
q are nonzero, then the particle is of infinite order. The only exception to this classification scheme is a particle with states corresponding to all Young diagrams.
Parastatistics is an exotic type of exchange statistics beyond fermions and bosons. Paraparticles transform in higher dimensional representations of the exchange symmetry group, analogous to non-Abelian anyons, yet consistently defined in any dimension. Although paraparticles have long been proposed, they were widely believed to be physically equivalent to fermions or bosons. Nevertheless, a recent paper proposed a different theory, called
R-parastatistics, and demonstrated that nontrivial
R-paraparticles can emerge as quasiparticles in condensed matter systems, and are observably distinct from both fermions and bosons. This paper develops the theoretical foundation and several extensions of
R-parastatistics, with particular emphasis on its observable consequences. Central to this paper is a general theory of local observables extending the basic family introduced before. First, we define local observables that distinguish particle types. Second, we formulate local observables at special point defects that probe the internal indices of
R-paraparticles, crucial for observing
R-parastatistics and for the proposed applications in quantum information. Third, we introduce local observables that create or annihilate particle-antiparticle pairs, important for building a relativistic quantum field theory for
R-paraparticles. We further introduce generalized hidden symmetries that act on internal indices of
R-paraparticles while preserving the local observable algebra, providing a basis for proving local indistinguishability and for connecting to a categorical description of
R-paraparticles. This work sets a solid theoretical foundation for understanding the fundamental physical properties of
R-paraparticles and pave the way for finding them in nature.
A new formulation of the many-electron correlation problem is presented based on the use of parastatistics. It is shown that second-order para-Fermi creation and annihilation operators, which correspond to the creation and annihilation of spin-averaged paraparticles, occur naturally for the spin-independent many-electron problem. Moreover, the spin-independent many-electron Hamiltonian is directly expressible in terms of the parafield operators. The structure of the general para-Fermi algebra is also investigated from the viewpoint of the pseudo-orthogonal group O(2n+1, 1). Finally, an explicit matrix representation for the para-Fermi algebra of order 2, which enables one to handle even particle-number-nonconserving operators, is obtained in a canonical U(n)-adapted basis.
Paraparticles beyond bosons and fermions can be exchanged via either the braid group (anyons, existing up to
D=2 space dimensions) or the permutation group; in the latter case the space dimensions are not limited. Besides being predicted, anyons have been experimentally detected. The situation differs for paraparticles exchanged via the permutation group ("permutation-group parastatistics").The first test to detect their theoretical signature was published in 2021 (for
Z2×Z2-graded parafermions; it was soon followed by a second paper proving the detectability of
Z2×Z2-graded parabosons). Later on, two further papers proved theoretical signatures of permutation-group parastatistics. These works demonstrate that, in certain situations, a long-held belief on the "conventionality of parastatistics" argument can be evaded: some measurements of permutation-group paraparticles cannot be recovered from ordinary bosons/fermions. The main question now is how to experimentally detect or engineer in the laboratory such paraparticles. For this aim a minimal setup for the theoretical test is here provided: a Gedankenexperiment (a simplified version of the two tests published in 2021) which, essentially, is a flow chart of logical operations. The key point is to present, to experimentalists, the necessary steps to be simulated/realized in the laboratory (possibly, by manipulating qudits). In this minimal setup, the detection/engineering of paraparticles is mapped into a chirality test. The mathematical setting is based on
Z2×Z2-graded color Lie (super)algebras and derived mathematical structures.
Hartle and Taylor have shown that the cluster law imposes certain restrictions on the allowed symmetries of first-quantized systems of identical particles. We extend their results to show that all particles which are neither bosons nor fermions and which obey the cluster law can be divided into two classes, those of finite order, and those of infinite order. For every positive integer
p there are two types of finite-order particle, which we call parabosons and parafermions of order
p. Ordinary bosons and fermions can be fitted into this scheme as particles of order 1. We conjecture that the finite-order particles can be identified with the parafermions and parabosons of the second-quantized, parafield theory. Infinite-order particles would seem to have no analog in the second-quantized theory, as presently formulated.