Abstract It is commonly believed that there are only two types of particle exchange statistics in quantum mechanics, fermions and bosons, with the exception of anyons in two dimensions 1–5 . In principle, a second exception known as parastatistics, which extends outside two dimensions, has been considered 6 but was believed to be physically equivalent to fermions and bosons 7–9 . Here we show that non-trivial parastatistics inequivalent to either fermions or bosons can exist in physical systems. These new types of identical particle obey generalized exclusion principles, leading to exotic free-particle thermodynamics distinct from any system of free fermions and bosons. We formulate our theory by developing a second quantization of paraparticles that naturally includes exactly solvable non-interacting theories and incorporates physical constraints such as locality. We then construct a family of exactly solvable quantum spin models in one and two dimensions, in which free paraparticles emerge as quasiparticle excitations, and their exchange statistics can be physically observed and are notably distinct from fermions and bosons. This demonstrates the possibility of a new type of quasiparticle in condensed matter systems and—more speculatively—the potential for previously unconsidered types of elementary particle.
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.
Satyendra Nath Bose's attempt to describe the quantum statistical aspects of light consistently in terms of particles, and Einstein's generalisation, lead to the concept of Bosons as a class of quanta obeying `Bose-Einstein statistics'. Their identity as a class came in sharp contrast when the Pauli exclusion principle and the Dirac equation revealed the other class called Fermions, obeying `Fermi-Dirac statistics'. Spin, and spin alone, is the determining factor of the multiparticle behaviour of fundamental quanta. This is the basis of the Spin-Statistics Connection. While it is known that the overall theoretical picture is consistent, the physical reason for the connection is unknown. Further, the class difference is sensitive only to the total spin in a quantum aggregate, as spectacularly seen in superconductivity and superfluidity, and in the Bose-Einstein condensation of neutral atomic gas. Can we grasp the true reason behind the difference in the collective behaviour of Bosons and Fermions? An explorer's journey demanding logical and physical consistency of what we already know takes us to the hidden factors in the relation between spin and the statistics of quanta. The surprising answer is in the domain of gravity, that too, on a cosmic scale.
This chapter follows on from the previous chapter in quantum statistical mechanics but specialising on systems with identical particles. Using Gibbs prescritpion on generic states from Chapter 3, the occupation number representation is introduced. Constraints imposed on statistics by irreducible representations of the permutation group are discussed. These group-theoretic considerations are used to justify the use of Gentile’s parastatistics. Fermions and bosons are introduced as special cases of Gentile’s statistics, corresponding to the trivial representation of the permutation group for bosons, and the sign representation of the permutation group for fermions. Basic applications to fermions and bosons is given, including the Fermi–Dirac and Bose–Einstein statistics. A detailed expose of why photons are said to have zero chemical potential is also proposed.