We introduce color Heisenberg-Lie (super)algebras graded by the abelian groups
Z32,
Z2p×Z32 for
p=1,2,3, and investigate the properties of their associated multi-particle quantum paraoscillators. In the Rittenberg-Wyler's color Lie (super)algebras framework the above abelian groups are the simplest ones which induce mixed brackets interpolating commutators and anticommutators. These mixed brackets allow to accommodate two types of parastatistics: one based on the permutation group (beyond bosons and fermions in any space dimension) and an anyonic parastatistics based on the braid group. In both such cases the two broad classes of paraparticles are given by parabosons and parafermions. Mixed-bracket parafermions are created by nilpotent operators; they satisfy a generalized Pauli exclusion principle leading to roots-of-unity truncations in their multi-particle energy spectrum (braided Majorana qubits and their Gentile-type parastatistics are recovered in this color Lie superalgebra setting). Mixed-bracket parabosons do not admit truncations of the spectrum; the minimal detectable signature of their parastatistics is encoded in the measurable probability density of two indistinguishable parabosonic oscillators in a given energy eigenstate.
Abstract The recent surge of interest in Z 2 × Z 2 -graded invariant mechanics poses the challenge of understanding the physical consequences of a Z 2 × Z 2 -graded symmetry. In this paper it is shown that non-trivial physics can be detected in the multiparticle sector of a theory, being induced by the Z 2 × Z 2 -graded parastatistics obeyed by the particles. The toy model of the N = 4 supersymmetric/ Z 2 × Z 2 -graded oscillator is used. In this set-up the one-particle energy levels and their degenerations are the same for both supersymmetric and Z 2 × Z 2 -graded versions. Nevertheless, in the multiparticle sector, a measurement of an observable operator on suitable states can discriminate whether the system under consideration is composed by ordinary bosons/fermions or by Z 2 × Z 2 -graded particles. Therefore, Z 2 × Z 2 -graded mechanics has experimentally testable consequences. Furthermore, the Z 2 × Z 2
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.
The aim of this article is to introduce into quantum field theory,
Zn×Zn graded quantum stochastic calculus with the aim of generalizing supersymmetric quantum stochastic calculus to situations where rather than just two kinds of particles, Bosons and Fermions, we can have particles of
n2 kinds graded by
Zn×Zn rather than by
Z2. Following the suggestions at the end of the book [1] by Timothy Eyre, we introduce the
Zn×Zn graded tensor product by means of a bi-character constructed from a primitive
nth root of unity and then proceed to grade both system and noise operators in Boson Fock space. This enables us to construct consistent Lie algebra supercommutation relations graded by
Zn×Zn for appropriately defined graded quantum stochastic processes. Using the
Zn×Zn graded quantum noise for driving the Hudson-Parthasarathy noisy Schrodinger equation combined with an appropriate non-demolition counting measurement process, we formulate the problem of
Zn×Zn graded quantum filtering as a generalization of the Belavkin filter. After that, we discuss how to construct Bosonic quantum noise out of these graded quantum noise processes by tensoring with appropriately graded system operators and then how to add such noise to Bosonic field theories including quantum gravity. We also discuss quantum noise in string theories and a little bit of string field theories wherein the action for the string field is chosen based on the BRST charge quantization process that defines the condition for a state to be physical in the absence of ghosts. The addition of higher degree terms in the string field action is then based on the condition that the action should satisfy the quantum master equation for invariance of the matrix elements under the choice of the gauge fixing functional. We suggest methods by which quantum noise fields can also be taken into account in string field theories. The entire aim of adding quantum noise to a quantum field theory is motivated by the fact that the resulting noisy Hamiltonian should describe a Hudson-Parthasarathy noisy Schrodinger equation with unitary evolution on the joint System-Bath Hilbert space so that after tracing out over the bath, one obtains a quantum dynamical semigroup of TPCP maps describing system evolution alone for open quantum systems. For noisy quantum field theories, finally, we explain how to compute propagator corrections caused by noise.
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.
Recently, we have observed that the bosonized supersymmetric quantum mechanics leads naturally to the R-deformed Heisenberg algebra (RDHA) related to parabosons and parafermions. Here we show that RDHA, in turn, gives rise not only to the bosonized supersymmetry associated with Witten's supersymmetric quantum mechanics but also supplies us with a supersymmetry characterized by a nonlinear superalgebra. Such nonlinear supersymmetry may be specified alternatively by a central extension of the usual N=1 superalgebra with grading operator included nontrivially as an even generator. The systems under consideration are described by the Hamiltonians of the simplest form
Hn=a+a− and
Ha=a−a+ and reveal supersymmetries at the special values of the deformation parameter corresponding to parabosons of odd order. The peculiar nature of parabosonic supersymmetries is encoded in essentially nonlinear structure of their supercharges realized in terms of parabosonic creation-annihilation operators $a^\\pm$. At the values of deformation parameter corresponding to the even order parabosons the spectra of the Hamiltonians
Hn and
Ha have no supersymmetry but reveal `holes' in comparison with bosonic spectra, whose number is correlated with the parastatistics' order. It is shown that RDHA and the associated linear and nonlinear supersymmetries are realizable in the system of two identical fermions. The nonlinear supersymmetry of pure parabosonic systems possesses the structure of a reduced parasupersymmetry and may also be realized by appropriate modification of the classical analog of Witten's supersymmetric quantum mechanics.