Inequivalent Z2n-graded brackets, n-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics

Given an associative ring of $Z_2^n$-graded operators, the number of inequivalent brackets of Lie-type which are compatible with the grading and satisfy graded Jacobi identities is $b_n=n+\lfloor n/2\rfloor +1$. This follows from the Rittenberg-Wyler and Scheunert analysis of “color” Lie (super)algebras which is revisited here in terms of Boolean logic gates.

The inequivalent brackets, recovered from $Z_2^n\times Z_2^n\to Z_2$ mappings, are defined by consistent sets of commutators/anticommutators describing particles accommodated into an n-bit parastatistics (ordinary bosons/fermions correspond to 1 bit). Depending on the given graded Lie (super)algebra, its graded sectors can fall into different classes of equivalence expressing different types of particles (bosons, parabosons, fermions, parafermions). As a consequence, the assignment of certain “marked” operators to a given graded sector is a further mechanism to induce inequivalent graded Lie (super)algebras (the basic examples of quaternions, split-quaternions and biquaternions illustrate these features).

As a first application we construct $Z_2^2$ and $Z_2^3$-graded quantum Hamiltonians which respectively admit $b_2=4$ and $b_3=5$ inequivalent multiparticle quantizations (the inequivalent parastatistics are discriminated by measuring the eigenvalues of certain observables in some given states). The extension to $Z_2^n$-graded quantum Hamiltonians for $n>3$ is immediate.

As a main physical application we prove that the $N$-extended, one-dimensional supersymmetric and superconformal quantum mechanics, for $N=1,2,4,8$, are respectively described by $s_N=2,6,10,14$ alternative formulations based on the inequivalent graded Lie (super)algebras. The $s_N$ numbers correspond to all possible “statistical transmutations” of a given set of supercharges which, for $N=1,2,4,8$, are accommodated into a $Z_2^n$-grading with $n=1,2,3,4$ (the identification is $N=2^{n-1}$).

In the simplest $N=2$ setting (the 2-particle sector of the de Alfaro-Fubini-Furlan deformed oscillator with $sl\left(2\right|1)$ spectrum-generating superalgebra), the $Z_2^2$-graded parastatistics imply a degeneration of the energy levels which cannot be reproduced by ordinary bosons/fermions statistics.