[Paper Review] Inequivalent $Z_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics

M. M. Balbino, Isaque P. de Freitas+2 more|arXiv (Cornell University)|Sep 2, 2023

Algebraic structures and combinatorial modelsMathematics3 citations

TL;DR

This paper establishes that ${\mathbb{Z}}_{2}^{n}$-graded Lie (super)algebras admit $b_n = n + \lfloor n/2\rfloor + 1$ inequivalent brackets compatible with graded Jacobi identities, generalizing ordinary statistics to $n$-bit parastatistics. It applies this framework to show that ${\cal N}$-extended one-dimensional supersymmetric quantum mechanics for ${\cal N} = 1,2,4,8$ admits $s_{\cal N} = 2,6,10,14$ distinct statistical transmutations via inequivalent parastatistical realizations, leading to novel energy level degeneracies not realizable with standard bosons or fermions.

ABSTRACT

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 $b_n= n+\lfloor n/2 floor+1$. 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 $Z_2^n imes Z_2^n ightarrow Z_2$ 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 (para)bosons and/or (para)fermions. 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). As a main physical application we prove that the $N$-extended, $1D$ 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. These 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 DFF deformed oscillator with $sl(2|1)$ 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.

Motivation & Objective

• To classify all inequivalent ${\mathbb{Z}}_{2}^{n}$-graded Lie (super)algebras compatible with graded Jacobi identities and associative ring multiplication.
• To generalize ordinary statistics (1-bit) to $n$-bit parastatistics, where particles are classified as bosons, parabosons, fermions, or parafermions based on their graded sector assignment.
• To apply the formalism to ${\cal N}$-extended supersymmetric quantum mechanics, identifying how different statistical transmutations arise from inequivalent graded brackets.
• To demonstrate that for ${\cal N} = 1,2,4,8$, the number of inequivalent quantizations is $s_{\cal N} = 2,6,10,14$, respectively, corresponding to $n=1,2,3,4$ bits.
• To show that these transmutations lead to distinct energy level degeneracies in the ${\cal N}=2$ case that cannot be reproduced by standard statistics.

Proposed method

• Derive the number $b_n = n + \lfloor n/2\rfloor + 1$ of inequivalent ${\mathbb{Z}}_{2}^{n}$-graded brackets via Rittenberg-Wyler and Scheunert's analysis of color Lie superalgebras.
• Reformulate the brackets using Boolean logic gates to classify consistent commutator/anticommutator rules for $n$-bit parastatistics.
• Construct ${\mathbb{Z}}_{2}^{n}$-graded quantum Hamiltonians for $n=2,3$ with $b_2=4$ and $b_3=5$ inequivalent multiparticle quantizations.
• Assign supercharges and empty slots to different graded sectors to generate inequivalent graded Lie (super)algebras, with explicit examples from quaternions, split-quaternions, and biquaternions.
• Analyze the $\mathcal{N}=2$ de Alfaro-Fubini-Furlan oscillator model to show that its ${\mathbb{Z}}_{2}^{2}$-graded structure leads to energy level degeneracies absent in standard statistics.
• Use explicit (anti)commutator relations for $3_{5,i}$, $3_{5,ii}$, and $3_{5,iii}$ subcases of biquaternions to demonstrate distinct real forms of graded superalgebras with different diagonal signatures.

Experimental results

Research questions

• RQ1 How many inequivalent ${\mathbb{Z}}_{2}^{n}$-graded Lie brackets satisfy the graded Jacobi identity and are compatible with the associative ring structure of $n$-bit parastatistics?
• RQ2 What is the minimal number $b_n$ of inequivalent graded Lie (super)algebras induced by a ${\mathbb{Z}}_{2}^{n}$-graded operator algebra, and how does it grow with $n$?
• RQ3 How do different assignments of 'marked' operators to graded sectors generate additional inequivalent graded Lie (super)algebras beyond the $b_n$ base count?
• RQ4 Can ${\cal N}$-extended supersymmetric quantum mechanics for ${\cal N}=1,2,4,8$ be realized through multiple inequivalent parastatistical quantizations, and if so, how many?
• RQ5 Do these inequivalent quantizations lead to physically distinct energy spectra, particularly in the ${\cal N}=2$ case, that cannot be reproduced by standard boson/fermion statistics?

Key findings

• The number of inequivalent ${\mathbb{Z}}_{2}^{n}$-graded brackets compatible with graded Jacobi identities is $b_n = n + \lfloor n/2\rfloor + 1$, with $b_2 = 4$ and $b_3 = 5$.
• For $n=2$, the quaternion algebra yields $c_2 = 4 = b_2$ inequivalent graded Lie superalgebras, while split-quaternions yield $c_2 = 6 > b_2$, and biquaternions yield $c_3 = 16 > b_3 = 5$, showing that marked operators increase the number of inequivalent realizations.
• The ${\cal N}=2$ de Alfaro-Fubini-Furlan oscillator model with $sl(2|1)$ spectrum-generating algebra exhibits energy level degeneracies under ${\mathbb{Z}}_{2}^{2}$-graded parastatistics that are not realizable with standard bosons or fermions.
• For ${\cal N}=1,2,4,8$, the number of inequivalent statistical transmutations of the supercharges is $s_{\cal N} = 2,6,10,14$, respectively, corresponding to $n=1,2,3,4$ bits via the relation ${\cal N} = 2^{n-1}$.
• Explicit (anti)commutator relations for three distinct $\mathbb{Z}_2^3$-graded superalgebras derived from biquaternions—$3_{5,i}$, $3_{5,ii}$, and $3_{5,iii}$—demonstrate different real forms with distinct diagonal signatures, such as $(-1,+1,+1,+1)$ and $(-1,-1,+1,+1)$, confirming their inequivalence.
• The $3_{5,i}$ and $3_{5,iii}$ subcases differ in the diagonal signature of parafermionic generators, confirming they represent distinct real forms of the same graded superalgebra, with $n_B = 16$ total inequivalent graded Lie (super)algebras for biquaternions.