Transmuted spectrum-generating algebras and detectable parastatistics of the Superconformal Quantum Mechanics

In a recent paper (Balbino-de Freitas-Rana-FT, arXiv:2309.00965) we proved that the supercharges of the supersymmetric quantum mechanics can be statistically transmuted and accommodated into a Z2n -graded parastatistics. In this talk I derive the 6 = 1 + 2 + 3 transmuted spectrum-generating algebras (whose respective Z2n gradings are n = 0, 1, 2) of the 𝒩 = 2 Superconformal Quantum Mechanics. These spectrum-generating algebras allow to compute, in the corresponding multiparticle sectors of the de Alfaro-Fubini-Furlan deformed oscillator, the degeneracies of each energy level. The levels induced by the Z 2 × Z 2-graded paraparticles cannot be reproduced by the ordinary bosons/fermions statistics. This implies the theoretical detectability of the Z 2 × Z 2-graded parastatistics.


Introduction
In recent years the Z n 2 -graded, color Lie algebras and superalgebras introduced in 1978 by Rittenberg-Wyler [1,2] (see also the 1979 paper by Scheunert [3]) received a boost of attention by both physicists and mathematicians due to several different developments.In particular it was shown that color superalgebras appear as dynamical symmetries of known physical systems as the Lévy-Leblond spinors [4,5], while a systematic construction of Z 2  2 -graded invariant classical [6,7] and quantum [8,9,10] models started (more information and references on recent developments are presented in [11] and [12]).For many years progress in this field was hampered by a misconception.Since Z n 2 -graded color Lie (super)algebras can be reconstructed via Klein's operators (for a recent account see e.g.[13]), they were dismissed as not having a direct physical relevance (the same argument, applied to ordinary fermions which, in lower dimensions, can be obtained via bosonization, would imply that fermions are not physically relevant either!).The connection of Z n 2 -graded Lie (super)algebras with a certain special type of parastatistics had been investigated in several works [14,15,16,17].Till recently, on the other hand, the question of whether these parastatistics imply inequivocal different results not reproducible by the ordinary statistics involving bosons and fermions was left unanswered.This question became urgent when the first quantum model invariant under a Z 2 2 -graded worldline superPoincaré algebra was presented by Bruce and Duplij in [8].Since the Hamiltonian of that model is also an example of an ordinary N = 2 supersymmetric quantum mechanics, the physical relevance of the Z 2 2 -graded parastatistics was unclear.A positive answer to this question was finally produced in [18] (for theories involving Z 2 2 -graded parafermions) and [19] (for theories involving Z 2 2 -graded parabosons).It was shown, in a controlled setup, that the eigenvalues of certain observables acting in the multiparticle sectors of such theories allow to determine whether the system under consideration is composed by ordinary particles or by Z 2 2 -graded paraparticles.The [18,19] works employed the Majid's framework [20] which encodes parastatistics within a graded Hopf algebra endowed with a braided tensor product (the traditional approach to parastatistics makes use of the Green's trilinear relations [21]; the connection between the two approaches has been discussed in [22,23]).The theoretical detectability of Z 2 2 -graded parastatistics becomes particularly interesting in the light of the recent experimentalists'advances in either simulating [24] or engineering in the laboratory [25] certain types of parastatistics.
The [12] paper presents several results concerning the classification of Z n 2 -graded Lie (super)algebras and associated parastatistics, their construction in terms of Boolean logic gates, invariant hamiltonians under Z n 2 -graded (super)algebras, the statistical transmutations of the supercharges of the supersymmetric quantum mechanics.It was further shown that Z 2 2 -graded paraparticles directly affect (contrary to the previous models discussed in [18,19]) the energy spectrum of the superconformal quantum mechanics.In this talk I discuss a side line which was not addressed in [12], namely the derivation of the energy spectra of the 2-particle statistical transmutations of the N = 2 de Alfaro-Fubini-Furlan deformed oscillator [26] as induced by the corresponding transmuted spectrum-generating algebras.There are six different cases: the two pairs of N = 2 creation/annihilation operators can be assumed to be 2 fermions (2F ), 1 fermion and 1 boson (1F + 1B), 2 bosons (2B), or Z 2 2 paraparticles, respectively given by 2 parafermions (2P F ), 1 parafermion and 1 paraboson (1P F + 1P B ), 2 parabosons (2P B ).The (para)fermions satisfy the Pauli exclusion principle.The original (not transmuted) spectrumgenerating superconformal algebra corresponds to the 2F case and is given by sl(2|1) (references for the not-transmuted de Alfaro-Fubini-Furlan N = 2 superconformal quantum mechanics are [27,28,29].

The N = 2 Superconformal Quantum Mechanical model
In terms of the 2 × 2 matrices −1 0 , the N = 2 differential matrix representation of sl(2|1) is given by where R is the R-symmetry generator.The operators are Hermitian and β is an arbitrary real parameter.The de Alfaro-Fubini-Furlan [26] Hamiltonian H DF F , introduced through the position corresponds to a β-deformation of a matrix quantum oscillator.The j = 1, 2 pairs of creation/annihilation operators a † j , a j , defined through satisfy

The multiparticle sectors
Following [20,18,19], the Z n 2 -graded parastatistics of a multiparticle sector is encoded in a graded Hopf algebra endowed with a braided tensor product.
Let A, B, C, D be Z n 2 -graded operators whose respective n-bit gradings are α, β, γ, δ.The braided tensor product, conveniently denoted as "⊗ br ", is defined to satisfy the relation in terms of a (−1) β,γ sign.For Z 2 2 , the 0, 1 values of β, γ are read from the tables ( 5) and ( 6).The coproduct ∆ is the relevant Hopf algebra operation which allows, in physical applications, to construct multiparticle states.For a Universal Enveloping Algebra U ≡ U (G) of a graded Lie algebra G the coproduct map, given by satisfies the coassociativity property and the comultiplication The action of the coproduct on the identity 1 ∈ U (G) and on the primitive elements g ∈ G is In physical applications, typical primitive elements are the Hamiltonians and the creation/annihilation operators.For β > − 1 2 the single-particle Hilbert space H β is spanned by repeatedly applying the a † 1 , a † 2 creation operators on the normalized single-particle Fock vacuum Ψ β (x) ≡ |vac 1 : Similarly, the 2-particle Hilbert space H β is spanned by repeatedly applying the ∆(a † 1 ), ∆(a † 2 ) creation operators on the normalized 2-particle Fock vacuum Ψ β (x, y) ≡ |vac 2 : where ρ 1 is the 16-component vector with entry 1 in the first position and 0 otherwise.The single-particle and 2-particle vacuum energy are respectively 1 2 + β and 1 + 2β: 5. The six transmuted spectrum-generating graded(super)algebras Let us set for convenience P = a † 1 , Q = a † 2 (in the single-particle sector) and 2 ) (in the 2-particle sector).The asterisk in the coproduct denotes one of the six (para)statistics 2B, 1B + 1F, 2F, 2P B , 1P B + 1P F , 2P F defined by the respective signs entering (7).The construction of P , Q is not affected by the signs.The signs specify how P , Q are interchanged.The coproduct guarantees the homomorphism of the graded Lie algebras (defined in terms of (anti)commutators) of the single-particle and 2-particle sectors.We present the six transmuted spectrum-generating algebras in terms of the P , Q 2-particle generators.In the 2B and 1B + 1F cases they close as non-linear (super)algebras satisfying (anti)symmetry and graded Jacobi identities.In the remaining cases the (super)algebras close linearly on a finite number of generators.The spectrum-generating (super)algebras are defined by the following sets of (anti)commutation relations: for arbitrary u, z values.A convenient choice is to set u = z = 0.
1B + 1F : for arbitrary values of u; P is a fermionic generator, while Q is bosonic.
where P is the parafermionic operator.
We now compute the degeneracies of the 2-particle energy levels implied by the above 6 graded (super)algebras.