On supersymmetric quantum mechanics Maurice Kibler, Mohammed Daoud To cite this version: Maurice Kibler, Mohammed Daoud. On supersymmetric quantum mechanics. E. Brandas and E.S. Kryachko. Fundamental World of Quantum Chemistry, A Tribute to the Memory of Per-Olov Lowdin, Volume 3, Springer-Verlag, pp.67-96, 2004. hal-00002942 HAL Id: hal-00002942 https://hal.archives-ouvertes.fr/hal-00002942 Submitted on 25 Sep 2004 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ON SUPERSYMMETRIC QUANTUM MECHANICS M.R. KIBLER Institut de Physique Nucl´eaire de Lyon IN2P3-CNRS et Universit´eClaude Bernard F-69622 Villeurbanne Cedex, France and M. DAOUD D´epartement de Physique Facult´edes Sciences Universit´eIbnou Zohr B.P. 28/S, Agadir, Morocco Abstract. This paper constitutes a review on = 2 fractional supersymmetric Quantum Mechanics of order k. The presentationN is based on the introduction of a generalized Weyl-Heisenberg algebra Wk. It is shown how a general Hamil- tonian can be associated with the algebra Wk. This general Hamiltonian covers various supersymmetrical versions of dynamical systems (Morse system, P¨oschl- Teller system, fractional supersymmetric oscillator of order k, etc.). The case of ordinary supersymmetric Quantum Mechanics corresponds to k = 2. A connection between fractional supersymmetric Quantum Mechanics and ordinary supersym- metric Quantum Mechanics is briefly described. A realization of the algebra Wk, of the = 2 supercharges and of the corresponding Hamiltonian is given in terms of deformed-bosonsN and k-fermions as well as in terms of differential operators. Review paper to be published in: Fundamental World of Quantum Chemistry A Tribute to the Memory of Per-Olov L¨owdin, Volume 3 E. Brandas and E.S. Kryachko (Eds.), Springer-Verlag, Berlin, 2004. ccsd-00002942, version 1 - 25 Sep 2004 ON SUPERSYMMETRIC QUANTUM MECHANICS M.R. KIBLER Institut de Physique Nucl´eaire de Lyon IN2P3-CNRS et Universit´eClaude Bernard F-69622 Villeurbanne Cedex, France and M. DAOUD D´epartement de Physique Facult´edes Sciences Universit´eIbnou Zohr B.P. 28/S, Agadir, Morocco Abstract. This paper constitutes a review on = 2 fractional supersym- metric Quantum Mechanics of order k. The presentationN is based on the introduction of a generalized Weyl-Heisenberg algebra Wk. It is shown how a general Hamiltonian can be associated with the algebra Wk. This general Hamiltonian covers various supersymmetrical versions of dynamical systems (Morse system, P¨oschl-Teller system, fractional supersymmetric oscillator of order k, etc.). The case of ordinary supersymmetric Quantum Mechan- ics corresponds to k = 2. A connection between fractional supersymmetric Quantum Mechanics and ordinary supersymmetric Quantum Mechanics is briefly described. A realization of the algebra Wk, of the = 2 super- charges and of the corresponding Hamiltonian is given in termsN of deformed- bosons and k-fermions as well as in terms of differential operators. 1 Introducing supersymmetry Supersymmetry (SUperSYmmetry or SUSY) can be defined as a symmetry be- tween bosons and fermions (as considered as elementary particles or simply as degrees of freedom). In other words, SUSY is based on the postulated existence of operators Qα which transform a bosonic field into a fermionic field and vice versa. In the context of quantum mechanics, such symmetry operators induce a Z2-grading of the Hilbert space of quantum states. In a more general way, frac- tional SUSY corresponds to a Zk-grading for which the Hilbert space involves both bosonic degrees of freedom (associated with bosons) and k-fermionic degrees of freedom (associated with k-fermions to be described below) with k N 0, 1, 2 ; the case k = 2 corresponds to ordinary SUSY. ∈ \{ } The concept of SUSY is very useful in Quantum Physics and Quantum Chem- istry. It was first introduced in elementary particle physics on the basis of the unification of internal and external symmetries [1,2]. More precisely, SUSY goes back to the sixtees when many attemps were done in order to unify external sym- metries (described by the Poincar´egroup) and internal symmetries (described by gauge groups) for elementary particles. These attempts led to a no-go theorem by Coleman and Mandula in 1967 [1] which states that, under reasonable assumptions concerning the S-matrix, the unification of internal and external symmetries can be achieved solely through the introduction of the direct product of the Poincar´e group with the relevant gauge group. In conclusion, this unification brings nothing new since it simply amounts to consider separately the two kinds of symmetries. A way to escape this no-go theorem was proposed by Haag, Lopuszanski and Sohnius in 1975 [2]: The remedy consists in replacing the Poincar´egroup by an extended Poincar´egroup, the Poincar´e supergroup (or Z2-graded Poincar´e group). Let us briefly discuss how to introduce the Poincar´esupergroup. We know that the Poincar´egroup has ten generators (six Mµν Mνµ and four Pµ with µ, ν ≡− ∈ 0, 1, 2, 3 ): Three Mµν describe ordinary rotations, three Mµν describe special { } Lorentz transformations and the four Pµ describe space-time translations. The Lie algebra of the Poincar´egroup is characterized by the commutation relations [Mµν ,Mρσ]= i(gµρMνσ + gνσMµρ gµσMνρ gνρMµσ) − − − [Mµν , Pλ]= i(gµλPν gνλPµ), [Pµ, Pν ]=0 − − where (gµν ) = diag(1, 1, 1, 1). The minimal extension of the Poincar´egroup − − − into a Poincar´esupergroup gives rise to a Lie superalgebra (or Z2-graded Lie alge- bra) involving the ten generators Mµν and Pµ plus four new generators Qα (with α 1, 2, 3, 4 ) referred to as supercharges. The Lie superalgebra of the Poincar´e supergroup∈{ is} then described by the commutation relations above plus the addi- tional commutation relations i [M ,Q ]= ([γ ,γ ]) Q , [P ,Q ]=0 µν α 4 µ ν αβ β µ α and the anticommutation relations µ Qα,Qβ = 2 (γ C) Pµ { } − αβ where the γ’s are Dirac matrices, C the charge conjugation matrix and α, β 1, 2, 3, 4 . ∈ { What} is the consequence of this increase of symmetries (i.e., passing from 10 to 14 generators) and of the introduction of anticommutators ? In general, an increase of symmetry yields an increase of degeneracies. For instance, in condensed matter physics, passing from the tetragonal symmetry to the cubical symmetry leads, from a situation where the degrees of degeneracy are 1 and 2, to a situation where the degrees of degeneracy are 1, 2 and 3 (in the absence of accidental degeneracies). Another possible consequence of the increase of symmetry is the occurrence of new states or new particles. For instance, the number of particles is doubled when going from the Schr¨odinger equation (with Galilean invariance) to the Dirac equation (with Lorentz invariance): With each particle of mass m, spin S and charge e is associated an antiparticle of mass m, spin S and charge e. In a similar way, when passing from the Poincar´egroup to the Poincar´esupergroup,− we associate with a known particle of mass m, spin S and charge e a new particle ′ ′ 1 of mass m , spin S = S 2 and charge e. The particle and the new particle, called a sparticle or a particlino± (according to whether as the known particle is a fermion or a boson), are accommodated in a given irreducible representation of the Z2-graded Poincar´egroup [3]. Consequently, they should have the same mass. However, we have m′ = m because SUSY is a broken symmetry. In terms of field 6 theory, the sparticle or particlino field results from the action of a supercharge Qα on the particle field (and vice versa): Qα : fermion sfermion = boson 7→ Qα : boson bosino = fermion 7→ (see also Refs. 3 and 4). We thus speak of a selectron (a particle of spin 0 associated 1 with the electron) and of a photino (a particle of spin 2 associated with the photon). The experimental evidence for SUSY is not yet firmly established. Some ar- guments in favour of SUSY come from: (i) condensed matter physics with the fractional quantum Hall effect and high temperature superconductivity [5]; (ii) nuclear physics where SUSY could connect the complex structure of odd-odd nu- clei to much simpler even-even and odd-A systems [6]; and (iii) high energy physics (especially in the search of supersymmetric particles and the lighter Higgs boson) where the observed signal around 115 GeV/c2 for a neutral Higgs boson is com- patible with the hypotheses of SUSY [7]. It is not our intention to further discuss SUSY from the viewpoint of condensed matter physics, nuclear physics and elementary particle physics. We shall rather focus our attention on supersymmetric Quantum Mechanics (sQM), a supersym- metric quantum field theory in D = 1+0 dimension [8] that has received a great deal of attention in the last twenty years and is still in a state of development. In recent years, the investigation of quantum groups, with deformation parameters taken as roots of unity, has been a catalyst for the study of fractional sQM which is an extension of ordinary sQM. This extension takes its motivation in the so-called intermediate or exotic statistics like: (i) anyonic statistics in D = 1+2 dimensions connected to braid groups [5,9-11], (ii) para-bosonic and para-fermionic statistics in D = 1 + 3 dimensions connected to permutation groups [12-15], and (iii) q- deformed statistics (see, for instance, [16,17]) arising from q-deformed oscillator algebras [18-23].