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arXiv:2003.11377 · Europhysics Letters (EPL)

Supersymmetric quantum mechanics requires g=2 for vector bosons

Georg Junker

Published May 29, 2020• 2 citations
arXivHTMLPDF

Abstract

Relativistic arbitrary spin Hamiltonians are shown to obey the algebraic structure of supersymmetric quantum system if their odd and even parts commute. This condition is identical to that required for the exactness of the Foldy-Wouthuysen transformation. Applied to a massive charged spin-1 particle in a constant magnetic field, supersymmetric quantum mechanics necessarily requires a gyromagnetic factor g = 2.

DOI
10.1209/0295-5075/130/30003

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Relativistic arbitrary spin Hamiltonians are shown to obey the algebraic structure of supersymmetric quantum system if their odd and even parts commute. This condition is identical to that required for the exactness of the Foldy-Wouthuysen transformation. Applied to a massive charged spin-1 particle in a constant magnetic field, supersymmetric quantum mechanics necessarily requires a gyromagnetic factor g = 2.
We find that in a supersymmetric quantum mechanics (SUSY QM) system, in addition to supersymmetric algebra, an associated SU(2) algebra can be obtained by using semiunitary (SUT) operator and projection operator, and the relevant constants of motion can be constructed. Two typical quantum systems are investigated as examples to demonstrate the above finding. The first example is the quantum system of a nonrelativistic charged particle moving in x-y plane and coupled to a magnetic field along z-axis. The second example is provided with the Dirac particle in a magnetic field. Similarly there exists an SU τ (2) ⊗ SU σ (2) symmetry in the context of the relativistic Pauli Hamiltonian squared. We show that there exists also an SU(2) symmetry associated with the supersymmetry of the Dirac particle.
The class of relativistic spin particle models reveals the `quantization' of parameters already at the classical level. The special parameter values emerge if one requires the maximality of classical global continuous symmetries. The same requirement applied to a non-relativistic particle with odd degrees of freedom gives rise to supersymmetric quantum mechanics. Coupling classical non-relativistic superparticle to a `U(1) gauge field', one can arrive at the quantum dynamical supersymmetry. This consists in supersymmetry appearing at special values of the coupling constant characterizing interaction of a system of boson and fermion but disappearing in a free case. Possible relevance of this phenomenon to high-temperature superconductivity is speculated.
Using supersymmetry quantum mechanics, we explain the covariance of the eigenvalue equation for the Jaynes-Cummings (JC) Hamiltonian under the particle-hole transformation. The JC Hamiltonians for the original spin-1/2 ``particle'' and the transformed partner (the ``hole'') are shown to be the fermionic and the bosonic parts of a supersymmetric Hamiltonian. The Hamiltonian, unlike ordinary one-dimensional supersymmetric quantum-mechanical models, can have a degenerate zero-energy ground state despite exact supersymmetry. The degeneracy is shown to arise from the hidden particle-hole exchange symmetry as well as supersymmetry in the Hamiltonian.
The (group and spin space) matrix Hamiltonian describing the dynamics of a nonrelativistic spin 1/2 particle moving in a static, but spatially dependent, non-Abelian magnetic field in two spatial dimensions is shown to take the form of an anticommutator of a nilpotent operator and its hermitian conjugate. Consequently, the (group space) matrix Hamiltonians for the two different spin projections form partners of a supersymmetric quantum mechanical system. The resulting supersymmetry algebra is exploited to explicitly construct the exact zero energy ground state wavefunction(s) for the system. The remaining eigenstates and eigenvalues of the two partner Hamiltonians form positive energy degenerate pairs.
The (group and spin space) matrix Hamiltonian describing the dynamics of a non-relativistic spin-½ particle moving in a static, but spatially dependent, non-Abelian magnetic field in two spatial dimensions is shown to take the form of an anticommutator of a nilpotent operator and its Hermitian conjugate. Consequently, the (group space) matrix Hamiltonians for the two different spin projections form partners of a supersymmetric quantum mechanical system. The resulting supersymmetry algebra is exploited to explicitly construct the exact zero energy ground state wave function(s) for the system. The remaining eigenstates and eigenvalues of the two partner Hamiltonians form positive energy degenerate pairs.
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−H_n=a^+a^-Hn​=a+a− and Ha=a−a+H_a=a^-a^+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 HnH_nHn​ and HaH_aHa​ 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.