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Register a new userOn the supersymmetry of the Klein-Gordon oscillator
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Authors
Georg Junker
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The three-dimensional Klein-Gordon oscillator is shown to exhibit an algebraic structure known from supersymmetric quantum mechanics. The supersymmetry is found to be unbroken with a vanishing Witten index, and it is utilized to derive the spectral properties of the Klein-Gordon oscillator, which is closely related to that of the non-relativistic harmonic oscillator in three dimensions. Supersymmetry also enables us to derive a closed-form expression for the energy-dependent Greens function.
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We consider the spectral problem associated with the Klein-Gordon equation for unbounded electric potentials. If the spectrum of this problem is contained in two disjoint real intervals and the two inner boundary points are eigenvalues, we show that these extremal eigenvalues are simple and possess strictly positive eigenfunctions. Examples of electric potentials satisfying these assumptions are given.
We present an elementary proof based on a direct calculation of the property of completeness at constant time of the solutions of the Klein-Gordon equation for a charged particle in a plane wave electromagnetic field. We also review different forms of the orthogonality and completeness relations previously presented in the literature and we discuss the possibility to construct the Feynman propagator for the particle in a plane-wave laser pulse as an expansion in terms of Volkov solutions. We show that this leads to a rigorous justification for the expression of the transition amplitude, currently used in the literature, for a class of laser assisted or laser induced processes.
The dynamical symmetries of the two-dimensional Klein-Gordon equations with equal scalar and vector potentials (ESVP) are studied. The dynamical symmetries are considered in the plane and the sphere respectively. The generators of the SO(3) group corresponding to the Coulomb potential, and the SU(2) group corresponding to the harmonic oscillator potential are derived. Moreover, the generators in the sphere construct the Higgs algebra. With the help of the Casimir operators, the energy levels of the Klein-Gordon systems are yielded naturally.
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We revise the unireps. of $U(2,2)$ describing conformal particles with continuous mass spectrum from a many-body perspective, which shows massive conformal particles as compounds of two correlated massless particles. The statistics of the compound (boson/fermion) depends on the helicity $h$ of the massless components (integer/half-integer). Coherent states (CS) of particle-hole pairs (excitons) are also explicitly constructed as the exponential action of exciton (non-canonical) creation operators on the ground state of unpaired particles. These CS are labeled by points $Z$ ($2times 2$ complex matrices) on the Cartan-Bergman domain $mathbb D_4=U(2,2)/U(2)^2$, and constitute a generalized (matrix) version of Perelomov $U(1,1)$ coherent states labeled by points $z$ on the unit disk $mathbb D_1=U(1,1)/U(1)^2$. Firstly we follow a geometric approach to the construction of CS, orthonormal basis, $U(2,2)$ generators and their matrix elements and symbols in the reproducing kernel Hilbert space $mathcal H_lambda(mathbb D_4)$ of analytic square-integrable holomorphic functions on $mathbb D_4$, which carries a unitary irreducible representation of $U(2,2)$ with index $lambdainmathbb N$ (the conformal or scale dimension). Then we introduce a many-body representation of the previous construction through an oscillator realization of the $U(2,2)$ Lie algebra generators in terms of eight boson operators with constraints. This particle picture allows us for a physical interpretation of our abstract mathematical construction in the many-body jargon. In particular, the index $lambda$ is related to the number $2(lambda-2)$ of unpaired quanta and to the helicity $h=(lambda-2)/2$ of each massless particle forming the massive compound.
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