Supersymmetric Klein–Gordon and Dirac oscillators

Abstract We have recently shown that the space of initial data (covariant phase space) of the relativistic oscillator in Minkowski space $$\mathbb {R}^{3,1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> is a homogeneous Kähler–Einstein manifold $$Z_6=\textrm{AdS}_7/\textrm{U}(1) =\textrm{U}(3,1)/\textrm{U}(3)\times \textrm{U}(1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>6</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mtext>AdS</mml:mtext> <mml:mn>7</mml:mn> </mml:msub> <mml:mo>/</mml:mo> <mml:mtext>U</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mtext>U</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mtext>U</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:mtext>U</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . It was also shown that the energy eigenstates of the quantum relativistic oscillator form a direct sum of two weighted Bergman spaces of holomorphic (particles) and antiholomorphic (antiparticles) square-integrable functions on the covariant phase space $$Z_6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> of the classical oscillator. Here we show that the covariant phase space of the supersymmetric version of the relativistic oscillator (oscillating spinning particle) is the odd tangent bundle of the space $$Z_6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> . Quantizing this model yields a Dirac oscillator equation on the phase space whose solution space is a direct sum of two spinor spaces parametrized by holomorphic and antiholomorphic functions on the odd tangent bundle of $$Z_6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> . After expanding the general solution in Grassmann variables, we obtain components of the spinor field that are holomorphic and antiholomorphic functions from Bergman spaces on $$Z_6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> with different weight functions. Thus, the supersymmetric model under consideration is exactly solvable, Lorentz covariant and unitary.