Quantization and construction of second order invariants for two dimensional super systems
The authors treat the following action in a dynamical system; \[ A=\int dt\quad L=\int dt d\theta_ i\overline{d\theta_ i}\quad {\mathfrak L}(z_ i,\quad D_ iz_ i,\quad \bar D_ iz_ i) \] with the superposition variable \(z_ i\) as a function of supertime variable, t and the Grassmann variables \(\theta_ i\) and \({\bar \theta}{}_ i\); \[ z_ i=z_ i(q_ i(t),\theta_ i,{\bar \theta}_ i)=q_ i(t)+i{\bar \theta}_ i\psi_ i(t)+i\theta_ i{\bar \psi}_ i(t)+\theta_ i{\bar \theta}_ iA_ i(t),\quad i=1,2, \] where \(\psi_ i\), \({\bar \psi}{}_ i\) and \(q_ i\), \(A_ i\) represent respectively the fermionic and bosonic spin degrees of freedom. Then by applying the Dirac's constraints the system can be reduced to a time-independent one. Thus the authors claim that the invariants in various orders of velocities and positions can be constructed. The authors construct the second order invariants involving the position, velocity and Grassmann degrees of freedom for the superharmonic oscillator and Kepler potentials.
scientific article; zbMATH DE number 4193182