J. Niederle

Galilei invariant theories: I. Constructions of indecomposable finite-dimensional representations of the homogeneous Galilei group: directly and via contractions

All indecomposable finite-dimensional representations of the homogeneous Galilei group which when restricted to the rotation subgroup are decomposed to spin-0, -1/2 and -1 representations are constructed and classified. These representations are also obtained via contractions of the corresponding representations of the Lorentz group. Finally, the obtained representations are used to derive a general Pauli anomalous interaction term as well as to deduce wave equations which describe Darwin and spin–orbit couplings of a Galilei particle interacting with an external electric field.

Extended supersymmetries for the Schrödinger–Pauli equation

It is argued that extended, reducible, and generalized supersymmetry (SUSY) are common in many systems of standard nonrelativistic quantum mechanics. For example, it is proved that a well-studied quantum mechanical system of a spin-12 particle interacting with constant and homogeneous magnetic field admits the N=4 SUSY and has the internal symmetry so(3,3). Then an approach of energy spectra of a SUSY nature is presented and developed. It is applied to a wide class of systems described by the Schrodinger–Pauli equation admitting N=3, N=4, and N=5 SUSY. Some of these supersymmetries have a very peculiar property—their supercharges are realized without usual fermionic variables. It is shown that for them, the usual extension N=3 to N=4 SUSY is no longer guaranteed.

Relativistic wave equations for interacting, massive particles with arbitrary half-integer spins

New formulation of relativistic wave equations (RWE) for massive particles with arbitrary half-integer spins s interacting with external electromagnetic fields are proposed. They are based on wave functions which are irreducible tensors of rank

Involutive symmetries, supersymmetries and reductions of the Dirac equation

n (

Galilean equations for massless fields

n=s-\frac12

The relativistic Coulomb problem for particles with arbitrary half-integer spin

) antisymmetric w.r.t. n pairs of indices, whose components are bispinors. The form of RWE is straightforward and free of inconsistencies associated with the other approaches to equations describing interacting higher spin particles.

Galilei-invariant equations for massive fields

A new algebra of involutive symmetries of the Dirac equation is found. This algebra is used to reduce the Dirac equation for a charged particle, interacting with an external field and to describe hidden supersymmetries of this equation. Reducibility of a class of equations of supersymmetric quantum mechanics is established.

Maxwell-Chern-Simons Models: Their Symmetries, Exact Solutions and Non-relativistic Limits

Galilei-invariant equations for massless fields are obtained via contractions of relativistic wave equations. It is shown that the collection of non-equivalent Galilei-invariant wave equations for massless fields with spin equal to 1 and 0 is very rich and corresponds to various contractions of the representations of the Lorentz group to those of the Galilei ones. It describes many physically consistent systems, e.g., those of electromagnetic fields in various media or Galilean Chern–Simons models. Finally, classification of all linear and a big group of nonlinear Galilei-invariant equations for massless fields is presented.

Irreducible representations of the extended Poincaré parasuperalgebra

Using relativistic tensor-bispinorial equations proposed in Niederle and Nikitin (2001 Phys. Rev. D 64 125013) we solve the Kepler problem for a charged particle with arbitrary half-integer spin interacting with the Coulomb potential.

Biharmonic superspace for N = 4 mechanics

Galilei-invariant equations for massive fields with various spins have been found and classified. They have been derived directly, i.e., by using requirement of the Galilei invariance and various facts on representations of the Galilei group deduced in the paper written by de Montigny, Niederle and Nikitin (2006 J. Phys. A: Math. Gen. 39 1–21). A completed list of non-equivalent Galilei-invariant wave equations for vector and scalar fields is presented. It shows two things. First that the collection of such equations is very broad and describes many physically consistent systems. In particular it is possible to describe spin–orbit and Darwin couplings in frames of a Galilei-invariant approach. Second, these Galilei-invariant equations can be obtained either via contraction of known relativistic equations or via contractions of quite new relativistic wave equations.