To restore invariance relatively to rotations (Lorentz-invariance) we propose two new algebras (nonrelativistic and relativistic) with spin noncommutativity of coordinates. It is shown, that in space with these algebras the minimal length is present. Its value is found for both versions of noncommutative algebras. For a relativistic algebra we propose a scheme of constructing a noncommutative function. This scheme is an alternate to the Moyal star product, which is used in space with canonical noncommutativity. Also mathematical properties of a product of such functions are investigated.
In space with spin noncommutativity of coordinates a harmonic oscillator is solved exactly. It is shown, that spin noncommutativity breaks degeneracy of energy levels relatively to a sum (2n+l) of principal and orbital quantum numbers.
Within the perturbation theory a spectrum of the Hydrogen atom in space with spin noncommutativity of coordinates is studied. It is shown, that for levels with l≠0 spin noncommutativity breaks a degeneracy of energy levels relatively to orbital quantum level. To find corrections to s-levels a modified perturbation theory is developed.
Time evolution of a quantum particle in an attractive inverse square potential is investigated. It is found, that quantum mechanical average 〈r^2〉 evolves as quadratic polynomial with time. Based on obtained results, conditions of falling into the attractive potential are found. It is well known, that an attractive inverse square potential does not produce stationary states, but in current work we show, that some quasi-stationary states exist. Such states evolve with 〈r^2〉 being constant in time, namely they neither fall into the center nor escape from it. An example of such a quasi-stationary state is given.
It is shown, that in quantum case a limit of falling exists – a particle cannot fall into an attractive inverse square potential for coupling constants smaller than some critical value.
An attractive inverse square potential is considered in space with different versions of spin noncommutative algebra. It is shown, that potential, and as conclusion total, energy of a particle in an inverse square potential in space with spin noncommutativity has lower limit. From the other hand, using the variational method, an upper estimation for a ground state energy is found. Thereby it is shown, that for sufficiently large coupling constants, in an attractive inverse square potential instead of falling of a particle into the center bound states appear.
Also a theory of electromagnetic field in space with spin noncommutativity of coordinates is built. Tensor of the electromagnetic field, the Lagrange function and its action are found. The constructed action is Lorentz- and C-, P-, T- invariant. Moreover, it is invariant relatively to some gauge transformations, which were found from the condition of gauge invariance of covariant derivative. From the expression for gauge transformation, using the Noether method a modified conservation law of electrical current is found. From the least action principle exact field equations are received. These equations are nonlinear and contain derivatives of all orders, nevertheless they are exact.
Within the considered electrodynamics several electromagnetic systems are considered. It is shown, that spin noncommutativity does not affect an electrostatic field of a single point charge. Nevertheless, because of nonlinearity of considered theory, the electrostatic field of the point charge interacts with an external magnetic field. Such an interaction leads to anisotropic screening of the charge by magnetic field. Also an opposite effect is present – the electric field of the point charge effectively decreases a strength of the magnetic field in the vicinity of the charge.
Nonlinearity of the electrodynamics leads to interaction of two plane waves. To consider the problem we generalized the well-known from classical mechanics Bogolyubov-Krylow method on field theory. A consideration within this method shows, that, besides generation of the higher harmonics, spin noncommutativity modifies a wave vector of some components of the interacting waves.
Despite the complexity of obtained field equations, we find an exact solution of a problem of a plane wave propagation in constant electric and magnetic fields. External fields only change a dispersion law, but do not produce higher harmonics and birefringence. It is interesting, that in this problem, the wave does not modify external fields, how it is in the case of a point charge in an external magnetic field.
