AS we were teaching a multivariate statistics course for doctoral students, one of the students in the class asked,"What are degrees of freedom? I know it is not good to lose degrees of freedom, but what are they?" Other students in the class waited for a clear-cut response. As we tried to give a textbook answer, we were not satisfied and we did not get the sense that our students understood. W...

Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalised complex geometry interpolates between the two of them. On the other hand, the classical and quantum mechanics of a finite number of degrees of freedom are respectively described by a symplectic structure and a complex structure on classical phase space. In this letter we analyse the role pl...

We examine the mechanical matrix model that can be derived from the SU (2) Yang-Mills light-cone field theory by restricting the gauge fields to depend on the light-cone time alone. We use Dirac's generalized Hamiltonian approach. In contrast to its well-known instant-time counterpart the light-cone version of SU (2) Yang-Mills mechanics has in addition to the constraints, generating the SU (2)...

The quantum mechanical concepts of states and operators arise naturally from the question how expectation values of observables can be computed in classical statistical systems where only incomplete local information about the probability distribution is available. The notions of evolution and time are related to the translation of the available information between neighboring local regions. Th...

Gunnar Nordström died just a few years before the advent of quantum mechanics. It is is very interesting, but of course futile to speculate how he would have contributed to the new subject. His early attempts to increase the number of space dimensions [1] could very well have been very important for him when the spin degrees of freedom were understood. The birth of quantum mechanics in the 1920...

Classical model SDcl of the Dirac particle SD is constructed. SD is the dynamic system described by the Dirac equation. For investigation of SD and construction of SDcl one uses a new dynamic method: dynamic disquantization. This relativistic purely dynamic procedure does not use principles of quantum mechanics. The obtained classical analog SDcl is described by a system of ordinary differentia...

Motor systems often require that superfluous degrees of freedom be constrained. For the oculomotor system, a redundancy in the degrees of freedom occurs during visually guided eye movements and is solved by implementing Listing's law and the half-angle rule, kinematic constraints that limit the range of eye positions and angular velocities used by the eyes. These constraints have been attribute...

We here put forward a simple argument for Born’s rule based on the requirement that transition probabilities in quantum mechanics should not be function of number degrees freedom system, or dimension Hilbert-space, respectively. The presented can interpreted as deriving from symmetry requirement.

We discuss quantum dynamics in multi-dimensional non-linear systems. It is well-known that wave function is localized in a single kicked rotor. However, coupling with other degrees of freedom breaks the localization. In order to clarify the difference in the quantum dynamics, we use rigid quantum trajectories, which is accompanied with the de BroglieBohm interpretation of the quantum mechanics....