Yang-Mills Equations of General Connections and a Certain Solutions in the Quaternionic Hopf Fibration Over Four-Sphere

We investigate a version of Yang-Mills theory by means of general connections. In order to deduce a basic equation, which we regard as a version of Yang-Mills equation, we construct a self-action density using the curvature of general connections. The most different point from the usual theory is that the solutions are given in pairs of two general connections. This enables us to get nontrivial solutions as general connections. Especially, in the quaternionic Hopf fibration over four-sphere, we demonstrate that there certainly exist nontrivial solutions, which are made by twisting the well-known BPST anti-instanton.

Algebra of Superalgebraic Spinors as Algebra of Second Quantization of Fermions

We developed the theory of superalgebraic spinors, which is based on the use of Grassmann densities and derivatives with respect to them in a pseudo-continuous space of momenta. The algebra that they form corresponds to the algebra of second quantization of fermions. We have constructed a vacuum state vector and have shown that it is symmetric with respect to $P$, $CT$ and $CPT$ transformations. Operators $C$ and $T$ transforms the vacuum into an alternative one. Therefore, time inversion $T$ and charge conjugation $C$ cannot be exact symmetries of the spinors.

Multicomponent Nonlinear Evolution Equations of the Heisenberg Ferromagnet Type: Local Versus Nonlocal Reductions

This work is dedicated to systems of matrix nonlinear evolution equations related to Hermitian symmetric spaces of the type $\mathbf{A.III}$. The systems under consideration generalize the $1+1$ dimensional Heisenberg ferromagnet equation in the sense that their Lax pairs are linear bundles in pole gauge like for the original Heisenberg model. Here we present certain local and nonlocal reductions. A local integrable deformation and some of its reductions are discussed as well.

The Transformation of Commutative Phase Space to Noncommutative One, and its Lorentz Transformation-Like Forms

Noncommutative phase space of arbitrary dimension is discussed. We introduce momentum-momentum noncommutativity in addition to co-ordinate-coordinate noncommutativity. We find an exact form for the linear transformation which relates a noncommutative phase space to the corresponding ordinary one. By using this form, we show that a noncommutative phase space of arbitrary dimension can be represented by the direct sum of two-dimensional noncommutative ones. In two-dimension, we obtain the transformation which relates a noncommutative phase space to commutative one. The transformation has the Lorentz transformation-like forms and can also describe the Bopp's shift.

On the Noncommutative Feynman Problem

We extend the Feynman derivation of the Maxwell-Lorentz equations to the case in which coordinates do not commute, adding significantly to previous results. New dynamics is pinned down precisely both at the level of the homogeneous equations and for the Lorentz force, for which a complete derivation is given for the first time.

Quantum Stochastic Products and the Quantum Convolution

A quantum stochastic product is a binary operation on the space of quantum states preserving the convex structure. We describe a class of associative stochastic products, the twirled products, that have interesting connections with quantum measurement theory. Constructing such a product involves a square integrable group representation, a probability measure and a fiducial state. By extending a twirled product to the full space of trace class operators, one obtains a Banach algebra. This algebra is commutative if the underlying group is abelian. In the case of the group of translations on phase space, one gets a quantum convolution algebra, a quantum counterpart of the classical phase-space convolution algebra. The peculiar role of the fiducial state characterizing each quantum convolution product is highlighted.

Lie Theory Suffices to Resolve the Local Classical Problem of Time

The problem of time - a foundational question in quantum gravity - is due to conceptual gaps between GR and physics' other observationally-confirmed theories. Its multiple facets originated with Wheeler-DeWitt-Dirac over 50~years ago. They were subsequently classified by Kucha\v{r}-Isham, who argued that most of the problem is facet interferences and posed the question of how to order the facets. We show the local classical level facets are two copies of Lie theory with a Wheelerian two-way route therebetween. This solves facet ordering and facet interference. Closure by a Lie algorithm generalization of Dirac's algorithm is central.

Star Mittag-Leffler Function

Star product for functions of one variable is given. A deformation of the Mittag-Leffler functions is suggested by means of the star product.

On Coordinate Expressions of Jet Groups and their Representations

A detailed derivation of the jet composition in local coordinates for jet (differential) groups is presented. A suitable faithful representation in matrix groups is demonstrated. Furthermore, Toupin subgroups which occur in continuum mechanics are demonstrated as an example in which representations can be used effectively.

An Integral Formula for a Riemannian Manifold with $k>2$ Singular Distributions

Mathematicians have shown interest in manifolds endowed with several distributions, e.g., webs composed of different regular foliations and multiply warped products, as well as distributions having variable dimensions (e.g., singular Riemannian foliations). In this paper, we extend our previous study of the mixed scalar curvature of two orthogonal singular distributions for the case of $k>2$ singular (or regular) pairwise orthogonal distributions, prove an integral formula with this kind of curvature, and illustrate it by characterizing autoparallel singular distributions.