We discuss deformation quantization in quantum mechanics and quantum field theory. We begin with a discussion of the mathematical question of deforming the commutative algebra of functions on a manifold into a non-commutative algebra by use of an associative product. We then apply these considerations to the commutative algebra of observables of a classical dynamical system, which may be deformed to the non-commutative algebra of quantum observables. This is the process of deformation quantization, which provides a canonical procedure for finding the measurable quantities of a quantum system. The deformation quantization approach is illustrated, first for the case of a simple harmonic oscillator, then for an oscillator coupled to an external source, and finally for a quantum field theory of scalar bosons, where the well-known formula for the number of quanta emitted by a given external source in terms of the Poisson distribution is reproduced.

The relation of the star product method to the better-known methods involving the representation of observables as linear operators on a Hilbert space, or the representation of expectation values as functional integrals, is analyzed. The final lecture deals with a remarkable formula of Cattaneo and Felder, which relates Kontsevich’s star product to an expectation value of a product of functions on a Poisson space, and indicates how this formula may be interpreted.

We introduce orbital functionals $\int \beta$ simultaneously for each commensurability class of orbital surfaces. They are realized on infinitely dimensional orbital divisor spaces spanned by (arithmetic-geodesic real 2-dimensional) orbital curves on any orbital surface. We discover infinitely many of them on each commensurability class of orbital Picard surfaces, which are real 4-spaces with cusps and negative constant Kähler–Einstein metric degenerated along an orbital cycle. For a suitable (Heegner) sequence $\int \mathbf{h}_N, N \in \mathbb{N}$, of them we investigate the corresponding formal orbital $q$-series $\sum_{N=0}^{\infty} (\int \mathbf{h}_N)q^N$. We show that after substitution $q = {\rm e}^{2 \pi \mathbf{i} \tau}$ and application to arithmetic orbital curves Ĉ on a fixed Picard surface class, the series $\sum_{N=0}^{\infty}(\int_{Ĉ} {\mathbb{h}_N})e^{2 \pi \mathbf{i} \tau}$ define modular forms of well-determined fixed weight, level and Nebentypus. The proof needs a new orbital understanding of orbital heights introduced in [12] and Mumford–Fulton’s rational intersection theory on singular surfaces in Riemann–Roch–Hirzebruch style. It has to be connected with Zeta and Theta functions of hermitian lines, indefinite quaternionic fields and of a matrix algebra along a research marathon over 75 years represented by Cogdell, Kudla, Hirzebruch, Zagier, Shimura, Schoeneberg and Hecke. Our aim is to open a door to an effective enumerative geometry for complex geodesics on orbital varieties with nice metrics.

Equivariant cohomology in general and the equivariant localization theorems in particular have taken on a role of increasing significance in theoretical physics of late (see e.g. [3], [4] and [10]). These lectures are an attempt to provide a self-contained and elementary introduction to the Cartan model of equivariant cohomology, a complete proof of the simplest of the localization theorems, and, as an application, a proof of the famous Duistermaat–Heckman theorem on exact stationary phase approximations.

If $M$ is a Riemannian space we denote by $g$ its Riemannian metric and by $\nabla$ its Riemannian connection. Additionally we assume that $M$ admits a non-symmetric metric connection $\nabla^{\ast}$. Let $C: x^i = x^{i}(s)$ be a curve in $M_n (\nabla, g)$ parametrized by the arc length. This curve can also be considered to be in $M_n (\nabla^{\ast},g)$. Using the normal curvature of $C$ in $M_{n}(\nabla^{\ast}, g)$ one can obtain the asymptotic lines of $M_{n}(\nabla^{\ast},g)$.

A curve in a hypersurface is defined to be an asymptotic line if the normal curvature along the curve vanishes identically. Here we investigate the special nets which are formed by the tangent fields of asymptotic lines in $M_{n}(\nabla^{\ast},g)$ from that of the nets in $M_{n}(\nabla,g)$ and by using the coefficients of $\nabla^{\ast}$ and ${\nabla}$.

In the Batalin–Vilkovisky field-antifield formalism a classical mechanical system is described by a solution of the classical master equation. The quantization of this general gauge theory in the Lagrangian approach can be accomplished in closed form. The AKSZ-formalism is a geometrical construction of such a solution as a $QP$-manifold. This can be extended and applied to topological quantum field theories.

We propose to determinate the nonadiabatic Hannay’s angle of spin one half in a varying external magnetic field, by using an averaged version of the variational principal. We also show how the evolution and this nonadiabatic Hannay’s angle is associated with the evolution of Grassmannian invariant-angle coherent states.

Let $L$ defines a regular problem in the calculus of variations on supermanifolds. The necessary conditions for a piecewise superdifferentiable supercurve $C$ in sense of Rogers to be a weak local minimum relative to two sub-supermanifolds are given.

We study reparametrization-invariant systems, mainly the relativistic particle and its $D$-dimensional extended object generalization $d$-brane. The corresponding matter Lagrangians naturally contain background interactions, like electromagnetism and gravity. For a $d$-brane that doesn’t alter the background fields, we define non-relativistic equations assuming integral sub-manifold embedding of the $d$-brane. The mass-shell constraint and the Klein–Gordon equation are shown to be universal when gravity-like interaction is present. Our approach to the Dirac equation follows Rund’s technique for the algebra of the $\gamma$-matrices that doesn’t rely on the Klein–Gordon equation.

Let $\mathcal{I}$ be Frobenius system of even rank. Consider a closed two-form $\Pi \in \mathcal{I} \bigwedge \mathcal{I}$ of maximal rank. A vector field $X$such that $L_X \Pi = 0$ is called a symmetry of $\Pi$. We determine the relationship between the solvable Lie group of symmetries of $\Pi$ and the rank of the reduced system obtained from $\mathcal{I}$ by Lie reduction. For an Euler–Lagrange system of ODE’s with the corresponding Lagrangian $L$, $\Pi$ can be taken to be the differential of the Poincaré–Cartan form $\eta_L$. A symmetry of $\Pi = \mathbf{d}\eta_L$ is a variational symmetry of the Lagrangian $L$. A proof of Noether’s theorem for Frobenius systems of even rank is provided.

The image $B_{p^{\sigma},q}$ of a product of balls $B_p \times B_q$ under a compression $c_{\sigma}(X,V) = (X,V(1 - ^t(\bar{X}X)^{\frac{\sigma}{2})}$ is called acompressed product of balls of exponent $\sigma \in \mathbb{R}$. The present note obtains the group Aut($B_{p^{\sigma},q}$) of the holomorphic automorphisms and the Aut($B_{p^{\sigma},q}$)-orbit structure of $B_{p^{\sigma},q}$ and its boundary ${\partial}B_{p^{\sigma},q}$ for $\sigma \gt 1$. The Bergman completeness of $B_{p^{\sigma},q}$ is verified by an explicit calculation of the Bergman kernel. As a consequence, local lower boundary estimates on the Bergman kernels of the bounded pseudoconvex domains are obtained, which are locally inscribed in $B_{p^{\sigma},q}$ at a common boundary point.

We construct explicit parametrizations in terms of elliptic functions for unduloids depending on a parameter. We then use these parametrizations to study geodesics on unduloids. In particular, we use Maple to find interesting closed geodesics on unduloids.

The Dirac problem for the charged heavy top is discussed and some of its properties are pointed out.

In this paper, we show that a totally umbilical hypersurface of a recurrent Weyl space is conformally recurrent. Also, while a totally umbilical hypersurface of a recurrent Weyl space is conharmonically recurrent or conharmonically Ricci-recurrent, theorems concerning some special nets are proved.

The stability problem and the existence of periodic orbits for the heavy top dynamics are discussed and some of its properties are pointed out.

In this paper we study harmonic forms on compact symplectic nilmanifolds. We consider harmonic cohomology groups of dimension 3 and of codimension 2 for 2-step nilmanifolds and give examples of compact 2-step symplectic nilmanifolds $G / \Gamma$ such that the dimension of harmonic cohomology groups varies.

We investigate the fundamental system of equations in the theory of conformal geometry, whose coefficients are considered as the conformal Schwarzian derivative. The integrability condition of the system is obtained in a simple method, which allow us to find a natural geometric structure on the solution space. From the solution spaces, using this geometric structure, we get a transformation whose Schwarzian derivative is equal to the given coefficients of the equation.

For any Lie algebra $\mathfrak{g}$ we introduce the notion of $\mathfrak{g}$-symplectic structures and show that every orbit of a principal $G$-bundle carries a natural $\mathfrak{g}$-symplectic form and an associated momentum map induced by the Maurer–Cartan form on $G$. We apply this to the BRST bicomplex and show that the associated momentum map is a solution of the Wess–Zumino consistency condition for the anomaly.

A difference $q$-analogue of the dressing chain is considered in this paper. The relation between the discrete and continuous models is also discussed.

In this work we present a brief summary of the “geometry of mass”. We show how the notion of mass of elementary particles is related to the geometrical concept of curvature. In particular, the bosonic mass matrices are related to the extrinsic curvature of specific sub-manifolds of the Higgs bundle and the underlying gauge bundle. In contrast, the mass matrix of the fermions is related to the intrinsic curvature of bundles that geometrically represent “free fermions” within the context of spontaneously broken Yang–Mills gauge theories

Since Pryce’s 1948 study of the relativistic center of momentum it is believed that the classical notion of the center of mass can be extended to special relativity theory only in “at least one Lorentz frame” (in Goldstein’s words). The aim of this work is, accordingly, to study the relativistic center of mass in a way fully analogous to its Newtonian counterpart.

Some results on global symplectic forms defined by local Lagrangians of a tangent manifold, studied earlier by the author, are summarized without proofs.

The Moyal product is considered on the complex plane $\mathbb{C}^2$. Path integral representation of $\ast$-exponential function is given for a quadratic form on $\mathbb{C}^2, H = ax^2 + 2bxy +cy^2$ for $(x,y) \in \mathbb{C}^2$, where $a,b,c \in \mathbb{C}$.