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A quantization method (strictly generalizing the Kostant-Souriau theory) is defined, which may be applied in some cases where both Kostant-Souriau prequantum bundles and metaplectic structures do not exist. It coincides with the Czyz theory for compact Kähler manifolds with locally constant scalar curvature. Quantization of dynamical variables is defined without use of intertwining operators, extending...
I. Galilei invariant quantization The geometric quantization scheme enables one to quantize time dependent dynamics without introducing any reference frame. The resulting theory is equivalent to the one described by a time-dependent Schroedinger equation. II. Quantization of non-relativistic dynamics with spin Geometric quantization of a classical model of a particle with...
A quantization procedure is proposed starting with the Lie algebra $$\mathbb{G}$$ of infinitesimal symmetries of a system and a $$\mathbb{G}$$ -action on a principle bundle . For quasi-complete $$\mathbb{G}$$ -actions the constructed vector field operators are essentially skew adjoint and can be interpreted as canonical momentum observables of a local Heisenberg system. Integrability of general...
Recent work on formulating relativistic quantum mechanics on stochastic phase spaces is described. Starting with a brief introduction to the mathematical theory of stochastic spaces, an account is given of non-relativistic quantum mechanics on stochastic phase space. The relativistic theory is introduced by constructing certain classes of representations of the Poincaré group on phase space, obtaining...
The mathematical description of quantum mechanics of charged particles in the field of magnetic monopoles seems to reach a rather consistent level. As we have seen, there might even be a natural mechanism which provides the confinement of magnetic poles. In principle, this could be checked by solving the Dirac equation for an electron in the field of a monopole-antimonopole pair.
Using the analogy between the spectrum-generating SU(n) approach in particle physics and the dynamical group approach in atomic and molecular physics, we outline the basic ideas behind this alternative to broken-symmetry SU(n) approaches. We review various tests of dynamical SU(3) and SU(4) method, and discuss in particular two crucial tests of the fundamental assumptions.
We consider nonlinear σ-models, gauge theories and general relativity as three classes of models of field theory which are of an intrinsically geometric nature as well as (possibly) topologically nontrivial, and explore the role of instantons as the basic tool for new perturbative schemes in these models. In particular, we emphasize the close analogy between nonlinear σ-models and pure gauge theories...
Salam's SL(6,ℂ) gauge theory of strong interactions is generalized to one having GL(2f,ℂ) ⊗ GL(2c,ℂ or the affine extension thereof as structure group. The concept of fibre bundles and Lie-algebra-valued differential forms are employed in order to exhibit the geometrical structure of this gauge-model. Its dynamics is founded on a gauge-invariant Einstein-Dirac-type Lagrangian. The Heisenberg-Pauli-Weyl...
Describing the projective structure P (given by the set of “freely falling particles”) and the conformal (light cone) structure ℂ of space time via subbundles of second order frame bundles, we investigate the existence and uniqueness of a Weyl geometry compatible with P and ℂ We first review some basic notions concerning the fibre bundle description of geometric structures on differentiable...
The first result of this paper is that the total Gaussian curvature of a compact Lorentz manifold vanishes (Theorem I, sec. 8). The argument in the proof is different from those given in [1] and [2]. In fact, we make extensive use of the existence of a global line field on a Lorentz manifold, rather than reducing the problem to the classical case of a Riemannian manifold. In §4 the index of...
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