Dirac fundamental quantization of gauge theories is natural way of reference frames in modern physics

We analyse two principal approaches to the quantization of physical models worked out to date. There are the Faddeev-Popov "heuristic" approach, based on fixing a gauge in the FP path integrals formalism, and the "fundamental" approach by Dirac based on the constraint-shell reduction of Hamiltonians with deleting unphysical variables. The relativistic invariant FP "heuristic" approach deals with the enough small class of problems associated with S-matrices squared taking on-shell of quantum fields. On the other hand, the "fundamental" quantization approach by Dirac involves the manifest relativistic covariance of quantum fields survived the constraint-shell reduction of Hamiltonians. This allows to apply this approach for the more broad class of problems than studying S-matrices. Researches about various bound states in QED and QCD are patterns of such applications. In the present study, with the example of the Dirac "fundamental" quantization of the Minkowskian non-Abelian Higgs model (us studied in its historical retrospective), we make sure in obvious advantages of this quantization approach. The arguments in favour of the Dirac fundamental quantization of physical model as a way of Einstein and Galilei relativity in modern physic will be presented.Comment: v5; the specific approach to the mass gab problem in the Minkowskian BPS monopole model quantized by Dirac was pointed (see p.37 of the present issue); some references are adde

BPS ansatzes as electric form-factors

We argue that BPS ansatzes, entering manifestly vacuum BPS monopole solutions to equations of motion in the (Minkowskian) non-Abelian Higgs model play the role of some electric form-factors and that this implies (soft) violating the CP-invariance of the mentioned model, similar to taking place in the Euclidian Yang-Mills (YM) theory with instantons, generating the $\theta$-term in the appropriate effective Hamiltonian.Comment: v4. important discussion, at the end of Section 2, about dyonic YM vacuum BPS monopole solution

Superfluid properties of BPS monopoles

This paper is devoted to demonstrating manifest superfluid properties of the Minkowskian Higgs model with vacuum BPS monopole solutions at assuming the "continuous" $\sim S^2$ vacuum geometry in that model. It will be also argued that point hedgehog topological defects are present in the Minkowskian Higgs model with BPS monopoles. It turns out, and we show this, that the enumerated phenomena are compatible with the Faddeev-Popov "heuristic" quantization of the Minkowskian Higgs model with vacuum BPS monopoles, coming to fixing the Weyl (temporal) gauge $A_0=0$ for gauge fields $A$ in the Faddeev-Popov path integral.Comment: 19pp; minor correction

Minkowskian Yang-Mills vacuum

The well-known Bogomol'nyi-Prasad-Sommerfeld (BPS) monopole is considered in the limit of the infinite mass of the Higgs field as a basis for constructing the Yang-Mills vacuum with the finite energy density. In this limit the Higgs field disappears at the spatial infinity, but it leaves, nevertheless, its trace as vacuum Yang-Mills BPS monopoles transformed into Wu-Yang monopoles obtained in the pure Yang-Mills theory by a spontaneous scale symmetry breaking in the class of functions with zero topological charges. The topological degeneration of a vacuum BPS monopole manifests itself via Gribov copies of the covariant Coulomb gauge in the form of the time integral of the Gauss law constraint. We also show that, in the considered theory, there is a zero mode of the Gauss constraint involving an "electric" monopole and the additional mass of the $\eta'$-meson in Minkowskian QCD. The consequences of the Minkowskian physical monopole vacuum: rising "golden section" potential and topological confinement, are studied in the framework of the perturbation theory. An estimation of the vacuum expectation value of the square of the magnetic tension is given through the $\eta'$-meson mass, and arguments in favour of the stability of the monopole vacuum are considered. We also discuss why all these "smiles" of the Cheshire cat are kept by the Dirac fundamental quantization, but not by the conventional Faddeev-Popov integral.Comment: LaTeX file, 56 pages, the TEX correction

Fermions with spin 1/2 as global SO(3) vortices

In this paper we show that the nontrivial fundamental group $\pi_1 SO(3) ={\Bbb Z}_2$ for the group SO(3) of global proper rotations of a four-dimensional Euclidian space (when a spin structure is introduced preliminarily in that space) implies always fermions as global SO(3) vortices, while bosons can be reduced to trivial lines (contracted into a point) in the SO(3) group space.Comment: 26 pages, 7 eps.figures, LaTe

Whether the vacuum manifold in the Minkowskian non-Abelian model quantized by Dirac can be described with the aid of the superselection rules?

We intend to show that the vacuum manifold inherent in the Minkowskian non-Abelian model involving Higgs and Yang-Mills BPS vacuum modes and herewith quantized by Dirac can be described with the help of the superselection rules if and only if the "discrete" geometry for this vacuum manifold is assumed (it is just a necessary thing in order justify the Dirac fundamental quantization scheme applied to the mentioned model) and only in the infinitely narrow spatial region of the cylindrical shape where topologically nontrivial vortices are located inside this discrete vacuum manifold.Comment: Some improvements and corrections. arXiv admin note: text overlap with arXiv:hep-th/070109

Superfluidity of Minkowskian Higgs vacuum with BPS monopoles quantized by Dirac may be described as Cauchy problem to Gribov ambiguity equation

We show that manifest superfluid properties of the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac may be described in the framework of the Cauchy problem to the Gribov ambiguity equation. The latter equation specifies the ambiguity in choosing the covariant Coulomb (transverse) gauge for Yang-Mills fields represented as topological Dirac variables, may be treated as solutions to the Gauss law constraint at the removal of temporal components of these fields. We demonstrate that the above Cauchy problem comes just to fixing the covariant Coulomb gauge for topological Dirac variables in the given initial time instant $t_0$ and finding the solutions to the Gribov ambiguity equation in the shape of vacuum BPS monopoles and excitations over the BPS monopole vacuum referring to the class of multipoles. The next goal of the present study will be specifying the look of Gribov topological multipliers entering Dirac variables in the Minkowskian Higgs model quantized by Dirac, especially at the spatial infinity, $| {\bf x} | \to \infty$ (that corresponds to the infrared region of the momentum space).Comment: New, the covering description for topological Dirac variable

Lectures on Topological Aspects of Theoretical Physics

This series of lectures is planned as a generalization of author's large (more than fifteen years) experience of work in the theoretical physics. The modern theoretical physics is based on the group-theoretical approach which generates the formalism of the principal fibre bundles and the instanton approach. The latter is based on the Pontrjagin's degree of map theorem and this theorem is the original ``bridge'' between homology and cohomology theories. The author plans to devote his two first lectures to fibre bundle theory: this is the foundation on which the modern physics rests -- the theory of gauge groups and the Yang-Mills fields. The idea of connection and curvature (first of all for the principal fibre bundles) will be given also. The lectures are devoted to the Pontrjagin's degree of map theorem, to the theories of monopoles and instantons, to the theory of the topological index of the elliptical operator. The information accumulated to this moment allows us to apply these theories to some questions of conformal anomalies and to the topological aspects of QCD. More comprehensive questions of modern topology (for example the algebraic (co)homology theories, the theory of the spectral sequences) will be expounded in the further lectures.Comment: AMSTEX, 57 page

Nontrivial topological dynamics in Minkowskian Higgs model quantized by Dirac

We study the nontrivial topological dynamics inherent in the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac. It comes to persistent collective solid rotations inside the physical BPS monopole vacuum, accompanied by never vanishing vacuum "electric" fields (vacuum monopoles) $\bf E$. The enumerated rotary effects inside the physical BPS monopole vacuum suffered the Dirac fundamental quantization are the specific display of the Josephson effect, whose nature will be reveal in the present study.Comment: 43 pages; it is argued that the nontrivial topological dynamics in the discussed model is possible only at bypassing the Haag's theore

The one example of Lorentz group

The aim of this work is to show, on the example of the behaviour of the spinless charged particle in the homogeneous electric field, that one can quantized the velocity of particle by the special gauge fixation. The work gives also the some information about the theory of second quantisation in the space of Hilbert- Fock and the theory of projectors in the Hilbert space. One consider in Appendix the theory of the spinless charged particle in the homogeneous addiabatical changed electrical field.Comment: LaTeX file, 20 page