arXiv subjects
Publications and source records attributed to Alexander Dynin.
Massless Dirac equation for spinor multiplets is minimally coupled with a unitary representation of an arbitrary compact semisimple gauge group. The spectrum of the second quantized interaction Hamiltonian has a positive mass gap running along the classical energy scale.
A mathematically rigorous relativistic quantum Yang-Mills theory with an arbitrary semisimple compact gauge Lie group is set up in the Hamiltonian canonical formalism. The theory is non-perturbative, without cut-offs, and agrees with the causality and stability principles. This paper presents a fully revised, simplified, and corrected version of the corresponding material in the previous papers DYNIN[11] and [12]. The principal result is established anew: due to the quartic self-interaction term in the Yang-Mills Lagrangian along with the semisimplicity of the gauge group, the quantum Yang-Mills energy spectrum has a positive mass gap. Furthermore, the quantum Yang-Mills Hamiltonian has a countable orthogonal eigenbasis in a Fock space, so that the quantum Yang-Mills spectrum is point and countable. In addition a fine structure of the spectrum is elucidated. KEYS: Millennium Yang-Mills problem; Finite propagation speed; Sobolev inequalities; Nuclear vector spaces; Infinite-dimensional holomorphy; Friedrichs operator extensions; Variational spectral principle; Symbols and spectral theory of pseudo-differential operators.
Inspired by F. Wilczek's QCD Lite, quantum Yang-Mills-Weyl Dynamics (YMWD) describes quantum interaction between gauge bosons (associated with a simple compact gauge Lie group $\mathbb{G}$) and larks (massless chiral fields colored by an irreducible unitary representation of $\mathbb{G}$). Schroedinger representation of this quantum Yang-Mills-Weyl theory is based on a sesqui-holomorphic operator calculus of infinite-dimensional operators with variational derivatives. The spectrum of the quantum YMWD, with initial data in the central euclidean ball of a radius $0<R<+\infty$, is self-similar in the inverse proportion to $R$. The spectrum is a sequence of eigenvalues convergent to $+\infty$. The eigenvalues have finite multiplicities with respect to a von Neumann algebra with a regular trace. The same holds for the quantum self-interaction of vector Yang-Mills bosons (Theorem 4.1). Furthermore, the fundamental vacuum eigenvalue is a simple zero (Appendix A). Presumably, this is a solution of the existence problem for a quantum Yang-Mills theory that implies a positive spectral mass gap. The rigorous mathematical theory is non-perturbative with a running coupling constant as the only ad hoc parameter. The application of the first mathematical principles depends essentially on the properties of the compact simple Lie group $\mathbb{G}$.
A non-perturbative and mathematically rigorous quantum Yang-Mills theory on 4-dimensional Minkowski spacetime is set up in the functional framework of a complex nuclear Kree-Gelfand triple. It involves a symbolic calculus of operators with variational derivatives and a new kind of infinite-dimensional ellipticity. In the temporal gauge and Schwinger first order formalism, Yang-Mills equations become a semilinear hyperbolic system for which the general Cauchy problem is reduced to initial data with compact supports. For a simple compact Yang-Mills gauge group and the anti-normal quantization of Yang-Mills energy-mass functional of initial data in a box, the quantum energy-mass spectrum is a sequence of non-negative eigenvalues converging to infinity. In particular, it has a positive mass gap. Furthermore, the energy-mass spectrum is self-similar (including the mass gap) in the inverse proportion to an infrared cutoff of the classical energy scale.
A non-perturbative quantization of the Yang-Mills energy-mass functional with a compact semi-simple gauge group entails an infinite discrete energy-mass spectrum of gauge bosons. The bosonic spectrum is bounded from below, and has a positive mass gap due to the quartic self-interaction term of pure Yang-Mills Lagrangian (with no Higgs term involved). This quantization is based on infinite-dimensional analysis in Kree nuclear triple of sesqui-holomorphic functionals of initial data for the the non-linear classical Yang-Mills equations in the temporal gauge.
A non-perturbative anti-normal quantization of relativistic Yang-Mills fields with a compact semisimple gauge group entails an infinite discrete bosonic energy-mass spectrum of gauge bosons in the framework of Gelfand nuclear triples. The quantum spectrum is bounded from below and has a positive mass gap. The spectrum is both Poincare and gauge invariant.
A bosonic-fermionic correspondence allows an analytic definition of functional super derivative, in particular, and a bosonic functional calculus, in general, on Bargmann- Gelfand triples for the second super quantization. A Feynman integral for the super transformation matrix elements in terms of bosonic anti-normal Berezin symbols is rigorously constructed.
Schrödinger equations in \emph{functional derivatives} are solved via quantized Galerkin limit of antinormal functional Feynman integrals for Schrödinger equations in \emph{partial derivatives
We consider functional Schrödinger equations associated with a wide class of Hamiltonians in all Fock representations of the bosonic canonical commutation relations, in particular the Cook-Fock, Friedrichs-Fock, and Bargmann-Fock models. An infinite-dimensional symbolic calculus allows us to prove the convergence of the corresponding Hamiltonian Feynman integrals for propagators of coherent states.
Functional Schrödinger equations for interacting fields are solved via rigorous non-perturbative Feynman type integrals.
Our rigorous path integrals costruction for the evolution operators is extended to metric-affine manifolds.
We propose a new rigorous time-slicing construction of the phase space Path Integrals for propagators both in Quantum Mechanics and Quantum Field Theory for a fairly general class of quantum observables (e.g. the Schroedinger hamiltonians with smooth scalar potentials of any power growth). Moreover we allow time-dependent hamiltonians and a great variety of discretizations, in particular, the standard, Weyl, and normal ones.