Feynman integral for functional Schr\"{o}dinger equations

We consider functional Schr\"{o}dinger 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.Comment: This paper is dedicated to M. I. Vishik and his Seminar at the Moscow State University. LaTex, 16 page

Functional derivatives, Schr\"{o}dinger equations, and Feynman integration

Schr\"{o}dinger equations in \emph{functional derivatives} are solved via quantized Galerkin limit of antinormal functional Feynman integrals for Schr\"{o}dinger equations in \emph{partial derivativesComment: 18 pages, a typo is correcte

A Rigorous Path Integral Construction in any Dimension

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.Comment: 17 page

Quantum Yang-Mills-Weyl Dynamics in Schroedinger paradigm

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 $\mathbb{G}$.Comment: Subections 1.1, 3.1, and 3.2 are revised. Proposition 3.2 is added. More typos are corrected. The main theorem are unchange

Mathematical quantum Yang-Mills theory revisited

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.Comment: Minor changes; misprints are correcte

Feynman Equation in Hamiltonian Quantum Field Theory

Functional Schr\"{o}dinger equations for interacting fields are solved via rigorous non-perturbative Feynman type integrals.Comment: 20 page

Rigorous Covariant Path Integrals

Our rigorous path integrals costruction for the evolution operators is extended to metric-affine manifolds.Comment: Submitted to the Proceedings of the 6th International Conference on Path-Integrals from peV to teV: 50 Years from Feynman's Paper, World Scientific, Singapore, 4 page

Quantum energy-mass spectra of relativistic Yang-Mills fields in a functional paradigm

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.Comment: Revised and corrected version. Mainly of Lemma 3.1 and its proof. arXiv admin note: text overlap with arXiv:1005.377

Energy-mass spectrum of Yang-Mills bosons is infinite and discrete

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.Comment: Thoroughly revised version. New introduction and discussion. The meaning of the bosonic spectrum is clarifie

Quantum energy-mass spectrum of Yang-Mills bosons

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.Comment: 17 pages. Abstract, Introduction, and Subsection on operator symbols are revised. arXiv admin note: substantial text overlap with arXiv:0903.472