Canonical Path Integral Quantization Research Articles

Degenerate Plebanski sector and spin foam quantization

We show that the degenerate sector of Spin(4) Plebanski formulation of four-dimensional gravity is exactly solvable and describes covariantly embedded SU(2) BF theory. This fact ensures that its spin foam quantization is given by the SU(2) Crane–Yetter model and allows to test various approaches of imposing the simplicity constraints. Our analysis strongly suggests that restricting representations and intertwiners in the state sum for Spin(4) BF theory is not sufficient to get the correct vertex amplitude. Instead, for a general theory of Plebanski type, we propose a quantization procedure which is by construction equivalent to the canonical path integral quantization and, being applied to our model, reproduces the SU(2) Crane–Yetter state sum. A characteristic feature of this procedure is the use of secondary second class constraints on an equal footing with the primary simplicity constraints, which leads to a new formula for the vertex amplitude.

Faddeev–Jackiw canonical path integral quantization for a general scenario, its proper vertices and generating functionals

We generalize the Faddeev-Jackiw canonical path integral quantization for the scenario of a Jacobian with J=1 to that for the general scenario of non-unit Jacobian, give the representation of the quantum transition amplitude with symplectic variables and obtain the generating functionals of the Green function and connected Green function. We deduce the unified expression of the symplectic field variable functions in terms of the Green function or the connected Green function with external sources. Furthermore, we generally get generating functionals of the general proper vertices of any n-points cases under the conditions of considering and not considering Grassmann variables, respectively; they are regular and are the simplest forms relative to the usual field theory.

Canonical Path Integral Quantization of Einstein's Gravitational Field

The connection between the canonical and the path integral formulations of Einstein's gravitational field is discussed using the Hamilton Jacobi method. Unlike conventional methods, its shown that our path integral method leads to obtain the measure of integration with no δ-functions, no need to fix any gauge and so no ambiguous determinants will appear.

Quantization of the O(N) Nonlinear Sigma Model

The Hamilton–Jacobi method of quantizing singular systems is discussed. The equations of motion are obtained as total differential equations in many variables. It is shown that if the system is integrable, one can obtain the canonical phase space coordinates and set of canonical Hamilton–Jacobi partial differential equations without any need to introduce unphysical auxiliary fields. As an example we quantize the O(2) nonlinear sigma model using two different approaches: the functional Schrödinger method to obtain the wave functionals for the ground and the exited state and then we quantize the same model using the canonical path integral quantization as an integration over the canonical phase-space coordinates.

The Canonical Path Integral Quantization of CP1 Model

The Hamilton-Jacobi method of quantizing singular systems is discussed. The equations of motion are obtained as total differential equations in many variables. It is shown that if the system is integrable, then one can obtain the canonical phase space coordinates and the set of the canonical Hamilton-Jacobi partial differential equations without any need to introduce unphysical auxiliary fields. As an example we quantize the CP1 model using the canonical path integral quantization formalism to obtain the path integral as an integration over the canonical phase-space coordinates.

SECOND CLASS CONSTRAINTS IN THE BATALIN-VILKOVISKY LAGRANGIAN BRST QUANTIZATION

The antibracket-antifield BRST formalism developed by Batalin and Vilkovisky is applied to constrained Hamiltonian systems with second class constraints. We derive an effective path integral in which first class and second class constraints are incorporated in a BRST invariant gauge fixed action. Full explicit agreement is found with the canonical path integral quantization of systems with second class constraints and, consequently, with the Lagrangian and Hamiltonian BRST quantization of first order Hamiltonian systems where the second class constraints have been eliminated by introducing the Dirac bracket.

Canonical path-integral quantization of yang-mills field theory with arbitrary external sources

Some modification of source terms is proposed for gauge field theories. In theSU(2) Yang-Mills theory with arbitrary external sources a canonical quantization procedure leads to a Lorentz-invariantS-matrix only when Fermi statistics is imposed on ghost fields. The usual source terms lead to a result that breaks Lorentz invariance and is singular when external chargesJskin0↠ vanish. The cases of the Abelian scalar electrodynamics and theSU(2) Yang-Mills field with external currents (Jskino↠=0,Jskini↠ ≠ 0) are also discussed.

Canonical path integral quantization of gauge systems

It is shown that the (possibly explicity time-dependent) gauge fixing conditions z a ( q, p, t ) = 0, invoked for the canonical quantization of gauge theories via the Faddeev path integral formulation, need not satisfy the restrictions { z a , z b } = 0. As a preparation, we reexamine and clarify how the canonical structure on the large (partly unphysical) phasespace Γ induces, through either Poisson brackets or Lagrange brackets or Dirac brackets, the canonical structure on the physical space of observables. The realizability of some particularly convenient coordinate systems on Γ is also proved.

Canonical path integral quantization of gauge systems: P. Ditsas. Physics Department, University of Crete, Iraklion, Greece and CERN, Geneva

Canonical path integral quantization of gauge systems: P. Ditsas. Physics Department, University of Crete, Iraklion, Greece and CERN, Geneva