Equivalence of Dirac quantization and Schwinger's action principle quantization

We show that the method of Dirac quantization is equivalent to Schwinger's action principle quantization. The relation between the Lagrange undetermined multipliers in Schwinger's method and Dirac's constraint bracket matrix is established and it is explicitly shown that the two methods yield identical (anti)commutators. This is demonstrated in the non-trivial example of supersymmetric quantum mechanics in superspace.     

Geometry and quantization

Gravity is treated geometrically in terms of nonlinear realizations ofGL(4, ) with particular reference to almost complex structures. This approach is used to carry out a Bargmann-Segal type quantization of space-time via the vector and spinor structures of the tangent space that formulates the theory of measurement as a quantum theory quantized in terms of a basic unit of length that appears in a new uncertainty relation. The theory is also used to discuss the gauge conditions for quantum gravity and the Kostant theory of quantization applied using a line bundle with structure groupGL(2, )/SL(2, ).     

The Bohr-Sommerfeld quantization rule and charge quantization in field theory

The “Bohr-Sommerfeld quantization rule” is shown to be, under certain circumstances, equivalent to the quantization of the charge. Charge conservation happens to imply the stability of the semiclassical spectrum of the massive Thirring model.     

Stochastic quantization and regularization

In this paper the method of stochastic quantization introduced by Parisi and Wu is extended to field theories that include fermions and are supersymmetric. A new non-perturbative regulator based on stochastic quantization is introduced. This regulator preserves all the symmetries of the lagrangian, including gauge, chiral, and supersymmetries, at the expense of introducing non-locality.     

Canonical quantization

The dynamical variables of a classical system form a Lie algebra , where the Lie multiplication is given by the Poisson bracket. Following the ideas ofSouriau andSegal, but with some modifications, we show that it is possible to realize as a concrete algebra of smooth transformations of the functionals on the manifold of smooth solutions to the classical equations of motion. It is even possible to do this in such a way that the action of a chosen dynamical variable, say the Hamiltonian, is given by the classical motion on the manifold, so that the quantum and classical motions coincide. In this realization, constant functionals are realized by multiples of the identity operator. For a finite number of degrees of freedom,n, the space of functionals can be made into a Hilbert space using the invariant Liouville volume element; the dynamical variablesF become operators in this space. We prove that for any hamiltonianH quadratic in the canonical variablesq ₁...q _n ,p ₁...p _n there exists a subspace ₁ which is invariant under the action of and , and such that the restriction of to ₁ form an irreducible set of operators. Therefore,Souriau's quantization rule agrees with the usual one for quadratic hamiltonians. In fact, it gives the Bargmann-Segal holomorphic function realization. For non-linear problems in general, however, the operators form a reducible set on any subspace of invariant under the action of the Hamiltonian. In particular this happens for . Therefore,Souriau's rule cannot agree with the usual quantization procedure for general non-linear systems.The method can be applied to the quantization of a non-linear wave equation and differs from the usual attempts in that (1) at any fixed time the field and its conjugate momentum may form a reducible set (2) the theory is less singular than usual.For a particular wave equation , we show heuristically that the interacting field may be defined as a first order differential operator acting onc -functions on the manifold of solutions. In order to make this space into a Hilbert space, one must define a suitable method of functional integration on the manifold; this problem is discussed, without, however, arriving at a satisfactory conclusion.On leave from Physics Department, Imperial College, London SW7.Work partly supported by the Office of Scientific Research, U.S. Air Force.     

Understanding quantization

Foundations of Physics - The metric known to be relevant for standard quantization procedures receives a natural interpretation and its explicit use simultaneously gives both physical and...     

Classical formalisms and quantization

The essential structural features held in common by various classical formalisms are examined. Particular emphasis is placed on those aspects which survive in the transition to a corresponding quantum theory. In particular we develop in some detail the less familiar constraint formalism.     

Topological quantization and cohomology

The relationships between topological charge quantization, Lagrangians and various cohomology theories are studied. A very general criterion for charge quantization is developed and applied to various physical models. The relationship between cohomology and homotopy is discussed.This work was supported in part by the National Science Foundation under Contracts PHY 81-18547; and by the Director, Office of High Energy and Nuclear Physics of the U.S. Department of Energy under Contracts DE-AC03-76SF00098Alfred P. Sloan Foundation Fellow     

Gauge algebra and quantization

Quantization of a general gauge theory in the lagrangian approach is accomplished in closed form. The generating equation is found, containing all the relations of the open gauge algebra. A new class of diagrams is revealed, required by BRS-symmetry, but completely definable only from the requirement of unitarity.     

Scale-covariant quantization

It is argued that conventional operator field equations derived from a stationary action principle are improper when local operator products involve infinite multiplicative renormalizations. An alternative, scale-covariant operator field equation is derived by making scale variations of the field, variations which are everywhere proportional to the field itself. The resultant equation of motion is the same as that previously found on the basis of augmented field theory.     

Algebraic quantization

The different correspondences (or orderings) used in quantum mechanics and the associated deformations, are both seen from an algebraic viewpoint. The deformations which are compatible with the diagonal map (the ₀-deformations) are introduced and connected to the formal groups. A very straighforward example of a ₀-deformation (the multiplicative deformation) appears in the normal quantization of the harmonic oscillator.     

Secondary Quantum Hamiltonian Reductions

Recently, it has been shown how to perform the quantum hamiltonian reduction in the case of general embeddings into Lie (super)algebras, and in the case of general embeddings into Lie superalgebras. In another development it has been shown that when and are both subalgebras of a Lie algebra with , then classically the algebra can be obtained by performing a secondary hamiltonian reduction on . In this paper we show that the corresponding statement is true also for quantum hamiltonian reduction when the simple roots of can be chosen as a subset of the simple roots of . As an application, we show that the quantum secondary reductions provide a natural framework to study and explain the linearization of the algebras, as well as a great number of new realizations of algebras. Received: 18 May 1995 / Accepted: 16 January 1996     

Interplay of topology and quantization: topological energy quantization in a cavity

The interplay between quantization and topology is investigated in the frame of a topological model of electromagnetism proposed by the author. In that model, the energy of electromagnetic radiation in a cubic cavity is where d is a topological integer index equal to the degree of a map between two orbifolds.     

Classical intertwiner space and quantization

Given two symplectic realizations, a symplectic manifold called the classical intertwiner space is introduced as a classical analogue of an intertwiner space of representations of an associative algebra. We describe explicitly how a quantum data on realizations induces a quantum data on their classical intertwiner space.Research partially supported by NSF Grant DMS92-03398 and at MSRI supported by NSF Grant DMS90-22140     

Coherent states and coordinate-free quantization

The usual quantization procedures interpret canonical transformations in an active way linking them with unitary transformations, while the quantization procedure offered by coherent states completely separates classical canonical transformations and unitary operator transformations. By exploiting this property, along with a physically motivated shadow metric, it is seen how to realize the quantization process in as coordinate-free a form as holds in classical mechanics.