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arXiv:math-ph/0512089

Quantum Systems with Linear Constraints and Quadratic Hamiltonians

O. Yu. Shvedov

Published Dec 28, 2005• 0 citations
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Abstract

Quantum systems with constraints are often considered in modern theoretical physcics. All realistic field models based on the idea of gauge symmetry are of this type. A partial case of constraints being linear in coordinate and momenta operators is very important. Namely, when one applies semiclassical methods to an arbitrary constrained system, the constraints in "general position case" become linear. In this paper, different mathematicals constructions for the Hilbert space space for the constraint system are discussed. Properties of Gaussian and quasi-Gaussian wave functions for these systems are investigated. An analog of the notion of Maslov complex germ is suggested. Properties of Hamiltonians being quadratic with respect to the coordinate and momenta operators are discussed. The Maslov theorem (it says that there exists a Gaussian eigenfunction of the quantum Hamiltonian iff the classical Hamiltonian system is stable) is generalized to the constrained systems. The case of infinite number degrees of freedom (constrained Fock space) is also discussed.

DOI
10.48550/arxiv.math-ph/0512089

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Quantum systems with constraints are often considered in modern theoretical physcics. All realistic field models based on the idea of gauge symmetry are of this type. A partial case of constraints being linear in coordinate and momenta operators is very important. Namely, when one applies semiclassical methods to an arbitrary constrained system, the constraints in "general position case" become linear. In this paper, different mathematicals constructions for the Hilbert space space for the constraint system are discussed. Properties of Gaussian and quasi-Gaussian wave functions for these systems are investigated. An analog of the notion of Maslov complex germ is suggested. Properties of Hamiltonians being quadratic with respect to the coordinate and momenta operators are discussed. The Maslov theorem (it says that there exists a Gaussian eigenfunction of the quantum Hamiltonian iff the classical Hamiltonian system is stable) is generalized to the constrained systems. The case of infinite number degrees of freedom (constrained Fock space) is also discussed.
Semiclassical mechanics of systems with first-class constraints is developed. Starting from the quantum theory, one investigates such objects as semiclassical states and observables, semiclassical inner product, semiclassical gauge transformations and evolution. Quantum mechanical semiclassical substitutions (not only the WKB-ansatz) can be viewed as "composed semiclassical states" being infinite superpositions of wave packets with minimal uncertainties of coordinates and momenta ("elementary semiclassical states"). Each elementary semiclassical state is specified by a set (X,f) of classical variables X (phase, coordinates, momenta) and quantum function f ("shape of the wave packet" or "quantum state in the background X"). A notion of an elemantary semiclassical state can be generalized to the constrained systems, provided that one uses the refined algebraic quantization approach based on modifying the inner product rather than on imposing the constrained conditions on physical states. The inner product of physical states is evaluated. It is obtained that classical part of X the semiclassical state should belong to the constrained surface; otherwise, the semiclassical state (X,f) will have zero norm for all f. Even under classical constraint conditions, the semiclassical inner product is degenerate. One should factorize then the space of semiclassical states. Semiclassical gauge transformations and evolution of semiclassical states are studied. The correspondence with semiclassical Dirac approach is discussed.
This dissertation addresses the problem of constructing true degrees of freedom in quantum mechanical systems with constraints. The method developed relies on assuming the behavior of the system of interest is ``nearly classical', or semiclassical, in a precise quantitative sense. The approximation is formulated as a perturbative hierarchy of the expectation values of quantum observables and their non-linear combinations, such as spreads and correlations. We formulate the constraint conditions and additional quantum gauge transformations that arise directly on the aforementioned quantities. We specialize this framework to several situations of particular interest. The first situation considered is the case of a constraint that commutes with all quantum observables, which is appropriate for quantization of a Lie algebra with a single Casimir polynomial. Through explicit order-by order counting argue that the true degrees of freedom are captured correctly at each order of the approximation. The rest of the models considered are motivated by the canonical approach to quantizing general relativity. The homogeneous sector of general relativity splits into classes of models, according to topology, which are described by a finite number of degrees of freedom and can therefore be quantized as quantum mechanical systems, making our construction directly applicable. In the Hamiltonian formulation the dynamics of these systems is governed by the Hamiltonian constraint, which, in the quantum theory, gives rise to several conceptual and technical issues collectively known as the Problem of Time. One of the aspects of this problem that does not possess a general solution is the dynamical interpretation of the theory. We use our construction, truncated at the leading order in quantum corrections, together with the intuition gained from dealing with constrained systems in classical mechanics, to define local notion of dynamics relative to a chosen configuration clock variable. We consider a class of models, where the chosen clock in not globally valid, eventually leading to singular dynamics. Within these models we construct an explicit transformation between dynamical evolution relative to two distinct clocks, thus providing a consistent local dynamical interpretation of the true degrees of freedom of the quantum theory.
This thesis is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real
We consider quantum mechanics on constrained surfaces which have non-Euclidean metrics and variable Gaussian curvature. The old controversy about the ambiguities involving terms in the Hamiltonian of order ${\ensuremath{\Elzxh}}^{2}$ multiplying the Gaussian curvature is addressed. We set out to clarify the matter by considering constraints to be the limits of large restoring forces as the constraint coordinates deviate from their constrained values. We find additional ambiguous terms of order ${\ensuremath{\Elzxh}}^{2}$ involving freedom in the constraining potentials, demonstrating that the classical constrained Hamiltonian or Lagrangian cannot uniquely specify the quantization: the ambiguity of directly quantizing a constrained system is inherently unresolvable. However, there is never any problem with a physical quantum system, which cannot have infinite constraint forces and always fluctuates around the mean constraint values. The issue is addressed from the perspectives of adiabatic approximations in quantum mechanics and Feynman path integrals, and semiclassically in terms of adiabatic actions.
A quantization scheme based on the extension of phase space with application of constrained quantization technique is considered. The obtained method is similar to the geometric quantization. For constrained systems the problem of scalar product on the reduced Hilbert space is investigated and a possible solution to this problem is given. Generalization of the Gupta–Bleuler-like conditions is done by minimization of quadratic fluctuations of quantum constraints. The scheme for the construction of generalized coherent states is considered and relation with Berezin quantization is found. The quantum distribution functions are introduced and their physical interpretation is discussed.