The issue of non‐perturbative background independent quantization of matrix models is addressed. The analysis is carried out by considering a simple matrix model which is a matrix extension of ordinary mechanics reduced to 0 dimension. It is shown that this model has an ordinary mechanical system evolving in time as a classical solution. But in this treatment the action principle admits a natural modification which results in algebraic relations describing quantum theory. The origin of quantization is similar to that in Adler’s generalized quantum dynamics. The problem with extension of this formalism to many degrees of freedom is solved by packing all the degrees of freedom into a single matrix. The possibility to apply this scheme to various matrix models is discussed.
The issue of non‐perturbative background independent quantization of matrix models is addressed. The analysis is carried out by considering a simple matrix model which is a matrix extension of ordinary mechanics reduced to 0 dimension. It is shown that this model has an ordinary mechanical system evolving in time as a classical solution. But in this treatment the action principle admits a natural modification which results in algebraic relations describing quantum theory. The origin of quantization is similar to that in Adler’s generalized quantum dynamics. The problem with extension of this formalism to many degrees of freedom is solved by packing all the degrees of freedom into a single matrix. The possibility to apply this scheme to various matrix models is discussed.
Despite the seminal connection between classical multiply periodic motion and Heisenberg matrix mechanics and the massive amount of work done on the associated problem of semiclassical Einstein-Brillouin-Keller (EBK) quantization of bound states, we show that there are, nevertheless, a number of previously unexploited aspects of this relationship that bear on the quantum-classical correspondence. In particular, we emphasize a quantum variational principle that implies the classical variational principle for invariant tori. We also expose the more indirect connection between commutation relations and quantization of action variables. In the special case of a one-dimensional system a different and succinct algebraic derivation of the WKB quantization rule for bound states is given. With the help of several standard models with one or two degrees of freedom, we then illustrate how the methods of Heisenberg matrix mechanics described in this paper may be used to obtain quantum solutions with a modest increase in effort compared to semiclassical calculations. We also describe and apply a method for obtaining leading quantum corrections to EBK results. Finally, we suggest several modified applications of EBK quantization. \textcopyright{} 1996 The American Physical Society.
The problem of semiclassical quantization of nonseparable systems with a finite number of degrees of freedom is studied within the framework of Heisenberg matrix mechanics, in extension of previous work on one-dimensional systems. The relationship between the quantum theory and multiply-periodic classical motions is derived anew. A suitably averaged Lagrangian provides a variational basis not only for the Fourier components of the semiclassical equations of motion, but also for the general definition of action variables. A Legendre transformation to the Hamiltonian verifies that these have been properly chosen and therefore provide a basis for the quantization of nonseparable systems. The problem of connection formulas is discussed by a method integral to the present approach. The action variables are shown to be adiabatic invariants of the classical system. An elementary application of the method is given. The methods of this paper are applicable to nondegenerate systems only.
After a brief review of the extensive work done on the theory of invariant tori and their quantization, we show that nevertheless an important connection between the quantum and classical theories remains to be exploited. This is the relationship between matrix elements of operators in the energy diagonal representation and Fourier components of the corresponding classical dynamical variables that was the basis for Heisenberg's invention of quantum mechanics. We describe a number of previously unknown or little-known aspects of this relationship, with special emphasis on variational principles and the connection between commutation relations and quantization of action variables. As a single illustration of the utility of these ideas we show that it is possible to obtain approximate solutions to the quantum scheme that are more accurate than the semiclassical approximation with little additional effort compared to the latter.
A theory of quantization is given, for a model with symmetry, as the inverse of dequantization. Dequanrization is defined as tin� process of extracting the classical mechanics of a model from its quantal counterpart. It is shown that classical dynamics is embedded in quantum dynamics as a constrained subsystem. Thus, classical mechanics is obtained from quantum mechanics without taking limits of large-dimensional representations or letting h �? 0. The embedding shows that, under appropriate conditions, a quantal density can be identified with a classical state. The results are consistent with the conjecture that an algebraic model is quantized by constructing a unitary irrep of its spectrum generating algebra. Quantization of operators not in this algebra is determined by compatibility with dequantization: this method resolves the Dirac problem in a way that avoids the ambiguities of canonical quantization. Geometric quantization is used to express the vector coherent state inducing construction as a theory of quantization. The result provides physically useful quantum models for systems with an algebraic structure and intrinsic degrees of freedom. The results are illustrated by application to the S i (3) model of a rotor. An algebraic model of a rigid rotor is presented, anti the techniques of vector coherent state quantization are applied to give a quantization, including intrinsic degrees of freedom, that is compatible with the geometric structure of the corresponding classical model. An explicit dequantization of the asymmetric top, leading to Euler�s equations, illustrates the consistency of this quantization.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
We state in the following a structure theory of the density matrices from the standpoint of secind quantization. The difficulties involved in the quantum-mechanical many body problem have so far proved to be so tremendous that as yet some have been obliged to stop before exact treatments and others have retired to phenomenological model theories. There are two main tools to approach the problem, the theory of second quantization and the theory of density matrix. The former, in spite of its generality and compactness, has never revealed us its adaptability to the practical problems of quantum statistical mechanics. In fact, about the exchange effect, namely the statistical correlation of particles, the latter informs us the detailed features of it by application of the group theory, while the former lacks, at present, such an exposition. For a long time a more general and simpler formulation than the current group theoretic method has been desired in the theoretical researches of spectroscopy. A distinguished contribution has been made by P. Jordan (1934) and H. Ostertag (1937) from the standpoint of second quantization. H. Ostertag made use of the theory of hypercomplex ring and introdeced the original concept of the elementary algebra. (Part II) Since the latter part of his formulation, however, seems to be complicated, it will be better, for clearness' sake, to avoid this part. Accordingly the concepts of the elementary algebra and the symmetric algebra will be introduced as a remedy for it. Then we shall complete our theory to include the whole theory of symmetric group. These concepts will serve to clarify and generalize the previous formulations and to unify the several group theoretic standpoints of the spectroscopy. In §1 and §2 the representation of the quantized density of Bose or Fermi particles will reveal itself as the density matrix of Bose or Fermi particles or as the density matrix of many electrons with a certain term of spin multiplets. Indeed, the theory of the statistical operator, or the density matrix of J. von Neumann and P.A.M. Dirac, the equilibrium properties of which have been fully examined by one of the authors (1940), is the natural formalism of dealing with quantum gases. A contribution has been made by W. Kofink (1938) to introduce the intrinsic degrees of freedom of particles. This attempt has resulted in a fair seccess. His formulation consists in making use of the initial stages of the theory of spectroscopy, which will be reexamined in the second section and Part II. In §3, as an example at zero temperature, the Hartree-Fock equation for a prescribed orbital and spin configuration will be derived, and in §4 some properties of the reduced density matrix will be illustrated.