A finite-dimensional system with a quadratic Hamiltonian constraint is Dirac quantized in holomorphic, antiholomorphic, and mixed representations. A unique inner product is found by imposing Hermitian conjugacy relations on an operator algebra. The different representations yield drastically different Hilbert spaces. In particular, all the spaces obtained in the antiholomorphic representation violate classical expectations for the spectra of certain operators, whereas no such violation occurs in the holomorphic representation. A subset of these Hilbert spaces is also recovered in a configuration space representation. A propagation amplitude obtained from an (anti)holomorphic path integral is shown to give the matrix elements of the identity operators in the relevant Hilbert spaces with respect to an overcomplete basis of representation-dependent generalized coherent states. The relation to quantization of spatially homogeneous cosmologies is discussed in view of the no-boundary proposal of Hartle and Hawking and the new variables of Ashtekar.
A finite-dimensional system with a quadratic Hamiltonian constraint is Dirac quantized in holomorphic, antiholomorphic, and mixed representations. A unique inner product is found by imposing Hermitian conjugacy relations on an operator algebra. The different representations yield drastically different Hilbert spaces. In particular, all the spaces obtained in the antiholomorphic representation violate classical expectations for the spectra of certain operators, whereas no such violation occurs in the holomorphic representation. A subset of these Hilbert spaces is also recovered in a configuration space representation. A propagation amplitude obtained from an (anti)holomorphic path integral is shown to give the matrix elements of the identity operators in the relevant Hilbert spaces with respect to an overcomplete basis of representation-dependent generalized coherent states. The relation to quantization of spatially homogeneous cosmologies is discussed in view of the no-boundary proposal of Hartle and Hawking and the new variables of Ashtekar.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as [Formula: see text], the true meaning and significance of the so-called charge operators [Formula: see text] and the [Formula: see text]-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.
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
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as PT, the true meaning and significance of the charge operators C and the CPT-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos, and biophysics.
This chapter introduces the holomorphic representation of quantum mechanics, because it allows a study of the properties of boson systems both from the point of view of evolution and of quantum statistical physics (in the so-called second quantization formalism). Path integrals then take the form of integrals over trajectories in phase space in a complex parametrization. A parallel formalism, based on integration over anti-commuting type or Grassmann variables, makes it possible to handle fermions in a way quite analogous to bosons. The corresponding path integral representation of the statistical operator is then derived. The holomorphic formalism is specially well adapted to a study of the harmonic oscillator and, more generally, of perturbed harmonic oscillators. As an illustration, the formalism is applied to the Bose-Einstein condensation. This chapter also considers complex integrals and Wick's theorem, kernel of operators, general Hamiltonians, partition function, and the quantum Bose gas.
We construct explicitly the quantization of classical linear maps of SL(2,{\\Bbb R}) on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of SL(2,{\\Bbb Z}) to the subgroup SL(2,{\\Bbb Z})/\\Gamma_l, \\Gamma_l being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, O_{l}(2)\\subset SL(2,l), for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.
We explicitly construct the quantization of classical linear maps of on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that finite quantum mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of to the subgroup , being the principal congruent subgroup , on a finite dimensional Hilbert space. The generators of the `rotation group' , , for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.