Canonical transformations to action and angle variables and their representation in quantum mechanics IV. Periodic potentials

In the previous papers of this series we discussed the representation in quantum mechanics of canonical transformations leading to action and angle variables, for Hamiltonians with bounded or unbounded orbits, i.e., whose spectra is either discrete, continuous, or mixed. In the present paper the results are extended to Hamiltonians with periodic potentials which have a band spectra. Again the canonical transformations are non-linear and non-bijective and the classical analysis shows that the angle variable φ (always in the interval 0≤φ≤2π) and action J can be defined for energies both below and above the maximum height of the potential. In all of the original phase space the variables (q, p) are then periodic functions of φ. Inversely, because of the invariance of the Hamiltonian under translations q → q + a, the (φ, J) are also periodic functions of q. thus to recover bijectiveness we require an infinite sheet structure in both the (q, p) and (φ, J) phase spaces. In turn the sheet structure can be replaced by appropriate ambiguity groups and spins, with the help of which we propose an explicit expression for the representation in quantum mechanics of the canonical transformation, and recover the latter when we pass to the classical limit with the help of the WKB approximation. The present analysis corroborates the previous surmise that the nature of the spectra of a quantum mechanical Hamiltonian, i.e., continuous, discrete, mixed, or of bands, is related to the ambiguity group and spin of the problem. As the latter originates in classical mechanics when we discuss the canonical transformations from (q, p) to (φ, J), we conclude that some quantum features, such as the nature of spectra of operators, are already implicit in the classical picture.     

Canonical transformations to action and angle variables and their representation in quantum mechanics II. The coulomb problem

The ambiguities in the classical canonical transformation leading to action and angle variables of the attractive Coulomb Problem (suitably generalized for positive energies) and the phase space structure they entail are analyzed. The ambiguity group is found to be different for positive and negative energies. Nevertheless it continues to be the essential concept for the construction of the quantum mechanical representation of the classical transformation, which we explicitly obtain.     

Canonical transformations in quantum mechanics

This paper presents the general theory of canonical transformations of coordinates in quantum mechanics. First, the theory is developed in the formalism of phase space quantum mechanics. It is shown that by transforming a star-product, when passing to a new coordinate system, observables and states transform as in classical mechanics, i.e., by composing them with a transformation of coordinates. Then the developed formalism of coordinate transformations is transferred to a standard formulation of quantum mechanics. In addition, the developed theory is illustrated on examples of particular classes of quantum canonical transformations.     

Parametrization of Ordering Ambiguity and Its Representation in Path Integral Formalism

ln this paper, we establish three different types (type I, type Ⅱ and type Ⅲ) of pararnctrization of ordering ambiguity in operator formalism of quantum mechanics and discuss their nontrivial relations. Some commonly occurred bewildcrment about ordcring problem is made clear. We also show that the ordering ambiguity exists in path intergral formalism and the representation of the ordering ambiguity in the path integral formalism (discrete schemes ) has a one-to-one correspondence with type Ⅱ parametrization of the ordering ambiguity in the operator formalism. As a result, we show that to select a physically a misled atempt,Instead, we have to work with type Ⅲ parametrization in the operator forrnalisim to reduce the ordering ambiguity by physical requirements.     

Squeezing and Quantum Canonical Transformations

In this paper we present an approach to quantum mechanical canonical transformations. Our main result is that time-dependent quantum canonical transformations can always be cast in the form of squeezing operators. We revise the main properties of these operators in regard to its Lie group properties, how two of them can be combined to yield another operator of the same class and how can also be decomposed and fragmented. In the second part of the paper we show how this procedure works extremely well for the time-dependent quantum harmonic oscillator. The issue of the systematic construction of quantum canonical transformations is also discussed along the lines of Dirac, Wigner, and Schwinger ideas and to the more recent work by Lee. The main conclusion is that the classical phase space transformation can be maintained in the operator formalism but the construction of the quantum canonical transformation is not clearly related to the classical generating function of a classical canonical transformation. We hereby propose the much more efficient method given by the squeezing operators. This method has also been proved to be very useful, by one of the authors, in the framework of the dynamical symmetries (Cerveró, J. M. (1999). International Journal of Theoretical Physics 38, 2095–2109).     

Standard quantum mechanics featuring probabilities instead of wave functions

A new formulation of quantum mechanics (probability representation) is discussed. In this representation, a quantum state is described by a standard positive definite probability distribution (tomogram) rather than by a wave function. An unambiguous relation (analog of Radon transformation) between the density operator and a tomogram is constructed both for continuous coordinates and for spin variables. A novel feature of a state, tomographic entropy, is considered, and its connection with von Neumann entropy is discussed. A one-to-one map of quantum observables (Hermitian operators) on positive probability distributions is found.     

Operator ordering and supersymmetry

We investigate supersymmetric quantum mechanics in curved space. In particular we discuss, in the framework of the canonical formulation, the role of supersymmetry (SUSY) and of invariance under general coordinate transformations (GCT) in solving the operator ordering ambiguity. It is shown that GCT invariance is instrumental to achieve such result. We also examine the connection with the Weyl ordering of operators and the relevance of our results for the path integral approach.     

Relativistic quantum mechanics of supersymmetric particles

The quantum mechanics of an electron in an external field is developed by Hamiltonian path integral methods. The electron is described classically by an action which is invariant under gauge supersymmetry transformations as well as worldline reparametrizations. The simpler case of a spinless particle is first reviewed and it is pointed out that a strictly canonical approach does not exist. This follows formally from the gauge invariance properties of the action and physically it corresponds to the fact that particles can travel backwards as well as forward in coordinate time. However, appropriate application of path integral techniques yields directly the proper time representation of the Feynman propagator. Next we extend the formalism to systems described by anticommuting variables. This problem presents some difficulty when the dimension of the phase space is odd, because the holomorphic representation does not exist. It is shown, however, that the usual connection between the evolution operator and the path integral still holds provided one indludes in the action the boundary term that makes the classical variational principle well defined. The path integral for the relativistic spinning particle is then evaluated and it is shown to lead directly to a representation for the Feynman propagator in terms of two proper times, one commuting, the other anticommuting, which appear in a symmetric manner. This representation is used to derive scattering amplitudes in an external field. In this step the anticommuting proper time is integrated away and the analysis is carried in terms of one (commuting) proper time only, just as in the spinless case. Finally, some properties of the quantum mechanics of the ghost particles that appear in the path integral for constrained systems are developed in an appendix.     

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method

By virtue of the new technique of performing integration over Dirac’s ket–bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel–Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.     

Stochastic physical origin of the quantum operator algebra and phase space interpretation of the Hilbert space formalism: The relativistic spin zero case

Based on a recent association of quantum observable algebra with stochastic processes in the frame of the causal stochastic interpretation of quantum mechanics, a relativistic Hilbert space is defined for the Klein-Gordon case. It is demonstrated that unitary transformations in Hilbert space reflect canonical transformations in the associated phase space, manifesting thus an underlying symplectic structure.     

Canonical averaging of the Schrödinger equation

The representation of the Schrödinger equation in the form of a classical Hamiltonian system makes it possible to construct a unified perturbation theory that is based on the theory of canonical transformations and covers both classical and quantum mechanics. Also, the closeness of the exact and approximate solutions of the Schrödinger equation can be approximately estimated with such a representation.     

Nonperturbative canonical formulation and Ward-Takahashi relations for quantum many-body systems at finite temperatures

A nonperturbative canonical field-theoretic approach to Matsubara's finite temperature quantum many-body theory is presented. The finite temperature Ward-Takahashi relations for the phase transformation, the Galilean transformation and general time-independent continuous transformations are derived for both fermions and bosons with interactions. The finite temperature Ward-Takahashi relations for nonconserved currents and the finite temperature Goldstone's theorem are also discussed.     

Quantum-mechanical representations of canonical transformations and adiabatic invariance

A representation of the canonical transformation to action-angle variables is obtained using the (classical) adiabatic invariance of the actions. It is shown that in one dimension this is in fact the exact quantum-mechanical formula for this representation and hence provides a formally exact expression for the eigenfunctions. For higher dimensions some interesting limitations of the method are also discussed and shown to be related to the adiabatic theorem in quantum mechanics. They cause the representation and thus the eigen-functions of integrable systems in more than one dimension to be given correctly to all orders of .     

A probabilistic approach to quantum mechanics based on ‘tomograms’

It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical‐like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.     

规范变换和量子相位因子

本文从历史发展的和几何的角度说明规范变换,相位因子和规范场等物理概念的关系。它是作者一组关于规范场理论记述[1～6]的后续和补充,特别是从规范的历史发展和相位因子几何概念初步去理解杨—米尔斯规范理论的渊源。本文只是从初等水平去说明,不去触及纤维丛等数学,以避免需要拓扑学的预备知识。     

Canonical quantization of gravity and quantum geometrodynamics

A representation in the form of the Faddeev-Popov path integral is constructed for solving the equations of quantum geometrodynamics (QGD). It is shown that QGD is equivalent to canonical quantization of gravity in a unitary gauge. Given the state of the gravitational field on the initial Cauchy hypersurface, a wave function of closed universe is constructed so that it satisfies the QGD equations. Using the principles of canonical quantization, a probabilistic interpretation of this wave function is constructed in a fashion close to Everett's concepts of quantum mechanics.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 37–51, March, 1986.     

Quantum theory of rotation angles: The problem of angle sum and angle difference

We reconsider the problem of the sum and difference of two angle variables in quantum mechanics. The spectra of the sum and difference operators have widths of , but angles differing by are indistinguishable. This means that the angle sum and difference probability distributions must be cast into a range. We obtain probability distributions for the angle sum and difference and relate this problem to the representation of nonbijective canonical transformations. Received: 6 December 1997 / Revised: 15 April 1998 / Accepted: 7 May 1998     

Dynamical symmetries in mechanics

The object of this review is to discuss methods that enable one to trace the origin of symmetries and conservation laws in mechanics to geometrical symmetries of space-time. Starting with the basic Newtonian assumptions on absolute space and time classical mechanics is developed in configuration space and phase space independently together with the related structures such as force-less mechanics. Heuristic considerations on geometric symmetries in configuration space reveal their intimate relation to conservation laws. Using the methods of differential geometry this relationship is put on a formal footing and symmetry groups of all spherically symmetric single term potentials are classified. The method of infinitesimal canonical transformations is presented as an alternative method of deducing dynamical symmetries of an arbitrary system in phase space. These methods also apply to non-relativistic quantum theory. Possible extension to special and general relatively is also discussed.     

级联光束分离器的纠缠特性及其应用

光束分离器是一个具有广泛应用的线性光学器件,它在非经典量子态特别是纠缠态的制备中具有重要作用.基于单个光束分离器的表象表示,本文进一步考察多个级联光束分离器的纠缠特性,特别是结合有序算符内的积分技术推导了级联光束分离器的正规乘积、紧指数表示及级联算符的表象表示.作为应用,本文利用两个级联光束分离器获得了量子力学表象及其Schmidt分解,并结合量子条件测量制备了qubit态的叠加态.本文的研究方法已被直接推广至多个光束分离器级联情况,相关研究内容为多模纠缠态、多模qubit态的制备提供了一种有效的途径,且为由光束分离器组成的线性器件系统总作用的算符正规乘积及其紧指数表示提供了一般方法.     

General quantum mechanical canonical point transformations

Problems related to the operator form of the generalized canonical momenta in quantum mechanics are resolved by use of the general quantum mechanical canonical point transformation method. This method can be applied to any general canonical point transformation irrespective of the relationship between the domains of the original and transformed variables. The differential representation of the original canonical momenta p_i in the original coordinate space is ?i \(\begin{array}{*{20}c} / \\ h \\ \end{array}\) ?/?x _i and of the transformed canonical momentap _i ′ in the transformed coordinate space is ?i \(\begin{array}{*{20}c} / \\ h \\ \end{array}\) ?/?x _i ′. Relationships are derived between the eigenvalues of the original and transformed momenta in either space. The ordering problem for general point transformations is discussed and is shown to be solved. As an example of the generality of the method, it is demonstrated on the point transformation in three dimensions from Cartesian rectilinear to spherical rectilinear coordinates.