Exact Solutions of Non-Autonomous Quantum Systems with Quadratic Hamiltonian

Based on algebraic dynamics, we present an algorithm to obtain exact solutions of the Schrodinger equation of non-autonomous quantum systems with Hamiltonian expressed in quadratic function of creation and annihilation operators of bosons. The Hamiltonian is treated as a linear function of generators of a symplectic group. Similar to the canonical transformation of classical dynamics, we employ a set of gauge transformations to gradually transform the Hamiltonian to a linear function of Cartan operators. The exact solutions are obtained by inverse gauge transformations. When the system is autonomous, this algorithm can obtain the normal mode of the Hamiltonian, as well as the eigenstates and eigenvalues.     

Quantum Systems Connected by a Time-Dependent Canonical Transformation

We study both classical and quantum relation between two Hamiltoniansystems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other istime-dependent Hamiltonian system. The quantum unitary operatorrelevant to classical canonical transformation between the two systems are obtained through rigorous evaluation. With the aid of the unitary operator, we have derived quantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservative system. The invariant operators of the two systems are presented and the relation between them are addressed. We showed that there exist numerous Hamiltonians, which gives the same classical equation of motion. Though it is impossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics, they can be distinguishable quantum mechanically.     

Phase space representation of quantum dynamics

We discuss a phase space representation of quantum dynamics of systems with many degrees of freedom. This representation is based on a perturbative expansion in quantum fluctuations around one of the classical limits. We explicitly analyze expansions around three such limits: (i) corpuscular or Newtonian limit in the coordinate-momentum representation, (ii) wave or Gross-Pitaevskii limit for interacting bosons in the coherent state representation, and (iii) Bloch limit for the spin systems. We discuss both the semiclassical (truncated Wigner) approximation and further quantum corrections appearing in the form of either stochastic quantum jumps along the classical trajectories or the nonlinear response to such jumps. We also discuss how quantum jumps naturally emerge in the analysis of non-equal time correlation functions. This representation of quantum dynamics is closely related to the phase space methods based on the Wigner-Weyl quantization and to the Keldysh technique. We show how such concepts as the Wigner function, Weyl symbol, Moyal product, Bopp operators, and others automatically emerge from the Feynmann's path integral representation of the evolution in the Heisenberg representation. We illustrate the applicability of this expansion with various examples mostly in the context of cold atom systems including sine-Gordon model, one- and two-dimensional Bose-Hubbard model, Dicke model and others.     

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.     

The Quantal Canonical Symmetries for a Singular Lagrangian System with Subsidiary Constraints

The quantization for a system containing subsidiary constraints (in configuration space) with a singular Lagrangian is studied, in certain case which can be brought into the theoretical framework of constrained Hamiltonian system. A modified Dirac-Bergmann algorithm for the calculation of all phase-space constraints in those systems is derived. The path integral quantization is formulated by using the Faddeev-Senjanovic scheme. The classical and quantum canonical symmetries (Noether theorem in canonical formalism) are established for such a system. An example is given to illustrate that the connection between the symmetry and conservation law in classical theory are not always validity in the quantum theory.     

Scalar Charged Particle in Weyl–Wigner–Moyal Phase Space. Constant Magnetic Field

A relativistic phase-space representation for a class of observables with matrix-valued Weyl symbols proportional to the identity matrix (charge-invariant observables) is proposed. We take into account the nontrivial charge structure of the position and momentum operators. The evolution equation coincides with its analog in relativistic quantum mechanics with nonlocal Hamiltonian under conditions where particle-pair creation does not take place (free particle and constant magnetic field). The differences in the equations are connected with the peculiarities of the constraints on the initial conditions. An effective increase in coherence between eigenstates of the Hamiltonian is found and possibilities of its experimental observation are discussed.     

Theory of extreme correlations using canonical Fermions and path integrals

The  t$t$–J$J$  model is studied using a novel and rigorous mapping of the Gutzwiller projected electrons, in terms of canonical electrons. The mapping has considerable similarity to the Dyson–Maleev transformation relating spin operators to canonical Bosons. This representation gives rise to a non Hermitian quantum theory, characterized by minimal redundancies. A path integral representation of the canonical theory is given. Using it, the salient results of the extremely correlated Fermi liquid (ECFL) theory, including the previously found Schwinger equations of motion, are easily rederived. Further, a transparent physical interpretation of the previously introduced auxiliary Greens function and the ‘caparison factor’, is obtained.     

Universal invariants in quantum mechanics and physics of optical and particle beams

We give a brief review of the theory of quantum universal invariants and their counterparts in the physics of light and particle beams. The invariants concerned are certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators, or the transverse phase-space coordinates of the paraxial beams of light or particles. They are conserved in time (or along the beam axis) independently of the concrete form of the coefficients of the Schrödinger-like equations governing the evolution of the systems, provided that the effective Hamiltonian is either a generic quadratic form of the generalized coordinate-momenta operators or a linear combination of generators of some finite-dimensional algebra (in particular, any semisimple Lie algebra). Using the phase space representation of quantum mechanics (paraxial optics) in terms of the Wigner function, we elucidate the relation of the quantum (optical) invariants to the classical universal integral invariants of Poincaré and Cartan. The specific features of Gaussian beams are discussed as examples. The concept of the universal quantum integrals of motion is introduced, and examples of the “universal invariant solutions” to the Schrödinger equation, i.e., self-consistent eigenstates of the universal integrals of motion, are given.     

Representations of the CAR generated by representations of the CCR. Fock case

We present a method of constructing the Fock representation of the canonical anti-communtation relations in the Fock representation of the canonical commutation relations. An explicit formula for Fermi creation and annihilation operators in terms of Bose ones is given.     

Unambiguous Quantization from the Maximum Classical Correspondence that Is Self-consistent: The Slightly Stronger Canonical Commutation Rule Dirac Missed

Dirac’s identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac’s canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of Poisson brackets between any linear function of phase space and the sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E.H. Kerner. Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical “shadow world” which can manifest negative densities in phase space.     

Quantum Symmetries in the Maxwell–Chern–Simons Theory Coupled to Scalar Fields

The Maxwell-Chern-Simons gauge theory coupled to a complex scalar field is quantized in the Becchi-Rouet-Stora-Tyutin (BRST) path integral formalism. On the basis of the symmetries of a constrained canonical (Hamiltonian) system, we get the quantal conserved angular momentum of the system under the global symmetry transformation. It is shown that fractional spin also appears at the quantum level. The canonical Ward identities for this system are derived under local gauge transformation.     

The symmetries in the Maxwell－Chern－Simons theory coupled to matter fields

The Maxwell－Chern－Simons gauge theory coupled to a complex scalar field is quantized in the Becchi－Rouet－Stora－Tyutin path integral formalism. Based on the symmetries of a constrained canonical (Hamiltonian) system, we obtain the quantal conserved angular momentum of the system under the global symmetry transformation. It is shown that fractional spin also appears at the quantum level. The canonical Ward identities for this system are derived under local gauge transformation.     

Quantum field theory of binary alternatives

We describe a framework for quantum field theory that is based on quantized binary alternatives. We discuss the relation of the dynamics of quantum fields and the time evolution of the Heisenberg operators defined by the creation and annihilation operators of the alternatives. An explicit expression for the vacuum vector of the quantum fields is derived. Finally we discuss eigenstates of the momentum operators.     

Phase space propagators for quantum operators

The Heisenberg evolution of a given unitary operator corresponds classically to a fixed canonical transformation that is viewed through a moving coordinate system. The operators that form the bases of the Weyl representation and its Fourier transform, the chord representation are, respectively, unitary reflection and translation operators. Thus, the general semiclassical study of unitary operators allows us to propagate arbitrary operators, including density operators, i.e., the Wigner function. The various propagation kernels are different representations of the super-operators which act on the space of operators of a closed quantum system. We here present the mixed semiclassical propagator, that takes translation chords to reflection centres, or vice versa. In contrast to the centre-centre propagator that directly evolves Wigner functions, they are guaranteed to be caustic free, having a simple WKB-like universal form for a finite time, whatever the number of degrees of freedom. Special attention is given to the near-classical region of small chords, since this dominates the averages of observables evaluated through the Wigner function.     

高阶微商场论中奇异拉氏量系统的量子正则对称性

给出了高阶徽商场论中奇异拉氏量系统规范生成元的构成.从相空间中Green函数的生成泛函出发,导出了约束Hamilton系统正则形式的Ward恒等式.指出该系统的量子正则方程与由Dirac猜想得到的经典正则方程不同.给出了与Chern-Simons理论等价的一个广义动力学系统的量子化.将正则Ward恒等式初步应用于该系统,不作出对正则动量的路径积分,也可导出场的传播子与正规顶角之间的某些关系.     

Quantum Dynamics of Systems Connected by a Canonical Transformation

Quantum Hamiltonian systems corresponding to classical systems related by a general canonical transformation are considered. The differential equation to find the unitary operator, which corresponds to the canonical transformation and connects quantum states of the original and transformed systems, is obtained. The propagator associated with their wave functions is found by the unitary operator. Quantum systems related by a linear canonical point transformation are analyzed. The results are tested by finding the wave functions of the under-, critical-, and over-damped harmonic oscillator from the wave functions of the harmonic oscillator, free-particle system, and negative harmonic potential system, using the unitary operator to connect them, respectively.     

Discrete symmetries and the propagator approach to coupled fermions in Quantum Field Theory. Generalities: The case of a single fermion-antifermion pair

Starting from Wigner’s symmetry representation theorem, we give a general account of discrete symmetries (parity P, charge conjugation C, time-reversal T), focusing on fermions in Quantum Field Theory. We provide the rules of transformation of Weyl spinors, both at the classical level (grassmanian wave functions) and quantum level (operators). Making use of Wightman’s definition of invariance, we outline ambiguities linked to the notion of classical fermionic Lagrangian. We then present the general constraints cast by these transformations and their products on the propagator of the simplest among coupled fermionic system, the one made with one fermion and its antifermion. Last, we put in correspondence the propagation of C eigenstates (Majorana fermions) and the criteria cast on their propagator by C and CP invariance.