Uninvertible linear canonical transformations

Reducible quasi-Fock representations of canonical commutation relations (CCR) are studied which result from uninvertible linear canonical transformations of the Fock representation.Necessary and sufficient conditions are proved for: (i) invertibility of the linear canonical transformation, (ii) quasi-Fock character of the obtained representation of CCR, (iii) coincidence of the vacuum subspaces of two representations of this type, (iv) coincidence of W¹-algebras of operators of these representations.The results obtained are interesting for constructive quantum field theory with asymptotic fields giving rise to the quasi-Fock representations of CCR.     

Adiabatic anholonomy and canonical transformations

Biswas and Soni [4] have surmised a semiclassical formula for Berry’s phase in terms of a generating function. We derive this formula apart from showing that it is not true in general and investigate its domain of validity. We also derive transformation formulae for Berry’s phase (Hannay’s angle) under general canonical transformations. A simpler proof for total angle invariance than hitherto available, is given.     

Continual integrals and canonical transformations

Continual integrals are considered for the matrix elements of an evolution operator of quantum-mechanical systems. It is shown that a continual integral can be reduced to a finite multiple integral by using canonical transformations. The method developed is illustrated by an example of a homogeneous force field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 16–19, May, 1982.     

Representations of canonical anticommutation relations and implementability of canonical transformations

It is proved that irreducible representations of CAR are determined by the groups of implementable automorphisms of the correspondingC*-algebra. This is done by a study of implementable canonical transformations. Some results in the same directions for factor representations are given.     

Path integrals and linear canonical transformations

For a system withN bosonic or fermionic degrees of freedom I calculate the coherent state propagator, i.e. the matrix element between coherent states of the evolution operator, for a general quadratic Hamiltonian plus a source term, using the holomorphic form of the path integral. The analysis and the result obtained are used to discuss the transformation properties of the path integral for linear canonical transformations (Bogoliubov-Valatin trfs), a preliminary to the formulation of a geometric theory of path integral quantization.

     

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.     

Point canonical transformations and the path integral

It is shown that the covariance arguments under point canonical transformations serve to remove the ambiguity that is inherent in ordering variables and in choosing an approximate form for the action.     

S-duality and canonical transformations in string theory

The symmetries of the tree level string effective action are discussed. An appropriate effective action is constructed starting from the manifestly SL(2,R) invarint form of string effective action introduced by Schwarz and Sen. The conserved charges are derived and generators of infinitesimal transformations are obtained in the Hamiltonian formalism. Some interesting consequences of the canonical transformations are explored.     

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 .     

Invariant expression for tunneling rate under canonical transformations

The original formula to calculate the tunneling rate through event horizons is apparently dependent on the type of coordinates used.In this paper,we propose an invariant expression under canonical transformations to study the tunneling effect.Moreover,the problem of factor 2 is solved naturally.As an application of this expression,we obtain the same tunneling rate both in the Schwarzschild and the Painleve′coordinates.It is shown that once the suitable formula to calculate tunneling rate is correctly identified,the tunneling method is manifestly covariant.     

Groups of canonical transformations and the virial-Noether theorem

In Hamiltonian mechanics a characterization of the infinitesimal generator of one-parameter Lie Groups of non-univalent canonical transformations is given. The result is used to derive a general form of the virial theorem, which has Noether's theorem as a special case. The theory is applied to the Toda lattice system.     

Schwinger action principle via linear quantum canonical transformations

We have applied the Schwinger action principle to general one-dimensional (1D), time-dependent quadratic systems via linear quantum canonical transformations, which allowed us to simplify the problems to be solved by this method. We show that while using a suitable linear canonical transformation, we can considerably simplify the evaluation of the propagator of the studied system to that for a free particle. The efficiency and exactness of this method is verified in the case of the simple harmonic oscillator. This technique enables us to evaluate easily and immediately the propagator in some particular cases such as the damped harmonic oscillator, the harmonic oscillator with a time-dependent frequency, and the harmonic oscillator with time-dependent mass and frequency, and in this way the propagator of the forced damped harmonic oscillator is easily calculated without any approach. PACS 02.30.Xx, 03.65.-w, 03.65.Ca     

Unitary transformations,Weyl's association and the role of canonical transformations

Quantum mechanical operators can be associated with functions of p, q through the Weyl or Wigner transform. In this paper we develop alternative associations through the use of unitary transformations, and study the relation between unitary transformations and canonical transformations of the p, q labels.     

一类广义Birkhoff系统的广义正则变换

研究一类广义Birkhoff系统的广义正则变换.建立这类广义Birkhoff系统的运动微分方程,得到了该系统的广义正则变换以及保持广义正则变换的条件.最后,举例说明结果的应用.