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.

     

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.     

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.     

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.     

Anomalous scale transformations and functional integrals

A method is presented to analyse euclidean field theories starting from small volume subsystems. It bears resemblance with the real space renormalization group, but it is not limited to the lattice cutoff. An intuitive picture emerges for the role of perturbative results in asymptotically free models. There it is shown how to derive the one-loop β-function directly as a jacobian under scale transformations in a functional integral.     

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.     

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.     

Space-time transformations in radial path integrals

Nonlinear space-time transformations in the radial path integral are discussed. A transformation formula is derived, which relates the original path integral to the Green function of a new quantum system with an effective potential containing an observable quantum correction ≈?². As an example the formula is applied to spherical brownian motion.     

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 .     

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.     

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.     

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.     

Definition of path integrals and rules for non-linear transformations

Functional integrals are defined as the limit of multidimensional integrals based on fundamental generating distributions. The “lattice choice” is put into a suitable functional form. The independence of the particular choice and the necessity of this fact are shown. Various forms of the path integrals are derived and discussed. The relation to the traditional ordering problem is pointed out. The mechanism of non-linear transformations of variables is investigated and rules are given. In the case of fields it turns out that the path integrals can also be considered for space translations.     

Evaluation of multidimensional canonical integrals in semiclassical collision theory

The evaluation of the multidimensional canonical integrals that occur in the uniform asymptotic representations of the S matrix in the semiclassical theory of inelastic and reactive molecular collisions is considered. For the non-separable two-dimensional canonical integral considered earlier, an exact series expansion is obtained with the help of convergence factors. This method avoids the complex variable techniques used previously. The uniform asymptotic formulae derived by Miller and Marcus are discussed, and compared with the approach adopted in the present paper.