Canonical transformation, quantum

S. R. White, Numerical canonical transformation approach to quantum many-body problems. J. Chem. Phys. 117, 7472 (2002). 

Arlen Anderson, Quantum canonical transformations and integrability beyond unitary transformations. Phys. Lett. B 319, 157 (1993). 

Consider the effective Hamiltonian (47) of a driven quantum harmonic oscillator. Since it is not diagonal, it may be suitable to diagonalize it with the aid of a canonical transformation that will affect it or its equivalent form (50), but not that of (46) or its equivalent expression (49), which is yet to be diagonal [13]. 

For this purpose, consider selective canonical transformations leading to a new quantum representation that we name /// in order to diagonalize the effective Hamiltonian corresponding to k = 1, without affecting that corresponding to k = 0. This may be performed on the different effective operators B dealing with k = 0,1 with the aid of... 

The state of a classical system is specified in terms of the values of a set of coordinates q and conjugate momenta p at some time t, the coordinates and momenta satisfying Hamilton s equations of motion. It is possible to perform a coordinate transformation to a new set of ps and qs which again satisfy Hamilton s equation of motion with respect to a Hamiltonian expressed in the new coordinates. Such a coordinate transformation is called a canonical transformation and, while changing the functional form of the Hamiltonian and of the expressions for other properties, it leaves all of the numerical values of the properties unchanged. Thus, a canonical transformation offers an alternative but equivalent description of a classical system. One may ask whether the same freedom of choosing equivalent descriptions of a system exists in quantum mechanics. The answer is in the affirmative and it is a unitary transformation which is the quantum analogue of the classical canonical transformation. 

In terms of these results it can be shown that the value of a Poisson bracket is invariant under a canonical transformation of the coordinates. Like the corresponding commutator relationships in quantum mechanics, to which it is related by the expression... 

The analoguous behiaviour of the Poisson bracket and the commutator has been used to establishcorrespondence between classical and quantum mechanics. It is, however/shown in the next section following the derivation of Schwinger s quantum action principle that the correspondence goes deeper and that the analogous behaviour of the Poisson bracket and commutator is a consequence of the properties of infinitesimal canonical transformations which are common to both mechanics. 

Using the invariabihty of the trace, we perform the canonical transformation in expression (217) by means of quantum mechanical thermal averaging over Fermi operators with an accuracy to terms linear in the carrier concentration, and then averaging over the phonon bath, we obtain [165]... 

Earlier in this section it was commented on how the minimal-coupling QED Hamiltonian is obtained from fhe classical Lagrangian function. A few words are in order regarding the derivation of the multipolar Hamiltonian (6). One method involves the application of a canonical transformation to the minimal-coupling Hamiltonian [32]. In classical mechanics, such a transformation renders the Poisson bracket and Hamilton s canonical equations of motion invariant. In quantum mechanics, a canonical transformation preserves both the commutator and Heisenberg s operator equation of motion. The appropriate generating function that converts H uit is propor-... 

One thus needs a prescription for constructing the classical-limit approximation to quantum mechanical amplitudes or transformation elements. This is given most generally by establishing the correspondence of canonical transformations between various coordinates and momenta in classical... 

We must now prove that the conditions (A), (B), and (C) really suffice for the logical applications of quantum conditions in the form (1) we carry out the proof by finding the most general canonical transformation... 

It is only the multiply-pcriodic solutions which are of importance for the quantum theory. The methods which we shall employ for their deduction in what follows are essentially the same as those which Poincard has treated in detail in his Methodes nouvdles de la Mdtanique celeste,2 By a solution we mean, as usual, the discovery of a principal function S which generates a canonical transformation,... 

L. Mercaldo, I. Rabuffo, and G. Vitiello, Canonical Transformations in Quantum Field Theory and Solitons, Nucl. Phys. 188B, 193-204 (1981). 

Within the framework of nonrelativistic quantum electrodynamics, the emission in electric-dipole transitions can be treated using two alternative Hamiltonians for field-matter interaction, i.e. a multipolar Hamiltonian and a minimal-coupling (p ) Hamiltonian, since the two are related by a canonical transformation . In what follows, the results concerning motional effects on the emission will be discussed and checked by showing that they are obtainable from both Hamiltonians. 

As shown previously analogous equations can be derived in a statistical framework both for localized fermions in a specific pairing mode and/or for bosons subject to a quantum transport environment [7]. The second interconnection regarding the relevance of the basis f is related to the fact that a transformation of form (20) connects canonical Jordan blocks to convenient complex symmetric forms. This will not be explicitly discussed and analysed here except pointing out the possible relationship between temperature scales and Jordan block formation by thermal correlations (see e.g. [7-9,14], for more details). 

In summary, the model allows for two types of interactions between the mirror spaces, the weak kinematical perturbation and the adiabatic and sudden limits equivalent to Eq. (17) or Eqs. (29)-(34). The overwhelming rate of particles over antiparticles in the Universe is inferred in this picture once the particular particle state has been selected. The Minkowski metric of the special theory of relativity is represented here by a non-positive definite metric, Eq. (8), bringing about a quantum model with a complex symmetric ansatz. Although the latter permits general symmetry violations, it is nevertheless surprising that fundamental transformations between complex symmetric representations and canonical forms come out unitary. 

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