Canonical transformation physical equivalence

The second axiom, which is reminiscent of Mach s principle, also contains the seeds of Leibniz s Monads [reschQl]. All is process. That is to say, there is no thing in the universe. Things, objects, entities, are abstractions of what is relatively constant from a process of movement and transformation. They are like the shapes that children like to see in the clouds. The Einstein-Podolsky-Rosen correlations (see section 12.7.1) remind us that what we empirically accept as fundamental particles - electrons, atoms, molecules, etc. - actually never exist in total isolation. Moreover, recalling von Neumann s uniqueness theorem for canonical commutation relations (which asserts that for locally compact phase spaces all Hilbert-space representations of the canonical commutation relations are physically equivalent), we note that for systems with non-locally-compact phase spaces, the uniqueness theorem fails, and therefore there must be infinitely many physically inequivalent and... 

We may recall and emphasize that the autocorrelation function obtained in the three representations I, II, and III must be equivalent, from the general properties of canonical transformation which must leave invariant the physical results. Thus, because of this equivalence, the spectral density obtained by Fourier transform of (43) and (45) will lead to the same Franck-Condon progression (51). 

The two pictures above (where U is viewed as acting on the wavefunction or acting on the Hamiltonian) are clearly mathematically equivalent. However, it is worth considering their physical equivalence in the language of canonical transformations. (A similar discussion of this issue may also be found in White [22].) In the first picture the Hamiltonian H, wavefunctions to and and transformation U are associated with particles defined by the operators c, Cj thus... 

If there are n0 open channels at energy E, there are n linearly independent degenerate solutions of the Schrodinger equation. Each solution is characterized by a vector of coefficients aips, for i = 0,1, defined by the asymptotic form of the multichannel wave function in Eq. (8.1). The rectangular column matrix a consists of the two n0 x n0 coefficient matrices ao, < i Any nonsingular linear combination of the column vectors of a produces a physically equivalent set of solutions. When multiplied on the right by the inverse of the original matrix a0, the transformed a-matrix takes the canonical form... 

It should be noticed that a((3) and a((3) satisfy the same algebraic relation as those given in Eq. (3), and also that a(/3) 0(/ )) = a(j3) 0(/ )) = 0. Then the thermal state 0(/3)) is a vacuum for a((3) and a(/3) (otherwise, 0,0) is the vacuum for the operators a and a). As a result, the thermal vacuum average of a non-thermal operator is equivalent to the Gibbs canonical average in statistical physics. As a consequence, the thermal problem can be treated by a Bogoliubov transformation, such that the thermal state describes a condensate with the mathematical characteristics of a pure state. 

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