K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys. 35, 265 (1974).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The two parameterizations are related by s = t, with a slow frequency the adiabaticity parameter. The epoch energy uncertainty then takes the form s e and so arbitrarily small in the adiabatic limit. Coherent states provide a convenient basis to analyze the semi classical limit [12, 6]. Semi classical analysis is traditionally about the 0 limit, but is equally valid when is xed (and henceforth set equal to one) and 0. Here we introduce coherent states labelled by points in the time energy plane, with time being the scattering time. As we shall see, the frozen S ....

K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys. 35, 265 (1974).

....is to produce accurate, computable approximations as h goes to zero, for as long a time interval as possible. In scattering situations the time interval is the whole real line. There are several results concerning the propagation of certain coherent states for finite time intervals. Early results [14, 7] constructed approximate solutions that were accurate up to O(h 1 2 ) errors. Later approximations were constructed with O(h l 2 ) errors for any l [8, 9, 5, 16] Very recently, approximations were constructed with errors of exponential order O(e # h ) with # 0 in [11] see also [22] ....

Hepp, K.: The Classical Limit for Quantum Mechanical Correlation Functions. Commun. Math. Phys. 35, 265--277 (1974).

....of this approach are therefore similar to the WKB approach. It is maybe interesting to note that the propagator itself does not have a classical limit in our approach, whereas the time evolution it implements on observables does (see Section 4.5) E) Limits of coherent states. In the papers [Hep,Hag] it is shown that in the limit h 0 the time evolution of a coherent state, which is initially concentrated near a given point in phase space, is well approximated by another coherent state, concentrated at the classically evolved point. This statement is essentially what one gets in the ....

....as well. Note, however, that a version of the Evolution Theorem can only hold if the classical time evolution exists for all times, so some restrictions on H h are always needed. A good way of handling unbounded Hamiltonians is also to study the dynamics in the norm limit of states (see (2) and [Hep,Hag] In the deformation quantization approach, dynamics was recently discussed in [Ri3] 4) Classical trajectories The Evolution Theorem does not explain how, in the classical limit, a description of the systems in terms of trajectories becomes possible. The statistics of trajectories 28 ....

K. Hepp: "The classical limit for quantum mechanical correlation functions", Commun.Math.Phys. 35(1974) 265--277.

....x 2 ) j = 0: 3.42) 29 Since x 1 and x 2 were arbitrary (a.e. this proves (3.28) Note that a rather different approach to the classical limit of the dynamics (3. 26) is presented in [40] the first rigorous results on the classical limit of quantum correlation functions were obtained by Hepp [19]. He used coherent states, of which our classical germs are a generalization. To sum up, we have a satisfactory quantum dynamics on A defined by (3.27) We may now ask if and how ff h t is implemented in the irreducible representations of A (cf. subsect. 3.2) That is, we look for a ....

K. Hepp, The classical limit of quantum mechanical correlation functions, Commun. Math. Phys. 35, 265-277 (1974).

....equation as h 0, it does not give a satisfactory explanation why in this limit the non commutativity of quantum observables suddenly turns into the commutativity of classical observables. The same is true of approaches based on Feynman integrals [AHK] and on the limits of coherent states [Hep, Hag]. An approach to the classical limit emphasizing the limit of observables and their algebraic structure has recently been developed in [We2] compare also [Rie, Em1] This approach makes rigorous the intuitive criterion for deciding which observables in quantum theory may effectively be treated ....

K. Hepp:"The classical limit for quantum mechanical correlation functions ", Commun.Math.Phys. 35(1974) 265--277.

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