By Alfredo M. Ozorio de Almeida
This advent to the idea of Hamiltonian chaos outlines the most ends up in the sphere, and is going directly to ponder implications for quantum mechanics. The examine of nonlinear dynamics, and specifically of chaotic structures, is likely one of the quickest transforming into and best components in physics and utilized arithmetic. In its first six chapters, this well timed publication introduces the speculation of classical Hamiltonian structures. the purpose isn't to be entire yet, particularly, to supply a mathematical trunk from which the reader might be in a position to department out. the focus is on periodic orbits and their neighbourhood, as this method is mainly compatible as an creation to the consequences of the speculation of chaos in quantum mechanics, that are mentioned within the final 3 chapters.
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Extra resources for Hamiltonian systems: chaos and quantization
23d) which substituting Eq. 23e) s in agreement with Eq. 23a). 24a) and the corresponding trace dynamics equation of motion of Eq. 15b). Thus, for Weyl-ordered Hamiltonians formed with c-number coefficients, we conclude that the trace dynamics equations of motion generated by H agree with the Heisenberg picture equations of motion generated by H , on an initial time slice on which the phase space variables are canonical. It is also evident that on this time slice [H, i] = 0. 24b) But since Eq. 24b) guarantees that the Heisenberg picture equations of motion preserve the canonical algebra on the next time slice, integrating forward in time step by step then implies that trace dynamics agrees with Heisenberg picture dynamics at all subsequent times, and therefore can be extended to a unitary dynamics in this case.
The example of Eq. 20c) required the use of two pairs of canonical variables q1,2 , p1,2 ; using our Weyl-ordering result, we shall now show that any trace dynamics for a single 36 Trace dynamics pair of bosonic variables q, p, when extended to the canonical algebra [q, p] = i, always can be represented as a unitary Heisenberg evolution. We shall proceed by induction, and assume that the result has been proved for any trace dynamics generated by a trace Hamiltonian H = TrH of degree n or less in p and q.
We conclude that as of this writing the localization model is favored, both because the assumptions needed to derive it within our framework are more robust, and because there are unresolved problems with the mechanisms that have been proposed to explain reduction in the energy-driven model. Finally, in Chapter 7 we indicate how our proposal for an emergent quantum theory addresses the motivational questions raised above in Section 3, and discuss some of the issues that will be relevant for future developments.