Abstract
Ensembles on configuration space have wide applicability. They may be used to describe classical, quantum and hybrid quantum-classical systems, physical systems that are deterministic or subject to uncertainty, discrete systems, particles and fields. They also lead to novel reconstructions of quantum theory from physical and geometric axioms. We introduce the basic elements of the theory, discuss a number of classical and quantum examples, and provide an overview of the many generalizations and applications that form the subjects of later chapters. The approach introduces very few physical and mathematical assumptions. The basic building blocks are the configuration space of the physical system, an ensemble of configurations, and dynamics generated from an action principle. An important role is played by the ensemble Hamiltonian which determines the equations of motion. It must satisfy certain requirements which we discuss in detail. We provide examples of classical and quantum systems and show that the primary difference between quantum and classical evolution lies in the choice of the ensemble Hamiltonian.
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Notes
- 1.
For readers unfamiliar with variational derivatives, more details are given in Appendix A of this book. It is sufficient to recall here that for \(F=\int dx\,g(x,f,\nabla f)\), one has \(\delta F/\delta f=\partial g/\partial f - \nabla \cdot \partial g/\partial (\nabla f)\). The case of discrete configuration spaces is mathematically simpler, as discussed in Sect. 1.3.
- 2.
For continuous configuration spaces P and S are functions, and hence one should more properly write the ensemble Hamiltonian as a functional, H[P, S], in this case. However, it is convenient to use the notation H(P, S) when referring to the general case.
- 3.
Here the defining property of the variational derivative, \(F[f+\delta f] -F[f]= \int dx ({\delta F}/{\delta f})\delta f\) for arbitrary infinitesimal variations \(\delta f\), has been used (see Appendix A of this book).
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Hall, M.J.W., Reginatto, M. (2016). Introduction. In: Ensembles on Configuration Space. Fundamental Theories of Physics, vol 184. Springer, Cham. https://doi.org/10.1007/978-3-319-34166-8_1
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