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Suppose you have dynamics of a coherent state. The state presents a normal distribution of finding the particle. Does anyone know of any attempts to connect modern advances in the probability theory to quantum dynamics of a coherent state.

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Do you mean like stochastic calculus? – Fonon Mar 9 '12 at 5:25
if you mean application of stochastic calculus to quantum dynamics than yes. but I don't mean using monte-carlo integration. – kirill_igum Mar 9 '12 at 5:41
@kirill_igum: The Feynman Kac Formula contains many buzz-words of interest here. Moreover, there are many connections between statistics and the path integral formulation of quantum mechanics. – NikolajK Mar 9 '12 at 8:18
As Arnold says in his Answer, you might clarify why you particularly reference coherent states. – Peter Morgan Mar 9 '12 at 12:53
may be it's of interest to others, currently DMC ( uses stochastic processes to find ground state energies in molecular-size system. It's something else I'm looking into. – kirill_igum Mar 9 '12 at 19:13
up vote 4 down vote accepted

There are tons of papers on the connection between quantum processes and probability theory (though I don't understand why you single out coherent states - they don't play a special role in this connection).

The theory of stochastic processes and the theory of quantum processes are the commutative and noncommutative side of the same coin, with many similarities.

See, e.g., the books by Gardiner (Handbook of stochastic processes) or Barndorff-Nielsen (Quantum independent increment processes: Structure of quantum Lévy processes, classical probability, and physics)

Online is the following article by Barndorff-Nielsen

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I work on semi-classical methods in quantum chemistry. most of them base on propagation of a coherent state because it is the closest equivalent to classical point in phase space (due to minimum uncertainty). – kirill_igum Mar 9 '12 at 18:10
yes, but doesn't approximation by coherent states eliminate the probabilistic aspect? What do you aim at with your question? – Arnold Neumaier Mar 9 '12 at 18:18
I'm not familiar with the probabilistic aspect beyond knowing that wave function (wf) is a probability amplitude. Originally I thought that since a wf is a probability amplitude, one can switch to stochastic processes. So I asked the question to see if there are advantages. One advantage that I'm looking for, is an ability to trace only partial quantum information along a single classical trajectory so that dynamics has linear cost with time that is similar to classical dynamics. To refine my question I need to look through the literature that you and others mentioned. – kirill_igum Mar 9 '12 at 19:05
Probably one reason to use coherent states is that they have a positive Wigner function, like all the Gaussian statest, including the squeezed states. If you are only interested in the quadratures (e.g. position and momentum), then the Wigner function is like a usual probability distributions. Things start to break down when the state is non-Gaussian (locally negative Wigner function) or if you are interested in other observables, like the energy, which have non-Gaussian eigenstates. – Frédéric Grosshans Mar 9 '12 at 19:25
@Arnold I would describe coherent states as still probabilistic, but perhaps one might say that they are not immediately quantum, in the sense, as Frédéric Grosshans notes, that their Wigner functions are positive definite (hence allowing a naive interpretation as a probability density; noting, however, that positive-definiteness of a Wigner function characterizes the state, but does not characterize the observables used). Superpositions of coherent states, however, allow the reconstruction of any state (coherent vectors in the Hilbert space constitute an over-complete basis). – Peter Morgan Mar 10 '12 at 13:01

Many different types of connection can be made between stochastic states over commutative algebras of observables and quantum states over noncommutative algebras of observables. As Arnold says, there is a substantial literature.

One approach is to construct both classical and quantum models in a formalism that accommodates both; within the structured environment provided by the joint formalism, one can hope to characterize the difference between classical and quantum. One example of that approach is Lucien Hardy's "Quantum Theory From Five Reasonable Axioms", which, although unpublished in that form and despite its failings, has had a significant impact. I could also self-advertize (again) my approach of constructing classical random fields in a quantum field theoretic formalism (which also, surprise, has failings, but it is rather different from Lucien's and from the Barndorff-Nielsen). Andrei Khrennikov has been trying to construct random field models in a more traditionally stochastic formalism, again with quite different ideas about what he's trying to achieve and making quite different choices of assumptions to modify or to drop.

All the approaches I've touched on above, and that you will find by working through the citations in Andrei Khrennikov's papers, say, are essentially speculative, whereas the Barndorff-Nielsen is more solid as mathematics (his co-author, Richard Gill, is also always interesting when writing alone). What you might find useful depends on your tastes and on what you really intend by your Question.

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