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It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in this conjecture (p. 30):

[Quantum Brownian Motion Conjecture]: For small [disorder] $\lambda$ and [dimension] $d \ge 3$, the location of the electron is governed by a heat equation in a vague sense: $$\partial_t \big|\psi_t(x)\big|^2 \sim \Delta_x \big|\psi_t(x)\big|^2 \quad \Rightarrow \quad \langle \, x^2 \, \rangle_t \sim t, \quad t \gg 1.$$ The precise formulation of the first statement requires a scaling limit. The second statement about the diffusive mean square displacement is mathematically precise, but what really stands behind it is a diffusive equation that on large scales mimics the Schrödinger evolution. Moreover, the dynamics of the quantum particle converges to the Brownian motion as a process as well; this means that the joint distribution of the quantum densities $\big|\psi_t(x)\big|^2$ at different times $t_1 < t_2 < \dots < t_n$ converges to the corresponding finite dimensional marginals of the Wiener process.

This is the Anderson model in $\mathbb R^d$ with disordered Hamiltonian $H = -\Delta + \lambda V$. The potential $V$ is disordered, and is generated by i.i.d. random fields; the parameter $\lambda$ controls the scale of the disorder.

Classical Brownian motion admits many characterizations and generalizations. For example, Wiener measure leads to the construction of an abstract Wiener space, which is the appropriate setting for the powerful Mallivin calculus. The structure theorem of Gaussian measures says that all Gaussian measures are abstract Wiener measures in this way. I would love to know what all this theory looks like in the language of non-commutative probability theory.

The QBM Conjecture states roughly that a quantum particle in a weakly disordered environment should behave like a quantum Brownian motion. This is an important open problem, but it doesn't quite capture what a QBM is, nor what different types of QBM may exist. Thus my question:

What kind of precise mathematical object is a quantum Brownian motion?

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I think the review by Erdos has nothing to do with non-commutative probability. The goal is to show that in some limit the quantum particle in a disordered medium behaves according to classical Brownian motion, with emphasis on the word "classical". –  Abdelmalek Abdesselam Feb 21 '13 at 13:31
@Abdelmalek Abdesselam: to clarify, while the review has nothing do with a non-commutative probabilistic description of quantum Brownian motion, I think that non-comm. prob. theory might be one framework in which to precisely describe QBM. I make no claims that this approach is necessary but it may be useful. –  Tom LaGatta Feb 21 '13 at 16:18

1 Answer 1

No answer, just another question:

Is quantum Brownian motion related to Quantum Noise or the quantum Wiener process? I think these notions have a well-established mathematical theory, e.g. there are quantum stochastic integrals defined for them.

For a more physical approach see

Gardiner, Zoller, Quantum Noise, Springer, 2004,

for more mathematical literature see, e.g.,

K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Springer, 1992,

P.A. Meyer, Quantum Probability for Probabilists, Lect. Notes in Math. 1538, Springer, 1995.

The quantum Wiener process has applications to quantum filtering, see, e.g.

L. Bouten, R. van Handel, M. James, An introduction to quantum filtering, http://arxiv.org/abs/math/0601741.

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Good point, @Uwe Franz. I just found another MathOverflow question on the quantum Wiener process: mathoverflow.net/questions/15973/… –  Tom LaGatta Feb 21 '13 at 16:06

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