# What is quantum Brownian motion?

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

A. To the extent that you think of Brownian motion as a random walk, the natural quantum extension is the quantum random walk. For a physics perspective, see Quantum random walks - an introductory overview, but you might prefer the more math-oriented exposition of Martin boundary theory of some quantum random walks and On algebraic and quantum random walks.

We give a concise prescription of the concept of a quantum random walk (QRW), using the example of QRW on integers as paradigm. It briefly explains the notion of quantum coin system and the coin tossing map, and summarizes two emblematic properties of that walk, namely the quadratic enhancement of its diffusion rate due to quantum entanglement between the walker and the entropy increase without majorization effect of its probability distributions. We conclude with a group theoretical scheme of classification of various known QRW's.

B. Concerning the relation between Wiener processes and quantum Brownian motion: A quantum version of the wavelet expansion of a Wiener process has been developed in A Levy-Cielsielski expansion for quantum Brownian motion and the construction of quantum Brownian bridges.

Classical Brownian motion has a delightful wavelet expansion obtained by combining the Schauder system with a sequence of i.i.d. standard normals. Our main technical result is to obtain a quantum version of this expansion and so construct quantum Brownian motion in Fock space. Consequently, only the discrete skeleton provided by a "quantum random walk" is required to generate the continuous time process. Our result seems easier to establish than the classical one of Lévy-Cielsielski as we don’t require logarithmic growth estimates on the squares of i.i.d. Gaussians, thanks to the nice action of annihilation operators on exponential vectors.

C. Concerning a mathematical description of the physical phenomenon of Brownian motion: We are then concerned with the effect of an environment having a large (infinite) number of degrees of freedom on the dynamics of a particle with a few degrees of freedom. So we are seeking a quantum theory of friction, diffusion, and thermalization. The seminal paper here is the path integral theory of Caldeira and Leggett. The literature is very extensive, an older but still relevant review is Quantum Brownian Motion: The Functional Integral Approach.

The quantum mechanical dynamics of a particle coupled to a heat bath is treated by functional integral methods and a generalization of the Feynman-Vernon influence functional is derived. The extended theory describes the time evolution of nonfactorizing initial states and of equilibrium correlation functions. The theory is illuminated through exactly solvable models.

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Thanks Carlo. Althoug I certainly want to understand the connection to random walks, I'm more interested in actual Brownian motion (continuous time, continuous space, and dynamic that rigorously arise from microscopic collisions). –  Jess Riedel Sep 20 at 15:02
added some more viewpoints beyond random walks. –  Carlo Beenakker Sep 20 at 21:45

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