It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is [this one](http://arxiv.org/pdf/1009.0843v1.pdf) 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](http://en.wikipedia.org/wiki/Brownian_motion) admits many characterizations and generalizations. For example, [Wiener measure](http://en.wikipedia.org/wiki/Wiener_measure) leads to the construction of an [abstract Wiener space](http://en.wikipedia.org/wiki/Abstract_Wiener_space), which is the appropriate setting for the powerful [Mallivin calculus](http://en.wikipedia.org/wiki/Malliavin_calculus). The [structure theorem of Gaussian measures](http://en.wikipedia.org/wiki/Structure_theorem_for_Gaussian_measures) says that *all* [Gaussian measures](http://en.wikipedia.org/wiki/Gaussian_measure#Gaussian_measures_on_infinite-dimensional_spaces) are abstract [Wiener measures](http://ncatlab.org/nlab/new/Wiener+measure) in this way. I would love to know what all this theory looks like in the language of [non-commutative probability theory](http://www.mitchener.staff.shef.ac.uk/free.pdf).


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? 
> 
> 
> 



What is quantum Brownian motion?
(In words explained below:) **Quantum Brownian motion (QBM) is a class of possible dynamics for an open, quantum, continuous degree of freedom in which the reduced dynamics are specified by a quadratic Hamiltonian and linear Lindblad operators in the phase-space variables $x$ and $p$.**


Consider the arbitrary time-evolution of a system's density matrix when it is in contact with an environment:
\begin{align}
\rho = \rho\_{\mathcal{S}}(t) = \mathrm{Tr}\_{\mathcal{E}} [U\_t \sigma^0\_{\mathcal{SE}} U\_t^\dagger],
\end{align}
where $\sigma^0\_{\mathcal{SE}}$ is the initial global state (both $\mathcal{S}$ and $\mathcal{E}$) and $U\_t$ is the unitary governing the global evolution. Then the system is said to evolve according to a special case of quantum Brownian motion — a QBM [quantum dynamical semigroup](https://en.wikipedia.org/wiki/Quantum_dynamical_semigroup) — when the evolution of its density matrix obeys a [Lindblad master equation](http://en.wikipedia.org/wiki/Lindblad_equation)
\begin{align}
\partial\_t \rho = -i [\hat{H},\rho] + \sum\_i \left(V\_i \rho V\_i^\dagger - \frac{1}{2} \{V\_i^\dagger V\_i, \rho\} \right),
\end{align}
generated by a time-independent Hamiltonian that is a quadratic polynomial in $x$ and $p$
\begin{align}
\hat{H} = \frac{1}{2m}\hat{p}^2 + \frac{\mu}{2} \{\hat{x},\hat{p}\} + \frac{m\omega^2}{2} \hat{x}^2,
\end{align}
with $\mu$, $m$, and $\omega^2$ real, and by time-independent Lindblad operators that are linear polynomials in the same
\begin{align}
V\_i = a\_i \hat{p} + b\_i \hat{x}, \qquad (i=1,2)
\end{align}
with $a\_i$ and $b\_i$ complex. The master equation can be re-written as
\begin{align}
\partial\_t \rho = -i &[\hat{H},\rho] + i (\lambda/2) [\hat{p},\{\hat{x},\rho\}] - i (\lambda/2) [\hat{x},\{\hat{p},\rho\}] \\
&- D\_{pp}[\hat{x},[\hat{x},\rho]] - D\_{xx}[\hat{p},[\hat{p},\rho]] + D\_{xp}[\hat{p},[\hat{x},\rho]] + D\_{px}[\hat{x},[\hat{p},\rho]]
\end{align}
with coefficients
\begin{align}
D\_{xx} &= \frac{\vert a\_1 \vert^2 + \vert a\_2 \vert^2}{2} \quad , \quad & D\_{pp} &= \frac{\vert b\_1 \vert^2 + \vert b\_2 \vert^2}{2},\\
D\_{xp} &= D\_{px} = -\mathrm{Re} \frac{a\_1^\* b\_1 + a\_2^\* b\_2}{2} \quad , \quad & \lambda &= \mathrm{Im} (a\_1^\* b\_1 + a\_2^\* b\_2),
\end{align}


**More generally, we say a system undergoes *quantum Brownian motion* when it evolves according to the above master equation**, regardless of whether it forms a quantum dynamical semigroup. If it obeys the master equation with time-independent coefficients then the QBM is *time-homogeneous* (in the sense of a Markov process); otherwise it is *time-inhomogeneous*. The class of all possible instantaneous QBM dynamics is parameterized by $\mu$, $m$, $\omega^2$, $a\_i$, and $b\_i$.


The resulting dynamics take a particularly beautiful form in the [Wigner representation](https://en.wikipedia.org/wiki/Wigner_quasiprobability_distribution). The above master equation for $\rho$ is equivalent to the following dynamical equation for the Wigner function $W(x,p)$:
\begin{align}
\partial\_t W = -\frac{p}{m}\partial\_x W + m\omega^2 & x \partial\_p W + (\lambda - \mu)\partial\_x (x W) + (\lambda + \mu)\partial\_p (p W)\\
&+D\_{pp} \partial^2\_x W + D\_{xx} \partial^2\_p + (D\_{xp}+D\_{px}) \partial\_x \partial\_p W.
\end{align}
More compactly:
\begin{align}
\partial\_t W (\alpha) &= \left[ F\_{ab} \partial\_a \alpha\_b + D\_{ab} \partial\_a \partial\_b \right] W(\alpha)
\end{align}
where 
\begin{align}
F\_{ab} = \left( \begin{array}{cc} \lambda - \mu & -1/m \\ m \omega^2 & \lambda+\mu \end{array} \right) \quad, \quad D\_{ab} = \left( \begin{array}{cc} D\_{xx} & D\_{xp} \\ D\_{px} & D\_{pp} \end{array} \right) 
\end{align}
are matrices with real elements. Above, the phase-space indices $a,b$ take the values $x,p$, with Einstein summation assumed, so that $\alpha\_a$ is a vector in phase space. (The directional derivative $\partial\_a$ is just shorthand for $\partial\_{\alpha\_a}$.)


This is *identical* in form to a Klein-Kramers equation (more generally a [Fokker-Planck](https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation)-type equation) for the phase-space probability distribution of a *classical* point particle undergoing Brownian motion. 


This is remarkable because such equations were originally derived for true probability distribution, but they also apply to the Wigner function. As a bonus, this gives us an immediate and simple physical interpretation for each of the terms in the QBM master equation


The best comprehensive modern statement of the above definition is probably


* Isar et al. "[Open Quantum Systems](http://www.worldscientific.com/doi/abs/10.1142/S0218301394000164)"


which includes comparisions to important special cases discussed by other authors. Here are some more references I found useful in compiling the above: 


* Alicki and Lendi, [arXiv:quant-ph/0205188](http://arxiv.org/abs/quant-ph/0205188).
* Alicki, [Quantum Dynamical Semigroups and Applications](http://rads.stackoverflow.com/amzn/click/354070860X).
* Lindblad, G. (1976). "[On the generators of quantum dynamical semigroups](http://link.springer.com/article/10.1007%2FBF01608499)". Commun. Math. Phys. 48 (2) 119.
* Breuer and Petruccione, [The Theory Open Quantum Systems](http://rads.stackoverflow.com/amzn/click/0198520638).
* C. Caves, [Completely positive maps, positive maps, and the Lindblad form](http://info.phys.unm.edu/~caves/reports/lindblad.pdf).
* G. Lindblad, "[Brownian motion of a quantum harmonic oscillator](http://www.sciencedirect.com/science/article/pii/003448777690029X)".
* H. Dekker, "[Quantization of the linearly damped harmonic oscillator](http://journals.aps.org/pra/abstract/10.1103/PhysRevA.16.2126)"
* A. Sandulescu and H. Scutaru, "[Open quantum systems and the damping of collective modes in deep inelastic collisions](http://www.sciencedirect.com/science/article/pii/000349168790162X)".
* A.O. Caldeira and A.J. Leggett, "[Path integral approach to quantum Brownian motion](http://www.sciencedirect.com/science/article/pii/0378437183900134)".
* B. Hu, J. Paz, and Y. Zhang, "[Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.45.2843)".
* J. Halliwell and T. Yu, "[Alternative derivation of the Hu-Paz-Zhang master equation of quantum Brownian motion](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.53.2012)".
* W Zurek, "[Decoherence, einselection, and the quantum origins of the classical](http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.75.715)".


