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?
Best Answer
(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 — when the evolution of its density matrix obeys a Lindblad master 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. 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-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
which includes comparisions to important special cases discussed by other authors. Here are some more references I found useful in compiling the above: