What does a coordinate representation of density matrix mean?

Question

A coordinate representation of density matrix $\rho$ is defined as

$$ \rho (x, x') \equiv \left<x\right| \rho \left|x'\right> .$$

When $x = x'$, this expresses a probability where a particle is in the state $\left|x\right>$.

Question: what does that mean when $x \neq x'$? Is that related to some probability?

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According to Feynman (Statistical Mechanics a set of lectures, (p.72)),

$$ I \equiv \int dx _1 \int dx _2 \cdots \int dx _{n - 1} \left<x\right| \rho \left|x_1\right> \left<x_1\right| \rho \left|x_2\right> \cdots \left<x_{n-1}\right| \rho \left|x'\right>$$

can be interpreted as that "the particle travels from $x'$ to $x$ through a series of intermediate steps, $x_1, x_2, \cdots, x_{n-1}$, which define a path". I don't understand this statement.

Answer 1

The diagonal entries of the density matrix are called populations and provide information about the probability density of the particles (described by the density matrix), i.e. their probability of "being found" in real space.

This is easily seen from a density matrix $\rho = |\Psi\rangle \langle \Psi|$, and $$\rho(x,x) = \langle x|\rho|x\rangle = \langle x|\Psi\rangle \langle \Psi|x\rangle = | \langle x |\Psi\rangle|^2 = |\psi(x)|^2,$$ which is the usual quantum mechanical probaibility density.

The off diagonal entries of the density matrix are called coherences and provide information about the phase coherence of the system described by $\rho$ between two positions $x$ and $x'$. Is there a fixed phase relationship between $x$ and $x'$, especially as $|x-x'|\rightarrow \infty$? I.e. will constructive & coherent interference occur over a large distance or will it be washed out?

The most famous application of the off diagonal elements of the density matrix is in Off-Diagonal Long-Range Order (ODLRO), which is what manifests in Bose-Einstein Condensates or Superfluids. These are phases where the system breaks a $U(1)$ phase and the wavefunction "picks" a specific phase $\theta$.

The "broken" phase is distinguished from the unbroken phase because it obeys: $$ \lim_{|x-x'|\rightarrow \infty} \rho(x,x') \rightarrow n_0 \neq 0,$$ i.e. phase coherence is preserved over arbitrarily long distances.