Quantum correlation function

Let $H$ be the Hamiltonian (quantum Hamiltonian ) on a finite-dimensional Hilbert space, and let $\tau_t$ denote the Heisenberg time evolution on observables (observable algebra ),

Let $\rho$ be a density operator state (density-operator state ). The (real-time) two-point correlation function of observables $A,B$ in state $\rho$ is

In thermal equilibrium at inverse temperature $\beta>0$ (inverse temperature β ), one uses the Gibbs state (quantum Gibbs state ) $\rho_\beta$ and writes

A commonly used connected (or “cumulant”) correlator subtracts the product of one-point functions:

where $\langle\cdot\rangle_\beta$ is the thermal quantum expectation value .

This is the quantum analogue of the classical two-point correlation function .

Physical interpretation

Two-point correlation functions quantify:

• Fluctuations: how deviations of $A$ from its mean co-vary with $B$ at later times.
• Dynamical memory: how quickly correlations decay (relaxation and mixing).
• Response: via linear response theory, commutator-based correlators determine susceptibilities and transport coefficients.

In equilibrium, correlation functions are time-translation invariant and obey characteristic “imaginary-time” relations captured by the KMS condition (finite) .

Key properties

1. Spectral (energy-eigenbasis) representation.
   If $H|n\rangle=E_n|n\rangle$ and $A_{nm}=\langle n|A|m\rangle$, then for the Gibbs state,

   $$G_{A,B}(t)\;=\;\frac{1}{Z(\beta)}\sum_{n,m} e^{-\beta E_n}\,e^{it(E_n-E_m)}\,A_{nm}\,B_{mn},$$

   where $Z(\beta)$ is the quantum partition function .

2. Equilibrium (stationarity).
   In the Gibbs state, $G_{A,B}(t)$ depends only on the time difference because the state is invariant under $\tau_t$. In particular, $\langle \tau_t(A)\rangle_\beta=\langle A\rangle_\beta$.

3. KMS (imaginary-time shift) relation.
   The Gibbs state is a $\beta$-KMS state (KMS condition (finite) ). One consequence is that the function

   $$F_{A,B}(t)\;=\;\langle A\,\tau_t(B)\rangle_\beta$$

   extends to complex time and satisfies the boundary identity

   $$\langle A\,\tau_{t+i\beta}(B)\rangle_\beta \;=\;\langle \tau_t(B)\,A\rangle_\beta.$$

   This is the precise statement of thermal “imaginary-time periodicity” (up to exchanging operator order).

4. Imaginary-time (Matsubara) correlators.
   For $\tau\in[0,\beta]$, define the imaginary-time correlator

   $$G_{A,B}(\tau)\;=\;\operatorname{Tr}\!\big(\rho_\beta\,\tau_{-i\tau}(A)\,B\big).$$

   The KMS condition determines how $G_{A,B}(\tau)$ behaves at the endpoints $\tau=0$ and $\tau=\beta$ via the order-exchange relation above.

5. Fourier-domain detailed-balance form (finite systems).
   If one defines a frequency-space correlation spectrum by Fourier transforming the real-time correlator, the KMS condition implies a thermal detailed-balance relation of the schematic form

   $$S_{A,B}(-\omega)\;=\;e^{-\beta\omega}\,S_{B,A}(\omega),$$

   encoding the asymmetry between absorption and emission at temperature $T$.

6. Special cases and reductions.

   • If $A$ commutes with $H$, then $\tau_t(A)=A$ and $G_{A,B}(t)$ is constant in time.
   • If $A=B$ is self-adjoint, then $G^{\mathrm c}_{A,A}(0)$ is the thermal variance of $A$, a static fluctuation measure.

7. Connection to thermodynamics.
   Static (equal-time) correlations and susceptibilities often control derivatives of thermodynamic potentials such as the Helmholtz free energy (Helmholtz free energy ) with respect to parameters that couple to observables.