Correlation Functions

Purpose

A correlation function is not specified by the symbol $\langle A B\rangle$ alone. One must declare:

1. the state or ensemble;
2. the operator pair and spatial labels;
3. the time-evolution generator;
4. the operator order or contour;
5. whether one-point products are subtracted;
6. bosonic or fermionic parity where ordering exchanges operators;
7. the Fourier sign, normalization, and spectral variable;
8. any volume, site-count, density, or degeneracy normalization.

This card uses one internally compatible convention set for quick calculation. The full cross-convention ledger is Correlation Function Definitions, and the physical map is Correlation Functions Overview.

At a glance

Object	Definition in this card	Main use
Greater	$C_{AB}^{>}(t)=\langle A(t)B(0)\rangle$	Ordered transitions
Lesser	$C_{AB}^{<}(t)=\langle B(0)A(t)\rangle$	Reversed order
Connected	$C_{AB}^{c}(t)=\langle\delta A(t)\delta B(0)\rangle$	Fluctuations about means
Symmetrized	$C_{AB}^{\mathrm{sym}}=\frac12\langle\{\delta A(t),\delta B(0)\}\rangle$	Two-order fluctuation average
Commutator	$C_{AB}^{\mathrm{com}}=\langle[A(t),B(0)]\rangle$	Causal response input
Time ordered	$C_{AB}^{\mathrm T}=\langle\mathcal T A(t)B(0)\rangle$	Real-time perturbation theory
Retarded	$\chi_{AB}^{\mathrm R}=(i/\hbar)\theta(t)\langle[A(t),B]\rangle$	Response to a source
Matsubara	$\mathcal C_{AB}(\tau)=s_{AB}\langle\mathcal T_\tau A(\tau)B(0)\rangle_\beta$	Thermal imaginary time
Ordered spectrum	$\mathcal S_{AB}=(2\pi)^{-1}\int dt\,e^{i\omega t}C_{AB}^{>}(t)$	Transition strengths
Structure factor	$S^c(\mathbf q,\omega)$	Momentum-resolved fluctuations

These objects are related, but they are not interchangeable.

State and time evolution

For a normalized density operator $\rho$,

$$\langle X\rangle_\rho \equiv \operatorname{Tr}(\rho X).$$

Real-time Heisenberg evolution under $H$ is

$$A(t) = e^{iHt/\hbar} A e^{-iHt/\hbar}.$$

If

$$[\rho,H]=0,$$

the state is stationary and

$$\langle A(t)B(t')\rangle_\rho = \langle A(t-t')B(0)\rangle_\rho.$$

Only in this stationary case does one relative time and one frequency contain the complete two-time dependence. For a nonstationary state, retain center and relative times:

$$T = \frac{t+t'}2, \qquad t_{\mathrm r} = t-t'.$$

In a grand-canonical state,

$$\rho_\beta = \frac{ e^{-\beta(H-\mu N)} }{ \operatorname{Tr}e^{-\beta(H-\mu N)} },$$

stationarity under $H$ holds if $[H,N]=0$. Number-changing propagators are also often evolved with $H-\mu N$, measuring energies relative to $\mu$. That generator choice shifts spectral labels and must be stated.

Equal-time and connected functions

An equal-time correlator is

$$C_{AB}(\mathbf x,\mathbf y;t) = \langle A(\mathbf x,t) B(\mathbf y,t) \rangle.$$

Equal time does not imply equal position, commutativity, normal ordering, connected subtraction, or stationarity.

Common one- and two-body field correlators are

$$\Gamma^{(1)}(\mathbf x,\mathbf y) = \left\langle \psi^\dagger(\mathbf x) \psi(\mathbf y) \right\rangle,$$ $$n(\mathbf x) = \Gamma^{(1)}(\mathbf x,\mathbf x),$$

and

$$G^{(2)}(\mathbf x,\mathbf y) = \left\langle \psi^\dagger(\mathbf x) \psi^\dagger(\mathbf y) \psi(\mathbf y) \psi(\mathbf x) \right\rangle.$$

Where the densities are nonzero, normalized coherence functions in this ordering are

$$g^{(1)}(\mathbf x,\mathbf y) = \frac{ \Gamma^{(1)}(\mathbf x,\mathbf y) }{ \sqrt{ n(\mathbf x)n(\mathbf y) } },$$

and

$$g^{(2)}(\mathbf x,\mathbf y) = \frac{ G^{(2)}(\mathbf x,\mathbf y) }{ n(\mathbf x)n(\mathbf y) }.$$

Normal ordering in $G^{(2)}$ excludes the self-contact term that appears in $\langle n(\mathbf x)n(\mathbf y)\rangle$. Lattice and continuum contact conventions must therefore not be mixed casually.

Define fluctuation operators

$$\delta A \equiv A-\langle A\rangle, \qquad \delta B \equiv B-\langle B\rangle.$$

The connected two-point function is

$$\begin{aligned} C_{AB}^{c}(t) &= \langle\delta A(t)\delta B(0)\rangle \\ &= \langle A(t)B(0)\rangle - \langle A\rangle\langle B\rangle \end{aligned}$$

for a stationary state. Connected does not mean symmetrized, retarded, diagrammatically irreducible, or entangled.

Greater, lesser, and conjugation

This card defines raw ordered products with no hidden factor of $i$ or $\hbar$:

$$\begin{aligned} C_{AB}^{>}(t) &\equiv \langle A(t)B(0)\rangle, \\ C_{AB}^{<}(t) &\equiv \langle B(0)A(t)\rangle. \end{aligned}$$

For a stationary state,

$$C_{AB}^{<}(t) = C_{BA}^{>}(-t).$$

Complex conjugation gives

$$\left[ C_{AB}^{>}(t) \right]^* = C_{B^\dagger A^\dagger}^{>}(-t).$$

For $B=A^\dagger$,

$$\left[ C_{AA^\dagger}^{>}(t) \right]^* = C_{AA^\dagger}^{>}(-t),$$

so its Fourier transform is real as a distribution. The labels greater and lesser refer to operator order, not numerical magnitude.

Symmetrized and commutator channels

For fluctuation operators,

$$C_{AB}^{\mathrm{sym}}(t) \equiv \frac12 \left\langle \left\lbrace \delta A(t),\delta B(0) \right\rbrace \right\rangle.$$

The commutator channel is

$$C_{AB}^{\mathrm{com}}(t) \equiv \left\langle [A(t),B(0)] \right\rangle.$$

Equivalently,

$$C_{AB}^{\mathrm{sym}}(t) = \frac12 \left[ C_{AB}^{>,c}(t) + C_{AB}^{<,c}(t) \right],$$

while

$$C_{AB}^{\mathrm{com}}(t) = C_{AB}^{>}(t) - C_{AB}^{<}(t).$$

Connected subtraction leaves the commutator unchanged because scalar one-point functions commute. The symmetrized function measures a two-order fluctuation average; the commutator supplies causal response. Their equilibrium relation is the fluctuation–dissipation theorem, not an identity in an arbitrary state.

Real-time Fourier convention

For a stationary function $F(t)$, this card uses angular frequency:

$$\begin{aligned} F(\omega) &= \int_{-\infty}^{\infty} dt\, e^{i\omega t} F(t), \\ F(t) &= \int_{-\infty}^{\infty} \frac{d\omega}{2\pi}\, e^{-i\omega t} F(\omega). \end{aligned}$$

The energy variable is

$$E=\hbar\omega.$$

If $S_\omega$ is a density per unit angular frequency and $S_E$ is the same weight per unit energy,

$$S_E(E) = \frac1\hbar S_\omega(E/\hbar).$$

Changing the horizontal axis without this Jacobian changes integrated spectral weight. Other valid conventions move factors of $2\pi$ or reverse the Fourier sign; translate all formulas together.

Time ordering

Let $p_A,p_B\in\{0,1\}$ denote fermion parity. Graded real-time ordering is

$$\begin{aligned} \mathcal T A(t)B(t') ={}& \theta(t-t')A(t)B(t') \\ &+ (-1)^{p_Ap_B} \theta(t'-t)B(t')A(t). \end{aligned}$$

Therefore

$$\begin{aligned} C_{AB}^{\mathrm T}(t) &\equiv \langle\mathcal T A(t)B(0)\rangle \\ &= \theta(t)C_{AB}^{>}(t) + (-1)^{p_Ap_B} \theta(-t)C_{AB}^{<}(t). \end{aligned}$$

For two odd fermionic operators, the reversed term acquires a minus sign. At $t=0$, canonical commutators or anticommutators can produce a jump or contact term, so a chosen $\theta(0)$ does not replace one-sided limits.

Many-body propagators often multiply the ordered expectation by $-i/\hbar$. Time ordering itself is only the ordering rule; the prefactor is a separate convention.

Retarded and advanced response

Adopt the source coupling

$$H_{\mathrm{pert}}(t) = - f(t)B.$$

Linear response is

$$\delta\langle A(t)\rangle = \int_{-\infty}^{\infty} dt'\, \chi_{AB}^{\mathrm R}(t-t') f(t'),$$

with

$$\chi_{AB}^{\mathrm R}(t) = \frac{i}{\hbar} \theta(t) \langle[A(t),B(0)]\rangle.$$

The matching advanced function is

$$\chi_{AB}^{\mathrm A}(t) = - \frac{i}{\hbar} \theta(-t) \langle[A(t),B(0)]\rangle.$$

Thus

$$\chi_{AB}^{\mathrm R}(t)=0 \quad(t<0),$$

and

$$\chi_{AB}^{\mathrm A}(t)=0 \quad(t>0).$$

Their difference is

$$\chi_{AB}^{\mathrm R}(t) - \chi_{AB}^{\mathrm A}(t) = \frac{i}{\hbar} \langle[A(t),B(0)]\rangle.$$

The sign of the retarded kernel follows from the minus sign in $H_{\mathrm{pert}}$. A text using $+fB$ or $-i\theta/\hbar$ must transform the response law consistently. A time-ordered function is not retarded merely because both formulas contain step functions.

Single-particle Green functions

For canonical annihilation operators, define

$$\eta = \begin{cases} +1, & \text{bosons},\\ -1, & \text{fermions}, \end{cases} \qquad [X,Y]_\eta \equiv XY-\eta YX.$$

A common normal convention is

$$\begin{aligned} G_{ab}^{>}(t) &= - \frac{i}{\hbar} \langle d_a(t)d_b^\dagger(0)\rangle, \\ G_{ab}^{<}(t) &= - \frac{i\eta}{\hbar} \langle d_b^\dagger(0)d_a(t)\rangle. \end{aligned}$$

Then

$$\begin{aligned} G_{ab}^{\mathrm R}(t) &= \theta(t) \left[ G_{ab}^{>}(t)-G_{ab}^{<}(t) \right] \\ &= - \frac{i}{\hbar} \theta(t) \left\langle [d_a(t),d_b^\dagger]_\eta \right\rangle. \end{aligned}$$

This retarded single-particle Green function uses a statistics-dependent bracket and a conventional negative prefactor. It is not the observable susceptibility above. Number-changing Lehmann sectors, chemical-potential shifts, anomalous propagators, and sum rules belong to Green Functions in Many-Body QM.

Imaginary time and Matsubara transform

Let

$$\mathcal K = H-\mu N, \qquad L_\tau = \beta\hbar.$$

Imaginary-time evolution is

$$A(\tau) = e^{\tau\mathcal K/\hbar} A e^{-\tau\mathcal K/\hbar}.$$

A channel-dependent Matsubara correlator is

$$\mathcal C_{AB}(\tau-\tau') = s_{AB} \left\langle \mathcal T_\tau A(\tau)B(\tau') \right\rangle_\beta.$$

The fixed sign $s_{AB}$ is definitional. The exchange sign inside $\mathcal T_\tau$ follows from fermion parity. Normal boson and fermion propagators often use $s_{AB}=-1$, while density and coordinate correlators often use $s_{AB}=+1$.

For transported parity $p$,

$$\mathcal C_p(\tau+\beta\hbar) = (-1)^p \mathcal C_p(\tau).$$

Even channels are periodic and odd single-particle channels are antiperiodic. Using Matsubara energies $\zeta_n$,

$$\begin{aligned} \mathcal C(i\zeta_n) &= \frac1\hbar \int_0^{\beta\hbar} d\tau\, e^{i\zeta_n\tau/\hbar} \mathcal C(\tau), \\ \mathcal C(\tau) &= \frac1\beta \sum_n e^{-i\zeta_n\tau/\hbar} \mathcal C(i\zeta_n). \end{aligned}$$

The grids are

$$\zeta_n^{\mathrm B} = \frac{2\pi n}{\beta}, \qquad \zeta_n^{\mathrm F} = \frac{(2n+1)\pi}{\beta}.$$

The corresponding angular frequencies are $\omega_n=\zeta_n/\hbar$. Analytic continuation requires a valid spectral representation; substituting $i\zeta_n\to E+i0$ into an arbitrary numerical fit is not a controlled continuation.

Ordered spectrum and Lehmann form

Define the ordered transition spectrum by

$$\mathcal S_{AB}(\omega) \equiv \frac1{2\pi} \int_{-\infty}^{\infty} dt\, e^{i\omega t} C_{AB}^{>}(t).$$

For

$$\rho = \sum_n p_n \lvert n\rangle\langle n\rvert, \qquad H\lvert n\rangle = E_n\lvert n\rangle,$$

the Lehmann representation is

$$\begin{aligned} \mathcal S_{AB}(\omega) ={}& \sum_{n,m} p_n A_{nm}B_{mn} \\ &\times \delta \left( \omega - \frac{E_m-E_n}{\hbar} \right). \end{aligned}$$

For $B=A^\dagger$,

$$\mathcal S_{AA^\dagger}(\omega) \geq0$$

as a spectral measure. The zeroth moment is

$$\int_{-\infty}^{\infty} d\omega\, \mathcal S_{AB}(\omega) = \langle AB\rangle.$$

At thermal equilibrium,

$$\mathcal S_{BA}(-\omega) = e^{-\beta\hbar\omega} \mathcal S_{AB}(\omega).$$

Detailed balance compares reversed operator order and reversed frequency. It does not say that an arbitrary cross-spectrum is even or nonnegative.

Response density and fluctuation–dissipation preview

For the observable-response convention above, define

$$\rho_{AB}(\omega) \equiv \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty} dt\, e^{i\omega t} \langle[A(t),B(0)]\rangle.$$

It is the signed difference

$$\rho_{AB}(\omega) = \frac1\hbar \left[ \mathcal S_{AB}(\omega) - \mathcal S_{BA}(-\omega) \right].$$

The retarded–advanced discontinuity is

$$\chi_{AB}^{\mathrm R}(\omega) - \chi_{AB}^{\mathrm A}(\omega) = 2\pi i\, \rho_{AB}(\omega).$$

For a conjugate Hermitian equilibrium channel in this convention,

$$\operatorname{Im} \chi_{AA}^{\mathrm R}(\omega) = \pi\rho_{AA}(\omega).$$

Detailed balance yields

$$\rho_{AB}(\omega) = \frac{ 1-e^{-\beta\hbar\omega} }{\hbar} \mathcal S_{AB}(\omega).$$

If

$$\mathcal S_{AB}^{\mathrm{sym}}(\omega) \equiv \frac12 \left[ \mathcal S_{AB}(\omega) + \mathcal S_{BA}(-\omega) \right],$$

then

$$\mathcal S_{AB}^{\mathrm{sym}}(\omega) = \frac{\hbar}{2} \coth \left( \frac{\beta\hbar\omega}{2} \right) \rho_{AB}(\omega).$$

These are equilibrium relations with the displayed normalization. The full zero-frequency, matrix-channel, and nonequilibrium cautions belong to the Fluctuation–Dissipation Theorem.

Single-particle spectral function

For a normal fermionic Green function written per unit energy, define

$$\mathcal A_{ab}(E) \equiv \frac{i}{2\pi} \left[ G_{ab}^{\mathrm R}(E) - G_{ab}^{\mathrm A}(E) \right].$$

For a diagonal channel,

$$\mathcal A_{aa}(E) = - \frac1\pi \operatorname{Im} G_{aa}^{\mathrm R}(E).$$

Canonical anticommutation gives

$$\int_{-\infty}^{\infty} dE\, \mathcal A_{aa}(E) = 1$$

for a complete normalized orbital. This positive single-particle spectral function is not the signed observable response density $\rho_{AB}(\omega)$, despite both being retarded–advanced discontinuities.

Static and dynamic structure factors

For a Hermitian lattice operator $O_j$, choose

$$O_{\mathbf q} \equiv \sum_{j=1}^{L} e^{-i\mathbf q\cdot\mathbf r_j} O_j,$$

and

$$\delta O_{\mathbf q} \equiv O_{\mathbf q} - \langle O_{\mathbf q}\rangle.$$

A connected static convention is

$$S_O^c(\mathbf q) \equiv \frac1L \left\langle \delta O_{\mathbf q} \delta O_{-\mathbf q} \right\rangle.$$

If $O_{-\mathbf q}=O_{\mathbf q}^\dagger$, then

$$S_O^c(\mathbf q)\geq0.$$

The compatible dynamic convention is

$$\begin{aligned} S_O^c(\mathbf q,\omega) \equiv{}& \frac1{2\pi L} \int_{-\infty}^{\infty} dt\, e^{i\omega t} \\ &\times \left\langle \delta O_{\mathbf q}(t) \delta O_{-\mathbf q}(0) \right\rangle. \end{aligned}$$

Its zeroth moment is

$$\int_{-\infty}^{\infty} d\omega\, S_O^c(\mathbf q,\omega) = S_O^c(\mathbf q).$$

At $\mathbf q=0$, a density structure factor measures total-number fluctuations under this normalization. It vanishes in an exact fixed-number state but need not vanish in a grand-canonical state. Scattering intensities also contain form factors, polarization, kinematics, resolution, and background; a structure factor is not by itself a complete cross section.

Spatial decay and correlation length

For a translationally invariant channel,

$$C_{AB}^c(\mathbf r) = \langle \delta A(\mathbf r) \delta B(\mathbf0) \rangle.$$

A common short-range asymptotic form is

$$C^c(r) \sim \frac{ e^{-r/\xi} }{ r^p },$$

where $\xi$ is a channel-dependent correlation length. At a critical point the exponential scale may diverge and algebraic decay can remain. A nonzero large-distance plateau indicates long-range order only after finite-size, symmetry-sector, connected-subtraction, and order-of-limits issues are resolved.

Clustering of connected local correlators is weaker than statistical independence and does not imply a product state at finite separation.

Validation checks

For an analytic or numerical correlator:

1. Check stationarity before reducing two times to one.
2. Verify the adjoint relation under $t\to-t$.
3. Confirm retarded support vanishes for negative time.
4. Integrate an ordered spectrum and recover the equal-time correlator.
5. Check positivity only in conjugate ordered or structure-factor channels.
6. Verify detailed balance only for a Gibbs equilibrium state.
7. Integrate the dynamic structure factor and recover the static one.
8. Check canonical equal-time commutator or anticommutator contact terms.
9. Preserve total spectral weight when introducing plotting broadening.
10. Report whether the horizontal variable is $E$ or $\omega$.

A finite-time transform has frequency resolution of order $1/t_{\max}$. Windowing trades ringing for broadening; a chosen Lorentzian width is not automatically a physical lifetime. Analytic continuation from imaginary time is ill-conditioned even when all exact identities are satisfied.

Common mistakes

• Calling every $\langle AB\rangle$ connected.
• Dropping operator order for noncommuting or fermionic insertions.
• Using a time-ordered propagator as a causal response.
• Mixing the observable commutator response with a statistics-dependent single-particle Green function.
• Comparing spectra without checking Fourier signs, $2\pi$, $\hbar$, volume, and per-energy versus per-frequency factors.
• Assuming every object called a spectral function is nonnegative.
• Treating detailed balance as valid in an arbitrary stationary state.
• Using the equilibrium fluctuation–dissipation theorem in a driven steady state without additional assumptions.
• Ignoring contact terms at coincident points or times.
• Treating the diagonal of a one-body density matrix as the full many-body state.
• Interpreting artificial broadening as a measured linewidth.
• Replacing a scattering forward model by a bare structure factor.
• Continuing Matsubara data by a formal substitution without a controlled spectral representation.

Exercises

1. Stationary reduction

Show that $[\rho,H]=0$ implies $\langle A(t)B(t')\rangle=\langle A(t-t')B(0)\rangle$.

Solution

Write the left side as

$$\operatorname{Tr} \left[ \rho e^{iHt/\hbar} A e^{-iHt/\hbar} e^{iHt'/\hbar} B e^{-iHt'/\hbar} \right].$$

Combine the middle exponentials and use cyclicity of the trace to move $e^{-iHt'/\hbar}$ to the front. Since $\rho$ commutes with $H$, it also commutes with this exponential. The result is

$$\operatorname{Tr} \left[ \rho e^{iH(t-t')/\hbar} A e^{-iH(t-t')/\hbar} B \right],$$

which is $\langle A(t-t')B(0)\rangle$.

2. Shift invariance of connected correlations

Let $A'=A+aI$ and $B'=B+bI$. Show that $\langle\delta A'\delta B'\rangle =\langle\delta A\delta B\rangle$.

Solution

The shifted fluctuation operator is

$$\delta A' = A+aI-\langle A+aI\rangle = A-\langle A\rangle = \delta A.$$

Likewise $\delta B'=\delta B$. Therefore their connected correlator is unchanged. The raw correlator generally changes, which is why the subtraction rule must be stated.

3. Positivity and zeroth moment

Use the Lehmann representation to show that $\mathcal S_{AA^\dagger}(\omega)$ is nonnegative and integrates to $\langle AA^\dagger\rangle$.

Solution

Setting $B=A^\dagger$ gives

$$\mathcal S_{AA^\dagger}(\omega) = \sum_{n,m} p_n \lvert A_{nm}\rvert^2 \delta \left( \omega-\frac{E_m-E_n}{\hbar} \right).$$

Every coefficient is nonnegative, so this is a nonnegative spectral measure. Integration over $\omega$ removes the delta function:

$$\begin{aligned} \int d\omega\, \mathcal S_{AA^\dagger}(\omega) &= \sum_{n,m} p_n \lvert A_{nm}\rvert^2 \\ &= \sum_n p_n \langle n\vert AA^\dagger \vert n\rangle \\ &= \langle AA^\dagger\rangle. \end{aligned}$$

4. Detailed balance to response

Assume thermal detailed balance and derive

$$\rho_{AB}(\omega) = \frac{ 1-e^{-\beta\hbar\omega} }{\hbar} \mathcal S_{AB}(\omega).$$

Solution

By definition,

$$\rho_{AB}(\omega) = \frac1\hbar \left[ \mathcal S_{AB}(\omega) - \mathcal S_{BA}(-\omega) \right].$$

Thermal detailed balance gives

$$\mathcal S_{BA}(-\omega) = e^{-\beta\hbar\omega} \mathcal S_{AB}(\omega).$$

Substitution yields

$$\rho_{AB}(\omega) = \frac{ 1-e^{-\beta\hbar\omega} }{\hbar} \mathcal S_{AB}(\omega),$$

with no additional assumption about time-reversal symmetry.

Canonical explanations

• Correlation Function Definitions
• Correlation Functions Overview
• Equal-Time Correlations
• Time-Dependent Correlations
• Connected Correlation Functions
• Structure Factors
• Green Functions in Many-Body QM
• Retarded and Advanced Response
• Spectral Functions
• Fluctuation–Dissipation Theorem
• Sum Rules

Related lookup pages

• Matsubara Frequency Table
• Linear Response Formula Sheet
• Second-Quantized One-Body Operator
• Green Functions Bridge
• Fourier Transform Conventions
• From Correlation Functions to QFT Observables

References

• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer, 2000.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview Press, 1998.
• R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I,” Journal of the Physical Society of Japan 12, 570–586 (1957).
• D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, CRC Press, 1990.