Correlation Function Definitions

A correlation function is not specified by the symbols $\langle A B\rangle$ alone. The state, operator order, time prescription, Fourier convention, normalization, statistics, and subtraction rule determine which physical object is being calculated.

This sheet compares the main families: ordered products describe fluctuations, commutators describe response, and time-ordered functions organize calculations. Their prefactors and units are stated explicitly so that one can translate between them.

Required background. Correlation functions introduce the physical questions; the Heisenberg picture supplies time evolution.

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. A two-time correlator then depends only on the time difference:

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

In a grand-canonical equilibrium state,

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

stationarity under $H$ still holds when $[H,N]=0$. Single-particle Green functions are also often evolved with $H-\mu N$ so that frequencies are measured relative to $\mu$. That choice must be stated rather than inferred.

For a nonstationary state, use center and relative times,

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

because a single-frequency transform in $t_{\mathrm r}$ does not capture the full $T$ dependence.

Raw Greater and Lesser Correlators

This sheet defines raw ordered products without hidden factors 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

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

The greater and lesser labels refer to operator order, not numerical magnitude. They remain distinct even when $A$ and $B$ are Hermitian.

Equal-Time Correlations

An equal-time correlator fixes both insertion times to one common value:

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

Equal time does not imply:

• equal position;
• commuting operators;
• connected subtraction;
• stationarity;
• normal ordering;
• a probability distribution.

Common lattice examples are

$$\begin{aligned} \Gamma_{ij}^{(1)} &= \langle d_i^\dagger d_j\rangle, \\ C_{nn}(i,j) &= \langle n_i n_j\rangle, \\ C_{SS}^{ab}(i,j) &= \langle S_i^a S_j^b\rangle, \\ C_{\Delta}(i,j) &= \langle\Delta_i^\dagger\Delta_j\rangle. \end{aligned}$$

They diagnose one-body coherence, density fluctuations, spin order, and pair coherence, respectively. The Equal-Time Correlations page explains their normalizations, contact terms, long-distance limits, and measurement interpretation.

Connected Correlations

Define fluctuation operators

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

The connected two-point function in the displayed order is

$$\begin{aligned} C_{AB}^{c} &\equiv \langle\delta A\,\delta B\rangle \\ &= \langle AB\rangle - \langle A\rangle \langle B\rangle. \end{aligned}$$

At unequal times in a stationary state,

$$C_{AB}^{c}(t) = \langle A(t)B(0)\rangle - \langle A\rangle \langle B\rangle.$$

Connected is not the same as symmetrized, retarded, irreducible in a diagrammatic channel, or entangled.

For a fixed three-operator order,

$$\langle ABC\rangle_c ={} \langle ABC\rangle - \langle A\rangle\langle BC\rangle - \langle B\rangle\langle AC\rangle - \langle C\rangle\langle AB\rangle + 2\langle A\rangle \langle B\rangle \langle C\rangle.$$

Higher connected functions subtract every set partition into lower moments. In noncommuting problems, the sequence and source-order convention must remain fixed. The canonical treatment is Connected Correlation Functions.

Symmetrized and Commutator Correlators

For fluctuation operators,

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

The commutator correlator is

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

Connected subtraction does not change a commutator because scalar one-point functions commute. The symmetrized correlator measures fluctuations with the two orders averaged; the commutator controls causal response. They are related at thermal equilibrium by the fluctuation–dissipation theorem, not by an identity valid in every state.

Real-Time Fourier Convention

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

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

Energy transfer is

$$E = \hbar\omega.$$

Changing from a spectrum per unit angular frequency to one per unit energy introduces a Jacobian:

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

This Jacobian applies to spectral densities whose integral is a weight. It does not apply to the susceptibility itself: $\chi_E(E)=\chi_\omega(E/\hbar)$.

Retarded Response

Adopt the source convention

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

The response of $A$ is

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

The retarded susceptibility is

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

The sign follows from the displayed minus sign in $H_{\mathrm{pert}}$. If a source is coupled as $+fB$, the response kernel changes sign.

Retarded means

$$\chi_{AB}^{\mathrm R}(t) = 0 \qquad (t<0).$$

It does not mean “any correlator with a small positive imaginary part,” and it is not the same object as a positive fluctuation spectrum.

Advanced Response

The matching advanced function is

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

It has support only for $t<0$. The retarded–advanced difference removes the step functions:

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

The Retarded and Advanced Response page explains adjoint identities, half-plane analyticity, dispersion relations, stability, and source-response derivations.

Time-Ordered Correlations

Let $p_A,p_B\in\{0,1\}$ be the fermion parities of parity-homogeneous operators. Real-time graded ordering is

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

The generic time-ordered expectation is

$$C_{AB}^{\mathrm T}(t-t') \equiv \left\langle \mathcal T A(t)B(t') \right\rangle.$$

For $t'=0$,

$$C_{AB}^{\mathrm T}(t) ={} \theta(t)C_{AB}^{>}(t) + (-1)^{p_Ap_B} \theta(-t)C_{AB}^{<}(t).$$

At $t=0$, the value of $\theta(0)$ does not replace the need for one-sided limits. Canonical (anti)commutators can produce a jump or contact term.

Many-body Green functions often multiply $C^{\mathrm T}$ by $-i/\hbar$; relativistic QFT and some condensed-matter texts use different prefactors. Time ordering itself is the operator-ordering rule, not that conventional prefactor.

Single-Particle Green-Function Family

For canonical annihilation operators $d_a$, define

$$\eta = \begin{cases} +1, & \text{bosons},\\ -1, & \text{fermions}, \end{cases}$$

and

$$[X,Y]_\eta \equiv XY-\eta YX.$$

A common normal convention is

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

Then

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

and

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

These single-particle retarded functions use a statistics-dependent bracket and a conventional $-i/\hbar$ prefactor. They should not be substituted into the observable-response formula without checking the channel and source convention. Green Functions in Many-Body QM develops the full family.

Imaginary Time and Matsubara Functions

Let

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

Imaginary-time evolution is

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

Graded imaginary-time ordering is

$$\mathcal T_\tau A(\tau)B(\tau') ={} \theta(\tau-\tau')A(\tau)B(\tau') + (-1)^{p_Ap_B} \theta(\tau'-\tau) {}\times B(\tau')A(\tau).$$

A channel-dependent convention is

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

The fixed sign $s_{AB}$ is a definition. The exchange sign $(-1)^{p_Ap_B}$ follows from fermion parity. For example:

• a normal fermion propagator often uses $s_{AB}=-1$;
• a normal boson propagator also often uses $s_{AB}=-1$;
• a density or coordinate correlator often uses $s_{AB}=+1$.

For a two-point function whose transported insertion has fermion parity $p$,

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

These boundary conditions use the full grand-canonical Fock-space trace and evolution with $\mathcal K$. In a fixed-$N$ trace they hold for sector-preserving observables, but cannot simply be applied to number-changing fields: cycling such a field through the sector projector changes the sector.

Matsubara Transform

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 allowed grids are

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

The corresponding angular frequencies are $\omega_n=\zeta_n/\hbar$. The canonical Bosonic and Fermionic Matsubara Frequencies page explains indexing, cutoff, and summation details.

Analytic continuation is not justified by replacing $i\zeta_n$ with $E+i0$ in an arbitrary fitted formula. A spectral representation and control of static bosonic terms are required.

Spectral Object Dictionary

The word spectral is overloaded:

Object	Defining ingredient	Typical sign property
ordered transition spectrum	Fourier transform of $\langle A(t)B(0)\rangle$	nonnegative for $B=A^\dagger$
response spectral density	Fourier transform of a commutator	generally signed
single-particle spectral function	retarded–advanced discontinuity of $G$	positive semidefinite in standard fermionic channels
dynamic structure factor	momentum-resolved ordered spectrum	nonnegative in a conjugate channel
density of states	trace or momentum sum of a single-particle spectrum	nonnegative in standard use
measured intensity	probe and detector forward model	nonnegative counts, but not a universal correlator

The formulas below use compatible conventions, but their units still differ.

Ordered Transition Spectrum

Define

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

Let

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

With $A_{nm}=\langle n\lvert A\rvert m\rangle$ and $B_{mn}=\langle m\lvert B\rvert n\rangle$,

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

If $B=A^\dagger$,

$$\mathcal S_{AA^\dagger}(\omega) \ge 0$$

as a spectral measure. Its zeroth moment is the equal-time ordered correlator:

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

For a Gibbs state with the same generator used in the time evolution, detailed balance is

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

This compares reversed order and frequency. In a grand-canonical state with $H$ evolution it applies as written to number-conserving channels; number-changing channels require chemical-work factors, or $\mathcal K=H-\mu N$ evolution with energy measured accordingly. One arbitrary cross-spectrum need not be even or positive.

Response Spectral Density

For the observable response convention above, define

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

It is the difference of ordered spectra:

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

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 sign convention,

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

Detailed balance gives the equilibrium preview

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

For a matrix of Hermitian observable channels, the general absorptive part is $\boldsymbol\chi''=(\boldsymbol\chi^{\mathrm R}-\boldsymbol\chi^{\mathrm A})/(2i)=\pi\boldsymbol\rho$, not the elementwise imaginary part. Also, any exactly zero-frequency fluctuation weight disappears from the commutator spectrum. It must be kept separately when reconstructing static fluctuations; dividing by $1-e^{-\beta\hbar\omega}$ cannot recover it.

The Fluctuation–Dissipation Theorem derives the conversion to symmetrized spectra and the zero-frequency limit.

Single-Particle Spectral Function

With $G^{\mathrm R}(E)=\int dt\,e^{iEt/\hbar}G^{\mathrm R}(t)$ and the earlier $-i/\hbar$ time prefactor, define the normal fermionic spectral density per unit energy:

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

For a diagonal channel,

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

Canonical anticommutation gives

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

for a complete normalized orbital. This positive single-particle spectrum is not the signed response density $\rho_{AB}$, even though both are retarded–advanced discontinuities.

Static Structure Factor

For a Hermitian lattice operator $O_j$, define

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

With

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

one connected static convention is

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

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

$$S_O^c(\mathbf q) \ge 0.$$

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 ensemble.

Dynamic Structure Factor

A compatible angular-frequency convention is

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

Its zeroth-moment relation is

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

The factor $1/L$, the sign in $O_{\mathbf q}$, and the factor $1/(2\pi)$ vary across disciplines. Scattering cross sections also include probe form factors, polarization tensors, kinematic prefactors, resolution, and background. The canonical home is Structure Factors.

Matrix-Valued Correlators

With internal indices, a spectrum is a matrix:

$$\mathbf S(\omega) = \left[ S_{ab}(\omega) \right].$$

Positivity means

$$\mathbf v^\dagger \mathbf S(\omega) \mathbf v \ge 0$$

for every vector $\mathbf v$ in a conjugate ordered channel. It does not require each off-diagonal entry to be real or positive.

Hermiticity, reciprocity, time reversal, inversion, and point-group symmetry impose different index and momentum relations. Apply only the symmetries actually possessed by the state, Hamiltonian, operators, and boundary conditions.

Equal-Time Contact Checks

For canonical modes,

$$\begin{aligned} \langle a_i a_j^\dagger\rangle &= \delta_{ij} + \langle a_j^\dagger a_i\rangle, \\ \langle c_i c_j^\dagger\rangle &= \delta_{ij} - \langle c_j^\dagger c_i\rangle. \end{aligned}$$

These relations fix jumps in one-sided single-particle Green functions. In a continuum,

$$\delta_{ij} \longrightarrow \delta^{(d)}(\mathbf x-\mathbf y),$$

so coincident-point expressions can be distributional or ultraviolet divergent. Normal ordering removes specified contractions; it is not a universal instruction to discard every contact term.

Common Channel Examples

Channel	Operators	Typical object
density	$\delta n_{\mathbf q}$, $\delta n_{-\mathbf q}$	structure factor and density response
spin	$S_{\mathbf q}^a$, $S_{-\mathbf q}^b$	magnetic structure factor and susceptibility
current	$J^a$, $J^b$	conductivity kernel with possible contact term
single particle	$d_{\mathbf k}$, $d_{\mathbf k}^\dagger$	Green function and addition/removal spectrum
pair	$\Delta_{\mathbf q}$, $\Delta_{\mathbf q}^\dagger$	pair susceptibility and coherence
order parameter	$\delta O_{\mathbf q}$, $\delta O_{-\mathbf q}$	static scaling and collective spectrum

The same operator pair can generate a raw correlator, response, Matsubara function, or structure factor. The time prescription and normalization select the physical question.

Conversion and Limit Warnings

• Static limit: $\lim_{\omega\to0}\chi(\mathbf q,\omega)$ can depend on whether $\mathbf q\to0$ is taken first.
• Equal-time limit: $t\to0^+$ and $t\to0^-$ can differ by a canonical contact term.
• Analytic continuation: finite Matsubara data do not determine a stable real-frequency spectrum without additional information.
• Thermodynamic limit: finite systems have discrete lines and recurrences; continua and irreversible widths emerge only after appropriate limits or coupling to an environment.
• Connected subtraction: removing $\langle A\rangle\langle B\rangle$ eliminates a zero-frequency elastic piece only when the state is stationary and the normalization is matched.
• Broadening: replacing $\delta$ functions by Lorentzians or Gaussians is a visualization or physical-resolution model, not an exact identity.
• Energy origin: number-changing spectra shift when evolution changes between $H$ and $H-\mu N$.

Exercises

1. Show shift invariance of a connected correlator

Let $A'=A+aI$ and $B'=B+bI$. Show that

$$\langle A'B'\rangle_c = \langle AB\rangle_c.$$

Solution

The shifted fluctuation operators are

$$\begin{aligned} \delta A' &= A+aI-\langle A+aI\rangle \\ &= A-\langle A\rangle = \delta A, \\ \delta B' &= \delta B. \end{aligned}$$

Therefore

$$\begin{aligned} \langle A'B'\rangle_c &= \langle\delta A'\delta B'\rangle \\ &= \langle\delta A\,\delta B\rangle \\ &= \langle AB\rangle_c. \end{aligned}$$

Connected two-point functions are insensitive to additive scalar offsets.

2. Relate greater and lesser orderings

In a stationary state, prove

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

Then write the time-ordered correlator for two odd fermionic operators.

Solution

By definition,

$$C_{BA}^{>}(-t) = \langle B(-t)A(0)\rangle.$$

Stationarity allows both times to be shifted by $t$:

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

If $p_A=p_B=1$, exchanging the operators under time ordering contributes a minus sign:

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

3. Prove positivity and the zeroth moment

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

Solution

Set $B=A^\dagger$ in the ordered spectrum:

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

Every $p_n$ and $|A_{nm}|^2$ is nonnegative, so the spectral measure is nonnegative.

Integrating over frequency removes the delta function:

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

4. Recover the static structure factor

Show that the dynamic definition on this page satisfies

$$\int d\omega\, S_O^c(\mathbf q,\omega) = S_O^c(\mathbf q).$$

Solution

Insert the dynamic definition and use

$$\int_{-\infty}^{\infty} \frac{d\omega}{2\pi} e^{i\omega t} = \delta(t).$$

Then

$$\begin{aligned} \int d\omega\, S_O^c(\mathbf q,\omega) &= \frac1L \int dt\, \delta(t) \times \left\langle \delta O_{\mathbf q}(t) \delta O_{-\mathbf q}(0) \right\rangle \\ &= \frac1L \left\langle \delta O_{\mathbf q} \delta O_{-\mathbf q} \right\rangle \\ &= S_O^c(\mathbf q). \end{aligned}$$

References

• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003) — Lehmann representations, propagators, response, and sum rules.
• R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I”, Journal of the Physical Society of Japan 12, 570–586 (1957) — linear response and fluctuation relations.
• S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Vol. 1, Clarendon Press (1984) — structure factors, response channels, and experimental normalization.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000) — real-time Green functions, response, spectral functions, and many-body conventions.
• P. C. Martin and J. Schwinger, “Theory of Many-Particle Systems. I”, Physical Review 115, 1342–1373 (1959) — equilibrium Green-function hierarchy and thermal boundary relations.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview Press (1998) — imaginary-time functions, Matsubara transforms, and functional methods.
• L. Van Hove, “Correlations in Space and Time and Born Approximation Scattering in Systems of Interacting Particles”, Physical Review 95, 249–262 (1954) — dynamic correlations and scattering.