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 page is a convention ledger. It defines the principal correlator families side by side and gives the shortest reliable conversions among them. The linked canonical pages retain the derivations, asymptotic physics, analytic structure, measurement models, and numerical methods.

Minimum Definition Ledger

Before quoting a correlator, state:

1. the state or ensemble defining $\langle\cdot\rangle$;
2. the generator used for real or imaginary-time evolution;
3. the operators, their order, and their fermion parity;
4. whether one-point or other disconnected pieces are subtracted;
5. lattice, continuum, volume, and internal-index normalizations;
6. the real-time, imaginary-time, angular-frequency, or energy variable;
7. the Fourier-transform sign and factors of $2\pi$ and $\hbar$;
8. whether the object is raw, symmetrized, time ordered, retarded, advanced, or Matsubara;
9. whether a plotted “spectral function” is an ordered spectrum, response density, single-particle spectrum, or measured intensity;
10. the one-sided prescription at equal time and any coincident-point regulator.

Most disagreements between two correct formulas can be traced to one item in this ledger.

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 owns 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,

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

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:

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

Never change the horizontal axis from $\omega$ to $E$ without changing the units of the vertical axis.

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 owns 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

$$\begin{aligned} \mathcal T A(t)B(t') &={} \\[-0.2em] &\quad \theta(t-t')A(t)B(t') \\ &\quad+ (-1)^{p_Ap_B} \theta(t'-t) \\[-0.2em] &\qquad{}\times B(t')A(t). \end{aligned}$$

The generic time-ordered expectation is

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

For $t'=0$,

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

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 owns 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

$$\begin{aligned} \mathcal T_\tau A(\tau)B(\tau') &={} \\[-0.2em] &\quad \theta(\tau-\tau')A(\tau)B(\tau') \\ &\quad+ (-1)^{p_Ap_B} \theta(\tau'-\tau) \\[-0.2em] &\qquad{}\times B(\tau')A(\tau). \end{aligned}$$

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 $p$,

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

Thus even channels are periodic and odd single-particle channels are antiperiodic around the thermal circle.

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 owns 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 $B_{mn}=\langle m\lvert B\rvert n\rangle$,

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

If $B=A^\dagger$,

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

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

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

At thermal equilibrium, detailed balance is

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

This relation compares reversed operator order and reversed frequency. It is not a statement that one arbitrary cross-spectrum is even or positive.

Response Spectral Density

For the observable response convention above, define

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

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

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

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

Single-Particle Spectral Function

For a normal fermionic Green function written 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

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

Its zeroth-moment 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 $H-\mu N$.

Fast Validation Checks

1. Set $t=0^\pm$ and recover the canonical (anti)commutator jump.
2. Integrate a dynamic spectrum and recover its equal-time correlator.
3. Check that a conjugate ordered spectrum and structure factor are nonnegative.
4. Verify detailed balance only in a thermal stationary state.
5. Confirm that the retarded kernel vanishes for negative time.
6. Confirm that the advanced kernel vanishes for positive time.
7. Check the dimensions after changing from $\omega$ to $E=\hbar\omega$.
8. Separate fixed convention signs from fermionic exchange signs.
9. Reconstruct a known free mode or two-level benchmark before analyzing interacting data.
10. Report finite-time windows, artificial broadening, and resolution functions.

Common Mistakes

• Calling every $\langle A(t)B(0)\rangle$ a Green function.
• Calling a time-ordered function retarded because both contain step functions.
• Using an anticommutator in an observable Kubo response merely because the microscopic fields are fermionic.
• Omitting the source sign when defining a susceptibility.
• Treating connected and symmetrized correlators as synonyms.
• Assuming a response spectral density is nonnegative at all frequencies.
• Comparing spectra per unit energy and per unit angular frequency without the factor of $\hbar$.
• Forgetting the fermionic sign in real or imaginary-time ordering.
• Setting $t=0$ without declaring a one-sided prescription.
• Applying thermal detailed balance to a quenched or driven state.
• Inferring every Hamiltonian eigenvalue from one operator-resolved spectrum.
• Interpreting broadened finite-size lines as intrinsic lifetimes without a scaling or resolution analysis.

Cross-Links

• Linear Response Formula Sheet
• Matsubara Frequency Table
• Correlation Functions Overview
• Equal-Time Correlations
• Time-Dependent Correlations
• Connected Correlation Functions
• Retarded and Advanced Response
• Green Functions in Many-Body QM
• Spectral Functions
• Structure Factors
• Fluctuation–Dissipation Theorem
• Sum Rules
• Thermal Green Functions
• Spectral Representation
• Bosonic and Fermionic Matsubara Frequencies
• Correlation Functions Formula Card

References

• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000) — real-time Green functions, response, spectral functions, and many-body conventions.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003) — Lehmann representations, propagators, response, and sum rules.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview Press (1998) — imaginary-time functions, Matsubara transforms, and functional methods.
• 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.
• 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.
• 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.
• S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Vol. 1, Clarendon Press (1984) — structure factors, response channels, and experimental normalization.

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:

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

3. Derive the retarded–advanced discontinuity

Starting from the time-domain definitions, show that

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

Solution

In time,

$$\begin{aligned} \chi_{AB}^{\mathrm R}(t) - \chi_{AB}^{\mathrm A}(t) &= \frac{i}{\hbar} \left[ \theta(t)+\theta(-t) \right] \\ &\quad\times \langle[A(t),B]\rangle. \end{aligned}$$

Away from the convention-dependent point $t=0$, the bracket of step functions is one. Hence

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

Fourier transformation and the definition

$$\rho_{AB}(\omega) = \frac{1}{2\pi\hbar} \int dt\, e^{i\omega t} \langle[A(t),B]\rangle$$

give the result.

4. 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:

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

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

Integrating over frequency removes the delta function:

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

5. 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

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

6. Derive the Matsubara grids

Let

$$\mathcal C(\tau+\beta\hbar) = \xi\mathcal C(\tau), \qquad \xi=\pm1.$$

Find the allowed Matsubara energies in $e^{-i\zeta_n\tau/\hbar}$.

Solution

The basis function must obey the same boundary condition:

$$e^{-i\zeta_n(\tau+\beta\hbar)/\hbar} = \xi e^{-i\zeta_n\tau/\hbar}.$$

After canceling the common factor,

$$e^{-i\beta\zeta_n} = \xi.$$

For $\xi=+1$,

$$\beta\zeta_n = 2\pi n,$$

so

$$\zeta_n^{\mathrm B} = \frac{2\pi n}{\beta}.$$

For $\xi=-1$,

$$\beta\zeta_n = (2n+1)\pi,$$

so

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