Many-Body and Quantum Statistical Mechanics Formulas

These cards support three recurring tasks:

1. compute equilibrium occupations of ideal bosonic or fermionic modes;
2. lift a one-particle operator into number-conserving Fock-space form;
3. identify and convert among equal-time, ordered, response, thermal, and spectral correlation functions.

They are lookup pages, not canonical derivations. Each card links to the page that owns the proof, interpretation, model scope, and specialist extensions.

Choose a card

Task	Card	First formula
Occupy an ideal fermionic mode	Fermi–Dirac Distribution	$f_i=[e^{\beta(\epsilon_i-\mu)}+1]^{-1}$
Occupy an ideal bosonic mode	Bose–Einstein Distribution	$b_i=[e^{\beta(\epsilon_i-\mu)}-1]^{-1}$
Convert a one-particle matrix to Fock space	Second-Quantized One-Body Operator	$d\Gamma(A)=\sum_{ij}A_{ij}d_i^\dagger d_j$
Select an equal-time or dynamical correlator	Correlation Functions	$C_{AB}^{>}(t)=\langle A(t)B(0)\rangle$

If the question asks for a Hamiltonian rather than a formula, use the Model Encyclopedia. If it asks for a theorem, use the Theorems and Results Index.

Shared notation

Symbol	Meaning
$T$	Absolute temperature
$\beta$	$1/(k_{\mathrm B}T)$
$\mu$	Chemical potential associated with a conserved number or charge
$\epsilon_i$	Energy of complete one-particle mode $i$
$d_i,d_i^\dagger$	Annihilation and creation operators in the declared mode basis
$\widehat n_i$	$d_i^\dagger d_i$
$\widehat N$	$\sum_i\widehat n_i$
$A_{ij}$	$\langle\varphi_i\vert A\vert\varphi_j\rangle$
$\langle X\rangle$	Expectation in the stated pure state, density operator, or ensemble
$D(\epsilon)$	One-particle density of states with its volume and degeneracy convention declared

The mode index can include position or momentum, spin, band, orbital, species, polarization, and branch labels. A suppressed label can turn a correct formula into a degeneracy error.

Equilibrium occupation layer

For a grand-canonical ideal mode,

$$\langle n_i\rangle = \frac{1}{ e^{\beta(\epsilon_i-\mu)} \mp 1 },$$

where the upper minus sign in the displayed denominator applies to bosons and the lower plus sign applies to fermions. Writing the two formulas separately is often safer:

$$b_i = \frac{1}{ e^{\beta(\epsilon_i-\mu)}-1 }, \qquad f_i = \frac{1}{ e^{\beta(\epsilon_i-\mu)}+1 }.$$

Their common classical limit is

$$b_i \simeq f_i \simeq e^{-\beta(\epsilon_i-\mu)}$$

when the relevant occupations are small.

Bose and Fermi comparison

Check	Bosons	Fermions
Allowed mode counts	$0,1,2,\ldots$	$0,1$
Mean range	$0\leq b_i<\infty$	$0\leq f_i\leq1$
Finite-mode convergence	$\mu<\epsilon_0$	No analogous upper bound
One-mode variance	$b_i(1+b_i)$	$f_i(1-f_i)$
Statistical correction to dilute occupation	Enhanced	Suppressed
Zero-temperature fixed-density structure	Possible condensate-mode separation	Fermi sea and surface
Typical nonconserved excitation	Photon or phonon with $\mu=0$	Not the standard setting

The plus or minus sign is not an effective attractive or repulsive force. It follows from the allowed occupation algebra in a thermal ensemble.

Number and energy sums

For independent modes,

$$N = \sum_i \langle n_i\rangle, \qquad U = \sum_i \epsilon_i \langle n_i\rangle.$$

In a continuum approximation,

$$N = \int D(\epsilon) \langle n(\epsilon)\rangle \,d\epsilon,$$

and similarly for $U$ with an extra factor of $\epsilon$. Before using the integral, check:

• whether $D$ is total or per volume;
• whether it includes spin, polarization, or branch degeneracy;
• whether a discrete bosonic ground mode must be separated;
• whether level spacing is small compared with the thermal scale;
• whether the dispersion and dimensionality match the quoted density of states.

At fixed $N$, the number equation determines $\mu(T)$. At fixed $\mu$, it determines the mean particle number. Those are different thermodynamic problems.

Operator layer

Choose an orthonormal one-particle basis $\{\lvert\varphi_i\rangle\}$. For

$$A_{ij} = \langle\varphi_i\vert A \vert\varphi_j\rangle,$$

the additive number-conserving lift is

$$\widehat A = d\Gamma(A) = \sum_{ij} A_{ij} d_i^\dagger d_j.$$

The bilinear removes one particle from $j$ and creates one in $i$. Diagonal terms weight occupations; off-diagonal terms transfer particles or internal quantum numbers.

Essential checks are

$$d\Gamma(I_1) = \widehat N,$$ $$d\Gamma(A)^\dagger = d\Gamma(A^\dagger),$$ $$[\widehat N,d\Gamma(A)] = 0,$$

and

$$[d\Gamma(A),d\Gamma(B)] = d\Gamma([A,B]).$$

For a one-body reduced density matrix defined by

$$\gamma_{ij} = \langle d_j^\dagger d_i\rangle,$$

one-body expectations are

$$\langle\widehat A\rangle = \operatorname{Tr}_1(A\gamma).$$

This does not imply that $\gamma$ determines $\operatorname{Var}(\widehat A)$ or the full many-body state; those generally contain two-body information.

Operator boundary

Form	Meaning
$A_{ij}d_i^\dagger d_j$	Number-conserving additive one-body lift
$\eta_i d_i^\dagger+\eta_i^*d_i$	Source term; changes particle number
$\Delta_{ij}d_i^\dagger d_j^\dagger+\mathrm{h.c.}$	Pairing term; quadratic but not an additive lift
$V_{ij;kl}d_i^\dagger d_j^\dagger d_l d_k$	Two-body interaction

Hopping between two sites is one-body because one particle moves. A density product is two-body because two occupations enter simultaneously.

Correlation layer

Correlation notation chooses a physical question as much as an algebraic expression. In the convention used by this category:

$$C_{AB}^{>}(t) = \langle A(t)B(0)\rangle, \qquad C_{AB}^{<}(t) = \langle B(0)A(t)\rangle.$$

The connected function subtracts one-point products:

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

For the source convention

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

the retarded response is

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

The real-time Fourier pair is

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

Do not combine a correlator from one convention with a response or spectral identity from another until factors of $i$, $\hbar$, $2\pi$, volume, and the Fourier sign have been translated.

Which correlation object?

Question	Object
Are fluctuations correlated beyond their means?	Connected correlator
Is there one-body coherence?	$\langle d_i^\dagger d_j\rangle$ or $\langle\psi^\dagger(x)\psi(y)\rangle$
What is the pair-coincidence or bunching signal?	Normally ordered $G^{(2)}$ or $g^{(2)}$
How does a source change an observable?	Retarded susceptibility
Which transitions carry ordered spectral weight?	Lehmann or ordered spectrum
What does a scattering probe resolve in momentum and energy?	Dynamic structure factor plus the probe forward model
What is computed in finite-temperature imaginary time?	Matsubara correlator
What adds or removes one particle?	Single-particle Green function

Connected, symmetrized, time ordered, retarded, and Matsubara are not synonyms.

End-to-end workflow

1. Identify the degrees of freedom

State whether $d_i$ describes exact particles, projected orbitals, quasiparticles, normal modes, or another effective basis. A formula diagonal in quasiparticles need not be diagonal in bare-particle operators.

2. Fix the state or ensemble

Declare pure state, canonical, grand canonical, generalized equilibrium, or nonequilibrium preparation. Equilibrium occupations and detailed-balance relations do not follow from stationarity alone.

3. Declare conserved quantities

Specify which number or charge couples to each chemical potential. For a number-changing operator, state whether time evolution uses $H$ or $H-\mu N$.

4. Declare basis and normalization

List all mode labels. State orthonormality, lattice versus continuum normalization, box volume, Fourier convention, and whether density of states or spectra are intensive.

5. Select ordering

Write the operator product explicitly before abbreviating it as a Green function or spectrum. Fermionic exchanges produce signs; observable response uses an ordinary commutator.

6. Run limiting checks

Useful checks include

$$0\leq f_i\leq1,$$ $$\mu<\epsilon_0 \quad \text{for a finite ideal Bose system},$$ $$d\Gamma(I_1)=\widehat N,$$ $$\chi^{\mathrm R}(t<0)=0,$$

and recovery of equal-time spectral weight by frequency integration.

Exact results and model assumptions

The following statements are structural:

• bosonic and fermionic canonical operator algebras;
• the additive lift $d\Gamma(A)$ in a declared orthonormal basis;
• identities following algebraically from a defined correlator;
• Lehmann representations for a specified Hamiltonian and stationary spectral decomposition.

The following require physical assumptions:

• Fermi–Dirac or Bose–Einstein occupations require thermal equilibrium and independent modes or a justified quasiparticle approximation;
• continuum densities of states require a thermodynamic or semiclassical replacement of discrete sums;
• fluctuation–dissipation requires Gibbs equilibrium and compatible conventions;
• quasiparticle poles, lifetimes, Wick factorization, and mean-field bilinears require their own approximation regimes;
• condensation formulas depend on dimension, dispersion, geometry, interactions, ensemble, and order of limits.

Compact notation should not hide which of these levels is being used.

Common mistakes

• Using the Fermi or Bose distribution for an arbitrary nonequilibrium mode population.
• Counting spin or polarization both in the mode sum and in a degeneracy factor.
• Treating a mean mode occupation as an allowed fractional number eigenvalue.
• Setting bosonic $\mu=\epsilon_0$ inside a finite grand partition function.
• Mixing one-particle matrix elements with ladder operators from another basis.
• Calling every quadratic expression a one-body lift.
• Assuming a one-body density matrix determines pair correlations.
• Dropping connected subtraction when testing clustering or structure factors.
• Calling a time-ordered propagator retarded.
• Comparing spectral plots without converting normalization and energy-frequency units.
• Treating artificial broadening as a physical lifetime.
• Inferring that an ideal Bose condensate is automatically a superfluid.

Canonical explanations

Quantum statistics

• Quantum Statistics Overview
• Fermi–Dirac Statistics
• Bose–Einstein Statistics
• Ideal Fermi Gas
• Ideal Bose Gas

Second quantization

• Creation and Annihilation Operators
• One-Body Operators
• Field Operators
• Two-Body Operators

Correlations and response

• Correlation Function Definitions
• Correlation Functions Overview
• Structure Factors
• Green Functions in Many-Body QM
• Retarded and Advanced Response
• Fluctuation–Dissipation Theorem

Related reference pages

• Many-Body and Quantum Statistical Mechanics Reference for the volume-wide route across conventions, formulas, Hamiltonians, glossaries, and canonical owners
• Quantum Gas Formula Sheet
• Fermi Gas Formula Sheet
• Bose Gas Formula Sheet
• Linear Response Formula Sheet
• Matsubara Frequency Table
• Many-Body QM Crosswalk
• Second Quantization Bridge

References

• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer, 2000.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview Press, 1998.
• M. Kardar, Statistical Physics of Particles, Cambridge University Press, 2007.
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 4th ed., Academic Press, 2021.