Diagrammatic Methods Preview

A many-body diagram is a compact representation of one algebraic term or one class of terms in an expansion of a correlation function, energy, free energy, response, or effective interaction. Lines encode specified two-point functions or interactions. Vertices encode specified couplings. Topology organizes contractions, signs, conservation laws, and repeated structures.

A diagram is not:

• a literal trajectory followed by a microscopic particle;
• a complete physical process independent of its mathematical rules;
• an approximation merely because it has been drawn;
• uniquely defined without an ordering convention, reference state, and observable;
• interchangeable across real time, imaginary time, stationary perturbation theory, and QFT.

The basic chain is

$$\text{ordered operator expansion} \longrightarrow \text{Wick contractions} \longrightarrow \text{algebraic terms} \longrightarrow \text{diagram topologies}.$$

The value lies in the last step: many long sums differ only by labels while sharing the same topology. Once the rules are fixed, diagrams expose which terms are connected, reducible, repeated, dominant, or required by symmetry.

Helpful background. Wick contractions generate the graphs, and many-body perturbation theory explains the reference and volume counting.

Fermion propagators and density vertices

A line acquires a meaning from the Hamiltonian, reference and ordering convention. In the example below a solid directed line is a fermion propagator and a dashed line is a bare instantaneous interaction. Other texts may use the same styles differently.

Working Matsubara convention

Use $\hbar=k_{\mathrm B}=1$ and a normal-state quadratic fermion reference at $T>0$, with $\beta=1/T$. The reference expectation is the full grand-canonical Fock trace of $e^{-\beta K_0}$, $K_0=H_0-\mu N$. A finite fixed-number projection is generally non-Gaussian and needs modified contraction rules. Define

$$G_0(1,2) = -\left\langle \mathcal T_\tau \psi(1) \psi^\dagger(2) \right\rangle_0.$$

For a translation-invariant diagonal reference,

$$G_0(\mathbf k,i\omega_n) = \frac{1}{ i\omega_n-\xi_{\mathbf k} },$$

where

$$\omega_n = (2n+1)\pi T$$

is a fermionic Matsubara frequency and

$$\xi_{\mathbf k} = \epsilon_{\mathbf k}-\mu.$$

This formula defines the solid directed line used below. It is a reference propagator, not yet a statement about the exact interacting spectrum.

Instantaneous density interaction

Take a two-body interaction

$$V = \frac{1}{2\mathcal V} \sum_{\mathbf k,\mathbf k',\mathbf q} \sum_{\sigma,\sigma'} v_{\mathbf q} \times c_{\mathbf k+\mathbf q,\sigma}^\dagger c_{\mathbf k'-\mathbf q,\sigma'}^\dagger c_{\mathbf k',\sigma'} c_{\mathbf k,\sigma}.$$

One may represent $v_{\mathbf q}$ by a dashed interaction line joining two density vertices. Alternatively, one may use one antisymmetrized four-fermion vertex. These are different bookkeeping conventions; exchange must not be built into the vertex and then added again as a separate exchange graph.

From the Ordered Expansion to Diagrams

For an imaginary-time observable $O$,

$$\langle O\rangle = \frac{ \left\langle \mathcal T_\tau O_I \exp\!\left[ -\lambda \int_0^\beta d\tau\, V_I(\tau) \right] \right\rangle_0 }{ \left\langle \mathcal T_\tau \exp\!\left[ -\lambda \int_0^\beta d\tau\, V_I(\tau) \right] \right\rangle_0 }.$$

Expanding the numerator gives

$$\langle O\rangle_{\mathrm{num}} = \sum_{n=0}^{\infty} \frac{(-\lambda)^n}{n!} \times \int_0^\beta d\tau_1\cdots d\tau_n \left\langle \mathcal T_\tau O_I V_I(\tau_1)\cdots V_I(\tau_n) \right\rangle_0.$$

If the reference is Gaussian, Wick’s theorem rewrites each ordered expectation value as a sum over complete contraction patterns. Every contraction contributes a reference two-point function. Every insertion of $V$ contributes a vertex. A diagram records the connectivity of one contraction pattern after equivalent labelings are grouped.

Why the denominator is present

The denominator contains diagrams with no connection to the external operator $O$. These vacuum pieces multiply every numerator contribution. Dividing removes them.

Equivalently, the logarithm of a partition or projection amplitude retains connected vacuum diagrams. Vacuum normalization and connectedness are related but distinct operations:

• the denominator removes vacuum factors from normalized correlators;
• selecting connected diagrams removes pieces disconnected from the external insertions;
• taking a logarithm selects connected vacuum contributions to free energies.

The factorial and symmetry factors

The factor $1/n!$ compensates permutations of $n$ identical interaction insertions. Many permutations generate the same labeled contraction topology and cancel part or all of this factorial. A residual symmetry factor remains when a diagram has automorphisms that leave it unchanged.

One cannot assign a symmetry factor from the picture alone unless the vertex normalization, line types, and whether labels are distinguishable have been specified.

The Elementary Dictionary

For the working convention:

Graphical object	Algebraic meaning
directed solid internal line	$G_0(\mathbf k,i\omega_n)$ or a declared dressed $G$
dashed interaction line	$v_{\mathbf q}$, including its normalization and sign convention
interaction vertex	coupling factor plus momentum, frequency, spin, and flavor conservation
external solid leg	field insertion defining a one-particle correlator
external source leg	density, spin, current, or other operator insertion
closed fermion loop	internal momentum-frequency sum and one extra factor $-1$
repeated identical topology	symmetry or combinatorial factor

Each internal fermion line contributes its propagator factor. Each independent loop variable, after vertex conservation laws are imposed, contributes a momentum-frequency sum or integral. For one such variable the measure and a line factor look like

$$T \sum_{\omega_n} \int_{\mathbf k} G_0(\mathbf k,i\omega_n),$$

only as part of the complete product dictated by the graph. Here

$$\int_{\mathbf k} \equiv \int_{\mathrm{BZ}} \frac{d^d k}{(2\pi)^d}$$

for a lattice Brillouin zone, or a regulated continuum integral.

Conservation at a vertex

For a translation-invariant instantaneous two-body vertex,

$$\mathbf k_1+\mathbf k_2 = \mathbf k_3+\mathbf k_4$$

and

$$\omega_1+\omega_2 = \omega_3+\omega_4.$$

On a crystal, momentum conservation is modulo a reciprocal-lattice vector, allowing Umklapp processes. Spin or flavor is conserved or transformed according to the interaction tensor. Drawing an unlabeled vertex does not remove these Kronecker or Dirac delta functions; it suppresses them visually.

Arrows

For a number-conserving fermion propagator, an arrow records the orientation of fermionic contraction and conserved charge flow. In a hole interpretation, the same line segment can be read relative to a filled reference. In Nambu notation, line and arrow conventions change because normal and anomalous propagators are combined.

An arrow is not a classical path through space.

Two density insertions join the fermion propagators $G_0(k+q)$ and $G_0(k)$. Here $k=(\mathbf k,i\omega_n)$ and $q=(\mathbf q,i\nu_m)$. There is one independent internal momentum-frequency sum, even though the graph has two internal lines. The solid arrows follow the fermion contractions; the external arrows show transferred momentum and frequency.

Perturbative Order and Power Counting

If every interaction insertion carries $\lambda$, a diagram with $V_d$ interaction vertices is nominally order

$$\lambda^{V_d}.$$

That statement is only the first layer. The actual size also depends on:

• powers of volume from loop sums and line normalization;
• spin, flavor, or component sums;
• energy denominators or low-frequency propagators;
• small external momentum or frequency;
• long-range interaction kernels;
• occupation factors and Fermi-surface geometry;
• symmetry factors and cancellations;
• logarithms generated by broad scale intervals.

A high-order diagram can dominate a lower-order one in an infrared-enhanced channel. Conversely, a nominally allowed diagram can vanish by symmetry, kinematics, or antisymmetry.

Large-N counting

If a closed matter loop sums over $N$ flavors, it can contribute a factor $N$. A vertex may carry $1/N$. Counting loops and vertices then organizes diagrams by powers of $1/N$.

Large-N and Saddle-Point Methods Preview explains the model scaling and fluctuation hierarchy. The diagram by itself does not reveal which $N$ is large.

Volume counting

Each independent internal momentum sum contributes a power of volume in finite normalization; each translation-invariant interaction matrix element carries compensating inverse powers. Connected vacuum or energy diagrams should combine to an extensive result when infrared and ultraviolet behavior is regular.

Disconnected products can carry higher powers of volume. Their cancellation belongs to the linked-cluster structure developed in Perturbation Theory in Many-Body Systems.

Fermionic Signs

Fermionic signs come from anticommuting operators into the order needed to form contractions.

Closed-loop sign

Each closed fermion loop contributes an additional factor

$$-1.$$

For density $n_{\mathbf q}=\sum_{\mathbf k\sigma}c_{\mathbf k\sigma}^\dagger c_{\mathbf k+\mathbf q,\sigma}$, define the connected Matsubara susceptibility per volume by

$$\chi_0(\mathbf q,\tau)=\frac1{\mathcal V} \langle\mathcal T_\tau\delta n_{\mathbf q}(\tau)\delta n_{-\mathbf q}(0)\rangle_0.$$

Its regular retarded continuation is the physical susceptibility for a source $K'=-f_{\mathbf q}n_{-\mathbf q}$, with $\chi^R=+i\theta(t)\langle[n_{\mathbf q}(t),n_{-\mathbf q}]\rangle/\mathcal V$. Wick contraction gives

$$\chi_0( \mathbf q, i\nu_m ) = -g T \sum_{\omega_n} \int_{\mathbf k} G_0( \mathbf k+\mathbf q, i\omega_n+i\nu_m ) \times G_0( \mathbf k, i\omega_n ),$$

where $g$ is a spin or flavor degeneracy when all components have the same propagator.

The loop minus sign gives a positive static susceptibility for the stable ideal gas. The polarization for a potential-energy source $+Un$ is $\Pi_0=-\chi_0$, as in RPA. This response-definition sign is separate from the fermion-loop sign.

Exchange sign

Interchanging two external or internal fermionic labels introduces an antisymmetry sign. One may handle exchange by:

• drawing direct and exchange contractions separately with their signs; or
• using antisymmetrized vertices that combine them.

Doing both double counts exchange.

Loop count is not the whole sign

The full sign can also depend on:

• the ordering of external fermionic operators;
• anomalous Nambu conventions;
• how an antisymmetrized matrix element is defined;
• permutation of identical external legs;
• contour-branch indices in nonequilibrium theory.

“Minus one per loop” is a rule within a complete convention, not a substitute for deriving that convention.

Connected, Reducible, and Skeleton Diagrams

Several topological classifications answer different questions.

Class	Definition	Typical role
disconnected	graph has more than one connected component	cancels from normalized connected correlators or exponentiates in $Z$
connected	every part is linked to every other part	contributes to connected correlators
one-particle reducible	cutting one internal single-particle line disconnects the graph	generated by repeated self-energy insertion
one-particle irreducible	cannot be disconnected by cutting one internal single-particle line	belongs to the proper self-energy or 1PI vertex
amputated	external propagator factors have been removed	yields a vertex function, which need not be irreducible
skeleton	built from dressed lines with no explicit self-energy insertion subgraph	used in self-consistent functionals

Connected does not imply one-particle irreducible. A chain of two proper self-energy insertions on one propagator is connected but one-particle reducible.

Two-particle reducibility is channel dependent

A four-point graph can be reducible by cutting two lines in:

• the direct particle–hole channel;
• the crossed particle–hole channel;
• the particle–particle channel.

The same graph can be irreducible in one channel and reducible in another. Bethe–Salpeter and parquet methods organize this channel dependence. Their full equations lie beyond this preview.

Self-Energy

The proper self-energy $\Sigma$ is the sum of one-particle-irreducible two-point insertions in a declared expansion.

The Dyson equation is

$$G = G_0 + G_0\Sigma G.$$

In a translation-invariant scalar notation,

$$G^{-1}( \mathbf k, i\omega_n ) = G_0^{-1}( \mathbf k, i\omega_n ) -\Sigma( \mathbf k, i\omega_n ).$$

Iterating once gives

$$\begin{aligned} G ={}& G_0 + G_0\Sigma G_0 + G_0\Sigma G_0\Sigma G_0 + \cdots. \end{aligned}$$

Thus $\Sigma$ is not merely an energy shift. It is the irreducible kernel whose repeated insertion dresses propagation.

Hartree and Fock contributions

For a two-body fermion interaction, first-order self-energy terms commonly include:

• a Hartree direct term determined by an average density;
• a Fock exchange term produced by antisymmetry.

For an instantaneous interaction and a stationary normal reference, these terms can be frequency independent. Hartree–Fock Approximation derives these fields variationally.

Higher-order self-energies can depend on momentum and frequency. After a justified analytic continuation, their real and imaginary parts affect dispersion and damping. Green Functions in Many-Body QM develops the pole and Lehmann interpretation. Spectral Functions explains how the self-energy is converted into quasiparticle positions, residues, linewidths, and line shapes.

Reference subtraction

If $G_0$ already contains a Hartree–Fock potential, the perturbation must include a counterterm that subtracts it. Otherwise the first-order self-energy is counted once in $G_0$ and again as an insertion.

This is the diagrammatic form of the repartitioning rule:

$$H = \left( H_{\mathrm{free}}+U_{\mathrm{ref}} \right) + \left( H_{\mathrm{int}}-U_{\mathrm{ref}} \right).$$

Bubble Diagrams

A bubble is a closed loop made from two single-particle propagators joined by two operator or interaction vertices. In a density channel, it describes the independent propagation of a particle–hole pair between two density insertions.

The Matsubara sum gives

$$\begin{aligned} \chi_0( \mathbf q, i\nu_m ) &= g \int_{\mathbf k} \frac{ n_{\mathrm F}( \xi_{\mathbf k+\mathbf q} ) -n_{\mathrm F}( \xi_{\mathbf k} ) }{ i\nu_m -\xi_{\mathbf k+\mathbf q} +\xi_{\mathbf k} }. \end{aligned}$$

For coincident energies and $\nu_m=0$, take the divided-difference limit: the integrand is $-n_{\mathrm F}'(\xi)>0$. For exactly coincident energies and $\nu_m\ne0$, it is zero. In particular, the conserved total number has a static Matsubara fluctuation term at $m=0$ but no nonzero-frequency retarded commutator response. Such exactly zero-energy equilibrium terms must be separated before continuing the regular dynamic part; they cannot be recovered by blindly substituting into a zero denominator.

What one bubble contains

One bubble contains:

• Pauli occupation factors;
• the independent particle–hole continuum;
• momentum and energy transfer from the external source;
• spin or flavor multiplicity;
• the reference dispersion.

It does not by itself contain:

• screening to all orders;
• self-energy broadening;
• vertex corrections;
• multiple particle–hole pairs;
• a collective pole outside the independent continuum.

Bubble chains

For repulsive $v$, the induced Hartree potential opposes the physical chemical-potential source:

$$\delta n=\chi_0(f-v\delta n),\qquad \chi_{\mathrm{RPA}}=\frac{\chi_0}{1+v\chi_0}.$$

Equivalently, $\Pi_{\mathrm{RPA}}=\Pi_0/(1-v\Pi_0)$ with $\Pi_0=-\chi_0$. RPA derives this feedback, screening and collective poles. A bubble is the two-propagator building block; the self-consistent chain is the additional approximation.

Not every two-line loop is the same channel

Pairing diagrams contain two fermion lines but organize particle–particle propagation rather than particle–hole propagation. Nambu bubbles can include anomalous propagators. A visual loop is not enough to identify the channel; arrows, external labels, and momentum routing matter.

Vertex Corrections

A vertex correction changes the coupling between an external source or interaction and the propagating particles. It can encode:

• exchange between a particle and a hole;
• repeated scattering in one channel;
• screening or local-field effects;
• coupling to collective modes;
• constraints required by a conservation law.

Let $\Gamma^\mu$ be the full vertex for a conserved $U(1)$ current. A Ward identity has the structural form

$$q_\mu \Gamma^\mu(k+q,k) = G^{-1}(k+q) -G^{-1}(k),$$

with components, factors of charge, $i$, and metric conventions fixed by the real- or imaginary-time formulation.

This relation says that self-energy and vertex corrections are not independent decorations. Changing $G$ changes the vertex structure required by charge conservation.

Bare bubble with dressed lines

Replacing $G_0$ by a dressed $G$ inside a bubble while keeping the bare density vertex can violate a Ward identity. It may be a useful approximation in a specified regime, but it is not automatically conserving.

The required correction depends on how the self-energy was generated. There is no universal vertex patch that repairs every dressed propagator.

Bare, Dressed, and Skeleton Expansions

Bare expansion

A bare expansion uses:

• fixed $G_0$ lines;
• fixed bare interactions;
• all diagrams through a declared order.

Its order counting is transparent, but a low-order truncation may poorly describe shifted poles or collective screening.

Dressed-line expansion

One may solve a Dyson equation and use $G$ in internal lines. This resums infinitely many bare self-energy insertions. If the original insertion diagrams are then retained explicitly, they are double counted.

Skeleton expansion

A skeleton expansion uses dressed propagators but retains only diagrams without explicit self-energy insertion subgraphs. The self-energy is obtained self-consistently from those skeletons.

This can improve consistency, but:

• self-consistency does not guarantee accuracy;
• multiple solutions can occur;
• vertex corrections must still be matched;
• nominal bare perturbative order is no longer the only ordering variable.

Functional consistency preview

In a conserving construction, one builds a functional $\Phi[G]$ from selected closed skeleton diagrams and defines

$$\Sigma = \frac{\delta\Phi}{\delta G}$$

up to convention-dependent signs and factors. Further derivatives generate compatible vertex kernels.

This Baym–Kadanoff or Luttinger–Ward logic explains why a coherent family of diagrams can preserve conservation laws better than an arbitrary list of individually plausible graphs. Its existence, uniqueness, and practical validity are nontrivial and belong to advanced many-body field theory.

Goldstone, Matsubara, and Real-Time Diagrams

Goldstone diagrams

Stationary many-body perturbation theory often uses Goldstone diagrams:

• time flows in one direction;
• particle and hole lines are defined relative to a reference determinant;
• intermediate states carry explicit energy denominators;
• each time ordering can be displayed separately.

They are natural for ground-state energies and wavefunction amplitudes.

Matsubara diagrams

Equilibrium finite-temperature diagrams use imaginary time:

• fermionic frequencies are $(2n+1)\pi T$;
• bosonic frequencies are $2m\pi T$;
• internal frequencies are summed;
• time orderings are combined into propagators;
• analytic continuation may later connect to real-frequency functions.

One Matsubara or Feynman graph can combine several Goldstone time orderings. Their numerical factors and denominators should not be transferred by visual analogy.

Real-time time-ordered diagrams

Zero-temperature time-ordered propagators use real frequencies with causal prescriptions. Retarded, advanced, lesser, greater, and time-ordered functions are distinct. A time-ordered self-energy cannot be inserted blindly into a retarded Dyson equation without the correct continuation or contour relation.

Nonequilibrium contours

Driven or transient systems often require a forward-and-backward time contour. Every line and vertex then carries a contour index or Keldysh component. Real-Time Thermal Dynamics Preview explains the initial-state and contour motivation; full Schwinger–Keldysh rules require a dedicated nonequilibrium field-theory treatment.

Analytic Continuation

A Matsubara function is known initially at discrete imaginary frequencies. For a function with the required analyticity and spectral behavior, the retarded quantity is obtained by continuation

$$i\nu_m \longrightarrow \omega+i0^+.$$

Several cautions matter:

• perform internal Matsubara sums and derive the analytic function before substituting;
• preserve branch cuts and thresholds, not only poles;
• continue the complete conserving combination when cancellations matter;
• numerical continuation from noisy data is ill conditioned;
• a finite broadening $\eta$ is not automatically a physical lifetime.

Retarded and Advanced Green Functions explains causal boundary values in the one-particle setting. Green Functions in Many-Body QM develops the normal many-body retarded and Matsubara spectral relation, including the ill-conditioned numerical-continuation warning.

Short-distance and low-energy limits

Diagrams expose divergences; they do not remove them.

Ultraviolet

Continuum loops can diverge at large internal momentum or frequency. A calculation must state:

• regulator;
• bare parameters;
• matching or renormalization prescription;
• counterterms;
• which observable is cutoff independent.

Normal ordering can remove selected vacuum contractions but does not renormalize every local product.

Infrared

Gapless lines, Fermi surfaces, Goldstone modes, or long-range interactions can enhance low-energy regions. Look for:

$$\ln\!\left( \frac{\Lambda}{E} \right), \qquad E^{-\alpha}, \qquad \ln L, \qquad T^{-\gamma}.$$

If a family of diagrams has the same enhanced scaling at all orders, fixed-order truncation is not controlled. One must identify and resum the relevant channel.

External limits

For response functions,

$$\lim_{\omega\to0} \lim_{\mathbf q\to0}$$

can differ from

$$\lim_{\mathbf q\to0} \lim_{\omega\to0}.$$

The diagram formula does not choose the physical limit. The measurement or thermodynamic question does.

Exercises

1. Evaluate the Matsubara bubble sum. Starting from

   $$\chi_0(\mathbf q,i\nu_m) = -gT \sum_{\omega_n} \int_{\mathbf k} \frac{1}{ i\omega_n+i\nu_m-\xi_{\mathbf k+\mathbf q} } \frac{1}{ i\omega_n-\xi_{\mathbf k} },$$

   use the standard fermionic sum to obtain the occupation-number form.

Solution

Use

$$T \sum_{\omega_n} \frac{1}{ (i\omega_n-a) (i\omega_n+i\nu_m-b) } = \frac{ n_{\mathrm F}(a)-n_{\mathrm F}(b) }{ i\nu_m+a-b }.$$

Set

$$a=\xi_{\mathbf k}, \qquad b=\xi_{\mathbf k+\mathbf q}.$$

Including the closed-loop minus sign gives

$$\chi_0( \mathbf q, i\nu_m ) = g \int_{\mathbf k} \frac{ n_{\mathrm F}( \xi_{\mathbf k+\mathbf q} ) -n_{\mathrm F}( \xi_{\mathbf k} ) }{ i\nu_m -\xi_{\mathbf k+\mathbf q} +\xi_{\mathbf k} }.$$

This is the susceptibility for $-fn$. The potential-energy polarization is its negative. When $a=b$ and $\nu_m=0$, the sum is $n_{\mathrm F}^\prime(a)$, so the loop minus sign gives $-n_{\mathrm F}^\prime(a)>0$.

2. Classify topology. A two-point graph consists of an external propagator, then a proper self-energy block, another propagator, a second proper self-energy block, and a final external propagator. Is the whole graph connected? Is it one-particle irreducible? Should it be included in $\Sigma$?

Solution

The graph is connected because every piece belongs to one component between the two external insertions.

It is one-particle reducible: cutting the propagator between the two self-energy blocks disconnects the graph. Therefore the full chain does not belong to the proper self-energy. It is generated automatically by iterating Dyson’s equation with each proper block included in $\Sigma$.

Putting the whole chain into $\Sigma$ and also solving Dyson’s equation would double count it.

3. Bare versus dressed double counting. Suppose

   $$G = G_0+G_0\Sigma^{(1)}G_0+O(\lambda^2).$$

   A calculation uses $G$ on every internal line and also adds every explicit first-order self-energy insertion graph. Show why an $O(\lambda)$ term is counted twice.

Solution

Replacing an internal bare line by $G$ already gives

$$G = G_0 + G_0\Sigma^{(1)}G_0 + O(\lambda^2).$$

Thus the dressed-line diagram contains the bare topology plus all versions in which one internal line carries the first-order insertion. Adding those insertion graphs explicitly contributes the same

$$G_0\Sigma^{(1)}G_0$$

terms again. A skeleton scheme avoids this by using dressed lines and omitting explicit self-energy insertion subgraphs.

4. Ward-identity limit. Use the structural Ward identity

   $$q_\mu\Gamma^\mu(k+q,k) = G^{-1}(k+q)-G^{-1}(k)$$

   to explain qualitatively why a frequency-dependent self-energy generally requires a corrected density vertex in the small-frequency limit.

Solution

After the appropriate analytic continuation, take the dynamic small-frequency limit at zero spatial momentum. This is distinct from the strictly static thermal derivative. Then

$$G^{-1}(i\omega+i\nu)-G^{-1}(i\omega) \simeq i\nu \frac{\partial G^{-1}}{ \partial(i\omega) }.$$

More explicitly,

$$\frac{ G^{-1}(i\omega+i\nu)-G^{-1}(i\omega) }{ i\nu } \longrightarrow \frac{\partial G^{-1}}{ \partial(i\omega) }.$$

Since

$$G^{-1} = G_0^{-1}-\Sigma,$$

the derivative contains

$$-\frac{\partial\Sigma}{ \partial(i\omega) }.$$

The density vertex must contain the corresponding correction. Using a frequency-dependent dressed $G$ with an unchanged bare vertex generally fails this relation.

References

• A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover (1975).
• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press (2010).
• G. Baym and L. P. Kadanoff, “Conservation Laws and Correlation Functions”, Physical Review 124, 287–299 (1961).
• G. Baym, “Self-Consistent Approximations in Many-Body Systems”, Physical Review 127, 1391–1401 (1962).
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015).
• F. J. Dyson, “The Radiation Theories of Tomonaga, Schwinger, and Feynman”, Physical Review 75, 486–502 (1949).
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
• R. P. Feynman, “Space-Time Approach to Quantum Electrodynamics”, Physical Review 76, 769–789 (1949).
• J. Goldstone, “Derivation of the Brueckner Many-Body Theory”, Proceedings of the Royal Society A 239, 267–279 (1957).
• L. Hedin, “New Method for Calculating the One-Particle Green’s Function with Application to the Electron-Gas Problem”, Physical Review 139, A796–A823 (1965).
• J. M. Luttinger and J. C. Ward, “Ground-State Energy of a Many-Fermion System. II”, Physical Review 118, 1417–1427 (1960).
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000).
• T. Matsubara, “A New Approach to Quantum-Statistical Mechanics”, Progress of Theoretical Physics 14, 351–378 (1955).
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview Press (1998).
• G. C. Wick, “The Evaluation of the Collision Matrix”, Physical Review 80, 268–272 (1950).