Green Functions from QM to QFT

The phrase Green function survives the passage from quantum mechanics to quantum field theory, but the object carrying that name changes with the question. In nonrelativistic one-particle quantum mechanics it may be an evolution kernel, a resolvent kernel, or a retarded response kernel. In many-body theory it is commonly a state-dependent correlation function of creation, annihilation, density, or spin operators. In relativistic QFT it is usually a correlation function of local fields.

The continuity is structural:

• a linear operator is inverted with a boundary prescription;
• poles and discontinuities encode spectral information;
• sources generate responses or insertions;
• ordering specifies which physical question is being asked.

The discontinuity is equally important: a QFT propagator is not generally a one-particle wavefunction kernel. Keeping the defining operator and state explicit makes it possible to compare conventions without confusing different experiments.

Green Functions in Nonrelativistic QM

Three objects establish the vocabulary.

Evolution kernel

The position-space kernel of unitary time evolution is

$$K(x_f,t_f;x_i,t_i) = \langle x_f\rvert U(t_f,t_i) \lvert x_i\rangle.$$

It evolves an initial wavefunction and composes through intermediate positions. Its definition and evolution equation are developed in Propagator Kernel.

Energy resolvent kernel

The coordinate kernel of a resolvent boundary value is

$$G^{\pm}(x,x';E) = \left\langle x\left\rvert \frac{1}{E-H\pm i0} \right\lvert x'\right\rangle.$$

It solves an energy-domain source equation and selects incoming or outgoing spatial behavior. Energy Green Function derives its differential equation and free examples.

Retarded response kernel

For a perturbation $H_{\mathrm{pert}}=-f(t)B$, a susceptibility has the form

$$\chi_{AB}^R(t,t') = \frac{i}{\hbar} \theta(t-t') \langle[A_H(t),B_H(t')]\rangle.$$

It maps a source coupled to $B$ into the first-order change of an observable $A$. The source convention and derivation are fixed in Green Functions and Response Preview.

These objects are related by spectral and Fourier transforms, but they are not interchangeable definitions.

QM object	Variables	Boundary or ordering data	Primary use
$K$	two times and two coordinates	initial-to-final evolution	propagate a state
$G^{\pm}(E)$	energy and coordinates	incoming or outgoing $i0$	stationary sources and scattering
$\chi^R$	two times or one frequency	retarded support	causal linear response

Correlation Functions in Many-Body QM

Many-body theory promotes operator correlations to central observables. For a state or ensemble $\rho$, a generic two-point function is

$$C_{AB}(t,t') = \operatorname{Tr} \left[ \rho A_H(t)B_H(t') \right].$$

The state is part of the definition. Ground-state, thermal, driven, and nonequilibrium correlators can differ even when the Hamiltonian and operators are unchanged.

Operator ordering is also part of the definition:

Correlator	Typical role
ordinary ordered $\langle A(t)B(t')\rangle$	transition weights and noise spectra
commutator with retarded support	response and dissipation
anticommutator or symmetrized product	fluctuations and detector noise
time-ordered product	perturbation theory and diagrammatic expansions
imaginary-time ordered product	thermal equilibrium and Matsubara methods

Diagrammatic Methods Preview fixes one explicit Matsubara convention and develops the associated line, vertex, self-energy, and bubble rules.

For nonrelativistic field operators, a one-particle Green function may be written schematically as

$$G(1,2) = -i \langle \mathcal T \psi(1)\psi^{\dagger}(2) \rangle,$$

with an additional minus sign under exchange for fermionic time ordering. Factors of $i$, $\hbar$, and the sign convention vary by subject. Green Functions in Many-Body QM sets out addition/removal, retarded, time-ordered, spectral, and Matsubara conventions; the compact lookup is Correlation Functions.

The spectral interpretation broadens as well. Isolated poles can identify sharp excitations, while continua and branch cuts encode multiparticle or incoherent spectral weight. Residues become overlap strengths, not merely normalized one-particle wavefunctions. Spectral Representation of Green Functions supplies the QM starting point for that language.

Propagators in Relativistic QFT

For a real scalar field, a standard QFT two-point function is the vacuum time-ordered correlator

$$D_F(x-y) = \langle0\rvert \mathcal T\left\{ \phi(x)\phi(y) \right\} \lvert0\rangle.$$

For a free scalar field, set $\hbar=c=1$, use $p^2=(p^0)^2-|\mathbf p|^2$, and write $D_F(x)=\int d^4p\,e^{-ip\cdot x}D_F(p)/(2\pi)^4$. Then

$$D_F(p) = \frac{i}{p^2-m^2+i0}.$$

Other conventions absorb the overall $i$ into the definition. The invariant content is the pole prescription and its relation to time ordering.

This object is a vacuum correlation function of field operators. It is not generally the amplitude for a permanently identifiable particle to travel from $y$ to $x$. In QFT:

• particle number need not be fixed;
• fields create and annihilate excitations;
• antiparticles and multiparticle states enter the spectrum;
• interacting two-point functions acquire shifted poles, residues, thresholds, and cuts;
• local fields are operator-valued distributions rather than ordinary functions.

The direct propagator comparison, including relativistic locality and pole interpretation, is developed in From Propagators in QM to Propagators in QFT.

Time Ordering and Feynman Propagators

For bosonic fields,

$$\mathcal T\{\phi(x)\phi(y)\} = \theta(x^0-y^0)\phi(x)\phi(y) + \theta(y^0-x^0)\phi(y)\phi(x).$$

Fermionic time ordering includes a minus sign when two fermionic operators are exchanged. Time ordering is therefore an algebraic prescription, not merely a relabeling of coordinates.

The Feynman $i0$ prescription tells a momentum integral how positive- and negative-frequency poles are bypassed. It implements time ordering and the in-out vacuum boundary condition used in standard perturbative amplitudes. It is not the same object as a retarded propagator, which is supported only after the source time.

This distinction matters for causality. A Feynman propagator need not vanish at spacelike separation. Relativistic microcausality makes physical local observables commute at spacelike separation. Fermionic fields, which are not themselves bosonic observables, obey the corresponding graded anticommutation rule. Causality, Support, and Interpretation in Nonrelativistic QM gives the careful comparison.

Sources and Generating Functionals

Sources unify response and correlation language. In a scalar field theory, introduce

$$Z[J] = \int\mathcal D\phi\, \exp\left[ iS[\phi] + i\int d^4x\,J(x)\phi(x) \right]$$

in units with $\hbar=1$. Functional derivatives insert fields:

$$\left. \frac{1}{i^n} \frac{\delta^n Z[J]} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0}$$

is proportional to an $n$-point correlation function, with normalization and boundary prescription determined by the definition of $Z[J]$. Dividing by $Z[0]$ or differentiating a connected generator changes which vacuum contributions are included.

The formal resemblance to a driven quantum-mechanical system is real, but the generated object depends on context. A fixed-endpoint QM kernel, a vacuum in-out functional, a closed-time-path functional, a Euclidean thermal functional, and a real-time response functional do not generate the same ordering.

The source calculus and normalization details are developed in From Sources in QM to Generating Functionals in QFT. The configuration-space replacement from paths $q(t)$ to fields $\phi(x)$ is explained in Path Integrals from QM to QFT.

What Changes When Fields Replace Particles

The following dictionary marks both the continuity and the new structure.

Quantum mechanics	Quantum field theory	Essential change
coordinate $q(t)$	field $\phi(x)$	one history becomes a spacetime configuration
wavefunction $\psi(q,t)$	state in a field-theory Hilbert space	the field is not generally a many-particle wavefunction
finitely many canonical pairs	local field operators and conjugate momenta	infinitely many degrees of freedom before regulation
position-space evolution kernel	field correlation function	propagation language becomes operator and state dependent
discrete levels and scattering continuum	particle poles, multiparticle thresholds, and cuts	spectral structure reflects variable particle number
Galilean kinematics	Poincaré covariance	space and time enter relativistic spacetime symmetry
ordinary products of operators	products of operator-valued distributions	coincident points require ultraviolet control
parameter fitting in a Hamiltonian	regularization and renormalization	short-distance definitions affect parameter relations

Free bosonic fields resemble collections of harmonic oscillators labeled by momentum, but this analogy is a starting point rather than a definition of interacting QFT. Locality, Lorentz symmetry, spin-statistics structure, gauge redundancy, vacuum structure, and renormalization introduce genuinely new constraints.

Common Mistakes

• Calling every object named a propagator the same mathematical kernel.
• Treating a QFT field $\phi(x)$ as a single-particle wavefunction.
• Interpreting the Feynman propagator as a causal signal-response kernel.
• Expecting the Feynman propagator itself to vanish at spacelike separation.
• Omitting the state or ensemble from a many-body correlation-function definition.
• Comparing formulas without checking factors of $i$, $\hbar$, metric signature, and Fourier convention.
• Assuming every spectral singularity is a stable one-particle pole.
• Forgetting the fermionic sign in time ordering.
• Differentiating a generating functional without specifying its normalization and contour.
• Treating ultraviolet regularization and renormalization as optional notation changes.

References

• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Kluwer Academic/Plenum, 2000.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.
• M. Srednicki, Quantum Field Theory, Cambridge University Press, 2007.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.

Exercises

1. Classify each object as an evolution kernel, energy resolvent, response function, or field correlator: $\langle x_f\rvert U\lvert x_i\rangle$, $\langle x\rvert(E-H+i0)^{-1}\lvert x'\rangle$, $\theta(t)\langle[A(t),B(0)]\rangle$, and $\langle0\rvert\mathcal T\phi(x)\phi(y)\lvert0\rangle$.

Solution

The classifications are, respectively:

1. a quantum-mechanical time-evolution kernel;
2. an outgoing energy-resolvent kernel;
3. a retarded response correlator, up to its convention-dependent prefactor;
4. a QFT time-ordered field correlator, commonly called the Feynman propagator after fixing normalization.

Their names overlap in the literature, so the defining formula is more reliable than the word propagator.

2. Why does time ordering not make a Feynman propagator identical to a retarded propagator?

Solution

Time ordering includes both temporal orderings:

$$\mathcal T\{\phi(x)\phi(y)\} = \begin{cases} \phi(x)\phi(y), & x^0\gt y^0,\\ \phi(y)\phi(x), & y^0\gt x^0. \end{cases}$$

A retarded propagator instead contains a step function multiplying a commutator and vanishes when the response time precedes the source time. The two functions can be related through spectral data, but they implement different boundary and ordering prescriptions.

3. Explain why a relativistic field operator is not generally a single-particle wavefunction.

Solution

A wavefunction is a state amplitude in a chosen configuration representation. A quantum field is an operator-valued distribution acting on a Hilbert space that can contain vacuum, one-particle, and multiparticle sectors. The field can create or annihilate excitations, while the state is a separate object. Matrix elements of the field can sometimes behave like wavefunctions in restricted sectors, but that does not identify the operator with the state.

4. Show schematically how two source derivatives generate a two-point function.

Solution

The source-dependent exponential contains

$$\exp\left[ i\int d^4x\,J(x)\phi(x) \right].$$

One functional derivative brings down $i\phi(x)$, and a second brings down $i\phi(y)$. Therefore

$$\left. \frac{1}{i^2} \frac{\delta^2Z[J]} {\delta J(x)\delta J(y)} \right|_{J=0}$$

generates a two-field insertion. Whether the result is time ordered, Euclidean ordered, contour ordered, or another correlator depends on the definition and contour of $Z[J]$. Normalization by $Z[0]$ may also be required.

5. What spectral features replace the simple isolated poles of a finite-dimensional resolvent in an interacting field theory?

Solution

Stable one-particle excitations can still produce isolated poles. Interacting theories also contain multiparticle thresholds and continua, represented by branch cuts or continuous spectral weight. Unstable resonances correspond to poles reached by analytic continuation rather than normalizable physical-sheet states. Pole residues encode overlap strengths, and ultraviolet behavior requires renormalized definitions. Thus the spectrum is richer than a finite sum over discrete denominators.