Green Functions

Green functions are response kernels, resolvents, or correlation functions depending on context. The bridge to field theory is powerful because the same words appear in several related but distinct roles.

Quantum Mechanics Starting Point

The time-evolution kernel is

K(x,t;x',0) = \langle x\vert e^{-iHt/\hbar}\vert x'\rangle.

It propagates wavefunctions:

\psi(x,t) = \int dx'\, K(x,t;x',0)\psi(x',0).

The energy-domain resolvent is

G(E) = \frac{1}{E-H+i0},

with position-space kernel

G(x,x';E) = \langle x\vert G(E)\vert x'\rangle.

For the specific way $+i0$ becomes outgoing spatial radiation and then an on-shell amplitude, see Green Function for Scattering. The distinction from time ordering is central to the bridge below.

Field-Theory Continuation

In field theory, a central object is the time-ordered two-point function

\Delta_F(x-y) = \langle 0\vert T\{\phi(x)\phi(y)\} \vert 0\rangle.

For a free scalar field, this object is the Feynman propagator. With sources, functional derivatives of a generating functional produce correlation functions. The word “propagator” therefore shifts from a one-particle kernel to a vacuum correlation function of fields.

Dictionary

Quantum Mechanics	Field Theory
time-evolution kernel	propagator or transition amplitude
resolvent $(E-H+i0)^{-1}$	momentum-space propagator denominator
retarded response	retarded Green function
spectral representation	Källén–Lehmann and spectral densities
source response	generating functional derivatives
poles of resolvent	particle masses, bound states, or resonances depending on context

Cautions

The same symbol $G$ can mean:

a resolvent,
a retarded Green function,
a Feynman propagator,
an imaginary-time correlator,
a many-body single-particle Green function,
a full interacting correlation function.

Always state the time ordering, boundary condition, and normalization. Retarded, advanced, Feynman, Euclidean, and Wightman functions are not interchangeable.

Common Mistakes

Calling every correlation function a propagator without specifying ordering.
Forgetting the $i0$ prescription in scattering and resolvent formulas.
Confusing a one-particle wavefunction kernel with a field correlation function.
Treating Euclidean and real-time Green functions as automatically equivalent.
Comparing many-body and high-energy sign conventions without translation.

Canonical Links

References

R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Westview Press, 1995.
M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.

Exercises

Why is the $i0$ prescription part of the definition rather than decoration?

Solution

The resolvent has singularities on the spectrum of $H$ . The infinitesimal imaginary part specifies how the singularity is approached and therefore which boundary condition or time ordering is selected. In scattering, this choice distinguishes incoming and outgoing wave boundary conditions.