From Propagators in QM to Propagators in QFT

The word “propagator” means different but related things in quantum mechanics and quantum field theory. In nonrelativistic quantum mechanics, the basic propagator is a kernel of the time-evolution operator:

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

In QFT, the most common propagator is a time-ordered two-point function of fields:

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

Both objects describe how disturbances, amplitudes, or correlations connect spacetime points, but they are not the same kind of mathematical object. This page is a translation guide.

QM Propagator as a Particle Amplitude

For a single nonrelativistic particle, the coordinate-space propagator kernel is

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

It evolves wavefunctions:

$$\psi(x_f,t_f) = \int dx_i\, K(x_f,t_f;x_i,t_i)\psi(x_i,t_i).$$

For a time-independent Hamiltonian with energy eigenfunctions $\psi_n(x)$,

$$K(x_f,t;x_i,0) = \sum_n \psi_n(x_f)\psi_n^*(x_i) e^{-iE_nt/\hbar},$$

with continuum contributions added when the spectrum is not purely discrete.

This object is an amplitude kernel. It is not itself a probability:

$$|K(x_f,t_f;x_i,t_i)|^2$$

is not generally a normalized transition probability density unless a specific preparation and measurement protocol has been specified.

The canonical quantum-mechanics page is Propagator Kernel.

Green Functions in Quantum Mechanics

Quantum mechanics also uses Green functions as inverse kernels. For a time-independent Hamiltonian,

$$G(z) = (z-H)^{-1}$$

is the resolvent. Its position-space kernel is

$$G(x,x';z) = \langle x|(z-H)^{-1}|x'\rangle.$$

Boundary values such as

$$G^\pm(E) = \lim_{\epsilon\to0^+} \frac{1}{E-H\pm i\epsilon}$$

select incoming or outgoing scattering prescriptions. Time-domain retarded, advanced, and time-ordered Green functions answer different response or ordering questions.

Thus even inside ordinary quantum mechanics, “propagator” can mean at least two related objects:

Object	Typical formula	Main role
Time-evolution kernel	$\langle x_f	U(t_f,t_i)
Energy Green function	$\langle x	(E-H\pm i0)^{-1}

The distinction is developed in What Is a Green Function?.

QFT Propagator as a Two-Point Function

In relativistic scalar QFT, a standard Feynman propagator is the vacuum time-ordered two-point function

$$\Delta_F(x-y) = \langle0| \mathcal T\{\phi(x)\phi(y)\} |0\rangle.$$

For a free real scalar field, a common convention writes

$$\Delta_F(x-y) = \int\frac{d^4p}{(2\pi)^4} \frac{i\,e^{-ip\cdot(x-y)}} {p^2-m^2+i0},$$

using the mostly-minus convention

$$p^2=(p^0)^2-\mathbf p^2.$$

Some texts absorb factors of $i$ into the definition of $\Delta_F$ or define $i\Delta_F$ as the propagator. The convention must be checked before comparing Feynman rules.

The conceptual shift is the important part: the QFT propagator is not the matrix element of a one-particle position-basis time-evolution operator. It is a field correlation function in a specified state, usually the vacuum. It becomes a particle-propagation object only after additional interpretation, such as free-field mode expansion or scattering theory with asymptotic particle states.

Time Ordering and the Feynman Prescription

The time-ordering symbol means

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

for bosonic fields, ignoring equal-time distribution subtleties. The Feynman $i0$ prescription in momentum space selects how the poles are bypassed:

$$\frac{1}{p^2-m^2+i0}.$$

This prescription is the analytic imprint of time ordering and vacuum boundary conditions. It is not interchangeable with the retarded prescription. A retarded propagator instead vanishes outside the future response region:

$$G_R(x-y)=0 \qquad \text{for }x^0\lt y^0.$$

Feynman propagators are central in perturbative scattering amplitudes. Retarded propagators are central in causal response. Euclidean propagators are central in imaginary-time and statistical formulations. They are related by analytic continuation under appropriate assumptions, but they answer different questions.

From Euclidean Time to Euclidean QFT develops the thermal-circle and Matsubara structure of the Euclidean case.

Relativistic Locality

Relativistic QFT imposes locality conditions that have no direct analogue in single-particle nonrelativistic quantum mechanics. Causality, Support, and Interpretation in Nonrelativistic QM compares the Schrödinger kernel, commutator distribution, retarded response, Feynman propagator, and Wightman function in detail. For a bosonic scalar field, microcausality states

$$[\phi(x),\phi(y)]=0 \qquad \text{when }(x-y)^2\lt0.$$

This says spacelike separated field measurements are compatible. It does not mean the Feynman propagator itself vanishes at spacelike separation. The Feynman propagator is a time-ordered correlation function, not the commutator that directly measures causal influence.

The retarded commutator-type response is the object tied most directly to causal propagation:

$$G_R(x-y) = \theta(x^0-y^0) \langle0|[\phi(x),\phi(y)]|0\rangle$$

up to conventional factors. The commutator vanishes at spacelike separation in a local relativistic theory.

Particle Interpretation Caveats

In a free scalar theory, the propagator has a pole at

$$p^2=m^2,$$

which matches the relativistic mass shell for a particle of mass $m$. In interacting theories, the full two-point function has a richer spectral structure. A schematic Källén–Lehmann representation is

$$\Delta_F(p) = \int_0^\infty d\mu^2\, \frac{i\rho(\mu^2)} {p^2-\mu^2+i0}.$$

An isolated pole can represent a stable particle; branch cuts represent multiparticle continua; resonances appear through analytic continuation. A field need not create a single particle cleanly, and a particle need not correspond to a unique elementary field.

Therefore the safest statement is:

$$\text{QFT propagators encode field correlations and spectral data.}$$

Particle language is often correct and extremely useful in scattering regimes, but it is not the definition.

Translation Table

Quantum-mechanical object	QFT analogue	Warning
$K(x_f,t_f;x_i,t_i)$	transition kernels or path-integral amplitudes	QFT usually uses field configurations, not one particle coordinate
$(E-H\pm i0)^{-1}$	momentum-space denominators and spectral Green functions	$i0$ prescriptions encode boundary conditions
time-ordered correlator	Feynman propagator	convention-dependent factors of $i$
retarded response	retarded field Green function	causal response differs from Feynman ordering
spectral poles	particle masses or bound-state energies	interacting spectra also have cuts and resonances
source response	generating functional derivatives	fields, sources, and normalization conventions must be fixed

Common Mistakes

• Treating a QFT Feynman propagator as a literal probability amplitude for a classical particle path.
• Forgetting that $K$, $G_R$, $G_F$, Euclidean correlators, and resolvents are different objects.
• Dropping the $i0$ prescription as if it were optional notation.
• Assuming the Feynman propagator must vanish outside the light cone.
• Comparing QFT and many-body propagator conventions without checking factors of $i$, signs, and Fourier transforms.
• Ignoring the state: vacuum, thermal, in-in, and scattering correlators are not the same.

Cross-Links

• Why Dynamics Matters for QFT gives the chapter-level bridge.
• From Evolution Operators to Time-Ordered Products explains why the QFT propagator appears inside ordered field products.
• From Euclidean Time to Euclidean QFT explains Euclidean and thermal field correlators.
• From Correlation Functions to QFT Observables explains how two-point and higher-point functions encode masses, response, and scattering.
• Propagator Kernel defines the QM time-evolution kernel.
• Causality, Support, and Interpretation in Nonrelativistic QM separates instantaneous Schrödinger tails from relativistic causal response.
• What Is a Green Function? separates inverse kernels, response functions, and boundary prescriptions.
• Green Functions from QM to QFT gives the chapter-closing translation map.
• Correlation Functions in Path Integrals explains time-ordered and response correlators in QM path integrals.
• Green Functions gives a reference-library bridge entry.
• Path Integrals connects propagators to field functional integrals.

References

• 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.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.
• A. Zee, Quantum Field Theory in a Nutshell, 2nd ed., Princeton University Press, 2010.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

Exercises

1. Show how the quantum-mechanical propagator kernel evolves a wavefunction.

Solution

Start from

$$\lvert\psi(t_f)\rangle = U(t_f,t_i)\lvert\psi(t_i)\rangle.$$

Insert a position resolution of identity at the initial time:

$$I=\int dx_i\,|x_i\rangle\langle x_i|.$$

Taking the final position amplitude gives

$$\psi(x_f,t_f) = \int dx_i\, \langle x_f|U(t_f,t_i)|x_i\rangle \psi(x_i,t_i).$$

Thus

$$K(x_f,t_f;x_i,t_i) = \langle x_f|U(t_f,t_i)|x_i\rangle$$

is the evolution kernel.

2. Why is a QFT Feynman propagator not simply a one-particle wavefunction kernel?

Solution

The Feynman propagator is a vacuum expectation value of time-ordered field operators:

$$\Delta_F(x-y) = \langle0|\mathcal T\{\phi(x)\phi(y)\}|0\rangle.$$

It is a correlation function of fields in a specified state. A one-particle kernel instead evolves a wavefunction between position eigenstates. In free or scattering regimes the field correlator has a particle interpretation, but that interpretation is additional structure, not the definition.

3. Which object is more directly tied to causal response: the Feynman propagator or the retarded propagator?

Solution

The retarded propagator is more directly tied to causal response because it vanishes before the source acts:

$$G_R(x-y)=0 \qquad \text{for }x^0\lt y^0.$$

The Feynman propagator is time ordered and is used in perturbative amplitudes. It is not the same as the causal response function.

4. What does the $i0$ prescription do in a propagator denominator?

Solution

The $i0$ prescription specifies how singularities are approached in the complex energy or momentum plane. In

$$\frac{1}{p^2-m^2+i0},$$

it tells how the poles are displaced and encodes the Feynman time-ordering boundary condition. Different prescriptions give different Green functions, such as retarded or advanced ones.

5. What spectral information is suggested by a pole at $p^2=m^2$?

Solution

For a free scalar field, a pole at

$$p^2=m^2$$

matches the relativistic mass shell of a particle with mass $m$. In an interacting theory, an isolated pole can indicate a stable particle state, while cuts and more complicated analytic structures represent multiparticle continua or resonances.