Correlation Functions in Path Integrals

A correlation function asks how quantum quantities at one time, position, or spacetime point are statistically and dynamically related to quantities at another. In path-integral language, correlation functions are represented by inserting variables into the sum over histories:

$$\int \mathcal D x\, x(t_1)x(t_2)e^{iS[x]/\hbar}.$$

That expression is only meaningful after specifying the state, boundary conditions, ordering prescription, and normalization. The word “correlation” covers several different objects: equal-time correlations, time-ordered correlators, connected correlators, retarded response functions, Euclidean correlators, and many-body Green functions. They are related, but they are not interchangeable.

This page uses one quantum-mechanical coordinate $x(t)$ as the model case. The same distinctions become essential in QFT, where correlation functions are often the primary observables.

What a Correlation Function Measures

For two operators $A$ and $B$ in a state $\rho$, an ordinary equal-time correlation is

$$\langle AB\rangle_\rho = \operatorname{Tr}(\rho AB).$$

The connected part subtracts the product of one-point functions:

$$C_{AB} = \langle AB\rangle_\rho - \langle A\rangle_\rho\langle B\rangle_\rho.$$

Connected correlations measure failure of factorization. If $A$ and $B$ are local observables on two subsystems, a nonzero connected correlation may come from entanglement, classical mixture, thermal correlations, or dynamical constraints; it is not by itself a complete entanglement witness.

For time-dependent operators, one must specify the time arguments and the ordering:

$$C_{AB}(t,t') = \langle A_H(t)B_H(t')\rangle_\rho.$$

If $A_H(t)$ and $B_H(t')$ do not commute, then

$$\langle A_H(t)B_H(t')\rangle_\rho \neq \langle B_H(t')A_H(t)\rangle_\rho$$

in general. The order is part of the definition.

Operator Formulation

In the Heisenberg picture,

$$A_H(t)=U^\dagger(t,t_0)A_SU(t,t_0),$$

for a time-independent or suitably defined reference evolution. Common two-time objects include the greater and lesser correlators

$$G^{\gt}_{AB}(t,t') = \langle A_H(t)B_H(t')\rangle, \qquad G^{\lt}_{AB}(t,t') = \langle B_H(t')A_H(t)\rangle.$$

The time-ordered correlator is

$$G^T_{AB}(t,t') = \langle \mathcal T\{A_H(t)B_H(t')\} \rangle.$$

For two bosonic operators this means

$$G^T_{AB}(t,t') = \theta(t-t')\langle A_H(t)B_H(t')\rangle + \theta(t'-t)\langle B_H(t')A_H(t)\rangle.$$

The retarded response function for a perturbation $H_{\rm pert}(t)=-f(t)B_H(t)$ is commonly written

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

It answers a different question from the time-ordered correlator: it gives the linear response of $\langle A(t)\rangle$ to the applied force $f(t')$. Many-body texts often absorb signs and factors of $i$ into definitions of Green functions, so the convention must be checked before comparing formulas.

Path-Integral Formulation

For a fixed state or boundary prescription, a path integral with coordinate insertions has the schematic form

$$\langle \mathcal T x(t_1)\cdots x(t_n)\rangle = \frac{1}{Z[0]} \int \mathcal D x\, x(t_1)\cdots x(t_n) e^{iS[x]/\hbar}.$$

The time-ordering symbol on the left is not decoration. In real-time path integrals derived from transition amplitudes, the time-sliced construction follows the order of operator evolution. Coordinate insertions in the path integral correspond to time-ordered coordinate operators in the operator formulation, provided the source, contour, and boundary prescription match.

The source method is the cleanest way to remember the normalization and factors of $i$. With

$$Z[J] = \int\mathcal D x\, \exp\left[ \frac{i}{\hbar} \left( S[x]+\int dt\,J(t)x(t) \right) \right],$$

one obtains

$$\langle \mathcal T x(t_1)\cdots x(t_n)\rangle = \frac{1}{Z[0]} \left( \frac{\hbar}{i} \right)^n \frac{\delta^n Z[J]} {\delta J(t_1)\cdots\delta J(t_n)} \bigg\rvert_{J=0}.$$

The page Sources and Generating Functionals in QM is the canonical home for the source-derivative derivation. Here the main point is interpretive: the path-integral insertion formula is a correlation-function formula only after the same object has been specified on both sides.

Fixed Endpoints Versus Vacuum Correlators

A fixed-endpoint kernel computes a transition amplitude:

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

With insertions, it gives

$$\langle x_f\rvert \mathcal T\{ x_H(t_1)\cdots x_H(t_n) \} U(t_f,t_i) \lvert x_i\rangle,$$

up to the chosen Heisenberg-reference convention. This is not the same as a vacuum expectation value.

A vacuum correlator is instead

$$\langle 0\rvert \mathcal T\{ x_H(t_1)\cdots x_H(t_n) \} \lvert 0\rangle.$$

In a path-integral construction, it is usually obtained by an infinite-time projection, an $i\epsilon$ prescription, or Euclidean preparation of the ground state. The distinction matters: a fixed-endpoint path integral can depend strongly on $x_i$ and $x_f$, while a vacuum correlator is defined by the ground state and the operator dynamics.

Thermal correlators use another prescription:

$$\langle A\rangle_\beta = \frac{1}{Z_\beta} \operatorname{Tr}\left(e^{-\beta H}A\right).$$

Their Euclidean path integrals have periodic or antiperiodic imaginary-time boundary conditions depending on the variables involved. That thermal structure should not be inferred from a zero-temperature transition kernel.

Connected Correlators

The full two-point function contains disconnected pieces whenever the one-point function is nonzero:

$$\langle \mathcal T x(t_1)x(t_2)\rangle = \langle \mathcal T x(t_1)x(t_2)\rangle_c + \langle x(t_1)\rangle\langle x(t_2)\rangle.$$

For higher $n$, the full correlator decomposes into sums of connected clusters. The logarithm of the generating functional isolates those connected pieces:

$$\mathcal W[J] = \frac{\hbar}{i}\log Z[J].$$

With the conventions of the source page,

$$\langle \mathcal T x(t_1)x(t_2)\rangle_c = \frac{\hbar}{i} \frac{\delta^2\mathcal W[J]} {\delta J(t_1)\delta J(t_2)} \bigg\rvert_{J=0}.$$

Connected correlators are important because they isolate nonfactorizing statistical dependence and, in QFT, correspond to connected Feynman diagrams before further amputation or Legendre transformation. The general cumulant and clustering analysis belongs to Connected Correlation Functions.

Time Ordering and Contact Terms

For coordinate-only correlators, time ordering is often straightforward. With momenta, velocities, or composite operators, equal-time limits can carry contact terms. For example, differentiating a time-ordered two-point function can differentiate the step functions hidden inside $\mathcal T$:

$$\frac{d}{dt} \left[ \theta(t-t')A(t)B(t') + \theta(t'-t)B(t')A(t) \right]$$

contains a term proportional to $\delta(t-t')$ when $A$ and $B$ have a nonzero commutator. In field theory, analogous contact terms appear in Ward identities and operator-product relations.

This is one reason path-integral manipulations with derivatives of fields or momenta require more care than insertions of $x(t)$ alone. The time-sliced definition, midpoint choices, and operator ordering can all matter.

Harmonic Oscillator Example

For the harmonic oscillator ground state,

$$x_H(t) = \sqrt{\frac{\hbar}{2m\omega}} \left( a e^{-i\omega t} +a^\dagger e^{i\omega t} \right).$$

The ordinary ordered correlator for $t\gt t'$ is

$$\langle 0\rvert x_H(t)x_H(t')\lvert 0\rangle = \frac{\hbar}{2m\omega} e^{-i\omega(t-t')}.$$

The time-ordered, or Feynman, two-point function is therefore

$$G_F(t,t') = \langle 0\rvert \mathcal T\{ x_H(t)x_H(t') \} \lvert 0\rangle = \frac{\hbar}{2m\omega} e^{-i\omega |t-t'|}.$$

The equal-time value is the ground-state variance:

$$G_F(t,t) = \langle 0\rvert x^2\lvert 0\rangle = \frac{\hbar}{2m\omega}.$$

The linear-response function for a force coupled as $H_{\rm pert}=-f(t)x$ is

$$\chi_{xx}^{R}(t,t') = \frac{i}{\hbar}\theta(t-t') \langle 0\rvert [x_H(t),x_H(t')]\lvert 0\rangle = \theta(t-t') \frac{\sin\omega(t-t')}{m\omega}.$$

Notice the difference. $G_F$ encodes vacuum fluctuations and time ordering; $\chi_{xx}^{R}$ encodes causal response. They are related through spectral information, but they are not the same function.

Because the oscillator is Gaussian, all higher ground-state time-ordered correlators are sums of products of $G_F$. For example,

$$\begin{aligned} &\langle 0\rvert \mathcal T\{ x_1x_2x_3x_4 \} \lvert 0\rangle \\ &\quad = G_F(t_1,t_2)G_F(t_3,t_4) +G_F(t_1,t_3)G_F(t_2,t_4) +G_F(t_1,t_4)G_F(t_2,t_3), \end{aligned}$$

where $x_j$ abbreviates $x_H(t_j)$. Interactions produce nonzero connected correlators beyond the two-point function.

Euclidean Correlators

Imaginary-time correlation functions are built from Euclidean evolution:

$$\langle x(\tau_1)x(\tau_2)\rangle_E = \frac{1}{Z_E[0]} \int\mathcal D x\, x(\tau_1)x(\tau_2)e^{-S_E[x]/\hbar}.$$

For the harmonic oscillator ground state,

$$G_E(\tau,\tau') = \frac{\hbar}{2m\omega} e^{-\omega|\tau-\tau'|}.$$

This is the analytic-continuation cousin of the real-time Feynman correlator, but one should not treat analytic continuation as automatic in every problem. Spectral conditions, boundary conditions, finite temperature, and singularities matter. The Euclidean construction is explained in Euclidean and Imaginary-Time Path Integrals.

QFT Continuation

In scalar QFT, the coordinate $x(t)$ is replaced by a field $\phi(x)$, and correlation functions become vacuum or thermal expectation values of field products:

$$G_n(x_1,\ldots,x_n) = \langle 0\rvert \mathcal T\{ \phi(x_1)\cdots\phi(x_n) \} \lvert 0\rangle.$$

Path integrals represent them as

$$G_n(x_1,\ldots,x_n) = \frac{1}{Z[0]} \int\mathcal D\phi\, \phi(x_1)\cdots\phi(x_n)e^{iS[\phi]/\hbar},$$

with the same caveat as in quantum mechanics: the boundary condition, contour, and normalization define which correlator is meant. In QFT, these correlation functions carry particle poles, scattering information, symmetry constraints, and renormalization-scale dependence. They are not merely decorative additions to a wavefunction formalism.

The bridge page Path Integrals from QM to QFT gives the broader translation from coordinate histories to field configurations. The reference page Green Functions explains common QFT naming conventions.

The complementary bridge From Evolution Operators to Time-Ordered Products derives the same field insertions from interaction-picture evolution and previews their Wick expansion.

From Euclidean Time to Euclidean QFT explains how vacuum and thermal Euclidean correlators acquire their boundary conditions and discrete frequencies.

From Correlation Functions to QFT Observables maps ordered, connected, retarded, spectral, and Euclidean correlators to masses, response, scattering, and lattice measurements.

Common Mistakes

• Calling every two-point object a Green function without specifying ordering and normalization.
• Treating a fixed-endpoint insertion as a vacuum correlator.
• Forgetting the connected subtraction when comparing correlations across states.
• Using a time-ordered correlator when a causal response function is required.
• Ignoring contact terms that arise from differentiating time-ordered products.
• Assuming Euclidean correlators can always be analytically continued without checking the spectrum and boundary conditions.
• Comparing many-body and QFT conventions without checking factors of $i$, $\hbar$, and signs.

Cross-Links

• Correlation Functions Overview
• Connected Correlation Functions
• Time-Dependent Correlations
• Green Functions in Many-Body QM
• Sources and Generating Functionals in QM
• Time Ordering
• From Evolution Operators to Time-Ordered Products
• From Euclidean Time to Euclidean QFT
• From Correlation Functions to QFT Observables
• Path Integral Conventions
• Euclidean and Imaginary-Time Path Integrals
• What Is a Green Function?
• Green Functions and Response Preview
• Wigner Function
• Correlation Functions
• Green Functions
• Path Integrals from QM to QFT

References

• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
• J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005.
• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• M. Srednicki, Quantum Field Theory, Cambridge University Press, 2007.

Exercises

1. Show that the harmonic-oscillator equal-time two-point function equals the ground-state position variance.

Solution

Using

$$x = \sqrt{\frac{\hbar}{2m\omega}} \left(a+a^\dagger\right),$$

and $a\lvert 0\rangle=0$,

$$\langle 0\rvert x^2\lvert 0\rangle = \frac{\hbar}{2m\omega} \langle 0\rvert \left(a+a^\dagger\right)^2 \lvert 0\rangle.$$

Only the term $aa^\dagger$ contributes:

$$\langle 0\rvert aa^\dagger\lvert 0\rangle=1.$$

Therefore

$$\langle 0\rvert x^2\lvert 0\rangle = \frac{\hbar}{2m\omega}.$$

This is also $G_F(t,t)$.

2. Derive the oscillator retarded response function from the commutator.

Solution

Write

$$x_H(t) = x\cos\omega t + \frac{p}{m\omega}\sin\omega t.$$

Using $[x,p]=i\hbar$ gives

$$[x_H(t),x_H(t')] = -\frac{i\hbar}{m\omega} \sin\omega(t-t').$$

For a force coupled as $H_{\rm pert}=-f(t)x$, the response convention used on this page is

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

Substitution gives

$$\chi_{xx}^{R}(t,t') = \theta(t-t') \frac{\sin\omega(t-t')}{m\omega}.$$

3. Explain why the connected two-point function equals the full two-point function for the centered oscillator ground state.

Solution

The connected two-point function is

$$\langle \mathcal T x(t)x(t')\rangle_c = \langle \mathcal T x(t)x(t')\rangle - \langle x(t)\rangle\langle x(t')\rangle.$$

For the harmonic oscillator ground state,

$$\langle 0\rvert x_H(t)\lvert 0\rangle=0,$$

because $x_H(t)$ is linear in $a$ and $a^\dagger$, and both one-point matrix elements vanish in the number-state ground state. Therefore the subtraction term is zero, so the connected and full two-point functions agree.

4. Why does a time-ordered correlator not automatically describe causal response?

Solution

A time-ordered correlator arranges operators according to their time labels:

$$\langle\mathcal T\{A(t)B(t')\}\rangle.$$

It contains both orderings, depending on whether $t\gt t'$ or $t'\lt t$. A causal response function must vanish when the response time is earlier than the perturbation time. That causality is enforced by a step function multiplying a commutator:

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

The two objects are related by spectral information, but they answer different questions.

5. In a path integral with fixed endpoints, what does inserting $x(t_1)x(t_2)$ compute?

Solution

It computes an insertion inside the fixed-endpoint transition amplitude:

$$\int_{x_i}^{x_f}\mathcal D x\, x(t_1)x(t_2)e^{iS[x]/\hbar}.$$

In operator language this corresponds to a matrix element with time-ordered coordinate operators between endpoint states, not automatically to a vacuum expectation value. To obtain a vacuum correlator, one needs a ground-state projection, an $i\epsilon$ prescription, or another boundary condition that prepares the vacuum.