Functional integrals and correlators

The free scalar can be described by operators acting on a vacuum or by a source-dependent Gaussian integral. When the regulator, state, boundary conditions, and normalization agree, the two descriptions produce the same time-ordered two-point function. This page derives that result rather than treating $\int\mathcal D\phi$ as a self-explanatory symbol, then shows how one generating functional organizes all free correlators.

The running example is a real massive scalar. Euclidean signature supplies a convergent finite Gaussian; Lorentzian signature uses the vacuum in–out contour and the Feynman $+i0$ prescription. Interacting measures, gauge fixing, composite operators, and non-vacuum contours require additional work.

Required background. Canonical quantization and the free scalar fixes the field normalization, vacuum, and oscillator algebra used in the cross-check below. You should also be able to invert a quadratic form and to distinguish a Green operator from its boundary condition. If either operation is uncertain, use the focused reviews of Fourier transforms, distributions, and Green functions or quantum states and operators.

The finite Gaussian contains the essential calculation

Let $\varphi\in\mathbb R^N$ collect finitely many real variables and let $K_E=K_E^{\mathsf T}>0$. These hypotheses specify both the integration cycle and convergence. With $J\in\mathbb R^N$,

$$\begin{aligned} \mathcal Z_E[J] &= \int_{\mathbb R^N}\frac{\mathrm d^N\varphi}{(2\pi)^{N/2}} \exp\!\left( -\frac12\varphi^{\mathsf T}K_E\varphi +J^{\mathsf T}\varphi \right) \\ &=(\det K_E)^{-1/2} \exp\!\left( \frac12J^{\mathsf T}K_E^{-1}J \right). \end{aligned}$$

The calculation is just completion of the square:

$$-\frac12\varphi^{\mathsf T}K_E\varphi+J^{\mathsf T}\varphi = -\frac12(\varphi-K_E^{-1}J)^{\mathsf T} K_E(\varphi-K_E^{-1}J) +\frac12J^{\mathsf T}K_E^{-1}J.$$

Normalize by the zero-source integral,

$$Z_E[J]\equiv\frac{\mathcal Z_E[J]}{\mathcal Z_E[0]} =\exp\!\left(\frac12J^{\mathsf T}C_EJ\right), \qquad C_E\equiv K_E^{-1}.$$

If $\langle\cdots\rangle_J$ denotes expectation with this source-dependent, normalized weight, differentiation gives

$$\langle\varphi_i\rangle_J =\frac{\partial\log Z_E}{\partial J_i} =(C_EJ)_i,$$

and

$$\langle\varphi_i\varphi_j\rangle_J -\langle\varphi_i\rangle_J\langle\varphi_j\rangle_J = \frac{\partial^2\log Z_E}{\partial J_i\partial J_j} =(C_E)_{ij}.$$

Thus the inverse quadratic kernel is the covariance. This statement is exact at finite $N$. The determinant cancels from normalized correlators only because numerator and denominator contain the same kernel, domain, and measure. It remains physical in free energies, vacuum amplitudes, determinant ratios, and comparisons between boundary conditions. The Gaussian and its source derivatives are developed systematically in Zinn-Justin 2021, §§ 1.1–1.4, pp. 1–7, and §§ 7.1–7.3, pp. 126–131.

A functional integral is a regulated family

On a finite spacetime lattice, or after retaining finitely many spacetime modes, the components of $\varphi$ are ordinary integration variables. A spatial cutoff alone leaves continuous-time paths and is not yet a finite-dimensional integral. The matrix $K_E$ discretizes the differential operator $-\partial_E^2+m^2$ together with its boundary conditions. Factors of the lattice spacing can be kept in the discrete inner products or absorbed into the variables and sources; one choice must be used consistently.

For every member of the regulated family, record the following data:

Datum	What it fixes	A useful check
Variables and domain	Sites or modes, finite volume, and the real or complex integration cycle	Can a point in the integration domain be described explicitly?
Measure	All finite-dimensional factors and any dimensionful reference scale	Does a change of variables transform both measure and domain?
Kernel	Regulator, mass, boundary conditions, and treatment of zero modes	Does the proposed inverse satisfy the defining equation on that domain?
State or boundary preparation	Vacuum projection, endpoint wave functions, thermal periodicity, or another contour	Which ordered object will source differentiation produce?
Limits	Lattice spacing, volume, mode cutoff, time extent, and pole prescription	Has the order of limits been stated?

The continuum notation

$$Z_E[J] =\frac{1}{\mathcal Z_E[0]} \int\mathcal D\phi\, \exp\!\left[ -S_E[\phi]+\int\mathrm d^d x_E\,J\phi \right]$$

summarizes such a family and a proposed limit. It does not define a flat Lebesgue measure on an infinite-dimensional space. For the massive free field, the regulated covariances have the formal continuum kernel

$$C_E(x_E-y_E) = \int\frac{\mathrm d^d p_E}{(2\pi)^d} \frac{e^{ip_E\cdot(x_E-y_E)}}{p_E^2+m^2}.$$

This is a distribution: its action on smeared fields is the primary object, and its coincident-point value can be ultraviolet singular. A zero eigenvalue of $K_E$ is a different problem. Then $K_E^{-1}$ and the normalized Gaussian above do not exist until the zero direction is removed, constrained, or treated as a separate collective variable. The detailed finite-to-continuum construction is given in Regulated Bosonic Field Integrals and Gaussian Fields and Sources.

Lorentzian sources select vacuum time ordering

Use the mostly-minus metric and Fourier convention inherited from Conventions and normalizations. For a real free scalar,

$$S_0[\phi] =\int\mathrm d^d x\, \left( \frac12\partial_\mu\phi\,\partial^\mu\phi -\frac12m^2\phi^2 \right) =-\frac12\phi\mathbin{\cdot}P\mathbin{\cdot}\phi, \qquad P=\Box+m^2,$$

where the final form assumes that the relevant boundary term vanishes. At a finite regulator, take the discretized $P$ to be real symmetric and add the damping prescription $P_\epsilon=P-i\epsilon I$ with $\epsilon>0$ and define $G_\epsilon=P_\epsilon^{-1}$. The ordinary oscillatory Gaussian then gives

$$\begin{aligned} Z_{0,\epsilon}[J] &= \frac{ \int\mathrm d^N\phi\, \exp\!\left[ -\frac{i}{2}\phi^{\mathsf T}P_\epsilon\phi +iJ^{\mathsf T}\phi \right] }{ \int\mathrm d^N\phi\, \exp\!\left[ -\frac{i}{2}\phi^{\mathsf T}P_\epsilon\phi \right] } \\ &= \exp\!\left( \frac{i}{2}J^{\mathsf T}G_\epsilon J \right). \end{aligned}$$

In the infinite-time vacuum limit, the same damping selects the Feynman boundary value. Write

$$G_F=P_F^{-1}, \qquad D_F(x-y) \equiv \langle0|\mathrm T\phi(x)\phi(y)|0\rangle, \qquad G_F=iD_F.$$

The distinction between $G_F$ and $D_F$ is important: the first is the delta-normalized inverse of $P_F$, while the second is the time-ordered vacuum correlator. Explicitly,

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

so

$$P_FG_F=I, \qquad P_FD_F=-iI.$$

The normalized generating functional can therefore be written in either of two equivalent forms,

$$Z_0[J] =\exp\!\left( \frac{i}{2}J\mathbin{\cdot}G_F\mathbin{\cdot}J \right) =\exp\!\left( -\frac12J\mathbin{\cdot}D_F\mathbin{\cdot}J \right).$$

Each derivative of $Z_0$ brings down an insertion of $i\phi$. In particular,

$$\left. \frac{\delta^2Z_0[J]} {\delta J(x)\delta J(y)} \right|_{J=0} =-D_F(x-y),$$

and multiplication by $(-i)^2=-1$ returns the correlator. This two-minus-sign check catches a common accidental mixture of Euclidean and Lorentzian source conventions.

The canonical calculation supplies an independent check

Let $z=x-y$ and $E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$. Performing the $p^0$ contour integral in $D_F$ gives

$$\begin{aligned} D_F(z) =\int\frac{\mathrm d^{d-1}\mathbf p}{(2\pi)^{d-1}} \frac{1}{2E_{\mathbf p}} \bigl[ &\theta(z^0)e^{-iE_{\mathbf p}z^0+i\mathbf p\cdot\mathbf z} \\ &+\theta(-z^0)e^{+iE_{\mathbf p}z^0-i\mathbf p\cdot\mathbf z} \bigr]. \end{aligned}$$

For $z^0>0$, this is $\langle0|\phi(x)\phi(y)|0\rangle$ computed from the canonical mode expansion. For $z^0<0$, the operator order reverses. The pole positions encode the vacuum and time ordering, while the residue $1/(2E_{\mathbf p})$ checks the canonical field normalization. Differentiating the step functions also reproduces $P_FD_F=-i\delta^{(d)}$. These mutually independent checks agree with the canonical and source derivations in Schwartz 2014, § 6.2, pp. 75–77, and §§ 14.3.1–14.3.2, pp. 262–263 and Weinberg 1995, § 6.2, pp. 274–277.

This agreement is conditional, not automatic. The canonical and functional calculations use the same field normalization, vacuum, finite-volume or mode regulator, boundary data, and Feynman prescription. Changing any one of those can change the correlator without making either formulation inconsistent.

Euclidean time continues in two ordered branches

The continuation can be checked explicitly at fixed spatial momentum. The Euclidean energy integral has poles at $p_E^0=\pm iE_{\mathbf p}$; closing above for $\tau>0$ and below for $\tau<0$ gives

$$\widehat C_E(\tau,\mathbf p) =\int\frac{\mathrm dp_E^0}{2\pi} \frac{e^{ip_E^0\tau}}{(p_E^0)^2+E_{\mathbf p}^2} =\frac{e^{-E_{\mathbf p}|\tau|}}{2E_{\mathbf p}}.$$

The absolute value is not a holomorphic function of complex time. Continue the positive- and negative-$\tau$ branches separately. For $\eta>0$ their boundary values are

$$\begin{aligned} F_+(\eta+it,\mathbf p) &=\frac{e^{-E_{\mathbf p}\eta-iE_{\mathbf p}t}}{2E_{\mathbf p}}, \\ F_-(-\eta+it,\mathbf p) &=\frac{e^{-E_{\mathbf p}\eta+iE_{\mathbf p}t}}{2E_{\mathbf p}}. \end{aligned}$$

Here $F_+(z,\mathbf p)=e^{-E_{\mathbf p}z}/(2E_{\mathbf p})$ for $\operatorname{Re}z>0$, and $F_-(z,\mathbf p)=e^{E_{\mathbf p}z}/(2E_{\mathbf p})$ for $\operatorname{Re}z<0$. Selecting $F_+$ for $t>0$ and $F_-$ for $t<0$, then taking $\eta\downarrow0$, recovers

$$\widehat D_F(t,\mathbf p) =\frac{e^{-iE_{\mathbf p}|t|}}{2E_{\mathbf p}}.$$

The normalization and the two operator orders therefore agree with the canonical calculation. After restoring the spatial Fourier integral, the limit is understood as a distribution, or after smearing, rather than as a coincident-point value. This establishes the continuation for the massive free vacuum used here. An arbitrary interacting theory, state, or regulator requires its own analyticity and limit argument. The energy-contour and complex-time versions of this check are developed together in Wick Rotation and Analytic Continuation.

Full and connected correlators carry different information

For the normalized Lorentzian functional at nonzero source, define the full time-ordered correlators by

$$G_J^{(n)}(x_1,\ldots,x_n) = \frac{(-i)^n}{Z[J]} \frac{\delta^nZ[J]} {\delta J(x_1)\cdots\delta J(x_n)}.$$

The connected generating functional is

$$W[J]=-i\log Z[J], \qquad Z[J]=e^{iW[J]},$$

and its derivatives give the connected correlators,

$$G_{c,J}^{(n)}(x_1,\ldots,x_n) =(-i)^{n-1} \frac{\delta^nW[J]} {\delta J(x_1)\cdots\delta J(x_n)}.$$

For the centered free scalar,

$$W_0[J] =\frac12J\mathbin{\cdot}G_F\mathbin{\cdot}J =\frac{i}{2}J\mathbin{\cdot}D_F\mathbin{\cdot}J.$$

It follows that $G_c^{(2)}=D_F$ and every connected correlator of order $n\ge3$ vanishes. The full higher correlators do not vanish, because differentiating the exponential reconstructs products of connected pieces. For example,

$$\begin{aligned} \langle0|\mathrm T\phi_1\phi_2\phi_3\phi_4|0\rangle =\;&D_F(1,2)D_F(3,4) \\ &+D_F(1,3)D_F(2,4) \\ &+D_F(1,4)D_F(2,3). \end{aligned}$$

This is Wick factorization for a centered free bosonic Gaussian. It is not an identity for an arbitrary interacting theory. Also, “connected” here refers to the cumulant decomposition; it does not by itself imply causal support or large-distance clustering. See The Generating Functional, Connected Correlators and Cumulants, and Wick’s Theorem and Free Gaussian Factorization for the general combinatorics and their qualifications.

Where the Gaussian argument stops

Several limits of the calculation are scientifically consequential.

A formal symbol is not a construction. A Euclidean interacting measure may require a lattice or another nonperturbative definition. A Lorentzian functional integral is oscillatory and is often used as a regulated perturbative generating object rather than as an ordinary probability measure.

An inverse requires a domain and boundary condition. Physical zero modes, unfixed gauge directions, and changing boundary data are different sources of singularity and require different remedies. Gauge fixing and physical-state conditions enter on the vector-field branch.

Coincident insertions add ultraviolet questions. Source differentiation at distinct points generates ordinary field correlators. Local products such as $\phi^2(x)$ generally require their own regulator and composite-operator renormalization.

Normalization does not erase determinants universally. Dividing by the same zero-source functional removes a common vacuum factor. Determinants return when the kernel, geometry, sector, or boundary conditions differ.

Continuation needs analytic information. A Euclidean answer does not automatically define a unitary Lorentzian theory, and an in–out Feynman correlator is not the same object as a retarded response or an in–in expectation value.

Exercises

1. Let

   $$K_E= \begin{pmatrix} a&c\\ c&b \end{pmatrix}, \qquad a>0, \qquad ab-c^2>0.$$

   Compute $Z_E[J]$, the source-dependent mean, and the connected covariance. What happens as $ab-c^2\to0^+$?

Solution

The inverse is

$$K_E^{-1} =\frac{1}{ab-c^2} \begin{pmatrix} b&-c\\ -c&a \end{pmatrix}.$$

For $J=(J_1,J_2)^{\mathsf T}$,

$$Z_E[J] =\exp\!\left[ \frac{bJ_1^2-2cJ_1J_2+aJ_2^2} {2(ab-c^2)} \right].$$

The mean is $K_E^{-1}J$, and the connected covariance is the matrix $K_E^{-1}$ itself, independent of $J$. As the determinant approaches zero from above, at least one eigenvalue of $K_E$ approaches zero and the covariance along its eigenvector diverges. The correct conclusion is that the Gaussian has lost confinement in that direction; the singular inverse cannot simply be used at the endpoint.

2. Starting from the momentum-space expression for $D_F(z)$, perform the $p^0$ integral for $z^0>0$ and $z^0<0$. Identify which pole contributes in each case and explain how the result checks the canonical normalization.

Solution

The poles lie at $p^0=+E_{\mathbf p}-i0$ and $p^0=-E_{\mathbf p}+i0$. For $z^0>0$, the factor $e^{-ip^0z^0}$ permits closure in the lower half-plane. The contour is clockwise, and the positive-energy pole gives

$$\int\frac{\mathrm dp^0}{2\pi} \frac{i\,e^{-ip^0z^0}} {(p^0)^2-E_{\mathbf p}^2+i0} =\frac{e^{-iE_{\mathbf p}z^0}}{2E_{\mathbf p}}.$$

For $z^0<0$, close in the upper half-plane; the negative-energy pole gives $e^{+iE_{\mathbf p}z^0}/(2E_{\mathbf p})$. Restoring the spatial Fourier factors yields the two terms displayed in the canonical cross-check. Their $1/(2E_{\mathbf p})$ residue is exactly the factor produced by two canonical mode coefficients $1/\sqrt{2E_{\mathbf p}}$.

3. Differentiate $Z_0[J]=\exp[-J\mathbin{\cdot}D_F\mathbin{\cdot}J/2]$ four times and set $J=0$. Recover the free four-point function, then explain why its connected part is zero even though the full answer is not. Use the same pairing rule to compute $\langle\varphi_1^2\varphi_2^2\rangle$ at zero source for the two-coordinate Euclidean Gaussian in Exercise 1.

Solution

Only the term $\tfrac{1}{2!}(-J\mathbin{\cdot}D_F\mathbin{\cdot}J/2)^2$ survives four derivatives at $J=0$. For each pairing, the two quadratic factors can be assigned in $2!$ ways, and the two slots in each factor in $2^2$ ways. These eight assignments cancel the coefficient $1/(2!2^2)$. There are three distinct pairings, so after the source factors $(-i)^4=1$ are included,

$$G^{(4)}(1,2,3,4) =D_F(1,2)D_F(3,4) +D_F(1,3)D_F(2,4) +D_F(1,4)D_F(2,3).$$

By contrast, $W_0=-i\log Z_0=iJ\mathbin{\cdot}D_F\mathbin{\cdot}J/2$ is quadratic. Its fourth derivative therefore vanishes, so $G_c^{(4)}=0$. The full correlator contains disconnected products assembled by exponentiating the connected two-point function.

For the Euclidean Gaussian, use $C_E=K_E^{-1}$ and ordinary source derivatives, without Lorentzian factors of $i$. The pairing $(11)(22)$ contributes $(C_E)_{11}(C_E)_{22}$, while $(12)(12)$ occurs twice. Thus

$$\langle\varphi_1^2\varphi_2^2\rangle =(C_E)_{11}(C_E)_{22}+2(C_E)_{12}^2 =\frac{ab+2c^2}{(ab-c^2)^2}.$$

When $c=0$, the two coordinates are independent and this reduces to $\langle\varphi_1^2\rangle\langle\varphi_2^2\rangle=1/(ab)$.

Continue from the shared Gaussian

The canonical free scalar and this functional calculation should now agree on the pole prescription, residue, and time-ordered two-point function. If they do not, compare the source sign, field normalization, state, and boundary data before taking any continuum limit.

For the worked scalar capstone, continue directly to the scalar derivation in Perturbative expansion and Feynman rules. It uses the same Gaussian and requires no spin or gauge machinery.

The full graduate core also needs two extensions. Continue to Fermions, spin, and anticommutation to replace ordinary sources by Grassmann sources and learn the origin of fermionic signs. In parallel, study Vector fields and gauge redundancy to see why a gauge-redundant quadratic kernel cannot be inverted before its redundancy and physical state space are handled. Both branches feed the later development of symmetry identities and perturbation theory.

References

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, doi:10.1017/9781139540940.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, doi:10.1017/CBO9781139644167.
• Jean Zinn-Justin, Quantum Field Theory and Critical Phenomena, 5th ed., Oxford University Press, 2021, doi:10.1093/oso/9780198834625.001.0001.