From Path Integrals to Generating Functionals

A source derivative inserts a field, but the state and boundary prescription determine which matrix element it inserts. In particular, differentiating a vacuum in/out functional can produce a complex Feynman-boundary solution of the sourced wave equation. It need not produce the causal expectation value after an applied force. The general construction belongs to From Sources in QM to Generating Functionals in QFT. This page makes the distinction explicit with the relativistic scalar conventions and one driven mode.

Required background. The source-functional owner provides differentiation and connected generators; Field Path Integrals explains the configuration space and regulator; From Propagators to Correlators fixes the state, ordering, and inverse conventions.

Helpful background. Schwinger–Keldysh Bridge supplies the in/in construction for a prepared initial state.

A source functional includes its boundary data

Use $\hbar=c=1$, metric $(+---)$, and a real massive scalar field. With surface terms controlled by the chosen boundary prescription, write

$$\begin{aligned} S_J[\phi]&=-\frac12\int d^4x\,\phi L\phi +\int d^4x\,J\phi,\\ L&=\Box+m^2. \end{aligned}$$

The stationary sourced equation is $L\phi=J$. This equation alone supplies neither the state nor the inverse used to solve it.

Object	Boundary or state data	What a source insertion means
Fixed-endpoint kernel	Initial and final configurations	A field insertion inside a transition amplitude
Vacuum in/out functional	Vacuum boundary conditions at early and late times	A vacuum time-ordered matrix element
In/in functional	An initial density operator and forward/backward evolution	A prepared-state expectation or response

The replacement of particle paths by field histories does not erase these distinctions. In a regulated bosonic integral, a history assigns a field coordinate to each spatial site and time slice; the integration variables are not a list of particle trajectories. The canonical field-path-integral bridge develops that construction.

For a vacuum with zero one-point function, normalize the time-ordered generator as

$$\begin{aligned} Z[J]&=\langle0|T\exp(i\!\int d^4x\,J\widehat\phi)|0\rangle,\\ Z[0]&=1. \end{aligned}$$

Here the operator definition also specifies the vacuum prescription for any path-integral representation. The insertion rule is

$$\begin{aligned} &\langle0|T\widehat\phi(x_1)\cdots\widehat\phi(x_n)|0\rangle\\ &\quad=i^{-n} \left.\frac{\delta^n Z} {\delta J(x_1)\cdots\delta J(x_n)}\right|_{J=0}. \end{aligned}$$

It generates full correlators. With $W=-i\log Z$, connected two-point insertions instead obey $\langle T\phi(x)\phi(y)\rangle_{c,J} =-i\,\delta^2W/\delta J(x)\delta J(y)$. These factors follow from the specified source coupling, not from the letter used to name a propagator.

The vacuum derivative selects a Feynman inverse

For the free vacuum define $D_F(x-y)=\langle0|T\widehat\phi(x)\widehat\phi(y)|0\rangle$. The Gaussian result derived on the source-functional owner takes the form

$$Z_0[J]= \exp\!\left[-\frac12\int d^4x\,d^4y\, J(x)D_F(x-y)J(y)\right].$$

The convention check specific to this volume is

$$LD_F=-i\delta^{(4)},\qquad G_F=iD_F.$$

Therefore the one-point derivative is

$$\begin{aligned} \Phi_{\rm out/in}(x) &\equiv\frac{\delta W}{\delta J(x)}\\ &=\int d^4y\,G_F(x-y)J(y), \end{aligned}$$

and it satisfies $L\Phi_{\rm out/in}=J$. The same derivative is the normalized insertion

$$\Phi_{\rm out/in}(x)= \frac{\langle0|T\{\widehat\phi(x)e^{i\int J\widehat\phi}\}|0\rangle} {\langle0|T e^{i\int J\widehat\phi}|0\rangle}.$$

This is a transition-matrix-element ratio with a final vacuum condition. It is not generally a Hermitian expectation value in a state evolved from the initial vacuum. Even for real $J$ it can be complex. Feynman boundary conditions, rather than initial-value conditions, are part of what it computes (Beisert, 2017, chapter 2).

By contrast, apply a real physical source through $H_{\rm int}(t)=-\int d^3x\,J(t,\mathbf x)\widehat\phi(t,\mathbf x)$ to an initially prepared state. Its linear response is

$$\delta\langle\widehat\phi(x)\rangle =\int d^4y\,G_R(x-y)J(y),$$

where

$$G_R(x-y)=i\theta(x^0-y^0) \langle[\widehat\phi(x),\widehat\phi(y)]\rangle.$$

For this free, linearly driven field the mean shift is exactly linear in $J$. It also solves the sourced wave equation, now with retarded initial data. For interacting theories the displayed formula is a linear-response statement, and the response kernel generally depends on the initial state.

One impulse separates the two answers

Consider a regulated oscillator mode of frequency $E>0$ with the normalization used in the occupied-mode correlator example. Its vacuum time-ordered function is

$$D_F^{\rm mode}(t)=\frac{e^{-iE|t|}}{2E}.$$

Use an idealized impulse $J(t)=j\delta(t)$, with real $j$. This is the distributional limit of a short smooth pulse in this finite-mode problem. The Feynman one-point derivative is

$$\begin{aligned} \Phi_{\rm out/in}(t) &=\frac{ij}{2E}e^{-iE|t|}\\ &=\frac{j}{2E}\sin(E|t|) +\frac{ij}{2E}\cos(Et). \end{aligned}$$

The physically driven vacuum instead has

$$\langle q(t)\rangle_J =\frac{j}{E}\theta(t)\sin(Et).$$

The second expression vanishes before the pulse and has a momentum jump $j$. The first has nonzero values on both sides and an imaginary part. Both have a first-derivative jump $j$ and satisfy

$$(\partial_t^2+E^2)\Phi=j\delta(t).$$

Their difference is the homogeneous solution

$$\begin{aligned} &\Phi_{\rm out/in}(t)-\langle q(t)\rangle_J\\ &\quad=\frac{j}{2E}\bigl[-\sin(Et)+i\cos(Et)\bigr]. \end{aligned}$$

Thus checking the differential equation and delta-source strength does not select the physical initial-value answer. The pre-pulse part of an in/out ratio is not an observable signal sent backward in time.

What the functional notation does not supply

A formal Minkowski weight $e^{iS}$ is not a positive probability distribution over field histories. The regulator, vacuum or endpoint conditions, normalization, and limiting prescription remain necessary. For fields with interactions, defining composite insertions also requires ultraviolet care.

For a fermion field, commuting sources do not reproduce odd-field statistics. Grassmann sources require an ordering and a declared left- or right-derivative convention. The scalar differentiation formula cannot simply be copied with $\phi$ replaced by $\psi$. Beisert (2017, sections 2.3–2.5) develops that extension.

For a physical finite-time expectation value, continue to the canonical Schwinger–Keldysh bridge. For vacuum correlators used to extract scattering amplitudes, the next step is LSZ reduction, with its separate external-state and pole assumptions.

Exercises

Change the sign of the source. If the action contains $-\int J\phi$ while all other conventions are retained, what changes in the equation of motion and insertion rule?

Solution

The sourced equation becomes $L\phi=-J$. The source exponential is $e^{-i\int J\widehat\phi}$, so an $n$-field insertion uses $(-i)^{-n}$ rather than $i^{-n}$. The quadratic free $Z_0[J]$ is unchanged because it is even in $J$, but $\delta W/\delta J$ now equals the negative of the normalized field insertion.

Check the impulse without a contour integral. Differentiate the two mode solutions on each side of zero. Verify the common source and the homogeneous difference.

Solution

Each solves the homogeneous oscillator equation for $t\ne0$. Both are continuous at zero. The retarded derivative changes from $0$ to $j$; the Feynman derivative changes from $-j/2$ to $j/2$. Each jump gives $j\delta(t)$ on the second derivative. Their difference has no jump and is the global sine/cosine combination shown above.

Restore the action unit. Reinstate $\hbar$ in the normalized vacuum generator and Gaussian, keeping $D_F=\langle T\phi\phi\rangle$ as the actual correlator.

Solution

The source exponential becomes $\exp[(i/\hbar)\int J\widehat\phi]$; the insertion factor is $(\hbar/i)^n$. The Gaussian exponent is $-\int JD_FJ/(2\hbar^2)$, and $W=(\hbar/i)\log Z$. There is no extra factor of $i$ hidden in the definition of $D_F$.

References

• Beisert, Niklas. Quantum Field Theory II. ETH Zurich lecture notes, spring semester 2017, chapter 2. Lecture notes. Vacuum field integrals, connected generators, and Grassmann sources.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press (2014). doi:10.1017/9781139540940. Source functionals, time ordering, and perturbative correlations.