From Sources in QM to Generating Functionals in QFT

Sources are bookkeeping fields that generate correlation functions. In quantum mechanics, one introduces a function $J(t)$ coupled to a coordinate $q(t)$:

$$S_J[q] = S[q] + \int dt\,J(t)q(t).$$

In field theory, the same idea becomes a spacetime source $J(x)$ coupled to a field $\phi(x)$:

$$S_J[\phi] = S[\phi] + \int d^d x\,J(x)\phi(x).$$

Functional derivatives with respect to the source insert the corresponding quantum variable. That single idea is the seed of generating functionals, propagators, Wick contractions, Feynman rules, response functions, and connected correlators.

The detailed quantum-mechanical derivation is given in Sources and Generating Functionals in QM. This page is the bridge from that derivation to QFT notation.

Source Terms in Quantum Mechanics

For a coordinate path integral, a source-dependent object has the schematic form

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

The key identity is

$$\frac{\delta}{\delta J(t)} \exp\left[ \frac{i}{\hbar} \int dt'\,J(t')q(t') \right] = \frac{i}{\hbar}q(t) \exp\left[ \frac{i}{\hbar} \int dt'\,J(t')q(t') \right].$$

Therefore source derivatives insert coordinates:

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

provided the boundary conditions, state, and ordering prescription defining $Z[J]$ are the same as those defining the brackets. This caveat is essential: a fixed-endpoint transition kernel, a vacuum expectation value, an in-in expectation value, and a thermal trace have different generating functionals.

Functional Derivatives

The source trick works because of the continuum delta-function rule

$$\frac{\delta J(y)} {\delta J(x)} = \delta^{(d)}(y-x).$$

For a source coupling,

$$F[J] = \int d^d y\,J(y)\phi(y),$$

one obtains

$$\frac{\delta F}{\delta J(x)} = \phi(x).$$

Repeated source derivatives insert repeated fields. The notation is developed in Functional Derivatives; the physics is that a source lets one probe how the theory responds to local disturbances.

Generating Correlation Functions

For a normalized interacting vacuum (denoted by $0$ here), the normalized QFT generating functional is

$$\mathcal Z[J] = \langle0| \mathcal T \exp\left[ \frac{i}{\hbar} \int d^d x\,J(x)\phi_H(x) \right] |0\rangle.$$

An unnormalized path integral with the same vacuum boundary prescription is

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

with a regulator understood. Its ratio $Z[J]/Z[0]$ equals $\mathcal Z[J]$, assuming $Z[0]\ne0$. Then

$$\langle0| \mathcal T\{ \phi(x_1)\cdots\phi(x_n) \} |0\rangle = \left( \frac{\hbar}{i} \right)^n \frac{1}{Z[0]} \frac{\delta^n Z[J]} {\delta J(x_1)\cdots\delta J(x_n)} \bigg\rvert_{J=0}.$$

This is the direct field-theory analogue of the quantum-mechanical coordinate formula.

Normalized and Connected Generators

It is often convenient to normalize

$$\mathcal Z[J] = \frac{Z[J]}{Z[0]}, \qquad \mathcal Z[0]=1.$$

Full correlators come from $\mathcal Z[J]$. Connected correlators come from the logarithm:

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

With this convention, the first derivative gives the source-dependent one-point function,

$$\frac{\delta W[J]}{\delta J(x)} = \langle\mathcal T\phi(x)\rangle_J,$$

and the second derivative gives the connected two-point function up to the standard factor:

$$\langle \mathcal T\phi(x)\phi(y) \rangle_{c,J} = \frac{\hbar}{i} \frac{\delta^2W[J]} {\delta J(x)\delta J(y)}.$$

In probability language, connected correlators are cumulants. In QFT language, they are the pieces that do not factor into products of lower-point correlators. Division by $Z[0]$ has already removed the vacuum-bubble factor. The logarithm additionally removes factorization among the external insertions. At nonzero source these in-out brackets are normalized transition insertions; they need not be expectation values in a single physical evolving state. For such measurements use an in-in contour.

Harmonic Oscillator with Source

The harmonic oscillator is the clean quantum-mechanical model for QFT sources because its path integral is Gaussian. With the ground-state time-ordering prescription, define

$$G_F(t,t') = \langle0| \mathcal T\{q_H(t)q_H(t')\} |0\rangle.$$

For a centered oscillator, the normalized source functional can be written in the convention used in the QM source page as

$$\mathcal Z_0[J] = \exp\left[ -\frac{1}{2\hbar^2} \int dt\,dt'\, J(t)G_F(t,t')J(t') \right].$$

Then

$$\left( \frac{\hbar}{i} \right)^2 \frac{\delta^2\mathcal Z_0[J]} {\delta J(t)\delta J(t')} \bigg\rvert_{J=0} = G_F(t,t').$$

Higher derivatives give Wick’s theorem. For example, the four-point function is a sum of products of two-point functions:

$$\langle0| \mathcal T\{ q(t_1)q(t_2)q(t_3)q(t_4) \} |0\rangle = G_F(t_1,t_2)G_F(t_3,t_4) + G_F(t_1,t_3)G_F(t_2,t_4) + G_F(t_1,t_4)G_F(t_2,t_3).$$

The exact Gaussian form is special. Interactions such as $\lambda q^4$ create nonzero connected correlators beyond the two-point function.

Field-Theory Generalization

For a free scalar field, the oscillator formula becomes the same Gaussian statement with time replaced by spacetime:

$$\mathcal Z_0[J] = \exp\left[ -\frac{1}{2\hbar^2} \int d^d x\,d^d y\, J(x)\Delta_F(x-y)J(y) \right],$$

where

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

in this convention. Source differentiation gives

$$\left( \frac{\hbar}{i} \right)^2 \frac{\delta^2\mathcal Z_0[J]} {\delta J(x)\delta J(y)} \bigg\rvert_{J=0} = \Delta_F(x-y).$$

Thus the free-field generating functional encodes all free-field time-ordered correlators. Wick’s theorem follows from the Gaussian.

An interacting scalar theory may be written schematically as

$$S[\phi] = S_0[\phi] - \int d^d x\, \frac{\lambda}{4!}\phi^4(x).$$

Perturbation theory expands around the Gaussian source functional. Functional derivatives bring down fields, and the Gaussian two-point function supplies contractions. This is the source-functional route to Feynman diagrams.

Diagrammatic Methods Preview gives the complementary operator-expansion grammar for propagators, vertices, self-energies, bubbles, and diagram classification in nonrelativistic many-body theory.

What Changes in QFT

The dictionary is compact:

Quantum mechanics	Field theory
coordinate history $q(t)$	field configuration $\phi(x)$
source $J(t)$	spacetime source $J(x)$
$\int dt\,J(t)q(t)$	$\int d^d x\,J(x)\phi(x)$
oscillator Green function $G_F(t,t')$	field propagator $\Delta_F(x-y)$
ordinary path integral over histories	functional integral over fields
connected correlators from $\log Z[J]$	connected diagrams from $\log Z[J]$

The complications are also important:

• the field has infinitely many degrees of freedom;
• ultraviolet divergences require regularization and renormalization;
• gauge theories require sources compatible with gauge structure or gauge fixing;
• real-time, Euclidean, in-out, in-in, and thermal generating functionals differ;
• normalization and factors of $i$ vary across conventions.

From Euclidean Time to Euclidean QFT specifies the vacuum and thermal Euclidean boundary conditions behind two of these generating functionals.

From Correlation Functions to QFT Observables explains how full and connected source derivatives become spectral, response, scattering, or Euclidean data.

Common Mistakes

• Treating $J$ as a physical force in every use, rather than an auxiliary source often set to zero.
• Forgetting the factor $(\hbar/i)^n$ attached to $n$ source derivatives in the convention used here.
• Mixing fixed-endpoint transition kernels with vacuum generating functionals.
• Using $\log Z[J]$ and expecting full correlators rather than connected correlators.
• Comparing source conventions without checking signs in $S+\int J\phi$ versus $S-\int J\phi$.
• Ignoring the distinction between real-time, Euclidean, thermal, and in-in generating functionals.

References

• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.
• A. Zee, Quantum Field Theory in a Nutshell, 2nd ed., Princeton University Press, 2010.
• J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005.

Exercises

1. Show that differentiating a source exponential inserts the variable coupled to the source.

Solution

For

$$F[J] = \exp\left[ \frac{i}{\hbar} \int dt'\,J(t')q(t') \right],$$

the functional derivative gives

$$\frac{\delta F[J]}{\delta J(t)} = \frac{i}{\hbar}q(t)F[J].$$

Thus multiplying by $\hbar/i$ inserts $q(t)$.

2. Translate the quantum-mechanical source term to scalar-field notation.

Solution

The quantum-mechanical source term is

$$\int dt\,J(t)q(t).$$

For a scalar field, the history variable is a spacetime field $\phi(x)$, and the source is a spacetime function $J(x)$. The source term becomes

$$\int d^d x\,J(x)\phi(x).$$

3. Why does $\log Z[J]$ generate connected correlators?

Solution

The logarithm of a moment-generating object produces cumulants. In field-theory language, cumulants are connected correlators. For example, the second derivative of $\log Z[J]$ subtracts the product of first derivatives, removing the disconnected product of one-point functions from the full two-point function.

4. Verify the two-point derivative for a centered Gaussian source functional.

Solution

Let

$$\mathcal Z_0[J] = \exp\left[ -\frac{1}{2\hbar^2} \int dx\,dy\,J(x)G(x,y)J(y) \right].$$

Assume the bosonic kernel is symmetric, $G(x,y)=G(y,x)$. The first derivative is proportional to $J$, so it vanishes at $J=0$. The second derivative at zero source is

$$\frac{\delta^2\mathcal Z_0[J]} {\delta J(x)\delta J(y)} \bigg\rvert_{J=0} = -\frac{1}{\hbar^2}G(x,y).$$

Multiplying by $(\hbar/i)^2=-\hbar^2$ gives

$$\left(\frac{\hbar}{i}\right)^2 \frac{\delta^2\mathcal Z_0[J]} {\delta J(x)\delta J(y)} \bigg\rvert_{J=0} = G(x,y).$$

5. Why must the boundary prescription be specified before interpreting derivatives of $Z[J]$?

Solution

The same formal source derivative inserts fields or coordinates, but the object being averaged depends on the definition of $Z[J]$. A fixed-endpoint kernel gives transition-amplitude insertions. A vacuum generating functional gives vacuum time-ordered correlators. A thermal generating functional gives thermal correlators. Without the boundary or state prescription, the brackets generated by the derivatives are ambiguous.