Sources and Generating Functionals in QM

A source is an auxiliary function coupled linearly to a quantum variable. In a coordinate path integral, the source $J(t)$ is introduced by replacing the action with

$$S_J[x] = S[x]+\int_{t_i}^{t_f}dt\,J(t)x(t).$$

The resulting source-dependent path integral is not usually introduced because $J(t)$ is a physical force that will remain in the final problem. It is introduced because differentiation with respect to $J(t)$ inserts factors of $x(t)$:

$$\frac{\delta}{\delta J(t)} e^{(i/\hbar)\int dt'\,J(t')x(t')} = \frac{i}{\hbar}x(t) e^{(i/\hbar)\int dt'\,J(t')x(t')}.$$

This small identity is the seed of generating functionals, correlation functions, Wick contractions, response theory, and the usual QFT source formalism. The quantum-mechanical case is the clean place to learn it because the integration variable is one coordinate history rather than a field over spacetime.

What the Source Couples To

For a particle with

$$S[x] = \int_{t_i}^{t_f}dt\, \left[ \frac{m}{2}\dot x^2-V(x) \right],$$

the sourced action is

$$S_J[x] = \int_{t_i}^{t_f}dt\, \left[ \frac{m}{2}\dot x^2-V(x)+J(t)x(t) \right].$$

If $J(t)$ is interpreted physically, it is an externally applied generalized force. In the generating-functional use, it is a bookkeeping variable set to zero after differentiation. The corresponding Hamiltonian has the sign

$$H_J(t)=H-J(t)x,$$

because adding $Jx$ to the Lagrangian subtracts $Jx$ from the Hamiltonian. This sign is a common source of confusion when translating between Lagrangian and Hamiltonian formulas.

For fixed endpoints, the sourced kernel is

$$K_J(x_f,t_f;x_i,t_i) = \int_{x(t_i)=x_i}^{x(t_f)=x_f} \mathcal D x(t)\, \exp\left[ \frac{i}{\hbar} \left( S[x]+\int_{t_i}^{t_f}dt\,J(t)x(t) \right) \right].$$

This object generates coordinate insertions inside a transition amplitude. It is not automatically a vacuum generating functional. Boundary conditions and external states determine what the derivatives mean.

Generating Functional Idea

A generating functional is a source-dependent object from which many related quantities are obtained by functional differentiation. For a fixed-endpoint kernel, one may use

$$Z_{fi}[J] = K_J(x_f,t_f;x_i,t_i).$$

Then

$$\frac{\delta Z_{fi}[J]}{\delta J(t_1)} = \frac{i}{\hbar} \int_{x_i}^{x_f}\mathcal D x\, x(t_1) e^{(i/\hbar)S_J[x]},$$

where the shorthand $\int_{x_i}^{x_f}\mathcal D x$ means the same fixed-endpoint path integral as above. Repeating the differentiation gives

$$\frac{\delta^n Z_{fi}[J]} {\delta J(t_1)\cdots\delta J(t_n)} = \left(\frac{i}{\hbar}\right)^n \int_{x_i}^{x_f}\mathcal D x\, x(t_1)\cdots x(t_n) e^{(i/\hbar)S_J[x]}.$$

At $J=0$,

$$\left( \frac{\hbar}{i} \right)^n \frac{\delta^n Z_{fi}[J]} {\delta J(t_1)\cdots\delta J(t_n)} \bigg\rvert_{J=0} = \int_{x_i}^{x_f}\mathcal D x\, x(t_1)\cdots x(t_n)e^{iS[x]/\hbar}.$$

Dividing by $Z_{fi}[0]$ produces an insertion normalized by the unsourced transition amplitude:

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

The notation $\langle\cdots\rangle_{fi}$ is only a compact symbol for a normalized fixed-endpoint path-integral insertion. It is not an expectation value in a normalizable state unless the endpoints and limits have been chosen to represent one.

Operator Meaning and Time Ordering

The same source can be described in operator language. Since $H_J(t)=H-J(t)x$, the source-dependent evolution operator may be written schematically as

$$U_J(t_f,t_i) = \mathcal T \exp\left[ -\frac{i}{\hbar} \int_{t_i}^{t_f}dt\, \bigl(H-J(t)x\bigr) \right],$$

with the usual qualifications about interaction-picture notation when $H$ and $x$ do not commute at different times. Differentiating with respect to $J$ inserts time-ordered Heisenberg-position operators:

$$\left( \frac{\hbar}{i} \right)^n \frac{\delta^n}{\delta J(t_1)\cdots\delta J(t_n)} \langle x_f\rvert U_J(t_f,t_i)\lvert x_i\rangle \bigg\rvert_{J=0} = \langle x_f\rvert \mathcal T\{ x_H(t_1)\cdots x_H(t_n) \} U(t_f,t_i) \lvert x_i\rangle,$$

with the placement of $U(t_f,t_i)$ understood according to the chosen Heisenberg-picture reference time. The important point is simpler than the notation: real-time source path integrals normally generate time-ordered insertions.

If the desired object is a vacuum correlator, one instead defines

$$Z_0[J] = \langle 0\rvert \mathcal T \exp\left[ \frac{i}{\hbar}\int dt\,J(t)x_H(t) \right] \lvert 0\rangle,$$

or realizes the same object by path-integral boundary conditions and an $i\epsilon$ prescription that projects onto the ground state. Then

$$\langle 0\rvert \mathcal T\{ x_H(t_1)\cdots x_H(t_n) \} \lvert 0\rangle = \left( \frac{\hbar}{i} \right)^n \frac{1}{Z_0[0]} \frac{\delta^n Z_0[J]} {\delta J(t_1)\cdots\delta J(t_n)} \bigg\rvert_{J=0}.$$

This is the quantum-mechanical prototype of the QFT formula for time-ordered vacuum correlation functions.

Normalized and Connected Generators

The raw generating functional $Z[J]$ contains all moments. It is often convenient to remove the $J=0$ normalization:

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

Then $\mathcal Z[0]=1$, and derivatives of $\mathcal Z[J]$ give normalized correlators. For example,

$$\langle \mathcal T x(t_1)x(t_2)\rangle = \left( \frac{\hbar}{i} \right)^2 \frac{\delta^2 \mathcal Z[J]} {\delta J(t_1)\delta J(t_2)} \bigg\rvert_{J=0},$$

where the state, boundary condition, or projection convention is part of the definition of the brackets.

Connected correlation functions are generated by the logarithm. Define

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

The first derivative gives the one-point function:

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

The second connected correlator is

$$\langle \mathcal T x(t_1)x(t_2)\rangle_{c,J} = \frac{\hbar}{i} \frac{\delta^2 \mathcal W[J]} {\delta J(t_1)\delta J(t_2)}.$$

At zero source, this becomes the usual connected part

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

Connected Correlation Functions owns the general moment–cumulant partition formula, clustering, and many-body subtraction diagnostics; this page retains the path-integral source derivation.

For a centered Gaussian system, the one-point function vanishes and the two-point function is already connected.

Finite-Dimensional Analogy

The functional formulas are the continuum version of a familiar finite-dimensional identity. Let $q=(q_1,\ldots,q_N)$ and introduce sources $J_i$:

$$Z(J) = \int d^Nq\, \exp\left[ \frac{i}{\hbar} \left( S(q)+\sum_{i=1}^N J_iq_i \right) \right].$$

Then

$$\frac{\partial Z}{\partial J_k} = \frac{i}{\hbar} \int d^Nq\, q_k \exp\left[ \frac{i}{\hbar} \left( S(q)+\sum_{i=1}^N J_iq_i \right) \right].$$

The path-integral formula is obtained by replacing the discrete label $k$ by a continuous time $t$:

$$J_kq_k \quad\longrightarrow\quad J(t)x(t),$$

and

$$\frac{\partial}{\partial J_k} \quad\longrightarrow\quad \frac{\delta}{\delta J(t)}.$$

The Functional Derivatives page develops this replacement carefully using delta-function identities.

Euclidean Source Conventions

In imaginary time, one commonly writes the Euclidean path integral as

$$Z_E[J] = \int\mathcal D x\, \exp\left[ -\frac{1}{\hbar}S_E[x] +\frac{1}{\hbar}\int d\tau\,J(\tau)x(\tau) \right].$$

With this convention,

$$\frac{\delta Z_E[J]}{\delta J(\tau)} = \frac{1}{\hbar} \int\mathcal D x\, x(\tau) \exp\left[ -\frac{1}{\hbar}S_E[x] +\frac{1}{\hbar}\int d\tau'\,J(\tau')x(\tau') \right],$$

so Euclidean correlators are generated by powers of $\hbar$, not by powers of $\hbar/i$:

$$\langle x(\tau_1)\cdots x(\tau_n)\rangle_E = \hbar^n \frac{1}{Z_E[0]} \frac{\delta^n Z_E[J]} {\delta J(\tau_1)\cdots\delta J(\tau_n)} \bigg\rvert_{J=0}.$$

Some books define $S_{E,J}=S_E+\int Jx$ instead. Then the derivative formulas acquire signs. This is not a physical difference; it is a convention for where the minus sign is placed in the Euclidean exponent. See Euclidean and Imaginary-Time Path Integrals for the Wick-rotation and ground-state-projection context.

Harmonic Oscillator with Source

The harmonic oscillator is the central example because its path integral is Gaussian. For

$$S[x] = \int dt\, \left[ \frac{m}{2}\dot x^2 - \frac{m\omega^2}{2}x^2 \right],$$

the sourced action is

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

The classical sourced equation follows from the Euler–Lagrange equation:

$$m\left( \frac{d^2}{dt^2}+\omega^2 \right)x_{\rm cl}(t) = J(t).$$

The real-time vacuum generating functional is fixed by the oscillator two-point function. With the ground-state prescription,

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

The normalized source functional is

$$\mathcal Z_0[J] = \frac{Z_0[J]}{Z_0[0]} = \exp\left[ -\frac{1}{2\hbar^2} \int dt\,dt'\, J(t)G_F(t,t')J(t') \right].$$

This compact formula encodes all vacuum time-ordered oscillator correlators. For example,

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

The fourth derivative gives Wick’s theorem:

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

The absence of higher connected correlators is the defining Gaussian feature. Interactions such as $\lambda x^4$ spoil this exact Gaussian form and are handled perturbatively, semiclassically, or numerically.

Relation to Green Functions

The source response is governed by an inverse differential operator. After integration by parts, the oscillator quadratic action can be written schematically as

$$S[x] = \frac12 \int dt\,x(t)D x(t), \qquad D=-m\left( \frac{d^2}{dt^2}+\omega^2 \right),$$

up to boundary terms fixed by the problem. The relevant Green function is an inverse of $D$ with a prescription:

$$D\,\Delta(t,t')=\delta(t-t').$$

Different prescriptions give different physical objects: fixed-endpoint kernels, retarded response functions, advanced functions, Euclidean Green functions, or Feynman time-ordered functions. Source methods do not eliminate the need to specify the boundary condition; they make that dependence explicit.

This is why the word “Green function” must be read with context. The What Is a Green Function? page separates inverse kernels, response functions, resolvents, and propagators.

Bridge to Field Theory

The field-theory replacement is structurally simple:

$$x(t) \longrightarrow \phi(x), \qquad J(t) \longrightarrow J(x), \qquad \int dt \longrightarrow \int d^{d+1}x.$$

A scalar-field generating functional has the schematic form

$$Z[J] = \int\mathcal D\phi\, \exp\left[ \frac{i}{\hbar} \left( S[\phi]+\int d^{d+1}x\,J(x)\phi(x) \right) \right].$$

Functional derivatives insert fields:

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

The analogy is exact at the level of notation, but QFT adds new issues: ultraviolet regularization, renormalization, local operator products, gauge redundancy, fermionic variables, and the definition of the continuum limit. The quantum-mechanical source formalism is therefore a bridge, not a substitute for a full QFT construction. See Path Integrals from QM to QFT for the broader translation map.

Common Mistakes

• Treating $J(t)$ as always physical. In generating functionals it is often only an auxiliary variable.
• Forgetting that fixed-endpoint source derivatives generate insertions inside a transition amplitude, not automatically vacuum expectation values.
• Dropping the normalization by $Z[0]$ when comparing correlators.
• Mixing real-time factors of $\hbar/i$ with Euclidean factors of $\hbar$.
• Assuming every Green function produced by a source is retarded. Real-time path-integral source functionals usually generate time-ordered functions unless a different contour or prescription is specified.
• Ignoring sign conventions in the Euclidean source term.
• Writing down QFT source formulas before specifying whether the fields are real, complex, fermionic, gauge-fixed, Euclidean, or Lorentzian.

Cross-Links

• Time Slicing
• Path Integral Conventions
• Free-Particle Path Integral
• Euclidean and Imaginary-Time Path Integrals
• Path Integrals from QM to QFT
• From Sources in QM to Generating Functionals in QFT
• Functional Derivatives
• Time Ordering
• What Is a Green Function?
• Path Integrals

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.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009.
• J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005.
• M. Srednicki, Quantum Field Theory, Cambridge University Press, 2007.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.

Exercises

1. Show directly that differentiating $Z[J]$ twice inserts two factors of $x(t)$.

Solution

Start from

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

The first derivative gives

$$\frac{\delta Z[J]}{\delta J(t_1)} = \frac{i}{\hbar} \int\mathcal D x\, x(t_1)e^{iS_J[x]/\hbar}.$$

Differentiating again,

$$\frac{\delta^2 Z[J]} {\delta J(t_1)\delta J(t_2)} = \left( \frac{i}{\hbar} \right)^2 \int\mathcal D x\, x(t_1)x(t_2)e^{iS_J[x]/\hbar}.$$

At $J=0$ and after multiplying by $(\hbar/i)^2$, this is the unnormalized two-point insertion.

2. Explain why adding $J(t)x(t)$ to the Lagrangian corresponds to subtracting $J(t)x$ from the Hamiltonian.

Solution

For a Lagrangian

$$L_J(x,\dot x,t)=L(x,\dot x,t)+J(t)x,$$

the canonical momentum is unchanged if the source contains no $\dot x$:

$$p=\frac{\partial L_J}{\partial \dot x} = \frac{\partial L}{\partial \dot x}.$$

The Hamiltonian is

$$H_J=p\dot x-L_J = p\dot x-L-J(t)x = H-J(t)x.$$

Thus the sign of the source term flips when moving from the Lagrangian to the Hamiltonian.

3. For the normalized oscillator functional $\mathcal Z_0[J]$, verify the two-point derivative formula.

Solution

Let

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

Because the exponent is quadratic in $J$, the first derivative vanishes at $J=0$. The second derivative at zero source is

$$\frac{\delta^2\mathcal Z_0[J]} {\delta J(t)\delta J(t')} \bigg\rvert_{J=0} = -\frac{1}{\hbar^2}G_F(t,t').$$

Therefore

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

because $(\hbar/i)^2=-\hbar^2$.

4. Why does $\log Z[J]$ generate connected correlators rather than all correlators?

Solution

The logarithm removes products of statistically independent pieces in the same way that the logarithm of an ordinary moment-generating function produces cumulants. For example,

$$\frac{\delta^2 \log Z}{\delta J(t_1)\delta J(t_2)} = \frac{1}{Z} \frac{\delta^2 Z}{\delta J(t_1)\delta J(t_2)} - \frac{1}{Z^2} \frac{\delta Z}{\delta J(t_1)} \frac{\delta Z}{\delta J(t_2)}.$$

After restoring the appropriate powers of $\hbar/i$, this subtracts

$$\langle x(t_1)\rangle\langle x(t_2)\rangle$$

from the full two-point function. Higher derivatives of $\log Z$ similarly produce higher cumulants, which are the connected correlators.

5. Translate the quantum-mechanical source term into scalar-field notation.

Solution

The quantum-mechanical source term is

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

For a scalar field, the coordinate history $x(t)$ is replaced by a field configuration $\phi(x)$ over spacetime, and the time integral is replaced by a spacetime integral:

$$\int d^{d+1}x\,J(x)\phi(x).$$

Functional derivatives with respect to $J(x)$ insert $\phi(x)$, just as derivatives with respect to $J(t)$ insert $x(t)$ in quantum mechanics.