Path Integral Conventions

This page fixes the path-integral conventions used in the Dynamics volume. It is a translation aid: when a textbook, paper, or neighboring page uses a different normalization, Fourier sign, source sign, or Euclidean convention, translate to these formulas before comparing results.

The central warning is simple. A path integral is not defined by the symbol $\int\mathcal Dq$ alone. The object being computed includes:

• the integration variable and configuration-space measure;
• endpoint, vacuum, thermal, or contour boundary conditions;
• the real-time or Euclidean weight;
• the time-slicing and operator-ordering convention;
• normalization factors and source signs.

Default Real-Time Kernel

For one coordinate $q(t)$ with fixed endpoints,

$$q(t_i)=q_i, \qquad q(t_f)=q_f,$$

the real-time propagator kernel is written formally as

$$K(q_f,t_f;q_i,t_i) = \int_{q_i}^{q_f}\mathcal Dq\, e^{iS[q]/\hbar},$$

where

$$S[q] = \int_{t_i}^{t_f}dt\,L(q,\dot q,t).$$

This is an amplitude. The factor $e^{iS[q]/\hbar}$ is oscillatory and should not be read as a probability weight. Probabilities are computed after amplitudes are combined and absolute squares, traces, or expectation values are formed according to the measurement question.

For

$$H=\frac{p^2}{2m}+V(q),$$

the Lagrangian convention is

$$L(q,\dot q) = \frac{m}{2}\dot q^2-V(q).$$

Time-Slicing Convention

Let

$$T=t_f-t_i, \qquad \epsilon=\frac{T}{N}, \qquad t_j=t_i+j\epsilon.$$

The endpoints are fixed:

$$q_0=q_i, \qquad q_N=q_f.$$

The intermediate variables $q_1,\ldots,q_{N-1}$ are integrated. With the left-endpoint potential convention used as the basic reference,

$$\begin{aligned} K(q_f,t_f;q_i,t_i) &= \lim_{N\to\infty} \left( \frac{m}{2\pi i\hbar\epsilon} \right)^{N/2} \int dq_1\cdots dq_{N-1} \\ &\quad\times \exp\left[ \frac{i}{\hbar}S_N(q_N,\ldots,q_0) \right], \end{aligned}$$

with

$$S_N = \sum_{j=0}^{N-1}\epsilon \left[ \frac{m}{2} \left( \frac{q_{j+1}-q_j}{\epsilon} \right)^2 -V(q_j) \right].$$

This convention is adequate for the standard coordinate path integral of $p^2/(2m)+V(q)$. More delicate Hamiltonians require more information. Different midpoint, left-point, right-point, or symmetric Trotter choices can encode different operator orderings. The canonical derivation page is Time Slicing.

Configuration-Space Measure

The compact notation

$$\int\mathcal Dq$$

means the limit of finite-dimensional integrals with the short-time normalization factors included. For the one-dimensional particle convention above,

$$\mathcal Dq \quad\text{stands for}\quad \lim_{N\to\infty} \left( \frac{m}{2\pi i\hbar\epsilon} \right)^{N/2} \prod_{j=1}^{N-1}dq_j,$$

inside a fixed-endpoint kernel. The prefactor is not optional. It is required for the equal-time delta-function limit and for the composition law of the propagator.

For several Cartesian coordinates $q^a$, the measure is the product of the corresponding coordinate measures. For curvilinear coordinates, constrained systems, spin coherent states, gauge systems, and fields, the correct measure may contain Jacobians, boundary terms, constraints, or gauge-fixing factors. Do not infer those cases from the flat Cartesian particle formula.

Phase-Space Convention

When it is better to keep momenta, a formal phase-space path integral uses

$$\int\mathcal Dp\,\mathcal Dq\, \exp\left[ \frac{i}{\hbar} \int dt\, \bigl(p\dot q-H(p,q,t)\bigr) \right].$$

A common time-sliced version is

$$\begin{aligned} K(q_f,t_f;q_i,t_i) &= \lim_{N\to\infty} \int \prod_{j=1}^{N-1}dq_j \prod_{j=0}^{N-1} \frac{dp_j}{2\pi\hbar} \\ &\quad\times \exp\left\{ \frac{i}{\hbar} \sum_{j=0}^{N-1} \left[ p_j(q_{j+1}-q_j) -\epsilon H(p_j,q_j,t_j) \right] \right\}. \end{aligned}$$

This formula also contains an ordering convention. Midpoint prescriptions are often preferred when translating Weyl-ordered Hamiltonians or when preserving coordinate-change behavior. The page Phase-Space Dynamics gives the neighboring phase-space language.

Euclidean Convention

Imaginary time is written

$$t=-i\tau.$$

For

$$L=\frac{m}{2}\dot q^2-V(q),$$

the Euclidean action convention is

$$S_E[q] = \int_{\tau_i}^{\tau_f}d\tau \left[ \frac{m}{2} \left( \frac{dq}{d\tau} \right)^2 +V(q) \right].$$

The Euclidean kernel is then

$$K_E(q_f,\tau_f;q_i,\tau_i) = \langle q_f|e^{-H(\tau_f-\tau_i)/\hbar}|q_i\rangle = \int_{q_i}^{q_f}\mathcal Dq\, e^{-S_E[q]/\hbar}.$$

The Euclidean weight can often be treated with tools closer to statistical mechanics, but it is still a quantum path integral with boundary conditions and normalization. The Wick rotation is a controlled analytic step only under assumptions about the spectrum, convergence, and singularities. The canonical page is Euclidean and Imaginary-Time Path Integrals.

Thermal Trace Convention

For inverse temperature $\beta$,

$$Z_\beta = \operatorname{Tr}e^{-\beta H}.$$

In Euclidean time the interval length is

$$\tau_f-\tau_i=\beta\hbar.$$

For an ordinary bosonic coordinate, the trace identifies the endpoints:

$$q(\tau_i)=q(\tau_i+\beta\hbar).$$

The formal path integral is

$$Z_\beta = \int_{\text{periodic}}\mathcal Dq\, e^{-S_E[q]/\hbar}.$$

The word “periodic” refers to the Euclidean-time boundary condition for the coordinate being traced. Fermionic coherent-state variables use antiperiodic thermal boundary conditions, and gauge systems require further constraint data.

Source Sign Convention

The real-time source convention in this volume is

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

Equivalently, in Hamiltonian language,

$$H_J(t)=H-J(t)q.$$

For a real-time generating functional

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

functional differentiation inserts factors of $q(t)$:

$$\frac{\delta Z[J]}{\delta J(t)} = \frac{i}{\hbar} \int\mathcal Dq\,q(t)\, e^{iS_J[q]/\hbar}.$$

Thus normalized time-ordered coordinate correlators use

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

with the state or boundary prescription included in the definition of $Z[J]$. See Sources and Generating Functionals in QM for the derivation.

For Euclidean source calculations, the convention used here is

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

Then

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

Some books absorb factors of $\hbar$ into $J$, set $\hbar=1$, or define the Euclidean source with the opposite sign. Translate before comparing correlation functions.

Normalization and Connected Generators

The unnormalized object $Z[J]$ includes the overall transition amplitude, vacuum persistence amplitude, or partition function. A normalized generator is

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

Connected correlators are generated by a logarithm. With the real-time convention above,

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

The logarithm removes disconnected vacuum factors in the usual perturbative setting. In Euclidean convention one often uses

$$W_E[J] = \hbar\log Z_E[J],$$

or a sign-shifted free-energy convention. The important habit is to state which convention is being used before identifying derivatives with connected correlators.

Fourier and i0 Conventions

For time-to-energy transforms in this volume, use

$$F(E) = \int_{-\infty}^{\infty}dt\, e^{iEt/\hbar}F(t),$$

and

$$F(t) = \int_{-\infty}^{\infty} \frac{dE}{2\pi\hbar}\, e^{-iEt/\hbar}F(E).$$

This convention matches the retarded Green-function prescription

$$G^R(E)=(E-H+i0)^{-1}$$

for

$$G^R(t) = -\frac{i}{\hbar}\theta(t)e^{-iHt/\hbar}.$$

For position-momentum Fourier transforms, use the site-wide convention in Fourier Transform Conventions and the table Fourier Transforms. When a source or kernel is transformed to frequency space, check both the exponential sign and the measure.

Boundary Prescriptions

The same formal action can define different path integrals:

Object	Boundary data	Weight	Typical result
Fixed-endpoint kernel	$q_i,q_f$ fixed	$e^{iS/\hbar}$	transition amplitude
Vacuum generator	ground-state projection or $i0$	$e^{iS/\hbar}$	time-ordered vacuum correlators
Euclidean kernel	$q_i,q_f$ fixed in imaginary time	$e^{-S_E/\hbar}$	heat-kernel amplitude
Thermal partition function	Euclidean-time trace	$e^{-S_E/\hbar}$	$Z_\beta$ and thermal correlators

Do not exchange these objects by changing notation alone. A fixed-endpoint insertion, a vacuum expectation value, and a thermal expectation value can have the same formal integral sign while computing different quantities.

Quick Translation Table

Quantity	Convention used here	Check when translating
Real-time weight	$e^{iS/\hbar}$	sign of the action and metric convention
Euclidean weight	$e^{-S_E/\hbar}$	whether source terms are included in $S_E$
Coordinate source	$S_J=S+\int Jq\,dt$	whether $H_J=H-Jq$ or $H+Jq$
Real-time derivative	$(\hbar/i)^n\delta^n/\delta J^n$	absorbed factors of $i$ and $\hbar$
Euclidean derivative	$\hbar^n\delta^n/\delta J^n$	source sign and $\hbar=1$ units
Energy transform	$F(E)=\int dt\,e^{iEt/\hbar}F(t)$	inverse measure $dE/(2\pi\hbar)$
Phase-space measure	$dp\,dq/(2\pi\hbar)$ per slice	ordering and midpoint prescriptions

Common Mistakes

• Treating $e^{iS/\hbar}$ as a probability weight.
• Dropping the short-time normalization factors in a kernel.
• Forgetting that $\mathcal Dq$ depends on the time-sliced construction.
• Comparing source derivatives without checking the sign of $Jq$.
• Calling every normalized insertion an expectation value in a state.
• Confusing fixed-endpoint kernels, vacuum correlators, and thermal traces.
• Ignoring the $i0$ or convergence prescription in real-time integrals.
• Assuming coordinate path-integral formulas automatically apply to constrained, gauge, spin, or curved configuration spaces.

Cross-Links

• Time Slicing
• Path Integral Formulation
• From Propagators to Path Integrals
• Euclidean and Imaginary-Time Path Integrals
• Sources and Generating Functionals in QM
• Correlation Functions in Path Integrals
• Common Pitfalls in Path Integrals
• Formula Sheet
• Propagator Table
• Fourier Transform Conventions
• Path Integrals

References

• R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Reviews of Modern Physics 20, 367-387, 1948.
• 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. Le Bellac, Quantum and Statistical Field Theory, Oxford University Press, 1991.

Exercises

1. In the time-sliced fixed-endpoint kernel with $N$ intervals, how many position integrations appear, and why?

Solution

There are $N-1$ position integrations. The endpoints $q_0=q_i$ and $q_N=q_f$ are fixed by the kernel. Only the intermediate positions $q_1,\ldots,q_{N-1}$ are summed over.

2. With the convention $S_J=S+\int dt\,J(t)q(t)$, show that one source derivative inserts $q(t)$ with a factor $i/\hbar$.

Solution

Differentiate the source-dependent exponential:

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

Integrating over paths gives

$$\frac{\delta Z[J]}{\delta J(t)} = \frac{i}{\hbar} \int\mathcal Dq\,q(t)e^{iS_J[q]/\hbar}.$$

3. Starting from $L=m\dot q^2/2-V(q)$, explain why the Euclidean action contains $+V(q)$.

Solution

Set $t=-i\tau$. Then $dt=-i\,d\tau$ and

$$\dot q=\frac{dq}{dt} = i\frac{dq}{d\tau}.$$

The real-time action becomes

$$S = \int dt \left[ \frac{m}{2}\dot q^2-V(q) \right] = i \int d\tau \left[ \frac{m}{2} \left( \frac{dq}{d\tau} \right)^2 +V(q) \right].$$

Thus $S=iS_E$ and $e^{iS/\hbar}$ becomes $e^{-S_E/\hbar}$.

4. Why does a thermal path integral for a coordinate use periodic Euclidean-time boundary conditions?

Solution

The thermal partition function is a trace:

$$Z_\beta = \operatorname{Tr}e^{-\beta H}.$$

In a coordinate basis, a trace integrates diagonal matrix elements:

$$Z_\beta = \int dq\, \langle q|e^{-\beta H}|q\rangle.$$

The initial and final coordinates are therefore identified. In Euclidean time, this becomes the boundary condition

$$q(\tau+\beta\hbar)=q(\tau)$$

for an ordinary bosonic coordinate.