Path Integrals

A path integral represents a quantum amplitude through histories weighted by the action. In ordinary quantum mechanics it is a representation of the same unitary evolution encoded by the Schrödinger equation and the evolution operator. In quantum field theory, the integration variable becomes an entire field configuration, and sources organize correlation functions.

The slogan “sum over paths” is useful only after its boundaries are clear. Real-time weights are complex phases rather than probabilities, the continuum measure is generally defined through a regulator or limiting procedure, and operator ordering can affect the resulting action and measure.

Quantum-Mechanical Starting Point

For $t_f>t_i$, define the position-space propagator

$$K(x_f,t_f;x_i,t_i) = \langle x_f\vert U(t_f,t_i) \vert x_i\rangle.$$

For a time-independent Hamiltonian,

$$U(t_f,t_i) = \exp\left[ -\frac{i}{\hbar}H(t_f-t_i) \right].$$

The propagator evolves a wavefunction:

$$\psi(x_f,t_f) = \int_{-\infty}^{\infty} K(x_f,t_f;x_i,t_i) \psi(x_i,t_i)\,dx_i.$$

It obeys the composition law

$$\begin{aligned} K(x_f,t_f;x_i,t_i) =\int_{-\infty}^{\infty}dx\, &K(x_f,t_f;x,t) \\ &\times K(x,t;x_i,t_i) \end{aligned}$$

for $t_i<t<t_f$, together with

$$\lim_{t_f\to t_i^+} K(x_f,t_f;x_i,t_i) =\delta(x_f-x_i).$$

The path integral is built by iterating this composition law over short time steps.

Time-Sliced Definition

Consider

$$H=\frac{p^2}{2m}+V(x,t), \qquad T=t_f-t_i, \qquad \epsilon=\frac{T}{N}.$$

Set $x_0=x_i$, $x_N=x_f$, and use midpoint coordinates $x_{j+1/2}=(x_{j+1}+x_j)/2$. A standard time-sliced expression is

$$\begin{aligned} K =\lim_{N\to\infty} \left( \frac{m}{2\pi i\hbar\epsilon} \right)^{N/2} \int\prod_{j=1}^{N-1}dx_j\, \exp\Bigg\{ \frac{i\epsilon}{\hbar} \sum_{j=0}^{N-1} \Bigg[ &\frac{m}{2} \left( \frac{x_{j+1}-x_j}{\epsilon} \right)^2 \\ &- V(x_{j+1/2},t_{j+1/2}) \Bigg] \Bigg\}. \end{aligned}$$

This limit motivates the formal notation

$$K(x_f,t_f;x_i,t_i) = \int_{x(t_i)=x_i}^{x(t_f)=x_f} \mathcal D x\, \exp\left[ \frac{i}{\hbar}S[x] \right],$$

where

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

The prefactor is part of the definition. Omitting it destroys the delta function limit, composition law, and dimensions of the kernel.

The midpoint choice is especially natural for Weyl-ordered Hamiltonians. For more general momentum dependence, curved coordinates, or constrained systems, the discretization rule carries operator-ordering information and can produce additional measure or action terms.

Phase-Space Form

Inserting both position and momentum resolutions of identity gives the formal phase-space path integral

$$\begin{aligned} K =\int\mathcal Dp\,\mathcal Dx\, \exp\Bigg\{ \frac{i}{\hbar} \int_{t_i}^{t_f} \Big[ p\dot x-H(p,x,t) \Big]dt \Bigg\}. \end{aligned}$$

Its time-sliced measure contains factors

$$\prod_{j=0}^{N-1} \frac{dp_j}{2\pi\hbar} \prod_{j=1}^{N-1}dx_j.$$

Integrating out $p$ gives the configuration-space form when the momentum integral is controlled, as for a standard quadratic kinetic term. For nonquadratic Hamiltonians, constraints, or nontrivial coordinate measures, that step is not a harmless substitution.

Two Exact Kernels

Free particle

For $T>0$,

$$K_0(x_f,T;x_i,0) = \sqrt{ \frac{m}{2\pi i\hbar T} } \exp\left[ \frac{im(x_f-x_i)^2}{2\hbar T} \right].$$

The square-root branch is fixed by the short-time prescription, often written with a small convergence factor. The kernel has dimensions $L^{-1}$ in one dimension and approaches $\delta(x_f-x_i)$ distributionally as $T\to0^+$.

Harmonic oscillator

For intervals away from caustics,

$$\begin{aligned} K_{\mathrm{ho}}(x_f,T;x_i,0) = \sqrt{ \frac{m\omega} {2\pi i\hbar\sin(\omega T)} } \exp\Bigg\{ \frac{im\omega} {2\hbar\sin(\omega T)} \Big[ &(x_f^2+x_i^2)\cos(\omega T) \\ &-2x_fx_i \Big] \Bigg\}. \end{aligned}$$

Crossing a zero of $\sin(\omega T)$ requires the correct branch and Maslov phase. The compact formula should not be continued through a caustic by treating the square root as an ordinary positive real number.

These examples provide normalization and semiclassical benchmarks for numerical or formal path-integral calculations.

Stationary Phase and the Classical Limit

Varying the action with fixed endpoints gives

$$\delta S[x_{\mathrm{cl}}]=0,$$

which is equivalent to the Euler–Lagrange equation under the usual regularity assumptions. Expand

$$x(t)=x_{\mathrm{cl}}(t)+\eta(t), \qquad \eta(t_i)=\eta(t_f)=0.$$

The linear fluctuation vanishes, and the quadratic fluctuation operator controls the leading prefactor. In one dimension, the semiclassical kernel has the Van Vleck form

$$\begin{aligned} K_{\mathrm{sc}} \simeq \left[ -\frac{1}{2\pi i\hbar} \frac{\partial^2S_{\mathrm{cl}}} {\partial x_f\,\partial x_i} \right]^{1/2} \exp\left[ \frac{i}{\hbar}S_{\mathrm{cl}} -\frac{i\pi\nu}{2} \right], \end{aligned}$$

with the branch and Maslov index $\nu$ determined by caustics.

Stationary phase does not say that only the classical path exists or contributes. It says that, in a controlled semiclassical regime, rapidly varying phases cancel away from stationary configurations and fluctuations around stationary paths can be organized systematically.

Euclidean Time

Under conditions that permit analytic continuation, set

$$t=-i\tau.$$

The Euclidean kernel is

$$K_E(x_f,\tau_f;x_i,\tau_i) = \langle x_f\vert e^{-H(\tau_f-\tau_i)/\hbar} \vert x_i\rangle$$

and has the formal representation

$$K_E = \int\mathcal D x\, \exp\left[ -\frac{1}{\hbar}S_E[x] \right],$$

where for the standard particle

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

The Euclidean weight is real and damping when the action is bounded below, which connects the construction to heat kernels and, for suitable potentials, the Feynman–Kac formula.

Ground-state projection

Insert energy eigenstates:

$$K_E(x_f,T;x_i,0) = \sum_n e^{-E_nT/\hbar} \psi_n(x_f)\psi_n^\ast(x_i).$$

If the ground state is isolated and has nonzero overlap with the boundary data, then at large $T$,

$$K_E \sim e^{-E_0T/\hbar} \psi_0(x_f)\psi_0^\ast(x_i).$$

This is the basis of imaginary-time ground-state projection methods.

Thermal trace

For inverse temperature $\beta$,

$$Z(\beta)=\operatorname{tr}e^{-\beta H}.$$

The path integral identifies the endpoints and is periodic over Euclidean time length $\hbar\beta$:

$$Z(\beta) = \int_{x(\hbar\beta)=x(0)} \mathcal D x\, e^{-S_E[x]/\hbar}.$$

When $\hbar=1$, field-theory texts usually call the Euclidean period simply $\beta$.

From Paths to Fields

For a scalar field, the transition is

Quantum mechanics	Quantum field theory
Coordinate history $x(t)$	Field history $\phi(x)$ over spacetime
Endpoint positions $x_i,x_f$	Boundary field configurations
Action $S[x]$	Field action $S[\phi]$
Time-sliced integral	Spacetime-regulated functional integral
Propagator $K$	Vacuum amplitude or transition functional
Source $J(t)x(t)$	Local source $\int d^4x\,J(x)\phi(x)$
Correlators of $x(t)$	Time-ordered field $n$-point functions
Classical path	Classical field configuration

A schematic generating functional is

$$Z[J] = \int\mathcal D\phi\, \exp\left\{ \frac{i}{\hbar} \left[ S[\phi] +\int d^4x\,J(x)\phi(x) \right] \right\}.$$

With this source-sign convention, normalized time-ordered correlations are generated by

\begin{aligned} \langle0\vert T\phi(x_1)\cdots\phi(x_n) \vert0\rangle = \left. \frac{1}{Z[0]} \left( \frac{\hbar}{i} \right)^n \frac{\delta^n Z[J]} {\delta J(x_1)\cdots\delta J(x_n)} \biggr\rvert_{J=0}. \end{aligned}

Changing the sign of the source term changes the derivative factors. Record the convention before using a generating-functional identity.

What a Regulator Does

A symbolic continuum expression

$$\int\mathcal D\phi$$

does not by itself define a measure. Common regulated meanings include:

• a finite time slicing in quantum mechanics;
• a spatial or spacetime lattice;
• a finite mode cutoff;
• dimensional or analytic regularization within perturbation theory;
• Euclidean measure constructions for suitable theories;
• an oscillatory-integral prescription with an $i\epsilon$ boundary condition.

The regulator makes the number of integration variables or perturbative integrals controllable. Removing it can require renormalization, and the continuum limit may fail to exist for some proposed models.

Additional Field-Theory Structures

Interactions and diagrams

Split

$$S[\phi]=S_0[\phi]+S_{\mathrm{int}}[\phi].$$

If $S_0$ is quadratic, its functional integral is Gaussian. Expanding $e^{iS_{\mathrm{int}}/\hbar}$ generates perturbative contractions and Feynman diagrams. The free inverse kernel determines the propagator, while interaction terms determine vertices.

Fermions

Fermionic path integrals use Grassmann-valued fields. Gaussian integration produces determinants rather than inverse square-root determinants, and reordering Grassmann variables changes signs.

Gauge fields

Integrating naively over all gauge-related configurations overcounts descriptions of the same physical configuration. Gauge fixing, constraints, and the associated determinant or ghost structure are required.

Nonequilibrium real time

Ordinary in–out generating functionals calculate transition amplitudes. Expectation values for an initial density operator and real-time evolution generally require a closed time contour, as in the Schwinger–Keldysh formalism.

Anomalies and measure Jacobians

A classical change of variables or symmetry can alter the regulated functional measure. A nontrivial Jacobian can survive regulator removal and contribute an anomaly. Formal claims that the measure is invariant must therefore be checked against the regulator.

Wick-Rotation Cautions

Wick rotation is not merely replacing every $t$ by $-i\tau$ inside an arbitrary answer. One must track:

• singularities crossed in the complex energy plane;
• the $i\epsilon$ prescription;
• boundary and initial-state conditions;
• convergence at large fields or long times;
• reflection positivity and reconstruction for Euclidean field theories;
• chemical potentials, real-time backgrounds, or curved geometries that can obstruct a simple continuation.

Euclidean and Lorentzian formulations are deeply related, but the relation is a controlled analytic statement, not typographical substitution.

How to Use This Bridge

1. Start from a clearly normalized operator kernel.
2. Choose a time slicing and state its operator ordering.
3. Verify the short-time delta limit and composition law.
4. Test the normalization against the free particle or oscillator.
5. State whether the integral is real time, Euclidean, thermal, or Schwinger–Keldysh.
6. For fields, identify the regulator, source convention, boundary condition, and normalization by $Z[0]$.
7. Separate formal continuum notation from a mathematically defined limit.

Common Mistakes

• Treating $e^{iS/\hbar}$ as a probability weight.
• Omitting the time-slice normalization.
• Saying the integral contains only classical paths.
• Ignoring operator ordering in the Hamiltonian-to-action map.
• Assuming all contributing real-time paths are differentiable.
• Continuing through oscillator caustics without the Maslov phase.
• Treating Wick rotation as automatic.
• Forgetting that thermal bosonic and fermionic fields have different Euclidean boundary conditions.
• Differentiating $Z[J]$ with source factors from a different sign convention.
• Writing a continuum field measure without a regulator or definition.
• Integrating gauge redundancy as if every configuration were physically distinct.
• Using an in–out functional for a nonequilibrium expectation-value problem.

Exercises

Exercise 1: Short-time normalization

Show why the free-particle prefactor is needed to recover the delta function as $T\to0^+$.

Solution

For a smooth test function $f$,

$$\begin{aligned} \int dx_i\, \sqrt{\frac{m}{2\pi i\hbar T}} \exp\left[ \frac{im(x_f-x_i)^2}{2\hbar T} \right] f(x_i) \longrightarrow f(x_f) \end{aligned}$$

as an oscillatory Gaussian limit with the prescribed branch. Without the square-root factor, the integral scales as $\sqrt T$ and vanishes rather than approaching the identity operator. Thus the prefactor fixes both the distributional normalization and the kernel dimension.

Exercise 2: Euclidean ground-state extraction

Suppose $E_0<E_1\le E_2\le\cdots$ and $\psi_0(x_i)\psi_0(x_f)\ne0$. Estimate the leading relative correction to the large-$T$ ground-state form of $K_E$.

Solution

Factor out the ground-state term:

$$\begin{aligned} K_E =e^{-E_0T/\hbar} \Big[ &\psi_0(x_f)\psi_0^\ast(x_i) \\ &+ e^{-(E_1-E_0)T/\hbar} \psi_1(x_f)\psi_1^\ast(x_i) +\cdots \Big]. \end{aligned}$$

The leading relative correction is of order

$$e^{-(E_1-E_0)T/\hbar}$$

times the corresponding ratio of endpoint wavefunctions. The spectral gap controls the projection rate.

Exercise 3: Source derivatives

For the source convention on this page, differentiate $Z[J]$ once and explain the factor needed to generate $\langle\phi(x)\rangle$.

Solution

Differentiating the exponent gives

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

Therefore the normalized expectation is

$$\langle\phi(x)\rangle_J = \frac{1}{Z[J]} \frac{\hbar}{i} \frac{\delta Z[J]}{\delta J(x)}.$$

Each additional source derivative contributes another factor $\hbar/i$ under the same convention.

Canonical Links

• Why Path Integrals
• Propagators to Path Integrals
• Time Slicing
• Free-Particle Path Integral
• Harmonic-Oscillator Path Integral
• Stationary Phase and Classical Limit
• Euclidean and Imaginary-Time Path Integrals
• Sources and Generating Functionals
• Correlation Functions from Path Integrals
• From QM to Field Path Integrals
• From Euclidean Time to Euclidean QFT
• Functional Derivatives
• Green Functions

Continue in Field Theory

Continue with Gaussian functional integrals, generating functionals, Wick’s theorem, Feynman rules, gauge fixing, fermionic Grassmann integrals, regularization, and renormalization at QFT.org.

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. Chaichian and A. Demichev, Path Integrals in Physics, Vol. I, Institute of Physics Publishing, 2001.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.