Euclidean and Imaginary-Time Path Integrals

Euclidean or imaginary-time quantum mechanics replaces the unitary real-time evolution operator

$$e^{-iHT/\hbar}$$

by the nonunitary damping operator

$$e^{-H\tau/\hbar},$$

where $\tau$ is imaginary time. In the path-integral language, this changes the oscillatory weight

$$e^{iS[x]/\hbar}$$

into the Euclidean weight

$$e^{-S_E[x]/\hbar}.$$

The change is not merely cosmetic. Real-time path integrals are amplitudes with interference; Euclidean path integrals are damping integrals that often behave more like statistical-mechanical partition sums. This is why imaginary time is central in ground-state projection, finite-temperature quantum mechanics, instanton estimates, lattice field theory, and the bridge from quantum mechanics to Euclidean QFT.

Imaginary Time owns the direct spectral-semigroup definition, heat-equation form, trace-class thermal use, and open-versus-closed operator dictionary. Path Integrals for Statistical Mechanics owns the closed thermal trace, cyclic discretization, permutation sectors, ring-polymer map, and periodic oscillator determinant. This page owns Wick rotation of kernels, the canonical projection derivation needed for the Euclidean construction, and the time-sliced construction of an open coordinate kernel.

Imaginary-Time Evolution

For a time-independent Hamiltonian, the real-time kernel is

$$K(x_f,T;x_i,0) = \langle x_f\rvert e^{-iHT/\hbar}\lvert x_i\rangle.$$

The formal Wick rotation

$$T=-i\tau, \qquad \tau\gt0,$$

suggests the imaginary-time kernel

$$K_E(x_f,\tau;x_i,0) = \langle x_f\rvert e^{-H\tau/\hbar}\lvert x_i\rangle.$$

The subscript $E$ stands for Euclidean, because the same analytic continuation turns Lorentzian time intervals into positive-definite Euclidean intervals in many field-theory settings. In ordinary nonrelativistic quantum mechanics, it is often enough to view $K_E$ as the coordinate-space kernel of the heat-type operator $e^{-H\tau/\hbar}$.

This operator is not unitary. It damps high-energy components faster than low-energy components. That is the essential mechanism behind imaginary-time projection.

Ground-State Projection

Let the Hamiltonian have a discrete spectrum bounded below:

$$H\lvert n\rangle=E_n\lvert n\rangle, \qquad E_0\leq E_1\leq E_2\leq\cdots.$$

For an initial state

$$\lvert\psi\rangle = \sum_n c_n\lvert n\rangle,$$

imaginary-time evolution gives

$$e^{-H\tau/\hbar}\lvert\psi\rangle = \sum_n c_n e^{-E_n\tau/\hbar}\lvert n\rangle.$$

If $c_0\neq 0$ and the ground state is nondegenerate, then after normalization the state approaches $\lvert 0\rangle$ as $\tau\to\infty$. More generally, it approaches the projection of $\lvert\psi\rangle$ onto the ground-state subspace.

It is often useful to factor out the ground energy:

$$e^{-H\tau/\hbar}\lvert\psi\rangle = e^{-E_0\tau/\hbar} \left[ c_0\lvert 0\rangle + \sum_{n\gt0}c_n e^{-(E_n-E_0)\tau/\hbar}\lvert n\rangle \right].$$

The overall factor $e^{-E_0\tau/\hbar}$ changes the norm but not the normalized state. Energy gaps control the rate of convergence.

Ground-state expectation values can be extracted by evolving on both sides of an operator:

$$\langle O\rangle_0 = \lim_{\tau\to\infty} \frac{ \langle\psi\rvert e^{-H\tau/(2\hbar)} O e^{-H\tau/(2\hbar)}\lvert\psi\rangle }{ \langle\psi\rvert e^{-H\tau/\hbar}\lvert\psi\rangle },$$

provided the trial state overlaps the ground state and the relevant limits exist. This formula is the operator version of many projector Monte Carlo and Euclidean path-integral methods.

Euclidean Action

For

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

the real-time action is

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

Set

$$t=-i\tau, \qquad dt=-i\,d\tau,$$

and write $x'(\tau)=dx/d\tau$. Since

$$\frac{dx}{dt} = i\,\frac{dx}{d\tau},$$

one obtains formally

$$S[x]=iS_E[x],$$

where

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

Therefore

$$e^{iS[x]/\hbar} \longrightarrow e^{-S_E[x]/\hbar}.$$

For potentials bounded below, the Euclidean weight suppresses paths with large Euclidean action. This is the basic reason Euclidean path integrals are often better behaved than real-time path integrals. It is not, by itself, a rigorous existence theorem: the measure, continuum limit, domains, and potential behavior still matter.

Euclidean Path Integral

For a particle with Hamiltonian

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

the imaginary-time kernel is formally

$$K_E(x_f,\tau_f;x_i,\tau_i) = \int_{x(\tau_i)=x_i}^{x(\tau_f)=x_f} \mathcal D x(\tau)\, e^{-S_E[x]/\hbar},$$

with

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

The time-sliced version has ordinary Gaussian damping factors rather than Fresnel phases. For a short Euclidean step $\epsilon_\tau$, the kinetic factor has the schematic form

$$\left( \frac{m}{2\pi\hbar\epsilon_\tau} \right)^{1/2} \exp\left[ - \frac{m(x_{j+1}-x_j)^2} {2\hbar\epsilon_\tau} \right],$$

with the potential contributing damping factors such as

$$\exp\left[ - \frac{\epsilon_\tau V(x_j)}{\hbar} \right].$$

Compare this with the real-time construction in Time Slicing, where the same short-time structure carries phases and normalization factors involving $i$.

Statistical-Mechanics Connection

The canonical thermal density operator is

$$\rho_\beta = \frac{e^{-\beta H}}{Z(\beta)}, \qquad Z(\beta)=\operatorname{Tr}e^{-\beta H}.$$

This has exactly the imaginary-time form if

$$\tau=\beta\hbar.$$

The density-matrix kernel is

$$\rho(x_f,x_i;\beta) = \frac{1}{Z(\beta)} \langle x_f\rvert e^{-\beta H}\lvert x_i\rangle = \frac{1}{Z(\beta)} K_E(x_f,\beta\hbar;x_i,0).$$

The partition function is the trace:

$$Z(\beta) = \int dx\, K_E(x,\beta\hbar;x,0).$$

In path-integral notation this becomes an integral over closed imaginary-time paths:

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

Thus finite temperature corresponds to a compact imaginary-time interval of length $\beta\hbar$. For ordinary coordinate path integrals, the trace imposes periodic boundary conditions. In coherent-state path integrals for fermions, antiperiodic imaginary-time boundary conditions appear; that belongs to the many-body and QFT continuation, not to the elementary coordinate kernel itself.

Harmonic Oscillator Example

The real-time harmonic-oscillator propagator has an exact analytic continuation. For $\tau\gt0$, the Euclidean kernel is

$$\begin{aligned} K_E(x_f,\tau;x_i,0) &= \left( \frac{m\omega} {2\pi\hbar\sinh\omega\tau} \right)^{1/2} \\ &\quad \times \exp\left[ - \frac{m\omega} {2\hbar\sinh\omega\tau} \left( (x_f^2+x_i^2)\cosh\omega\tau -2x_f x_i \right) \right]. \end{aligned}$$

This formula is useful because it shows both meanings of imaginary time. For small $\tau$, it behaves like a heat kernel. For large $\tau$, the spectral expansion gives

$$K_E(x_f,\tau;x_i,0) = \sum_{n=0}^{\infty} \psi_n(x_f)\psi_n^*(x_i) e^{-E_n\tau/\hbar},$$

so the leading term is

$$K_E(x_f,\tau;x_i,0) \sim \psi_0(x_f)\psi_0^*(x_i)e^{-\omega\tau/2}.$$

The Euclidean kernel therefore contains the ground-state wavefunction in its long-time endpoint dependence. This is the same projection mechanism described above, now seen directly in coordinate space.

The thermal oscillator partition function follows from the spectrum:

$$Z(\beta) = \sum_{n=0}^{\infty} e^{-\beta\hbar\omega(n+1/2)} = \frac{1}{2\sinh(\beta\hbar\omega/2)}.$$

The Euclidean trace of the oscillator kernel gives the same result.

Boundary Conditions

Different Euclidean objects use different boundary conditions:

Object	Operator expression	Euclidean boundary condition
Transition kernel	$\langle x_f\rvert e^{-H\tau/\hbar}\lvert x_i\rangle$	fixed endpoints $x(0)=x_i$, $x(\tau)=x_f$
Ground-state projection	$e^{-H\tau/\hbar}\lvert\psi\rangle$	endpoints weighted by the trial state or projected source data
Thermal trace	$\operatorname{Tr}e^{-\beta H}$	closed paths $x(\beta\hbar)=x(0)$
Thermal correlator	$\operatorname{Tr}(e^{-\beta H}T_\tau O(\tau)O(0))$	closed paths with operator insertions

The boundary condition is part of the definition of the Euclidean quantity. It determines whether the path integral computes a kernel, a ground-state matrix element, a partition function, or a correlation function.

Returning to Real Time

Euclidean methods are powerful, but they do not automatically answer every real-time question. Analytic continuation back to real time may be obstructed or ill-conditioned because singularities, branch cuts, and spectral data control the continuation.

For example, an imaginary-time correlation function can encode spectral information, but extracting a real-time response function requires the correct continuation and boundary prescription. The distinction is the same kind of care needed when comparing resolvents, retarded Green functions, and time-ordered propagators in Spectral Representation of Green Functions.

This is the core warning: Euclidean path integrals are not a replacement for real-time quantum mechanics. They are a different representation that is especially natural for ground states, thermal traces, tunneling estimates, and field-theory bridges.

From Euclidean Time to Euclidean QFT carries these operator results to field wavefunctionals, thermal circles, Matsubara modes, and Lorentzian reconstruction.

Common Mistakes

• Treating $e^{-S_E/\hbar}$ as a probability density without normalization, measure, and boundary conditions.
• Forgetting that $\tau=\beta\hbar$ in the thermal trace.
• Assuming every real-time expression can be Wick-rotated by symbol substitution.
• Confusing a fixed-endpoint Euclidean kernel with a thermal partition function.
• Ignoring ground-state degeneracy in imaginary-time projection.
• Using Euclidean damping arguments for Hamiltonians that are not bounded below.

Cross-Links

• Thermal Density Operators
• Time Slicing
• Path Integral Conventions
• Free-Particle Path Integral
• Harmonic Oscillator Propagator
• Liouville–von Neumann Equation
• Spectral Representation of Green Functions
• Barrier Penetration and Tunneling
• Euclidean Time and Imaginary-Time Action
• Instantons in Quantum Mechanics Preview
• Common Pitfalls in Path Integrals
• Why Dynamics Matters for QFT
• From Euclidean Time to Euclidean QFT
• Path Integrals
• Statistical Mechanics Checklist
• Imaginary-Time Projection Notebook

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.
• J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Addison-Wesley, 1988.

Exercises

1. Show explicitly how imaginary-time evolution projects onto the ground state for a two-level Hamiltonian with energies $E_0\lt E_1$.

Solution

Let

$$\lvert\psi\rangle = c_0\lvert 0\rangle+c_1\lvert 1\rangle.$$

Then

$$e^{-H\tau/\hbar}\lvert\psi\rangle = e^{-E_0\tau/\hbar} \left[ c_0\lvert 0\rangle +c_1e^{-(E_1-E_0)\tau/\hbar}\lvert 1\rangle \right].$$

If $c_0\neq 0$, the relative excited-state amplitude is

$$\frac{c_1}{c_0} e^{-(E_1-E_0)\tau/\hbar},$$

which tends to zero as $\tau\to\infty$. After normalization, the state tends to $\lvert 0\rangle$ up to the phase of $c_0$.

2. Derive the Euclidean action for $L=m\dot x^2/2-V(x)$ using $t=-i\tau$.

Solution

Since $dt=-i\,d\tau$ and $dx/dt=i\,dx/d\tau$,

$$\begin{aligned} S &= \int dt\, \left[ \frac{m}{2} \left( \frac{dx}{dt} \right)^2 -V(x) \right] \\ &= -i\int d\tau\, \left[ - \frac{m}{2} \left( \frac{dx}{d\tau} \right)^2 -V(x) \right] \\ &= i\int d\tau\, \left[ \frac{m}{2} \left( \frac{dx}{d\tau} \right)^2 +V(x) \right]. \end{aligned}$$

Thus $S=iS_E$, with

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

Therefore $e^{iS/\hbar}=e^{-S_E/\hbar}$.

3. Use the spectral representation of the Euclidean kernel to explain why large imaginary time reveals the ground-state wavefunction.

Solution

Insert energy eigenstates:

$$K_E(x_f,\tau;x_i,0) = \sum_n \langle x_f\vert n\rangle \langle n\vert x_i\rangle e^{-E_n\tau/\hbar}.$$

Writing $\psi_n(x)=\langle x\vert n\rangle$ gives

$$K_E(x_f,\tau;x_i,0) = \sum_n \psi_n(x_f)\psi_n^*(x_i) e^{-E_n\tau/\hbar}.$$

Factoring out $e^{-E_0\tau/\hbar}$ leaves excited terms suppressed by

$$e^{-(E_n-E_0)\tau/\hbar}.$$

If the ground state is nondegenerate, the endpoint dependence at large $\tau$ is proportional to $\psi_0(x_f)\psi_0^*(x_i)$.

4. Why does the finite-temperature trace impose closed imaginary-time paths?

Solution

The trace of an operator is obtained by integrating its diagonal coordinate kernel:

$$\operatorname{Tr}e^{-\beta H} = \int dx\, \langle x\rvert e^{-\beta H}\lvert x\rangle.$$

Since $e^{-\beta H}=e^{-H(\beta\hbar)/\hbar}$, the diagonal kernel is a Euclidean kernel whose initial and final coordinates are the same:

$$x(0)=x(\beta\hbar).$$

In the path integral, summing over the diagonal coordinate $x$ integrates over all paths closed in imaginary time.