Euclidean Time and Imaginary-Time Action

Euclidean time turns selected quantum amplitudes into damping problems. For a time-independent Hamiltonian, the basic operator is

$$e^{-\hat H T/\hbar}, \qquad T\gt0,$$

and for a particle with real-time Lagrangian

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

the corresponding Euclidean action is

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

Here a dot denotes $d/dt$, a prime denotes $d/d\tau$, and the contour convention is $t=-i\tau$. The positive sign of the Euclidean kinetic term is essential.

The tunneling questions are which boundary conditions select a saddle and whether its exponent describes an amplitude, a splitting, or a decay rate.

Required background. Euclidean path integrals gives the kernel and thermal trace; classical action and phase introduces stationary action; energy eigenstates supplies the spectral expansion.

The Rotation t → −iτ

Real-time evolution is generated by

$$U(t) = e^{-i\hat Ht/\hbar}.$$

At the imaginary-time endpoint $t=-iT$,

$$U(-iT) = e^{-\hat H T/\hbar}.$$

A useful way to picture the continuation is to rotate the time contour continuously,

$$t = e^{-i\theta}s, \qquad 0\leq\theta\leq\frac{\pi}{2}, \qquad s\geq0.$$

The Lorentzian contour is $\theta=0$; the Euclidean contour is $\theta=\pi/2$. The word Wick rotation refers to this contour deformation, not merely to replacing one symbol by another.

If $\hat H$ is self-adjoint and bounded below, define $E_0=\inf\sigma(\hat H)$. Then the operator $e^{-\hat H T/\hbar}$ is well defined for $T\gt0$, with

$$\left\| e^{-\hat H T/\hbar} \right\| = e^{-E_0T/\hbar}.$$

The overall norm can grow when $E_0\lt0$, but high-energy components are still damped relative to low-energy components. Shifting $\hat H$ by a constant changes only the common normalization factor.

Three statements that must be separated

The following statements are related, but they are not interchangeable:

1. Operator continuation: define the semigroup $e^{-\hat H T/\hbar}$ by spectral calculus.
2. Kernel continuation: analytically continue a real-time kernel in a domain free of obstructing singularities.
3. Path-integral deformation: deform an integration cycle so that $e^{iS_M/\hbar}$ becomes a convergent or asymptotic Euclidean integral.

The first can be meaningful even when no simple real-time path integral is available. The third requires information about integration contours and singularities that is not contained in the symbolic substitution $t=-i\tau$.

From Lorentzian to Euclidean Action

Start with

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

Under

$$t=-i\tau, \qquad dt=-i\,d\tau, \qquad \dot x=i x',$$

the action becomes

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

Therefore

$$\frac{i}{\hbar}S_M = -\frac{1}{\hbar}S_E, \qquad e^{iS_M/\hbar} \longrightarrow e^{-S_E/\hbar}.$$

For several coordinates $q^a$ with a positive configuration-space metric $G_{ab}(q)$,

$$L_M = \frac12 G_{ab}(q)\dot q^a\dot q^b - V(q)$$

continues to

$$S_E[q] = \int d\tau \left[ \frac12 G_{ab}(q)q^{\prime a}q^{\prime b} + V(q) \right].$$

Positivity of the kinetic form matters. A nonstandard kinetic term, constrained system, or momentum-dependent interaction requires its own continuation rather than this template.

The time contour rotates from the positive real axis to $t=-iT$. The Euclidean equation of motion is equivalent to ordinary motion in the inverted potential $-U(x)$, but the Euclidean saddle is not a hidden real-time trajectory.

For a position-dependent metric, this is the classical action. Defining the quantum kernel additionally requires a configuration-space measure and an operator-ordering prescription; possible curvature terms are not fixed by the classical Wick rotation alone.

Checking the signs

Three quick checks catch most sign errors:

• The free-particle Euclidean action must contain $+m(x')^2/2$.
• A stable harmonic potential $V=m\omega^2x^2/2$ must contribute $+m\omega^2x^2/2$.
• Varying $S_E$ must give $mx''=V'(x)$, which is motion in the inverted potential $-V$.

The potential is inverted in the equation-of-motion analogy, not in $S_E$. Writing $S_E$ with $-V$ would destroy the damping structure of the stable oscillator.

Potential shifts and background subtraction

Suppose a reference configuration $x_{\mathrm{ref}}$ has energy

$$V_{\mathrm{ref}} = V(x_{\mathrm{ref}}).$$

Define

$$U(x) = V(x)-V_{\mathrm{ref}}.$$

Then

$$S_E[x] = V_{\mathrm{ref}}T + \int d\tau \left[ \frac{m}{2}(x')^2 + U(x) \right].$$

The constant $V_{\mathrm{ref}}T$ matters for an absolute kernel, but it cancels from normalized ratios and from a saddle action measured relative to the reference background. For tunneling on an infinite Euclidean interval, the finite quantity is normally

$$\begin{aligned} B[x] &\equiv S_E[x]-S_E[x_{\mathrm{ref}}] \\ &= \int_{-\infty}^{\infty} d\tau \left[ \frac{m}{2}(x')^2 + U(x) \right]. \end{aligned}$$

Failing to subtract the background can turn a finite tunneling exponent into an irrelevant infinite vacuum contribution.

Velocity-linear terms

Euclidean actions need not be real. For example, in several dimensions,

$$L_M = \frac{m}{2}\dot{\mathbf x}^{\,2} + q\mathbf A(\mathbf x)\cdot\dot{\mathbf x} - V(\mathbf x)$$

gives

$$S_E = \int d\tau \left[ \frac{m}{2}(\mathbf x')^2 + V(\mathbf x) \right] - iq\int d\tau\, \mathbf A(\mathbf x)\cdot\mathbf x'.$$

The magnetic line integral becomes an imaginary part of $S_E$. The real part can still damp paths, while the imaginary part carries phase information. Thus “Euclidean” does not always mean “positive real weight.”

Imaginary-Time Schrödinger Evolution

The real-time Schrödinger equation is

$$i\hbar \frac{\partial}{\partial t} \vert\psi(t)\rangle = \hat H\vert\psi(t)\rangle.$$

Since $t=-i\tau$ implies $\partial_t=i\partial_\tau$, imaginary-time evolution obeys

$$\hbar \frac{\partial}{\partial\tau} \vert\psi(\tau)\rangle = -\hat H\vert\psi(\tau)\rangle.$$

For

$$\hat H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x),$$

the wavefunction satisfies the diffusion-reaction equation

$$\hbar \frac{\partial\psi}{\partial\tau} = \frac{\hbar^2}{2m} \frac{\partial^2\psi}{\partial x^2} - V(x)\psi.$$

This equation is parabolic rather than unitary. It smooths short-wavelength structure while damping regions with positive potential relative to the chosen energy reference. The connection to Wiener measure and diffusion is formalized, under suitable hypotheses, by the Feynman–Kac formula.

Normalized imaginary-time flow

The unnormalized norm is not conserved. If the state is normalized after each Euclidean-time step, write it as $\vert\phi(\tau)\rangle$. Its evolution is

$$\hbar \frac{\partial}{\partial\tau} \vert\phi\rangle = - \left( \hat H-\langle\hat H\rangle_\phi \right) \vert\phi\rangle.$$

For a time-independent self-adjoint Hamiltonian and a normalized state in $D(\hat H)$ with finite energy variance,

$$\frac{d}{d\tau} \langle\hat H\rangle_\phi = - \frac{2}{\hbar} \left[ \langle\hat H^2\rangle_\phi - \langle\hat H\rangle_\phi^2 \right] \leq0.$$

Normalized imaginary-time evolution therefore lowers the energy expectation monotonically, stopping only when the state has support inside an energy eigenspace. This is a computational descent principle, not physical dissipative time evolution.

Ground-State Projection and Its Scales

Let

$$\hat H\vert n\rangle = E_n\vert n\rangle, \qquad E_0\leq E_1\leq\cdots,$$

and

$$\vert\psi\rangle = \sum_n c_n\vert n\rangle.$$

Then

$$e^{-\hat H T/\hbar}\vert\psi\rangle = e^{-E_0T/\hbar} \Bigg[ c_0\vert0\rangle + \sum_{n\gt0} c_ne^{-(E_n-E_0)T/\hbar} \vert n\rangle \Bigg].$$

If $c_0\ne0$ and the ground state is nondegenerate, normalization leaves $\vert0\rangle$ as $T\to\infty$. If the ground space is degenerate, the limit is the normalized projection into that subspace.

The relative contamination of level $n$ is

$$r_n(T) = \frac{c_n}{c_0} e^{-(E_n-E_0)T/\hbar}.$$

To make the first excited contribution smaller than a target $\delta$, it is sufficient that

$$T \geq \frac{\hbar}{E_1-E_0} \log\left( \frac{|c_1|} {\delta|c_0|} \right),$$

when the logarithm is positive. Projection time is controlled by the gap and by trial-state overlap, not by a universal number of time steps.

The double-well hierarchy

In a deep symmetric double well, the tunneling splitting $\Delta_{\mathrm{tun}}$ can be exponentially smaller than the intrawell excitation scale $\hbar\omega$. Two Euclidean-time requirements are then parametrically different:

$$T \gg \frac{1}{\omega}$$

suppresses higher intrawell excitations, whereas

$$T \gg \frac{\hbar}{\Delta_{\mathrm{tun}}}$$

is needed to isolate one parity eigenstate from the low-energy doublet.

The intermediate window

$$\frac{1}{\omega} \ll T \ll \frac{\hbar}{\Delta_{\mathrm{tun}}}$$

projects efficiently onto the two-dimensional low-energy subspace without resolving its exponentially small splitting. This distinction is central in numerical tunneling calculations.

Tunneling Splittings turns this hierarchy into a practical Euclidean-kernel estimator for the doublet gap.

Extracting the ground energy

For boundary states with nonzero ground-state overlap, define

$$Z_{\chi\psi}(T) = \langle\chi\vert e^{-\hat H T/\hbar} \vert\psi\rangle.$$

At large $T$,

$$E_0 = - \hbar \frac{d}{dT} \log Z_{\chi\psi}(T) + O\left( (E_1-E_0) e^{-(E_1-E_0)T/\hbar} \right),$$

provided the leading coefficient does not vanish through symmetry or cancellation. Ratios at nearby $T$ values are often more stable than the exponentially small kernel itself.

Varying the Euclidean Action

For the background-subtracted action

$$B[x] = \int_{\tau_i}^{\tau_f} d\tau \left[ \frac{m}{2}(x')^2 + U(x) \right],$$

the first variation is

$$\delta B = \left[ mx'\delta x \right]_{\tau_i}^{\tau_f} + \int_{\tau_i}^{\tau_f} d\tau \left[ -mx'' + U'(x) \right]\delta x.$$

For fixed endpoints, $\delta x(\tau_i)=\delta x(\tau_f)=0$, so stationarity gives

$$mx'' = U'(x) = V'(x).$$

This is Newton’s equation in the inverted potential $-U(x)$:

$$mx'' = - \frac{d}{dx} \left[ -U(x) \right].$$

The analogy helps solve the boundary-value problem. It does not mean that the quantum particle follows $x(\tau)$ in real time.

Boundary terms are physical data

Different observables eliminate or modify the endpoint term in different ways:

Euclidean object	Endpoint data
Fixed-endpoint kernel	$\delta x(\tau_i)=\delta x(\tau_f)=0$
Thermal trace	$x(\tau_f)=x(\tau_i)$ and matching Euclidean momentum
Instanton	endpoints approach distinct degenerate minima as $\tau\to\pm\infty$
Bounce	both ends approach the same metastable minimum
Wavefunction-weighted endpoint	boundary variation is balanced by the logarithmic derivative of the endpoint wavefunction

Choosing a saddle before choosing these data reverses the logic. The boundary conditions define the observable; only then can one identify the relevant stationary configurations.

Conserved Euclidean energy

If $U$ has no explicit $\tau$ dependence, the Euclidean Noether energy is

$$\mathcal E_E = x' \frac{\partial L_E}{\partial x'} - L_E = \frac{m}{2}(x')^2 - U(x).$$

It is constant along a saddle:

$$\frac{d\mathcal E_E}{d\tau} = 0.$$

For a finite-action path approaching a reference minimum with

$$U(x_{\mathrm{ref}})=0, \qquad x'\to0,$$

one has $\mathcal E_E=0$. Therefore

$$\frac{m}{2}(x')^2 = U(x).$$

For a monotone path between degenerate minima,

$$\begin{aligned} B &= \int d\tau \left[ \frac{m}{2}(x')^2 + U(x) \right] \\ &= \int_{x_-}^{x_+} dx\, \sqrt{2mU(x)}. \end{aligned}$$

This action identity is the bridge to the WKB forbidden-region exponent. The explicit instanton profile, collective coordinate, and determinant belong to Instantons in Quantum Mechanics.

Exponential approach to a minimum

Near a nondegenerate minimum $x_\star$, write $x=x_\star+\eta$ and

$$U(x) = \frac{m\omega_\star^2}{2}\eta^2 + O(\eta^3).$$

The linearized Euclidean equation is

$$\eta'' = \omega_\star^2\eta.$$

A finite-action tail must select the decaying branch,

$$\eta(\tau) \propto e^{-\omega_\star|\tau-\tau_0|}.$$

This tail supplies practical boundary data for numerical shooting and explains why truncation errors on a finite Euclidean interval are exponentially small in the endpoint distance from the saddle center.

Fixed-Time and Fixed-Energy Exponents

A common source of factor and sign errors is to confuse a kernel at fixed Euclidean duration with a tunneling amplitude at fixed energy.

At fixed $T$, a saddle contributes schematically

$$K_E(T) \sim A(T) e^{-S_E(T)/\hbar}.$$

Define the resolvent kernel, away from the spectrum, by

$$G_E(E;x_f,x_i) \equiv \langle x_f\vert (\hat H-E)^{-1} \vert x_i\rangle.$$

The following Laplace representation converges for $\operatorname{Re}E<\inf\sigma(\hat H)$:

$$G_E(E;x_f,x_i) = \frac{1}{\hbar} \int_0^\infty dT \times e^{ET/\hbar} K_E(x_f,T;x_i,0).$$

At a bound-state energy the integral generally diverges; extracting poles or resonances requires analytic continuation and the appropriate boundary prescription. The saddle argument below identifies the local forbidden-region action, and does not justify evaluating the uncontinued integral at such an energy. Its exponent is

$$S_E(T)-ET.$$

Stationarity with respect to $T$ selects a trajectory with Euclidean energy

$$\mathcal E_E = -E,$$

so

$$\frac{m}{2}(x')^2 = V(x)-E.$$

The fixed-energy abbreviated action is the Legendre transform

$$W(E) = S_E(T)-ET.$$

Between turning points $x_1(E)$ and $x_2(E)$,

$$\begin{aligned} W(E) &= \int d\tau \left[ \frac{m}{2}(x')^2 + V(x)-E \right] \\ &= \int_{x_1(E)}^{x_2(E)} dx\, \sqrt{2m[V(x)-E]}. \end{aligned}$$

Thus the forbidden-region wavefunction amplitude carries

$$e^{-W(E)/\hbar},$$

while a leading one-dimensional transmission probability carries

$$e^{-2W(E)/\hbar}.$$

The factor of two distinguishes an amplitude exponent from a probability exponent. Barrier Penetration and Tunneling derives the connection formulas and transmission probability.

When Wick Rotation Is Not Automatic

The formal continuation can fail or become incomplete for several reasons.

Singularities and contour obstructions

A kernel or integrand may have poles, branch cuts, or essential growth in the region swept out by the contour. Deforming across a singularity changes the answer by residues or discontinuities. The correct real-time boundary prescription must be carried through the deformation.

Hamiltonians not bounded below

If the spectrum extends to $-\infty$, $e^{-\hat H T/\hbar}$ amplifies increasingly negative energies and need not define a bounded semigroup. A Euclidean kinetic term that looks positive does not repair an unstable quantum Hamiltonian.

Complex Euclidean actions

Magnetic terms, Berry phases, chemical potentials, real-time sources, and complex saddles can make $S_E$ complex. The integral then retains a phase problem and may require a deformed integration cycle.

Explicit time dependence

For $V(x,t)$, the continuation produces $V(x,-i\tau)$. Analyticity and the chosen contour become part of the problem, and the Euclidean action may be complex. This local substitution alone does not generate a nonlocal action; nonlocality can arise after eliminating other degrees of freedom.

Real-time observables

Euclidean correlators can encode spectra, but recovering retarded response, transport, or late-time oscillations requires analytic continuation with the correct boundary values. Numerically, that inverse problem can be severely ill-conditioned.

Metastability

A metastable state is not an exact normalizable energy eigenstate with a real discrete eigenvalue. Its decay rate is extracted through analytic continuation, resonance boundary conditions, and the negative mode of a bounce. False Vacuum Decay in Quantum Mechanics develops the corresponding resonance width, survival probability, and nonexponential time regimes. Simple ground-state projection does not by itself produce a decay law.

Open and non-Hermitian dynamics

A Lindblad generator or non-Hermitian effective Hamiltonian does not generally become an equilibrium Euclidean action under $t=-i\tau$. Its spectrum, left and right eigenvectors, and contour structure must be treated on their own terms.

Bridge to Euclidean QFT

For a real scalar field in $d$ spatial dimensions,

$$\begin{aligned} S_M[\phi] &= \int dt\,d^d x\, \mathcal L_M, \\ S_E[\phi] &= \int d\tau\,d^d x\, \mathcal L_E. \end{aligned}$$

The Lorentzian and Euclidean densities are

$$\begin{aligned} \mathcal L_M &= \frac12(\partial_t\phi)^2 - \frac12(\boldsymbol\nabla\phi)^2 - V(\phi),\\ \mathcal L_E &= \frac12(\partial_\tau\phi)^2 + \frac12(\boldsymbol\nabla\phi)^2 + V(\phi). \end{aligned}$$

The Euclidean field equation is

$$- \partial_\tau^2\phi - \boldsymbol\nabla^2\phi + V'(\phi) = 0.$$

Quantum mechanics is the $0+1$-dimensional case: there is no spatial gradient term, and the field configuration on a time slice reduces to one or finitely many coordinates. In field theory, the saddle is a function of Euclidean spacetime, its finite-action boundary conditions are imposed at spatial and temporal infinity, and fluctuations are differential operators in $d+1$ dimensions.

The translation is structural, not automatic. Gauge fixing, topology, renormalization, fermion zero modes, reflection positivity, and Lorentzian reconstruction add genuinely field-theoretic issues. From Euclidean Time to Euclidean QFT explains vacuum functionals, thermal circles, Matsubara frequencies, and reconstruction. Bridge to QFT Instantons focuses on how finite-action quantum-mechanical saddles generalize to fields.

Exercises

1. Resolve a doublet, or only its subspace

A deep double well has tunneling splitting $\Delta_{\mathrm{tun}}$ and an intrawell gap approximately $\hbar\omega$, with $\Delta_{\mathrm{tun}}\ll\hbar\omega$.

1. Give a Euclidean-time regime that suppresses intrawell excitations but does not resolve the doublet.
2. Give the regime needed to isolate the unique parity ground state.
3. Explain what happens if the trial state has odd parity.

Solution

Intrawell excitations are suppressed by factors of order $e^{-\omega T}$, while the unwanted partner in the doublet is suppressed only by

$$e^{-\Delta_{\mathrm{tun}}T/\hbar}.$$

Therefore

$$\frac{1}{\omega} \ll T \ll \frac{\hbar}{\Delta_{\mathrm{tun}}}$$

projects onto the low-energy doublet without selecting one member. Isolating the even ground state requires

$$T \gg \frac{\hbar}{\Delta_{\mathrm{tun}}}.$$

If the Hamiltonian preserves parity and the trial state is purely odd, its overlap with the even ground state vanishes. Imaginary-time evolution preserves the odd sector and projects onto the lowest odd state instead.

2. Test the resolvent with a free particle

For $E<0$ and $\kappa=\sqrt{-2mE}$, use

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

to identify the stationary duration in the Laplace integral and its action. Check the exact resolvent normalization independently from its derivative jump.

Solution

For $d=|x_f-x_i|>0$, minimize $md^2/(2T)+|E|T$. The stationary duration is $T_*=md/\kappa$ and the exponent is $\kappa d/\hbar$. The exact result is

$$G_E(E;x_f,x_i)=\frac{m}{\hbar\kappa}e^{-\kappa d/\hbar}.$$

Away from $x_f=x_i$ it solves $(H-E)G_E=0$. Its first derivative jumps by $-2m/\hbar^2$, giving the delta function required by $(H-E)G_E=\delta(x_f-x_i)$. The energy condition $E<0$ is also the convergence condition of the original Laplace integral.

References

• S. Coleman, “The Uses of Instantons”, in Aspects of Symmetry, Cambridge University Press, 1985, pp. 265–350. Canonical pedagogical treatment of Euclidean saddles and tunneling.
• S. Coleman, “Fate of the False Vacuum: Semiclassical Theory”, Physical Review D 15, 2929–2936 (1977), with erratum in 16, 1248 (1977). Canonical Euclidean bounce framework and a clear example of why contour and negative-mode information matter.
• R. P. Feynman, “Space-Time Approach to Non-Relativistic Quantum Mechanics”, Reviews of Modern Physics 20, 367–387 (1948). Original path-integral formulation and action-based quantum evolution.
• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, emended ed., Dover, 2010. Standard path-integral treatment and imaginary-time continuation.
• M. Kac, “On Distributions of Certain Wiener Functionals”, Transactions of the American Mathematical Society 65, 1–13 (1949). Foundational relation between diffusion semigroups and exponentially weighted path functionals.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009. Broad reference on Euclidean kernels, fluctuations, and semiclassical approximation.
• R. Rajaraman, Solitons and Instantons, North-Holland, 1982. Finite-action solutions, boundary conditions, and fluctuation spectra.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981. Detailed treatment of propagators, continuation, and semiclassical methods.
• B. Simon, Functional Integration and Quantum Physics, 2nd ed., AMS Chelsea, 2005. Mathematical treatment of Euclidean functional integration and Schrödinger semigroups.
• J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005. Systematic account of Euclidean quantum mechanics, saddle expansions, and instantons.