Euclidean path integral ($Z=\int \mathcal{D}\varphi e^{-S_E[\varphi ]}$)

A path integral with weight $e^{-S_E}$ uses the Euclidean action obtained by Wick rotation. Bosonic convergence requires appropriate positivity and boundary conditions.

For positive mass $m$ and angular frequency $\omega$, the Euclidean action $S_E=\frac{m}{2}{\int }_0^{\mathcal{T}}({\dot{x}}^2+{\omega }^2{x}^2)d\tau$ with $x(0)=x(\mathcal{T})=x$ has the unique minimizing path displayed. The Euler-Lagrange equation is ${\ddot{x}}_0={\omega }^2{x}_0$, and the positive second variation proves uniqueness. On this path the integrand is $\frac{m}{2}d(x_0{\dot{x}}_0)/d\tau$, giving $S_E[x_0]=m\omega x^2\tanh (\omega \mathcal{T}/2)$. Increasing the interval of imaginary time lets its interior approach the minimum of the potential energy even while both endpoints remain fixed.

A local $d$-dimensional quantum field at finite temperature can often be represented by an anisotropic classical-looking action in $d+1$ dimensions, with a finite extra imaginary-time extent. The zero-temperature limit makes that extent infinite. A real nonnegative statistical weight is an additional requirement: fermionic signs and imaginary terms can obstruct interpretation as an ordinary classical probability distribution.

Taking the trace of the position-space imaginary-time kernel identifies its endpoints and integrates over the common position. The resulting closed paths have imaginary-time length $\beta \hslash$. Bosonic coherent fields are periodic, while fermionic coherent fields are antiperiodic. This temporal condition follows from the trace, independently of the spatial boundary conditions.

For $H=p^2/(2m)+V(x)$ and a smooth potential on scales larger than the thermal de Broglie wavelength ${\lambda }_T=\sqrt{2\pi \beta {\hslash }^2/m}$, the diagonal thermal kernel approaches $⟨x|e^{-\beta H}|x⟩\simeq {\lambda }_T^{-d}{e}^{-\beta V(x)}$. Integration over its diagonal gives the classical phase space integral with one factor $(2\pi \hslash {)}^{-1}$ per coordinate-momentum pair. Exchange and boundary corrections must be small; a positive imaginary-time weight by itself does not make an arbitrary quantum system classical.

An instanton is a localized finite-action classical solution in Euclidean spacetime, used in a semiclassical description of quantum tunnelling and other nonperturbative effects. In one-dimensional quantum mechanics it can connect distinct degenerate potential minima over infinite Euclidean time. A quartic double-well instanton is an elementary example; a Yang-Mills instanton is a gauge-field example with different geometric structure.

A Euclidean bounce is a finite-action Euclidean saddle associated with decay of a false vacuum. It returns to the same metastable vacuum at the two ends of Euclidean time, instead of joining two degenerate stable vacua. Its characteristic negative fluctuation direction produces the imaginary part associated with a decay rate; the leading semiclassical suppression is $e^{-S_B/\hslash }$. Translation zero modes are treated as collective coordinates. It should not be confused with the stable double-well tunnelling instanton whose repeated crossings give a real level splitting.

The orientation-reversed Euclidean trajectory of an instanton. In a symmetric double well it goes from the right vacuum to the left vacuum and has the same positive action as the forward crossing. Its topological orientation changes sign, not its Euclidean action.

When individual instantons are much narrower than their typical separation, multi-crossing paths factor at leading semiclassical order. Integrating $n$ ordered centers over time $\tau$ contributes ${\tau }^n/n!$. In a double well, crossings alternate direction and even/odd counts yield hyperbolic cosine/sine amplitudes. This approximation requires large action relative to $\hslash$, matched local well structure and control of corrections from interactions and local anharmonicity.

For a one-dimensional quantum tunnelling instanton, the prefactor combines nonzero Gaussian fluctuation eigenvalues with the translation zero-mode Jacobian. Here $L_I=-{\partial }_t^2+V^{\prime \prime }(q_I)/m$, $L_0=-{\partial }_t^2+{\omega }^2$, and the prime excludes the zero mode. Common endpoint regularization and reference normalization define the determinant ratio. The resulting $K$ has frequency units, while $K{e}^{-S_I/\hslash }$ is the dilute crossing rate. This determinant/collective-coordinate structure is also discussed in Dunne, Sulejmanpasic and Ünsal's treatment; unequal neighbouring well frequencies require care and need not share the symmetric-double-well result.

Differentiating the Euclidean classical equation for an instanton gives a zero eigenfunction proportional to its time derivative. Its norm satisfies $\mu \int {\dot{q}}_I^{2}d\tau =S_I$, so replacing the Gaussian zero-mode integral by the center coordinate gives $\sqrt{S_I/(2\pi \hslash )}d{\tau }_0$. The nonzero modes belong in the primed functional determinant. This removes the false divergence without deleting the center's integration volume.

For $V(q)=m{\omega }^2(q^2-a^2{)}^2/(8a^2)$, the zero Euclidean-energy condition is $m{\dot{q}}^2/2=V(q)$. Integrating gives the displayed instanton from $-a$ to $a$. Its action is ${\int }_{-a}^a\sqrt{2mV(q)}dq=2m\omega a^2/3$. The arbitrary center $t_0$ supplies a translation zero mode, treated as a collective coordinate in the instanton fluctuation prefactor.

Return amplitudes sum even numbers of well-to-well events, while transfer amplitudes sum odd numbers. A dilute event gas gives cosh and sinh factors and a doublet splitting $2\hslash A{e}^{-S_0/\hslash }$.

The quartic double-well instanton has $L_I=-{\partial }_{\tau }^2+{\omega }^2[1-3{\mathrm{sech}}^2(\omega \tau /2)/2]$. For a positive regulator $\epsilon$ and $r=2\sqrt{1+\epsilon /{\omega }^2}$, the common infinite-line determinant ratio is $(r-1)(r-2)/[(r+1)(r+2)]$. Its zero-mode factor is $\epsilon /(12{\omega }^2)$ to leading order, giving the displayed primed ratio. Therefore the instanton fluctuation prefactor is $K=\omega \sqrt{6S_I/(\pi \hslash )}$ and the double-well tunneling splitting is $2\hslash K{e}^{-S_I/\hslash }$.