Fluctuation Determinants Preview

Fluctuation determinants are the prefactors that multiply the leading instanton exponential. The instanton action gives the dominant factor

$$e^{-S_0/\hbar},$$

but a quantitative tunneling splitting or decay rate also depends on the Gaussian integral over small fluctuations around the Euclidean saddle.

Required background. Instantons supplies the saddle and translation mode; Sturm–Liouville theory supplies the fluctuation spectrum; semiclassical propagators provides the Gaussian-saddle interpretation. The discussion below treats one real coordinate with constant positive mass.

Quadratic Expansion

Let $x_{\mathrm{cl}}(\tau)$ be a Euclidean saddle point of

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

Write a nearby path as

$$x(\tau) = x_{\mathrm{cl}}(\tau)+\eta(\tau).$$

With fixed endpoints, impose $\eta=0$ at both ends. The linear term vanishes by the saddle equation and these boundary conditions; the equation of motion alone does not remove boundary variations. The action expands as

$$S_E[x_{\mathrm{cl}}+\eta] = S_E[x_{\mathrm{cl}}] + \frac12 \int d\tau\, \eta(\tau) \mathcal M \eta(\tau) + \cdots,$$

where the fluctuation operator is

$$\mathcal M = - m\frac{d^2}{d\tau^2} + V''(x_{\mathrm{cl}}(\tau)).$$

The omitted terms are cubic and higher in $\eta$.

Gaussian Integral

Formally, the Gaussian path integral over fluctuations gives

$$\int\mathcal D\eta\, \exp\left[ - \frac{1}{2\hbar} \int d\tau\, \eta\mathcal M\eta \right] \propto \left(\det\mathcal M\right)^{-1/2}.$$

This is the origin of the determinant prefactor. For a tunneling quantity, one usually needs a ratio such as

$$\left[ \frac{\det\mathcal M_0} {\det\mathcal M} \right]^{1/2},$$

where $\mathcal M_0$ is the fluctuation operator around the reference vacuum or harmonic minimum. Ratios remove many normalization factors that are common to the numerator and denominator.

The determinant here is not a finite matrix determinant. It is a determinant of a differential operator, defined through a regulated product of eigenvalues or an equivalent method.

Spectral Meaning

If

$$\mathcal M\eta_n = \lambda_n\eta_n,$$

with suitable boundary conditions, then formally

$$\det\mathcal M \sim \prod_n\lambda_n.$$

This product is infinite and must be regularized. In one-dimensional quantum mechanics, determinant ratios can often be computed by spectral comparison or by the Gel’fand–Yaglom method, which replaces a determinant ratio by a solution of an associated ordinary differential equation.

For $\mathcal L=-d^2/d\tau^2+U(\tau)$ and $\mathcal L_0=-d^2/d\tau^2+U_0(\tau)$ on the same finite interval $[0,T]$, with Dirichlet endpoints and neither operator having a zero eigenvalue, the Gel’fand–Yaglom ratio is

$$\frac{\det\mathcal L}{\det\mathcal L_0}=\frac{y(T)}{y_0(T)}, \qquad \mathcal L y=0,\quad y(0)=0,\quad y'(0)=1,$$

with the analogous initial-value problem for $y_0$. The shared normalization cancels. This simple formula changes with the boundary conditions; it must not be copied directly to periodic paths or to a primed zero-mode determinant. Dunne’s review gives extensions.

Zero Modes

An instanton in an infinite Euclidean time interval can be translated without changing its action. If

$$x_{\mathrm{inst}}(\tau-\tau_0)$$

is a one-parameter family of saddles, then differentiating with respect to $\tau_0$ gives a zero mode:

$$\eta_0(\tau) \propto \frac{dx_{\mathrm{inst}}}{d\tau}.$$

Therefore

$$\mathcal M\eta_0=0.$$

An ordinary determinant would then vanish. The correct procedure is not to set the instanton contribution to infinity or zero. Instead, one removes the zero eigenvalue from the determinant,

$$\det{}'\mathcal M,$$

and integrates over the collective coordinate $\tau_0$.

The mass convention matters once an eigenvalue is omitted. With the displacement norm $\int\eta_n^2d\tau=1$ used for $\mathcal M$, the zero-mode Jacobian is $\sqrt{S_0/(2\pi m\hbar)}\,d\tau_0$. Equivalently set $\xi=\sqrt m\,\eta$ and $\mathcal L=\mathcal M/m$. In this mass-normalized convention the one-event factor is

$$\sqrt{\frac{S_0}{2\pi\hbar}} \left(\frac{\det\mathcal L_0}{\det{}'\mathcal L}\right)^{1/2} d\tau_0\,e^{-S_0/\hbar}.$$

The determinant ratio now has units of inverse time squared, because the denominator lacks one eigenvalue. Its square root supplies the inverse time needed to make the event factor dimensionless. This fixes a normalization that a merely schematic determinant cannot determine.

On a large finite interval, fixed endpoints usually lift translation invariance. One must remove the mode tending to zero in a controlled large-interval limit, not discard an arbitrary small numerical eigenvalue.

Negative Modes

Some Euclidean saddles have one negative fluctuation eigenvalue. This is typical for a bounce describing decay of a metastable state. A negative mode means the saddle is not a minimum of the Euclidean action; it is a saddle in the ordinary sense. Bounce Solutions derives the classical saddle and uses the Sturm–Liouville node theorem to identify this mode.

In false-vacuum decay, the negative direction requires a contour prescription or analytic continuation of the metastable amplitude. It is this prescribed continuation that produces the imaginary resonance energy; a negative determinant by itself is not a definition of a decay rate. False Vacuum Decay in Quantum Mechanics fixes the energy-width convention and explains when that rate controls a survival law. This is different from the double-well instanton, where the central issue is a zero mode from translation symmetry.

Relation to Van Vleck Prefactors

The determinant in an instanton calculation is the Euclidean analogue of the stability prefactor in the semiclassical propagator. The Van Vleck Determinant comes from a Gaussian expansion around a real-time classical path. The instanton fluctuation determinant comes from a Gaussian expansion around a Euclidean saddle.

Both are quadratic fluctuation determinants. The difference is the boundary-value problem, the signature of the action, and the treatment of zero or negative modes.

Exercises

1. Compute the Dirichlet determinant ratio for $\mathcal L_\omega=-d^2/d\tau^2+\omega^2$ against the free operator $\mathcal L_f=-d^2/d\tau^2$ on $[0,T]$. Check the small-$\omega T$ limit.

Solution

The initial-value solutions are $y_\omega(\tau)=\sinh(\omega\tau)/\omega$ and $y_f(\tau)=\tau$, so

$$\frac{\det\mathcal L_\omega}{\det\mathcal L_f} =\frac{\sinh(\omega T)}{\omega T} =1+\frac{(\omega T)^2}{6}+O((\omega T)^4).$$

The ratio tends to one, as it must when the two operators coincide. Both are strictly positive for Dirichlet endpoints; the free operator has no constant zero mode on this domain.

2. Verify directly that differentiating the instanton equation yields the translation zero mode. Why is it not automatically an allowed zero mode on a finite Dirichlet interval?

Solution

Differentiating $mx''=V'(x)$ gives $mx^{(3)}=V''(x)x'$, hence $\mathcal M x'=0$. On the infinite line $x'$ decays and is normalizable. At finite fixed endpoints it generally does not vanish exactly, so it fails the fluctuation boundary conditions. The differential expression and the operator domain are both needed to identify an eigenmode.

References

• S. Coleman, “The Uses of Instantons”, in Aspects of Symmetry, Cambridge University Press, 1985, pp. 265–350.
• G. V. Dunne, “Functional Determinants in Quantum Field Theory”, Journal of Physics A: Mathematical and Theoretical 41, 304006 (2008).
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009.
• R. Rajaraman, Solitons and Instantons, North-Holland, 1982.
• J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005.