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.

This page is a preview. It explains what the determinant means, why zero modes require special treatment, and why prefactors are technically harder than exponents. Full determinant technology belongs in a more mathematical treatment of functional determinants and spectral theory.

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

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

Because $x_{\mathrm{cl}}$ satisfies the Euclidean equation of motion, the linear term vanishes. 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 Gelfand-Yaglom method, which replaces a determinant ratio by a solution of an associated ordinary differential equation.

For the purposes of this volume, the key point is conceptual: the prefactor knows about the full spectrum of small oscillations around the saddle, not only about the classical action.

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 $\tau_0$.

For a single translational zero mode, this replacement produces a factor of the schematic form

$$\left( \frac{S_0}{2\pi\hbar} \right)^{1/2} \int d\tau_0,$$

up to convention-dependent normalization factors. The prime on the determinant and the collective-coordinate measure are two parts of the same correction.

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 mode is responsible for the imaginary part of the metastable energy and hence for the 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.

Why Prefactors Are Harder Than Exponents

The exponent usually requires only the classical Euclidean solution and its action. The prefactor requires:

• the full fluctuation operator,
• boundary conditions,
• determinant regularization,
• zero-mode normalization,
• negative-mode interpretation when present,
• matching to the normalization of the quantity being computed.

This is why many semiclassical estimates quote the exponent first and treat the prefactor separately. The exponent gives the dominant dependence when $S_0/\hbar\gg1$, but the prefactor is necessary for a normalized rate or splitting. Tunneling Splittings shows where that normalized one-instanton density enters the spectral gap.

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.

Common Mistakes

• Treating $e^{-S_0/\hbar}$ as the complete answer for a splitting or rate.
• Including a zero eigenvalue in an ordinary determinant.
• Forgetting that determinant ratios depend on boundary conditions and normalization.
• Interpreting a negative mode as an error rather than as part of metastable decay physics.
• Assuming that a prefactor computed in one convention can be copied into another without checking units and state normalization.

Exercises

1. Derive the fluctuation operator for the Euclidean action.

Solution

Expand

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

The potential gives

$$V(x_{\mathrm{cl}}+\eta) = V(x_{\mathrm{cl}}) + V'(x_{\mathrm{cl}})\eta + \frac12V''(x_{\mathrm{cl}})\eta^2 + \cdots.$$

The kinetic term gives

$$\frac{m}{2}(\dot x_{\mathrm{cl}}+\dot\eta)^2 = \frac{m}{2}\dot x_{\mathrm{cl}}^2 + m\dot x_{\mathrm{cl}}\dot\eta + \frac{m}{2}\dot\eta^2.$$

After integrating the cross term by parts, the linear terms vanish by the Euclidean equation of motion. The quadratic part is

$$\frac12 \int d\tau \left[ m\dot\eta^2 + V''(x_{\mathrm{cl}})\eta^2 \right].$$

Integrating the kinetic term by parts gives

$$\frac12 \int d\tau\, \eta \left( - m\frac{d^2}{d\tau^2} + V''(x_{\mathrm{cl}}) \right) \eta.$$

2. Explain why the translated instanton produces a zero mode.

Solution

If $x_{\mathrm{inst}}(\tau-\tau_0)$ is a family of saddles with the same action, then moving along the parameter $\tau_0$ does not change the action. The tangent to this family is

$$\frac{\partial}{\partial\tau_0} x_{\mathrm{inst}}(\tau-\tau_0) = - \frac{dx_{\mathrm{inst}}}{d\tau}.$$

Since the second variation of the action along an exactly flat direction vanishes, this tangent is an eigenfunction of the fluctuation operator with eigenvalue zero.

3. Why are determinant ratios more meaningful than isolated determinants?

Solution

Each determinant is an infinite product of eigenvalues and contains divergent normalization factors. In a ratio between the instanton background and a reference background, many common factors cancel or can be regularized in a controlled way. The ratio is also the physical object that compares fluctuations around two saddles in the same path integral normalization.

References

• S. Coleman, Aspects of Symmetry, Cambridge University Press, 1985.
• R. Rajaraman, Solitons and Instantons, North-Holland, 1982.
• 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.