Fluctuation Operators and Determinant Ratios

The one-loop prefactor around a nontrivial saddle is the Gaussian measure of its nonzero fluctuations, normalized to the same measure in a reference sector and renormalized with the same boundary conditions and subtraction scheme. It is therefore a determinant ratio, not an isolated formal product of eigenvalues. This page derives that ratio, states when the Gel’fand–Yaglom method computes it, and evaluates the fluctuation operator of the quartic-double-well instanton.

Required background. Saddles, control parameters, and loop counting supplies the rescaling that isolates the Hessian and the criterion selecting a contributing saddle.

Helpful background. Heat kernels, zeta functions, and spectral determinants supplies regulator-independent definitions of determinant ratios and their ultraviolet expansion; second variation, Hessians, and Jacobi fields supplies the boundary-value problem defined by the quadratic action.

From the second variation to a determinant ratio

Let $\phi_\sigma$ be a Euclidean saddle of a regulated action $\mathcal S/g$. For a bosonic fluctuation $\eta$ obeying the linearized boundary conditions,

$$\mathcal S[\phi_\sigma+\sqrt g\,\eta] =\mathcal S_\sigma +\frac g2\langle\eta,M_\sigma\eta\rangle +O(g^{3/2}),$$

where

$$M_\sigma(x,y) =\left.\frac{\delta^2\mathcal S} {\delta\phi(x)\delta\phi(y)}\right|_{\phi_\sigma}.$$

In a finite spectral regulator with strictly positive eigenvalues $\lambda_n^\sigma$, the Gaussian integral is proportional to

$$\prod_n(\lambda_n^\sigma)^{-1/2} =(\det M_\sigma)^{-1/2}.$$

The normalization of the functional measure cancels only in a ratio. Relative to a reference saddle with operator $M_{\rm ref}$,

$$\mathcal A_\sigma^{(1)} =\left(\frac{\det M_\sigma}{\det M_{\rm ref}}\right)^{-1/2}.$$

If $M_\sigma$ has $k$ zero modes, they must be omitted and replaced by collective-coordinate integrals:

$$\mathcal A_\sigma^{(1)} =\left(\frac{\det{}' M_\sigma}{\det M_{\rm ref}}\right)^{-1/2}.$$

The prime has operational content: it specifies a projection, reduced determinant, or limiting prescription. It must not mean “discard whichever eigenvalues make the answer inconvenient.” Negative eigenvalues likewise cannot be hidden inside an absolute value; their phase is fixed by the integration cycle.

For a positive elliptic operator with eigenvalues of mass dimension two, zeta regularization introduces a reference scale $\mu$ and defines

$$\zeta_M(s;\mu)=\sum_n\left(\frac{\lambda_n}{\mu^2}\right)^{-s}, \qquad \log\det\!\left(\frac{M}{\mu^2}\right)=-\zeta_M'(0;\mu).$$

Equivalently,

$$\log\frac{\det M_\sigma}{\det M_{\rm ref}} =-\int_0^\infty\frac{\mathrm dt}{t}\, \operatorname{Tr}\!\left(e^{-tM_\sigma}-e^{-tM_{\rm ref}}\right),$$

after the required small-$t$ subtractions. In quantum mechanics the difference is often finite directly. In QFT the small-$t$ terms are local ultraviolet divergences and must be canceled by counterterms fixed in the reference vacuum; the residual $\mu$ dependence of the regulated determinant cancels against those counterterms and the running parameters through the retained order. A determinant value without its regulator, boundary conditions, zero-mode prescription, and renormalization scale is not a reproducible prefactor. Dunne 2008, §§2–4, printed pp. 3–8 reviews these complementary definitions and their domain of validity.

Gel’fand–Yaglom on a finite interval

For scalar Sturm–Liouville operators on $[a,b]$,

$$M=-\frac{d^2}{d\tau^2}+U(\tau), \qquad M_0=-\frac{d^2}{d\tau^2}+U_0(\tau),$$

with the same Dirichlet boundary conditions, solve

$$My=0,\quad y(a)=0,\quad y'(a)=1,$$

and the analogous problem for $y_0$. When neither operator has a zero mode,

$$\frac{\det M}{\det M_0}=\frac{y(b)}{y_0(b)}.$$

This formula is powerful because it replaces an infinite spectral product by an initial-value problem. Its hypotheses are equally important:

• the operators have the same leading symbol and compatible boundary conditions;
• the comparison uses the same interval and normalization;
• a zero mode is treated by a reduced-determinant limit;
• matrix-valued problems use a fundamental solution matrix and its determinant;
• ultraviolet counterterms are still required when transverse momenta or higher-dimensional fields are present.

One reduced-determinant prescription is to shift $M\mapsto M+\lambda$, compute the ratio, and remove the simple zero:

$$\det{}'M =\lim_{\lambda\to0} \frac{\det(M+\lambda)}{\lambda}.$$

On a finite interval an infinite-volume translation zero mode is usually lifted to a small eigenvalue. One must project the known mode or take the correlated large-interval and zero-eigenvalue limit; simply deleting the numerically smallest eigenvalue can remove a physical soft mode instead. Mariño 2015, §1.6, pp. 30–35 derives the reduced Gel’fand–Yaglom construction for instanton operators.

Pöschl–Teller fluctuations of the double well

For the action

$$S_E[x]=\frac1g\int \mathrm d\tau\, \left[\frac12\dot x^2+\frac12(x^2-1)^2\right]$$

and instanton $x_I(\tau)=\tanh(\tau-\tau_0)$, the Hessian is

$$M_I=-\frac{d^2}{d\tau^2} +4-6\,\operatorname{sech}^2(\tau-\tau_0).$$

The vacuum operator at $x=\pm1$ is

$$M_0=-\frac{d^2}{d\tau^2}+4.$$

This is the reflectionless $l=2$ Pöschl–Teller problem. Its discrete spectrum consists of

$$\lambda_0=0,\qquad \lambda_1=3,$$

with continuum $\lambda(k)=k^2+4$. The normalized zero mode is proportional to $\operatorname{sech}^2(\tau-\tau_0)$; there is no negative eigenvalue.

The spectral calculation can be made explicit. For the reflectionless family

$$M_{\ell,m}=-\frac{\mathrm d^2}{\mathrm d\tau^2} +m^2-\ell(\ell+1)\operatorname{sech}^2\tau,$$

the Jost factors give

$$\frac{\det M_{\ell,m}}{\det M_{0,m}} =\prod_{j=1}^{\ell}\frac{m-j}{m+j}.$$

At $m=\ell=2$, the factor $m-2$ marks the translation eigenvalue. The actual eigenvalue in this family is $m^2-4=(m-2)(m+2)$, so dividing by it before taking $m\to2$ gives

$$\frac{\det{}'M_I}{\det M_0} =\lim_{m\to2} \frac{1}{m^2-4} \prod_{j=1}^{2}\frac{m-j}{m+j} =\frac{2^2-1^2}{(2+1)^2(2+2)^2} =\frac1{48}.$$

An initial-value calculation reaches the same number without multiplying the spectrum. Put the operators on $[-L,L]$ with Dirichlet boundaries. For $M_Iy_I=0$, $y_I(-L)=0$, and $y_I'(-L)=1$, reduction of order from $u_0=\operatorname{sech}^2\tau$ gives

$$y_I(L)=2\operatorname{sech}^4L\,I(L), \qquad I(L)=\int_0^L\cosh^4t\,\mathrm dt =\frac{3L}{8}+\frac{\sinh2L}{4}+\frac{\sinh4L}{32}.$$

The reference solution is $y_0(L)=\sinh(4L)/2$, so Gel’fand–Yaglom gives

$$r_L\equiv\frac{\det M_{I,L}}{\det M_{0,L}} =\frac{4\operatorname{sech}^4L\,I(L)}{\sinh4L} =2e^{-4L}\bigl[1+o(1)\bigr].$$

The coefficient of the lifted mode can also be read from the even shooting solution. Write $u=u_0+\lambda u_1+O(\lambda^2)$ with $u(0)=1$, $u'(0)=0$, and $M_Iu_1=u_0$. Variation of parameters with the second zero-energy solution $v=u_0I$ gives

$$u_1(L)=-\frac{e^{2L}}{24}\bigl[1+o(1)\bigr], \qquad u_0(L)=4e^{-2L}\bigl[1+o(1)\bigr].$$

The Dirichlet condition $u(L)=0$ therefore lifts the translation mode to $\lambda_0(L)=96e^{-4L}[1+o(1)]$. Projecting precisely that mode yields

$$\lim_{L\to\infty}\frac{r_L}{\lambda_0(L)}=\frac1{48},$$

in agreement with the spectral calculation. This comparison also shows why deleting an arbitrary small numerical eigenvalue is not an acceptable reduced-determinant prescription.

Before nondimensionalization, $\det{}'M_I/\det M_0$ has dimensions of time squared because one inverse-time-squared eigenvalue has been removed. Its inverse square root supplies one inverse time. In the normalization above the Jacobian $J=(\mathcal S_I/2\pi g)^{1/2}$ is dimensionless, while $J\,\mathrm d\tau_0$ carries one time; after extracting $\int\mathrm d\tau_0=T$, the remaining fugacity has dimensions of inverse time. The value $1/48$ is tied to the displayed operator normalization; rescaling $\tau$ rescales a primed determinant differently from an unprimed one. Mariño 2015, §§1.5–1.8, pp. 26–42 derives both evaluations.

The complete one-instanton factor is not just $\sqrt{48}$. It has the form

$$\kappa =\left(\frac{\mathcal S_I}{2\pi g}\right)^{1/2} \left(\frac{\det{}'M_I}{\det M_0}\right)^{-1/2} e^{-\mathcal S_I/g} \bigl[1+O(g)\bigr], \qquad \mathcal S_I=\frac43,$$

up to the stated normalization of the Euclidean time kernel. The first factor is the dimensionless translation Jacobian, the second supplies the inverse-time dimension of the event fugacity, and the exponential is the classical weight. The separate integral over $\tau_0$ produces the total-time factor. Their separation prevents a zero eigenvalue from being counted twice.

Shared calculation. The saddle-contribution anatomy shows where this determinant sits relative to the contour phase, collective measure, counterterms, and observable. Shared comparison. The canonical comparison table records the corresponding prescription and failure boundary for several standard saddles.

Renormalization and failure tests

In a field theory, write the one-loop contribution schematically as

$$\Gamma_\sigma^{(1)} =\frac12\log\det{}'M_\sigma -\frac12\log\det M_{\rm ref} +S_{\rm ct}^{(1)}[\phi_\sigma] -S_{\rm ct}^{(1)}[\phi_{\rm ref}].$$

The same renormalization conditions must fix both sectors. A successful calculation passes four checks:

1. all ultraviolet regulator dependence cancels through the claimed order;
2. zero and negative mode counts agree between spectral and variational analyses;
3. the result has the dimension required by the moduli measure and observable;
4. an independent method, such as phase shifts versus Gel’fand–Yaglom, agrees.

The ordinary determinant expansion is not uniform when a nonzero eigenvalue becomes comparable to the interaction corrections. Near such a soft mode, integrate that coordinate non-Gaussianly or use a uniform approximation. It also fails if the reference operator uses different boundary data, if an infinite-volume limit is taken before projecting the zero mode, or if a QFT determinant is quoted before counterterm subtraction.

Common pitfalls

Taking the determinant of a differential expression without a domain. Boundary conditions are part of the operator. Changing Dirichlet to periodic data changes both the spectrum and the determinant.

Deleting the smallest numerical eigenvalue automatically. Finite-volume translation modes are small but nonzero, and other physical modes may also soften. Project the analytically identified zero mode and test stability as the volume changes.

Using an absolute determinant for a negative mode. This destroys the contour phase that distinguishes a decay saddle from a stable one. Continue the Gaussian along the prescribed steepest direction instead.

Exercises

1. Derive the double-well fluctuation potential $4-6\,\operatorname{sech}^2\tau$.

Solution

For $V(x)=\tfrac12(x^2-1)^2$,

$$V''(x)=6x^2-2.$$

Substituting $x_I=\tanh\tau$ and using $\tanh^2\tau=1-\operatorname{sech}^2\tau$ gives

$$V''(x_I)=6(1-\operatorname{sech}^2\tau)-2 =4-6\,\operatorname{sech}^2\tau.$$

2. Why does $\det{}'M_I/\det M_0$ acquire dimensions even if the unprimed ratio is dimensionless?

Solution

Both unprimed determinants contain the same number of eigenvalues, so their dimensions cancel in a common regulator. Removing one eigenvalue from the numerator removes one factor of inverse time squared. The primed ratio therefore has dimensions of time squared. Its inverse square root has dimensions of inverse time and combines with the collective-coordinate integral to give the required dimension.

3. Let $M_\epsilon=M+\epsilon$ and suppose $M$ has one normalized zero mode and otherwise positive spectrum. Show that $\det M_\epsilon=\epsilon\,\det{}'M+O(\epsilon^2)$.

Solution

In a spectral regulator,

$$\det M_\epsilon =\epsilon\prod_{n\geq1}(\lambda_n+\epsilon) =\epsilon\left(\prod_{n\geq1}\lambda_n\right) \bigl[1+O(\epsilon)\bigr].$$

The product over nonzero eigenvalues is $\det{}'M$, which proves the stated limit. The same conclusion follows after a consistent zeta or determinant-ratio regularization.

Continue from spectra to the remaining saddle data

• To replace each identified zero eigenvalue by an invariant integration measure, continue to Zero Modes, Collective Coordinates, and Moduli Measures.
• If the Hessian has a negative eigenvalue, do not take an absolute determinant first; continue to Negative Modes and Instability Indices to fix its contour phase.
• For determinant theory beyond this saddle application, use Heat Kernels, Zeta Functions, and Spectral Determinants; to combine the renormalized ratio with an observable, continue to Renormalized Saddle Contributions and Validity Tests.

References

• Dunne, Gerald V. “Functional Determinants in Quantum Field Theory.” Journal of Physics A: Mathematical and Theoretical 41 (2008): 304006. DOI.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015, ch. 1, pp. 3–61. DOI.