Tunneling determinants and zero modes

Take a particle with unit mass and Euclidean action equal to the integral of one half velocity squared plus a smooth nonnegative double-well potential. The degenerate minima have zero potential. A finite-action instanton approaches one minimum at negative infinite Euclidean time and the other at positive infinite time. Its action is the integral of the square root of twice the potential between the minima. This relation uses the zero Euclidean first integral selected by those endpoints.

Write a nearby path as the saddle plus a fluctuation. The quadratic term contains a second-order differential operator with potential given by the second derivative of the mechanical potential along the saddle. Its spectrum depends on boundary conditions. On a long finite interval one may impose fixed endpoint fluctuations and compare with a vacuum operator using the same regulator. The exact whole-line translation mode can become a near-zero mode at finite interval, so a finite-size determinant must be matched carefully to the collective-coordinate limit.

Translation of the instanton center tau zero gives a fluctuation proportional to its time derivative. Differentiating the saddle equation shows that this is a zero mode. The unnormalized translation tangent is the instanton velocity. Its squared L2 norm is the integral of the instanton velocity squared, equal to S zero in the unit-mass convention because kinetic and potential contributions are equal on the solution. Changing variables from that mode’s Gaussian coefficient to the center coordinate produces a Jacobian containing the square root of action divided by two pi hbar.

A prime on the instanton determinant removes the translation eigenvalue, while the collective-coordinate measure restores the integration along that direction. The reference vacuum determinant contains its full spectrum. Removing one inverse-time-squared eigenvalue gives the determinant ratio a net inverse-time-squared dimension, so its square root supplies an inverse time. Normalization constants must be compared with the same measure and regulator; a bare infinite product of eigenvalues is not a complete observable.

For positive Gaussian eigenvalues, the real Euclidean fluctuation integral converges and the determinant square root has the positive real branch. A zero eigenvalue signals a flat direction and requires separate treatment. A negative eigenvalue makes a real Gaussian divergent; taking the absolute determinant discards the phase information rather than solving that problem. The appropriate deformed contour or analytic continuation must follow from the observable and the definition of the metastable or oscillatory integral.

The Gel’fand–Yaglom method can obtain a one-dimensional determinant ratio from initial-value differential equations under specified boundary conditions, avoiding an explicit product over eigenvalues. It does not remove the need to handle zero and negative modes or renormalization in higher-dimensional field theories. In gauge theories gauge fixing and ghost determinants enter as well. Keeping these ingredients explicit prevents a real tunneling amplitude, a stable level splitting, and a false-vacuum imaginary contribution from being treated as interchangeable numbers.