Gel'fand–Yaglom Formula Overview

• The Gel'fand–Yaglom formula is a technique that transforms the computation of functional determinants into solving an initial-value problem with prescribed boundary conditions.
• It is extended via discrete formulations and matrix-valued recursions to handle various boundary conditions and higher-dimensional operators.
• The method underpins key calculations in quantum mechanics, quantum field theory, and statistical mechanics, such as vacuum energies and instanton fluctuations.

The Gel'fand–Yaglom formula is a central result in the computation of functional determinants of one-dimensional second-order differential operators, notably within quantum mechanics, quantum field theory, and statistical mechanics. By transforming infinite-dimensional spectral products into the value of an initial-value problem, the Gel'fand–Yaglom method provides a powerful and practical technique for regularizing and computing the determinants underpinning fluctuation determinants, one-loop effective actions, vacuum energies, and transition rates.

1. Classical Formulation of the Gel'fand–Yaglom Formula

In its archetypal setting, the Gel'fand–Yaglom theorem considers the Sturm–Liouville operator on a finite interval,

$L = -\frac{d^2}{dx^2} + V(x)$

acting on functions with Dirichlet boundary conditions. The determinant of $L$ (zeta or heat kernel regularized) is expressed not as an infinite product over eigenvalues but as the value at the interval endpoint of the unique solution to the associated homogeneous initial-value problem: $L y(x) = 0,\quad y(0) = 0,\; y'(0) = 1$ The functional determinant is then given by

$\det(-\frac{d^2}{dx^2} + V(x)) = y(L)$

For general boundary conditions, including Robin and mixed, a straightforward extension exists: if $y_\alpha(x)$ solves %%%%2%%%% with $y_\alpha(0) = 1$, $y'_\alpha(0) = \alpha$, then for Robin data $y'(L) = \beta y(L)$,

$\det(L) = 2(y'_\alpha(L) - \beta y_\alpha(L))$

The derivative of the logarithm of this determinant generates Euler–Rayleigh sums over the inverse powers of eigenvalues, establishing an analytic bridge to spectral zeta regularization and vacuum expectation values (Ttira et al., 2011, Shea, 2020).

2. Discrete Gel'fand–Yaglom Formula

By discretizing the interval into $v+2$ points and considering the second-order finite-difference operator,

$$y(j+1) + [1 - V(j) - 2]\,y(j) + y(j-1) = 0, \qquad 1 \le j \le v$$

the determinant can be computed using a transfer matrix formalism. Encoding the system as a first-order recursion,

$\mathbf{Y}(j) = M(j, \lambda) \mathbf{Y}(j-1)$

with $M(j, \lambda)$ the $2 \times 2$ tridiagonal transfer matrix, the full transfer from $j = 0$ to $j = v$ yields

$K(\lambda; v) = M(v)M(v-1)\ldots M(1)$

For Dirichlet boundary conditions, the discrete determinant becomes

$\Det(-\Delta+W)\bigr|_D = h^{2v} y(v+1;0)$

where $y(v+1;0)$ is extracted as a component of $K(0;v)$ acting on an initial vector set by the boundary (Dowker, 2011, Kan et al., 2017). General Robin data utilize boundary vectors and construct $$P(\lambda) = \mathbf{Y}_{\mathrm{out}}(v) K(\lambda; v) \mathbf{Y}_{\mathrm{in}}(0)$$, leading to

$\Det(-\Delta + W)\bigr|_{\mathrm{Robin}(a,b)} = h^{2v} P(0)$

The discrete GY theorem recovers the continuum result in the $h \to 0$ limit (grid spacing vanishing at fixed $L$).

3. Generalizations: Higher Dimensions, Matrix Extensions, and Stochastic Systems

To handle operators in higher dimensions,

$H = -\Delta_d + V(\mathbf{r})$

one generalizes the GY approach by considering recursion in one “distinguished” direction and treating the remaining transverse directions via matrix-valued recursions: $$-\left(Y_{n+1} - 2Y_n + Y_{n-1}\right) + \left(-\Delta_{d-1} + V_n\right)Y_n = 0$$ with initial conditions $Y_0 = 0$, $Y_1 = I$. The determinant becomes

$\det H = \det Y_N$

in the discrete setting or

$$\det(-\Delta_d + V(\mathbf{r})) = \det Y(L) = \exp\biggl\{ \int_0^L \operatorname{Tr}\mathcal{Z}(x)\,dx \biggr\}$$

in the continuum, where $\mathcal{Z}(x)$ solves a matrix Riccati equation (Ossipov, 2018).

For quantum mechanical Hamiltonian systems with Lagrangian boundary data, the GY approach gives direct lattice regularizations of determinants, matching the results of $\zeta$-function techniques in the continuum limit (Shea, 2020). In stochastic systems, Gel'fand–Yaglom-type equations have been extended to fluctuation determinants around instantons via the recursion and Riccati methods, with prefactors controlled by the matrix evolution equation for the fluctuation operator (Schorlepp et al., 2021).

4. Boundary Conditions, Explicit Examples, and Special Potentials

The GY method seamlessly accommodates Dirichlet, Neumann, Robin, and periodic boundary conditions by adjusting the initial data and the algebraic constraints in the transfer-matrix approach (Dowker, 2011, Shea, 2020). For instance, with Robin boundary conditions defined via a scaled finite-difference, the determinant formula is

$\Det(-\Delta+W)\bigr|_{\mathrm{Robin}(a,b)} = h^{2v} P(0)$

Chebyshev polynomials $U_n$ arise for constant potentials, serving as propagators in the transfer-matrix recursion.

For a delta potential localized at a single lattice site, the determinant is an explicit function of the Chebyshev polynomial evaluated at $1$ plus a perturbative correction: $\Det(-\Delta + v_k \delta_{jk}) = h^{2v} [ U_{v+1}(1) + 2 v_k U_{v-1}(1) ]$ This explicit construction, combined with analytic expansion of $\ln P(\lambda)$, allows for the computation of all spectral sums (Euler–Rayleigh) associated with the eigenvalues.

5. Applications: Quantum Field Theory, Vacuum Energies, and Instantons

The primary physical application of the GY formula is the reduction of functional determinants in field theory path integrals to initial-value problems. This includes:

• Vacuum energy and Casimir force: The GY formula enables analytic progress in computing the Lifshitz formula for Casimir forces, expressing the regularized energy as a sum over logarithms of GY determinants, constructed from transfer matrices encoding scattering or reflection coefficients (Ttira et al., 2011). Higher-dimensional corrections enter only through the dimensions of the momentum integrals.
• Instantons and fluctuation determinants: In semiclassical transitions, the fluctuation determinant is computed via the GY formula in each partial wave. In cases with nontrivial topological sectors, such as the Abelian Higgs instanton, the naive GY prescription fails in the $s$-wave, requiring a modified normalization and spectral cutoff procedure to isolate and remove spurious divergences. Agreement between GY and Green's function methods is observed to better than 1% after such regularization (0803.4333).
• Holographic Wilson loops: Calculation of 1-loop string partition functions and ratio of determinants in $AdS_5 \times S^5$ for circular and latitude Wilson loops is made tractable with the GY formalism, including modifications to project out zero modes and matrix-valued extensions (Botao et al., 2020).

6. Analytic Structure, Trace and Born Series, and Discrete–Continuum Correspondence

A trace formula for the GY function can be constructed from the transfer matrix and boundary vectors,

$$P(\lambda) = \operatorname{Tr}\bigl[ K(\lambda; v)\, (\mathbf{Y}_{\mathrm{in}} \otimes \mathbf{Y}_{\mathrm{out}}) \bigr ]$$

This formulation admits a finite-order Born series expansion, connecting each perturbation in the transfer matrix to corresponding spectral corrections (Dowker, 2011).

The continuum limit, taken by scaling the lattice spacing $h \rightarrow 0$ while keeping the total interval fixed, reproduces established analytic expressions for regularized determinants. For example, with constant potential and Robin boundary conditions,

$\Det_{\rm Robin} \xrightarrow{h\to 0} \frac{(a+b)\cosh(pL) + (ab + p^2)\sinh(pL)}{p}$

as expected from continuum spectral theory.

Zero modes require special care: in their presence, determinant ratios must be defined via derivatives with respect to spectral parameters or through projected determinants, with explicit formulas for the ratio of determinants omitting zero modes (Botao et al., 2020, 0803.4333).

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References:

(Dowker, 2011, Ossipov, 2018, Ttira et al., 2011, Botao et al., 2020, 0803.4333, Shea, 2020, Kan et al., 2017, Schorlepp et al., 2021)