Reduce a regulated Gaussian to its eigenmodes

Let x contain n real, dimensionless variables and let A be a real symmetric positive-definite matrix. Work in units where ℏ=1 and the Euclidean action is xᵀAx/2. An orthogonal change of variables diagonalizes A and has absolute Jacobian one. Each eigenmode contributes the familiar one-dimensional Gaussian integral, so the product is controlled by the product of eigenvalues. This is the origin of the inverse square root determinant for real bosonic variables.

A general nonorthogonal coordinate change has a nontrivial Jacobian, which must accompany the transformed quadratic form. Keeping only the changed determinant would make the answer coordinate dependent. Overall measure conventions can supply additional constants, so state whether each integration uses dx or dx/√(2π). Ratios may cancel shared constants, but that cancellation is justified only when numerator and denominator use compatible measures and the same regulated degrees of freedom.

Normalize the Gaussian by its partition integral. A small linear source can be removed by completing the square; differentiating its resulting exponential gives the moments. The mean at zero source vanishes and the two-point covariance is A inverse. Off-diagonal entries can be negative even when A is positive definite: positivity constrains the variance of every linear combination, not the sign of every cross correlation. Higher even moments follow by pairing, while odd centered moments vanish.

A time-sliced quadratic action supplies a particular matrix A whose entries depend on spacing and endpoint conditions. Its inverse is a regulated Green function. Dirichlet, periodic, or other boundary conditions generally change its spectrum and hence its determinant and inverse. Calling the Green function merely an inverse differential operator without specifying the domain leaves the problem incomplete. An operator with a constant zero mode, for example, has no inverse on that full domain.

With more lattice sites or retained modes, the finite determinant becomes a growing product. Its continuum limit commonly diverges, even when suitably normalized correlations have sensible limits. A cutoff, determinant ratio, or another stated regularization must define the expression, followed by any required counterterms. Zero modes require separate treatment through constraints, collective coordinates, or a reduced determinant. Gauge redundancies are an especially important source of such modes, not extra physical Gaussian fluctuations to integrate repeatedly.

The matrix lab checks a change-of-variables illustration. Reset and set diagonal scale to 1 and shear to 0; predict final-A determinant 1 and singular values 1,1. Reset and set scale to 2 and shear to 0; predict determinant 2 and singular values 2,1. Readouts describe the selected final A. Geometry starts at identity at Reset, and Play or Step interpolates it toward A and back. This triangular transformation family does not compute a quadratic-action spectrum or functional determinant.