Gaussian mode geometry

Profiles on ordinary spatial coordinates

On $\mathbb R^d$ let a normalized density have mean $X$ and positive definite covariance $\Sigma$:

$$\rho(x)=\frac{\exp[-(x-X)^T\Sigma^{-1}(x-X)/2]} {(2\pi)^{d/2}\sqrt{\det\Sigma}}, \qquad \psi(x)=\sqrt{\rho(x)}.$$

The coordinates $x$ describe ordinary spatial directions of a sector in the nested physical manifold. $X$ and $\Sigma$ label field profiles on that space.

For parameters $\theta^i$, define the projected derivative tensor

$$Q_{ij}=\langle\partial_i\psi, (I-|\psi\rangle\langle\psi|)\partial_j\psi\rangle, \qquad g_{ij}=\operatorname{Re}Q_{ij}.$$

This convention measures the leading loss of squared overlap: $1-|\langle\psi(\theta),\psi(\theta+d\theta)\rangle|^2 =g_{ij}d\theta^i d\theta^j+o(|d\theta|^2)$.

Derivation from Gaussian moments

Put $y=x-X$ and $A=\Sigma^{-1}$. Logarithmic differentiation gives

$$d\log\psi= \frac12 dX^TAy +\frac14\left[y^TA(d\Sigma)Ay-\operatorname{tr}(A\,d\Sigma)\right].$$

Its expectation against $\rho$ is zero. Normalization and reality also give $\langle\psi,d\psi\rangle=0$, so the projector subtraction vanishes. The linear and centered quadratic terms have zero cross moment by odd Gaussian parity. For symmetric $B,C$, Gaussian fourth moments imply

$$\mathbb E[(y^TBy-\operatorname{tr}(B\Sigma)) (y^TCy-\operatorname{tr}(C\Sigma))] =2\operatorname{tr}(B\Sigma C\Sigma).$$

One proof writes $y=\Sigma^{1/2}z$ and uses independent standard normal coordinates, whose fourth moment is $\mathbb E[z_i z_j z_k z_l]= \delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}$. The resulting line element is

$$\boxed{ds^2=\frac14dX^T\Sigma^{-1}dX +\frac18\operatorname{tr} (\Sigma^{-1}d\Sigma\,\Sigma^{-1}d\Sigma).}$$

Because $d\log\psi=\tfrac12d\log\rho$, this is one quarter of the classical Fisher information metric of the Gaussian density family. The mean/covariance form of that statistical metric is also given in Lawson et al., §2.

Berry connection and oscillator width

With convention $\mathcal A=-i\langle\psi,d\psi\rangle$, the connection vanishes throughout this globally positive real family. Consequently its Berry curvature $d\mathcal A$ is zero. The metric is nonzero because neighboring normalized profiles differ even when their relative phase is fixed. Multiplying every profile by a smooth parameter-dependent common phase changes the connection by an exact differential and leaves both metric and curvature unchanged.

For the ground state of $H=-\Delta+\lambda|x|^2$, $\Sigma=I/(2\sqrt\lambda)$. At fixed center,

$$\Sigma^{-1}d\Sigma=-\frac{d\lambda}{2\lambda}I, \qquad g_{\lambda\lambda}=\frac{d}{32\lambda^2}.$$

If a specified self-consistency model imposes $\lambda=(g/2)^{2/3}$ with $g>0$, the chain rule gives

$$g_{gg}=\frac{d}{72g^2}=\frac{d}{288\lambda^3}.$$

This is a metric on the chosen family of normalized profiles. A kinetic term for time-dependent width or a physical response coefficient must be derived from the action and its other modes.

Verification and related

spectral_analysis_checks.py checks the metric with direct Gaussian quadrature and finite differences of independently integrated overlaps, including a correlated covariance and simultaneous center/width changes.

• Gaussian contact profiles — uses the same covariance convention for spatial density overlap.
• Confining oscillator domain — defines the oscillator and its normalization.
• Spectral analysis tools — navigation and scope.