Projective Hilbert Space: Topology and Tangent Geometry

Projective Hilbert space $\mathbb P(\mathcal H)$ is the quotient of nonzero vectors by nonzero complex scalars. Its set-theoretic identification with pure states is only the first step: quantum limits and differentiable motion require a topology and a local geometry. With the norm quotient topology, it is a complex Hilbert manifold, including when $\mathcal H$ is infinite dimensional. This page constructs that local structure. Finite-dimensional coordinate examples, the Bloch sphere, and the introductory Fubini–Study interpretation remain in Core Formalism.

Required background. Physical States as Rays establishes the equivalence of rays, normalized phase classes, and rank-one projectors.

The quotient topology and phase-independent convergence

Let $S(\mathcal H)$ denote the unit sphere and $q:S(\mathcal H)\to\mathbb P(\mathcal H)$ its quotient by $U(1)$. The quotient topology declares $U\subseteq\mathbb P(\mathcal H)$ open exactly when $q^{-1}(U)$ is open in the sphere. We use the Hilbert norm, not the weak topology.

For unit representatives, a useful metric is

$$d_{\rm ray}([\psi],[\phi]) =\inf_{\alpha\in\mathbb R}\|\psi-e^{i\alpha}\phi\| =\sqrt{2-2|\langle\psi,\phi\rangle|}.$$

The second equality follows by expanding the squared norm and maximizing the real part of the overlap. The expression vanishes precisely on the same ray. Its triangle inequality follows by choosing phases for two successive comparisons and using the vector triangle inequality. It induces the quotient topology.

Consequently, rays $[\psi_n]$ converge to $[\psi]$ exactly when their unit representatives can be rephased to converge in norm to $\psi$. The original representatives need not converge: $(-1)^n\psi$ describes a constant ray. Equivalently, the absolute overlaps tend to one.

The angle $\arccos|\langle\psi,\phi\rangle|$ is another metric with the same topology. Metric normalizations vary in geometric quantum mechanics, so an overall factor must be specified before comparing geometric lengths or speeds.

Local coordinates without a finite-dimensional restriction

Fix a unit vector $e$ and consider

$$U_e=\{[\psi]:\langle e,\psi\rangle\ne0\}.$$

Each ray in $U_e$ has a unique representative of the form $e+z$, where $z\in e^\perp$. Indeed,

$$z=\frac{\psi}{\langle e,\psi\rangle}-e, \qquad \langle e,z\rangle=0.$$

This coordinate does not depend on which nonzero multiple of $\psi$ was used. Conversely, every $z\in e^\perp$ gives the ray $[e+z]$. Hence $U_e$ is modeled on the Hilbert space $e^\perp$.

A normalized local section is

$$s_e(z)=\frac{e+z}{\sqrt{1+\|z\|^2}},$$

whose overlap with $e$ is positive real. On overlapping charts, the change of coordinates is obtained by dividing $e+z$ by its nonzero overlap with the new reference vector and subtracting that vector. These maps are smooth and holomorphic in the complex chart variables. No coordinate sum needs to be finite for this construction.

The chart misses all rays orthogonal to $e$. A coordinate divergence can therefore indicate departure from a chart, not a singular physical state. A collection of such charts covers the full projective space.

Tangent vectors and removal of phase velocity

For a differentiable unit-vector curve $\psi(t)$,

$$\operatorname{Re}\langle\psi,\dot\psi\rangle=0.$$

This is the tangent-space condition on the real unit sphere. Pure phase motion has velocity $ia\psi$, with real $a$, and contributes no motion of the ray. Removing it gives the horizontal representative

$$v=(I-P_\psi)\dot\psi, \qquad \langle\psi,v\rangle=0.$$

The tangent space to projective space at $[\psi]$ can thus be identified locally with $\psi^\perp$. Under a change of phase representative, the corresponding horizontal vectors change by the same phase, preserving their inner products.

We choose the metric normalization

$$ds^2=\|d\psi\|^2-|\langle\psi,d\psi\rangle|^2$$

for unit representatives. Its value is the squared norm of the horizontal displacement. The real and imaginary parts of the horizontal Hilbert inner product give compatible metric and symplectic structures. Ashtekar and Schilling explain the resulting Kähler formulation; the local construction here makes explicit which topology and tangent vectors are being used.

Unbounded generators do not define smooth motion everywhere

For a self-adjoint Hamiltonian $H$, the unitary group $e^{-itH/\hbar}$ acts continuously on every ray. A norm derivative of the lifted curve exists when $\psi\in D(H)$. For such a unit vector,

$$v=-\frac{i}{\hbar}(H-\langle H\rangle_\psi)\psi, \qquad \|v\|^2=\frac{\|(H-\langle H\rangle_\psi)\psi\|^2}{\hbar^2}.$$

The right side is the spectral variance divided by $\hbar^2$. Writing $\langle\psi,H^2\psi\rangle$ literally would require $\psi\in D(H^2)$; the squared-norm or spectral-integral expression needs only $\psi\in D(H)$.

This is an important infinite-dimensional boundary. A bounded Hamiltonian produces a smooth vector field everywhere. An unbounded Hamiltonian has a proper generator domain, so one must not silently regard its expectation as an everywhere-smooth Hamiltonian function. The exact differentiability statement is supplied by Stone’s Theorem.

Exercise: a chart and a horizontal speed

For orthonormal $e_0,e_1$, let $\psi(t)=\cos t\,e_0+\sin t\,e_1$. Find its coordinate in $U_{e_0}$ and its projective speed in the metric above. What happens at $t=\pi/2$?

Solution

When $\cos t\ne0$, the chart coordinate is $z(t)=\tan t\,e_1$. Meanwhile $\langle\psi,\dot\psi\rangle=0$ and $\|\dot\psi\|=1$, so the horizontal and projective speed is one. At $t=\pi/2$ the coordinate diverges, but $\psi=e_1$ is perfectly regular and lies in $U_{e_1}$. The curve has no physical or manifold singularity there.

References

• A. Ashtekar and T. A. Schilling, “Geometrical Formulation of Quantum Mechanics”, in On Einstein’s Path, edited by A. Harvey, Springer, 1999, pp. 23–65 — Hilbert-space quotient geometry, metric, and Hamiltonian flow.
• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013 — state spaces, self-adjoint generators, and the operator-domain background.