Hamiltonian Flow on Projective Hilbert Space

Pure-state Schrödinger evolution can be written as an ordinary Hamiltonian flow, but its phase space is not the classical space of positions and momenta. It is the projective Hilbert space $\mathbb P(\mathcal H)$ of quantum rays. The Hamiltonian function is the energy expectation value, and the symplectic form is inherited from the imaginary part of the Hilbert-space inner product.

This page derives that statement with one fixed convention:

$$\omega_{\rm FS}(u,v) = 2\hbar\, \operatorname{Im}\langle u\vert v\rangle, \qquad \iota_{X_f}\omega_{\rm FS}=df.$$

Here $u$ and $v$ are horizontal representatives of tangent vectors to projective Hilbert space. With this normalization, the Hamiltonian vector field is exactly the phase-free part of the Schrödinger velocity and

$$\{f_A,f_B\}_{\rm FS} = \frac{1}{i\hbar}f_{[A,B]}.$$

The ray construction itself belongs to Projective Hilbert Space, while distances and quantum speed belong to Fubini–Study Geometry.

Tangent Vectors and the Symplectic Form

Let $\lvert\psi\rangle$ be a normalized representative of a ray:

$$\langle\psi\vert\psi\rangle=1.$$

A variation $\lvert u\rangle$ contains a vertical component parallel to $\lvert\psi\rangle$, which changes only the representative’s phase. Its horizontal part is

$$\lvert u_\perp\rangle = \left( I-\lvert\psi\rangle\langle\psi\rvert \right) \lvert u\rangle,$$

and satisfies

$$\langle\psi\vert u_\perp\rangle=0.$$

Horizontal vectors represent genuine tangent directions at $[\psi]\in\mathbb P(\mathcal H)$. Two useful bilinear forms descend from the inner product:

$$g_{\rm FS}(u,v) = \operatorname{Re} \langle u_\perp\vert v_\perp\rangle,$$

and

$$\omega_{\rm FS}(u,v) = 2\hbar\, \operatorname{Im} \langle u_\perp\vert v_\perp\rangle.$$

The first is the Fubini–Study metric in the distance convention used throughout this chapter. The second is the Hamiltonian-normalized symplectic form. It is closed and nondegenerate on projective space. Before the physical factor $2\hbar$ is inserted, the metric, symplectic form, and multiplication by $i$ are the compatible parts of the Kähler structure; the factor is an overall rescaling used to normalize time evolution.

The factor $2\hbar$ is conventional but useful. Rescaling a symplectic form rescales every Hamiltonian vector field. This choice makes the geometric evolution parameter equal to physical time and makes commutators appear with the familiar factor $1/(i\hbar)$.

Expectation Value as Hamiltonian Function

Every self-adjoint operator $A$ defines a real function on ray space:

$$f_A([\psi]) = \frac{ \langle\psi\vert A\vert\psi\rangle }{ \langle\psi\vert\psi\rangle }.$$

For normalized representatives,

$$f_A([\psi]) = \langle\psi\vert A\vert\psi\rangle.$$

The function is well defined on rays because numerator and denominator scale by the same factor under $\lvert\psi\rangle\mapsto\lambda\lvert\psi\rangle$.

For dynamics, use the expectation of the Hamiltonian operator:

$$h([\psi]) = f_H([\psi]) = \langle H\rangle_\psi.$$

Let $\lvert u\rangle$ be horizontal. Differentiating $h$ in that direction gives

$$\begin{aligned} dh(u) &= \langle u\vert H\vert\psi\rangle + \langle\psi\vert H\vert u\rangle\\ &= 2\operatorname{Re} \langle u\vert \left( H-\langle H\rangle_\psi \right) \vert\psi\rangle. \end{aligned}$$

The expectation value may be subtracted because $\langle u\vert\psi\rangle=0$. This already shows that adding a scalar to $H$ does not change the differential of $h$:

$$H\longmapsto H+cI \quad\Longrightarrow\quad h\longmapsto h+c, \qquad dh\longmapsto dh.$$

As in classical mechanics, an additive constant changes the Hamiltonian function but not its vector field.

Deriving the Schrödinger Flow

The Hamiltonian vector field $X_h$ is defined by

$$\omega_{\rm FS}(X_h,u) = dh(u)$$

for every tangent direction $u$. Consider

$$\lvert X_h(\psi)\rangle = -\frac{i}{\hbar} \left( H-\langle H\rangle_\psi \right) \lvert\psi\rangle.$$

This vector is horizontal:

$$\langle\psi\vert X_h(\psi)\rangle=0.$$

Using self-adjointness of $H$,

$$\begin{aligned} \omega_{\rm FS}(X_h,u) &= 2\hbar\, \operatorname{Im} \langle X_h(\psi)\vert u\rangle\\ &= 2\operatorname{Re} \langle u\vert \left( H-\langle H\rangle_\psi \right) \vert\psi\rangle\\ &= dh(u). \end{aligned}$$

Therefore

$$X_h([\psi]) \quad\text{is represented by}\quad -\frac{i}{\hbar} \left( H-\langle H\rangle_\psi \right) \lvert\psi\rangle,$$

where the hat over the equality reminds us that the vector on the right is a horizontal representative of a projective tangent vector.

The full Schrödinger velocity is

$$\lvert\dot\psi\rangle = -\frac{i}{\hbar}H\lvert\psi\rangle.$$

It differs from $X_h$ by

$$-\frac{i}{\hbar} \langle H\rangle_\psi \lvert\psi\rangle,$$

which is purely vertical phase motion. Both velocities therefore project to the same curve of rays. Schrödinger evolution on $\mathcal H$ and Hamiltonian flow on $\mathbb P(\mathcal H)$ are the same physical pure-state dynamics.

The phase-free projector form is

$$\dot\Pi = -\frac{i}{\hbar}[H,\Pi], \qquad \Pi=\lvert\psi\rangle\langle\psi\rvert.$$

This equation depends only on the ray and is unchanged by $H\mapsto H+cI$.

Poisson Brackets and Commutators

For a real function $f$ on a symplectic manifold, let $X_f$ satisfy

$$\iota_{X_f}\omega_{\rm FS}=df.$$

Define the Poisson bracket by

$$\{f,g\}_{\rm FS} = \omega_{\rm FS}(X_f,X_g).$$

For expectation-value functions of self-adjoint operators,

$$\lvert X_{f_A}\rangle = -\frac{i}{\hbar} \left( A-\langle A\rangle_\psi \right) \lvert\psi\rangle.$$

A direct calculation gives

$$\begin{aligned} \{f_A,f_B\}_{\rm FS} &= 2\hbar\, \operatorname{Im} \langle X_{f_A}\vert X_{f_B}\rangle\\ &= \frac{1}{i\hbar} \langle\psi\vert[A,B]\vert\psi\rangle\\ &= \frac{1}{i\hbar}f_{[A,B]}. \end{aligned}$$

This is an exact identity inside quantum mechanics. It is not the classical-limit correspondence that replaces commutators by Poisson brackets on ordinary classical phase space. The bracket here lives on quantum pure-state space.

Along Hamiltonian flow,

$$\frac{d}{dt}f_A = \{f_A,h\}_{\rm FS} = \frac{1}{i\hbar} \langle[A,H]\rangle_\psi.$$

This is the expectation-value form of the Heisenberg equation for an operator with no explicit time dependence.

The metric contains the complementary symmetric information. Writing

$$\Delta_\psi A = A-\langle A\rangle_\psi I,$$

one finds

$$g_{\rm FS} \left( X_{f_A},X_{f_B} \right) = \frac{1}{2\hbar^2} \left\langle \{ \Delta_\psi A, \Delta_\psi B \} \right\rangle_\psi.$$

Thus the symplectic form encodes commutators, while the metric encodes symmetrized covariances. The two parts of the Hilbert-space inner product recover the antisymmetric and symmetric operator products.

Canonical Coordinates Before Reduction

The Hamiltonian structure can also be seen by treating $\mathcal H$ temporarily as a real vector space. Choose an orthonormal basis and write

$$\psi^k = \frac{q^k+ip_k}{\sqrt{2\hbar}}.$$

The imaginary part of the Hilbert-space inner product gives the canonical symplectic form

$$\Omega = \sum_kdq^k\wedge dp_k.$$

For the quadratic function

$$\mathcal H(q,p) = \langle\psi\vert H\vert\psi\rangle,$$

Hamilton’s equations are

$$\dot q^k = \frac{\partial\mathcal H}{\partial p_k}, \qquad \dot p_k = -\frac{\partial\mathcal H}{\partial q^k}.$$

Together these equations are equivalent to

$$i\hbar\dot\psi=H\psi.$$

Physical pure-state space is obtained in two steps. Normalization restricts to

$$\sum_k \left[ (q^k)^2+p_k^2 \right] = 2\hbar,$$

and quotienting the common phase removes the $U(1)$ orbit. This symplectic reduction produces projective Hilbert space with its Fubini–Study symplectic form.

This coordinate description explains why Schrödinger’s equation can resemble a large classical Hamiltonian system. It does not identify the variables $q^k,p_k$ with a particle’s classical position and momentum. They are real and imaginary parts of Hilbert-space amplitudes.

Qubit Example

Let

$$H = cI + \frac{\hbar\Omega}{2} \mathbf n\cdot\boldsymbol{\sigma}, \qquad \lVert\mathbf n\rVert=1.$$

A pure qubit ray corresponds to a unit Bloch vector $\mathbf r$, and the Hamiltonian function is

$$h(\mathbf r) = c + \frac{\hbar\Omega}{2} \mathbf n\cdot\mathbf r.$$

Choose $\mathbf n$ along $z$. In Bloch angles,

$$\mathbf r = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta),$$

and

$$\omega_{\rm FS} = \frac{\hbar}{2} \sin\theta\, d\theta\wedge d\phi.$$

Because

$$h = c+\frac{\hbar\Omega}{2}\cos\theta,$$

the equation $\iota_{X_h}\omega_{\rm FS}=dh$ gives

$$\dot\theta=0, \qquad \dot\phi=\Omega.$$

In coordinate-free form,

$$\dot{\mathbf r} = \Omega\, \mathbf n\times\mathbf r.$$

The ray follows a circle around the Hamiltonian axis. The scalar $c$ changes only the vector’s overall phase and disappears from the Bloch motion. This is the geometric form of spin precession.

Stationary Rays and Critical Points

A ray is stationary exactly when its Hamiltonian vector field vanishes:

$$X_h([\psi])=0.$$

From the explicit vector field,

$$\left( H-\langle H\rangle_\psi \right) \lvert\psi\rangle =0.$$

Thus stationary rays are energy eigenrays:

$$H\lvert\psi\rangle = E\lvert\psi\rangle, \qquad E=\langle H\rangle_\psi.$$

Equivalently, they are critical points of the expectation-value function $h$. If an eigenvalue is degenerate, every ray in the corresponding eigenspace is fixed, so the critical set is a projective subspace rather than an isolated point.

The local curvature of $h$ near a critical ray reflects energy gaps. This observation connects geometric dynamics with stability analysis, variational methods, and quantum speed, although those developments require more machinery than the fixed-point criterion itself.

Comparison with Classical Hamiltonian Mechanics

The structural dictionary is exact at the symplectic level:

Classical Hamiltonian mechanics	Geometric pure-state quantum mechanics
state $x\in M$	ray $[\psi]\in\mathbb P(\mathcal H)$
symplectic form $\omega$	$\omega_{\rm FS}$
observable $f(x)$	expectation function $f_A([\psi])$
Hamiltonian vector field $X_H$	projected Schrödinger vector field $X_h$
$\dot f=\{f,H\}$	$\dot f_A=\{f_A,h\}_{\rm FS}$
canonical flow	unitary projective flow

The differences are equally important.

First, standard quantum observables form a restricted family of functions on $\mathbb P(\mathcal H)$. Not every smooth real function has the form $f_A$. In finite dimensions, operator expectation functions are distinguished geometrically by Hamiltonian flows that also preserve the Fubini–Study metric.

Second, quantum phase space has a compatible metric and complex structure, not merely a symplectic form. The metric carries transition probabilities and uncertainties.

Third, evaluating $f_A([\psi])$ gives an expectation value, not a deterministic measurement outcome. The Born rule and state-update theory are not replaced by Hamiltonian geometry.

Finally, $\mathbb P(\mathcal H)$ is generally not the classical phase space obtained in the limit $\hbar\to0$. A particle on a line has infinite-dimensional projective Hilbert space, while its classical phase space is the two-dimensional $(x,p)$ plane.

Applications and Limitations

The Hamiltonian-flow viewpoint is useful in several settings:

• Symmetries: unitary group actions on ray space are symplectic and isometric, and expectation values act as momentum-map components.
• Variational dynamics: restricting the quantum action to a parameterized state manifold produces Hamiltonian-like equations when the pulled-back symplectic form is nondegenerate; see Time-Dependent Variational Principle.
• Coherent states: suitable coherent-state submanifolds can approximate classical phase spaces and support semiclassical reduced flows.
• Quantum control: reachable pure-state motion can be studied through Hamiltonian vector fields generated by available controls.
• Stationary-state analysis: eigenstates become fixed points or critical manifolds of the energy expectation.

The formulation also has clear limits:

• It directly describes pure states and reversible closed dynamics, not general mixed states or noisy channels.
• An arbitrary Hamiltonian function on projective space can generate nonlinear dynamics that is not standard linear quantum mechanics.
• Infinite-dimensional projective Hilbert spaces require functional-analytic care, especially for unbounded Hamiltonians and their domains.
• A variational ansatz may have redundant parameters or a degenerate pulled-back two-form.
• Measurement probabilities and state updates require more than the symplectic flow.
• Geometrizing quantum dynamics does not by itself solve the quantum-to-classical limit.

Common Mistakes

• Using the classical particle phase space in place of projective Hilbert space.
• Omitting the subtraction of $\langle H\rangle_\psi$ in a horizontal representative of the ray velocity.
• Mixing Fubini–Study normalization conventions and introducing an incorrect factor of two.
• Calling the projective Poisson bracket a classical-limit approximation. Its relation to the commutator is exact.
• Assuming every smooth projective-space function is a standard quantum observable.
• Interpreting expectation values as deterministic measurement outcomes.
• Forgetting that adding $cI$ changes vector phase but not projective dynamics.
• Applying the pure-state Hamiltonian picture directly to dissipative open-system evolution.
• Ignoring domains when $H$ is unbounded.

References

• T. W. B. Kibble, “Geometrization of quantum mechanics,” Communications in Mathematical Physics 65, 189–201, 1979.
• A. Ashtekar and T. A. Schilling, “Geometrical formulation of quantum mechanics,” in On Einstein’s Path, Springer, 1999; arXiv:gr-qc/9706069.
• A. Heslot, “Quantum mechanics as a classical theory,” Physical Review D 31, 1341–1348, 1985.
• J. Anandan and Y. Aharonov, “Geometry of quantum evolution,” Physical Review Letters 65, 1697–1700, 1990.
• D. C. Brody and L. P. Hughston, “Geometric quantum mechanics,” Journal of Geometry and Physics 38, 19–53, 2001.
• I. Bengtsson and K. Życzkowski, Geometry of Quantum States, 2nd ed., Cambridge University Press, 2017.

Exercises

1. Derive the horizontal Hamiltonian vector field from the symplectic form.

Solution

For a horizontal variation $u$,

$$dh(u) = 2\operatorname{Re} \langle u\vert \left( H-\langle H\rangle_\psi \right) \vert\psi\rangle.$$

Set

$$X_h = -\frac{i}{\hbar} \left( H-\langle H\rangle_\psi \right) \lvert\psi\rangle.$$

It is horizontal because its overlap with $\lvert\psi\rangle$ vanishes. Moreover,

$$\begin{aligned} \omega_{\rm FS}(X_h,u) &= 2\hbar\, \operatorname{Im} \langle X_h\vert u\rangle\\ &= 2\operatorname{Re} \langle u\vert \left( H-\langle H\rangle_\psi \right) \vert\psi\rangle\\ &= dh(u). \end{aligned}$$

Hence $\iota_{X_h}\omega_{\rm FS}=dh$.

2. Show that the Poisson bracket of expectation functions reproduces the commutator.

Solution

Let

$$\Delta A=A-\langle A\rangle_\psi I, \qquad \Delta B=B-\langle B\rangle_\psi I.$$

Then

$$X_{f_A} = -\frac{i}{\hbar}\Delta A\lvert\psi\rangle, \qquad X_{f_B} = -\frac{i}{\hbar}\Delta B\lvert\psi\rangle.$$

Therefore

$$\begin{aligned} \{f_A,f_B\}_{\rm FS} &= 2\hbar\, \operatorname{Im} \langle X_{f_A}\vert X_{f_B}\rangle\\ &= \frac{2}{\hbar} \operatorname{Im} \langle\psi\vert\Delta A\Delta B\vert\psi\rangle\\ &= \frac{1}{i\hbar} \langle\psi\vert[A,B]\vert\psi\rangle. \end{aligned}$$

Constants cancel from the commutator, so $[\Delta A,\Delta B]=[A,B]$.

3. Prove that stationary rays are precisely energy eigenrays.

Solution

A stationary ray has

$$X_h([\psi])=0.$$

Using the horizontal representative,

$$\left( H-\langle H\rangle_\psi \right) \lvert\psi\rangle =0.$$

Thus $\lvert\psi\rangle$ is an eigenvector of $H$ with eigenvalue $\langle H\rangle_\psi$.

Conversely, if $H\lvert\psi\rangle=E\lvert\psi\rangle$, then $\langle H\rangle_\psi=E$ and $X_h=0$. Schrödinger evolution changes only the representative’s phase:

$$\lvert\psi(t)\rangle = e^{-iEt/\hbar}\lvert\psi(0)\rangle.$$

4. Derive the Bloch-sphere flow for

$$H=\frac{\hbar\Omega}{2}\sigma_z.$$

Solution

The Hamiltonian function and symplectic form are

$$h = \frac{\hbar\Omega}{2}\cos\theta, \qquad \omega_{\rm FS} = \frac{\hbar}{2} \sin\theta\, d\theta\wedge d\phi.$$

Write

$$X_h = \dot\theta\,\partial_\theta + \dot\phi\,\partial_\phi.$$

Then

$$\iota_{X_h}\omega_{\rm FS} = \frac{\hbar}{2}\sin\theta \left( \dot\theta\,d\phi - \dot\phi\,d\theta \right),$$

while

$$dh = -\frac{\hbar\Omega}{2} \sin\theta\,d\theta.$$

Matching coefficients gives

$$\dot\theta=0, \qquad \dot\phi=\Omega.$$

The Bloch vector therefore precesses as

$$\dot{\mathbf r} = \Omega\,\hat{\mathbf z}\times\mathbf r.$$

5. Show directly that the real amplitude coordinates obey Hamilton’s equations.

Solution

Write

$$\psi^k = \frac{q^k+ip_k}{\sqrt{2\hbar}},$$

and let

$$\mathcal H(q,p) = \langle\psi\vert H\vert\psi\rangle.$$

For a Hermitian matrix $H$, differentiating the real quadratic form and using

$$\dot q^k = \frac{\partial\mathcal H}{\partial p_k}, \qquad \dot p_k = -\frac{\partial\mathcal H}{\partial q^k}$$

combines into

$$\dot q^k+i\dot p_k = -\frac{i}{\hbar} \sum_jH_{kj} \left( q^j+ip_j \right).$$

Dividing by $\sqrt{2\hbar}$ gives

$$\dot\psi^k = -\frac{i}{\hbar} \sum_jH_{kj}\psi^j,$$

or

$$i\hbar\dot\psi=H\psi.$$

Normalization fixes a sphere in the $(q,p)$ space, and quotienting the common phase gives the projective Hamiltonian system.