User:Linas/Arbatsky's principle unmasked

Arbatsky's principle unmasked
Arbatsky's principle sheds geometrical light on the nature of the exp map in mathematics. The exp map occurs in many contexts: it is the map that takes elements of a Lie algebra to the Lie group, it is the map that defines the flow of geodesics on Riemannian manifolds, and, in quantum mechanics, it is the map that takes a Hermitian operator to a unitary operator. Arbatsky's principle works in all of these contexts. It relates the motion of a vector under the exp map to the root-mean-square expectation value of the generator of that motion. Specifically, it states that the angular speed of the motion is exactly equal to the RMS or standard deviation of the generator. The idea of "angular speed" is a direct physical/geometrical notion. The idea of the standard deviation of a generator is somewhat algebraically abstract; standard deviations occur in many places (including the uncertainty principle), but can sometimes be intuitively opaque (especially in quantum mechanics).

The original statement is here. Below follows the restatement of the principle and its proof.

Arbatsky's principle
Let $$x_0$$ be a unit-length tangent vector on a (finite dimensional) manifold, or a vector in finite or infinite-dimensional Hilbert space, if you wish. Let $$G$$ be an element of an algebra, such as a Lie algebra. Alternately, let $$G$$ be a Hermitian operator on the Hilbert space, if you wish. Let $$U(t)=\exp (tG)$$ be the one-parameter group of motions generated by $$G$$ and the real parameter "time" $$t$$. In the case of Hilbert space, one writes $$U(t)=\exp itG$$ so that $$U(t)$$ is unitary.

Define $$x(t)=U(t) x_0$$ to be the motion of the vector $$x_0$$ under the influence of the generator $$G$$. Let $$v(t)=\dot{x}(t)=dx(t)/dt$$ be the velocity of the vector. Note that the velocity is just another vector. The component of the velocity that is perpendicular to $$x$$ is


 * $$v_\perp = v - x (x \cdot v)$$

where $$(x \cdot v)$$ is the dot product of $$x$$ and $$v$$ (the inner product for Hilbert spaces; the inner product on the tangent space induced by the metric for manifolds). The length of $$v_\perp$$ is given by


 * $$||v_\perp|| = \sqrt {v\cdot v - (x \cdot v)^2} $$

Arbatsky's principle states that the scalar $$||v_\perp||$$ is equal to the angular speed of $$x(t)$$, and is equal to the standard deviation of the generator $$G$$. To see the latter, we write


 * $$x(t)=e^{tG}x_0$$

so that the velocity is


 * $$v(t) = Gx(t)$$

This leads trivially to


 * $$(x \cdot v) = x \cdot Gx$$

and


 * $$v^2 = (Gx) \cdot (Gx) = x \cdot G^TGx$$

so that


 * $$||v_\perp||=\sqrt{ x \cdot G^TG x - (x \cdot Gx)^2}$$

Switching to quantum mechanical bra-ket notation, one writes $$|\psi\rangle=x_0$$ so that


 * $$||v_\perp||=\sqrt{\langle\psi |G^TG|\psi\rangle -

\langle\psi |G|\psi\rangle^2 }$$

which is clearly the standard deviation of $$G$$. That this deserves to be called an angular speed follows from the properties of the dot product. For two unit-length vectors a and b, their dot product is just the cosine of the angle between them:


 * $$\theta = \arccos (a \cdot b)$$

This angle in fact defines a metric on projective Hilbert space and is known as the Fubini-Study metric. Let $$a=x(t)$$ and $$b=x_0$$. The angular speed is then


 * $$\omega = \frac{d\theta}{dt}$$

and its not hard to show that
 * $$\omega = ||v_\perp||$$

(Caution: state vectors in quantum mechanics are usually normalized to unit length; a normalization factor may be missing in this last formula.)

Thus we have the desired relationship: the angular speed of a vector being transported by a one-parameter group generated by G is the standard deviation of G (with respect to the vector).

To gain the physical intuition of this statement, pick your favorite exp map, be it on a Lie group or a Hilbert space, and plug in some examples. Arbatsky made a particularly interesting choice for G: he picked the Hamiltonian H, which is the time-evolution operator in quantum mechanics. The result is that the angular speed of any given state vector is given by the uncertainty in energy of that state.