15.1 Vector fields and the exponential map

15.1 Vector fields and the exponential map

Concept links · terms present in this machine draft; source roles are unverified: vector space · Lie algebra · Lie group · Lie bracket

$A$ vector field on $M={\mathbf{R}}^2$ can be thought of as a choice of a two dimensional vector at each point in ${\mathbf{R}}^2$ , so given by a vector-valued function

$$\mathbf{F}(q,p)=\left(\genfrac{}{}{0ex}{}{F_q(q,p)}{F_p(q,p)}\right)$$

Such a vector field determines a system of diferential equations

$$\frac{dq}{dt}=F_q,\frac{dp}{dt}=F_p$$

Once initial conditions

$$q(0)=q_0,p(0)=p_0$$

are specified, if $F_q$ and $F_p$ are diferentiable functions these diferential equations have a unique solution $q(t),p(t)$ , at least for some neighborhood of $t=0$ (from the existence and uniqueness theorem that can be found for instance in [48]). These solutions $q(t),p(t)$ describe trajectories in ${\mathbf{R}}^2$ with velocity vector $\mathbf{F}(q(t),p(t))$ and such trajectories can be used to define the “flow” of the vector field: for each t this is the map that takes the initial point $(q(0),p(0))\in {\mathbf{R}}^2$ to the point $(q(t),p(t))\in {\mathbf{R}}^2$

Another equivalent way to define vector fields on ${\mathbf{R}}^2$ is to use instead the directional derivative along the vector field, identifying

$$\mathbf{F}(q,p)\leftrightarrow \mathbf{F}(q,p)\cdot \nabla =F_q(q,p)\frac{\partial }{\partial q}+F_p(q,p)\frac{\partial }{\partial p}$$

The case of F a constant vector is just our previous identification of the vector space $M$ with linear combinations of $\frac{\partial }{\partial q}$ and $\frac{\partial }{\partial p}$

An advantage of defining vector fields in this way as first-order linear diferential operators is that it shows that vector fields form a Lie algebra, where one takes as Lie bracket of vector fields $X_1,X_2$ the commutator

$$[X_1,X_2]=X_1{X}_2-X_2{X}_1\text{\hspace{1em}(15.1)}$$

of the diferential operators. The commutator of two first-order diferential operators is another first-order diferential operator since second-order derivatives will cancel, using equality of mixed partial derivatives. In addition, such a commutator will satisfy the Jacobi identity.

Given this Lie algebra of vector fields, one can ask what the corresponding group might be. This is not a finite dimensional matrix Lie algebra, so exponentiation of matrices will not give the group. The flow of the vector field $X$ can be used to define an analog of the exponential of a parameter $t$ times X:

Definition (Flow of a vector field and the exponential map)

The flow of the vector field $X$ on $M$ is the map

$${\Phi }_X:(t,m)\in \mathbf{R}\times M\to {\Phi }_X(t,m)\in M$$

satisfying

$$\frac{d}{dt}{\Phi }_X(t,m)=X({\Phi }_X(t,m))$$ $${\Phi }_X(0,m)=m$$

In words, ${\Phi }_X(t,m)$ is the trajectory in $M$ that passes through $m\in M$ at $t=0$ with velocity vector given by the vector field $X$ evaluated along the trajectory.

The flow can be written as a map

$$\exp (tX):m\in M\to {\Phi }_X(t,m)\in M$$

called the exponential map.

If the vector field $X$ is diferentiable (with bounded derivative), exp(tX) will be a well-defined map for some neighborhood of $t=0$ , and satisfy

$$\exp (t_{1}X)\exp (t_{2}X)=\exp ((t_1+t_2)X)$$

thus providing a one-parameter group of maps from M to itself, with derivative X at the identity.

Digression. For any manifold $M$, there is an infinite dimensional Lie group, the group of invertible maps from M to itself, such that the maps and their inverses are both diferentiable. This group is called the difeomorphism group of M and written Dif(M). Its Lie algebra is the Lie algebra of vector fields.

The Lie algebra of vector fields acts on functions on $M$ by diferentiation. This is the diferential of the representation of Dif(M) on functions induced in the usual way (see equation 1.3) from the action of Dif(M) on the space $M$. This representation however is not one of relevance to quantum mechanics, since it acts on functions on phase space, whereas the quantum state space is given by functions on just half the phase space coordinates (positions or momenta).

Chapter contents · Next section: 15.2 Hamiltonian vector fields and canonical transformations · 中文版本

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