Phase Spaces and Symplectic Geometry

03Dec10

These are notes from my Monday nite Musings on October 25, 2010.

What are phase spaces? There are many different usages of this term but generally they can be said to be simply state spaces. Then the next question is what are states? A good brief discussion can be found in Isham’s nifty book. Enough to say that states are essentially the optimal information regarding a physical system and this is very much model-dependent. It’s optimal in the sense that it is not superfluous but enough to generate other information of the system. Note that in general, states are often described by the “properties” of a system and they are necessarily expressed in numbers. Most obvious case is the ON-OFF state but this often given the numerical representation 1, 0. Pushing to the extremes, one can talk for example about the “mental states” of happy, sad or indifferent which certainly have no obvious numerical representations. The advantage of using numbers is the properties that go with them e.g. ordering, performing calculus. A good high-school example is the state of an ideal gas, which can be any two of the observables pressure $P$ , volume $V$ , and temperature $T$ , all of which are related by the following equation of state:

$\cfrac{PV}{T} = \textrm{constant}\quad.$

In fact, in many cases, equations of state are actually sought for, to help define the state of the physical system. Another important property of the states is the fact that the knowledge of one state at an initial time $t_0$ allows access of information of states at other times $t$ . This is often possible through equations of motion.

Let us consider the familiar Newtonian mechanics with the celebrated second law for the equation of motion:

$\underline{F} = \cfrac{d\underline{p}}{dt}\quad,$

best written in this form

$\cfrac{d\underline{r}}{dt} = \cfrac{\underline{p}}{m} = \cfrac{\partial}{\partial\underline{p}}\left(\cfrac{\underline{p}^2}{2m}\right)\quad ;$

$\cfrac{d\underline{p}}{dt} = - \underline{\nabla} V(\underline{r}) = - \cfrac{\partial}{\partial\underline{r}} V(\underline{r})\quad .$

The classical mechanical state is now the pair $(\underline{r}, \underline{p})$ , which can be solved for at any time $t$ .

Alternatively, one could also use the Lagrangian mechanics formalism with Lagrangian function

$L = \frac{1}{2} m\dot{\underline{r}}\cdot\dot{\underline{r}} -V(\underline{r})$

with equations of motion

$\cfrac{d\underline{\dot{r}}}{dt} = \cfrac{\partial L}{\partial\underline{r}} = - \cfrac{\partial V}{\partial\underline{r}}\quad ;$

$\cfrac{d\underline{r}}{dt} = \cfrac{\partial L}{\partial\dot{\underline{r}}} = \cfrac{\partial}{\partial\dot{\underline{r}}} \left(\frac{1}{2} m\dot{\underline{r}}\cdot\dot{\underline{r}}\right)\quad .$

Note that in this case, the classical mechanical state is the pair $(\underline{r},\dot{\underline{r}})$ and the “symplectic nature” is no longer obvious or available directly (To resolve one needs to define the energy function in addition to the Lagrangian function).

What does one mean by the “symplectic nature” in the above? Let us rewrite our earlier equation of motion in matrix form:

$\cfrac{d}{dt} \left( \begin{matrix} \underline{r}\\ \underline {p} \end{matrix}\right) = \left( \begin{matrix}{\cfrac{\partial}{\partial\underline{p}} (\underline{p}\cdot\underline{p})/(2m)}\\ - \cfrac{\partial}{\partial\underline{r}} V(\underline{r}) \end{matrix} \right)\quad .$

If $H= (\underline{p}\cdot\underline{p})/(2m) + V(\underline{r})$ , then can write

$\cfrac{d}{dt} \left( \begin{matrix} \underline{r}\\ \underline{p} \end{matrix}\right) = \left( \begin{matrix} \cfrac{\partial H}{\partial\underline{p}}\\ - \cfrac{\partial H}{\partial\underline{r}} \end{matrix}\right) = \left( \begin{matrix} 0&I\\-I&0 \end{matrix}\right) \left(\begin{matrix} \cfrac{\partial H}{\partial\underline{r}}\\ \cfrac{\partial H}{\partial\underline{p}} \end{matrix} \right)\quad .$

Note that the square matrix $J$ on the RHS is important in matching the $\underline{r}$ and $\underline{p}$ -components on both sides and getting the right signs. The matrix $J$ can be called symplectic form – a more technical version will be given later. Here, in fact, if one assumes that the phase space vectors are in place, then $J$ encodes (in a way) the equations of motion.

Generally, the matrices $M$ satisfying $M^t JM= J$ are called symplectic matrices in general. From the condition, we also have that symplectic matrices have the property $\det M = \pm 1$ .The matrix $J$ itself obeys this condition by noting $J^t = -J$ . An important case is when $M$ is a transformation matrix, say, for the transformation $(\underline{r}, \underline{p})\rightarrow (\underline{R},\underline{P})$ . One obvious example of such transformation is that arising from simple translations of $\underline{r}$ and $\underline{p}$ – these are simply linear ones. We can have however interesting nonlinear ones.

Consider transformation $Q = \frac{1}{2}(q^2 + p^2)$ and $P=\tan^{-1}(p/q)$ for the harmonic oscillator. Then

$\cfrac{\partial Q}{\partial q} = q\quad\quad ;\quad\quad \cfrac{\partial Q}{\partial p} = p\ .$

$\tan P = \cfrac{p}{q}=\cfrac{p/\sqrt{p^2+ q^2}}{q/\sqrt{p^2 + q^2}}\equiv\cfrac{\sin P}{\cos P}\ ;$

$\Rightarrow\quad\cfrac{\partial P}{\partial q}= -\cfrac{p}{q^2\sec^2 P}= -\cfrac{p}{p^2 + q^2}\ .$

$\Rightarrow\quad\cfrac{\partial P}{\partial p}= \cfrac{1}{q\sec^2 P}=\cfrac{q}{p^2 + q^2}\ .$

One easily checks that

$M = \left( \begin{matrix} q&p\\ \cfrac{-p}{q^2 + p^2}&\cfrac{q}{q^2 + p^2}\end{matrix} \right)$

satisfies the symplectic condition $M^t J M = J$ and $\det M = 1$ . This is in fact the transformation from the position representation to Fock or number representation.

So far $\underline{r},\ \underline{p}$ belong to vector spaces. In general however, $\underline{r}$ can belong to a nonlinear space $Q$ . To enable the advantage of the earlier linear structure, one creates vectors on $Q$ by taking tangent vectors to points in $Q$ . In some local neighbourhood, in fact, one can set up local coordinates, making $Q$ looks like $\mathbb{R}^n$ . We can thus defne vectors at $q\in Q$ by

$v= v^i \cfrac{\partial}{\partial x^i}\in T_q Q\ .$

Einstein summation convention implied there. Note that one can easily check that vector space operations hold for such objects. So at each point $q$ , we have tangent space $T_q Q$ , which is a vector space. The collection of all such vector spaces at all points give rise to the tangent bundle $TQ=\bigcup_{q\in Q} T_q Q$ .

Dual to vectors are covectors which are really linear functionals of vectors, mapping vectors into the reals. Thus one can form the cotangent space $T_q ^* Q$ , dual to $T_q Q$ . A covector is written as $l = l_i\ dx^i\ \in\ T_q^* Q$ . The duality can be established from defining the contraction of their bases:

$\left\langle dx^i\ ,\ \cfrac{\partial}{\partial x^j} \right\rangle_q = \delta^i_j\ .$

Note that the space $T_q^* Q$ is a vector space and by the same token, the collection of all such vector spaces form the cotangent bundle $T^* Q=\bigcup_{q\in Q} T_q^* Q$ . From here, one can initiate the tensor industry, forming much richer structure through the use of tensor product $\otimes$ (will be skipped here).

Of interest here, is a special tensor product of covectors, that is antisymmetric. This is best demonstrated from the definition of a wedge product between two bases of covector:

$dx^i\wedge dx^j = dx^i \otimes dx^j - dx^j\otimes dx^i = - dx^j\wedge dx^i\ .$

Contracted with the basis of vectors gives

$dx^i\wedge dx^j\ \left(\cfrac{\partial}{\partial x^k}\ ,\ \cfrac{\partial}{\partial x^l}\right)=\ \delta^i_k \delta^j_l - \delta^j_k\delta^i_l\ .$

The product $dx^i\wedge dx^j$ forms the basis for the space of two-forms $\Lambda^2(Q)$ which are really antisymmetric tensors of covariant rank two. One can generalize this further by defining the space of k-forms $\Lambda^k(Q)$ such that

$\Lambda^k(Q)\ni\chi :\ T_q Q\times \cdots\textrm{k times}\cdots\times T_q Q\ \rightarrow\ \mathbb{R}\ .$

We will use these k-forms later to define a special structure for phase space.

We return now to our constructed phase spaces. In the Lagrangian formalism, the phase space is the tangent bundle $TQ$ that can be coordinatized by $\{ x^i, \cdot{x}^i\}$ . In the Hamiltonian formalism, it is cotangent bundle $T^*Q$ that can be coordinatized by $\{ x^i,p_i\}$ . The use of Hamiltonian formalism is preferred due to its inherent symplectic nature. On the fibres of $T^* Q$ , we have the covectors $p_i dx^i$ . It is sometimes useful to take $p_i$ as the fibre coordinate function

$p_i\equiv\cfrac{\partial}{\partial x^i}\ ,$

analogous to the coordinate function $x^i(q) = q^i$ on the configuration space. This is so from the following action on the fibre

$p_i(l) =\cfrac{\partial}{\partial x^i}\ (l_j dx^i) = l_j\ \delta_i^j = l_i\ .$

It is interesting to note that this coordinate function is reminiscent of the quantum momentum operator

$\hat{p}_i = - i\hbar\cfrac{\partial}{\partial x^i}\ ,$

though we are not sure whether there is any deep meaning in this.

Going back to Hamilton’s equations

$\cfrac{d}{dt}\ x^i = \cfrac{\partial H}{\partial p_i}\quad\quad ;\quad\quad \cfrac{d}{dt}\ p_i = - \cfrac{\partial H}{\partial x^i}\ ,$

one can generalise these to any observables $f= f(x^i,p_j)$ :

$\cfrac{df}{dt} = \cfrac{\partial f}{\partial x^i}\cfrac{dx^i}{dt}\ +\ \cfrac{\partial f}{\partial p_i}\cfrac{dp_i}{dt} = \cfrac{\partial f}{\partial x^i}\cfrac{\partial H}{\partial p_i}\ -\ \cfrac{\partial f}{\partial p_i}\cfrac{\partial H}{\partial x^i}\equiv \{ f, H\}\ .$

What we have just defined is the Poisson bracket structure:

$\{ f,g\} = \cfrac{\partial f}{\partial x^i}\cfrac{\partial g}{\partial p_i}\ -\ \cfrac{\partial f}{\partial p_i}\cfrac{\partial g}{\partial x^i}\ .$

Note that in this case, it is the Poisson bracket encodes the equation of motion instead of the symplectic matrix $J$ .

So far we treated the phase space $T^* Q$ as a manifold and have not define any structures on it. We can do so now by writing $df/dt = \{ f, H\}$ as

$\left( \cfrac{\partial H}{\partial p_i} \cfrac{\partial}{\partial x^i} - \cfrac{\partial H}{\partial x^i} \cfrac{\partial}{\partial p_i}\right) f\ \equiv\ \xi_H f\ .$

Note that $\partial/ \partial x^i,\ \partial/\partial p_i$ are basis vectors on $T^* Q$ . Thus, $\xi_H$ belongs to the tangent space $T_{(x,p)}(T^* Q)$ (the tangent space to the cotangent bundle). Generalizing, we have in this tangent space

$\cfrac{\partial f}{\partial p_i}\cfrac{\partial}{\partial x^i} - \cfrac{\partial f}{\partial x^i}\cfrac{\partial}{\partial p_i}\ =\ \xi_f\ .$

Next, we dualize. Suppose $l\in T^*_{(x,p)}(T^*Q)$ , then we can write:

$l = l_i^{(x)} dx^i + l^i_{(p)} dp_i\ .$

Contracting with $\xi_f$ gives

$\xi_f (l) = l(\xi_f) = \cfrac{\partial f}{\partial p_i}\ l_i^{(x)} - \cfrac{\partial f}{\partial x^i}\ l^i_{(p)}\ .$

Consider now a two-form on $T^* Q$ i.e.

$\omega = \sum_i dx^i\wedge dp_i \in \Lambda^2 (T^* Q)\ .$

This two-form is known as the symplectic form on the phase space $T^* Q$ . Note that if one contracts it with a vector field, one has

$\omega(\xi_f) = \cfrac{\partial f}{\partial p_i}\ dp_i + \cfrac{\partial f}{\partial x^i}\ dx^i = df$

which is in $T^*_{(x,p)} (T^*Q)$ . Further contracting with another vector field gives

$\omega(\xi_f , \xi_g) = - \cfrac{\partial f}{\partial p_i}\ \cfrac{\partial g}{\partial x^i} + \cfrac{\partial f}{\partial x^i}\ \cfrac{\partial g}{\partial p_i}$

which is precisely the Poisson bracket $\{ f, g \}$ ! Thus the chain of concepts from “equations of motion” to “Poisson bracket” to “symplectic form”.

Recall earlier the idea of canonical transformations (eliminating dependence on the particular coordinates of phase space) preserving the equations of motion. This is now replaced by symplectomorphisms that preserve the symplectic form, providing the needed generalization to a more abstract geometry. A manifold $M$ with a symplectic form $\omega$ is called a symplectic manifold $(M,\omega)$ . The study of symplectic manifolds is called symplectic geometry and it is the modern geometric language for classical mechanics. As such, it is also the starting point for several geometric approaches to quantization where classical systems are converted to quantum ones.

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Phase Spaces and Symplectic Geometry

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