### Phase Spaces and Symplectic Geometry

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 , volume , and temperature , all of which are related by the following *equation of state*:

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 allows access of information of states at other times . 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:

best written in this form

The classical mechanical state is now the pair , which can be solved for at any time .

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

with equations of motion

Note that in this case, the classical mechanical state is the pair 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:

If , then can write

Note that the square matrix on the RHS is important in matching the and -components on both sides and getting the right signs. The matrix 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 encodes (in a way) the equations of motion.

Generally, the matrices satisfying are called *symplectic matrices* in general. From the condition, we also have that symplectic matrices have the property .The matrix itself obeys this condition by noting . An important case is when is a transformation matrix, say, for the transformation . One obvious example of such transformation is that arising from simple translations of and – these are simply linear ones. We can have however interesting nonlinear ones.

Consider transformation and for the harmonic oscillator. Then

One easily checks that

satisfies the symplectic condition and . This is in fact the transformation from the position representation to Fock or number representation.

So far belong to vector spaces. In general however, can belong to a nonlinear space . To enable the advantage of the earlier linear structure, one creates vectors on by taking tangent vectors to points in . In some local neighbourhood, in fact, one can set up local coordinates, making looks like . We can thus defne vectors at by

Einstein summation convention implied there. Note that one can easily check that vector space operations hold for such objects. So at each point , we have *tangent space* , which is a vector space. The collection of all such vector spaces at all points give rise to the *tangent bundle* .

Dual to vectors are covectors which are really *linear functionals* of vectors, mapping vectors into the reals. Thus one can form the *cotangent space* , dual to . A *covector* is written as . The duality can be established from defining the contraction of their bases:

Note that the space is a vector space and by the same token, the collection of all such vector spaces form the *cotangent bundle* . From here, one can initiate the tensor industry, forming much richer structure through the use of *tensor product* (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:

Contracted with the basis of vectors gives

The product forms the basis for the space of *two-forms* which are really antisymmetric tensors of covariant rank two. One can generalize this further by defining the space of *k-forms* such that

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 that can be coordinatized by . In the Hamiltonian formalism, it is cotangent bundle that can be coordinatized by . The use of Hamiltonian formalism is preferred due to its inherent symplectic nature. On the fibres of , we have the covectors . It is sometimes useful to take as the fibre coordinate function

analogous to the coordinate function on the configuration space. This is so from the following action on the fibre

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

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

Going back to Hamilton’s equations

one can generalise these to any observables :

What we have just defined is the *Poisson bracket* structure:

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

So far we treated the phase space as a manifold and have not define any structures on it. We can do so now by writing as

Note that are basis vectors on . Thus, belongs to the tangent space (the tangent space to the cotangent bundle). Generalizing, we have in this tangent space

Next, we dualize. Suppose , then we can write:

Contracting with gives

Consider now a two-form on i.e.

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

which is in . Further contracting with another vector field gives

which is precisely the Poisson bracket ! 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 with a symplectic form is called a *symplectic manifold* . 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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