## Angles in Quantum Mechanics

We know from intro quantum mechanics the positions and of a particle should be interpreted as operators. What happens when we transform to polar coordinates, , ? Since classically becomes a dynamical variable, we expect it to become an operator quantum mechanically. What happens when we do this? The result is a very basic consequence of operators on a Hilbert space, but comes as a bit of a surprise, since these types of issues are not normally discussed in quantum mechanics classes or texts.

We can define the angular momentum operator, . And we can find the commuter . We also know has integer eigenvalues. Hitting each side of the commuter with eigenstates, we have

.

Simple, right? No! This equation is false. When , it reads .

What has gone wrong?

To answer this, let’s look at another question. From the commutation relation above, we can, by using the generalized uncertainty principal, deduce,

.

However, eigenstates of have . Another contradiction! How to we resolve this?

Let’s recall our definition of . It is an operator in a Hilbert space which acts on functions. But which functions does it act on? It only acts on functions which are periodic in .^{1} So we have to be careful when acting on a function like , which is either multi-valued, or nonanalytic.

Because multivalued and nonanalytic operators are not allowed as observables, should not be thought of an observable over its whole range.

The obvious resolution to this is to only consider the analytic functions made from . So we can find the commutation relations

We can then find the uncertainty relations,

which is not a contradiction, because eigenstates of have .

^{1} How do we phrase this in terms of compactness? Functions that acts on take their values from a compact set, however the domain isn’t the set of functions whose domain is compact. So something else about the topology of the domain of functions that acts on is needed, but I don’t know enough functional analysis to have an obvious more elegant way to write this. We can phrase this in terms of when is self-adjoint, but I don’t know that this leads to a nicer description offhand.

## Cobordisms

A **cobordism **between two oriented manifolds and is an oriented manifold such that its boundary is , where indicates with reversed orientation.

Why do we have a reversed orientation? For the same reason we have . Which is a consequence of how the orientation of a manifold induces an orientation on the boundary. This is the same as the source of the minus sign in the fundamental theorem of calculus, .

We can make this into a stronger relation called an **h-cobordism**. An h-cobordism is a cobordism as defined above, where is homotopically . From this we can state,

**The h-Cobordism Theorem**

Let and be compact, simply connected, cobordant, oriented m-manifolds. If , then there is a diffeomorphism , implying and are diffeomorphic.

The above statement and proof won Smale a fields medal in the 1960s.

What about in fewer dimensions? In 2-dimensions, this is equivalent to the Poincare conjecture, which has been proven to be true. However in 4-dimensions the theorem is false.

Does this theorem have anything to do with physics? Can string interactions be thought of as cobordisms? From the picture we draw of string interactions, I would not think it was unreasonable if someone told me that interactions could be viewed as cobordisms where the in/out state manifolds aren’t diffeomorphic, but it is not obvious to me if this is actually the case, or if one can do anything useful with it.

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