# Principal Bundle

## Angles in Quantum Mechanics

Posted in Physics, Quanum Mechanics by principalbundle on February 28, 2009

We know from intro quantum mechanics the positions $x$ and $y$ of a particle should be interpreted as operators. What happens when we transform to polar coordinates, $x=r \cos\theta$, $y=r\sin\theta$? Since $\theta$ 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, $L_z = -i \frac{\partial}{\partial\theta}$. And we can find the commuter $[L_z,\theta]=-i$. We also know $L_z$ has integer eigenvalues. Hitting each side of the commuter with eigenstates, we have

$(m-m') \langle m | \theta | m' \rangle = -i\delta_{m,m'}$.

Simple, right? No! This equation is false. When $m=m'$, it reads $0=1$.

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,

$\Delta L_z \Delta\theta \geq \hbar/2$.

However, eigenstates of $L_z$ have $\Delta L_z = 0$. Another contradiction! How to we resolve this?

Let’s recall our definition of $L_z$. 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 $\theta$.1 So we have to be careful when acting on a function like $\theta$, which is either multi-valued, or nonanalytic.

Because multivalued and nonanalytic operators are not allowed as observables, $\theta$ 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 $\theta$. So we can find the commutation relations

$[L_z,\sin\theta] = -i \cos\theta$
$[L_z,\cos\theta] = +i \sin\theta$

We can then find the uncertainty relations,

$\Delta L_z \Delta \sin\theta \geq \frac{1}{2} \langle\cos\theta\rangle$
$\Delta L_z \Delta \cos\theta \geq \frac{1}{2} \langle\sin\theta\rangle$

which is not a contradiction, because eigenstates of $L_z$ have $\langle\sin\theta\rangle = \langle\cos\theta\rangle = 0$.

1 How do we phrase this in terms of compactness? Functions that $L_z$ 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 $L_z$ 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 $L_z$ is self-adjoint, but I don’t know that this leads to a nicer description offhand.

## Cobordisms

Posted in Geometry, Math, Topology by principalbundle on February 22, 2009

A cobordism between two oriented $m$ manifolds $M$ and $N$ is an oriented $(m+1)$ manifold $W$ such that its boundary is $\partial W = \bar{M} \cup N$, where $\bar{M}$ indicates $M$ with reversed orientation.

Why do we have a reversed orientation?  For the same reason we have $\partial (X \times [0,1]) = \bar{X} \times 0 \cup X \times 1$.  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, $\int_{[a,b]} df = f(b)-f(a)$.

We can make this into a stronger relation called an h-cobordism.  An h-cobordism is a cobordism as defined above, where $W$ is homotopically $M \times [0,1]$.  From this we can state,

The h-Cobordism Theorem

Let $M^n$ and $N^m$ be compact, simply connected, cobordant, oriented m-manifolds.  If $m>5$, then there is a diffeomorphism $W = M \times[0,1]$, implying $M$ and $N$ 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.