Tag Archives: Hilbert spaces

A Slapdash Introduction to TQFTs from the Ground Up, Part III.

In parts I and II, I defined the notions of manifolds and cobordisms which make up some of the mathematical machinery for the general relativity side of topological quantum field theory. In particular, two manifolds M and N may be regarded as representing states of the macroscale physical universe and a cobordism B between them may be thought of as a worldsheet between one state and the other. That is to say, a cobordism can be used to represent the part of spacetime that passes between two such states.

We now turn our attention to the quantum mechanical part of the TQFT picture and define Hilbert spaces. Let \mathcal{H} be an inner product space over a field \mathbb{K} – for our purposes, we will always have that \mathbb{K}=\mathbb{R} or \mathbb{K}=\mathbb{C}. We say that \mathcal{H} is a Hilbert space if it is also a complete metric space with respect to the metric induced by the inner product \mathcal{H}\times\mathcal{H}\rightarrow\mathbb{K}. That is to say that \mathcal{H} is a topological space equipped with the distance function d(x,y)=||x-y||, where the right hand side of this equation is the norm induced by the inner product such that every Cauchy sequence (see any textbook on real analysis) of points in \mathcal{H} converges to a point in \mathcal{H}.

In particular, the quantum states – which, for instance, can be used to describe the likelihood of a particle being in a given position at a given time – of quantum mechanics may be represented as unit vectors in a special type of Hilbert space known as a state space.

A crucial detail about Hilbert spaces is the fact that if one has a linear transformation T:\mathcal{H}\rightarrow{H}^\prime between Hilbert spaces \mathcal{H} and \mathcal{H}^\prime, then the inner products on \mathcal{H} and \mathcal{H}^\prime give us a canonical way to obtain an “oppositve” linear transformation T^\ast:\mathcal{H}^\prime\rightarrow\mathcal{H} called the adjoint of T. One accomplishes this by defining T^\ast to be the unique linear transformation such that \left< Tv,w\right>_{\mathcal{H}^\prime}=\left<v,T^\ast w\right>_{\mathcal{H}}.

Thus we come to our first interesting parallel between general relativity and quantum mechanics. Given a cobordism (B;M,N,e,e^\prime), we may define the adjoint cobordism (B^\ast;N,M,e^\prime,e) by swapping the roles of the past an the future in B.

In Part IV, we dive headlong into some elementary abstract nonsense and then finally define a topological quantum field theory,

 

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