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September 9, 2013 · 4:32 pm

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,

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

Topological quantum field theory is an interest of mine which has grown in intensity over the last year. I have, however, made the unfortunate mistake of attempting to describe some of these concepts to family and friends with results ranging from eyes-glazed-over incomprehension to “That’s really cool but my brain hurts now.” As a result of this, I have decided to try to improve my exposition by way of writing a cursory introduction to the subject requiring little more than a familiarity with sets, functions, and vector spaces.

Disclaimer: the notes presented herein should not be taken to be rigorous or comprehensive in any way (or, even, necessarily coherent). I refer the interested reader to V. Turaev’s Quantum Invariants of Knots and 3-Manifolds or any of John Baez’s excellent papers on the topic. Although this post is primarily mathematical in its intent, I will follow the latter’s approach to the heuristics of topological quantum field theory.

For the physically inclined, topological quantum field theory – or, more properly, the theory of topological quantum field theories, which I shall abbreviate as TQFTs – makes rigorous certain analogies between Einstein’s theory of general relativity (i.e. gravity) and quantum mechanics. Historically, topological quantum field theory arose out of attempts to make rigorous the notion of a Feynman path integral, a problem which has proven enormously difficult owing to the nonexistence of certain complex valued measures. (I may write a post on measure theory at some point, though I reserve the right not to.)

Formally a topological quantum field theory is a functor $\mathcal{F}:\boldsymbol{nCob}\rightarrow\boldsymbol{Hilb}$ from the category $\boldsymbol{nCob}$ of $n$ -cobordisms to the category $\boldsymbol{Hilb}$ of Hilbert spaces obeying certain axioms. However, this definition requires a great deal of machinery largely inaccessible to the layman and even to most undergraduate students of either mathematics or physics so we must first develop that here.

In order to accomplish this, corners must be cut and many essential theorems completely ignored but that is, sadly, the price one must pay for even the vaguest semblance of brevity.

We begin, then, with the notion of a topological space. Let $X$ be a set and denote by $2^X$ the set of subsets of $X$ . We say that $\mathcal{T}\subset 2^X$ is a topology on $X$ if it is closed under arbitrary unions and finite intersections and contains both the empty set $\varnothing$ and $X$ itself, and we refer to its elements as open sets. That is to say that if $\{U_\alpha\}_{\alpha\in A}\subset\mathcal{T}$ is a collection of open sets, then $\bigcup_{\alpha\in A}U_\alpha\in\mathcal{T}$ and if $U,V\in\mathcal{T}$ , then $U\cap V\in\mathcal{T}$ . A topological space is then an ordered pair $(X,\mathcal{T})$ consisting of a set $X$ and a topology $\mathcal{T}$ on $X$ – by abuse of notation, we will often write $X$ when we mean $(X,\mathcal{T})$ .

Let $X$ and $Y$ be topological spaces. We say that a function $f:X\rightarrow Y$ is continuous if, given an open set $U\subset Y$ , the preimage $f^{-1}(U):=\{x\in X:f(x)\in U\}\subset X$ of $U$ under $f$ is open.

A homeomorphism $f:X\rightarrow Y$ between topological spaces is a bicontinuous bijection from $X$ to $Y$ . This means that $f$ is continuous, as is its inverse, and we additionally have that $f(X)$ and if $f(x)=f(y)$ , then $x=y$ .

Tag Archives: TQFT

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

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

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