As the GRE mathematics subject test is being administered this coming Saturday, I must delay any further postings until after said event. However, that being said, I will most certainly conclude my lengthy and rambling exposition on TQFTs hopefully not long thereafter. That is all.
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I currently sit, well-fed, warm, and comfortable in a room at my father’s house – on Summer break, no less – so it seems rather remarkable that I should have anything to complain about and yet I do, albeit one that I have technically brought upon myself. In a violent fit of naïveté this past Spring, I arrived at the rather bizarre conclusion that signing up for sixteen units – eight of which being graduate coursework and four of which being honors units – while also having to fill out graduate school applications and study for the GREs, was somehow a fabulous idea.
Having found myself thoroughly entrenched in applications and a measure theory homework assignment that is likely to amass as many as thirty pages a full eleven days prior to the start of Fall quarter, I’m beginning to second guess the wisdom of my decisions.
In reflection upon the title of this post: I have concluded that it is mandatory that my future home have a system of pneumatic tubes…yes.
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 and may be regarded as representing states of the macroscale physical universe and a cobordism 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 be an inner product space over a field – for our purposes, we will always have that or . We say that is a Hilbert space if it is also a complete metric space with respect to the metric induced by the inner product . That is to say that is a topological space equipped with the distance function , 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 converges to a point in .
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 between Hilbert spaces and , then the inner products on and give us a canonical way to obtain an “oppositve” linear transformation called the adjoint of . One accomplishes this by defining to be the unique linear transformation such that .
Thus we come to our first interesting parallel between general relativity and quantum mechanics. Given a cobordism , we may define the adjoint cobordism by swapping the roles of the past an the future in .
In Part IV, we dive headlong into some elementary abstract nonsense and then finally define a topological quantum field theory,
Recalling from Part I the notions of topological space and homeomorphism, we will now use these concepts to rigorously define an -dimensional manifold. This is a notion central to numerous fascinating branches of mathematics and physics but we shall restrict out attention to the details which will allow us to define a topological quantum field theory.
We do, however, require one more detail before we can proceed. We say that is a compact subset of if every cover of by open subsets of admits a finite subcover. This means that if is a collection of open sets with , then there is a finite collection with . Note that the existence of a finite cover for a set does not imply compactness.
We say that a topological space is a -dimensional manifold if it is a second countable Hausdorff space which is locally homeomorphic to -dimensional Euclidian space . We will not go over the notion of second countability here, since it would merely distract from the point, but interested readers are urged to consult any available text on point set topology for the definition. A Hausdorff space is a topological space such that if are distinct points, then there exist open sets with and and . A continuous function between topological spaces and is called a local homeomorphism if, for any point , there exists an open set such that is open and the restriction is a homeomorphism.
Informally, what this means is that if one chooses a point in a manifold , then it can always be separated from any other point and, if one looks closely enough, there is a neighborhood of which “looks like” ordinary -dimensional space. A smooth -manifold is a manifold which “looks enough like ” that you can “do calculus” on it. Additionally, a (smooth) manifold with boundary is defined in the same way as above with the half-space replacing .
We are now equipped with enough information to define a -cobordism. Formally a –cobordism is a quintuple consisting of a -dimensional compact smooth manifold , two compact -manifolds and with boundary, and two embeddings (“ways of inserting”) and of and into the boundary of such that , where denotes the union of disjoint sets.
For the sake of intuition, one may think of a cobordism as a “soap bubble” between two manifolds represented visually as loops of wire.
Cobordisms, or more precisely the category of cobordisms, make up one half of the machinery we need.
In Part III, I will go over Hilbert spaces and basic category theory.
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 from the category of -cobordisms to the category 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 be a set and denote by the set of subsets of . We say that is a topology on if it is closed under arbitrary unions and finite intersections and contains both the empty set and itself, and we refer to its elements as open sets. That is to say that if is a collection of open sets, then and if , then . A topological space is then an ordered pair consisting of a set and a topology on – by abuse of notation, we will often write when we mean .
Let and be topological spaces. We say that a function is continuous if, given an open set , the preimage of under is open.
A homeomorphism between topological spaces is a bicontinuous bijection from to . This means that is continuous, as is its inverse, and we additionally have that and if , then .
Good morning and welcome to my blog. Firstly, I wish to make note of the fact that the title of this little soapbox of mine is not misspelled: the name ‘Plum de Nom’ bears no relation to French literary terms – except by mere resemblance – and should be thought of solely in terms of drupes of the genus prunus.
The forecast for the foreseeable future is as follows: sporadic mathematical and academic ramblings, occasional complaints, and a chance of culinary digressions.