/sci/ - Science & Math

>>9581789
Use proper mathematical rigour to understand the deeper physics of what you're doing. That's something we Mathematical Physicists practice in TQFT and related empirical fields. What you're saying sounds like what people would gather from skimming Griffith. Go read an actual QM book like Townsend, Sakurai or Landau-Lifshitz to obtain the physical intuitions you clearly lack at this point in your education.
I bet you're one of those people who think that what is essentially happening in TQFT is that we have an oversimplified toy model of an interaction on a small scale, but get all these interesting phenomena after re-normalizing to a larger scale. That's not really the case.
>>9581863
That's pretty cool. I've been doing similar stuff in TQFT lately, although it's obviously a lot more rigorous.

>>9481453
That's a new Yukari pic, mind if I save it?

>>9377447
It also needs to be modular and maps the empty cobordism to [math]\mathbb{C}[/math].
In general a TQFT is a tuple [math](\mathscr{T},\tau)[/math] where [math]\mathscr{T}[/math] is the functor you've mentioned and [math]\tau[/math] is the quantum invariant. [math]\mathscr{T}[/math] encodes all the categorical information (i.e. naturality and functoriality of glueing patterns) while [math]\tau[/math] makes sure the TQFT is a topological field theory (i.e. computes topological invariants with values in the R-module).
The category of 2-cobordisms can be finitely generated by cusps, pants and cylinders which satisfy certain algebraic relations, which is why 2D TQFTs are isomorphic to commutative Frobenius algebras. The quantum invariants is just the Euler characteristic.
In the case of 3D TQFTs more work is needed since the Euler characteristic is not a cobordism invariant. Knot theory and Dehn surgery theory has been used to embed ribbon graphs into [math]S^3[/math] and surgering 3-manifolds from it, which keeps track of 3-cobordisms in an invariant way. The quantum invariant in this case can then be defined via knot invariants.
4D TQFTs are yet harder, because there exists exotic 4-manifolds that are h-cobordant but does not admit diffeomorphic differential structures. This is a problem for studying 4D Yang-Mills from the TQFT/CFT point of view, and I'm my opinion presents one of the biggest roadblocks to the Yang-Mills gap problem. In fact it has been proven a few years back that 4D unitary TQFTs induce a non-positive definite inner product on [math]\mathbb{C}[/maths]-linear spaces, which is a fatal problem if you want your UTQFT to describe any physical theory. This is precisely due to the fact that the knot-based quantum invariants generalized from 3D TQFT is not a h-cobordism invariant on exotic Mazur 4-manifolds.