Fermi & Bose

Quantum mechanics reveals the world to be made of two types of objects, which we call Fermions after Enrico Fermi, and Bosons after Satyendra Nath Bose. These two species have radically different behavior, Fermions take up space (thus acting like matter), and Bosons do not - they can pile atop one another, magnifying their microscopic properies to macroscopic scales (underlying the fundamental forces of the world).

In physics, the difference between these two is often exemplified by giving a relation satisfied by multiparticle wavefunctions $\psi$ for a given particle species. A particle species is a boson if swapping two particles has no effect

$$\psi(y,x)=\psi(x,y)$$

and a particle is a fermion if swapping two particles results in a negative sign (more precisely, any odd permuation of particles contributes a sign change)

$$\psi(y,x)=-\psi(x,y)$$

That nature makes use of exactly these two radically different forms is apparent from both experience and experiment. But is this all she could have chosen? Or, in contrast, are there a multitude of mathematically possible particle species that are not realized in our world? Such things happen all the time - for instance, while mathematically the spin of an elementary particle can take on any value in $\frac{1}{2}\mathbb{Z}$, we seemingly find in our universe only the values $$0,\pm\frac{1}{2},\pm 1,\pm 2$$ (where the $0$ seems to only be realized by the Higgs, and $\pm 2$ is only conjectural, assuming the existence of the graviton).

The goal of this note is to show that in the case of particle statistics, nature truly does make use of all the variation available to her from mathematics: there are only two possible ways for particles to behave in three dimensional space. One thing we will not do in this note is show that these two behaviors (invariance of sign swapping upon permutation) entail the statistical behavior of exclusion or clumping we see in matter and forces - I still have to figure that out.

This note provides the second piece of a puzzle I’ve been trying to piece together over some time: understanding the spin-statistics theorem. A previous note deals with the phenomenology of spin - specifically, how the difference between representations and projective representations in math turns into the concept of integer spin and non-integer spin in quantum mechanics. In an aspirational third installment to this series, I’ll actually figure out how these two pieces are related! But that as well is for another time.

Geometric (Pre)-Quantization

• Describe the classical situation as a configuration space
• Goal: quantize the tangent bundle
• Functions vs sections of a line bundle
• Building a hilbert space
• Getting the operators (with the connection)

Identical Particles

• Unordered Configurations
• Line Bundles with perscribed curvature
• Classifying Flat Bundles and the Fundamental Group
• The special case of $n=2$ vs $n\geq 3$
• Two Line Bundles is all we have

Recovering a wave-function

How do we get a function that looks like the physicists, from our abstract line bundle picture?

The Necessity of Abstraction

Proof that we could not recover the Fermi statistics if we had not dealt with bundles, and had only thought about functions.

Anyons and Dimension 2

Work out what happens in dimension 2, when we have the braid group.

Appendix: Math

Line Bundles

Existence of a Line Bundle with a perscribed curvature: Weil 1958? If the curvature is a cohomology class, has to do with it being integral.

On a cotangent bundle, the cannonical symplectic form $\omega$ is the derivative of the symplectic potential $\omega = d\theta$, so it is exact. Thus $[\omega]=0$ in cohomology, so its integral and such a connection always exists.

How to find a connection $\nabla$ with $F_\nabla = \omega$? Start with some random connection $\tilde{\nabla}$, with $F_{\tilde{\nabla}}=\eta$. Because the space of connections is affine proof here.), we can write every other connection $\nabla$ as $\nabla = \tilde{\nabla}+\alpha$, and compute the curvature (copying this). This gives us an equation for $d\alpha$ in terms of $\eta$ and our desired curvature $\omega$ that we have to solve, but the existence of such a solution comes from cohomology.

Now, how many bundles are there with this curvature? You can tensor any flat bundle with it! Thus they are in 1-1 correspondence with flat bundles. These we can count, with the fundamental group representations into the fiber automorphism group.

How does tensoring work with connections?

https://diffgeom.subwiki.org/wiki/Tensor_product_of_connections

https://diffgeom.subwiki.org/wiki/Formula_for_curvature_of_tensor_product_of_connections

https://diffgeom.subwiki.org/wiki/Tensor_product_of_flat_connections_is_flat

Hilbert Spaces

The hilbert space tensor product is not the algebraic tensor product, but rather its completion. It also satisfies a universal property, but there’s a change in what the maps have to be (weakly Hilbert Schmidt). All here on wikipedia

To take the $n$-fold tensor product of a hilbert space with itself, we simply write

$$\bigotimes^n H$$

Inside such a tensor product we identify two subspaces via the natural action of the symmetric group $\mathrm{Sym}_n$ acting on the pure tensors by permuting indices. The symmetric algebra $S^nH$ are the vectors fixed by this map

$$S^n H = \left\{ v\in\bigotimes^n H \,\mid\, \sigma.v=v\,\,\forall \sigma\in\mathrm{Sym}_m \right\}$$

The alternating algebra is the set of vectors which transform under the sign representation of the symmetric group:

$$\Lambda^n H = \left\{ v\in\bigotimes^n H \,\mid\, \sigma.v=\mathrm{sgn}(\sigma)v\,\,\forall \sigma\in\mathrm{Sym}_m \right\}$$

Using the definition of the hilbert space tensor product, we can prove the following useful result about function spaces

$$L^2(X\times Y,\mathbb{C})\cong L^2(X,\mathbb{C})\otimes L^2(Y,\mathbb{C})$$

While most of the work here will actually be happening in $L^2$ of the space of complex sections to a bundle, I won’t need a theorem of this form in that langauge. We will first lift everything from the bundle over unordered configuration space to ordered configuration space. This space is simply connected! So any bundle with a flat connection over it is trivial.

Thus, we can chose a unit section once and for all, and identify all other sections with honest functions to $\mathbb{C}$. The remaining problem is that our space is not actually a product, but is a dense subset of a product (we’ve removed some hyperplanes). This (1) doesn’t have any affect on $L^2$ and (2) maybe we should think about continuous extension, if we were restricting to continuous things…. so either way we can replace unordered configurations with the full product

$$L^2(\mathrm{Conf}_n,\mathbb{C})\cong L^2(Q^n,\mathbb{C})$$ $$\cong\bigotimes^n L^2(Q,\mathbb{C})$$

Then we can look inside here for the wavefunctions that are the preimages of our two different bundles over unordered configuration space. One of them will be the alternating algebra, and the other is the symmetric algebra!