Wave Function As a Bundle
While going through some notes, I came across an interesting relation between the wave function and bundles.
Namely that: What one tends to call the wave function (in quantum mechanics), in a certain sense is not at all a function, rather it is a section of a ℂ-line bundle over physical space. It is only under the assumption that this ℂ-line bundle is locally trivial, and hence each section of this bundle can be represented by a map from the base space to the total space, which ultimately lets one call it a wave function. Now obviously this seems a rather unnecessary, highbrow notion and flys over ones head from unfamiliarity, but it is rather intriguing to poke into.
Wave
Firstly, the wave function 𝜓 is a (mathematical) description of a quantum state of a particle as a function of some observable, e.g., momentum, position. One can say that the |𝜓|² is a probability density of a particle being at a certain place, or having a certain momentum, at a given time.
In a naive standard sense, it is a complex valued function, a function on some physical space into the complex: (ℝ^d) → ℂ.
Bundle Stuff
As a quick run down, a bundle is a triple (E, 𝝅, M) where E and M are topological manifolds, E being the total space, M being the base space, and 𝝅 being a continuous surjective map from the total to the base space. Additionally, Fp is the fibre at the point p, and is the pre-image of p or 𝝅^-1(p), more intuitively, Fp are the set of points in E (in the above diagram, we have a line) attached to the point p, and the map 𝝅 sends all these points in E to the single point p in M. Lastly, we have the section, which is a map 𝛔, that sends the point p from M into the fibre Fp, such that 𝝅 ∘ 𝛔 is equal to the identity map on M (𝝅 ∘ 𝛔 sends the point back to the point).
Now to the notion of triviality. A bundle is trivial, if it is isomorphic to the product bundle, where the product bundle is just the triple (M×N, 𝝅, M), and M×N is the cartesian product of two manifolds M and N. On the other hand, a bundle is locally trivial, if it is locally isomorphic to a product bundle, where locally isomorphic means that for all points p in the base space M, one can find a neighbourhood U(p) such that we have 𝝅^-1(U(p)) → U(p), under the map 𝝅_[𝝅^-1(U(p))], is isomorphic to the product bundle (the neighbourhood restriction is called the restricted bundle). Put simply, one can take portions (hence local) over all of the base space M, which are isomorphic to the product bundle. Isomorphism between two bundles just means that the diagram below commutes (v ∘ 𝝅 = 𝝅’ ∘ u, and 𝝅 ∘ u^-1 = v^-1 ∘ 𝝅’ ).
Ride the Wave
With the mathematical machinery roughly outlined, we can now put it all together to view the wave function. A rough outline is shown in the diagram below.
Physical space is represented by ℝ^d, the base space. From this we create a vector fibre bundle, which is just a generalization of the concept from the previous section, namely that every fibre is a vector space, in this case a complex one. Hence, we can use the section 𝛔 to create the required wave function 𝜓.
Now (locally) this looks like a function over ℝ^d into ℂ, which is exactly what we had in the standard case.
Further
This whole notion can be further extended with the use of associated bundles, frame bundles and plenty other machinery to generalize and derive an actual benefit, mainly that when using a different coordinate system or coordinate chart the wave function does not transform like a complex valued function. Hence, one gets a more definitive idea of how quantum mechanics operates on curved spaces, and arbitrary coordinates, etc, etc.
