How to derive Quantum Mechanics in curved physical space?

[victorvmotti]

TL;DR Summary

We want to "lift" the pulled-back connection one-form on the base manifold to the total space, but how?!

I am following [this YouTube lecture by Schuller][1] where he finds the appropriate formalism for the quantum mechanics in the physical curved space.

Everything makes sense to me but at the very end I see that we find the pull backed connection one-form on the base manifold.

He says to the end of the lecture that later on we will see how we can find the connection one-form on the total space, the principal bundle, i.e., the frame bundle, level.

This is not shown, however, in the next video.

I wonder how can we calculate the connection one-form on the total space when we only have information about the pulled back of this form on the base manifold through a section map?

If we have a tangent vector on the base manifold clearly we can push it forward via the section map. But how about when we have defined a form, as we cannot push it forward from the base manifold to the total space via the section map.Is the following a good way to illustrate in the case of a flat base space how to derive the connection one-form on the total space from the given pulled-back connection one-form on the base manifold?

Consider the trivial bundle $\mathbb{R}^2 \times G \to \mathbb{R}^2$, where $G$ is a Lie group, and let $E$ denote the principal G-bundle with fiber $G$. We can think of $E$ as the bundle whose fiber over each point in $\mathbb{R}^2$ is just $G$.

Suppose we have a connection one-form $\widetilde \omega$ on the base manifold $\mathbb{R}^2$ that takes values in the Lie algebra of $G$. This connection one-form is given by a one-form on $\mathbb{R}^2$ whose values are elements of the Lie algebra of $G$ at each point.

Now suppose we have a section $s:\mathbb{R}^2 \to E$ of the principal G-bundle $E$. We can think of $s$ as a map that assigns to each point in $\mathbb{R}^2$ an element of $G$. We can also think of $s$ as a map that takes a point in $\mathbb{R}^2$ and "lifts" it to a point in $E$ by taking the point in $\mathbb{R}^2$ and mapping it to the point in the fiber over that point determined by $s$.

To derive the connection one-form on the total space, we want to "lift" the pulled-back connection one-form on the base manifold to $E$. We can do this as follows:

1/ Given a point $p \in E$, we can use the section map $s$ to identify the point $q=s^{-1}(p)$ in $\mathbb{R}^2$.

2/ We can then evaluate the pulled-back connection one-form $\widetilde{\omega} = s^* \omega$ on $\mathbb{R}^2$ at the point $q$. Since $\widetilde{\omega}$ takes values in the Lie algebra of $G$, this evaluation gives us an element of the Lie algebra of $G$.

3/ We can then "lift" this element to a Lie algebra-valued one-form on $E$ by extending it trivially in the direction transverse to the fiber. Specifically, for any vector $v$ tangent to $E$ at $p$, we can define the value of the lifted connection one-form at $p$ in the direction of $v$ to be the element of the Lie algebra of $G$ we obtained in step 2.

By repeating this process for all points in $E$, we obtain a connection one-form on $E$ that reduces to $\omega$ when restricted to any fiber of $E$. This connection one-form satisfies all the conditions required of a connection and can be used to define parallel transport and curvature on $E$.

[1]:

 

[strangerep]

Well, this is not really "QM in curved space", but rather how to do QM covariantly in general curvilinear coordinates.

In any case, you're seriously over-thinking it.

That lecture is a rather excruciatingly tedious way of discovering that the ordinary QM wave functions are not scalar-valued functions, but scalar-density-valued functions. (These lectures also show why one should lecture from notes, rather than trying to reproduce everything off the cuff - which leads to frequent mistakes on the board.)

Schuller's $\omega_\alpha$ is just $\Gamma^\lambda_{~\alpha\lambda}$, where $\Gamma$ is the usual Levi-Civita affine connection. Although the usual covariant derivative acts on a function $f$ as $\,\nabla_\alpha f = \partial_\alpha f$, it acts on a scalar-density $\psi$ as$\,\nabla_\alpha \psi = \partial_\alpha \psi + \Gamma^\lambda_{~\alpha\lambda} \psi\,$. (You can google for "covariant derivative of scalar density" for more detail on this.)

All the gumph about passing between a "section in the frame bundle" and a field on the base manifold is essentially just a high-falutin' way of obscuring the usual non-tensorial transformation rule for connection components.

 

[LittleSchwinger]

OP another word to look for is "line bundle". It's essentially the same thing as a scalar density bundle under a different name. Many abstract treatments of QM will talk about how the wavefunction is a section of a line bundle.

 

[dextercioby]

The question is so general, that it would take someone 15 pages to try to answer it. I urge the OP to get a textbook on this topic. Sniatycki, J. - Geometric Quantization And Quantum Mechanics (Springer, 1980).

 

[LittleSchwinger]

Another standard reference is:
Woodhouse N.M.J. "Geometric Quantization" 2nd edition, OUP

 

[dextercioby]

It's the so-called Kostant-Souriau formulation of QM. Schuller's presentation is nice, but did he put them in a book? I want to compare his text with the classics.

 

[victorvmotti]

Answer provided by Ted Shifrin:


You don't push forward forms, of course. Here's the idea: Cover $M$ by open sets $U_\alpha$ over which the frame bundle $P$ is trivial, and let $s_\alpha\colon U_\alpha\to P$ be sections for all $\alpha$. Define the projection $\psi_\alpha\colon P|_{U_\alpha}\to G$ for all $\alpha$. *Provided* that we have

$\omega_\beta = g_{\alpha\beta}^{-1}\omega_\alpha g_{\alpha\beta} + g_{\alpha\beta}^{-1}dg_{\alpha\beta} \quad\text{with } s_\beta=g_{\alpha\beta}\cdot s_\alpha \text{ on } U_\alpha\cap U_\beta,$

then we define the $\mathfrak g$-valued $1$-form $\omega$ on $P$ by

$\omega = (\text{Ad}\,\psi_\alpha^{-1})\pi^*\omega_\alpha + \psi_\alpha^*\phi,$

where $\phi$ is the left-invariant Maurer-Cartan form on $G$.

I leave you to check well-definedness. You can find this all done carefully in Kobayashi-Nomizu (section 1 of Chapter II).