# Geometric derivation of quantum mechanics from Lagrangian mechanics

I have used classical Lagrangian mechanics for quite a while, and what I like about it is that everything can be derived from a very small number of geometric principles. There are just three things you need to "take on faith":

1. That configuration space should be endowed with a Riemannian metric given by mass;

2. That forces arise from scalar potentials on configuration space;

3. Hamilton's principle of extremal action.

The last is perhaps most objectionable: why should the universe want to evolve in a way that minimizes the action? I have no idea, and my understanding is that nobody else does either. But if I assume these hypotheses, everything else follows mechanically: equations of motion, Noether's theorem, etc.

What I would like to do is to learn quantum mechanics, starting from similar geometric principles. For example, I might take as a key principle that configurations should be probability distributions over configuration space, rather than single points, but this doesn't lead to quantum mechanics; I also need (for some reason?) to instead take configurations to be complex-valued functions over configuration space. But even then I don't see how to get anything like Schroedinger's equation from a (modification of) Hamilton's principle.

I did some searching and I've found that there is indeed some variational foundation for quantum mechanics (and Feynman seemed to have championed this formulation) but the articles I've seen so far assume I already understand quantum mechanics.

My question: Is there a good reference that builds quantum mechanics from scratch, based on variational and geometric principles? If it's easier to instead start with a more general theory (e.g. quantum field theory) let me know as well.

• The proper setting for a geometric theory of quantum mechanics is the Hamiltonian, not the Lagrangian geometry. There you can do geometric quantization, but really, you don't learn what most physicists do in quantum mechanics from this formal approach. Commented Feb 14, 2016 at 20:57
• The first subquestion Why action principle? is a duplicate of physics.stackexchange.com/q/9/2451 , physics.stackexchange.com/q/15899/2451 and links therein. Commented Feb 14, 2016 at 22:32