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Thanks, this is very helpful. When I took undergrad quantum physics, we saw that the Schrödinger equation can be represented without complex numbers. But that
by civilized 3y ago
Thanks, this is very helpful.
When I took undergrad quantum physics, we saw that the Schrödinger equation can be represented without complex numbers. But that's for one particle, and it sounds like you're saying this somehow breaks down when you have multiple systems interacting, due to these tensor products not working as we think they should?
- eigenket 3y agoyes, specifically doing things without complex numbers breaks if you require composing systems to work like they do in normal quantum mechanics (composing with the tensor product). If you drop that assumption about how things compose then you can use something like the real matrix representation where 1 is a 2x2 identity matrix and i is some anti-symmetric 2x2 matrix.
- civilized 3y agoHelp a lapsed mathematician out here... the complex numbers can be represented as skew-symmetric 2x2 real matrices, right? And this is an isomorphism? So if the 2x2 real skew-symmetric matrices are "the same" as the complex numbers, how do we end up having this issue that comes up in quantum physics where you can only use the actual complex numbers and not the representation?
- eigenket 3y agoAny explanation I can give in a comment here will be pretty flawed, if you want more detail than my previous comment I think your best bet is to read the papers at the arxiv links I posted above.