## LESSWRONGLW

This might be a good time for me to respond to one of your earlier comments.

I was under the impression that it's a big mystery what complex numbers are doing (i.e., what is their physical function, what do they mean) in QM. I was under the impression that this is a matter of much speculative debate. Yet when I said that, I was downvoted a lot, and you implicitly alleged that I was obviously ignorant or misinformed in some way, and that we have a perfectly good understanding of what complex numbers are doing in the Dirac equation. Could you or someone please give me some background info, so that I can better understand the current state of understanding of the role of complex numbers in QM?

Note that complex numbers can be replaced with 2x2 real matrices, such as i=(0,-1;1,0), since multiplication by i is basically rotation by 90 degrees in the complex plane. Given that the Dirac equation is already full of matrices, does it make you feel better about it?

5Luke_A_Somers8yA momentary note on why the conversation went the way it did: Hopefully you can see why this looked like a rhetorical objection rather than a serious inquiry. ~~~~ So, what IS the i doing in Dirac's equation? Well, first let's look into what the i is doing in Schrödinger's equation, which is with H the 'Hamiltonian' operator: the operator that scales each component of psi by its energy (so, H has only real eigenvalues. Important!) The time-propagation operator which gives the solutions to this equation is To see how you can take e to the power of an operator, think of the Taylor expansion of the exponential. The upshot is that each eigenvector is multiplied by e to the eigenvalue. In this case, that eigenvalue is the eigenvector's energy, times time, times a constant that contains i. That i turns an exponential growth or decay into an oscillation. This means that he universe isn't simply picking out the lowest energy state and promoting it more and more over time - it's conserving overall state amplitude, conserving energy, and letting things slosh around based on energy differences. The i in the Dirac equation serves essentially the same purpose - it's a reformulation of Schrödinger's equation in a way that's consistent with relativity. You've got a field over spacetime, and the relation between space and time is that in every constant-velocity reference frame, it looks a lot like what was described in the previous paragraph. IF on the other hand you aren't bothered by that i, and you mean 'what are the various i doing in the alpha or gamma matrices', well, that's just part of making a set of matrices with the required relationships between the dimensions - i in that case is being used to indicate physical rotations, not complex phase. You can tell because each of those matrices is Hermitian - transposing it and taking the complex conjugate leaves it the same - so all of the eigenvalues are real. If it had anything to do with states' complex phase as
1Zack_M_Davis8yI did [http://lesswrong.com/lw/aha/rationality_quotes_march_2012/5zbz] (on the physical usefulness and non-mysteriousness of complex numbers, not quantum mechanics specifically).