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 [] (on the physical usefulness and non-mysteriousness of complex numbers, not quantum mechanics specifically).

How accurate is the quantum physics sequence?

by Paul Crowley 1 min read17th Apr 201268 comments


Prompted by Mitchell Porter, I asked on Physics StackExchange about the accuracy of the physics in the Quantum Physics sequence:

What errors would one learn from Eliezer Yudkowsky's introduction to quantum physics?

Eliezer Yudkowsky wrote an introduction to quantum physics from a strictly realist standpoint. However, he has no qualifications in the subject and it is not his specialty. Does it paint an accurate picture overall? What mistaken ideas about QM might someone who read only this introduction come away with?

I've had some interesting answers so far, including one from a friend that seems to point up a definite error, though AFAICT not a very consequential one: in Configurations and Amplitude, a multiplication factor of i is used for the mirrors where -1 is correct.

Physics StackExchange: What errors would one learn from Eliezer Yudkowsky's introduction to quantum physics?