A momentary note on why the conversation went the way it did:
what the hell are complex numbers doing in the Dirac equation?
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
ihbar;partial_{t}|psi>=H|psi>
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 a thing in itself instead of just any other element of the state it’s operating on, it would let some of the imaginary part of the matrix get out. Instead, it keeps it on the inside as an ‘implementation detail’.
A 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
ihbar;partial_{t}|psi>=H|psi>
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 a thing in itself instead of just any other element of the state it’s operating on, it would let some of the imaginary part of the matrix get out. Instead, it keeps it on the inside as an ‘implementation detail’.