The generators of the Lorentz group are given by $$M_{\mu \nu} = i( x_{\mu} \partial_{\nu} - x_{\nu} \partial_{\mu}).$$ They can be split into boost generators $$K_i = M_{0i}$$ and rotation generators $$J_i = - \frac{1}{2} \epsilon_i^{jk}M_{jk}.$$ The generators $J$ and $K$ should have different hermiticity properties. However, defined this way, both $J$ and $K$ are hermitian. This shouldn't be the case. If we use $J$ and $K$ to generate lorentz transformations, then both transformations come out unitary. This doesn't work if we want to conserve quantum mechanical probability as in the Dirac eq.: \begin{equation} P = \int \psi^{\dagger} \psi dV. \end{equation} Since boosts cause length contraction, the volume $dV$ changes, and so must the probability density $\psi^{\dagger} \psi$ to conserve total probability. If boost generators $K$ are hermitian, then boost operators $$S = exp\Big( -\frac{i}{2} \omega K \Big)$$ are unitary. A unitary transformation would lead to an invariant probability density, but then that violates conservation of probability. Is there anything wrong with my reasoning?
1 Answers
It's easiest to expose the weakness of your gambit first expression if you merely ignore $x^2$ & $x^3$ and only consider the boost $K_1$ rotating $x^0$ into and out of $x^1$. Because of the Minkowski metric dictating Lorentz-invariance of the interval $(x^0)^2- (x^1)^2$, you see the noncompactness manifesting itself, so $$ \begin{bmatrix} x^0\\ x^1 \end{bmatrix} \mapsto \begin{bmatrix} \cosh \xi & \sinh\xi \\ \sinh\xi &\cosh \xi \end{bmatrix} \begin{bmatrix} x^0\\ x^1 \end{bmatrix} , $$ where $\xi$ is the rapidity. So $K_1$ is not antisymmetric (antihermitean), and its exponential is not orthogonal, unlike the rotations, which are compact.
You appear to be misreading a book, but you desisted from referencing it. In this non-unitary finite dimensional representation (doublet simplifying the quartet of nature), follow the infinitesimal Lorentz invariance of the interval consistently with your gambit formula, and leave hermiticity transpositions for later!
To properly correct/appreciate your formula, go to Chapter 10 of Wu-Ki Tung's Group Theory in physics, ISBN 9971-966-56-5. Do you now see the point? Most students follow (3.16) and (3.18) of Peskin & Schroeder.
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