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So I was toying around attempting to simulate some relativistic wave equations for a recreational project. Now I have never studied spinors in dept and the knowledge I have is from reading online (apologies in advance if the question may be too obvious).

To keep things simple and have less terms to deal with, I simplified the Dirac Equation by assuming that the particle is massless to get the Weyl Equation $\sigma^\mu \partial_\mu \psi = 0$ where $\psi=(\psi_L,\psi_R)\in\mathbb{C}^2$ seems to be what is called a "spinor" with $L$ and $R$ describing spin direction, and $\sigma^\mu$ the Pauli matrices. Then I got two separate PDEs by multiplying out the matrices: $$\partial_t\psi_R + \partial_x\psi_L -i \partial_y\psi_L + \partial_z\psi_R = 0$$ $$\partial_t\psi_L + \partial_x\psi_R +i \partial_y\psi_R - \partial_z\psi_L = 0$$

Next, because most programming languages do not have built in support for complex numbers, I handwavingly took $\psi_k = u_k + i v_k$ and further broke down the 2 PDEs into 4 PDEs: $$\partial_tu_R + \partial_xu_L + \partial_yu_L + \partial_zu_R = 0$$ $$\partial_tv_R + \partial_xv_L - \partial_yu_L + \partial_zv_R = 0$$ $$\partial_tu_L + \partial_xu_R - \partial_yv_R - \partial_zu_L = 0$$ $$\partial_tv_L + \partial_xv_R + \partial_yu_R - \partial_zv_L = 0$$

Now my question is this:

  1. Firstly, is such an approach considered to be correct? Just double checking.

  2. Now for my real question: In QM, we have that $\mathbb{P}(x,t) = \psi \psi^* = |\psi(x,y)|^2$ as out probability density function. By extrapolating this, I am assuming that with spinors $\mathbb{P}_k(x,t) = \psi_k \psi_k^* = |\psi_k(x,y)|^2$ where $k$ is the particle spin direction. Now assuming that I don't care about the spin and just want the probability of finding a particle with any spin, would it be as simple as $\mathbb{P}(x,t) = \mathbb{P}_L(x,t) + \mathbb{P}_R(x,t) = \psi.\psi^* = \langle\psi|\psi\rangle$?

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