Free Hamiltonian problem for relativistic mechanics

[forhad_jnu]

I need to elaborate the equation ,and need to know what is the physical significance and how matrices will manipulate in the equation $$
\hat{H} = (\hat{\tau_3}+i\hat{\tau_2})\frac{\hat{p}^2}{2m_0}+ \hat{\tau_3}m_0 c^2 = \frac{\hat{p}^2}{2m_0}
\left| \begin{array}{ccc}
1 & 1 \\
-1 & -1 \\
\end{array}\right| \frac{\hat{p}^2}{2m_0} + \left| \begin{array}{ccc}
1 & 0 \\
0 & -1 \\
\end{array}\right| m_0 c^2
$$

Where $$\tau_1 , \tau_2,\tau_3
$$ are Pauli matrices and Hamiltonian comes from "Schrödinger form of the free Klein_Gordon equation
And also why did we added Pauli matrices in the free Hamiltonian ?

 

[Bill_K]

This equation arises from trying to make the second-order Klein-Gordon Equation look like the first-order Dirac Equation. Define a matrix Ψ = (u v) where u = φ - iħ/mc² ∂φ/∂t and v = φ + iħ/mc² ∂φ/∂t. Then Ψ obeys a matrix Schrödinger Equation, iħ∂Ψ/∂t = HΨ where the matrix Hamiltonian H is what you have written. Expressing it in terms of Pauli matrices is just for convenience.

The problem is that H is not Hermitian due to the iτ₂ term. What Feshbach and Villars did to fix this was again by analogy with the Dirac Equation. In the Dirac Equation we modified the definition of the adjoint, and instead of ψ† we use ψ ≡ ψ†γ⁰. Here we use φ ≡ φ†τ₃. In terms of this "metric" the norm of ψ is (ψ, ψ) ≡ ∫φ†τ₃φ d³x = ∫(u*u - v*v) d³x = iħ/mc² ∫ (φ* ∂φ/∂t - ∂φ*/∂t φ) d³x, which is the familiar conserved quantity.