- #1

HeavyMetal

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He uses the Hamiltonian [itex]H_{ab}=cP_{j}(\alpha^{\ j})_{ab}+mc^2(\beta)_{ab}[/itex]

So it is easy to see that [itex](H_{ab})^2=c^2P_{j}P_{k}(\alpha^{\ j}\alpha^{k})_{ab}+mc^{3}P_{j}(\alpha^{\ j}\beta+\beta\alpha^{\ j})_{ab}+m^{2}c^{4}(\beta^2)_{ab}[/itex]

He then explains that if we choose suitable matrices that satisfy a few equations, that we can obtain the momentum eigenstate in the correct momentum-energy form. These matrices satisfy:

{[itex]\alpha^{\ j},\alpha^{k}[/itex]}[itex]_{ab}=2\delta^{\ jk}\delta_{ab}[/itex]

{[itex]\alpha^{\ j},\beta[/itex]}[itex]_{ab}=0[/itex]

[itex](\beta^2)_{ab}=\delta_{ab}[/itex]

Where brackets represent the anticommutator.

Ultimately, after some arithmetic we find that:

[itex](H^2)_{ab}=\textbf{P}^{2}c^{2}\delta_{ab}+m^{2}c^{4}\delta_{ab}=(\textbf{P}^{2}c^{2}+m^{2}c^{4})\delta_{ab}[/itex]

I'm having trouble seeing the steps that it takes to get there. Substituting in the values obtained from the matrices, I can see that the middle term drops out because the value is zero, i.e. [itex]mc^{3}P_{j}(\alpha^{\ j}\beta+\beta\alpha^{\ j})_{ab}=mc^{3}P_{j}*0=0[/itex].

I can also see that the last term is included because [itex]m^{2}c^{4}(\beta^2)_{ab}=m^{2}c^{4}\delta_{ab}[/itex].

I cannot, however, see why [itex]c^{2}P_{j}P_{k}(\alpha^{\ j}\alpha^{k})_{ab}=\textbf{P}^{2}c^{2}\delta_{ab}[/itex].

I can see that [itex]c^{2}P_{j}P_{k}(\alpha^{\ j}\alpha^{k})_{ab}=c^{2}P_{j}P_{k}\frac{1}{2}[/itex]{[itex]\alpha^{\ j},\alpha^{k}[/itex]}[itex]_{ab}=c^2\textbf{P}^2\delta^{\ jk}\delta_{ab}[/itex]. I am confused as to where the [itex]\delta^{\ jk}[/itex] goes!

Is it simply the fact that [itex]\delta^{\ jk}[/itex] and [itex]\delta_{ab}[/itex] are Kronecker deltas? I reread the previous sections, but I don't think it was mentioned. If this is the case, then I understand where that [itex]\delta^{\ jk}[/itex] goes. More importantly, I tried to work out the three equivalencies above by using Pauli matrices, but I quickly get lost! When {[itex]\alpha^{\ j},\alpha^{k}[/itex]}[itex]_{ab}[/itex] is written, does it refer to a 2 x 2 matrix (a x b), in which each entry is a 2 x 2 Pauli matrix (j x k) and therefore is ultimately a 4 x 4 matrix?

Thanks in advance,

HeavyMetal \m/