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We follow the interpretation and notation given in a recent paper:Ann. of Phys.,71, 519 (1972).
Equation (2) without the «mass» term was first written down byY. Murai:Progr. Theor. Phys. (Kyoto),18, 109 (1955).
One possible representation ofβ a isβ u=γ u·⋌σ3,β 4=γ 5·⋌σ3,β 6=1·⋌σ1,which givesβ ϰ.
It is assumed thatp λ≠0 in the rest framep 1=p 2=p 3=0. Hence we must do a «tilt» to get rid of theΣ ϰλ term. In this space, since β0 is diagonal, the parity operatorP=β 0 β 4 β 6 is proportional to\(P \equiv \beta ^0 \beta ^4 \beta ^6 = \left\{ {_{\gamma ^0 \gamma ^5 0}^{0 \gamma ^0 \gamma ^5 } } \right\}\).
The positive definite scalar product for the 8-component equation is Lorentz invariant, but not conformally invariant. This is so because the little group we are using isSO 4,3 and unitary representations are infinite dimensional; therefore the μ states are unstable. The nonunitarity means that the decay products are not included in the description.
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Barut, A.O., Haugen, R.B. Solutions of the conformal invariant spinor equations and theory of the electron–muon system. Lett. Nuovo Cimento 7, 625–628 (1973). https://doi.org/10.1007/BF02728039
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DOI: https://doi.org/10.1007/BF02728039