To see the Lorentz invariance recall that for the Pauli matrices \u03c3 1 and \u03c3 3

To see the lorentz invariance recall that for the

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To see the Lorentz invariance, recall that for the Pauli matrices, σ 1 and σ 3 are real, and σ 2 is imaginary. σ 1 = parenleftbigg 0 1 1 0 parenrightbigg , σ 2 = parenleftbigg 0 i i 0 parenrightbigg , σ 3 = parenleftbigg 1 0 0 1 parenrightbigg (128) So, σ 1 = σ 1 , σ 2 = σ 2 , σ 3 = σ 3 (129) σ 1 T = σ 1 , σ 2 T = σ 2 , σ 3 T = σ 3 (130) This implies σ 1 T σ 2 = σ 1 σ 2 = σ 2 σ 1 , σ 3 T σ 2 = σ 3 σ 2 = σ 2 σ 3 and σ 2 T σ 2 = σ 2 σ 2 . That is, σ j T σ 2 = σ 2 σ j (131) and so δ ( ψ R T σ 2 ) = 1 2 ( j + β j ) ψ R T σ j T σ 2 = 1 2 ( j β j )( ψ R T σ 2 ) σ j (132) Combining this with the transformation of ψ R in Eq. (125) we see that L Maj in Eq. (127) is Lorentz invariant. Since σ 2 = parenleftbigg - i i parenrightbigg the Majorana mass can be expanding out to ψ R T σ 2 ψ R = ( ψ 1 ψ 2 ) parenleftbigg i i parenrightbiggparenleftbigg ψ 1 ψ 2 parenrightbigg = i ( ψ 1 ψ 2 ψ 2 ψ 1 ) (133) Thus we have shown that ψ 1 ψ 2 ψ 2 ψ 1 is Lorentz invariant. We often write this as ψ 1 ψ 2 ψ 2 ψ 1 = ψ α ψ β ε αβ , ε αβ = parenleftbigg 0 1 1 0 parenrightbigg (134) which avoids picking a σ 2 . There’s only one problem: if the fermion components commute ψ 1 ψ 2 ψ 2 ψ 1 = 0 ! For Majo- rana masses to be non-trivial, fermion components can’t be regular numbers, they must be anti- commuting numbers. Such things are called Grassmann numbers and satisfy a Grassmann algebra. Further explanation of why spinors must anticommute is given in the Lecture II-5, on the spin-statistics theorem. Grassmann numbers are discussed more in Lecture II-7 on the path integral. 7.2 Notation for Weyl spinors In QED, we will be mostly interested in Dirac spinors, like the electron. But since Weyl spinors correspond to irreducible representations of the Lorentz group, it is sometimes helpful to have concise notation for constructing products and contractions of Weyl spinors only. This notation is useful in many contexts besides gauge theories, such as supersymmetry. It is also related to the spinor-helicity formalism we will discuss in Lecture IV-3. If you’re just interested in QED, you can skip this part. 18 Section 7
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Let us write ψ for left-handed spinors and ψ ˜ for right-handed spinors. Sometimes the nota- tion ψ is used, especially in the contexts of supersymmetry, but this can be confused with the bar notation for a Dirac spinor, ψ ¯ ψ γ 0 , so we will stick with ψ ˜ . We index the two compo- nents of left-handed Weyl spinors with Greek indices from the beginning of the alphabet, i.e. ψ α . For right-handed spinors, we use dotted Greek indices, like ψ ˜ α ˙ . A Dirac spinor is ψ = parenleftBigg ψ α ψ ˜ β ˙ parenrightBigg (135) Conventionally, left-handed spinors (and right-handed anti-spinors) have upper undotted indices and right-handed spinors (and left-handed anti-spinors) have lower dotted indices.
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