From Special Relativity to Feynman Diagrams.pdf

As proven in appendix e in order to have finitely

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As proven in Appendix E, in order to have finitely many spin states, we need the two operators ˆ N a ˆ J 0 a ˆ J 1 a to vanish on the states | ¯ p , r , so that ˆ W a = 0 and we can effectively write: ˆ W μ = ˆ J 1 ¯ p μ = ˆ ¯ p μ , (9.127) where we have defined the helicity operator as ˆ ˆ J · ¯ p p | = ˆ J 1 . (9.128) In going from S 0 to any other frame S , ¯ p μ and ˆ W μ are four vectors transforming by the same Lorentz transformation, so that, in S , ˆ W μ = p μ ˆ . We conclude that ˆ is a Lorentz invariant operator. The condition that the single particle state transform in an irreducible representation of the little group further implies that there can be just two helicity states: ˆ | ˆ p , ± s = ± s | ˆ p , ± s , (9.129) The state of a single massless particle is completely defined by the value if its helicity , which is a Poincaré invariant quantity. 15 14 If we consider proper Lorentz transformations ( 0 0 0 , det = 1 , ) the sign of p 0 , eigenvalue of ˆ P 0 , is invariant as well. 15 Here we are restricting to proper Lorentz transformations. The parity transformation P : p 0 p 0 , p → − p reverses the sign of .
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9.5 A Note on Lorentz Invariant Normalizations 299 9.5 A Note on Lorentz Invariant Normalizations In this note we show that the normalization that we have adopted for single particle states is Lorentz invariant. To this end let us consider a particle of mass m and let S 0 denote its rest frame in which ¯ p = 0 and ¯ p 0 = mc . S 0 then moves, relative to a given RF S , at the corresponding velocity v of the particle. The relation between the four-momenta ¯ p and p of the particle in S 0 and S , respectively, is given by the Lorentz boost p . If we write ¯ p = 1 p p , expressing the transformation matrix in terms of v we have: ¯ p 0 = γ (v) p 0 v · p c = γ (v) p 0 v 2 c 2 p 0 = 1 γ p 0 , ¯ p i = p i + 1 ) v · p c 2 v i γ v c p 0 , (9.130) where we have used the relation p = p 0 v / c and v 2 ≡ | v | 2 (here, as usual, upper and lower indices for three-dimensional vectors are the same: v i = v i , p i = p i ). If we perturb the rest state of the particle in S 0 by an infinitesimal velocity, but keeping the relative motion between the two frames unchanged, the momentum p relative to S will vary by an infinitesimal amount p p + dp . We can relate the infinitesimal variation of ¯ p to that of p by computing the Jacobian matrix J p i j = ¯ p i p j : d ¯ p i = ¯ p i p j dp j = J p i j dp j . (9.131) This Jacobian is computed from the transformation law ( 9.130 ) by taking into account that p 0 is not independent of p , being p 0 = | p | 2 + m 2 c 2 . Using the property: p 0 p i = p i p 0 = v i c , (9.132) from ( 9.130 ) we find: J p i j = ¯ p i p j = δ i j + 1 ) v i v j v 2 γ v i c p 0 p j = δ i j + 1 γ 1 v i v j v 2 . (9.133) The reader can easily verify that the matrix J p has one eigenvalue 1 corresponding to the eigenvector v , and two eigenvalues 1 , corresponding to the two vectors per- pendicular to v . The determinant of this matrix, being the product of its eigenvalues, is therefore: det ( J p ) = 1 γ . (9.134)
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300 9 Quantum Mechanics Formalism
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