By H. Muirhead
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Extra info for Notes on Elementary Particle Physics
What are we going to write down for spin which makes sense for relativistic systems and transformations? We want something which is 1. Lorentz invariant; 2. commutes with P\ and Ρλ since we want to make simultaneous measurements on "spin" and momentum in order to fully define the particles properties; 42 RELATIVISTIC WAVE EQUATIONS A N D FIELDS [Ch. 3 3. reduces to the normal concept of spin in non-relativistic quantum mechanics in the limit ß -> 0. Experience has taught us that this concept is completely successful.
48) and since 2 = 1(1 + 1) we have a system for describing spin-one particles. 49) V2 so that (S" is the diagonalised operator) S'te± = + e ± S'ze0 = 0. 50) This is all very fine, but what we want is a four-vector and not a threevector , . _„ A Ψ^Φμ μ = 1,2, 3, 4 (3·5·51) 0μ = e/**· We then have the problem of dealing with the fourth component. Since φμ will behave like χμ9 then<£ = φ,φ 4 and under spatial notations φ behaves like a vector as before whilst φ^ behaves like a scalar, so that we have a mixture of spin 1 and 0 systems.
It is evident therefore that transformation x -* x' must reflect back into the interpretation of |y>>. 3) 38 RELATIVISTIC WAVE EQUATIONS AND FIELDS [Ch. 3 then the transformation implies that the value of the wave function of x is taken to Λχ and at the same time the value of the wave function at Λ~χχ moves to x. 4) ψ(Λ~1χ) -► ψ'(χ) —► x but if the value of the wave function remains invariant under the trans formation, then rA \ \ ,, \ /» ^ ^ f(A-xx) = ψ'(χ). 5) Let us firstly consider displacements Χμ-+χ'μ = χμ + % and denote the operator which transforms ψ to ψ' by U(a) \y>'}= U(a)\y>).