Notes on Elementary Particle Physics by H. Muirhead

By H. Muirhead

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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>).

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