On Generations
نویسنده
چکیده
The well known operator ordering ambiguity could motivate the existence of generations. This possibility is explored by exploiting the relationship between ordering and discretization rules. 1 Wilson fermion as analogy Non Commutative Geometry is not the only area knowing how to get mass terms from nowhere. Lattice theorists learnt time ago a nice trick from Wilson. Consider the forward and backward discrete derivatives, ∂+, ∂− as usual ∂+f(x) = f(x + )− f(x) , ∂−f(x) = f(x) − f(x − ) Suitable derivatives in the lattice can be implemented as any linear combination of both, 1 a+b (a∂+ + b∂−) Now, by noticing that ∂−(f(x + ) − f(x)) = (∂+ − ∂−)f(x) we can rewrite a∂+ + b∂− a+ b = 1 2 (∂+ + ∂−) + a− b a+ b (∂+ − ∂−) = 1 2 (∂+ + ∂−) + λ∂−∂+ where λ ≡ a−b a+b . Here Wilson stops and we follow up: If some technical or quantum effect avoids the limit λ going to zero, then the system taking account of the ambiguity should be written d dxf → ∂f + λ∂f And, if two derivatives are done, in order to account for the ambiguity we shall rewrite m d dx2 f → m(∂ + λ1∂)(∂ + λ2∂)f Just as a funny notation, define Ψe(x) ≡ f(x),Ψμ ≡ f ′,Ψτ ≡ f ′′, then: m df dx2 = M0∂ Ψe + M1∂ Ψμ + M2∂ Ψτ Note that M0,M1,M2 contain orders of , , 2 respectively. ∗somewhere in Kent, England. Email: [email protected]
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