Master's Thesis in Theoretical Physics - Universiteit Utrecht

and satisfy therefore {γ µ ,γ ν} {= e µ ae ν bγ a ,γ b} = −2g µν . (6.31)Now we use the conformal FLRW metric as the metric to describe our expanding universeg µν = a 2 (η)η µν . A comparison to Eq. (6.26) shows thate a µ = a(η)δ a µe µ a =1a(η) δµ ae µa = η ka e k µ = a(η)η µa. (6.32)Using the form of the Christoffel symbols with the metric (B.20) and the expressions abovewe find thatΓ µ = 1 a ′ [4 a η µb γ 0 ,γ b] . (6.33)We can now writeiγ µ ∇ µ ψ = 1 a iγc ∂ c ψ + i D − 1 a ′2 a 2 γ 0ψ= a − D+12 iγ c ∂ c (a D−12 ψ). (6.34)Therefore we can rewrite the action from Eq. (6.27) as∫S = d D xa D{ i2 [ψ D+1La− 2 γ c ∂ c (a D−12 ψL ) − ∂ c (a D−12 ψL )a − D+12 γ c ψ L ]=+ i 2 [ψ D+1Ra− 2 γ c ∂ c (a D−12 ψR ) − ∂ c (a D−12 ψR )a − D+12 γ c ψ R ]}−f u u L φ 1 u R − f d d L φ ∗ 2 d R − fu ∗ u Rφ ∗ 1 u L − f ∗ d d Rφ 2 d L∫ { id D D−1x [(a 2 ψL )γ c ∂ c (a D−12 ψL ) − ∂ c (a D−12 ψL )γ c (a D−12 ψL )] (6.35)2+ i D−1[(a 2 ψR )γ c ∂ c (a D−12 ψR ) − ∂ c (a D−12 ψR )γ c (a D−12 ψR )]2−f u a D u L φ 1 u R − f d a D d L φ ∗ 2 d R − f ∗ u aD u R φ ∗ 1 u L − f ∗ d aD d R φ 2 d L}. (6.36)We now define a new conformally rescaled fermion fieldχ = a D−12 ψ, (6.37)which allows us to write the action as∫ iS = d x{ D 2 [χ L γa ∂ a χ L − (∂ a χ L )γ a χ L ] + i 2 [χ R γa ∂ a χ R − (∂ a χ R )γ a χ R ]−af u u L φ 1 u R − af d d L φ ∗ 2 d R − af ∗ u u Rφ ∗ 1 u L − af ∗ d d Rφ 2 d L}. (6.38)We stress that the u and d quarks have also been conformally rescaled, a D−12 u → u. Thuswe see that by performing a conformal rescaling of the fermion field, the action is almostthe same as the flat Minkowski action from Eq. (6.23). The only difference is that themass term (the coupling of the fermions to the scalar field) is multiplied by the scale factor.In section 4.2 we made a statement that a massless fermion field is invariant under aconformal transformation, and this should be clear from the action (6.38). For a masslessfermion the action in a D dimensional FLRW universe is precisely the same as the actionin Minkowski space after a conformal rescaling of the fermion field.