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Quantum Phenomena in the Realm of Cosmology and Astrophysics

Quantum Phenomena in the Realm of Cosmology and Astrophysics

28 Part II. Dark Energy

28 Part II. Dark Energy from the Vacuum Energy of Quantum Fields Tr F denotes the trace over the D-dimensional functional space of the matrix G b . Thanks to the property of the logarithm, it is possible to rewrite the logarithm of the determinant of the Greens function as the negative of the logarithm of the determinant of the inverse Greens function, and thus not only avoid the problem of finding the Greens function by inverting G −1 , but also bring the expressions for the effective actions into similar shapes. b In the case of fermions, we have S (f) eff = −i ln det[G−1 f ] = − i 2 ln det [̸∂ 2 + m 2 f 1 ] = − i 2 Tr F,D ln [̸∂ 2 + m 2 f 1 ] . (3.33) For the fermions the trace denotes not only the integral over functional space, but also the trace over the Dirac indices µ. This is the same result as for the complex scalar field before, up to a multiplicative factor of −1/2, and a contraction of the four-derivative with the gamma matrices, which originates from Dirac theory for spinors. Taking the trace over Dirac space leads to an additional factor of four in the expression: S (f) eff = − i 2 Tr F,D ln [̸∂ 2 + m 2 f 1 ] = −2i Tr F ln [ ∂ 2 + m 2 f ] , (3.34) and thus we end up with the right multiplicative factor - neutral scalar fields have one degree of freedom, complex scalar fields possess two (for charge conjugation), and massive Dirac fermions have four degrees of freedom (two for the charge conjugation and two for the spin orientations). That means the result for the Dirac fermions has to have an additional factor of four with respect to neutral scalar particles and a factor of two for charged scalars, which we have been assuming above, and thus the results are in agreement. Summarising the results, we can write the partition function of the boson-fermion system as Z = e iS eff = exp [ −Tr F ln ( ∂ 2 + m 2 b) + 2 TrF ln ( ∂ 2 + m 2 f )] . (3.35) These calculations are to be modified when considering real scalar fields (uncharged bosons) or Majorana spinors (uncharged fermions, like presumably neutrinos), which will be necessary when calculating the vacuum energy of the standard model. In all cases there will be additional numerical factors to account for the corresponding number of degrees of freedom. A charge implies a factor of two, to consider particle and antiparticle. For a non-zero spin, the different possible spin orientations have to be included as numerical factors as well. This holds for all possible particles species, e.g. also for the vector bosons of the standard model, whose degrees of freedom manifest themselves as different polarisation states ϵ µ (ν) in the plane wave formulation of the field. For massive bosons with spin 1,

3. Vacuum energy of free bosons and fermions 29 there are two transversal and one longitudinal polarisation mode possible, implying three degrees of freedom, whereas for massless bosons, the longitudinal mode vanishes and two degrees of freedom remain. Apart from these particularities however, the general rule is that bosons and fermions enter the effective action with opposite signs, bosons having negative and fermions positive effective action. This fact can be employed in order to mutually balance the contributions of bosons and fermions with each other, and to achieve a complete cancellation of divergences and tune possible convergent factors in the effective action. The following calculations will consider only the first loop diagrams of the effective action, and disregard higher order interactions. To make the model more realistic, further diagrams could be included in the calculations, however, these will be neglected in this work. It is sufficient for the purpose of demonstrating the principle of cancellation to restrict ourselves to the one-loop effective action.

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