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Quantum Phenomena in the Realm of Cosmology and Astrophysics

Quantum Phenomena in the Realm of Cosmology and Astrophysics

26 Part II. Dark Energy

26 Part II. Dark Energy from the Vacuum Energy of Quantum Fields 3.2. Fermionic fields For a free complex massive fermionic field ψ, the relativistic Lagrangian density comes from Dirac theory, L f = ¯ψ i (iγ a ij∂ a − m f 1 ab )ψ j = ¯ψ(i̸∂ − m f 1)ψ , (3.19) where ̸∂ = γij∂ a a is the Feynman-slashed derivative operator, and the γ ij are the Dirac γ-matrices in flat spacetime. This Lagrangian leads to the functional integral ∫ [ ∫ Z f = D ¯ψ Dψ exp i d D x √ ] −η ¯ψ G −1 f ψ . (3.20) with the inverse Greens function, or propagator, G −1 f := i̸∂ − m f 1 . (3.21) For fermions the functional integral is calculated slightly differently, since fermions obey the Pauli principle, which forbids the existence of more than two fermions in the same place. The corresponding governing algebra is a Grassmannian one, i.e. the integral over a Grassmann variable is one, but the integral over the square of a Grassmann variable must vanish, since it is forbidden to have two fermions in the same state. The Gaussian integral over Grassmannian variables x, y is defined as ∫ dx dy exp [ x T Ay ] = det A , (3.22) and thus the partition function for a fermionic field reads (using the same substitution for the field into y as before and diagonalise the inverse Greens function) ∫ ∏ [ ∫ Z f = d ¯ψ k dψ k exp −i d D x √ ] −η ¯ψ G −1 f ψ k (3.23) ∝ det G −1 f , with the inverse of the Greens function here being G −1 f = i̸∂ − m f 1 . (3.24) This is at first sight quite different to the expression for the bosonic field - not only is it the inverse expression of the bosonic partition function, but also it contains a first order differential operator, and not one of second order, as for the bosons. We can convert the expression for the fermions to a form that is more similar to the bosonic formulation. The inverse Greens function can be rewritten by simply arguing that for fermions described by the Dirac equation the energies are symmetric, and so det [i̸∂ − m f 1] = det [−i̸∂ − m f 1] = det [+i̸∂ + m f 1] , (3.25)

3. Vacuum energy of free bosons and fermions 27 with the last equality following from the fact that for an even number of spacetime dimensions, an overall sign in the determinant does not matter. So we can write [√ ] Z f = det [i̸∂ − m f 1] = det (i̸∂ − mf 1)(i̸∂ + m f 1) = det[̸∂ 2 + m 2 f 1] 1/2 . (3.26) Consequently, the effective action is S (f) eff = −i ln Z = −i ln det G f = −i ln det[̸∂ 2 + m 2 f 1] 1/2 = − i 2 ln det[̸∂2 + m 2 f 1] . (3.27) For a real fermion field, i.e. uncharged spinors like Majorana fermions, the functional integral and thus the effective action again are modified by a factor of 1/2, i.e. which leads to an effective action for real spinor fields of S (f) eff Z f = det G −1/2 f , (3.28) = −i ln det G−1/2 f = −i ln det[̸∂ 2 + m 2 f 1] 1/4 = − i 4 ln det[̸∂2 + m 2 f 1] . (3.29) It becomes already clear that it is possible to cast the expressions for the effective action for bosonic and fermionic fields in quite similar shapes due to the advantageous properties of the logarithm. In the following, we will combine the expressions into a general effective action of a system of bosons and fermions. 3.3. Combining bosons and fermions So far we derived the 1-loop contributions of bosonic and fermionic fields to the effective action, or equivalently to the vacuum energy of the universe, assuming the simplest models of free particles without any interaction terms. We have seen that we can write the partition function as the exponent of an effective action S eff , which in the example of a complex bosonic field reads By the identity Z b = e iS eff = det G b = det[G −1 b ]−1 , (3.30) ln det A = ln ∏ a i = ∑ ln a i = Tr ln A , (3.31) i i which uses the fact that the determinant of a matrix A is given by the product of all eigenvalues a i to transform the logarithm of the determinant into the trace of the logarithm of the matrix, we can write the effective action for bosons, and thus their vacuum energy, as S (b) eff = i ln det G −1 b = i ln det [ ] ∂ 2 + m 2 b = iTr F ln [ ∂ 2 + m 2 b] . (3.32)

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