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Quantum Phenomena in the Realm of Cosmology and Astrophysics

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 **the**ory, 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 **in**tegral ∫ [ ∫ Z f = D ¯ψ Dψ exp i d D x √ ] −η ¯ψ G −1 f ψ . (3.20) with **the** **in**verse Greens function, or propagator, G −1 f := i̸∂ − m f 1 . (3.21) For fermions **the** functional **in**tegral is calculated slightly differently, s**in**ce fermions obey **the** Pauli pr**in**ciple, which forbids **the** existence **of** more than two fermions **in** **the** same place. The correspond**in**g govern**in**g algebra is a Grassmannian one, i.e. **the** **in**tegral over a Grassmann variable is one, but **the** **in**tegral over **the** square **of** a Grassmann variable must vanish, s**in**ce it is forbidden to have two fermions **in** **the** same state. The Gaussian **in**tegral over Grassmannian variables x, y is def**in**ed as ∫ dx dy exp [ x T Ay ] = det A , (3.22) **and** thus **the** partition function for a fermionic field reads (us**in**g **the** same substitution for **the** field **in**to y as before **and** diagonalise **the** **in**verse 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** **in**verse **of** **the** Greens function here be**in**g 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** **in**verse expression **of** **the** bosonic partition function, but also it conta**in**s 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 **in**verse Greens function can be rewritten by simply argu**in**g 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 follow**in**g from **the** fact that for an even number **of** spacetime dimensions, an overall sign **in** **the** determ**in**ant 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 sp**in**ors like Majorana fermions, **the** functional **in**tegral **and** thus **the** effective action aga**in** are modified by a factor **of** 1/2, i.e. which leads to an effective action for real sp**in**or 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** follow**in**g, we will comb**in**e **the** expressions **in**to a general effective action **of** a system **of** bosons **and** fermions. 3.3. Comb**in****in**g 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, assum**in**g **the** simplest models **of** free particles without any **in**teraction 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** determ**in**ant **of** a matrix A is given by **the** product **of** all eigenvalues a i to transform **the** logarithm **of** **the** determ**in**ant **in**to **the** trace **of** **the** logarithm **of** **the** matrix, we can write **the** effective action for bosons, **and** thus **the**ir 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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Bibliography 167 Bibliography [1] A

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Bibliography 169 [27] V. Vitagliano

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Bibliography 171 [58] L. E. Parker

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Bibliography 173 [87] R. R. Caldwel

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Bibliography 175 [116] R. C. Tolman

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Bibliography 177 [143] P. Oehberg a

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Acknowledgments 181 Acknowledgments