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Quantum Phenomena in the Realm of Cosmology and Astrophysics

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** determ**in**ant **of** **the** Greens function as **the** negative **of** **the** logarithm **of** **the** determ**in**ant **of** **the** **in**verse Greens function, **and** thus not only avoid **the** problem **of** f**in**d**in**g **the** Greens function by **in**vert**in**g G −1 , but also br**in**g **the** expressions for **the** effective actions **in**to 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** **in**tegral over functional space, but also **the** trace over **the** Dirac **in**dices 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 orig**in**ates from Dirac **the**ory for sp**in**ors. Tak**in**g **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** sp**in** 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 assum**in**g above, **and** thus **the** results are **in** agreement. Summaris**in**g **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 consider**in**g real scalar fields (uncharged bosons) or Majorana sp**in**ors (uncharged fermions, like presumably neutr**in**os), which will be necessary when calculat**in**g **the** vacuum energy **of** **the** st**and**ard model. In all cases **the**re will be additional numerical factors to account for **the** correspond**in**g number **of** degrees **of** freedom. A charge implies a factor **of** two, to consider particle **and** antiparticle. For a non-zero sp**in**, **the** different possible sp**in** orientations have to be **in**cluded as numerical factors as well. This holds for all possible particles species, e.g. also for **the** vector bosons **of** **the** st**and**ard model, whose degrees **of** freedom manifest **the**mselves as different polarisation states ϵ µ (ν) **in** **the** plane wave formulation **of** **the** field. For massive bosons with sp**in** 1,

3. Vacuum energy **of** free bosons **and** fermions 29 **the**re are two transversal **and** one longitud**in**al polarisation mode possible, imply**in**g three degrees **of** freedom, whereas for massless bosons, **the** longitud**in**al mode vanishes **and** two degrees **of** freedom rema**in**. Apart from **the**se particularities however, **the** general rule is that bosons **and** fermions enter **the** effective action with opposite signs, bosons hav**in**g negative **and** fermions positive effective action. This fact can be employed **in** order to mutually balance **the** contributions **of** bosons **and** fermions with each o**the**r, **and** to achieve a complete cancellation **of** divergences **and** tune possible convergent factors **in** **the** effective action. The follow**in**g calculations will consider only **the** first loop diagrams **of** **the** effective action, **and** disregard higher order **in**teractions. To make **the** model more realistic, fur**the**r diagrams could be **in**cluded **in** **the** calculations, however, **the**se will be neglected **in** this work. It is sufficient for **the** purpose **of** demonstrat**in**g **the** pr**in**ciple **of** cancellation to restrict ourselves to **the** one-loop effective action.

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10. Numerical analyses 85 10. Numer

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10. Numerical analyses 87 priors to

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10. Numerical analyses 89 z y1 y 4z

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10. Numerical analyses 91 Table 10.

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10. Numerical analyses 93 Model A M

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10. Numerical analyses 95 Table 10.

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10. Numerical analyses 97 Figure 10

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10. Numerical analyses 99 Table 10.

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10. Numerical analyses 101 Figure 1

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10. Numerical analyses 103 Table 10

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106 Part III. Dark Energy from an O

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12. Bose-Einstein condensates in As

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122 Part IV. Bose-Einstein condensa

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124 Part IV. Bose-Einstein condensa

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126 Part IV. Bose-Einstein condensa

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16. Numerical solution 145 16. Nume

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16. Numerical solution 147 16.3. Ou

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16. Numerical solution 149 T [K] 0

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16. Numerical solution 151 Figure 1

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16. Numerical solution 155 Figure 1

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16. Numerical solution 157 The best

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16. Numerical solution 159 Figure 1

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16. Numerical solution 161 The zero

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164 Part IV. Bose-Einstein condensa

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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