32 Part II. Dark Energy from the Vacuum Energy ofQuantum Fields That means, the effective action and thus the vacuum energy is proportional to the expression S eff ∝ = ∫ ∞ p c [ p 4 p 2 sin θ dφ dθ dp (2π) 3 4 + m2 p 3 6 − m4 8 p √ 1 + m2 p 2 ln p − m6 48p 2 − ... ] ∞ p c . (4.5) This expression diverges inthe ultraviolet limit ofthe momentum for the first three terms; and is moreover dependent on an infrared cutoff parameter p c . The cutoff p c has been introduced because the expansion (4.4) is only valid for large momenta, and we are not interested inthe behaviour for small momenta. The ultraviolet divergences pose a serious problem since there is no more elegant argumentation for thetheir renormalisation than the assumption oftheinvalidity of conventional quantum field theories at scales beyond the Planck scale. The usual way to deal with those infinities is thus to introduce another cutoff  in phase space, for example at the Planck scale, and disregard particles with momenta higher than that cutoff, leading to theinfamous energy density of ρ Λ,th ≃ 10 76 GeV 4 mentioned before. There are alternative renormalisation methods, such as dimensional regularisation , which however all serve the purpose of peeling out the singularities explicitly only to then disregard those terms or eliminate them by appropriate counterterms. The concept we would like to put forward in this work to deal with these divergences is to make the divergent contributions from bosons and fermions cancel exactly, so that only the finite parts oftheintegral remain, which then determines the vacuum energy. The way to achieve this is by considering the one other quantity occurring inthe effective action above, i.e. the mass ofthe particle. If we would like to cancel the divergences between two particles, and taking into account that bosons and fermions give opposite-sign contributions to the effective action, it is clear that by fine-tuning the masses ofthe particles inthe system accordingly, it is possible to get rid ofthe divergent parts. In order to have the first three terms inthe above expression for bosons and fermions cancel each other out there are some relations that have to be fulfilled. The total expression for the vacuum energy for i particle species is S eff ∝ ∑ [ ν i p 4 + m2 i p 3 4 6 i − m4 i 8 ln p − ... ] . (4.6) To cancel the first term inthe sums, we demand to have the same number of degrees of freedom on the bosonic and fermionic sector ofthe system, denoted by b and f, ∑ ν b = ∑ ν f . (4.7) b f
4. Vacuum energy in flat spacetimes 33 To cancel the other two divergent terms inthe sum, the masses ofthe bosons and fermions must obey the relations ∑ m 2 b = ∑ m 2 f , b f ∑ m 4 b = ∑ m 4 f . (4.8) b f Given the assumption that the masses of bosons and fermions can be chosen to fulfil these relations, this is the basic schedule that has to be obeyed in order to achieve the cancellation of divergences. These simple estimations have shown the principle that we employ. In the following sections, we will calculate the vacuum energy exactly, peel out the divergences with slightly different methods, and try to find the exact cancellation conditions that have to be fulfilled. We expect however that they will be in principle the same as what we obtained from these sketchy considerations. 4.1. Exact calculation ofthe vacuum energy Going back to theintegral in question, ∫ d D p (2π) D ln ( −p 2 + m 2) , (4.9) we will apply the technique of dimensional regularisation, which means that theintegral will be rewritten in terms of Γ-functions as a function ofthe number of spacetime dimensions D, which will then be set to a natural number plus an infinitesimal part, i.e. D = 4 − ϵ. The resulting expression will be expanded in terms ofthe small quantity ϵ, andinthe end the limit ϵ → 0 will be taken. From the above expression, we first carry out a Wick rotation, andthen calculate theintegral as (see e.g. ) ∫ d D p (2π) ln ( −p 2 + m 2) = −i Γ(−D/2) 1 . (4.10) D (4π) D/2 (m 2 ) −D/2 This integral is divergent because the Γ-function diverges for negative even values without any possibility of analytic continuation or such tricks, and for D = 4, we have Γ(−2) inthe numerator. The expression needs to be processed further using dimensional regularisation, i.e. we substitute D by D = 4 − ϵ, which will make it possible to explicitly isolate theinfinities in terms which diverge inthe limit ϵ → 0. The result oftheintegral contains ϵ twice: withinthe Γ-function, and as power ofthe mass. It is possible to expandthe Γ-function as  Γ(−n + ϵ) ≃ (−1)n n! [ 1 ϵ + ψ(n + 1) + ϵ ( )] π 2 2 3 + ψ2 (n + 1) − ψ ′ (n + 1) , (4.11)
The universe is not a world of separate things and events but is a cosmos that is connected, coherent, and bears a profound resemblance to the visions held in the earliest spiritual traditions in which the physical world and spiritual experience were both aspects of the same reality and man and the universe were one. The findings that justify this new vision of the underlying logic of the universe come from almost all of the empirical sciences: physics, cosmology, the life sciences, and consciousness research. They explain how interactions lead to interconnections that produce instantaneous and multifaceted coherence–what happens to one part also happens to the other parts, and hence to the system as a whole. The sense of sacred oneness experienced by our ancestors that was displaced by the unyielding material presumptions of modern science can be restored, and humanity can once again feel at home in the universe.