Re: Cosmological Constant

On May 2, 4:32 pm, Uncle Al <Uncle...@xxxxxxxxxxxxx> wrote:
If one allows zero point fluctuations of the vacuum, 1/2
photon/allowed EM mode to the Planck energy,

(rho)_vacuum ~ 10^92 erg/cm^3
(omega)_(lambda) ~ 10^120

Even theorists are embarrassed to disappear an omega that big to
exactly obtain the necessary fractional residue.

It need not be anything more than a self-constitency principle that
restricts what the actual spectrum of fermions and bosons must be. In
that case, this goes from a liability to a major asset.

Finiteness for a total ZPE can be obtained if
sum_b m_b^n = sum_f m_f^n
over all boson modes (b) and fermion modes (f), for n = 0 , 2, 4;
where m_b or m_f is the mass associated with the respective mode.
Making it small requires further constraining the respective mass
spectra.

If you explain away mass as a dynamic effect of the Higgs, then this
becomes a constraint on the couplings of the fermions and bosons to
the Higgs.

The cosmological constant already arises from non-Abelian Yang-Mills
fields through the contribution
Lambda = 1/4 sum f^{abc} f_{abc}
where the structure constants f_{ab}^c are with indices raised and
lowered by the gauge group metric k_{ab}.

In a U(1) theory, k_{ab} is just (epsilon_0 c), the vacuum
permittivity. If you assume k is constant, then the scale of the
Lambda contribution becomes fixed, and you're in the middle of the
"fine tuning problem".

If, on the other hand, k is variable; then this contribution can vary
and become asymptotically small or zero. The fine-tuning issue is
potentially evaded. Moreover, you also acquire extra contributions to
the Lagrangian involving the gradient of k -- "dark energy" terms.
Since you're lowering 2 indices, and raising only one, Lambda then
scales proportionally with k.

The k metric components are equivalent to the Jordan-Brans-Dicke
scalars; and also to dilatons (the dilaton is equivalent to the
logarithm of the determinant of the k metric). In terms of the Maxwell
U(1) field, k is just the dielectric coefficient of the vacuum; so
that the extra terms are none other than the representation of the
dielectric energy stored in the vacuum(!) I.e., a vindication of
Maxwell's notion of a universal dielectric medium.

Some (or maybe even most) quintessence models use Jordan-Brans-Dicke
scalars (or scalar-tensor-matter) as their basis. If you want to
revert this back to the Kaluza-Klein representation k = g_{55} or k =
(g_{ab};a,b=5,6,...), a confirmation of a link between the
cosmological constant and vacuum energy to the extra terms arising the
gauge group's metric would represent an indirect confirmation of the
geometric interpretation of the gauge fields as extra-dimensional
gravity; and of the general notion of physical extra dimensions.

The effective Lagrangians mentioned above are those which come out of
the total metric
h_{mn} = e^{2U} g_{mn} + k_{ab} A^a_m A^b_n
h_{mb} = k_{ab} A^a_m
h_{an} = k_{ab} A^b_n
h_{ab} = k_{ab}
where different choices of U can be used to define what the effective
"base space" metric is. The choice e^{2U} = k^{-1/2} for a 4-D base
space gives you an Einstein-Hilbert Lagrangian of the form root(|h|)
R_h = root(|g|) R_g + ... Otherwise, you get a power of k out in front
for the leading term (which is sometimes used to model a variable G).

.

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