I am not sure whether the wave physics in the paper by ov_WaveMotion_45_154_EvolutionWavePackets.pdf">Christov](https://www.fqxi.org/data/forum-attachments/2008CChrist
ov_WaveMotion_45_154_EvolutionWavePackets.pdf)
involves quantum mechanics. The dissipative equation has a solution with a complex frequency. An instanton or tunneling state has a frequency or wave number that is imaginary, eg multiplied by i = sqrt{-1}. However, a complex frequency ω = α + iβ means the system has dispersion and dissipative physics. This demolishes some foundations of quantum mechanics in a way which I find terribly problematic. So as a fundamental wave equation for a free particle this is not likely to be fundamental.
The wave equation is argued to be a source for the redshift of galaxies. A variant of this wave equation could be derived with general relativity for the de Sitter spacetime. The metric is
ds^2 = dt^2 - e^{sqrt{Λ/3}t}(dr^2 + r^2dΩ^2)
where we only consider the metric coefficient g_{rr} = -e^{sqrt{Λ/3}t} and g_{tt} = 1 to ignore rotational motion. The Christoffel symbols are
Γ^a_{bc} = (1/2)g^{ad}(g_{db,c} + g_{dc,b} - g_{bc,d})
which, as I calculate in my mind, leaves us with
Γ^t_{rr} = -(1/2)g^{tt}g_{rr,t} = (Λ/3)e^{sqrt{Λ/3}t}
This connection coefficient is used in covariant derivatives
Du^a/ds = du^a/ds + Γ^a_{bc}u^bu^c,
where for u^a = dx^a/ds this is the geodesic equation if the acceleration Du^a/ds is zero.
So for the vector valued wave function u^a the time derivative ∂u^a/∂r = ∂_ru^a requires a covariant derivative
D_ru^a = ∂_ru^a + Γ^a_{rc}u^c
which requires the index c -- > r and
D_ru^a = ∂_ru^a + Γ^t_{rr}u^t
= ∂_ru^a + (Λ/3)e^{sqrt{Λ/3}t}u^t.
The second derivative would then be
D^2_ru^a = = ∂^2_ru^a + 2(Λ/3)e^{sqrt{Λ/3}t}u^t + (Λ/3) e^{sqrt{2Λ/3}t}u^t.
That is rather messy, so I will Taylor expand the exponential and keep only factors O(Λ) and we get
D^2_ru^a = = ∂^2_ru^a + 2(Λ/3)u^t.
So the wave equation is adjusted to
0 = ∂^2_tu^a - D^2_ru^a = ∂^2_tu^a - ∂^2_ru^a - 2(Λ/3)u^t.
In now use the light cone condition that u^t = g_{rr}u^r for a massless particle and I have the wave equation
∂^2_tu^a - ∂^2_ru^a - 2(Λ/3)u^r = 0.
I now let u^r = a exp(ikr - iωt) and substitute this in to get the condition
-ω^2 + k^2 - 2i(Λ/3)k = 0
k = i(Λ/3) +/- sqrt{- i(Λ/3)^2 + ω^2}
The binomial theorem is used for ω^2 >> |(Λ/3)| and we get an approximate answer
k =~ {iΛ/6 - ω, iΛ/2 + ω}
The result is a dissipative dispersion relationship. The photon loses energy, it loses energy to this cosmological gravity field. This loss of energy manifests itself as the redshift. The second solution may be written as
ω = k - iΛ/2
and a wave function Gaussian wave packet
u^r = exp(β(k - k')^2/2)exp(ikr)
has δk/k = βΛ/2 which for β = 8πGt/3k and H^2 = 8πGΛ/3 this gives the time dependent spread of the wave packet with time.
In this approach, which I worked on screen and took far more time than I was wanting to spend, there is no need to posit a quantum field which has some dissipation term, which leads to non-unitarity in quantum mechanics. Non-unitary quantum mechanics would mean the universe is far less sensible. So based on what I have done here, modulo any math error which might have occurred due to working this in my mind, the expansion of space can quite adequately account for this red shift.
Cheers LC
