Differentiated Sakharov Equation for Vacuum Fluctuations

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We explore Sakharov's famous fluctuation equation where he gives corrections to the gravitational field. In this post, we apply a simple differentiation technique to obtain a totally new form.
To a non-physicist, I know some papers can appear very abstract, and Sakharovs equation was one of them. You can follow his ideas from various articles, here's a few to chew on

https://www.atticusrarebooks.com/pa...d-the-theory-of-gravitation-in-soviet-physics

https://www.researchgate.net/publication/326451872_Sakharov_Curvature_in_Rowlands_Duality_Spacetime_Do_vacuum_%27spacetime_forces%27_curve_matter/link/5c69b8baa6fdcc404eb73615/download

As abstract his equation may appear, I'll break it down so it can be umderstood. It won't take long as the premise is easy to grapple. In his paper, he was attempting to explain how early physics didn't believe in the existence of fluctuations, which is described by a divergent integral series where it is taken over the momentum of the ground state of the field and was presumed zero. In his approach, he showed that this not the case (today, we take the existence of these ground state particles now as a matter of experimental fact) but still remains a very hot topic in physics, because it lies now at the very nature of quantum mechanics, where these fluctuations really do come into and out of existence, governed by the creation and annihilation operators: It's a phenom so engrained in physics that Bogoluibov transformations take place in a wide range of physical systems, even black holes that preserve the idea of Unruh-Hawking radiation.

One way to produce particles in a quantum theory is by making the grabational potential an operator satisfying the creation amd anmihilation of particles, and look like

##\hat{\phi} \propto aa^{\dagger} + a^{\dagger}a##

Whether or not this was known to Sakharov at the time of his now dated paper, has relevance to the reason why he approached his theory. Even if gravitation is not a real field, it has been considered to unify in some way to the world of particles. Sakharov's natural insight allowed him to show that higher powers of the metric curvature could sometimes allow virual particles to become real. Maybe at the very heart of his theory, particles themselves owed their existence to curvature. Its a theory of how the background curvature contributes to the field of particles.

Salharov explained that you could expand a Lagrange equation in a geometric series that satisfies

##\mathcal{L} = C + A \int\ k\ dk \cdot R + B \int\ \frac{dk}{k} R^2##

It was so fruitful that he not only obtained a gravitational action

##G = - \frac{1}{16 \pi A \int\ k\ dk}##

Where ##A\propto 1## but that the geomeyric series which produced ##\frac{dk}{k} \propto 137## the infamous fine structure.It was further discovered that by intehration of the wave length yielded the value of ##k_0 \propto 10^{28}eV \propto 10^{33} cm^{-1}##, corresponded to the inverse of a Planck length, an indication that the wavelength of the particle was that corresponding to its smallest smearing in space.

In Sakharovs equation, he didn't specify all to clearly what the constants of ##A## and ##B## where, but its full translation has bern written out, the first authors I read who made this equation cear in the context of quantum mechanics came from Arun and Sivaram, they explain the nodes of the field will satisfy Sakharovs Langrangian in the following way

##\mathcal{L} = C + \hbar c \int\ k\ dk \cdot R + \hbar c \int\ \frac{dk}{k} R^2##

A quick inspection from dimenional analysis upholds this. If R is the curvature from Einsteins theory, then it has units of inverse length squared, so inspection from the first part

## \hbar c \int\ k\ dk \cdot R##

We see is like saying the energy is

##\frac{\hbar c}{length}##

With one extra wave length node ##k## and the curvature ##R## is equivslent to a density, so it has units of energy density, exactly what a Langrangian density should have. Now here is where I invite the "new" physics from some application of calculus.

Since we are blessed with the prospect of knowing that the nodes of the field are described by the relaively simple formula using ##k## as the wave lengths or nodes, the inverse can be characterized as the length of the wave, which in physics id given by the speed of the particle by a frequency term. As physicists, we often call it ##\lambda## and has units of length. We now say that the curvature is a function associated to these waves, so we write a general formula,

##R^n f(\lambda) = \frac{d}{dx}(\frac{d}{dx})f(\lambda)##

And its differentiation will yield

##= \frac{d}{dx}(n \lambda^{n-1}) = n(n -1)\lambda^{n-2}##

Physicists are often more than not pure mathematicians, we can be quite different animals, but the result is standard enough that to a mathematician, it should be clear that ##\lambda## becomes our length that has been differentiated like it was in respect to ##x##. Now we csn also define it as

##R^n f(\lambda)^n = \frac{d}{dx}(\frac{d}{dx}) f(\lambda)^n##

So long as we recognise that ##n=2## and we see now that Sakharovs equation satisfies the differentiation written as

##\hbar c \int\ \frac{dk}{k} R^n f(\lambda)^n ... + \hbar c\ d(log_k) n(n-1)\lambda^{n-2}##

Where we have rewritten ##\frac{dk}{k}## as the differential logarithm of the wave number, and we have replaced ##R^nf(\lambda)^n## with its differential equivalent form of ##n(n-1)\lambda^{n-2}## which we explained earlier was nothing but ##R^n\lambda^n## in a differential form of ##... = \frac{d}{dx}(n \lambda^{n-1}) = n(n -1)\lambda^{n-2}##
 
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