A Very Simple Theory of Gravity in the Hilbert Space

[Dubbelosix]

 

I'll be closing my studies on gravity in the Hilbert space soon now. I will finalise a more comprehensible paper from it as well, but i wanted to reserve for this post, the most interesting conclusions I made. There were many things I found while investigating this model, but the following will only be concerned with the dynamics of [math]L^2[/math] Cauchy-Schwarz space and the Hilbert space formalism. The ultimate goal was to find a Hilbert space theory of gravity which seemed (at least plausible) by linking it dynamically with the Schrodinger evolution. I achieved this but it needs so much more work that (maybe) someone else will take up as I will be going back into cosmology investigations. 

Working (back from where I ended up) actually would make more sense to the reader as I can keep it short and simple. The analogue of the geodesic equation found in GR:

[math]\frac{d^2x^{\mu}}{d\tau^2} + \Gamma^{\mu}_{\nu \lambda} \frac{dx^{\nu}}{d \tau} \frac{dx^{\lambda}}{d\tau} = 0[/math]

was in fact, satisfied in a very concise way using the bra-ket notation, 

[math]\nabla_n \dot{\gamma}(t) = \nabla_n\frac{dx^{\mu}}{d\tau} \equiv\ min\ \sqrt{<\dot{\psi}|[\nabla_j,\nabla_j]|\dot{\psi}>} = 0[/math]

There is classical commutation going on, on the RHS 

[math][\nabla_j, \nabla_j] = 0[/math]

which just means in the usual sense, that the order of the operators does not matter, for obvious reasons. This is why it should be noted, that space time non-commutivity is subtly hinted at when you consider connections with derivatives in both space and time. Anyway, moving on, from the second equation we can construct two solutions for the time-dependent Schrodinger equation which satisfies 

[math]\frac{1}{ i \hbar}H|\psi>\ = |\dot{\psi}>[/math]

When you consider the geometry related to the Hamiltonian in such a way:

[math]\sqrt{<\dot{\psi}|\dot{\psi}>} = \int \int\ |W(q,p)^2| \sqrt{<\psi|\Gamma^2|\psi>}\ dqdp \geq \frac{1}{\hbar}\sqrt{<\psi|H^2|\psi>}[/math]

Which was derived by myself using the Wigner function, then it becomes more apparent that it was possible to construct a wave equation encoding the rank 2 tensor dynamics of the connection and the stress energy tensor in the following way

[math]\nabla_n|\dot{\psi}>\ =  \frac{c^4}{8 \pi G} \int \int\ |W(q,p)^2|\ (\frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i})|\psi>\ dqdp \geq \int\  \frac{1}{\hbar}(\frac{d}{dx^n}T^{ij} + \Gamma^{i}_{n\rho} T^{\rho j} + \Gamma^{j}_{n \rho}T^{\rho}_{i})dV|\psi>[/math]

This is in fact, totally analogous to an acceleration/curve in the general theory of relativity, except this time, it satisfies an equality bound and it also satisfies the wave dynamics of a Schrodinger equation, albeit, a non-linear one. It is not immediately obvious there are terms in this last equation which can satisfy an inequality bound, but in my early investigation, the bounds where found as:

[math]\sqrt{|<\nabla^2_i><\nabla^2_j>|} \geq \frac{1}{2}<\psi|[\nabla_i, \nabla_j] |\psi>\ = \frac{1}{2} <\psi|R^2_{ij}| \psi>[/math]

[math]<\psi|[\nabla_i, \nabla_j] |\psi>\ =\ <\psi| R_{ij}| \psi>\ \leq 2 \sqrt{|<\nabla^2_i><\nabla^2_j>|}[/math]

Which is basically a mean deviation that can reach twice the classical upper bound. This bound holds importance for our equation which satisfied the geodesic equation

[math]\nabla_n \dot{\gamma}(t) = \nabla_n\frac{dx^{\mu}}{d\tau} \equiv\ min\ \sqrt{<\dot{\psi}|[\nabla_j,\nabla_j]|\dot{\psi}>} = 0[/math]

Which can have a Berry curve definition 

[math]\gamma = i \oint <n(\mathbf{R})\nabla_{\mathbf{R}}|n(\mathbf{R})>\ d\mathbf{R}[/math]

This formulation links the wave dynamics with the geometry and the energy associated to the geometry of the system. This geometric look at the Hilbert space has allowed us to find solutions to the geodesic equation, which appear at the surface, a simple argument. This ability to give a vector space curvature was really the most enlightening key of the investigation. The Hilbert space, even though is an abstract space, is one that actually works very well with quantum mechanics, for whatever unseen intrinsic reason there is for this. Equally, in such an abstract mathematical space, you can create mathematical objects to describe the necessary curvature required to describe the accelerations in the phase space. Since gravity is a pseudo force from the first principles of relativity, then it is unlikely quantization of the field into the usual spin 2 graviton gauge theory is a bit extreme - why should gravity get a mediator particle when the Coriolis force and the Centrifugal force do not require one? How is gravity a pseudo force?

Gravity is a pseudo force that can be understood in the following (neat) and (concise and short) way:

[math]\frac{d^2x^{\mu}}{d\tau^2} + \Gamma^{\mu}_{\nu \lambda} \frac{dx^{\nu}}{d \tau} \frac{dx^{\lambda}}{d\tau} = 0[/math]

where

[math]\Gamma^{\mu}_{\nu \lambda} = \frac{\partial x^{\mu}}{\partial \eta^{a}}\frac{\partial^2 \eta^a}{\partial x^{\nu}\partial x^{\lambda}}[/math]

or more compactly 

[math]\Gamma^{\mu}_{\nu \lambda} = J^{\mu}_{a} \partial_{\nu} J^{a}_{\lambda} = J^{\mu}_{a} \partial_{\lambda} J^{a}_{\nu} \equiv J^{\nu}_{a} J^{a}_{\nu \lambda}[/math]

which represents a pseudo force for gravity which makes it in the same league as the Coriolis and the Centrifugal forces.The previous equations also expose the inner structure of our investigation. Keep in mind for the Schrodinger Curve equation we derived, the covariant derivative acts on rank 2 tensors in the following way:

[math]\nabla_n\Gamma^{ij} = \frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i}[/math]

[math]\nabla_nT^{ij} = \frac{d}{dx^n}T^{ij} + \Gamma^{i}_{n\rho} T^{\rho j} + \Gamma^{j}_{n \rho}T^{\rho}_{i}[/math]

Where the Hamiltonian has been 'naively(?)' replaced with the stress energy tensor. In a much later study when I get interest in this again, I will try and formulate a theory of how temperature and gravity translate into each other, since it was an observation from early investigations that the curve of a metric is in fact related to the temperature of the system given as (and as you will see, the definition on the left imples a factor of one half attached to our metric):

[math]K_BT = \frac{1}{2}(\frac{dx^{\mu}}{d\tau} \cdot \frac{dx^{\mu}}{d\tau}) \equiv\ \frac{1}{2}<\dot{\psi}|\dot{\psi}>[/math]

This extra factor of [math]\frac{1}{2}[/math] is acceptable since one such term exists in the classical upper bound equation and [math]m=1[/math] for a constant mass.

That's the latex fixed, sorry about that.

I'd also like to say, Mordred helped me along with this. He/she did direct my in the right way a few times. One such example was finding the bound which I had not considered until they offered it. 

Edited November 22, 2017 by Dubbelosix