"I think that your idea is to look for equations that might reveal an information channel."

Dear Ch.Bayarsaikhan,

Thank you for your comment!

Yes, it does seem to me that the equation for the Born rule in David Bohm's book on Quantum Theory and the equation defining proper time in terms of coordinate time-- both-- support "local logic" for respective infomorphisms: the former involving an information channel from the nonStandard future into the nonStandard past, the latter an information channel from this process into a locally flat piece of spacetime and therefore into a local quantum field, the slight curvature between such local fields then being the result of escaped hidden energy from this process. It would be like a "thermodynamic Computer Automaton"-- as in this video about the CA Interpretation of Quantum Mechanics by G. 't Hooft. But instead of linking a CA to each location of space as 't Hooft describes, I would see a "thermodynamic CA" in each of these particle-related processes.

Then the particle-as-object is like the continuous flow that we see in a motion picture comprising discrete image frames. Everything we physically perceive is technically the past like this, not the present. Because it takes time for the brain to assemble and process the incoming information from the discrete frames. So all of our intuition is based on this image of the past, which we think of as the present. Hence discrete images from the thermodynamic CA create an image recorded in the past of a continuously existing object-- which we perceive as the classical, continuously existing "particle." The wave nature exists as a field of possibilities in the future, which we perceive only through our mathematical imagination, and not our physical perception as we do the past. In this way the "thermodynamic CA" is both wave and particle.

Granted, using the "hidden" character of this energy as it must be in order to support the game required, in terms of mathematical game theory, may be a stretch. But I suspect that we are approaching the limit of our ability to understand the Universe, and at this limit, the mathematical methods become sparse. This is my justification for saying that because this energy must be hidden for the mathematical game to work-- therefore it must be "dark" in terms of the current mathematical methods.

Very Best Regards!

L

13 days later

ADDENDUM--

It's been said that writing is learning. Hemingway re-wrote one of his endings 47 times and according to this idea, he was simply learning what he wanted to say. For me writing this essay has been learning about the Born infomorphism and the associated proper time for the individual particle. Then after writing the essay, writing comments has been learning about N particles. Specifically I have been learning about this question:

If there's a Born infomorphism for N particles, what physically specifies the worldline of the associated proper time?

In the Schrödinger equation for N particles, there is just one time for the system of N particles and not multiple proper times, one for each particle. Then-- is this one time in the Schrödinger equation the proper time for the system of N particles-- is it the proper time for the system of N particles, which holds the Born infomorphism for the system? For support I look to Richard Feynman:

In Space-Time Approach to Non-Relativistic Quantum Mechanics, Richard Feynman wrote, "The formulation is mathematically equivalent to the more usual formulations. There are, therefore, no fundamentally new results...The total contribution from all paths reaching x, t from the past is the wave function Ψ(x,t). This is shown to satisfy Schroedinger's equation."

Rather than finding something like "1+2 = 3 is the same as 2+1=3," Feynman seems to have discovered, here, an "information channel" from the entangled system of N particles to the their quantum fields as represented by a Feynman diagram of the N particles. But Feynman didn't use mathematics to describe the information channel he'd discovered-- instead he used natural language, as above.

In this essay, I used some mathematics to describe an information channel. To show that an information channel exists, I first have to show that an infomorphism exists. Which means showing there are two functions or arrows pointing in opposite directions, with heads connected to tails by each pointing to a respective situation, each situation supporting infons or elements of information from which the respective function or arrow originates-- altogether meaning that the information supported in the one situation is connected by these arrows to the information supported in the other situation. The arrows take you back and forth in a closed circuit to the same information, with two equivalent translations. (Please see the diagram attached to this comment.)

So from the proper time of the system of N entangled particles specified by the Schrödinger equation, there is an arrow to the SET (I must emphasize) of possible coordinate time frames in spacetime, where for each coordinate time frame, the usual defining equation of proper time holds.

To get the above function describing an infomorphism, an arrow must go from the proper time of the entangled system to the SET of possible coordinate time frames.

Because in the other direction, there is the arrow from spacetime to the wave function of the entangled system.

Because (recalling what I learned in the essay) the entangled wave function includes not only possibilities in space but also possibilities in spin (etc.).

Space is just a "part" of these possibilities.

In "part of" relations, there is a function mapping each part to the system of which it is a part.

Which determines the direction of the arrow between the two kinds situations in the infomorphism.

To get a function going in the opposite direction, as required to identify an informorphism, there must also be a function from the proper time of the entangled system of N particles to the SET of possible coordinate time frames in spacetime.

Then to obtain Bohr's correspondence principle, which ultimately enabled Newton to have calculated using centers of mass (which don't exist in relativity), the worldline associated with proper time of the entangled system must be attached to the Fokker-Pryce center of inertia. About which, Pryce wrote, "Of these only one is independent of the frame in which it is defined. It suffers from the disadvantage that its components do not commute (in classical mechanics, do not have zero Poisson brackets), and are therefore unsuitable as generalized co-ordinates in mechanics."

Which "unsuitability" seems compatible with the idea of space emerging or depending upon entangled possibilities-- and not the other way around.Attachment #1: 1_infomorphism.png

Applying G. 't Hooft's Cellular Automaton Interpretation of Quantum Mechanics

First I assume it can be inferred from 't Hooft's paper (above) that algorithms may underlie Hilbert Space.

Then for an algorithm implementing a stream, there would exist in the mathematical models two functions:

(1) a function from the stream to a set of types, which would be the possible types of an element of the stream.

For example--

aStream = (elementOfSomeType, aStream)

which produces the stream of elements:

elementOfSomeType_1, elementOfSomeType_2, elementOfSomeType_3 and so on

Which requires a function from aStream for example, to the set--

{type_a, type_b, type_c}

Where an element produced by the aStream could, then, be any of the three types-- type_a, type_b, or type_c-- as specified by the above set.

(2) As well-- for each cycle of aStream-- a function from the specific cycle to the actual element produced in that cycle. For example:

aStreamInCycle1 -> anElementOfType_a

aStreamInCycle2 -> anElementOfType_c

aStreamInCycle3 -> anElementOfType_b

and so on.

Then if we see these functions in the mathematical model, by the assumed inference from 't Hooft's work on Cellular Automata, we can infer that an algorithm exists-- in this case, an algorithm that generates a stream.

In the ADDENDUM (previous post here), "the Feynman infomorphism" exists (to give it a name). Which requires the existence of a function from proper time of the entangled system of particles to the SET of possible coordinate time frames that support the usual formula for defining proper time, applied to this proper time.

This set is then like the above set used to define the types of aStream.

And in that case, we are just one step away from finding an algorithm-- which from the assumed inference based on 't Hooft, would underlie Hilbert space.

The next step:

Find the second function, as above, which maps each cycle of the stream to an actual instance of type specified in the set.

(In the essay, this is apparent in the Born infomorphism for the 3-valued Chu space suggested by Abramsky.)

More later.

11 days later

The (perhaps) smallest possible connection between General Relativity and the Standard Model of Particle Physics

In previous posts-- to establish an infomorphism or information channel between the proper time of a system of entangled particles and coordinate times of spacetime, a function was said to exist from (a) this proper time to (b) a set of coordinate times from which this proper time can be defined.

In this context, it is possible to imagine this set to be associated with a set of Feynman diagrams. For example consider Compton scattering. In the Standard Model of Particle Physics, in Compton scattering there is a set of Feynman diagrams.

While in General Relativity, there is just the lightcone on which the photon in Compton scattering travels. In the diagram from GR, the proper time of the photon to be Compton scattered has an interval of zero time from creation in the experiment to destruction in the Compton scattering event (as in the Feynman diagrams of the scattering event).

Nonstandard analysis can be applied to connect this diagram in GR to its Feynman diagrams in the Standard Model.

On the GR diagram of the lightcone on which the photon to be scattered exists, for any point on the worldline of the photon on that light cone (starting from the initial event in the experiment and ending at the Compton scattering event), the photon at that point will have in front of it an amount of time left to travel the worldline that is infinitely close to zero. And behind it, an amount of time already travelled that is infinitely close to zero.

Then in this case, "zero time" for the photon to travel this worldline in the usual GR diagram comprises instead a "nonstandard past" and a "nonstandard future"-- in nonstandard analysis, both being intervals of time infinitely close to zero.

And in this nonstandard future, exists the set of possibilities diagrammed in the Standard Model by the Feynman diagrams of Compton scattering.

This connection of worldline diagram in GR to Feynam diagrams in the Standard Model (based on nonstandard analysis) might be the smallest possible connection that one could draw between GR and the Standard Model.

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