An algebraic approach to Koopman classical mechanics

That's not what the preview showed! Grrrrr. Aaaaannnnndddd I can't edit it.

Approximately what I said after the equation that screwed up was: The inner product (f,g) that's used in Eq. (12) can be replaced by any structure for which (f_i,f_j) is a positive semi-definite matrix, which doesn't necessarily have to be sesquilinear in f_i and f_j. If we can find a manifestly diffeomorphism invariant form (f,g), with f and g appropriately structured for a candidate geometry, then we have a candidate first approximation for a quantum gravity.

There are now three articles on The Quantum Daily:

Unifying Classical Physics and Quantum Physics Measurement

Unifying "Collapse" and "No-Collapse" Approaches to Quantum Physics

Breaking the Hold of Bell Inequalities

Hi Peter, happy to see you on FQXi and that you share your ideas, great

Dear Peter, have you already thought to play with these koopman algebras and the entropy ?

Dear Steve,

I generally prefer to work with the collection of probability distributions associated with a given collection of measurements, because many quantities such as entropy are undefined for nontrivial quantum or random fields. [One could perhaps work with quantities such as the relative entropy between different states, but I haven't. Entropy is problematic, in any case, I think, because it's a property of a state, not a measurement result.] I think it's really important that we work with test functions as descriptions of measurements that are performed and of how we modulate measurement results, not with what's really there (such as the entropy of the state), because many aspects of the latter are not well-defined, even though the measurement results are well-defined except at singular points.

In any case, I've started to work again on what I had been working on before AlgKoopman, the problem of renormalization, or, rather, on how to construct interacting quantum and random fields in a well-defined way. AlgKoopman came out of that research in a natural way, but it's not my principal research direction.

Dear Peter, I can understand seen that this entropy is indeed problematic and complex due to its property. I try personally to find a road for the non relativistic fields and the complexity with this dark matter encoded , I consider a different sense of rotation and the cold also , the clifford algebras and other mathematical tools for a kind of partition but it is not easy. Regards

I'm pleased to announce that Annals of Physics has selected "An algebraic approach to Koopman classical mechanics" as a Highlighted Article. There is a highlight post written by Rob Lea on the Annals of Physics website.

Congrats Peter, Frank Delplace of Facebook also for his article about the vicosity and several others also that I know, the annals of physics are well , you are good about the Koopman alg , regards

The only way you know you've made progress towards quantum gravity, is if you can come up with an experiment. But congratulations on your "algebraic approch to Koopman classical mechanics".

Thank you! I have to say that Quantum Gravity is definitely not my principal interest: I'm mostly interested in finding ways to rethink the conceptual issues in QM and in QFT, which I think AlgKoopman does in a more-or-less new way (if you read the introduction to AlgKoopman you'll see that almost everything in AlgKoopman has a precursor: Koopman, signal analysis, Wigner functions, Generalized Probability, contextuality, algebraic QM/QFT, ..., but I think they are put together in a way that I've not seen done elsewhere in the literature).

We'll hopefully know in five or ten years whether the way the ideas are put together in AlgKoopman help much or whether they just become more background noise. Still, my feeling is that anything that gives us a new way to understand the relationship between CM and QM might give someone a clue how to construct a different kind of quantum gravity, which hopefully we could test.

Hi Peter,

You should consider the possibility that the graviton really does exist. A graviton that expands from a point, at the speed of light, with a radius r = ct, would fill all space with spacetime geometry, inertial refernece frames, and virtual photons. You could say that you captured a graviton when you create quantum entangled photons. That would be your doorway to performing an experiment.

Best wishes,

JW

this quantum gravity is like I told difficult to renormalise in considering the bosonic fields, that cannot be quantified in this road, the photons encoded are not the solution apparently, we can utilise all the mathematial tools that we want and the non commutativity , we don t reach it, Connes has tried, Penrose and his twistors, the Lie E8 , the loops,the superstings or Mtheory or others, all the best thinkers have tried in fractalising the fields , they don t arrive, and verlinde also has tried with his entropical gravity, that does not solve, the problem foundamental is that the majority of thinkers have considered only these photons like primordial essence and that all comes from fields, and it is not proved , they have not renormalised it because there is a deeper secret to superimpose to unify this GR and QM simply , furthermore we need a balance for the heat and electromagnetism and to explain the anti matter also, the secret and I have reached it is to encode a deeper gravitation logic than just these photons oscillating differently, and so we change just the distances because the codes are farer and are gravitational and that this electromagnetism is just emergent. It is only simple than this, never in considering this weakest quantum force like an emergent electromagnetic force you arrive to renormalise even with all the geometrical algebras possible, the QFT is interesting but the gravitation is different than these bosons.It is necessary to reach this quantum gravitation to forget this GR and this photonic electromagnetism, the problem is easily solved when you think beyond the box.

your photonic gravitons so oscillating differently are not the solution, and are not the answer

the points also in 1D like foundamental objects are not proved and are not the answer even with adapted geometrodynamics and oscillations

Hi Jason, I think I'm only spinning my own wheels. They're mine to spin, I suppose, though perhaps I'm wrong about that as about many things. Like I say, my interests are not in quantum gravity, so I only have anything to say about QG in the very broadest terms.

It looks like this thread is spinning faster than I can keep up with it, so I guess I'll bow out.

Peter , sonme thinkers are good in details, others are good for the generality , it is just like this the life, we must adamit our limitations, I liked your koopman alg, they are good for a better understanding of this QFT, congrats, be the force with you Jedi of the Sphere, spherically and humbly yours

Hi Peter,

Forgive me for being the hare (the rabbit) that hops ahead and looks down the road to see what's there, while the tortoise (the hard workers) labor deeply on the mathematics.

I want the physics community to discover gravity field generators. I would do it myself if I had the talent and the energy needed.

I see the flaws in superstrings and quantum loop gravity because strings and loops just sit their and wiggle. But whatever the universe is made of, it has to explode and expand quickly (perhaps even faster than the speed of light) because that's what the big bang did.

I advocate for the idea that gravitons really do exist. But they begin as points (perhaps from the Planck scale) and they expand at the speed of light into spheres that overlap with other expanding gravitons. The idea is that all of these overlapping gravitons produce the effect of a spacetime continuum. I think that the Einstein equations are describing an equilibrium of expanding gravitons.

But this is the part of my idea that I'm quite proud of. Every quantum mechanics problems is solved by calculating the wave function. So does the wave function actually exist? Some people think it does; I do. But I think that the wave function is what happens when an expanding graviton collides with a particle, it becomes a trapped graviton.

Wave functions have operators for momentum p_x, position x, energy E, etc. In quantum mechanics, you can calculate the expectation value of momentum states, position states, etc...

I think that the graviton has quantum states for position, momentum, etc., built into the expanding graviton.

If you ask me what spacetime is ultimately made of, I would argue that it's made of gravitons which are made of quantum states for position/momentum, therefore, spacetime is made of quantum states for position/momentum.

There is an experiment that can be performed if you have an appetite for it.

JW

I got the idea of expanding gravitons from the derivation of special relativity.

https://physics.stackexchange.com/questions/134984/time-dilation-confusion

Each inertial reference frame is created by gravitons that are expanding from a point, to a sphere with radius r = ct.

If there are gravitons being created constantly, at every point in space, and for every possible velocity, AND those gravitons are creating space and time, then the speed of light is invariant.

The spacetime interval also looks like an instantaneous condition of two gravitons expanding, with relative velocity of 0.

Hi Steve,

Do you know how the wave function has a phase, e.g.,

[math]\Psi = Ae^{i(kx - \omega t)}[/math]

Like I've argued, wave functions are gravitons. Gravitons expand from a point, every point in space. They are continually being generated. Therefore, all space is being filled with wave functions, filled with gravitons. There are so many gravitons being generated, there is always a 100% chance that two wave functions, two gravitons will approach each other from opposite directions, opposite frequency (same frequency 180 degrees out of phase).

The result will be a probability density,

[math]\rho(x,t) = \frac{|\Psi(x,t)|^2}{\int _{-\infty}^{\infty} |\Psi(x,t)|^2 dx }[/math]

So where is the electron? It's somewhere are the x-axis. There are gravitons that are constantly bombarding it from both sides of the axis.