Mordred

The problem can easily get compounded when using AI to study physics. One example is having AI look up a specific equation. One example being if you were to look for relations specific to a metric or methodology. AI could very well return a relation specific to say a canonical treatment as opposed to a conformal treatment. With the FLRW metric it often confuses conformal coordinates as opposed to commoving coordinates. If the AI user isnt aware of these distinctions to recognize the AI mistakes they could easily get confused as well as get frustrated when they try to apply those equations. As AI looks through literature Ive seen it throw in cosmographic metrics as well and mix them with commoving metrics. (LOL the above can also be used to recognize someone relying too heavily on AI)

Forgot to add I don't see anything particularly wrong in your treatment above at the moment. In so far as the math relations involved. I would be curious though if you agree that direction would be an inherent degree of freedom of any underlying state/system being described. Where one state resides in relation to another obviously is related.

Thank you for the above, its a tremendous help in understanding the purpose of your article. Sorry I was being a bit of a stickler on material needed being presented here. I do have good reasons for that, lol lets just say I've come across one poster in the past that although his ideas were sound. He had dozens of different papers and articles he kept referring to and you literally had to go through them to get any sense of what he was doing in the first place.... That's not the reason of course but its a good extreme example. In the above you have a statement of avoiding any unnecessary complexity. Obviously scalar relations does indeed simplify the mathematics I would argue that requiring "direction of kinematic relations is a necessary complexity". Which direction an interaction (whatever that kinematic interaction represents) is just as important as the scalar relations. Obviously we all know any " Field treatment requires geometry" particularly for any mappings of particle or measured quantity distributions". Depending on what your after those mappings will also give a necessary complexity. Those are two aspects I would consider as being necessary ( for what I do in physics absolutely necessary) So the question of what is "necessary complexity" is something I think should be looked into in greater detail. Side note I will often post added Mainstream relations relating to a thread. I've found in the past this habit is an aid to other readers not involved in the conversation better understand what is being discussed as well as useful for comparisons between methodologies etc

agreed though propogators cannot be directly measured and include probability currents which are mathematical as well

Your list above is fairly accurate though some of the list is fairly broad. Light path deflection for example would include spectography redshifting. For example integrated Sache Wolfe effect as signals pass through the mass variation of DM halos as one example. If I think of anything not already covered on that list I will post it As far as the fine structure constant your methodology from what you described here sounds remarkably similar to whats done in BSBM model (Berkenstein Model ) a version of TeVeS MOND. The problem with coupling the fine structure constant is that you may find you would require a varying fine structure constant as per BSBM as well as the Hubble constant also varies over time. ( it's only constant everywhere at a given time slice. Ie today. If you would like to test it at different Z ranges I can give you the Hubble constant value at any given redshift value. The cosmocalc in my signature which I was involved with developing has the correct second order terms for when the recessive velocity exceeds c for redshift beyond 1.49 ( Hubble Horizon) to the particle horizon. the following below is for other readers to keep others at the same speed. The second order formula I'm referring to is the last formula on the list. The previous formulas is the mathematical proof using the equations of state and how they evolve over the universe expansion history. FLRW Metric equations \[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\] \[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\] \[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\] \[H^2=(\frac{\dot{a}}{a})^2=\frac{8 \pi G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^2}\] setting \[T^{\mu\nu}_\nu=0\] gives the energy stress mometum tensor as \[T^{\mu\nu}=pg^{\mu\nu}+(p=\rho)U^\mu U^\nu)\] \[T^{\mu\nu}_\nu\sim\frac{d}{dt}(\rho a^3)+p(\frac{d}{dt}(a^3)=0\] which describes the conservation of energy of a perfect fluid in commoving coordinates describes by the scale factor a with curvature term K=0. the related GR solution the the above will be the Newton approximation. \[G_{\mu\nu}=\eta_{\mu\nu}+H_{\mu\nu}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}\] Thermodynamics Tds=DU+pDV Adiabatic and isentropic fluid (closed system) equation of state \[w=\frac{\rho}{p}\sim p=\omega\rho\] \[\frac{d}{d}(\rho a^3)=-p\frac{d}{dt}(a^3)=-3H\omega(\rho a^3)\] as radiation equation of state is \[p_R=\rho_R/3\equiv \omega=1/3 \] radiation density in thermal equilibrium is therefore \[\rho_R=\frac{\pi^2}{30}{g_{*S}=\sum_{i=bosons}gi(\frac{T_i}{T})^3+\frac{7}{8}\sum_{i=fermions}gi(\frac{T_i}{T})}^3 \] \[S=\frac{2\pi^2}{45}g_{*s}(at)^3=constant\] temperature scales inversely to the scale factor giving \[T=T_O(1+z)\] with the density evolution of radiation, matter and Lambda given as a function of z \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] its other purpose was more my work testing the accuracy of the inverse relation to blackbody temperature. I rarely trust literature on any verbatim basis so often like to see how a statement such as temperature being the inverse of the scale factor is determined as being accurate. Sides its good practice lol ( above i had done previously in my Nucleosynthesis thread. ) the last formula the cosmocalc employs though has from version 1 of the cosmocalc well over a decade ago . specifically this formula will provide the Hubble constant value as a function of redshift \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] should note for others as well the GR statements are for the Newton approximation which the FLRW metric falls under just a side note the FLRW metric is not maximally symmetric where the Minkowskii metric under SR is. The use of the scale factor is one of the key issues with maximal symmetry (You can see this via the Christoffels for the FLRW metric ) or another way to learn this is through the Rayleigh equations.

In Feymann integrals the propogator ( a propogator propagates an operator) with the propogator being the internal lines and operator being the external solid lines ( observables ) ie real particles with internal often associated with virtual particles though its more accurate to just treat the propogator as field. You require one quanta of effective progator action to affect an operator . Thats about the only way one can potentially denotes some form of minimal threshold that I myself am aware of.

The writeup was likely using a Hilbert space common in QM treatments. A Hilbert space being defined from the inner product of a vector field. Its not the only class of wavefunctions. You can have wavefunctions that do not require a Hilbert space nor the inner product. Scalar field spaces being one example. You have no need for vectors nor inner or cross product. However you can still have a wavefunction relating to number density of photons as one example based on the amplitude of the probability current. Just an FYI. Lol one solid clue to keep track of the distinction. A function is a mathematical set of operations. The prefix of wave is simply naming the type of function. Same applies to correlation function for entanglement.

Lets straighten out the wavefunction being not physical. You develop the wavefunction using known properties of the particles state and apply it to the Schrodinger equation or Klein Gordon etc. You can also take into consideration the experimental apparatus, error margins etc. In QFT you can employ a probability current just a side note. Its simply our formulas employed with previous well tested studies of the particle properties, application of the appropriate formulas. Strictly determined via mathematics. Mathematics are not physical even though they may describe a physical state etc. Physical is what you have measured. You measure physical properties the mathematics only describe or predict what you will measure. That's a very important distinction

Well it may help to consider that its not necessarily the galaxy rotation curves themselves that provide the strongest support of DM being a particle. Consider the following if you take the FLRW metric and use the equations of state and apply the FLRW metric acceleration equations. Then remove the DM component and just apply baryonic matter of just 3% then there would never be enough matter in our universe for matter to become the dominant contributor to expansion. Instead of radiation era, matter era the Lambda era. You would only go from radiation directly to Lambda dominant. The Hubble constant would not have the value it does today. Matter radiation equality would never occur ( roughly when the universe is 7 Glyrs old.) Expansion rates themselves and it how it evolves over time would be completely different. Now as expansion occurs radiation diffuses more readily in an increased volume than matter so their densities evolve at different rates. Matter having an equation of state w=0 meaning it exerts no equivalent pressure term. This one can construe as being the primary evidence that influences the research more in favor of a particle constituent. Coupled with the detail that DM halos do cause gravitational lensing helps us confirm the density distributions. In point of detail Hubble telescope often makes use of these DM halos lenses to extend its range. Hope that helps if you like some of the related mathematics I can post them here. Lol wouldn't take any real effort as I have em handy in another thread. Edit correction on above the time frame was for matter lambda equality radiation/equality is sometime prior to Z=1150 depending on dataset used I would have to check later on. Zeq 3387 using Planck 2018+BAO dataset.

Bingo the one point all the crackpots miss lmao. Not stating anyone here is one lol.

Easy way is to consider a classic example if you determine some probability function for simplicity lets just use coin tosses but dropping a collection of coins in a given time frame. This forms a time or time independent wavefunction depending on drop rate. Once you make measurements ie number of coins with heads up as opposed to heads down. The original wavefunction isn't needed you have made determinations through observation and measurement you now have a determined wavefunction as opposed to a probability wave function. Some often refer to the latter as simply waveform to avoid confusion with the probability characteristic of a wavefunction.

Interesting article and proposal will be interesting what future findings on this will present itself for those wanting to look through the arxiv article itself https://arxiv.org/abs/2510.19087

This post had nothing to do with your article. I simply thought it was an idea you could make use of. Good luck. You want thinking outside the box its simple any mathematical methodology that can accurately describe a system or state has validity. You dont need tensors to do GR its simply another handy mathematical tool. You dont need to use 4 parameters to describe spacetime you can use parametric equations to reduce them. Thats my view point

Whatever as I mentioned before the rules state that there is no requirement to visit other sites or links and all pertinent information should be here. Im sorry you do not get that policy but its your full pdf on your opening page I have absolutely zero interest in opening up any other of your website links. So good luck with your work . Im done I have better things to do

Here's an interesting trick for you then take the equation of motion for a mechanical spring. If you compare the equation for the quantum harmonic oscillator you will find the precise same relations albeit a change in variables applied. The ratios of change are identical. If you study deep enough you will find a great deal of similarities between the seemingly complex equations have similar relations to many classical physics formulas commonly used in engineering. This is quite a bit more complex first and foremost the conservation rules require a closed system or a closed group. To go into greater detail would be more suitable to a seperate thread and such a discussion can get extensively lengthy. In some treatments involving spacetime one can define a conserved system usually ties into innvariance of a quantity. This is often done under local geometry in some mathematical space or manifolds. Anyways best left for a different thread