Dubbelosix

But maybe the thread should be closed. Unless you prove me wrong, can you try and present your thoughts a bit more clearly? I mean, I am hardly Einstein myself, but I do have some knowledge in physics, and your claim to be a chemist, surprises me, since even chemists are usually trained in the scientific discipline. Trust me you can get there, but effort is needed. Yes quite right. Even for me, in which I tend to use math, would still not dare do this in the mainstream until almost certain about something.

I don't know, maybe it is just me, but I have no idea what you are talking about. I'll leave it for others to decide this, but will urge its not a reason for a banning. I am only saying this because people with (deficits) in the ability to talk the english language fluently may be interpreted wrongly.

Strange, it actually wasn't private. I will iterate, that I used a word against you in the open forum which was unacceptable. The only private thing that has went on, was the correction made to me, and rightly so, from the moderator. It made me think about the situation, which has led to this one. I am sorry you feel embarrassed. I don't know why. You didn't do anything wrong. I did.

So yes... put more time into your posts if: 1. Your theory doesn't match current physics, certainly check it over 2. Make sure your theory is consistent - in both the wording/terminology... but also if you ever come to math, make sure you understand its complexities, because I will not lie, mathematics is a very hard subject - take Einstein for an example, he actually failed his primary math tests. His relativity was so mathematically incomplete, he required his wife to help him finish it, who had no less, a PhD in mathematics. 3. Never lose hope even if you are proven wrong. This is the way of science. Also, make sure a theory can be falsified and fits observational or experimental data. These would be good principles to hold by when talking about the scientific methodology with physics. What kind of career exactly? Are you talking about dispensing drugs to patients over the counter?

you can make errors all you want. I'll never hold you responsible for them unless you intend them. Many years before I am today, I was ridiculed badly for early idea's I had no capability to uphold with math. It actually... damaged me a bit, emotionally. But I realised, that I needed to prove those idea's so that I could in some way prove to people, I was not insane and actually capable of learning the math that the great prof. 's of yesteryear preached. I went to college. I became a drop-out in my third year. I realised I was suffering from bad mental health that was impeding my ability to continue with my studies, so I became a hermit for four years and studying in silence. I come here, I meet physicists elsewhere who have appreciated, I am not on the same score, but certainly on the same page (some of the time :p) and the result is... I am much happier I put this personal time into it because I learned much more than any college class threw at me.

May I ask you a question without sounded too .... well you know. What made you interested in talking about a complex subject like quantum gravity, when your roots are in chemistry? There is quite a difference between the two, and if you don't mind me saying, if you are a chemist, set an example so that you express your physics clearly. I wrote some work not long ago about a pre-big bang phase to the universe and actually involved a long study into chemistry. That might be a surprise, but we have to do it sometimes. I took chemistry at college along with biology, but sometimes, we need to go back to basics. The mind can only remember so much after a certain time. Unless you are Sheldon

Well if you want to get really technical, mass exists at the expense of the electroweak symmetry breaking in early cosmology. The Higgs boson can be thought of as the reason why a massless particle can obtain mass. In this picture, the Higgs mechanism can be thought of a mexican hat potential in which a Goldstone Boson deviates from the ground state (implying that mass has been added to the system) in a non-trivial way. Since most scientists think of pure energy fields leading to symmetry breaking, then the definition of mass in context of energy becomes unclear, because in a way, you can argue energy is actually more fundamental than mass.

Whatever. Have you ever heard of the idiom word salad? I'd be careful using terminology that doesn't make sense when stringed together. You can take more time to contemplate the meaning of what you post. It isn't a timed chess game

I couldn't word it better, but, I do take full responsibility in this matter. As was made clear, if I want to be scientist, I must take all criticism. How can I argue with this? I must accept when I was wrong. I will admit myself... ashamed when I looked back and read it. I was angry, for all the wrong reasons.

Many of us know of John A. Wheeler --- what many do not know, is that he approached de Witt in the 60's ... possibly the year of 68' .... in which he asked for his direct help in the quantization of gravity. In their early approach, which used early methods, they arrived at the Wheeler de Witt equation, which has caused much controversy since. Like Wheeler, I want to reach out to people like Mordred who puts time into contemplating theories, they may not even agree in. A wise man once said, it takes an educated mind to appreciate a theory you may not even agree with. (Mordred) Once again, thank you for your in-put and I will contemplate it for a bit before responding. I am meticulous this way, you know me Matti will be pleased you agree with him (ps.) I'm trying to write a thermodynamic theory now based on the above. We'll see how it goes.

Well to be honest strange, I think the real praise should go to you, when I was so ignorant in my use of words, you did not react. I took a real look at what was being said to me and so made this post

I'll let you know I understand energy. If I take relativity seriously, then they are both the same thing, but in what way do we mean this, really? What does [math]E = Mc^2[/math] actually mean, in this context? Or as you ask, what exactly is energy: 1) energy is nothing but a diffused form of matter 2) matter is the physics of condensed energy From this, many have concluded, matter is just a trapped form of energy, quite possibly light since most particles can and will reduce back to photon energy in matter-antimatter collisions. This may be a non-trivial relationship of light to matter. A group of Glaswegian scientists showed that matter could actually be trapped forms of light in tightly curved geodesics (knot theory). I actually found it, quite convincing. same thing* edited

After (what in hindsight) appears to be some trivial misunderstandings, it seems that ultmately my friend Matti P. (PhD) has finally accepted this space theory is acceptable (in context of the last posts concerning the action) . I asked his permission to relay a certain quote of his that I found, interesting and insightful: ''OK. We come back where we started. This is non-relativistic case and one considers kinetic energy. One can introduce in condensed matter physics mass tensor when the effective kinetic energy is not anymore mv^2/2 but more complex function m_ijv^i v^j/2. This is just effective description and non-relativistic. I would not introduce it in GRT since the mass is by definition Lorentz scalar in this context. I would speak about energy momentum tensor and introduce to it this kind of terms.'' I consider Matti, not just a friend, but also a mentor. He has been influential. I have made it clear to him, if he was within distance, I'd properly pay him for his assistence to take me on as a pupil. Matti is very intelligent, and while I do not agree with every aspect of his topological geometrodynamics theory, I think its one of the most promising theories on the market that really values an investigation. Ah, I just looked at the curve equation, it also predicts that temperature arises from the curvature of space. I think I just found my new favourite investigation. That really puzzled me, but can you guess which equation can be understood this way? Why should the temperature arise from the curvature of space? Well I will tell you why, because with curvature comes acceleration and with acceleration as a parameter to a condensed collection of systems results in increased thermodynamic properties. This feels promising. Thermodynamics is about the movements of the constituents it is made of. It seems curvature at the quantum scale could provide a thermodynamic translation into gravity (or geometry). This is totally unexpected, but I admit, actually makes sense for once.

There are only so many fundamental constants, if any of them change, either through the evolution of a universe and thus slight adjustments may appear depending on the theory which deals with those parameters. There may be evidence a fundamental constant has changed somewhere... go dig the internet if you can. Secondly, Nob. Prize win. Abdus Salam has noted that if gravity has a cosmological and quantum scaling, then gravity could explain the strong force as understood in terms of the quarks. Interestingly, [math]P^{-2}[/math] propagators have been shown to be totally describable by the spacetime curvature they inhabit. If in extended theories where the speed of light is connected to the speed of gravitons fundamentally, then you could in theory change the permeability and permittivity then you could alter the speed of light itself. Barrow and Tippler did hypothesise that the universe could have undergone early changes that satisfies the alteration of the universal speed limit over time. Phi is right, you must address people. Go check my own posts, you don't need to read my work, but will you have look at how I interact with people - I do myself no harm though, because I do make attempts to make sure such questions will not throw me off by excellent questions that can otherwise, come from the blue. May I suggest, until you are happy with a theory, do not post it - this is my very first key principle - it has to agree with a fuller understanding before I post anything, otherwise, if it is a subject I know nothing about, I tend to only say a few things then tend to be quiet. But you really should talk with others, you might learn a lot, like I did many years ago. and still today ( no less), I am still learning. Will always learn from others. Even my work, I dare not post in physics section so let that be a lesson for those who do not wish to be meticulous with their physics - I've been told I am when I write my stuff, but I still have to keep myself firmly on terra firma and know the place of where quantum gravity and speculations about it, sits. Until evidence points in the way of the theory, then it is falsifiable properly. Before any evidence, something remains hypothesis. Popper falsifiability is very important and not even taken that seriously by many scientists who like to write about subjects outside the experimental means.

After some talks with two moderators at the site, I quickly came to realise my terrninology was actually offensive under the surface, which was much deeper than I intended at the time. I won't defend myself again, this is a public post to say, I recognise that behaviour should not be tolerated and my words to you about not wanting to talk or commune over the subjects here, can come to an end. I am sorry and it won't happen again. regards. G. Lee

( really theoretical post) Let's go back for a moment because I want to apply some ''older knowledge'' I have on a possible continuation for the following (what I call) Hilbert-Wigner Distribution Inequality: [math]\sqrt{<\dot{\psi}|\dot{\psi}>} = |W(q,p)| \sqrt{<\psi|R_{ij}^2|\psi>} \geq \frac{1}{\hbar}\sqrt{<\psi|H^2|\psi>}[/math] Under the Jacobi formulation of the Maupertuis principle involving an action of the form [math]\int\ p \cdot dq[/math], the kinetic energy is related to the generalized velocities, [math]T = \frac{1}{2} \frac{dq}{dt} \cdot \mathbf{M} \cdot \frac{dq}{dt}[/math] where the not-so-frequently encountered mass tensor [math]\mathbf{M}[/math] features here. The key point is that it relates the temperature with the action in the following way [math]2T = p \cdot \dot{q}[/math] Alternative to the first equation, we can also find arguements for the metric of the form [math]ds^2 = dq \cdot \mathbf{M} \cdot dq[/math] and even maybe an argument for the kinetic energy written with a mass term (where now I fix the dimensions) [math]k_B T = \frac{1}{2}m(\frac{ds}{dt})^2[/math] Actually according to the Virial theorem or even from equipartition, you could even argue the quantity is more true for the kinetic motion of systems: [math]k_B T = \frac{3}{2}m(\frac{ds}{dt})^2[/math] Notice we have a curve argument in the term [math](\frac{ds}{dt})^2[/math]? From here we will use this physics to implement into our model, for a purely theoretical approach. Remember, the Wigner function plays a possible fundamental role in the notion of the curve in the Hilbert space, at least in theory, since that curve could be adjusted to suit a definition of the form [math]|W(q,p)| \sqrt{<\psi|R_{ij}|\psi>}[/math] - this curve was also set equal to the primary definition: [math]\sqrt{<\dot{\psi}|\dot{\psi}>}[/math]. It is also possible to retrive a half factor on the terms for the first equation, I just haven't derived it exlicitely but will do if asked. I just haven't to save time. So for equation 1, we have a number of simple procedures to get to where we want to be. Square the equation [math]<\dot{\psi}|\dot{\psi}> = |W(q,p)^2| <\psi|R_{ij}^2|\psi>[/math] Let the mass tensor now play a role and apply one half to the right hand side [math]<\dot{\psi}|\mathbf{M}|\dot{\psi}> = \frac{1}{2}|W(q,p)^2| <\psi|R_{ij}^{2'}|\psi>\ \geq \frac{1}{2(L \cdot S)}<\psi|H^{2'}\psi>[/math] Where we have used for the square of the action, the spin-orbit coupling factor. Where I have used the notation [math]R_{ij}^{2'}[/math] to say that the mass tensor is acting on this curvature tensor. I think of this latter relationship as (maybe being tied) to some way to describe the correlations between matter and geometry, a subject that has been taken of interest in emergent gravity theories (see Fotini Markoupolou) - in her theory it is an important feature that matter can entangle with geometry. Nothing in the laws of physics forbids it - in fact, matter can entangle with matter. Space can even entangle with space! And so, the laws of physics definitely shows it is possible to link the two through a dynamic entanglement of matter to geometry. I won't lie, there are some interesting but very important questions from this last equation, like, is the spin-orbit actually quantum in nature [math](L \cdot S)[/math] as it give rise to the inequality like we have done? Well I have known for a while that the mathematics of classical spin are actually identical the quantum spin dynamics, there is no difference between the two except notation, so yes, this is a quantum property if we wish to assume it is, like we do. From this I deduct we now have a reason why the curvature of the Hilbert space must be implied: Through the spin-orbit coupling, specifically which obviously must follow a curvilinear trajectory. This means, the curvature in the Hilbert space is not an observable of time per se, but can be though of that, but in this one, it seems the curve is brought on by the spin coupling of the system with the orbit. Sorry typos fixed. Also you can argue the mass tensor transforms the curvature tensor is such a way you end up with the modified Hamiltonian [math]H^{2'}[/math]. The spin-orbit coupling will also allow for energy shifts. I didn't realise you said this earlier, I think you are far too modest.You are in my opinion (one of the least likely) posters here who could accidentally add knowledge in any hope of derailing the thread. I am also going to apply my knowledge of electroweak symmetry breaking to the context above and see if there are any obvious reasons why mass should appear in the theory - for instance, can we work in a massless understanding of the theory as well? This will take longer than my usual posts and may be some time before I come to a conclusion.

Thank you Mordred. I have tried. It's a combination of idea's heavily laden with knowledge from papers and additional help from posters like you. Just a small note, not really for your benefit as I am sure you know how I have gone around most of these things, but for other posters, a quick explanation of the last inequalities.We used the form: [math]\frac{ds}{dt} = |W(q,p)| \sqrt{<\psi|R_{ij}^2|\psi>}\ \geq \frac{1}{\hbar} \sqrt{<\psi|H^2|\psi>}[/math] But when expressed fully in Dirac notation, I prefer it in the form: [math]\sqrt{<\dot{\psi}|\dot{\psi}>} = |W(q,p)| \sqrt{<\psi|R_{ij}^2|\psi>}\ \geq \frac{1}{\hbar} \sqrt{<\psi|H^2|\psi>}[/math] You can argue it satisfies the Schrodinger equation: [math]H |\psi> = i \hbar |\dot{\psi}>[/math] I'll be clear how the inequality arises also from the dimensions, the equation given in the reference below is [math]\frac{ds}{dt} = \frac{1}{\hbar} \sqrt{<\psi|H^2|\psi>}[/math] We noted in my work that [math]\frac{c^4}{8 \pi G}<\psi|R_{ij}|\psi> = <\psi|E|\psi>[/math] square the operators and take the square root of the components like : [math]\sqrt{\frac{c^8}{16 \pi^2 G^2}<\psi|R_{ij}^2|\psi>} = \sqrt{<\psi|E^2|\psi>}[/math] and so. [math]\frac{ds}{dt} = \frac{1}{ \pi \hbar} \sqrt{\frac{c^8}{16 G^2}<\psi|R_{ij}^2|\psi>} = \frac{1}{\hbar}\sqrt{<\psi|H^2|\psi>}[/math] Pull the pi out of the denominator and into the definition of the first inverse hbar and use natural units after identifying [math]\frac{1}{\pi \hbar}[/math] as the mean Wigner function |W(q,p)| [math]\frac{ds}{dt} = |W(q,p)| \sqrt{<\psi|R_{ij}^2|\psi>}[/math] and the inequality holds because of the quantization of hbar after identifying [math]\frac{ds}{dt} = \sqrt{<\dot{\psi}|\dot{\psi}>}[/math] we simply get, [math]\sqrt{<\dot{\psi}|\dot{\psi}>} = |W(q,p)| \sqrt{<\psi|R_{ij}^2|\psi>}\ \geq \frac{1}{\hbar} \sqrt{<\psi|H^2|\psi>}[/math] http://www.physik.uni-leipzig.de/~uhlmann/PDF/UC07

Good summary and worded in a fresh way.

''Side note your primarily using ODEs observable differential equations, the other being partial Pde,s'' Well spotted, I will need to tackle this. Thanks for the rest.

Assuming I have done all this right so far, I did find an interesting continuation which uses the Wigner function, which is why I explained them in more depth not long ago. The curve of the length can be related to an inequality by pulling out the terms to write a Wigner function. [math]\frac{ds}{dt} \equiv |W(q,p)| \sqrt{<\psi|R_{ij}^2\psi>}\ \geq \frac{1}{\hbar} \sqrt{<\psi|H^2|\psi>}[/math] due to the inequality relationship: [math]|W(q,p)| \geq \frac{1}{\pi \hbar}[/math] There are other ways to argue this inequality. For instance, some standard formula [math]\int <q|\rho|q>\ dq = \int <p|\rho|p>\ dp = Tr(\rho) = \mathbf{1}[/math] and [math]\int <q|\rho|q> <p|\rho|p>\ dqdp = Tr(\rho)^2 = \mathbf{1}[/math] Because our model deals with the phase space, you can argue there exists a Cauchy Schwarz inequality [math]2 \pi \hbar W(q,p)^2\ \leq\ <q|\rho|q><p|\rho|p>[/math] or [math]W(q,p)^2\ \geq\ \frac{1}{2 \hbar} <q|\rho|q><p|\rho|p>[/math] Again, to satisfy our Cauchy Schwarz space. The inequality satisfies the norm through [math]\int W(q,p)^2\ dqdp \geq \int \int <q|\rho|q><p|\rho|p>\ dqdp = Tr(\rho)^2 = \mathbf{1}[/math] There is an error in the last two equations of not the last post (which will merge with this) but the one before it, the Hamiltonian should be squared, which when [math] c=8 \pi G = 1[/math] [math]\frac{ds}{dt} = \frac{1}{\hbar} \sqrt{<\psi|R_{ij}^2|\psi>} = \frac{1}{\hbar} \sqrt{<\psi|H^2|\psi>}[/math] and so the length is [math]ds = \frac{1}{\hbar}(t_1 - t_2) \sqrt{<\psi|R_{ij}^2|\psi>} = \frac{1}{\hbar} (t_1-t_2)\sqrt{<\psi|H^2|\psi>}[/math] And again, all updates are wrote down here as well for a more compact collection of the work. http://www.physicsgre.com/viewtopic.php?f=10&t=127412&p=198855#p198855

Yes the sites latex is sensitive to quotes it seems. No I don't give up. I don't think I'll ever become bored of physics to be honest. Right now, I am writing up on why curvature in the Hilbert space is not very well defined - though I will offer suggestions. It became clear to me, it wasn't a simple topic in just adding it curvature to Hilbert space, but I did come across a great paper that will be linked after that helped this post. Does a Hilbert space have a curvature? No it doesn't - not intrinsically, but it does actually have a geometric structure, simply because it does possess a scalar product. I've shown the Hilbert space in the context of a geometric uncertainty relationship for space, known as the Cauchy Schwarz space. What is important, though not intrinsically possessing curvature, it may indeed have one and I see no reason why it can't. The linear Hilbert space will only satisfy the natural flat geometry. There is in fact, no easy way to give ''curvature'' to a Hilbert space, There is a way to describe that through the flat metric [math]d(\psi, \phi) = |\psi - \phi|[/math] There is a projective Hilbert space [math]\mathbf{C}P^n[/math] then we can consider what is known as the distance function [math]d_{proj}(\psi, \phi) = \min_{\alpha}|\psi - e^{i\alpha}\phi| = \sqrt{2 - 2|<\psi,\phi>|}[/math] This is often used to find the ground state of some Hamiltonian. Interestingly (something I learned) the distance function contains a singularity at a quantum phase transition. The distance function can be thought of as a metric - this may be a better alternative to the Bure's metric (which I have read some authors) claim that it is not actually a true metric: Neverthless, we have looked at a statistical theory of gravity and the Bure's metric has implication for quantum geometry information theory. So there are links here that should not really be ignored without investigation. How true the statement that the Bure metric is not a true metric is, I am unsure. What I can say is that the Bure metric is a ''metric'' which measures the infinitesimal distance between density operators which define the quantum states. The Bure's metric, can also be thought of as a ''statistical distance.'' This has formal similarity to a distance function of the form [math]d(\psi,\phi) = \arccos(|<\psi|\phi>|)[/math] This measures also, the shortest geodesic between any two states! It is an angle formula, we may consider it also in the form [math]d(\rho_1,\rho_2) = \arccos \sqrt{F(\rho_1,\rho_2)}[/math] Of course, (iff) the Bure's metric is not a true metric, then neither is this ''shortest geodesic distance'' equation. It will be interesting to read vigorous arguments for and against the Bure's metric as being a real metric or not. Now... we must go back quickly, and go back to the distance function. Using just single notation now, we have [math]|\psi - \psi'|[/math] This still doesn't yet describe curvilinear space - this is actually a Euclidean measure of the distance between two states. The length of a curve however can be given as: [math]\frac{ds}{dt} = \sqrt{<\dot{\psi}|\dot{\psi}>}[/math] This involves the tangent vector [math]\dot{\psi}[/math] has lengths that is the belocity which travels in the Hilbert space. I can construct a Schrodinger (like) equation from this. [math]\frac{ds}{dt} = \frac{1}{\hbar} \frac{c^4}{8 \pi G}\sqrt{<\psi|R_{ij}|\psi>} = \frac{1}{\hbar} \sqrt{<\psi|H|\psi>}[/math] and the metric of the solution is [math]ds = \frac{1}{\hbar}(t_1 - t_2)\sqrt{<\psi|H|\psi>}[/math] Anandan, a physicist whome I used an equation that described the difference of geometries (but made it within the context of the curvature tensor) has proposed that the Euclidean length is an intrinsic parameter of the Hilbert space. references http://www.physik.uni-leipzig.de/~uhlmann/PDF/UC07.pdf CURVATURE OF HILBERT SPACE AND q-DEFORMED 'QUANTUM MECHANICS' BINAYAK DUTTA-ROY Saha Institute of Nuclear Physics, Calcutta 700 064, India

I didn't note this before, but these are Wigner functions. It has relationships with the density operator [math]\int W(q,p)\ dp = <q|\rho|q>[/math] [math]\int W(q,p)\ dq = <p|\rho|p>[/math] [math]\int W(q,p)\ dq\ dp = Tr(\rho) = \mathbf{1}[/math] There is also a property which satisfies [math]|W(q,p)| \geq \frac{1}{\pi \hbar}[/math]

[math]\Delta <\mathbf{H}> = \sqrt{|H|^2 - <\psi|H|\psi>^2}[/math] which measures the uncertainty in the Hamiltonian density in the state [math]\psi[/math] for the system in the phase space. Then we can implement the survival probability in the context of the binding energy equation. With [math]c = 8 \pi G = 1[/math] we have [math]\Delta \mathbf{H} = \sqrt{|R_{ij}|^2 - <\psi|R_{ij}|\psi>^2} - \sqrt{|R_{ij}|^2 - <\psi|R_{ij}|\psi>^2}[/math] So the geometries themselves depend on the survival probabilities of each system. In this case, we cannot think of the curvature tensor [math]R_{ij}[/math] as the anti-symmetric relationship - this will be studied further because we have the ground work. The uncertainties in this last equation can be though of as consisting of two terms: [math]\sqrt{|\psi|^2 \ln[|\psi(p)|^2] - |\hat{\psi}|^2 \ln[|\hat{\psi}(p)|^2]}[/math] [math]\sqrt{|\psi|^2 \ln[|\psi(q)|^2] - |\hat{\psi}|^2 \ln[|\hat{\psi}(q)|^2]}[/math] So in terms of pure statistics, the Hamiltonian density can be expressible as [math]\sqrt{|\psi|^2 \ln[|\psi|^2] - |\hat{\psi}(p)|^2 \ln[|\hat{\psi}(p)|^2]} - \sqrt{|\psi|^2 \ln[|\psi|^2] - |\hat{\psi}(q)|^2 \ln[|\hat{\psi}(q)|^2]}[/math] This application can be found from Uncertainty relations for information entropy in wave mechanics'' (linked below). Don't be put off with a wave function existing for the first terms, when there are non featured in [math]|H^2|[/math] in this part. This is because it can actually be written as [math]\Delta <\mathbf{H}> = \sqrt{<\psi|H^2|\psi> - <\psi|H|\psi>^2}[/math] This just appears to be a matter of notation. https://link.springer.com/article/10.1007/BF01608825?no-access=true Of course, it should be noted just for clarity, this does not apply to an interpretation of an antisymmetric operator, but one rather that would have to apply to linear geometric operators - we will still look into the curvilinear models like Mordred has suggested and I was already on the fence about. It's just about finding the right model (that I am happy with). Fixed the dependencies on the statistics, think I had wrong, think I have it right now.

Thanks! This looks valuable as reading material as well http://www.ams.org/journals/tran/1960-094-03/S0002-9947-1960-0112025-3/S0002-9947-1960-0112025-3.pdf

I realised while attempting to sleep there is an important interpretation in the work and once again, its because you mentioned something relevant. I remembered that the property [math]<\psi|U A U|\psi> = <\psi'|A|\psi> = <\psi'|\psi'> = 1[/math] Holds if [math]A[/math] is a linear operator. So, while in my past work I considered the geometry as the observable, it would mean we'd have to consider a linear geometry in this case. Which related back to your mentioning, it may be best to look at the non-linear cases now for the more juicy structures of the theory. I had a look at some non-linear wave equations before already... but will take a much greater look into it and see if I can see anything.