Mordred

I would also say +1 to Dubbelesix who through diligence is arriving at a similar understanding. Glad to see how well your picking up on this as well Koti

I lost count of the number of errors just reading the English extract. Try learning physics. I bet you never included any of the relevant formulas in your book.

Indeed it is impossible to seperate the two to understand gravity as per curvature of spacetime.

In essence the above is accurate, gravity is best understood via spacetime curvature. No quantization is needed to accurately model gravity and indeed may or may not require a boson as a mediator. The more deep you get into the calculations of other fields involved in the same region. The more one starts to question whether or not gravity is a force as per a mediated field or if its simply the sum of all field interactions. Now if gravity does require a mediator, it makes sense why we cannot yet find the graviton, it would need to be a spin 2 boson and the heaviest boson at that. However one must be careful as to how understands what particles truly are... Field excitations is a fundamental key. So the best way I found to relate to this is to study wave formulas and constructive and destructive wave interference. This actually helps understand how virtual and real particle production occurs. Two overlapping waves that are in phase and of the same frequency will combine giving an increased ie increased amplitude. Which you already have the formula posted as to how this amplitude increase will affect particle number density. The same applies to destructive interference. The pointlike characteristics of a particle is in essence the Compton wavelength, this is sets the boundary for the confinement to the pointlike characteristic. Now with the above "Field excitations" it is easy to see that QFT promotes the field itself as the operators. This is where it changes from the QM treatment. (one of the articles I wrote on this site and linked to your thread previous mentions this) Look at the two slit experiment itself, and look at it specifically in terms of constructive and destructive wave interference. This will demonstrate the above, and indeed one can see that although wave particle duality still exists, " All particles exhibit pointlike and wavelike characteristics". The pointlike characteristic does not mean that particles are corpuscular. Indeed there is no corpuscular aspect to any particle.

no prob been extremely busy myself

Sorry been very busy with work, I might have time to study what you have added this weekend

not yet work has been keeping me hopping. Might have time this weekend

Been a bit busy lately, yes the stress tensor is far more complex than many realize. It isnt the mass term though, Think of it as your fluid relations. For example [latex]T^{00}[/latex] is a scalar field in a set region.( particles in a box). Your next diagonal is the pressure, (particle rate crossing an axis or hitting the sides of the box) QM wavefunctions applicable,,, Equations of state ie radiation, matter, Lambda. The other two diagonals involve flux and vorticity. Yes the stress tensor requires a metric (the box.) However it is not a metric. That is the metric tensor. In order for your fluidic formulas to work requires a coordinate basis. I recommend you examine your explanation, with regards to the differences between conformal vs canonical treatments. Though what you described isn't incorrect. It lacks the mathematical precision in the descriptive of the above two treatments. (Hamilton is canonical) I will be rather short on time this week, so won't be as thorough in my responses as I normally am. (That and very unreliable connections in the field lol) For the notation used in this thread vector calculus which obviously includes differential geometry, though largely were also using linearizations via linear algebra relations as per Clifford and Lie algebra. (linear is symmetric) so when non linear you linear approximate. (The two methods mentioned above) Both the QM and QFT above require the above two. edit more accurately the formulas of this thread involve Clifford and Lie as they establish the tensor relations of the Lie/Clifford group.( see previous summary on products)

I hadn't seen this paper in years... I read too much lol. Though the above paper came up on other forums I used to visit. I hadn't seen much continuation on the toroidal approach to the electron. Though never did particularly try to keep track of this aproach. It is a variation of the toriodal ring model which was later replaced with spin via Dirac and experimentation, looking for an internal structure to the electron. None so far has been detected, in terms of the toriodal ring. Wiki has a bit on it https://en.m.wikipedia.org/wiki/Toroidal_ring_model

Oh joy geometrodynamics reduced to a bowl of soup lol. Really hits the topic of G.U.T theoretically. Seriously though, matter, energy, mass, can all be described as properties of the same state. ie under action displacement, with time and time independent relations. This is fully applicable to any kinematics I mentioned above. Quite frankly the fastest way from my experience to understand all major topics of physics is to understand how to model geometric displacement via scalar and vectors. Then study how each of the more advanced models organize and reduce the complexity of multivector fields. Naturally engineering and physics both apply many of the same techniques and formulas. As I have my two grandaughters over this eve, I can't help but think of them as an example of condensed matter, but with seemingly unlimitted energy. Wonder what Grandma would think, if I tried mathematically modelling my granddaughters? Well as discussed it is a property of the state being described. Yes it does seem incomplete, as it is just one of many properties of the same state. In order to appreciate the definition, you need to study how that definition is applied with other related properties. The ability to perform work definition would be extremely difficult to beat considering the sheer and far ranging diversity of applications work is applicable to. I completely concur with the above

I was going to suggest the stress tensor approach as well. The time time component T^00 describes the massless stress components. ie pc^2. This is a scalar value as in the line element for seperation distance, a massless particle will have [math] ds^2=0.[/math]. If you like tonight I can post some of the essentials to the stress tensor for the other non reducible components. On this a also agree with Matti in applying the stress tensor. In the above via Kronecker delta connection, which you have already modelled, this is Euclidean and will follow the principle of equivalence. An easy way to understand Kronecker is parallel transport of two vectors. In Euclid geometry that parallel transport is preserved. Now under curvature of spacetime you will get a screw symmetry from the simultaneous affects upon the x and ct axis. (length contraction, time dilation.) However barring relativity simple curvature will cause a loss of parallel transport, the two vectors will either converge or diverge, Example Newtons laws with a central potential. So one can assign a vector mapping the seperation distance between those previous two parallel paths. This is the essence of the Principle of Covariance, Yes indeed curvature has definitive affects on temperature.

Lol, the book is one of my favourites, goes into a decent array of topics including the basics of refrigeration and equipment designs at that time.. The atom in this book simply has proton and electron. The Neutron being discovered later.

Indeed that very definition has withstood the test of time. If there is any definition that never changed since its application via Newtons laws its this one. More to the point fond of it because when teaching, one should stick with the proper definitions. (For one thing, you remind them of the kinematic lessons from earlier studies in school.) Here is some definitions from my 1919 physics book I happened upon. Energy " ability to perform work" (unchanged from today) Potential energy "Energy of position" kinetic energy "Energy of motion" I particularly liked how simply yet accurately expressed the latter two are. Yet apply accurately under any physics treatment. In all my studies of any topic those definitions apply. Particularly since every interaction can modelled via "action" and displacement. "Work" also works well when understanding the meaning of mass, as the above defintions arose from f=ma. In this book from 1919 "Work : the quantity obtained when we multiply the distance in the direction of the force through which it acts". Ie any vector quantity. Certainly shows the "Property" aspects mentioned above even then...Side note pricetag of textbook one dollar. Lol don't I wish

We really need to teach you how to latex lmao.

Thanks, granted any heuristic explanation is an oversimplification. Quite frankly the detail you placed in this thread has been an excellent in quality and paid close attention to all the essential aspects.

Np it was also for the benefict of other readers to help them follow the thread somewhat without derailing it. Lets summarize heuristically for other readers benefict. The OP is studying different techniques and specifics on how to mathematically define an object and system under Vector fields via symmetry relations that apply to Noethers theorem. First he went through the proofs defining a vector itself, ie the requirement of 2 units to describe. (magnitude and direction). Magnitude is a scalar quantity. Then he established that these two units are independant quantities. One can change in value without changing the other. Then he estsblished the boundary conditions of the magnitude as a normalized unit under a coordinate basis. ie the scale of the graph, to the ratio of change to the length of the vector 1 to 1 ratio to axis coordinates. Then he went further and applied Cauchy inequality, to a plane, which also shows the triangle inequality. This relates to i,j. Thus establishing an orthonormal and orthogonal basis. In order to define Hilbert space he had to apply all the above to apply the outer products of two vectors to close the Hilbert group. Now he is looking at how these two Hilbert spaces are applied under treatments. So here is a practicum question. Take right angle triangle ABC with identity connection i,j. Is C an independent unit ? Is it a Hilbert space?

Good on the above but lets be careful here. One can linearize nonlinear systems to good approximation which is essentially what you are doing above. In this there is huge variety of techniques. Two primary classes of field treatments apply to this. Conformal and canonical. Canonical it is the distances via end points that is the priori. On conformal it is the angles. So be careful to identify the treatment involved when jumping theories or mixing them... Also remember [latex]\mathcal{H}×\mathcal{H}=\mathbb{C}[/latex] "× " is The cross product. So these two Hilbert spaces. The two vector fields are perpindicular to one another in cross products. Angular momentum is a Cross product space https://en.m.wikipedia.org/wiki/Angular_momentum Side note your primarily using ODEs observable differential equations, the other being partial Pde,s

The River model at one time was one of the more popular models towards understanding GR. It was one of the earliest studies, that I personally took when I was first starting to learn GR. Nothing wrong with using this platform, to start wading into the depths of GR.

Here this will also help to know the commutations as applied to Hamiltons. https://www.google.ca/url?sa=t&source=web&rct=j&url=http://web.science.mq.edu.au/~chris/quad/CHAP04%20Quadratic%20Algebras.pdf&ved=0ahUKEwjX-ICd9fbWAhUDzGMKHV8nBfIQFgg3MAg&usg=AOvVaw0KDOtRQEktblBG0YfpB1Us Unfortunately the better explanations are typically in textbooks so I will keep digging for you. The first link is an example applied to a spacecraft. Second link has added details

mine too lol night mate. Here add this to your study list, you will find understanding the 4 quaturnion numbers incredibly useful to understand Hamilton and any QFT theory. https://www.google.ca/url?sa=t&source=web&rct=j&url=http://malcolmdshuster.com/Pubp_021_072x_J_JAS0000_quat_MDS.pdf&ved=0ahUKEwi5-J3R7PbWAhUH92MKHUO4DHsQFggwMAM&usg=AOvVaw2rwLoxBYHemplrpKOPlPpF Above once again can be applied to any number of dimensions including time. (basis behind the rotation matrixes) ie modelling a particle state under acceleration.

Mixed states will involve both linear and nonlinear treatments. Mixed states is interesting, I'm not steering you from mixed states but you will find understanding curvilinear a necessity to some of the mixed state examinations.

Glad to see the trace operator above. Been a long time since I last seen a discussion mention Bose-Hubbard. Some of the other models I don't recall. Looks like your understanding of QFT based papers is sufficient that you can easily process numerous different model dynamics and get a good strong understanding of those models. Glad to see the leap and bound change in your comprehension of these papers. Most of the work you have been doing are linear treatments. A good section to study would be curvilinear treatments, in particular Bevier curve approximations. As applied to a graph, this will relate to certain operators on curvilinear treatments. Once you have curvilinear ie [latex]SU^{n+1}[/latex] there won't be a field treatment that you wouldn't be able to understand a decent portion of. You already know the SU(N) aspects with the above largely. Once you have the first group you can understand any particle physics group of the SO(10). Also good catch noticing that one state does not need to have any interaction with another state in order to have a correlation function A does not need to have anything to do with B. That's one of the tricks of statistical math. If the two have similar trends then there is a correlation. (A couple of good statistical mechanics textbooks and also on Vector Calculus is your best aid to any QFT model)

Hrrm how to explain e-folding. It is in essence a doubling of volume with a time derivitave. The constant e is a derived constant. See here https://en.m.wikipedia.org/wiki/E_(mathematical_constant) in essence the universe increased in volume roughly [latex]10^{26}[/latex] to save you the calculations. Assuming 60 e-folds. More accurately distances increased by a factor 10^26.

By released, the density was such that prior to the "release". That the mean free path of photons was approximately [latex]10^{-32} [/latex] metres if I recall correctly. epoch commonly called dark ages under Cosmology. We cannot get messurements of nor past this epoch. Once atoms started forming the average density decreased enough that photons could travel far enough to reach us today. We can see the haze, via a combination of Hubble and strong gravitational lenses.

No its a bit more complex than that. After the first 10^-43 sec. Inflation occurs, this caused a rapid supercooling via the ideal gas laws. However inflation didn't instantly stop but slow rolls to a stop. This event caused a rapid reheating stage. Now the volume has increased exponentially so particles will become stable, ie drop out of thermal equilibrium. The temperature is low enough that atoms can now form. This removes a large percentage of free ptotons neutrons and electrons. Thus reducing the opacity of the CMB, this is the surface of last scattering. The uniformity we observe is due to the rapid expansion supercooling and slow roll reheating in such a short time frame, that temp/mass density anistropies didn't have time to redeveloped after being evenly redistributed.