Mordred

Here is a factor to consider. Humans need to carry their young. With chimps and apes they can cling to their mother thus freeing up the mothers hands for travel. Human babies don't have that kind of strength. The ability to carry resources as well as infants is one advantage of bipeds.

Both Orion and myself do, we're breaking apart the relations that went into the OP Langrene. Right now I'm trying to determine if it's canonical or conformal by looking at the EW Langrene through symmetry break via the Higgs. Which will confirm the Higgs and Yukawa couplings underbrace sections. We're both learning from this gives us a refreshing challenge. The Yukawa section is rather challenging. I've already confirmed the Higgs and Dirac covariant derivative forms.

[latex]\begin{pmatrix}\acute{d}\\\acute{s}\\\acute{b}\end{pmatrix}\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{pmatrix}\begin{pmatrix}d\\s\\b\end{pmatrix}[/latex] Electroweak correlations [latex]\mathcal{L}=\mathcal{L}_{gauge}+\mathcal{L}_f+\mathcal{L}_\phi+\mathcal{L}_{yuk}[/latex] Gauge sector [latex]\mathcal{L}_{gauge}=\frac{1}{4}W^i_{\mu\nu}W^{\mu\nu I}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}[/latex] Where [latex]W_{\mu\nu}[/latex] and [latex]B_{\mu\nu}[/latex] are the SU(2)and U(1) field strength tensors. [latex]W^i_{\mu\nu}=\partial_\mu^1-\partial W^i_\nu-\partial_\nu W^i_\mu-g\epsilon_{ijk}W_{\mu}^jW^k_\nu W^k_\nu[/latex] [latex] B^i_{\mu\nu}=\partial_\mu B_\nu-\partial_\nu B_\mu[/latex] [latex]\epsilon_{ijk}[/latex] group structure constants of SU(2) B of U(1) abelion group has no self interaction (gauge boson) [latex] \mathcal{f}\subset\Sigma(\bar{q}+\bar{\ell}i\displaystyle{\not}D \ell)[/latex] q is quark [latex]\ell[/latex] is leptons it sums over generations. The quage covariant derivative is [latex]D_q=(\partial_\mu+\frac{ig}{2}\vec{\tau}\cdot\vec{W}_\mu+i\acute{g}Y\cdot B_\mu)q[/latex] [latex]\displaystyle{\not}D=\gamma D^\mu[/latex] g and [latex]\acute{g}[/latex] are the gauge coupling constants of [latex]SU(2)_w[/latex] and [latex]U(1)_y[/latex] [latex]\vec{\tau}[/latex] refers to Pauli matrices. Y is hypercharge of U(1) the electric charge Q is [latex]Q=I_3+\frac{1}{2}Y[/latex] langrangian for complex scalar fields. [latex]\mathcal{L}_\phi=(D^\mu)^\dagger D_\mu\phi-V(\phi)[/latex] [latex]D_\mu \phi=(\partial_\mu+\frac{ig}{2}\vec{\tau}\cdot\vec{W}_\mu+\frac{i\acute{g}}{2}B_\mu)\phi[/latex] [latex]V(\phi)=\mu^2\phi^\dagger\phi+\lambda(\phi^\dagger\phi)^2[/latex] Lambda is the self interaction term

Added warning tensors calculus has two meanings for the word covariant you have covariant vectors=covector and covariance which is a principle similar meaning to the laws of physics is the same all reference frames Google Lorentz covariance. Then you also have the covariant derivatives and contravariant derivatives. (Just to make things confusing lmao)

NotReference one is a proof of Lorentz invariance using the inner products of the four momentum which is the expression you have above however those are the two transformation matrices of each of the two four momentum components as seen from two reference frames. S and S prime Now inner products are symmetric a lot of literature express this symmetry by this expression [math]\mu\cdot\nu=\nu\cdot\mu=\eta[/math] the author expressed the two four momentum to show this invariance instead. He then showed that one reference frame has the same equation in the other reference frame equating the axiom the laws of physics is the same in all reference frames. Arriving at the equation with the Kronecker delta relations g_{\mu\nu}\lambda^\mu_\alpha\Lambda^\nu_\beta=g_{\alpha\beta} Where Lambda is the rotation matrix. Often called a transformation matrix but a boost is a type of rotation. More on that equation can be found here https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/1103.0156&ved=2ahUKEwiBnZq_wYHkAhVrllQKHXKeDyM4ChAWMAJ6BAgCEAE&usg=AOvVaw3RMIagygRKMga1sXHDiY3V In essence this is one proof of Lorentz invariance of the Lorentz group which is a subgroup of the Poincare group. Other papers can show the Lorentz invariance by other vector components even using different vectors with different vector components. Though for Kronecker delta the components are typically i,j,k. This is a common route via Calculus 1. These being unit vectors. They are unitary =1. They typically use these to further describe the covariant and contravariant relations. The covariant vectors are perpendicular perdendicular projections on M The contravariant vectors are tangent to the local axis. Or parallel projections on a manifold M In tensors they are reciprocal. To get a rank two tensor you need the dyad product of a vector. https://www.google.com/url?sa=t&source=web&rct=j&url=https://web.stanford.edu/class/me331b/documents/VectorBasisIndependent.pdf&ved=2ahUKEwiIz6bsooHkAhWTAHwKHTRGCUsQFjAAegQIBxAB&usg=AOvVaw3B_c2AOC-H2pmeCptog-NE A tensor of rank zero is a scalar A tensor of rank one is a vector A tensor of rank two is a dyad of a vector A tensor of rank three is a triad of a vector and so forth. For tensors rank (m,n) for GR study this guide https://www.google.com/url?sa=t&source=web&rct=j&url=http://web.mit.edu/edbert/GR/gr1.pdf&ved=2ahUKEwjU9YyVqYHkAhXS0J8KHehACPQQFjACegQIAhAB&usg=AOvVaw3GlcNdoopYnvY-SOkLmLU5 PS you will also want to be familiar with Levi Cevitta coefficient for GR.

Lol one of my three rules of life. Tomorrow I will be a better man than I am today (always strive for self improvement ) Treat others as I wish to to treated If I'm not having fun pretend. ( learn to enjoy every moment no matter how daunting)

Game of thrones currently binge watching. Never watched while on TV. Now have all the DVD except last season.

Purpose of life to live and experience life.

In this case start with a large body survey questions. Design the question are with neurological processes in mind. Ie tingling sensations is a sign of oxygen deprivation, look into other side effects that can be induced due to reduced oxygen levels to vital organs such as the brain. Figure out which feelings can be eliminated due to oxygen deprivation. How is that for a start? Instead of immediately jumping into supernatural phenomena look into side effects such as the one mentioned above. (For the record I have practiced meditation for several decades. I can induce a wide range of feelings throughout my body. At one time I could slow my heart beat to 10 per minute) Not once did I attribute any feelings to a supernatural reason. Meditation has stress management health benefits. However one should never confuse self induced feelings with supernatural causes. The mind is cabaple of many different interpretations of signal responses from neurons. Lol side note it drives my wife crazy that no matter how many mosquitos bite me I don't itch. (I taught myself to ignore that sensation)

They can more readily follow conjecture and verbal word play than mathematics or proper scientific analysis. Over 3 decades experience to see that habit lmao

A scientific fact without reliable scientific data good luck. Much of what is described depends on the individual feelings. The descriptive are primarily interpretations so too are the descriptive.

Don't see anything wrong with the above I will still be editing last post, I want to keep the equations from that reference I am referencing in the same post. Want to fill as many of the connections to the OP equation. (will help ensure the accuracy of our derivatives). On a personal note I would like to take our examinations of the various Langrangian's to the SO(10) regime under Pati-Salam in the particle data group restricted to the extended SM (MSM) models with minimal correlations to the MSSM supersymmetric groups with the above. I would like this thread to eventually include the SO(10) terms, with examinations of each group. (its a good study thread) . personal reminder note (Clifford algebra on double cover Poincare group with operator [latex]\mathbb{Z}^2)[/latex]

Ah found a beautiful treatment with the [latex] D_\mu[/latex] operators under the U(1), SU(3) and SU(3) gauge fields. The layout representations of the Langrangian under each group irrep is excellently done. Several of these equations I am going to latex out as I have time now and I know I will be referring to them to break down the QED,QCD, and Higgs portions of the OP equation. They will all be from the same Dissertation paper. https://www-d0.fnal.gov/results/publications_talks/thesis/nguyen/thesis.pdf U(1) Langrangian single component fields. [latex] \mathcal{L}=\overline{\Psi}(i\gamma^\mu \partial_\mu-m)\Psi[/latex] [latex]U(1)_{em}[/latex] fermion gauge field transforms [latex] \Psi\rightarrow\acute{\psi}=e{i\alpha(x)Q}\Psi[/latex] [latex]\partial_\mu \rightarrow D_\mu ieQ A_mu[/latex] [latex] A_\mu \rightarrow A_\mu +\frac{1}{e\partial_mu \alpha}[/latex] under covariant derivative terms ([latex]D_\mu[/latex] massless [latex]\mathcal{L}=\underbrace{\overline{\Psi}(i\gamma^\mu \partial_\mu-m)\psi}_{free fermion}-\underbrace{e\overline{\Psi}\gamma^\mu Q\Psi A_\mu}_{interaction}[/latex] Field strength tensor [latex] F_{\mu\nu}=\partial_\mu \alpha_\nu-\partial_\nu \alpha_\mu[/latex] [latex]\mathcal{L}=\underbrace{\overline{\Psi}(i\gamma^\mu \partial_\mu-m)\Psi}_{free fermion}-\underbrace{e\overline{\Psi}\gamma^\mu Q\Psi A_\mu}_{interaction}-\underbrace{\frac{1}{4}F_{\mu\nu}F^{\mu\nu}}_{k.e. of A_\mu}[/latex] Electroweak 4 component two component on Majarona spinor basis later post for neutrino chirality Right hand rule terms further reference to CKM under particle data group treatment. RHS reminder signature (+,-,-,-). see equations 2.9 to 2.22 to arrive at QCD Langrangian. (Glashow) quaternion/verbian\tetrad under Gell Mann matrix treatments). Pertains to the Weyl Dirac and Majarona spinors. (recall Majorona must be charge neutral under CPT.) will latex in equations 2.22 later on (lol want to maximize future cut and paste operations in this thread) need a break lmao

Adding some relevant relations pertaining to the derivatives above for potential references and time saving. In a sense reminders... https://en.wikipedia.org/wiki/Wigner_D-matrix https://en.wikipedia.org/wiki/Clebsch–Gordan_coefficients https://en.wikipedia.org/wiki/Table_of_Clebsch–Gordan_coefficients "A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch-Gordan coefficients " see arxiv next link under group irrep. https://arxiv.org/pdf/1009.0437.pdf https://en.wikipedia.org/wiki/Self-adjoint_operator Hermitean operators are self adjoint operators. https://en.wikipedia.org/wiki/Hermitian_matrix self adjoint matrixes are self adjoint. I need to make a correction on my Conformal statement in the above post, the equations to the right of the Poincare group are under canonical basis not conformal.

Woah wait a minute, you have the correct metric signature {+,-,-,-} however you have the Lorentz group tensor field irrep [latex]\Lambda_\alpha^\mu \Lambda_\beta^\nu T^{\alpha\beta} (x)[/latex] as General Relativity spacetime metric and Planck quantum gravity identity yet the proof included in the reference you provided for that term doesn't include the quantum regime in terms of the Planck HUP. That formula is the macro scale not at the quantum regime ? Also describe [latex] g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}[/latex] as the GR weak field limit it is a specific class of solution 1 of three main class of solutions under GR. You have the vacuum, the weak field and the Schwartzchild metric as the main categories. The vacuum is your scalar field solution, the weak field is the Newton approximation format. Applies well in particle physics as gravity is weak at that scale however under strong relativistic effects the Schwarzschild metric is the better approximation. Now all these are inclusive under the tensor field representation [latex]\Lambda_\alpha^\mu \Lambda_\beta^\nu T^{\alpha\beta} (x)[/latex] However you require additional terms not in this group to handle the quantum harmonic oscillator and HUP. The three solutions above will fall under [latex] g_{\mu\nu}=g_{\mu\nu}+h_{\mu\nu}[/latex] this is what you can place under the title General relativity spacetime metric as it includes the full range of solutions. Anyways its Valentines day so will look at this further tomorrow. edit: never mind the concern I had on [latex]\Lambda_\alpha^\mu \Lambda_\beta^\nu T^{\alpha\beta} (x)[/latex]. Found the terms [latex] \Lambda=exp(-iw_{\mu\nu}J^{\mu\nu}/2[/latex] dang arbitrary matrix [latex] \Lambda[/latex] Let me look at this further you have the right approach but I need to rehash the spin 2 connections, This will become critical to maintain the chirality basis with the remainder of the equation. The iw term corresponds to the QFT operators so I'm good on that lol dang [latex] \Lambda[/latex]. The detail that the remaining portion of the original equation is already Poincare/Lorentz invariant so all this is already accounted for in their derivatives. I can't shake the feeling that all we need to apply is the spin connection itself common format [latex] D_\mu V_\nu^\alpha[/latex] see here https://en.wikipedia.org/wiki/Spin_connection

excellent much better, the only thing I would now look into is the technicality of formalism matching and sign convention matching so that the equation is the same in both throughout. The equations to the RHS of the relativity underbrace is using conformal as opposed to a canonical treatment with metric signature (+,-,-,-). It is also in Spinor representation form. Other than that I can see no other main issues as a standalone what you have under the QG underbrace is certainly accurate and includes the spin 2 statistics. So the above is just making sure we match the methodology to the remainder of the equation. Hence why I chose to use the Differential operators from that link. You might however consider staying in the weak field limit for particle to particle interactions [latex] g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}[/latex] this is mainly due to gravity being such as weak force on the particle to particle regime. Also the bulk of QFT operates in the Newton limit regime.

Anyways that should provide a good direction of approach. I've watched your other thread though I have to catch up to where your at with it when I get time lol to know how serious you approach a topic and have a measure of your skills in that topic. Quite frankly its fun to study with all the relevancy to the topic. Hope this helps... note to above yes graviton as a gauge boson would likely be massless. One still need to correlate the possibility under group. One also sees the [latex]D_\mu[/latex] above this is the Differential matrix , and it correlates the difference between the covariant and contravariant terms. It will change in values as per the application applied in particular in different field treatments.

Yes the four momentum is instrinsic in GR however it requires the use of indices that follow the Einstein summation for covariant and contravariant terms. see here to see how this can apply in the above https://en.wikipedia.org/wiki/Four-momentum the remainder of the equation that you are trying to fill in the relativity section is in the 4 momentum, 4 force and 3 velocity format. This way you have the same coordinate relations throughout the equation. Though it is assumed direction of motion would be in the [latex] x^1 [/latex] direction this isn't always the case. Now I am going to ask a related question :How much affect does gravity play on the path integral relations that the first equation describes in its particle to particle scattering ? (side note the other portions are already Lorentz invariant). https://en.wikipedia.org/wiki/Lorentz_covariance https://gdenittis.files.wordpress.com/2016/04/ayudantiavi.pdf see here for further details on the Lorentz group and Lorentz invariance with ( you will see that the RHS of the [latex]\underbrace{\mathbb{R}}_{relativity}[/latex] is already Lorentz invariant in the terms. (side note QFT does this via the Klein Gordon equation as opposed to the Schrodinger ) Which brings to mind another question one can ask. Why was the [latex]\underbrace{\mathbb{R}}_{relativity}[/latex] left unfilled ? personally my feelings on this is that we haven't got a working quantum theory of gravity that doesn't suffer the renormalization problem sufficient to extrapolate the hypothetical graviton interactions (the graviton isn't yet part of the standard model of particles). Although if such a graviton could be possible with the spin 2 being the more likely (under research) spin 0 is also plausible. This in turn affects the degrees of freedom required to determine the gauge group representations. One must account for all effective degrees of freedom. see here for an example equation 36 for an example of the QCD Langrangian. https://cds.cern.ch/record/935622/files/p27.pdf This articles has pertinent details to understand a large portion of the equation in the OP. (the last article should provide a sufficient answer to the question of whether your Langrangian attempts suffice). Keep in mind the line from the introductory, the remainder of the article should hone in on how complex it really is.... In final note an effective Langrangian for the relativity portion should correlate to all the effective degrees of freedom that define the spin 2 statistics. A source of observational evidence is the graviton waves (quadrupolar ). So I would start here. [latex] g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}[/latex] if you research these groups you will find much of the work in terms of the Langrenians is already done. see this example on massive spin 2. gives other spin examples also https://arxiv.org/pdf/hep-th/0609170.pdf equation 21. Which reflects that the [latex]\underbrace{\mathbb{R}}_{relativity}[/latex] will correspond to something similar to this. [latex]\underbrace{|D_{\mu} D_{\nu}|\upsilon_{\alpha}}_{relativity}[/latex] ROUGH EXAMPLE ONLY>>>>one lack being massless particles...

going straight from Newton to a relativistic Langrangian by simply adding gamma isn't sufficient by itself. You must also preserve Lorentz invariance which will require the use of proper time given as [latex]\tau [/latex]. Recall different observers will measure the variant quantities differently this includes time. Even then its not complete, as you will need to include the 4 momentum and 4 velocity. See here, it mentions some of the issues I didn't https://en.wikipedia.org/wiki/Relativistic_Lagrangian_mechanics Another related reference here http://fma.if.usp.br/~amsilva/Livros/Zwiebach/chapter5.pdf A primary goal is to ascertain from the Euler-Langrangian the geodesic equation. http://people.uncw.edu/hermanr/GRcosmo/euler-equation-geodesics.pdf This wiki link has a good breakdown of how to employ the Euler Langrangian to derive the geodesic https://en.wikipedia.org/wiki/Geodesics_in_general_relativity

LOL I created a monster... Unfortunately I do not have the pdf I got that formula from, I had a copy of the original pdf on my phone which I had lost a while back but had the equation written down. Lately I decided to break it down to identify the various terms in the equation and see how encompassing it is. Here is what I have thus far. I will simply cut and paste what I have however keep in mind much of it will be in a note to myself format including related formulas. (in a real sense personal study notes) lol I screwed up on the quote section ah well the details are there. Anyways in a sense I have been trying to in essence reverse engineer the equation. Some of the links you have posted may come in handy in my endeavor I have to agree with this, in so far as to locating any proof for the equation itself.

Several reasons for that, many are due to most ppl want removable legs for moving and storage purposes. The second being many ppl look for the easier and quick put together methods even to the point of sacrificing solidity. Still if you can do that joint the dovetail shouldn't present much more challenge and dado joints are always easy.

Excellent work, very well crafted you don't often see those joints used today but they are solid joints that last years.

Section is too tight for anything but a come-along tied to a fence grrr

lol

I much prefer to recieve my wifes rewards lol. This is her gift to fulfill a promise to make her one. I have enough material to make another to see if I can sell one. However my first priority is making raised planters to fix the yard of the place I moved to 4 months ago. Still digging up bush roots along the side of the house. Yeesh 10 hours digging for three bushes so far....