Mordred

you won't find that equation in a textbook, textbooks only show the basic equations in math speak in this case you would usually see the first order equation this delves into the second order. just as most textbooks won't show the equation \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] this shows the expansion rate H varies over time (it will also help to better understand the first equation as well as the Hogg paper I posted. now as you mentioned DM and DE one line of research is Higgs being responsible. Sterile neutrinos (right hand are heavier than left hand neutrinos ) antimatter and matter neutrinos. so the calculated abundance could fall into range \[\Omega_pdmh^2=\frac{G^{3/2}T_0^3h^2}{H_0\sigma v}=\frac{3*10-{27} cm^3s^{-1}}{\sigma v}\] research is still on going. Just as the equation of state for the Higgs field may explain inflation as well as the cosmological constant. That should sufficiently show that what really goes on in the professional circles isn't something one can simply google at best that just gives hints

no coordinate choice affects the mass distribution. I could describe the universe in numerous different coordinate choices example Euclidean, spherical cylindrical etc without causing any difference. It is precisely why we use invariance. The mathematics is set up that way so that we do not have any coordinate choice dependency. you know full well GR fully describes time dilation the FLRW metric is a GR solution. We don't arbitrarily choose DM and DE as the full explanation those two terms are simply placeholders until we can determine the cause of each. We still can measure their effects through indirect evidence. I rarely give downvotes so its someone else. As far as sampling range is concerned, redshift is only one of many pieces of evidence of an expanding universe. In point of detail its not even close to the strongest evidence. Its the one most ppl are familiar with but the real evidence comes from our thermodynamic laws in regards to temperature and how it influences the SM model of particles via processes such as BB nucleosynthesis in regards to the CMB. One danger of trying to understand cosmology by rote instead of learning the math is that too often you get incorrect information. I will give an example if I looked up hydrogen and its temperature it could form with stability a google search will state 3000 kelvin. However if one knows how to use the Saha equations that would reveal that value equates to 75 % of the potential hydrogen. Hydrogen can start to form as low as 6000 kelvin=25% 4000 kelvin for 50 %. That is just one example. however knowing this one can study the metallicity of our universe evolution via hydrogen, lithium, deuterium etc. So I just described another piece of evidence for expansion. In other words were not restricted to redshift to determine if our universe is expanding . In point of detail we do not rely on redshift in cosmology it is too full of other influences such as gravitational redshift, transverse redshift, Integrated Sache-Wolfe effect, Doppler redshift. etc etc. We examine all pieces of possible evidence to confirm the accuracy of cosmological redshift. Nor do we use the generic formula everyone sees on google. https://en.wikipedia.org/wiki/Redshift this formula only works for nearby objects it loses accuracy as near as one MPC. The full formula includes the influence of the evolution history of matter, radiation and Lambda. details can be found here "Distance measures in cosmology" David W. Hogg https://arxiv.org/abs/astro-ph/9905116 side note the paper also applies to luminosity distance we also have a different formula for Luminosity distance than what one would google. \[H_O dl=(1+z)|\Omega_k|^{-1/2}sinn[\Omega_k^{1/2} \int^z_o\frac{d\acute{z}}{\sqrt{(1+\acute{z})^2\Omega_R+(1+\acute{z}\Omega_m-\acute{z})(2+\acute{z})\Omega_\Lambda}}]\] What this equation shows is that matter, radiation and Lambda density not only influences expansion rates it also influences redshift and luminosity as well as any curvature term k

No there is no assumptions due to coordinate choice. You already know time dilation is a consequence of spacetime curvature or Relativistic inertia. The math and observational evidence shows us that there is no curvature term k=0. So where would you get time dilation ? This has already previously been mentioned. As massless particles travel at c we can ignore the inertial gamma factor. A higher density past the answer either. To go into greater detail if you take 3 time slices say time now, time at the CMB say z=1100. And a slice at say universe age 7 billion years old. If you describe the geometry of each slice. Each slice has a uniform mass distribution so no slice has a non uniform mass distribution to have a curvature term. Hint this is the real advantage of the scale factor a. No time slice has any change in geometry or curvature it's simply volume change between slices and density changes as a result of the ideal gas laws

Without looking at that link as the material needs to be posted here. The math done in that paper was done by your colleague correct ? By your statement above he refused to describe the mathematics in regards to quantum Strangeness so that paper wouldn't contain that detail with the needed math. Using toroids is nothing new in physics a cyclotron can be described using a toriod geometry. Yes you can mathematically describe any geometry in regards to an earlier comment of yours. Regardless if the person who did the math refused your conjecture then that wouldn't have the math beyond what the two of you were working on.

Add kinematics as it's used in every physics theory including the entirety of the SM model f=ma always applies for example.

Ok let's make it easier for when your looking at Higgs related papers E is energy , \(\rho\) is energy density, v is used for VeV. Think of VeV as a coupling constant for Higgs interactions a lot of the equations apply it in that manner. Energy is the ability to perform work. Energy density is the mean average over some field volume. Three distinct properties with distinctions in the mathematics Hope that helps

Correct the only direct confirmation is via LHC and Atlas etc. Studies are continually improving at each CERN LHC etc upgrade. We barely hit the required energy levels in 2012 so yes research is continually improving.

Usually first and second year QM. However I should mention those boring lessons your getting now will apply at every level of physics. In particular any physics involving kinematics. However this prevent you from learning QM early on.

inflationary gravity waves Weak field limit transverse , traceless components with \(R_{\mu\nu}=0\) \[h^\mu_\mu=0\] \[\partial_\mu h^{\mu\nu}=\partial_\mu h^{\nu\mu}=0\] \[R_{\mu\nu}=8\pi G_N(T_{\mu\nu}-\frac{1}{2}T^\rho_\rho g_{\mu\nu})\] vacuum T=0 so \(\square h_{\mu\nu}=0\) transverse traceless wave equation \[\nabla^2h-\frac{\partial^2h}{c^2\partial t^2}=\frac{16\pi G_N}{c^4}T\] inhomogeneous perturbations of the RW metric \[ds^2=(1+2A)dt^2-2RB_idtdx^i-R^2[(1+2C)\delta_{ij}+\partial_i\partial_j E+h_{ij}]dx^idx^j\] where A,B,E and C are scalar perturbations while \(h_{ij}\) are the transverse traceless tensor metric perturbations each tensor mode with wave vector k has two transverse traceless polarizations. \[h_{ij}(\vec{k})=h_\vec{k} \bar{q}_{ij}+h_\vec{k} \bar{q}_{ij}\] *+x* polarizations The linearized Einstein equations then yield the same evolution equation for the amplitude as that for a massless field in RW spacetime. \[\ddot{h}_\vec{k}+3H\dot{h}_\vec{k}+\frac{k^2}{R^2}h_\vec{k}=0\] https://pdg.lbl.gov/2018/reviews/rpp2018-rev-inflation.pdf

Just to add for acceleration involving change in direction will involve transverse redshift. Just to add some useful relations more for the benefit of any readers not familiar with the types of redshift. \[\frac{\Delta_f}{f} = \frac{\lambda}{\lambda_o} = \frac{v}{c}=\frac{E_o}{E}=1+\frac{hc}{\lambda_o} \frac{\lambda}{hc}\] Doppler shift \[z=\frac{v}{c}\] Relativistic Doppler redshift \[1+z=(1+\frac{v}{c})\gamma\] Transverse redshift \[1+z=\frac{1+v Cos\theta/c}{\sqrt{1-v^2/c^2}}\] If \(\theta=0 \) degrees this reduces to \[1+z=\sqrt\frac{1+v/c}{1-v/c}\] At right angles this gives a redshift even though the emitter is not moving away from the observer \[1+z=\frac{1}{\sqrt{1-v^2/c^2}}\] From this we can see the constant velocity twin will have a transverse Doppler even though the velocity is constant. The acceleration as per change in velocity is straight forward with the above equations as the redshift/blueshift will continously change with the change in velocity term. The equations in this link will help better understand the equivalence principle in regards to gravity wells such as a planet https://en.m.wikipedia.org/wiki/Pound–Rebka_experiment The non relativistic form being \[\acute{f}=f(1+\frac{gh}{c^2})\]

In essence that's correct without going too indepth on the differences between operators and propogators of QFT. You can accurately treat it as a fundamental constant of the Higgs field with regards to how the field couples to other particles for the mass term I really wouldn't trust Chatgp your far better off in this regard studying the standard model via the Lanqrangian equations. For the W boson it's the SU(2) group and U(1) groups for the relevant details with Higgs. It's also why I recommended starting with Quantum field theory Demystified as it's reasonably well explained for the laymen to grasp.

The VeV isn't an issue it's something you observe only during scatterrings via say a particle accelerator it's a local property at each particle such as the W boson simply put a coupling term. The probabilities are much the same as the probabilities associated with Feymann path integrals. It isn't the vacuum energy density itself so it's not anywhere near Like the vacuum catastrophe from QM.

We cross posted Migl but I included a primary missing detail in terms of VEV being a probability value much like a weighted sum in statistics.

Might be easier to understand that in statistical mechanics, QM and QFT the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement. What that statement tells us is that it includes all possible outcomes. It is a probability function. Expectational values is used regularly in statistical mechanics, QM and QFT. Path integrals also have probability weighted sums

If your certain of your equations and it's validity I'm sure your going to want to test them. If you think about it I provided the essential equations to do just that with a given dataset such as Planck. I certainly do when I model build or simply test and cross check any new relations/interactions. Those equations apply LCDM. to the cosmological redshift. As far as a new value of G well all I can say to that is good freaking luck on that score with what you have shown so far. this is a listing of the various types of studies and results form them for variations of G tests for spatial dependence is page 200 onward http://www2.fisica.unlp.edu.ar/materias/FisGral2semestre2/Gillies.pdf

fair enough, something to keep in mind if your looking at cosmological redshift is that the expansion rates are not linear. The equation above shows this as the resultant is to determine the Hubble value at a given Z compared to the value today. The relations under the square root is the evolution of the energy density for matter, radiation and Lambda. You can learn these here. https://en.wikipedia.org/wiki/Equation_of_state_(cosmology) this related to the FLRW acecleration equations. described here https://en.wikipedia.org/wiki/Friedmann_equations that link supplies some very useful integrals with regards to the scale factor the evolution of the scale factor "a" using the above relations gives \[\frac{\ddot{a}}{a}=-\frac{4G}{3}(\rho+3P)+\frac{\Lambda}{3}\] however to get the FLRW metric cosmological redshift equation you will also need the Newton weak field limit treatments as per GR. Particularly for curvature K=0 if your interested in that let me know and I'll provide more details

ok First off you have vacuum energy and vacuum energy density confused. The first case though not a useful form for energy density. The VeV is the vacuum expectation value VeV this isn't the density. This is a term describing the effective action https://en.wikipedia.org/wiki/Effective_action for Higgs the effective action is defined by the equation \[v-\frac{1}{\sqrt{\sqrt{2}G^0_W}}=\frac{2M_W}{g}\] here \(M_W\) is the mass of the W boson and \( G^0_W\) is the reduced Fermi constant. These are used primarily when dealing with Feymann path integrals in scatterings or other particle to particle interactions involving Higgs in particular dealing with the CKMS mass mixing matrix. So its not your energy density more specifically they describe CKMS mixing angles or Weinberg mixing angles. for the above without going into too much detail the mixing angles are \[M_W=\frac{1}{2}gv\] \[M_Z=\frac{1}{COS\Theta_W}\frac{1}{2}gv=\frac{1}{Cos\theta_W}M_W\] more details can be found here. Page three I'm starting to compile the previous pages now if you want the vacuum energy density the FLRW has a useful equation. \[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\] if you take the value of the Hubble constant today and plug it into that formula you will get approximately \(5.5\times 10^{-10} joules/m^3\) if you convert that over you will find your fairly close to 3.4 GeV/m^3 which matches depending on the dataset used for the Hubble constant. The confusion you had was simply not realizing the VeV isn't the energy density. hope that helps. I won't get into too many details of the quantum harmonic oscillator via zero point energy but if you take the zero point energy formula and integrate over momentum space d^3x you will end up with infinite energy. So you must renormalize by applying constraints on momentum space. However even following the renormalization procedure you still end up 120 orders of magnitude too high. There has been resolutions presented to this problem however nothing conclusive enough. Quantum field theory demystified by David Mcmahon has a decent coverage of the vacuum catastrophe edit forgot to add calculating the energy density for the cosmological constant uses the same procedure as per the critical density formula.

Who cares what wiki states it's never written by a professor in the field involved. It's never been nor will ever be an authority in physics or any other science. Garbage not even close to being accurate regardless if your describing LET or SR/GR.. Tell me do even understand what an inertial frame is as opposed to a non inertial frame ? It's no different in LET and you cannot even describe LET correctly if you don't know the difference. Tell me many more pages will it take before you realize that you never convince anyone that you are correct when you cannot describe the theories under discussions without being full of errors? Everyone is literally forced to correct your errors to the point where discussing the Pros and Cons between the two theories simply isn't happening.

This is the FLRW metric \[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2\] \[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\] This is the redshift equation(cosmological) that gets used at all ranges as it takes the evolution of matter, radiation and Lambda. \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\]

The redshift has little to do with gravitational constant and we have means of testing redshift by understanding the processes involved. We can for example examine hydrogen which is one of most common elements in our universe and using spectrography. There is nothing random that isn't cross checked by numerous means involving redshift. We don't even rely on it as our only means of distance calculation. Quite frankly no one method works for every distance range. A huge portion of papers can be found studying the accuracy of redshift at different ranges and those cross checks using other means such as interstellar parallax. Same applies to luminosity distance. By the way the redshift formula you find in textbooks is only useful at short distances cosmological scale.

This evening when I get home I will be able to run the formulas for you. Yes you can calculate the vacuum energy density per cubic metre. For that one can get a decent estimate using the critical density formula. (Assuming Lambda is the result of the Higfs field) one line of research. The calculations differ for the quantum harmonic oscillator contributions however that results in the vacuum catastrophe but I also have the related calculations for that as well.

As light climbs in and out of a gravity well it will blue shift or redshift. For example an outside observer looking at infalling material at the EH of a blackhole will see infinite redshift but an observer at the EH will see infinite blue shift. This is due to gravitational redshift The path will be determined by the Principle of least action which correlates the Potential energy and kinetic energy terms. What most ppl don't realize is that the path is never truly straight. That's just the mean average. If you consider all the little infinitesimal changes in direction (sometimes up/down left right etc) then it becomes much easier to understand. As Markus the geodesic equations are the extremum of all the miniscule deviations

How does a coupling constant appear smaller ? If you apply \(F=G\frac{m_1m_2}{r^2}\) the coupling constant remains constant what changes is the force exerted by the coupling between two masses as a function of radius. Not the coupling constant itself. We describe our observable universe itself in the FLRW metric we know the universe extends beyond that it could be finite or infinite as we can never measure beyond that we deal with what we can Observe and measure. (Region of shared causality)

yes I did understand that but I'm trying to ascertain your eventual goals with this to provide direction for improvement. If you think about we do much the same with the use of the scale factor under the FLRW However a key point is that G is a constant under the FLRW so your going to have to explain why you feel G would change as a result of change in radius ?

SU(2) \[{\small\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline Field & \ell_L& \ell_R &v_L&U_L&d_L&U_R &D_R&\phi^+&\phi^0\\\hline T_3&- \frac{1}{2}&0&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&0&0&\frac{1}{2}&-\frac{1}{2} \\\hline Y&-\frac{1}{2}&-1&-\frac{1}{2}&\frac{1}{6}&\frac{1}{6}& \frac{2}{3}&-\frac{1}{3}&\frac{1}{2}&\frac{1}{2}\\\hline Q&-1&-1&0&\frac{2}{3}&-\frac{1}{3}&\frac{2}{3}&-\frac{1}{3}&1&0\\\hline\end{array}}\] \(\psi_L\) doublet \[D_\mu\psi_L=[\partial_\mu-i\frac{g}{\sqrt{2}}(\tau^+W_\mu^+\tau^-W_\mu^-)-i\frac{g}{2}\tau^3W^3_\mu+i\acute{g}YB_\mu]\psi_L=\]\[\partial_\mu-i\frac{g}{\sqrt{2}}(\tau^+W_\mu^-)+ieQA_\mu-i\frac{g}{cos\theta_W}(\frac{t_3}{2}-Qsin^2\theta_W)Z_\mu]\psi_L\] \(\psi_R\) singlet \[D_\mu\psi_R=[\partial\mu+i\acute{g}YB_\mu]\psi_R=\partial_\mu+ieQA_\mu+i\frac{g}{cos\theta_W}Qsin^2\theta_WZ_\mu]\psi_W\] with \[\tau\pm=i\frac{\tau_1\pm\tau_2}{2}\] and charge operator defined as \[Q=\begin{pmatrix}\frac{1}{2}+Y&0\\0&-\frac{1}{2}+Y\end{pmatrix}\] \[e=g.sin\theta_W=g.cos\theta_W\] \[W_\mu\pm=\frac{W^1_\mu\pm iW_\mu^2}{\sqrt{2}}\] \[V_{ckm}=V^\dagger_{\mu L} V_{dL}\] The gauge group of electroweak interactions is \[SU(2)_L\otimes U(1)_Y\] where left handed quarks are in doublets of \[ SU(2)_L\] while right handed quarks are in singlets the electroweak interaction is given by the Langrangian \[\mathcal{L}=-\frac{1}{4}W^a_{\mu\nu}W^{\mu\nu}_a-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+\overline{\Psi}i\gamma_\mu D^\mu \Psi\] where \[W^{1,2,3},B_\mu\] are the four spin 1 boson fields associated to the generators of the gauge transformation \[\Psi\] The 3 generators of the \[SU(2)_L\] transformation are the three isospin operator components \[t^a=\frac{1}{2} \tau^a \] with \[\tau^a \] being the Pauli matrix and the generator of \[U(1)_\gamma\] being the weak hypercharge operator. The weak isospin "I" and hyper charge \[\gamma\] are related to the electric charge Q and given as \[Q+I^3+\frac{\gamma}{2}\] with quarks and lepton fields organized in left-handed doublets and right-handed singlets: the covariant derivative is given as \[D^\mu=\partial_\mu+igW_\mu\frac{\tau}{2}-\frac{i\acute{g}}{2}B_\mu\] \[\begin{pmatrix}V_\ell\\\ell\end{pmatrix}_L,\ell_R,\begin{pmatrix}u\\d\end{pmatrix}_,u_R,d_R\] The mass eugenstates given by the Weinberg angles are \[W\pm_\mu=\sqrt{\frac{1}{2}}(W^1_\mu\mp i W_\mu^2)\] with the photon and Z boson given as \[A_\mu=B\mu cos\theta_W+W^3_\mu sin\theta_W\] \[Z_\mu=B\mu sin\theta_W+W^3_\mu cos\theta_W\] the mass mixings are given by the CKM matrix below \[\begin{pmatrix}\acute{d}\\\acute{s}\\\acute{b}\end{pmatrix}\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{pmatrix}\begin{pmatrix}d\\s\\b\end{pmatrix}\] mass euqenstates given by \(A_\mu\) an \(Z_\mu\) \[W^3_\mu=Z_\mu cos\theta_W+A_\mu sin\theta_W\] \[B_\mu= Z_\mu sin\theta_W+A_\mu cos\theta_W\] \[Z_\mu=W^3_\mu cos\theta_W+B_\mu sin\theta_W\] \[A_\mu=-W^3_\mu\sin\theta_W+B_\mu cos\theta_W\] ghost field given by \[\acute{\psi}=e^{iY\alpha_Y}\psi\] \[\acute{B}_\mu=B_\mu-\frac{1}{\acute{g}}\partial_\mu\alpha Y\] [latex]D_\mu[/latex] minimally coupled gauge covariant derivative. h Higg's bosonic field [latex] \chi[/latex] is the Goldstone boson (not shown above) Goldstone no longer applies after spontaneous symmetry breaking [latex]\overline{\psi}[/latex] is the adjoint spinor [latex]\mathcal{L}_h=|D\mu|^2-\lambda(|h|^2-\frac{v^2}{2})^2[/latex] [latex]D_\mu=\partial_\mu-ie A_\mu[/latex] where [latex] A_\mu[/latex] is the electromagnetic four potential QCD gauge covariant derivative [latex] D_\mu=\partial_\mu \pm ig_s t_a \mathcal{A}^a_\mu[/latex] matrix A represents each scalar gluon field Single Dirac Field [latex]\mathcal{L}=\overline{\psi}I\gamma^\mu\partial_\mu-m)\psi[/latex] under U(1) EM fermion field equates to [latex]\psi\rightarrow\acute{\psi}=e^{I\alpha(x)Q}\psi[/latex] due to invariance requirement of the Langrene above and with the last equation leads to the gauge field [latex]A_\mu[/latex] [latex] \partial_\mu[/latex] is replaced by the covariant derivitave [latex]\partial_\mu\rightarrow D_\mu=\partial_\mu+ieQA_\mu[/latex] where [latex]A_\mu[/latex] transforms as [latex]A_\mu+\frac{1}{e}\partial_\mu\alpha[/latex] Single Gauge field U(1) [latex]\mathcal{L}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}[/latex] [latex]F_{\mu\nu}=\partial_\nu A_\mu-\partial_\mu A_\nu[/latex] add mass which violates local gauge invariance above [latex]\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^2A_\mu A^\mu[/latex] guage invariance demands photon be massless to repair gauge invariance add a single complex scalar field [latex]\phi=\frac{1}{\sqrt{2}}(\phi_1+i\phi_2[/latex] Langrene becomes [latex] \mathcal{L}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+|D_\mu \phi|^2-V_\phi[/latex] where [latex]D_\mu=\partial_\mu-ieA_\mu[/latex] [latex]V_\phi=\mu^2|\phi^2|+\lambda(|\phi^2|)^2[/latex] [latex]\overline{\psi}=\psi^\dagger \gamma^0[/latex] where [latex]\psi^\dagger[/latex] is the hermitean adjoint and [latex]\gamma^0 [/latex] is the timelike gamma matrix the four contravariant matrix are as follows [latex]\gamma^0=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}[/latex] [latex]\gamma^1=\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&0&-1&0\\-1&0&0&0\end{pmatrix}[/latex] [latex]\gamma^2=\begin{pmatrix}0&0&0&-i\\0&0&i&0\\0&i&0&0\\-i&0&0&0\end{pmatrix}[/latex] [latex]\gamma^3=\begin{pmatrix}0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{pmatrix}[/latex] where [latex] \gamma^0[/latex] is timelike rest are spacelike V denotes the CKM matrix usage [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] [latex]V_{ckm}=V^\dagger_{\mu L} V_{dL}[/latex] the CKM mixing angles correlates the cross section between the mass eigenstates and the weak interaction eigenstates. Involves CP violations and chirality relations. Dirac 4 component spinor fields [latex]\gamma^5=i\gamma_0,\gamma_1,\gamma_2,\gamma_3[/latex] 4 component Minkowskii with above 4 component Dirac Spinor and 4 component Dirac gamma matrixes are defined as [latex] {\gamma^\mu\gamma^\nu}=2g^{\mu\nu}\mathbb{I}[/latex] where [latex]\mathbb{I}[/latex] is the identity matrix. (required under MSSM electroweak symmetry break} in Chiral basis [latex]\gamma^5[/latex] is diagonal in [latex]2\otimes 2[/latex] the gamma matrixes are [latex]\begin{pmatrix}0&\sigma^\mu_{\alpha\beta}\\\overline{\sigma^{\mu\dot{\alpha}\beta}}&0\end{pmatrix}[/latex] [latex]\gamma^5=i{\gamma_0,\gamma_1,\gamma_2,\gamma_3}=\begin{pmatrix}-\delta_\alpha^\beta&0\\0&\delta^\dot{\alpha}_\dot{\beta}\end{pmatrix}[/latex] [latex]\mathbb{I}=\begin{pmatrix}\delta_\alpha^\beta&0\\0&\delta^\dot{\alpha}_\dot{\beta}\end{pmatrix}[/latex] Lorentz group identifiers in [latex](\frac{1}{2},0)\otimes(0,\frac{1}{2})[/latex] [latex]\sigma\frac{I}{4}=(\gamma^\mu\gamma^\nu)=\begin{pmatrix}\sigma^{\mu\nu\beta}_{\alpha}&0\\0&-\sigma^{\mu\nu\dot{\alpha}}_{\dot{\beta}}\end{pmatrix}[/latex] [latex]\sigma^{\mu\nu}[/latex] duality satisfies [latex]\gamma_5\sigma^{\mu\nu}=\frac{1}{2}I\epsilon^{\mu\nu\rho\tau}\sigma_{\rho\tau}[/latex] a 4 component Spinor Dirac field is made up of two mass degenerate Dirac spinor fields U(1) helicity [latex](\chi_\alpha(x)),(\eta_\beta(x))[/latex] [latex]\psi(x)=\begin{pmatrix}\chi^{\alpha\beta}(x)\\ \eta^{\dagger \dot{\alpha}}(x)\end{pmatrix}[/latex] the [latex](\alpha\beta)=(\frac{1}{2},0)[/latex] while the [latex](\dot{\alpha}\dot{\beta})=(0,\frac{1}{2})[/latex] this section relates the SO(4) double cover of the SU(2) gauge requiring the chiral projection operator next. chiral projections operator [latex]P_L=\frac{1}{2}(\mathbb{I}-\gamma_5=\begin{pmatrix}\delta_\alpha^\beta&0\\0&0\end{pmatrix}[/latex] [latex]P_R=\frac{1}{2}(\mathbb{I}+\gamma_5=\begin{pmatrix}0&0\\ 0&\delta^\dot{\alpha}_\dot{\beta}\end{pmatrix}[/latex] Weyl spinors [latex]\psi_L(x)=P_L\psi(x)=\begin{pmatrix}\chi_\alpha(x)\\0\end{pmatrix}[/latex] [latex]\psi_R(x)=P_R\psi(x)=\begin{pmatrix}0\\ \eta^{\dagger\dot{a}}(x)\end{pmatrix}[/latex] also requires Yukawa couplings...SU(2) matrixes given by [latex]diag(Y_{u1},Y_{u2},Y_{u3})=diag(Y_u,Y_c,Y_t)=diag(L^t_u,\mathbb{Y}_u,R_u)[/latex] [latex]diag(Y_{d1},Y_{d2},Y_{d3})=diag(Y_d,Y_s,Y_b)=diag(L^t_d,\mathbb{Y}_d,R_d[/latex] [latex]diag(Y_{\ell 1},Y_{\ell 2},Y_{\ell3})=diag(Y_e,Y_\mu,Y_\tau)=diag(L^T_\ell,\mathbb{Y}_\ell,R_\ell)[/latex] the fermion masses [latex]Y_{ui}=m_{ui}/V_u[/latex] [latex]Y_{di}=m_{di}/V_d[/latex] [latex]Y_{\ell i}=m_{\ell i}/V_\ell[/latex] Reminder notes: Dirac is massive 1/2 fermions, Weyl the massless. Majorona fermion has its own antiparticle pair while Dirac and Weyl do not. The RH neutrino would be more massive than the LH neutrino, same for the corresponding LH antineutrino and RH Neutrino via seesaw mechanism which is used with the seesaw mechanism under MSM. Under MSSM with different Higgs/higglets can be numerous seesaws. The Majorona method has conservation violations also these fermions must be electric charge neutral. (must be antiparticles of themselves) the CKM and PMNS are different mixing angels in distinction from on another. However they operate much the same way. CKM is more commonly used as its better tested to higher precision levels atm. Quark family is Dirac fermions due to electric charge cannot be its own antiparticle. Same applies to the charged lepton family. Neutrinos are members of the charge neutral lepton family Lorentz group Lorentz transformations list spherical coordinates (rotation along the z axis through an angle ) \[\theta\] \[(x^0,x^1,x^2,x^3)=(ct,r,\theta\phi)\] \[(x_0,x_1,x_2,x_3)=(-ct,r,r^2,\theta,[r^2\sin^2\theta]\phi)\] \[\acute{x}=x\cos\theta+y\sin\theta,,,\acute{y}=-x\sin\theta+y \cos\theta\] \[\Lambda^\mu_\nu=\begin{pmatrix}1&0&0&0\\0&\cos\theta&\sin\theta&0\\0&\sin\theta&\cos\theta&0\\0&0&0&1\end{pmatrix}\] generator along z axis \[k_z=\frac{1\partial\phi}{i\partial\phi}|_{\phi=0}\] generator of boost along x axis:: \[k_x=\frac{1\partial\phi}{i\partial\phi}|_{\phi=0}=-i\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0 \end{pmatrix}\] boost along y axis\ \[k_y=-i\begin{pmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0 \end{pmatrix}\] generator of boost along z direction \[k_z=-i\begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0 \end{pmatrix}\] the above is the generator of boosts below is the generator of rotations. \[J_z=\frac{1\partial\Lambda}{i\partial\theta}|_{\theta=0}\] \[J_x=-i\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&1\\0&0&-1&0 \end{pmatrix}\] \[J_y=-i\begin{pmatrix}0&0&0&0\\0&0&0&-1\\0&0&1&0\\0&0&0&0 \end{pmatrix}\] \[J_z=-i\begin{pmatrix}0&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&0 \end{pmatrix}\] there is the boosts and rotations we will need and they obey commutations \[[A,B]=AB-BA\] SO(3) Rotations list set x,y,z rotation as \[\varphi,\Phi\phi\] \[R_x(\varphi)=\begin{pmatrix}1&0&0\\0&\cos\varphi&\sin\varphi\\o&-sin\varphi&cos\varphi \end{pmatrix}\] \[R_y(\phi)=\begin{pmatrix}cos\Phi&0&\sin\Phi\\0&1&0\\-sin\Phi&0&cos\Phi\end{pmatrix}\] \[R_z(\phi)=\begin{pmatrix}cos\theta&sin\theta&0\\-sin\theta&\cos\theta&o\\o&0&1 \end{pmatrix}\] Generators for each non commutative group. \[J_x=-i\frac{dR_x}{d\varphi}|_{\varphi=0}=\begin{pmatrix}0&0&0\\0&0&-i\\o&i&0\end{pmatrix}\] \[J_y=-i\frac{dR_y}{d\Phi}|_{\Phi=0}=\begin{pmatrix}0&0&-i\\0&0&0\\i&i&0\end{pmatrix}\] \[J_z=-i\frac{dR_z}{d\phi}|_{\phi=0}=\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}\] with angular momentum operator \[{J_i,J_J}=i\epsilon_{ijk}J_k\] with Levi-Civita \[\varepsilon_{123}=\varepsilon_{312}=\varepsilon_{231}=+1\] \[\varepsilon_{123}=\varepsilon_{321}=\varepsilon_{213}=-1\] SU(3) generators Gell Mann matrix's \[\lambda_1=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}\] \[\lambda_2=\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}\] \[\lambda_3=\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}\] \[\lambda_4=\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}\] \[\lambda_5=\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}\] \[\lambda_6=\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}\] \[\lambda_7=\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}\] \[\lambda_8=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}\] commutation relations \[[\lambda_i\lambda_j]=2i\sum^8_{k=1}f_{ijk}\lambda_k\] with algebraic structure \[f_{123}=1,f_{147}=f_{165}=f_{246}=f_{246}=f_{257}=f_{345}=f_{376}=\frac{1}{2},f_{458}=f_{678}=\frac{3}{2}\] with Casimer Operator \[\vec{J}^2=J_x^2+J_y^2+j_z^2\]