Mordred

Yes I do work with these on a professional level. Its also one of my primary focusses in regards to my primary expertise in Cosmology. Feel free to ask any questions and I will be glad to help you on it. Most articles including dissertations on Higgs will likely have those equations. Its certainly covered under papers regarding CKMS mass mixing. So resources are readily available to learn how to eventually understand the above. A big step is knowing vector and spinor relations under math. LIttle hint every SM model for every particle interaction (Feymann path integral) applies the Principle of least action via the Langrangian. Which forms also applies to the Hamilton

This is where we have to be careful. You may recall that the VeV is a probability function correct? That probability function will have a probability current. The term "expectation" value denotes this. So it has a weighted average that is described by the 246 VeV value. The other important detail is the Higgs field isn't just one field it is an SU(2) doublet. \[\phi=\begin{pmatrix}\phi^+\\\phi^0\end{pmatrix}\] however these are complex fields \[\phi^+=\frac{1}{2}(\phi_1+i\phi_2)\] \[\phi^0=\frac{1}{2}(\phi_3+\phi_4\] now these two statements describe rotations ( matrix, tensor operations) simply put. Yes those two equations do form a matrix but we can ignore that for now. now fermions have a few properties the Higgs mediates with each has a mass contribution the Higgs relation to charge Q, weak isospin eigenvalue \(T_3\), and hypercharge Y is related for the Higgs by \[Q=(T_3+\frac{Y}{2})\phi_0=0\] only the \(\phi_0\) current that gets a VEV...a probability current giving the weighted average likelyhood the last equation directly relates to the W, Z and photons gaining mass or not. Unfortunately this is where I'm going to have to turn it up a notch or a dozen notches quarks and lepton fields are organized in left-handed doublets and right-handed singlets: Matter is left handed, antimatter is right handed 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 \(Z_\mu\) \[W^3_\mu=Z_\mu cos\theta_W+A_\mu sin\theta_W\] \[B_\mu= Z_\mu sin\theta_W+A_\mu cos\theta_W\] \[Z_\mu=W^3_\mu cos\theta_W+B_\mu sin\theta_W\] \[A_\mu=-W^3_\mu\sin\theta_W+B_\mu cos\theta_W\] this is how the mass terms are generated using eh CKMS mass mixing matrix above. Unfortunately this is a stage where I had to resort to under the math to be accurate enough on how the mass terms apply for W,Z, and why photons do not acquire mass. However this table may help visualize what is going on \[{\small\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline Field & \ell_L& \ell_R &v_L&U_L&d_L&U_R &D_R&\phi^+&\phi^0\\\hline T_3&- \frac{1}{2}&0&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&0&0&\frac{1}{2}&-\frac{1}{2} \\\hline Y&-\frac{1}{2}&-1&-\frac{1}{2}&\frac{1}{6}&\frac{1}{6}& \frac{2}{3}&-\frac{1}{3}&\frac{1}{2}&\frac{1}{2}\\\hline Q&-1&-1&0&\frac{2}{3}&-\frac{1}{3}&\frac{2}{3}&-\frac{1}{3}&1&0\\\hline\end{array}}\] in the above table you also have Yukawa couplings as well for example for a quark \[\mathcal{L}=q_d\overline{Q}_L\phi d_R+g_\mu \overline{Q}_L\phi_c U_R +h.c\] h.c. is the hermitean conjugate in QM don't worry about the last equation its just to show that the mass terms isn't strictly due to Higgs. Yukawa couplings also contributes and it uses the same table as above. Also I did not show the right handed singlets in the above. for antineutrinos they have different mixing angles as singlets and will involve Majarona what I have shown may likely make your head explode as is lol. oh forgot to add prior to symmetry breaking the SM model uses the Goldstone bosons

well your on the right path see section 4 https://cds.cern.ch/record/348154/files/9803257.pdf "(4) In the unitary gauge, the isodoublet is replaced by the physical Higgs eld ! [0; (v+H)=p 2], which describes the uctuation of the I3 = 1=2 component of the isodoublet eld about the ground-state value v=p 2. The scale v of the electroweak symmetry breaking is xed by the W mass, which in turn can be reexpressed by the Fermi coupling, v = 1=qp 2GF 246 GeV. The quartic coupling and the Yukawa couplings gf can be reexpressed in terms of the physical Higgs mass MH and the fermion mass" it doesn't copy over well from the document please note this is a pre Higgs discovery paper published prior to confirming the Higgs mass so some of the numbers will be off. but it explains the VeV and how its set.

nice I wish there was a way to show the propagators better in latex. As there is two symbols for propagators in the Feymann rules. Wavy line being one the other dotted line (ghost propagators) the problem isn't the horizontal but the diagonals for triple and quartic interactions. lecture_16.pdf (usp.br) second link has the ghost propagators https://arxiv.org/pdf/1209.6213

Are you familiar with spontaneous symmetry breaking and the Mexican hat potential of the Higgs field ? Yes the VeV can be described as a fundamental property of the Higgs field in so far as it sets the scale where spontaneous symmetry breaking occurs to give the particles it interacts with their mass terms. At a certain temperature is when the spontaneous symmetry breaking occurs ( the precise value depends on the model ) however wiki gives the value 159.5 GeV which I for one do not trust. ( the paper wiki used didnt include the U(1) gauge. The value I have commonly seen is roughly 10^15 GeV which make more sense. Regardless of the temperature value the VeV describes the temperature where spontaneous symmetry breaking (electroweak symmetry breaking) occurs. At a higher temperature all particles are massless. At that temperature spontaneous symmetry breaking occurs and particles acquire mass. Here is a link to spontaneous symmetry breaking and it shows the Mexican hat potential https://en.m.wikipedia.org/wiki/Spontaneous_symmetry_breaking https://en.wikipedia.org/wiki/Electroweak_interaction the second link describes the VeV "above the unification energy, on the order of 246 GeV,[a] they would merge into a single force. Thus, if the temperature is high enough – approximately 1015 K – then the electromagnetic force and weak force merge into a combined electroweak force. During the quark epoch (shortly after the Big Bang), the electroweak force split into the electromagnetic and weak force. " this link has the correct value. I didn't bother including the link with the incorrect value.

Blackholes could exist at the time of the CMB but the blackbody temperature Migl mentioned would still be far lower than the blackbody temperature at that same time period. So they would be growing and not dying. We really do not know what occurs beyond the Event horizon so any statement made would be nothing more than guess work There has been some research papers suggesting this as one possibility. They have even developed tests for this possibility. One of those tests directly relates to the article you posted concerning GW wave data.

\.begin{array}{rcl} a&b&c\\a&b&c\\a&b&c\end{array} \begin{array}{rcl} a&b&c\\a&b&c\\a&b&c\end{array} interesting the \begin{array} self activates f(z) = \left\{ \.begin{array}{rcl} a&b&c\\a&b&c\\a&b&c\end{array} \right . \[f(z) = \left\{ \begin{array}{rcl} a&b&c\\a&b&c\\a&b&c\end{array} \right .\]

\[\vec{v}_e+p\longrightarrow n+e^+\] \[\array{ n_e \searrow&&\nearrow n \\&\leadsto &\\p \nearrow && \searrow e^2}\]

your welcome glad to help

Correct now your getting it +1 on seeing that connection

It's a workable descriptive not completely accurate but sufficient for a layman understanding. Getting into the renormalization aspects would be a bit too advanced it's sufficient to accept that it's a renormalized value.

Well at least Chatgp got that part correct as that's precisely what it's used for. The VeV is used in a similar manner just an fyi

Here is the association of VeV to Fermi-constant Fermi's interaction - Wikipedia

A possible antineutrino cross section calculation massless case \[\vec{v}_e+p\longrightarrow n+e^+\] Fermi constant=\(1.1663787(6)*10^{-4} GeV^{-2}\) \[\frac{d\sigma}{d\Omega}=\frac{S|M|^2\acute{p}^2}{M_2|\vec{p_1}|2|\vec{p_1}|(E_1+m_2c^2)-|\vec{p_1}|\prime{E_1}cos\theta}\] Fermi theory \[|M|^2=E\acute{E}|M_0^2|=E\acute{E}(M_Pc^2)^2G^2_F\] \[\frac{d\sigma}{d\Omega}=(\frac{h}{8\pi}^2)\frac{M_pc^4(\acute{E})^2G^3_F}{[(E+M_p^2)-Ecos\theta]}\] \[\frac{d\sigma}{d\Omega}=(\frac{h}{8\pi}^2)\frac{M_pc^4(\acute{E})^2G^3_F}{M_pc^2}(1+\mathcal{O}(\frac{E}{M_oc^2})\] \[\sigma=(\frac{\hbar cG_F\acute{E}^2}{8\pi})^2\simeq 10^{-45} cm^2\] \

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.