Electric charge = (weak isotopic charge) + 0.5(hypercharge) [Equation 1]
or
Hypercharge = 2{(electric charge) – (weak isotopic charge)} [Equation 2]
Quark hypecharge is based on the Gell-mann-Nishijima formula which includes contributions from baryon number, strangeness, bottomness and topness. However, Wikipedia’s article on hypercharge is currently in error because it doesn’t allow for the chirality of hypercharge, e.g. it states hypercharges only for left-handed spinors as if they are absolute, and ignores the fact that right-handed spinors have different hypercharges! This is 100% consistent with the fact that the mainstream – which controls wikipedia – simply ignores or downplays anything important in order to defend status quo. Pathetic! Anyone “fact checking” this post against Wikipedia using the assumption that the groupthink authority of the latter is superior to everything will thus come to a false “conclusion”! Strange quarks are stated to have a hypercharge of -2/3 (with zero weak isospin, like a right-handed spinor), as compared to 1/3 for a downquark.
If you look at a physical mechanism for lepton-quark “unification” whereby vacuum polarization of the electric field creates virtual particles which are polarized radially thus absorbing energy from the electric field and converting it into virtual particles that include weak and – at high enough energy – strong interaction vector bosons, then Equation 1 in the standard model above makes sense because the electric charge is partially converted (by vacuum polarization) into the weak force field. Equation 2 is of interest if we consider hypercharge to be fundamental to understanding lepton-quark unification (other approaches taken by people over the past fifty years simply lead to a mess): notice that hypercharge isn’t directly observed so the factor of 2 in Equation 2 can now be considered as an arbitrary scaling factor. Hypercharge-based unification:
If we change Equation 2 by removing the factor of 2, then all of the hypercharges in the diagram above are halved, so each pair of particles (shown encapsulated) has a total hypercharge of -1/3. But what about the underlying theory for this unification mechanism? As stated in the previous post, the experimental evidence for the Omega Minus baryon, containing three strange quarks (they were called strange because they form particles of relatively high stability, i.e. relatively slow decay rate), is the key rosetta stone for understanding lepton-quark unification “mechanically” (i.e., by solid field energy conservation effects of vacuum polarization, ignored entirely as a taboo in present day QFT textbooks!). If you want to crack open nature, you find a clue and insert your crowbar and apply all the force you have. You don’t label it an anomaly or trivia to be swept under the carpet and ignored, and run away from the clue, unless you want to fail. We have to try things to see if they work, or we get nowhere. Another thing to consider above (because fermion numbers are conserved, unlike bosons) is that there should be an equality in the numbers of particle and anti-particles in defining such particles (unlike the Standard Model’s farce, which originated with Dirac’s gamma matrices bungling in 1928!).
ABOVE: of significance to the previous post here, in a new post at Not Even Wrong, “The Mystery of Spin” (referring at the start to a post by Dr Baez on the topic) Dr Woit states: “What’s a bit mysterious is not the above, but the fact that when we do quantum mechanics, we have to work with complex numbers [i], not quaternions. We then have to find a consistent way to replace quaternions by complex two by two matrices when they are rotations and and complex column vectors when they are spinors…”, before arguing that “multiplication of quaternions in the spinor story correspond correctly to multiplication of an element of 𝐶^2 by a matrix.” Quaternions obey:
What is interesting here is that there are different versions of complexified 2 x 2 matrices that correspond to Dirac’s four 4 x 4 gamma matrices (Weyl’s matrices, and various versions of Pauli’s SU(2) matrices, as indicated by Woit, above). If you look at our diagram summary above for the most stable low-energy generation of the standard model particles, the real “anti-particle” of the electron is the upquark, with a mechanistic explanation via vacuum polarization energy conservation for fractional quark charge!
There’s direct evidence for this in our weak decay route anomaly diagram, indicating that including weak bosons in beta decay in 1967 really demonstrates quark-lepton unification if you do a little critical analysis and fact-checking (beyond BBC style “fact checking” which is just the Orwellian four-legs-good-two-legs-bad bigoted dogmatic “authority”-biased chanting). What about the muon, tauon, and all the quarks of the two further generations? They are all only observed as or in highly unstable (radioactive) particles, and the basic mechanism for how they arise (briefly, before they decay!) is dealt with here, including comparisons of directly measured masses to data (the only masses for particles in string theory are semi-empirical Regge-trajectory correlations of hadronic string theory, in which the strong force is analogous to elastic bands aka hadronic string!).
Spin is related to relativity in the Dirac equation. Pauli in 1927 modified Schroedinger’s Hamiltonian by including spin: Pauli simply subtracted from the Hamiltonian energy (H) the term µB (i.e. the product of the electron’s magnetic moment µ and the magnetic field B). But Dirac the next year (1928) came up with the discovery that this was automatically done (and then some!) if you simply made the Hamiltonian relativistic, put putting space and time on an equal footing. Making the Hamiltonian relativistic introduced the four 4 x 4 Dirac gamma matrices, which predict electron spin effects and also antimatter. However, it became dogma once “verified” by the discovery of the positron in 1932. This had disastrous consequences for shutting down alternative options for unification. Superstring “theory” is one result!
Why? Dirac first said his gamma matrices anti-electron was the proton: causing the difficulty that the proton is more massive than the electron! Then he said it was the positron. End of story (as far as the unreasonable mainstream bigots are concerned). Problems however also arise for the positron being the “anti”-electron: (1) why the asymmetry matter and “anti-matter” in universe (there are more electrons than positrons, DUH!), and (2) in the 1950s (too late to influence Dirac’s and Pauli’s dogma) the asymmetry of fermion spin was demonstrated for the weak force coupling to only left-handed spinors! For another thing, there is now also evidence (that we can’t get published anywhere it should be) that lepton-quark unification can be done without the extravagant dead end extra dimensional multiverse hype from string “theory”. For instance, if QED contains SU(2) spin for fermions, why not use SU(2) for QED rather than using assuming it is U(1) (note here the problem that Yang-Mills theory is needed for a physical mechanistic explanation of how exchange of massless spin-1 gauge bosons correctly produces the electromagnetic forces, physically). These SU(2) matrices arise, as Dirac showed, from the effects of special relativity on spin in quantum mechanics (sometimes it is falsely “explained” by blather such as “nobody understands quantum mechanics”).
But you’d think resolving such details would be the number one priority for theoretical physicists. On page 49 of his 2002 paper on QFT and representation theory, Dr Woit states: “The motivating conjecture of this paper is that the quantum field theory underlying the Standard Model can be understood in terms of the representation theory of the automorphism group of some geometric structure.”
Woit on spin in his 2002 preprint QFT and Representation Theory p51:
This points the way towards a low-dimensional unification of quarks and leptons. There’s a discussion of the relationship between U(2) and its subgroup SU(2) here, which states:
On the topic of squaring negative numbers to get positive modulus, the fact that x^2 =+1 cannot be used to determine a single precise value for x (because x can be either +1 or -1 here), has always done in the naive Bohr-ing assumption that the universe is not deterministic or mechanical. Indeterminism ( i.e. that x = 1 or -1 in the equation x^2 =+1, and nobody can be more specific than that!), is intrinsic in the mathematical model of reality, whereby squaring of negative amplitudes (wavefunctions) is done to get real world (positive) modulus for probability. Thus, it’s mathematical modelling that produces indeterminism, not necessarily nature (which never exactly corresponds to the maths for physical reality anyway, e.g. calculus ignores the whole problem of discontinuities, like quantized fields). So I’m dismissive of long-winded abstract mathematical garbage that starts with some arbitrary mathematical axioms and ends up with the “mathematical” discovery that the universe is not deterministic…
According to Dr Wilson (https://robwilson1.wordpress.com/2024/06/08/looking-in-the-mirror/#comment-8235):
“The SM claims that the electro-weak gauge group is U(2) = (U(1) x SU(2))/Z_2. Yet if you look closely at the calculations that are actually done, you will see that the group that is actually used is U(1) x SU(2), without the quotient by Z_2. This is necessary in order to separate the scalar -1 in U(1), that negates all the spinors, from the scalar -1 in SU(2), that negates only the “left-handed” spinors. Hence the quotient by Z_2 is an obfuscation that prevents us from seeing the essential difference between LH and RH spinors.
“In the same way, the SM claims that the electro-strong gauge group is U(3) = (U(1) x SU(3))/Z_3. Again, this is an obfuscation that prevents us from seeing the essential difference between the three generations of fermions. By lifting to U(1) x SU(3) as I propose, we separate scalars of order 3 in U(1), which act on all spinors, from the scalars of order 3 in SU(3), which act only on the quarks. The latter is a “colour” symmetry, as described by the theory of quantum chromodynamics, leaving the former available as a “generation” symmetry for all elementary fermions.” [Emphasis added in bold.]
I have to go into this in detail. I am interested in the connection between Woit’s use of U(2) in 4-d to get the SM electroweak charges, and Wilson’s statement claiming that there’s an error in the SM’s U(2) = (U(1) x SU(2))/Z_2 concerning Z_2. I wonder also if Woit’s use of U(2) in 4-d to produce electoweak charges, can be extended to U(3) = (U(1) x SU(3))/Z_3, to produce QCD colour charges, etc?
According to Wikipedia:
GL(n, C) ∩ Sp(2n) = U(n),
or
GL(2, C) ∩ Sp(4) = U(2),
which may be related to the Woit idea of getting electroweak charges from U(2) in 4-d spacetime. It’s also of interest to see the effect of “relativity” or 4-d (complex) spacetime on the spin group, in comparison to 3-d (real) spacetime tabulated below:
The real mystery of spin is why it’s taboo to point out anomalies or resolve them. Celebrity hubris again?
UPDATE (16 June 2024):
Dr Woit has hosted comments from Dr Lisi and others regarding arXiv censorship (at least delayed preprint hosting for “five weeks”) of the technical paper now linked here (locally here). Dr Lisi’s paper states on p14: “In the complete Standard Model there are 8 fermion types: neutrinos, electron-type leptons, three colors of up-type quarks, and three colors of downtype quarks. In a unified theory (which includes right-handed neutrinos) these must correspond to 8 disjoint 24-cells. The only current proposal for a unified theory that meets this criteria, with a triality symmetry acting between 192 distinct fermion weights grouped into 8 disjoint 24-cells, is E8 Theory.”
IMHO, there are big problems with this approach (which includes a whole load of epicycle maths which explains nothing and predicts nothing), and also with the arbitrary way that arXiv (not to be confused with vixra) hubris works: no mechanism, no predictions of masses, no nothing except a kind of obsessive dogmatic effort to come up with a an epicycle-fit E8 system to “explain” the existing Standard Model (without first identifying and correcting errors in it). On the positive side, Dr Lisi has at least made an effort to address some issues with Dirac spinors, sensibly arguing on p5 that you should ignore fermion mass until after it is supplied by the Higgs mechanism: “A Dirac spinor corresponds to a fermion or anti-fermion with specific momentum and spin. Massive fermions can be treated as massless fermions interacting with a Higgs field. Massless fermions have definite helicity, with aligned or anti-aligned spin and momentum.”
Naturally, Dr Lisi ignores my 2011 paper‘s use of a correction for standard model mass for unification of fermions and bosons via correcting electromagnetic theory to Yang-Mills theory: charged massless boson exchanges explain the mechanism of electromagnetic interactions and are permitted (contrary to fake back-of-envelope no-go theorems) because the two-way (field equilibrium) exchange of charged massless bosons cancels the infinite magnetic self-inductance that prevents charged massless bosons going one-way. This hard scientific fact simply kills off so much of the blather in mainstream “unification” epicycle “theory”, that you can see why the mainstream – including mainstream side-shoots claiming to be independent but also tied to SM BS, e.g. Dr Woit and Dr Lisi – don’t succeed.
What’s needed, again, is objective censorship of crap, not elitist censorship of fact based corrections of SM errors; I’m not against censorship of errors – quite the opposite – I’m just against incorrect bigoted censorship based on fashion or celebrity endorsement or whatever, i.e. if you want to censor Dr Lisi you should do so for reasonable reasons!
Apart from that, the final sentence, “The only current proposal for a unified theory that meets this criteria, with a triality symmetry acting between 192 distinct fermion weights grouped into 8 disjoint 24-cells, is E8 Theory”, makes you laugh. It is clear Dr Lisi is being held up by arXiv for (a) that conclusion, and (b) for being high profile (having got publicity for E8 before Dr Distler rejected it). I submitted a paper (a forerunner of https://vixra.org/abs/1305.0012) to arXiv using my university email account in 2002, and it was displayed for just a few seconds before being deleted without explanation. Subsequent physics discussion attempts with arXiv’s string proponent Dr Distler of University of Texas (on his blog hosted by his university department) showed these guys are elistist who try to use back-of-the-envelope “fake no go theorems” against alternative ideas to mainstream, groupthink string nonsense, while not of course doing that with string theory. Pathetic! Notice this is about hypocrisy, since string “theory” stuff on arXiv is crap. I’m not saying Dr Lisi’s new paper it substantial: it sure looks like empty E8 propaganda. But if you have developed a dry sense of humor to survive the bigots of this world, it sure is a funny situation.
E.g., Dr Lisi has to be pressurised by David Roberts into accepting the wild idea of putting his own 5-weeks-arXiv-delayed preprint paper on his own website after mentioning it on Dr Woit’s blog comments, and at first writes: “The arXiv is where it belongs, and the most reasonable place people would see it and possibly cite it. If they reject it, which I see no good reason for, maybe I’ll retract all my papers in disgust.” Yeah, right. I feel your pain.
Quantum field theory 
