Woit’s Euclidean Twistor Unification slides at the Algebra, Particles and Quantum Theory Seminar, Feb. 14, 2022

Highlights of Woit’s Twistor Unification Theory

Over the past few years, Woit has been developing a twistor unification framework for the basic ideas he first published back in 1988, which I found interesting on page 23 of a 2011 paper: https://vixra.org/abs/1111.0111 – “For right-handed particles, Woit(30) shows how the most trivial possible Clifford algebra representation of U(2) spinors in Euclidean spacetime yields the chiral electroweak isospin and hypercharge law. This proves the claim that left-handed helicity electrons have a hypercharge of -1, and right-handed electrons have a hypercharge of -2 because, in a left-handed electron, half of the hypercharge field energy appears as weak isospin charge, which doesn’t happen in right-handed electrons. This is because left-handed electrons are not a mirror-image of right-handed electrons; there is no such thing as parity.”

In other words, already in 1988 Woit had shown how to get the observed left handedness of the weak nuclear force modelled by using a Clifford algebra U(2) in Euclidean spacetime, not Minkowski spacetime, and since Feynman’s discovery of the path integral which is Euclidean (not Minkowski), you’d think this would gain attention. Nope, because of string theory hype (political groupthink in modern physics). Anyway, now Woit has expanded his 1988 sketchy hypothesis on to broader canvas in the manner of Titian:

“One can do four-dimensional complex geometry by identifying C^4 with 2×2 complex matrices
(z0, z1, z2, z3) ↔ z = z01 − i(z1σ1 + z2σ2 + z3σ3)
and defining
|z|2 = det z
Pairs g_L, g_R ∈ SL(2, C) × SL(2, C) = Spin(4, C) act preserving |z| by
z → g_L z [g^−1]_R
We are interested in “real forms” of this (real 4d subspaces that give above after complexification).

Three real forms are:
(2, 2) signature inner product: Spin(2, 2) = SL(2, R) × SL(2, R),
using g_L, g_R ∈ SL(2, R).
(3, 1) signature inner product: Spin(3, 1) = SL(2, C), using
g_R = (g†_L)^−1
This is Minkowski space-time.
(4, 0) signature inner product: Spin(4, 0) = SU(2) × SU(2), using
gL, gR ∈ SU(2).
This is Euclidean space-time.
Our interest will be in the Minkowski and Euclidean cases, together with the analytic continuation relating them.

In Euclidean signature, can use quaternions instead of complex matrices
(x0, x1, x2, x3) ↔ x = x01 + x1i + x2j + x3k …

Going back from number theory to physics, the philosophy we will pursue is that fundamental theory should be defined in Euclidean signature space-time, our observed physical space time is an analytic continuation. On reason is that QFT has inherent definitional problems in Minkowski signature that don’t occur in Euclidean signature. …

There have been attempts to unify the weak interactions with gravity, using the chiral decomposition of the spin connection as above, with SU(2)_R a space-time symmetry giving a gravity theory, and SU(2)_L the internal symmetry of a Yang-Mills theory of the weak interactions. Our proposal is of this nature, but with the following different features:

Take the Euclidean signature QFT theory as fundamental, with Minkowski signature physics to be found later by analytic continuation. Note that in Euclidean QFT one component of the vierbein is
distinguished (the imaginary time direction). Use twistor geometry to get not just an SU(2)_L internal symmetry but the full electroweak SU(2)_L × U(1) electroweak internal symmetry, with the imaginary time component of the vierbein behaving like a Higgs field.

If one works on the projective twistor space PT, one can get the idea of gravi-weak unification to work (in its Euclidean form):

  • There is not just an SU(2) internal symmetry, but also a U(1), given by the complex structure specified by the point in the fiber. This complex structure picks out a U(2) ⊂ SO(4), the complex structure preserving orthogonal transformations of the tangent space to the point on the base S^4. This is the electroweak U(2) symmetry, to be gauged to get the standard electroweak gauge theory.
  • If one lifts the choice of vector in the imaginary time direction up to PT, it transforms like the Higgs field: it is a vector in C^2 (using the complex structure on the tangent space given by the point in the fiber). The U(2) act on this C^2 in the usual way. Each choice of Higgs field breaks the U(2) down to a U(1) subgroup, which will be the unbroken gauge symmetry of electromagnetism. [Emphasis added; Woit then goes on to SU(4) which acts on C^4, getting U(3), which includes an SU(3) and a U(1).]

A generation of SM [Standard Model of particle physics] matter fields has exactly the transformation properties under the SM gauge groups as maps from C^4 to itself. … In this proposal, there’s a profound reorganization of fundamental degrees of freedom. They now live on points of PT which one can think of as light-rays, rather than on points of space-time. Mathematically, one needs to find a formalism on PT that corresponds to the usual Yang-Mills formalism on the base S^4. Need to use holomorphicity on the CP^1 fibers to match degrees of freedom on S^4 and on PT. The Penrose-Ward correspondence does this for anti-self-dual connections. Similarly need to match the Dirac equation on S^4 and equations on PT. For bundles on the base with ASD connections, this is done by the Penrose-Ward correspondence, but the U(1) and SU(3) bundles are on PT, vary on a fiber. Have mostly just rewritten the usual electroweak and GR theory. One difference though is that one component of the vierbein is now the Higgs, which has the electroweak dynamics.”

  • Spinors are tautological objects (a point in space-time is a space of Weyl spinors), rather than complicated objects that must be separately introduced in the usual geometrical formalism.
  • Analytic continuation between Minkowski and Euclidean space-time can be naturally performed in twistor geometry.
  • Exactly the internal symmetries of the Standard Model occur.
  • The intricate transformation properties of a generation of Standard Model fermions correspond to a simple construction.
  • One gets a new chiral formulation of gravity, unified with the SM.
  • Conformal symmetry is built into the picture in a fundamental way.
  • Points in space time are described by the p = ∞ analog of the Fargues-Fontaine description of the “points” p of number theory – from Woit’s February 2022, “Euclidean Twistor Unification and the Twistor P^1”, Algebra, Particles and Quantum Theory Seminar, Feb. 14, 2022 slides, see also his blog post.

I’d like to again point out – see note 30 on page 63 of the 2011 paper, vixra.org/abs/1111.0111 – that Woit shows “that the most trivial possible Clifford algebra representation of U(2) spinors in Euclidean spacetime produces this chiral electroweak isospin and hypercharge association: P. Woit, “Supersymmetric quantum mechanics, spinors and the standard model,” Nuclear Physics, v. B303 (1988), pp. 329-42.”

The problem is that once “anomalies” in the Standard Model became encoded into a hardened empirical mixing angle and parameter set orthodoxy, theories like string set out to reproduce them (along with a landscape of 10^500 or more other variants), rather than to really challenge them, let alone explain them. Whenever a difficulty arises, you also get two psychological reactions from any groupthink infrastructure:

  1. Glass is half empty: give up the climb, accept the existing dogma as being the pinnacle, and kick away constructive criticisms into the long grass as being crackpot or outsider nonsense.
  2. Glass is half full: keep doing the same general thing (not a really radical change in direction).

What you don’t get from groupthink is radically objective but sensible, non-paranoid activity. This didn’t matter that much before say 1945, since there was no groupthink “big science” with the smell of money to corrupt mainstreams into a transistor style logic gate switch-over mentality, whereby all media ignore new ideas until they generate enough current to break down a barrier and activate a switch. In groupthink, “noise” is suppressed by a squelch circuit until the signal strength exceeds a threshold, which allows mainstream publicity. This slows down the emergence of new ideas into the mainstream, while accelerating the development of mainstream ideas. The older non-groupthink model of science was the opposite: far more “noise” (radical nascent ideas) was publishable, which speeded up the emergence of new ideas, but this came at the price of slowing down the development of mainstream ideas which have already emerged (there was little funding for them, as most of the money was spent generating “noise”, the radical nascent ideas). The more money you pump into something, the more red-tape groupthink.

A problem for people like Woit is that the sort of people interested in radical objective physics models today are not always the Directors of Stringy Groupthink Corporation, but outsiders.

We became interested in Woit’s work after he started blogging in 2004. We wanted a critical revision of the basis of the standard model of particle physics, rather than the usual stringy notion that what everybody needs is to assume as the 100% true Holy Grail the ad hoc parameter filled, standard model and existing quantum gravity speculation (i.e. all anomalies are assumed to be fundamental facets of nature, not merely indications of mathematical approximations that are incomplete or a boostrap fix), and also a subset of some bigger, more complex superstring framework in 10/11 dimensions.

For various reasons, e.g. family health issues, and also the groupthink demotivation of mainstream hype, we haven’t published physics papers on vixra since Massless Electroweak Field Propagator Predicts Mass Gap https://vixra.org/abs/1408.0151 which uses the Euclidean Laplace transform in 3-dimensions to understand masses, as opposed to the usual obfuscating Fourier transform. Don’t get us wrong: we don’t object to any particular mathematical tool in general, just the specific problem that trying to use a bulldozer to recover a china cup is not always the best selection of tool available. Sometimes, despite hype from funding obsessed fools, cheap, simpler tools have their uses and are most appropriate.

By obfuscating, we mean that while Fourier transforms have their uses (they create frequency spectra from waveforms, etc) but their problem of poles in complex space (integrations around the origin on Argand diagrams) means that nobody else notices that particle masses result from integrating cut-offs.

Another mathematician, besides Woit, has recently been coming up against the mainstream groupthink problem: see Professor Robert Arnott Wilson’s comments on being censored by arXiv for heretical papers submitted to maths journals. His London uni page is here. A pre-print of his January 2022 submission to the International Journal of Geometric Methods in Modern Physics is uploaded here, “Remarks on the group-Theoretical Foundations of Particle Physics” whose abstract states: “I propose using the group SL(4, R) as a generalisation of the Dirac group SL(2, C) used in quantum mechanics, in an attempt to match the symmetries of the ‘internal’ spacetime of elementary particles to the symmetries of the ‘external’ spacetime of general relativity”. On his blog, he writes: “the arXiv refused to post it”.

The problem is, this censorship makes the mainstream appear as it is: corrupt. It all recalls old 1930s poems about the degenerative censorship methods used against the various group of critics and outsiders, shutting down controversies about the beloved Leader. “First they came for those who objected to the Leader’s theory, then they came for those with alternative suggestions, then they rounded up and deported those all who reported what was really going on, then in the end they became so paranoid, deluded and contemptuous of objective, useful criticisms, and so submerged in their own one-sided hype and glory, that they lost sight of realistic objectives and methods, and lost their costly war.”

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