Axiomatic basis for quantum field theory: Part 1 (how 1st quantization went wrong)

The hard facts of how quantum field theory (2nd quantization) differs physically from populist 1st quantization, aka non-relativistic "quantum mechanics" with its single wavefunction fallacy

Hermann Weyl in 1918 quantized the gravity metric as a function of electromagnetism,

g_µν ’ = g_µν exp[∫{iq/(ℏc)} ∇✕A_µ dS] (From: https://vixra.org/pdf/1111.0111v1.pdf p7).

This is false (it was debunked by Einstein), but the key mathematical idea of Weyl, that you can quantize something, “s” (not necessarily “action”, S), by using a complex exponent, exp(is), survived when four years later, in 1922, Schroedinger applied it to “explain” quantized Bohr orbits, assuming that the real solutions are the discrete observed electron states, while the imaginary (complex) solutions are unobservable.

Hence, Schroedinger has his orbital electron wavefunction ψ ~ exp(iEt/ℏ), based on the same basic mathematical idea of Weyl’s fake quantized gravity; continuously variable solutions are in complex space and unobservable, while discrete (quantum) solutions are in real space. Now what I want to point out here immediately is that really Feynman (in his 1985 David Bohm-influenced book, QED, not his fake news earlier stuff) shows that the truth is that you have path integrals, with multiple real ψ ~ cos (Et/ℏ), a completely real wavefunction, one for each “virtual” path, and the multipath interference (not complex space) provides a physical mechanism for quantization.

Let’s start with ψ_x,t = ψ_x exp(iEt/ℏ), [Equation 1]

the solution to the differential Schroedinger equation:

idψ_x,t /dt = -{ℏ/(2m)}d^2 ψ_x,t /dt^2 [Equation 2]

To get the usual time-dependent form of Schroedinger’s equation,

iℏdψ_x,t /dt = Eψ_x,t, [Equation 3]

you put ψ_x = sin (kx) into Equation 1, giving:

ψ_x,t = sin (kx) exp(iEt/ℏ) [Equation 4]

notice the use of sin, rather than cos, which Feynman points out is appropriate for real solutions in path integrals, since exp(iS) = i sin S + cos S. Now differentiate:

iℏdψ_x,t /dt

= iℏd [ sin (kx) exp(iEt/ℏ) ] /dt

= iEℏ (-iE/ℏ) sin (kx) exp(iEt/ℏ)

= E sin (kx) exp(iEt/ℏ), which is simply E multiplied by Equation 4, so :

iℏdψ_x,t /dt = E ψ_x,t [Equation 5],

Schroedinger’s time-dependent equation!

Similarly,

dψ/dt = [sin (kx) exp(iEt/ℏ)]/dx

= k.cos(kx) exp(iEt/ℏ), so that

dψ^2 /dt^2 = -(k^2) sin(kx) exp(iEt/ℏ),

leading to the standard result E = (ℏk)^2 /(2m).

Now, consider the question of whether the “real” wavefunction of the time-independent Schroedinger type, dψ^2 /dt^2 = -(k^2)ψ has the real solution of sin(kx) or cos(kx). In fact, the rigorous solution to dψ^2 /dx^2 = -(k^2)ψ is actually a combination of both:

ψ_x = A sin(kx) + B cos (kx),

thus the answer is: “whatever works”, because you can set A = 0, or B = 0. QED. So there is nothing complex or magical here.

There is however a bit of mathematical magic in the way the Schroedinger equation is solved for the hydrogen atom, e.g. see pages 146-157 of David J. Griffiths Introduction to Quantum Mechanics, 2nd ed., which also has an amusing Preface, stating:

“… We encounter Legendre, Hermite, and Laguerre polynomials, spherical harmonics, Bessel, Neumann, and Hankel functions, Airy functions, and even the Riemann zeta function - not to mention Fourier transforms, Hilbert spaces, hermitian operators, Clebsch-Gordan coefficients, and Lagrange multipliers. Is all this baggage really necessary? Perhaps not, but physics is like carpentry: Using the right tool makes the job easier … teaching quantum mechanics without the appropriate mathematical equipment is like asking the student to dug a foundation with a screwdriver.”

The problem is, the heavy duty mathematical machinery covers up the dynamics when building theories, like using a bulldozer to “fix” a faulty circuit board. It sure as hell gets rid of the problem of understanding what is really wrong (never mind avoiding the work of repairing the errors).

But surely, you ask, 1st quantization is orthodoxy because Born discovered you square the wavefunction to get probability or particle (aka energy) density! Nope, in classical electromagnetic wave theory you get electric field oscillating as sin(kx), and you square this electric field strength (field amplitude!) to get its energy density, exactly as you do in QM! So Born’s allegedly “wonderful insight” is just classical physics! Duh:

dψ^2 /dx^2 = -(k^2)ψ of course (if negatives are squared) also has the complex solution:

ψ = c.exp(±ikx),

which differs from the Weyl/Schroedinger solution in that you have two wavefunctions, not one, due to the “±” in the exponent!

The point I’m making is this: you can pick out mathematical models to fit the facts, while ignoring physical dynamics, but you can’t do it rigorously. All such equations are ad hoc approximations or guesses. Only by looking for a consistent dynamics or mechanism, can you make real progress in this subject now. The mathematics alone can lead you in all kinds of directions even at the foundations, so you must be guided by experimental data, and you can’t make progress by rigorous mathematical axiomatic derivations without supplementing your equations using some additional dynamics to understand what you’re actually doing. Otherwise, you get BS, not physics (“the basis of QFT is annihilation-creation operators acting on Hilbert space, blah, blah, blah”). I’m arguing that we need to tell the pure mathematicians to look at the physical phenomena to make rapid progress in the axiomatic basis of QM, to reconcile it properly with QFT. (See p25 of https://vixra.org/pdf/1111.0111v1.pdf for discussion!)

Next: https://vixra.org/pdf/1111.0111v1.pdf at page 27 and onwards. Basically, fermions are radiating virtual particles at the rate given by Hawking’s black hole law, a tremendous number, and this provides the physical basis for quantum field theory, with most exchanges “cancelling out” in an equilibrium, and only the effects of accelerations being interpreted as “real photons”! And no, I don’t “like” this particularly, it’s just the theory which happens to predict the facts correctly and is checkable, unlike more “attractive” superstring BS which doesn’t work - except in Parkinson’s Law and Pournelle’s Iron Law of Bureaucracy!

Other problems with 1st quantization (non-relativistic quantum mechanics):

(1) the failure to “move on” from single-wavefunction indeterminancy BS to multi-wavefunction interference (path integrals) leads to a whole load of single-wavefunction interpretative BS from qubits in “quantum computing” to “entanglement” science fantasy. Shame on all the people who don’t point out the error!

(2) the successes really are in path integrals, i.e. relativistic 2nd quantiation (not non-relatistic QM), e.g. the perturbative corrections are where you get the highly accurate predictions of the Lamb shift, magnetic moments of leptons, etc., are all from Feynman’s perturbative expansion to the path integral! I.e., multipath interference, not single wavefunction BS. However, liars claim QM is justified by the success of QFT, without pointing out that these are totally different theories, totally incompatible unless you talk BS.

(3) QM’s key failure is clear when you see it relies on a classical Coulomb potential, making no allowance for the random field quanta exchanges that cause indeterminancy in QFT. The reason this delusion persists is entirely mathematical, not physical: it’s easy to use the delusional Schroedinger equation with the fake news classical Coulomb potential to get good approximations to single-wavefunction amplitudes in QM, including angular momentum energy and then finding a solution using Laguerre polynomials to get the electron’s supposed single wavefunction amplitude (a fiction) versus distance from the nucleus, you then square that to find relative probability. The problem is that these calculations cover up the failure implicit in the model, just like the “successes” of trigonometry were used to cover up errors of Ptolemy’s earth-centred epicycles universe!

In other words, you cook up an equation that models some stuff approximately, and you say, that mathematics proves those equations must be defensible physics! Only it doesn’t work, when you need a completely different model (e.g., QFT path integrals in our example, or Kepler’s ellipses in the solar system alternative to Ptolemy) to make more accurate predictions!

I remember a long argument with string theorist Lubos Motl about this on his blog (before he shut it down): he argued that Ptolemy’s model must be a valid rough approximation to Kepler’s elliptical orbits in order for it to have made any kind of accurate predictions at all. I pointed out no, the distance from earth to moon (and other planet’s distance) was wrong in Ptolemy’s model, which only approximately estimates the latitude and longitude of moon and stars, not their correct spatial location (which includes their distance from earth or sun!). He then deleted the comments when he realised he was wrong. (So it’s a waste of time arguing with such people, even if they do have “comments sections”, they’re delusional and fraudulent.)

In another argument, he promoted some arxiv QFT teaching notes from string theorists (employed as teachers) that falsely claimed the running coupling is a continuously variable logarithmic function of energy. This is usually what is depicted on graphs of “screened coupling unification” and is clearly not true, because the various virtual pairs all have threshold energies (the lowest being the IR cutof at 1.022 Mev for virtual electron + positron, then 211.4 Mev for muon + antimuon, etc, etc.), each one causing a discrete “step”, not a smooth curve in a plot of screened coupling versus energy. Motl then emailed the authors to change it, and they simply rewrote the equation as a summation for all pairs, with a footnote thanking Motl for pointing out a “heuristic error” in their previous version of the QFT lecture notes!

The problem here is, it’s not just a “heuristic error”, it’s fundamentally insane, and still today such people still don’t take this physical mechanism of vacuum polarization - the key discovery of QFT - seriously, e.g. the IR cutoff is ignored, and the UV cutoff is falsely taken to Planck’s numerological (dimensional analysis) energy, not the far higher, more fundamental and physically defensible, black hole scale energy - which really is proved to unify physics (see p32 of https://vixra.org/pdf/1111.0111v1.pdf ). They see the QFT facts as a mathematical game, and they see QM mathematical games as the facts. Pathetic and deluded! But how do you get a revolution?

ENDNOTE

The post above demonstrates some of the issues with technical arguments. How many people bother to read them? If you avoid them, bigots posing as “critics” can complain you’re not making a technical argument, and if you go technical, they can simply ignore it because everyone else in the public ignores it (so it doesn’t get any “oxygen of publicity” either way!). I’ve also revised the introduction to the nuclear effects and credible deterrence blog www.nukegate.org but again, the whole basis for the bigoted disinformation delivering BBC, Wikipedia, mass media, and other allegedly “fact checking” dictatorship is not fact checking at all, but Orwellian doublethink-style “conformity checking”, i.e. those who repeat lies are deemed “correct”, but those debunking lies and presenting the facts to replace them, are condemned by the liars and loons. This is why we don’t use tactical nuclear deterrence to end war since 1992, etc. I won’t go on any more here, but I think you get the drift. Fascism isn’t pointing out the truth, but is suppressing the truth using bigotry. That “subtle difference” is being papered-over.

PART 2 PREVIEW (coming next in a separate post):

http://www.quantumfieldtheory.org/ and nigecook.substack.com/p/coming-soon gives pointers of what’s coming. Basically, QFT got off to a bad start by being based by Dirac on simply changing the Hamiltonian energy operator into a relativistic “spinor” matrix while preserving the single wavefunction Schroedinger equation in complex space, ψ ~ exp(iHt/ℏ). (We’ll omit Dirac’s “bra and ket” notation and other maths cake icing since we’re interested in dynamics, not obfuscation of dynamics.) Since the field is contained in the energy Hamiltonian H, you can vaguely say that “Dirac quantized the field”, thus “quantum field theory”, and of course you can say he predicted antimatter in his spinor (although he had problems with mass, so there are various interpretations such as Dirac spinor, Weyl spinor, Majorana spinor, etc.

You also have the problem that Feynman identified, namely that the various possible discrete interactions of your fermion with its bosonic “quantized field” lead to a series of different ways a charge can interact with field quanta, so you need a Feynman “path integral” which Feynman changes (following 1970s discussions with David Bohm) to a non-calculus based, discrete, real space summation of simple phase amplitudes ∑ψ = ∑cos (S/ℏ) in his 1985 book QED, contrasted to his earlier complex space path integral, ∑ψ = ∫ exp(iS/ℏ) Dx, where the Hamiltonian energy H = ∑​pq ​− L, in action S = ∫​​Ldt.

Basically, as Feynman demonstrates in his 1985 book QED (widely ignored/gaslighted), you have to simplify the maths a trifle to understand the connection to the physics (anathema to Woit). I’m saying that just because of Euler’s exp(iS/ℏ) = i sin(S/ℏ) + cos(S/ℏ), you can replace exp(iS/ℏ) with its real-only component cos(S/ℏ) and ignore the complex component i sin(S/ℏ) simply because the role of the latter is hidden in complex space and gives you the complex vector direction, which is also along the real axis for the “path of least action” of the real world.

What I’m not saying is that complex analysis is rubbish, because you might want exp(iS/ℏ) because it provides more information for some purposes, e.g. a photon of light needs more than one number to describe it: it has polarization angle for its transverse oscillations, as well as frequency, which determines its “amplitude” and energy, and is a distinct property from the angle of polarization. Thus, cos(S/ℏ) does not contain all the information needed to describe a photon. However, you don’t necessarily need to describe the polarization angle and the amplitude using complex analysis, it’s one neat and useful tool, not proof photons really contain complex space!

The point again: getting bogged down in mathematical tools distracts attention from the physical dynamics which are needed to guide the theory accurately. There’s an analogy in electromagnetic theory, which went off into fairyland with Maxwell’s equations: the equations prevented dynamical theory development, so classical theory came to a dead end, instead of developing into quantum field theory. For details, see pages 20-29 of https://vixra.org/pdf/1111.0111v1.pdf for details; Maxwell’s actual “theory” leads to an SU(2) gauge theory not to the usual U(1) quantum electrodynamics which is due to a false assumption that massless gauge bosons don’t propagate, which is only true for the limited case of one-way propagation, not for the reality of exchange, as demonstrated by the reality of the charged capacitor with vacuum dielectric in Fig 21 of that paper. Basically, the folks who developed QFT were ignorant and took short cuts which then became hardened dogmatic orthodoxy. Nobody reviewed/corrected these early errors in QFT, so it’s now a quicksand foundation for a massive tower and changing the foundations are now taboo.

But there’s another huge error about “unification” dogma (discussed particularly in Fig 23, p30 of https://vixra.org/pdf/1111.0111v1.pdf and also the mechanism summary on p2 of https://vixra.org/pdf/1408.0151v1.pdf), whose possible solution became apparent to me when pure mathematician Dr Wilson kindly exchanged a few comments with me on his blog last year. This new step, which I’ve been worrying out for 15 years, is outlined at nigecook.substack.com/p/coming-soon: what we need to do is to go back to the Dirac Hamiltonian and unify the various kinds field energy (running couplings and masses, for different charges) at that point, in other words put some dynamics into the hamiltonian energy model, to allow electromagnetic fields to produce (via pair production) self-shielding virtual particles which contribute (1) mass (e.g. most of the mass of hadrons is from the “virtual” field, not the quark “core”; when the “virtual fermion pair” absorbs any energy in being polarized by the field it is created in, it’s lifespan increases beyond Heisenberg’s t ~ ℏ /E so it’s “less virtual” and has some real effect (maybe becoming structured like nucleon shells, by Pauli’s exclusion principle, thus contributing “quantized mass” as explained in https://vixra.org/pdf/1408.0151v1.pdf which gives a pretty clear picture of what’s going on), and (2) the “virtual particles” will include weak field bosons and (at very high energy or short ranges) strongly interacting particles, so that there is a conversion of some of the energy of the electromagnetic field into short-ranged fields due to this detailed dynamics of vacuum polarization. This is all being ignored by QFT guys like Woit, who instead head off on “more exciting” high-tech mathematical jaunts.

What I’m arguing specifically is, if the standard IR cutoff limit (low energy, screened, “Coulomb law”) electromagnetic coupling is α and the true (bare core) coupling is 1, then the total electromagnetic charge energy converted into short-range effects (mass and short-range nuclear fields, strong and weak) is simply 1 - α. So you can use “renormalization” to predict mass and other couplings, and also the interaction “mixing angles”, e.g. amplitudes for different decay routes (flavour mixing) as explained in fig 35 on p 45 of https://vixra.org/pdf/1111.0111v1.pdf).

So, the electric field energy per unit volume of an electron’s unrenormalized field (i.e. ignoring the key vacuum polarization shielding effect) is (1/2)εE^2 J/m^3, where E is electric field strength, and there’s also a similar magnetic field energy density of B^2/(2μ) J/m^3. In classical textbooks, you integrate these energy densities over radial distance from the classical electron radius to infinity. WTF isn’t this done on p1 of QFT textbooks, using the correctly renormalized running coupling for the electromagnetic field vacuum polarization shielding? Maths too hard? (More likely, math too easy for intellectually groupthink orientated “elitist snobs” who don’t like equations that yield checkable results.) But seriously, not everyone in QFT is in an ivory tower. Surely someone will someday do this. The only reason why this is not done is a taboo: physics has gone off into superstringy BS.

When I started in this, 40 years ago, I thought that the subject was corrupt at the top (superstring), but hoped it would be less corrupt at the lower levels. Not so! The whole filtering system of the educational establishment excludes anyone who is a realist (yours truly) and attracts BS lovers. So, sadly, I’m not likely to get any sensible collaboration, and so will have to fix all the flaws in the subject myself (not because I want to, because it’s the only way forward - having had childhood speech/hearing problems, I have something to prove here, just to get any sort of self confidence against groupthink bigotry!).

Unfortunately, like everybody else, my time for this is limited, which probably affects speed/quality and the need to be openly bitter in every other sentence. Maybe when the draft manuscript is send to Nika Talbot for editing, she will be able to delete some of my bitter comments, if/where necessary. (There’s a problem there, however, in that “speech filtering” - which I’ve also hated because it’s basically biased talk or lying that leaves you with a false impression - is going out of fashion now Trump is in office. I’ve always hated arrogant “personalities” like Trump, but he’s an exception because - along with making endless errors - he’s kicking bigoted self-righteous self-serving complacent groupthink mass media/fashion enforced orthodoxy blob in the teeth, and it’s about time someone did. So the question is, will groupthink physics crap someday fall like the liberal elite in politics? Clearly, the faster we can make progress, the faster we can get into a position to fight corrupt science effectively.)

Note that the paper https://vixra.org/pdf/1111.0111v1.pdf linked above is a nascent exploratory research paper (there are others, more recent at https://vixra.org/author/nigel_b_cook, but none are intended as a textbook; the plan outline for the latter is Quantum Gravity Demystified: nigecook.substack.com/p/coming-soon One big problem with this is presentation style. I’ve recently had a discussion by email with Joseph Friedlander (who kindly wrote about some of the www.nukegate.org information debunking deterrence propaganda years ago), about the problems of presentation. I’ll write about that someplace else, but my view is that this is not the usual “storm in a teacup debate”, but more like a revolutionary war…

(BTW, I recommend taking a look at Dr David Tong’s Cambridge uni physics stuff. His lecture on the Stark effect - the electric field-induced spectral line-splitting splitting analogue to the Zeeman effect for atoms in magnetic fields - https://www.damtp.cam.ac.uk/user/tong/aqm/aqmeight.pdf on pp222-3 states:

“8.1.3 A Little Nazi-Physics History. The Stark effect was discovered by Johannes Stark in 1913. For this he was awarded the 1922 Nobel prize. Stark was a deeply unpleasant man. He was an early adopter of the Nazi agenda and a leading light in the Deutsche Physik movement of the early 1930s whose primary goal was to discredit the Juedische Physik of Einstein’s relativity. Stark’s motivation was to win approval from the party and become the Fuehrer of German physics. Stark’s plans backfired when he tangled with Heisenberg who had the temerity to explain that, regardless of its origin, relativity was still correct. In retaliation, Stark branded Heisenberg a “white Jew” and had him investigated by the SS. Things came to a head when – and I’m not making this up – Heisenberg’s mum called Himmler’s mum and asked the Nazi party to leave her poor boy alone. Apparently the Nazi’s realised that they were better off with Heisenberg’s genius than Stark’s bitterness, and House Stark fell from grace.”

The point of this quotation is: politics has always played a role in physics, despite the facts being edited out of most textbooks. The politics has always been about no-platforming, ridicule, launching hate attacks on the proponents of rival theories, censorship, and fascism.)