Here’s the current solution to the old problem of whether
Haag’s theorem prevents axiomatic proof of the self-consistency of the
(essential) running charge cut-off (charge renormalization) in quantum field
theory:
Enough
about renormalization, please. At this level of arguing about whether
perturbative divergences in general are
1.
A serious problem indicating the theory is ill-defined
2. Things that can
be eliminated with mathematical machine X
3. Not there if you use the
renormalization group properly
the
discussion is stuck in a 40-50 year old time warp. I don’t think that
endlessly repeating geriatric arguments is fruitful. – Peter Woit
The argument Dr Woit was responding to was between Dr Chris Oakley and Dr
Igor Khavkine. The successive terms in a path
integral’s perturbative expansion each represent the magnitude of the
contribution from a successively more complex Feynman diagram, which
pictorially describes interactions between off-shell (virtual)
particles. Virtual fermions are polarized around a real charge,
absorbing energy from the field and reducing (shielding) the charge as seen
from a greater distance (i.e. a distance beyond the location of the
polarized pairs of virtual fermions, which extend out to the low-energy or IR
cutoff, the limit for spontaneous pair production in the vacuum given by
Schwinger).
There is a groupthink denial about the details of this physical
mechanism and the mathematics of renormalization procedures. The fashion
is wooden mathematics. Weyl in 1918 gave a flawed quantum gravity
gauge quantization by trying to quantize the metric of general
relativity, scaling it by a complex exponential function of the
electromagnetic field S using exp(iS). After Einstein pointed
out it was wrong, Schroedinger in 1922 modified Weyl’s idea into a new
mathematical “eigenvalue” model of the Bohr atom, changing the scaling from
the metric to the probability of the existence of a discrete energy level
existing as function of the electron’s orbital path, the periodic real
plane solutions to exp(iS) = cos S + i sin S represented the eigenvalues
for “stationary states” of orbital electrons. Finally, after de
Broglie’s particle-wave duality became fashionable, Schroedinger published the
famous complex plane time-dependent wave equation to which exp(iS)
is a solution.
In quantum field theory, as Dirac developed it, Schroedinger’s
time-dependent wave equation is supplied with a new Hamiltonian (Dirac’s
spinor) to make it treat space and time the same way, to meet relativistic
requirements. The new Hamiltonian, however, quantizes the field.
Instead of just having one one particle interacting and behaving unpredictably
with no mechanism (which is what the single wavefunction model in
Schroedinger’s equation or Heisenberg’s matrix says), in quantum field theory
you suddenly have a mechanism: lots of virtual particles deflecting an
electron whose path action is small compared to h bar. Each of
interactions between a virtual particle in the field and the electron has an
aplitude and thus a wavefunction. The 2nd quantization (QFT) path
integral in quantum field theory, as Feynman points out in his book QED
(1985) is now a physical sum of physical processes, so the 1st quantization
(non-relativistic QM) “uncertainty principle” is “not needed (Feynman).
Uncertainty is now not a metaphysical law from the mind of Heisenberg, it’s
good old “multipath interference”, exactly the effect that causes radio
interference.
So why exp(iS)? If we have the mechanism of multiple path
interference determining eigenvalues in 2nd quantization, why
use Schroedinger’s purely ad hoc complex wave equation, whose
complex Hilbert space defies a self-consistent axiomatic proof of
renormalization (Haag’s theorem)? Why not accept that exp(iS) and the complex
wave equation is a historical vestige? What do must replace it with is
real space: exp(iS) can be replaced simply with cos S, as Feynman demonstrates
graphically in his 1985 book, QED. Thinking physically (without the
wooden fuzziness of “believing” in ad hoc mathematical models as a religious
belief), you can see that the path integral is always giving a real plane
solution: the only variable is the amplitude not the direction of the arrow on
an Argand diagram. A path integral can either add up unit length arrows
with variable directions which the mainstream method today, using exp(iS), or
you can get precisely the same result by making the arrows all point in the
same direction (the real axis) but have varying lengths. The path
integral is always the same so far as observation is concerned: nobody can see
any non-real plane final arrows in the laboratory. Inteferences only
affect amplitudes on the real plane so far as we observe them. The
cross-sections and probabilities you get from the path integral are always
real numbers, never containing i. If that is true, exp(iS) = i sin S +
cos S can be replaced by dropping to i sin S to give cos S. This should
have been done by Dirac and Feynman when 2nd quantization was developed.
Instead, Hilbert space – despite Haag’s theorem – is a religion in quantum
field theory.
Update (15 february 2011): relevant extracts from an email on this subject
to Dr Mario Rabinowitz
From: Nige Cook
To: Mario Rabinowitz
Sent: Wednesday, February 15, 2012
5:15 PM
… “As you may recall I think that existing QM and GR are
presently mutually incompatible, being a deterrent to a consistent theory of
QG.”
Years ago, I read your excellent paper, “Deterrents to a theory of quantum
gravity”, which is very helpful and provides some vital insights. Your
approach defines QM by the mainstream Schroedinger 1926 equation of 1st
quantization:
i * {h-bar} * d{Psi}/dt = H *{Psi}
Any equation of this form (where the rate of charge of a variable, Psi is
directly proportional to Psi) will have an exponential solution, i.e.
{Psi}_t = {Psi}_0 exp(iHt).
This is what Dirac came up with in 1933. I’m sure you’re well aware of the
mathematics, but maybe the history and the physical interpretation are less
familiar:
1. Weyl came up with the complex exponent, exp(iX), in 1918 as a multiplying
factor for the metric of general relativity. This quantized the metric, the
original gauge theory of quantum gravity (references are in my paper). Weyl’s
factor X was a function of the electromagnetic field, so he claimed to unify
electromagnetism and gravity in his theory. Einstein pointed out that Weyl’s
1918 theory contradicted observed data (e.g. line spectra from stars with strong
gravitational fields).
2. In 1922, Schroedinger reapplied Weyl’s exp(iX) factor to model the
quantized electron energy levels in the atom (the
references are in my paper): exp(iX) is a cyclic function on an Argand
diagram (complex plane). Schroedinger’s 1922 paper defined the periodic real
plane of exp(iX) = i sin X + cos X as the observed electron states corresponding
to line spectra, so that exp(iX) was unity (probability of finding the electron
= 1) for “real” (observable) electron states.
This was a brilliant application of mathematical intuition to “explain” why
lines are quantized: the electron is in some sense in a complex plane
(unobservable) when inbetween discrete energy levels.
3. In 1926, after being asked to give a lecture on de Broglie’s wave particle
duality, Schroedinger presented his famous reverse-engineered wave equation, to
which his 1922 paper’s probability = exp(iX) is the solution. (Feynman claimed
in his Lectures on Physics that the wave-equation was a guess which came out of
the “mind of Schroedinger”. It actually came out of the mind of Weyl’s gauge
theory in 1918, but was changed by Schroedinger from scaling the gravitational
metric to scaling the wavefunction.)
4. In 1929, Dirac had to change the non-relativistic Hamiltonian to an SU(2)
matrix type spinor in order to make the Schroedinger theory relativistic. Dirac
found that this quantizes the field (2nd quantization).
5. in 1933, Dirac suggested following the wavefunction over a path by {Psi}_t
= {Psi}_0 exp(iHt). This is really a circular argument physically, since it is
what Schroedinger did in his 1922 paper.
My argument is that the amplitude exp(iHt) or its equivalent in the path
integral for least action, exp(iS), is only necessary in 1st quantization
quantum mechanics where you have a single wavefunction. In this case, you have
to rely on the complex conjugate to quantize phenomena.
In 2nd quantization, you have more than one wavefunction (the path integral,
one wavefunction for every path). All the many wavefunctions interfere to
produce probabilities. For classical situations (path actions minimal compared
to h-bar), exp(iS) ~ exp(i*0) ~ 1, so the classical path takes is roughly 100%
likely (thus all non-classical paths have trivial contributions).
Mathematically you don’t need the complex
wavefunction amplitude exp(iS) in the path integral (2nd quantization). It’s
just a vestige of 1st quantization. Use Euler’s equation for exp(iS) and drop
the complex terms which have no value: exp (iS) = i sin S + cos S. You can
replace exp(iS) with cos S in the path integral.
Benefits:
1. No more complex Hilbert space, and no more complex Schroedinger wave
equation. No complex space. No problems of Hilbert space in trying to reconcile
quantum mechanics and gravity!
2. No more problems axiomatically in quantum field theory! Haag’s disproof of
self-consistent axioms for renormalization is based on complex (Hilbert) space.
Drop complex (Hilbert) space, and self-consistent renormalization is no longer a
crack to be covered with renormalization group wall-paper.
3. Nobody has ever seen any need for a complex space in 2nd quantization. The
path integral’s outputs are alway real numbers: real plane cross-sections, and
real probabilities. If quantum mechanics is defended on the basis of empiricism
by Bohr and Heisenberg, why include non-observables like complex space, when
they are no longer needed. As Feynman states in his 1985 book QED, in the path
integral probabilities arise from multipath (multiple wavefunction)
interferences, just like the old HF sky-wave radio interference from partial
reflection of radio by several different (D, E, and F) layers in the ionosphere.
There is a physical mechanism in 2nd quantization so you no longer need exp(iS)
which is vital to explain energy level quantization if you only have a single
wavefunction (1st quantization)
Notice that if you replace the path wavefunction (amplitude) Psi = exp(iS)
with Psi = cos S (obviously S is in units of h bar), what you are doing in the
path integral looks a bit different graphically, but is an exact mathematical
duality for all real outputs (probabilities, cross-sections).
On an Argand diagram, using exp(iS) as a path’s amplitude means that to
determine the “sum over histories” (path integral) you add arrows of fixed
length for but variable direction for each path, and the path integral is then
the resultant arrow (the sum over the histories).
This sounds as if it involves a vector result, i.e. generates two variables:
the path integral (final arrow) has both length (amplitude) and direction.
However, although technically “true” in a “wooden” mathematical sense, it is
contrived sophistry in a physical sense, because the direction of the vector is
always zero, i.e. on the real (horizontal) axis. To repeat, the path integral
only produces scalar (not complex vector) probabilities and cross-sections,
since the direction of the final arrow is always real. There are only two axes
on the Argant diagram: real and imaginary (complex). The fact the path integral
is always on the real axis, allows us to replace exp(iS) with cos S (using
Euler’s identity), without loss of information. We’re not physically breaking
any mathematical rules by “replacing a vector with a scalar”. It’s a scalar at
output anyway.
In other words, the only true variable in every
experimentally checked and confirmed path integral is the amplitude of a path
along the real axis, i.e. cos S. So forget exp(iS), it’s unnecessary in 2nd
quantization where we’re calculating real numbers like real probabilities and
real cross-sections.
I’m arguing is that the whole of 1st quantization is rendered obsolete by 2nd
quantization, and by Haag’s theorem to achieve self-consistent axiomatic
renormalization we need to dump the Weyl/Schroedinger/Dirac/Feynman exp(iS)
wavefunction amplitude and move over to using cos S as its replacement.
This ends all the doublethink and mathematical duplicity that have held up
the development of quantum field theory for the past 80 years. Each path now has
no complex vector, just the scalar amplitude cos S. The path integral produces
precisely the same checkable cross-sections and probabilities with cos S as with
exp(iS). However, we are now dealing with real spacetime, not complex space, so
the mathematical barriers to axiomatical progress and unification with gravity
are eliminated.
The drawback is that there is a great deal of “genius” invested in exp(iS)
and the complex Schroedinger equation, and we can expect a great deal of
hostility to progress by replacing exp(iS) with cos S. Mathematical geeks
(Pythagorean cult worshippers) like Ed Witten will not find my humble suggestion
praiseworthy, but destructive to educational syllabuses, existing textbooks, and
the confusion of students. It would make physics less arcane, less mysterious,
less attractive to B grade pure mathematics students. You would get more
technician-calibre* Michael Faraday’s getting into the ivory towers and
upsetting status quo by making discoveries “out of turn”. Physics might start
making some real, revolutionary progress again, like it did in the 1920s.*
Tragic for the old guard.
_____
*Politically incorrect footnote: a tragedy nearly occurred with
Oliver Heaviside, who turned Maxwell’s differential equations into vector
calculus without bringing any kudos to Oxbridge (or any academia), but
fortunately Sir William Preece had Heaviside censored out when Heaviside started
including in published papers sarcastic “bitter” jokes at the expense of
perplexed leading Oxbridge educated academia. (Here’s
a beautiful specific example from a published article of Heaviside, reprinted in
his book Electromagnetic Theory, vol 1, 1893, p337: “Internal obstruction and
superficial construction … If you have got anything new, in substance or in
method, and want to propagate it rapidly, you need not expect anything but
hindrance from the old practitioner – even though he sat at the feet of Faraday.
Beetles could do that. Besides, the old practitioner [any so-called
“professional” scientist in general as well] is apt to measure the value of
science by the number of dollars he thinks it is likely to bring into his
pocket, and if he does not see the dollars, he is very disinclined to disturb
his ancient prejudices. But only give him plenty of rope, and when the new views
have become fashionably current, he may find it worth his while to adopt them,
though, perhaps, in a somewhat sneaky manner [plagiarism], not unmixed with
bluster, and make believe he knew about it when he was a little boy! He sees a
prospect of dollars in the distance, that is the reason. The perfect obstructor
[“peer”-review bias] having failed, try the perfect conductor. … Prof. Tait [the
famed quaternionic hyper] says he cannot understand my vectors, though he can
understand much harder things. But men who have no quaterionic prejudices can
understand them, and do.”) As another example, Dirac studied
electrical engineering at Bristol University (which also taught bricklaying and
shoemaking) before coming up with the Dirac spinor (the foundation of
quantum field theory), but despite his
arguments with Heisenberg over whether 1st quantization QM was a subject
“closed” for all time or not, at least he was politically correct enough to
end up Lucasian Professor of Mathematics at Cambridge from 1932 to 1969. The
brilliant new groupthink ideology is to encourage a diversity of ideas in
physics by eliminating anybody who doesn’t think within the (existing flawed
status quo) box. The elimination technique is based on mathematical sophistry.
If you accept superfluous unobservables and use them to hold back progress, all
is well. 
Further discussion:
From: Nige Cook
To: Mario Rabinowitz
Sent: Wednesday, February 15, 2012
8:28 PM
Subject: Re: I was just skimming your paper, “U(1) ´ SU(2) ´ SU(3)
quantum gravity successes.”
… My approach is that the Schroedinger equation is misleading because it only
has a single wavefunction and was an ad hoc model formulated before the path
integral. People cling on to vestiges long after the reason for them has
disappeared. In 2nd quantization you don’t need exp(iS), because cos S does the
same job, better, avoiding Hilbert space (Haag’s theorem). If you accept the
necessity for a path integral, then each path has a separate wavefunction, and
as Feynman explains in QED (his lucid 1985 book), multipath interference between
many wavefunctions – one for each path – produces all indeterminancy. There is
no intrinsic indeterminancy. All indeterminancy is due to multipath
interference. Keeping 1st quantization vestiges in place after 2nd quantization
had made them unnecessary obfuscations is like Copernicus’s attempt to retain
epicycles in the solar system: it is a half-baked mainstream theory.
Love is an ex-USAF pilot who has a maths PhD and he emailed me a paper called
“Towards an Einsteinian Quantum Theory”, which tries to replace the Standard
Model, however he doesn’t seem to find any problem with U(1) electrodynamics,
just replacing the SU(2) and SU(3) weak and strong gauge group symmetries.
My approach is the opposite. There is an enormous amount of evidence for
SU(2) weak and SU(3) strong symmetries. The problem, I find, is U(1)
electrodynamics which is really a disguised SU(2) Yang-Mills symmetry. You can
see the SU(2) nature of electrodynamics in both Dirac’s SU(2) spinor of
relativistic QED, and in the asymmetry in Maxwell’s vector calculus equations:
div.B = 0 is not matched by div.E = charge density per unit permittivity. It
really seems that magnetic fields are not a U(1) symmetry but an SU(2) symmetry,
deriving from spin. This lack of magnetic monopoles is an asymmetry between
electricity and magnetism, analogous to the left-handed asymmetry (parity
violation) in the weak interaction when electromagnetism is represented by a
massless boson SU(2) symmetry. Weyl actually predicted in 1929 that Dirac’s
spinor (Weyl’s spinor) breaks parity in electromagnetic interactions, although
he didn’t interpret this physically as the lack of magnetic monopoles in
Maxwell’s equations, and Pauli dismissed it. Parity conservation was only
confirmed for weak interactions (beta decay) in the late 1950s, nobody bothered
to see if electromagnetism could be derived from SU(2) Yang-Mills with massless
gauge bosons.
Instead of unifying electromagnetism and weak interactions by
electromagnetism an SU(2) Yang-Mills theory which reduces to an asymmetric U(1)
Maxwell theory due to the massless bosons in electromagnetism (which prevent the
charge transfer term in the Yang-Mills term from operating), the simplistic
mainstream (wooden mathematics) approach has been to “predict (non-observed)
magnetic monopoles”, and despite failing to discover magnetic monopoles in
searches, to continue looking and hyping the “prediction” (analogous to the
politically convenient “search” for cosmic strings). Maxwell’s original 1861
paper, “On Magnetic Lines of Force”, as quoted in my paper, argues that magnetic
fields are just the angular momentum of field quanta spin. Maxwell used vacuum
vortices, not field quanta, which did not arise until QED was developed to the
stage of Moller scattering theory due to virtual photon elechange. The virtual
photons will convey magnetic fields by spin angular momentum.
Update (1 March 2012)
Mathematician Dr Marni Sheppeard has closed her Arcadian
Pseudofunctor blog, a post after commenting (sarcastically) that the
scientific conference disclaimer: “In particular, no bona fide scientist
will be excluded from participation on the grounds of national origin,
nationality, or political considerations unrelated to science” is “cute”.
“Hubris” is perhaps the best word for the censorship of politically incorrect
nascent science by elite greasy pole climbing geniuses who use what they call
“science” to fill their wallets. Great, I say, just be careful to give honest
results in return for your wages. What is wrong is not just groupthink science
or the politics of science that comes from commercializing research with fancy
PR conferences, fancy brochure magazine journals, and other elitist advertising,
but the corruption of fashion and orthodoxy in frontier research which gradually
creeps into science indirectly as a result, and the labelling of the
corruption as “science methodology”. Like Orwellian big brother politics, once
you have an establishment which knows it’s heart is in the right place, it finds
making excuses for extending the corruption very easy, just as it finds it very
easy to keep making promises to discover new exciting epicycles if the taxpayer
or big business stumps up every more cash.
Eventually, it’s defensiveness in labelling all critics as conspiracy
theorists, merely for suggesting that the existing research directions are
failures which are being pursued because they bring in research grants from
deceived sponsors, starts to look like bitter paranoia, even to Brezhnev era
jobsworths who would rather be verbally crucified by long oppressed dissenters
and critics, than be disloyal to their dear Party Comrade. Another post
of Dr Sheppeard’s quotes a string theorist: “It is really the case that there
are brilliant loners out there and that there is some kind of conspiracy by the
physics “establishment” to prevent their voices being heard?” Again, too
much of this kind of defensiveness can eventually sound like bitter or paranoid
hubris. By analogy, if the medical establishment is reducing suffering in return
for taxpayer’s cash grants, then fine. But if were to go off into some kind of
alternative therapy for 30 years which failed to achieve any checkable evidence
in that time, and then started to burn critics merely for suggesting that
alternative nascent ideas exist that have been starved of funding, then the
credibility and respect of the public in that medical establishment might be
affected. “Weak point, shout louder” is advice that has a limited shelf-life,
then looks like propaganda, or even the dictatorship of a band of corrupted
self-deceived geniuses.
But maybe I’m completely wrong about this. I hope so.
