First, a warning. Most of what I post to this blog is
intended to be understandable to a lot of people. But this is a difficult
subject; even if I do my best to make the key points clearly, you may well be
confused or, worse, misunderstand.
In the past, I would not even bother to write down these
kinds of things when, for example, I was delving into the field of neural
networks, before that field existed as an organized academic and technological
enterprise. For example, sometimes I would see a very interesting idea, write
it down in my notebook, and analyze it further – ultimately finding out why it
was already incorporated into other things, or would not work. I have many
dozens of notebooks on neural networks and basic physics, with analysis I never
wrote down at all for others to read, in handwriting which even I have problems
reading years later. I am tempted to do the same today..
but some of the key points are too important to become
totally lost beyond all hope of recovery.
BACKGROUND
This week, I have been looking a bit more into the
properties of a new theory of physics, a new theory of how the universe might
work, which I posted this week at vixra.org. The theory certainly did not come
out of thin air; like the concepts of backpropagation and adaptive critics
which I started trying to explain and simplify many years ago, it was the
“final” outcome of exploring dozens upon dozens of different approaches and
questions, from many streams of literature and from things I could think of
myself.
The final outcome actually has two or three strands:
(The methods for working out the actual PDE, the
Lagrange-Euler equations for a Lagrangian, are relatively simple, and found in
many textbooks, such as chapter 2 of Quantum Field Theory by Mandl and Shaw.
They also show how to derive the Hamiltonian H, the energy function. Slightly
more advanced material, based on the famous theorem of Noether, shows how to
work out the momentum and the angular momentum implied by any Lagrangian.) In this posting, I also discuss a variation and a further method of creating variations of that Lagrangian.
(2)
http://www.scribd.com/doc/95547363/How-to-Quantize-June2012
shows “how to quantize” the theory. But the remarkable fact is that
“quantizing” does not mean changing the theory in this case! Quantizing
basically just means working out the emergent statistical properties of a
universe governed by these kinds of PDE, which generate “solitons” (aka “stable
lumps of energy”) and strange chaotic behavior.
Years ago, when quantum theory was first being developed,
Einstein claimed that the weird phenomena of quantum mechanics could be
explained someday as the emergent statistical outcome of something more
familiar. But how? People gradually agreed it would be impossible, because they
couldn’t figure out how. (The story of my invention of backpropagation was much the same; after
Minsky’s book Perceptrons was widely read, people agreed that it must be
impossible to train neural networks even to perform some relatively simple
tasks, and it become total heresy to get up and say “Hey, but I can do this,
here is how.” It took 15 years of pain and all-out effort to persuade several
groups to “reinvent” it at that time… but for the physics, it is objectively
more complicated.) Most people who have studied this issue today will tell
you that “Bell’s Theorem” has finally, decisively shown us that such an
explanation is impossible. But this is not correct; a more complete analysis,
written to be as simple as possible, is in the open access journal paper: http://www.springerlink.com/index/K6784172712U3007.pdf.
The key point in that paper is that we need to
understand the emergent behavior predicted by the PDE WITHOUT inserting
inappropriate classical ad hoc statistical assumptions such as “time forwards
statistics” to make it easier to do the calculations. But how can we work out the statistics
directly WITHOUT inserting such assumptions? That is the topic of the second
paper above, the scribd paper, which is by far the more difficult of the two.
But at the end of the day, one can still make some important testable
predictions without working out the formal quantum scattering statistics; for example, one can use PDE
methods – analysis and simulation – for masses and spectra, for particles and
even nuclei. The new understanding of nuclei may even play a crucial role in
helping us understand what is needed to develop the technology for complete
conversion of matter to energy.
Because physicists have already been willing to put considerable
effort into studying simpler
classical PDE systems, such as the skyrme model and the
“BPS monopole,” I hope that the realism and my personal expectations would not
get in the way of studying the PDE system in the vixra paper.
There is a third very important strand of work in this
approach – the empirical testing and connection. I have posted a number of
papers in various places discussing this important strand, but I do not have
any ONE reference. The IJTP paper (springerlink above) points towards Bell’s Theorem experiments with less accurate polarizers
as a testbed. A more audacious filing at arxiv about quantum separators
describes another. An arxiv paper addressing Schwinger’s “magnetic model of
matter” describes a third, though we also need the mathematical work to nail
down the relevant solitons and have the theory better prepared for that area.
And the high-energy (revealing details at 2 femtometers or less)
electron-electron scattering mentioned in the scribd paper may also be
decisive.
UPDATE ON RECENT DISCUSSIONS
At the present moment, I am growing more optimistic that
this simple Lagrangian may qualify logically as a “next standard model” – and,
more concretely, as an upgraded version of electroweak theory (EWT), the most
highly tested pillar of today’s standard model. The main new offering here is
the ability to explain the existence (and masses) of elementary particles, as
“solitons.” The most important
target for new mathematical work is to explain the electron, while making room
for the constituents of the proton and neutron (“modified quarks”).
One key mathematical task is to REALLY prove or disprove
the stability of the famous “BPS monopole” – not because it is a realistic
model (though many think it might be, for use in grand unification), but
because the general new mathematical tools would be useful for other models
such as my proposed new Lagrangian. I discussed this at the Midwest PDE seminar
two weeks ago in Memphis, and have hopes that they can work this out. Many
physicists will assure you that this has already been proven, because
“Bogolmonyi and Coleman proved it” or “E. Weinberg showed there is some
topology here; however, there are very large loopholes and gaps in logic which
need to be analyzed.
Another key mathematical task for now is to go ahead the
way physicists did with BPS,
and assume or hope that we can look for spherically
symmetric solutions, and that they will be stable if they obey the right
boundary conditions. In the vixra paper… this is something any student in my
old graduate courses could do in less than a week! First, get a copy of the Prasad and Sommerfield and
Julia/Zee and tHooft/Hasenfratz papers on BPS, as background, as an example to
copy. Second, work out the Hamiltonian for this Lagrangian, expressing it in
“three-dimensional language,” just as those guys did.
Third, assume all time derivatives are zero.
Fourth work out the Lagrange-Euler equations in THREE
dimensions to minimize this Hamiltonian. Fifth, substitute the values for the
fields given by the “ansatz” in my vixra paper, and work out the resulting ODE,
analogous to the Julia-Zee ODE which Prasad talks about. (Prasad’s paper is
only three pages long!) Finally, publish and analyze the properties of these
ODE, remembering that others may take that analysis further. The identification
of solitons
Is basically a matter of identifying solutions to these
ODE under appropriate boundary conditions.
Legitimate boundary conditions to be considered are the
null condition (where Q**2 and phi**2
always have their “infinite horizon” value, even at the
origin), and the condition where either or both goes to zero at the origin. A
fourth very important version is where the “epsilon” tensor
is givn the opposite sign for ONE of the fields but not
the other, and both Q**2 and phi**2 are zero at the origin. One of these
solutions may correspond to the electron, while others to modified quarks. Just
doing an electron is interesting enough.
Of course, this system would have lots of excited and
bound states as well, but nailing down the electron, proton and neutron is a
worthwhile enough goal for now.
Would we need yet more topological charges to do justice
to whatever the proton and neutron are really made of? Maybe. But I don’t yet
see any real reason why two should not be enough. In the worst case, exploring
two would be a good way station towards exploring more.
The scribd paper says a bit more about complete conversion
of matter to energy (if that link goes to the most recent version). If the
TOTAL Q charge and phi charge of the neutron are both zero,
as I would tend to expect, complete conversion would be
very much analogous to fusion itself,
something difficult but not impossible, requiring
exploitation of coherence effects for realistic technology. I do hope that the relevant
experiments will not be done on the surface of the earth,
and that we move faster to develop low-cost access to
space.
NEW CONSIDERATIONS
As we think about the possibility of spherically symmetric
solutions to explain the existence of the electron, an obvious question
appears: what about spin?
Do we need to look for axially symmetric solutions
instead, requiring a much more difficult hybrid of computer and analytic
techniques? The mathematical work proposed for the BPS monopole ties into this.
Maybe, but I hope not, and I think probably not…
Should we think of electrons as (chaotic) bound states of
TWO spherically symmetric solitons?
One of the neat things about skyrmions is that stable
“bound states” in the PDE simulations are really just “stable” states of the
PDE themselves. But then we would ask how far apart the two
“cores” of the two parts would be. Is there a quadrupole
moment we could use to test this? But again, it seems unlikely.
Intuitively – a key issue here is that spin, like electric
charge itself, seems universal in a way which would make an ad hoc explanation
based on the properties of just one of the solitons
questionable. It calls for a more universal kind of
explanation. For charge, there is topological charge, which I have accepted
emotionally mainly because charge is so universal (but also because it fits
Higgs terms and is tractable). (Nontopological solitons may yet be possible too
in relevant theories, but they are very hard to work with, with any known
tools. But let me not become a Minsky.) For spin, there are already some
emergent properties from this topology,
but probably not enough, no mater how we play with it.
For now, I tend to view the issue as follows. “Spin” is
really a matter of “quantized” angular momentum, not magnetic moment. (We
calculate the basic magnetic moment of the electron
like
(e/m)(1/2), dividing out the “mass-like” truly quantized quantity by m
and multiplying by e.)
That in turn is universal like momentum and energy itself,
a case where we are back to the old question “where does Planck’s constant come
in?” That is basically what the scribd paper addresses. It is an emergent
property. The oscillations can be thought of as “an artifact of using Fourier
analysis in the scattering equations,” though really they are an emergent
behavior resulting from what Fourier analysis tells us about that. Concretely,
I would for now adopt a “roller ball magnet” model of electric spin as
something very protean… as it is. The underlying
Soliton most likely really is just spherically symmetric.
Since the simplest version is also the most likely, it makes sense to pursue it
for as long as we can, in this generation.
Even so, this gave me pause. Would there be a way of
modifying the new EWT Lagrangian to
better ensure emergent spin? What of a Higgs field,
instead of Q and phi, which would be more pleasing in a way, like a two-by-two
complex matrix (“twistor” as in Penrose’s book), where the matrix approaches
uv* at the infinte horizon (giving u and v charges)? The Higgs field V is
all we need to ensure topology at the infinite horizon.
But: as I look closely at this, it doesn’t seem that it works. A “det” term in
V would push the twistor to approach uv* all right, but
separately making each u and v become unit vectors at the
horizon is not so easy or natural.
It’s easy to make the twistor approach a unitary matrix
(SU(2)), but that looks a lot like Skyrme model – interesting mathematically,
but only one charge. What of an SU(3) Higgs field? There my weakness in topology
is a barrier. Makahankov, Rybakov and Sanyuk (MRS) give a table on page 226 or
so, but it’s backwards – from spheres to fields, when we want the other way.
Still, even if I assume it’s invertible… I think of Manton’s last chapter.. I
simply cannot find or think of a form of the Higgs field which would yield BOTH
charges in such a unified way. MRS do note that this table was the result of a
huge amount of work, so perhaps the answer about such possibilities is unknown.
PERHAPS an alternate Lagrangian using a “matrix Higgs field” instead of two
vectors could yield two charges, but perhaps not. Perhaps a more clever
relativistic V term could fix this; perhaps not. For now, I will work with the
one form which I know does work,
with more sense of its likelihood of working out.
Well... for M a two-by-two complex matrix, I can imagine inserting M in place of Q, and M* in place of phi,
in the Lagrangian (with mutiplication in the coupling terms adapted in the usual way from 3-vectors to 2-spinors), and
V = (Det M)**2 +c(1-Tr(MM*)**2), where * is Hermitian conjugate.
At the infinite horizon, that does yield M going to uv*, and (|u|**2)(|v|**2) going to 1.
I would prefer it is |u|**2 and |v|**2 each had to go to 1 separately, but this still might be viable,
considering that the (|u|**2)/(|v|**2) ratio is really set by the universal horizon anyway. So this
actually is a decent alternative Lagrangian to consider if the present form doesn't work.
It is quite posisble that one would work but the other not, in matching empirical reality.
There are other physics issues which I am not even
touching here, like dark energy
and alternative forms of gravity and superweak
interactions, which are issues for the standard model as well. One step at a
time. There is enough to clean up here, and it is crucial to clean it up
in order to be flexible enough to deal with the other
issues.
A general impression – perhaps the “neutrino,” like the
photon,” is a mode of radiation, quantized by boundary conditions, and not a
soliton at all. A quasi-boson? But electrons first,
protons and neutrons and nuclei second…
===================
======================
The two alternative Lagrangians discussed here have another important property: in both cases,
everything which I have checked so far remains valid if V is replaced by f(V), where f
is a smooth monotonic function with the property f(0)=0.
This property would be extremely distressing to those who have faith in the power of pure reason to deduce the laws of physics, unaided by crass empirical reality, as was promoted by Aristotelian church physics
during the dark ages and to a greater extreme by superstring physics. However, I agree with Einstein
that Kant's Critique of Pure Reason is an important source even for physics. I would prefer to try to hang onto
the scientific method, which is actually refined to some extent by modern learning theory.
MOST of the two Lagrangians above is deeply rooted in centuries of empirical work by many players,
starting with Maxwell's Laws, proceeding to the early tests of QED (e.g. in Mandl and Shaw),
and the initial development of EWT itself following the overthrow of parity (one of the great esthetcaly satisfying Aristotelian principles in its time). Even the coupling terms in these new Lagrangians
is taken directly from the coupling of particles to W and B in electroweak theory, which need to
be retained when we propose that particles are solitons of the four or three bosonic fields here.
The new quantization assumptions are also based on very extensive empirical work in the area of "applied QED" summarized briefly in the IJTP paper (URL above). In essence, it is only the form of V itself (within the established Higgs type boundary conditions) which remains unknown in empirical reality.
This puts us in a situation analogous to the years just before Newton's theory of gravity, when the general idea of gravity as a universal force had already attracted great interest and support, and there was only an uncertainty about whether it would be an r**2 attraction or r**3 or something else. The requirement
then was for extensive calculation of the empirical implications of the alternatives, matching
them with empirical data which had already been expressed in clear form by Kepler
(a great heretic to the Aristotelians of his day). The task now is basically similar. For a spherically symmetric ansatz, at least, it should be easy enough to work out the ODE (analogous to the ODE
of Julia and Zee for the BPS system) for the general case of f(V) -- but with different choices of f,
the solutions are different to some degree.
It may be that we will still have a host of possibilities, in principle, which can explain the mass and electric charge of the electron, and provide a promising foundation for a new view of what the "quark" may really be. But it may be that PDE simulations of the nucleus and of nuclei, in the spirit of Manton's calculations, is necessary to really nail down the alternatives. New nuclear experiments, to nail down details discussed in my arxiv paper on Schwinger's concept of the nucleon, will also be important. (But note that the Lagrangian proposed here may end up somewhere in a spectrum between QCD and Schwinger's concept; we do not yet know, and we may even have choices for models of the quark to be resolved by empirical comparison, which is easier when we can use PDE simulations to do the calculations.) In Newton's time, the r**2 choice was one of the simple and obvious and appealing possibilities; likewise, it is quite possible that f(x)=x will turn out to be the right one here in the end anyway. But at this time we should not pretend that we know. The ODE need to be worked out and studied....
After that is all done, physics will have made a really great leap forward, but of course that would
not be the end of the process. I can think of many other things worth probing, not only in using the new Lagrangian, but in exploring possible alternatives, starting from a more powerful, realistic, simple and flexible foundation than what we have to build on today.
=========================================
An additional question which comes up when the ODE are known for the alternative new Lagrangians...
WHICH of the basic solitons would correspond to an electron? I would guess that a soliton with "Q"
charge of +1 and a phi charge of +1 would work. (Likewise, in tthe twistor variation(s),
a left-hand charge of +1 and a right-hand charge of +1.) This way, both Q and phi fields are in play,
and both coupling terms are in effect for electron-electron interactions, as we see in nature.
(Though more complex mechanisms could be in play.) A MIX of Q and phi charges gives a mixed source of W and B fields -- which is appropriate since the ordinary electric field is basically a MIX of W and B fields,
a mix which has long-range propagation more than W or B on their own. That's a basic property of EWT.
The quarks would be some mix of the OTHER solitons, bound together by forces (other mixes of W and B)
which are more short-range in nature.
In a sense, the idea here is that actual "gluons" are really another mix of W and B. We do not need so many quantum numbers for stable quarks as in QCD, because we now know that we do not need quarks to obey
the Pauli exclusion principle; QCD was derived in the old days, when it was believed that a bound fermion (like the proton or neutron) can only arise as a state of bound fermions (or with SOME component fermions at least), but extensive work on "bosonization' shows that this is not so. Again, the scribd paper whose URL is above gets into the quantization issues. The claim here is that we should be able to produce modified quarks
as emergent solitons, good enough to satisfy the important sliver of experiment which has been thoroughly checked so far in strong nuclear experiments, but computationally tractable enough to allow testing across a much broader range of nuclear phenomena (like fusion or models of nuclei) where it would be useful to improve on the very limited models (like the skyrme model which has no kind of quark at all) now available for such nuclear work.
=============================
There is a very basic question some might have. Am I really sure that either new
Lagrangian, the one based on Q and phi or the one based on M (which I will now call omega,
in a paper I have started to write on google drive), will generate two topological charges --
which is really most of the battle in establishing real variationally stable solitons.
(And that is all that has really been PROVEN for the BPS monopole, widely accepted by physics.)
For the Q/phi case, it is very straightforward. The "V" Higgs term strongly enforces
the requirement that both vectors in R3, go to a fixed length at the infinite horizon
of the particle; in other words, we have a mapping from S2 (the two dimensional surface of a sphere in three dimensions) to S2 (the horizon itself), EXACTLY as we do with the well-known BPS monopole. Two vectors, two mappings, two charges. In the M or omega case, it is tricky.
At the horizon we have uv*, enforced by the HIggs term. Each of these vectors, u and v,
is from C2, a two-component complex vector, with 4 degrees of freedom. The Higgs term
effectively constrains each of them to unit length (though only the matrix M actually exists,
we can represent it always as a product of unit-length u and unit-length v). That gives them the topology of the THREE-dimensional surface of a sphere in FOUR dimensions, S3. But ..
it turns out that the group SU(2), used in the skyrme model, also has the topology of S3.
From the Skyrmion work (or, more precisely, the work on topology which gave rise to it),
we know that the mapping from S3 to S2 works just as well.
In checking his, I have read parts of Manton's book I had not read before, and reread Makhankov, Rybakov and Sanyuk (MRS). I was somewhat surprised that the MRS treatment was mch clearer
on this particular point. At the end of the day, the topology of the skyrme model is based on a mapping from SU(2) to S2, period; it is that simple. Manton's notion of rational extended mappings
similar to some of the discussion in MRS) is basically a way of understanding the mapping -- which is useful, but should not obscure the simple fact that the topology of mapping from SU(2) to S2 is what matters here. And it does seem to work.
==============
=============
Thinking about this, I can't help speculating just a little further.
As I think about the tau-sub-mu group of FOUR matrices, versus the usual group of three Pauli matrices which generate SU(2)... it reminds me a lot of the relation between the three field components of the W fields in EWT versus the B component. Could it be that "W" and "B" are actually just ONE representation, one gauge, of a more truly four dimensional matrix
(well, four by four, over Minkowski space), which has a more truly four-dimensional
(Minkowski-like) gauge symmetry? If we can represent all of physics with just three objects --
a four-by-four metric tensor, a four-by-four augmented W tensor or isotensor,
and omega.... maybe it could be fit even more tightly together. Who knows?
For now... it's OK to work in one legitimate gauge. The two new Lagrangians are enough for
now as the next big step forward.
===========
=============
Added later: have worked out more on the new Lagrangians, pretty much proving that
solitons with two topological charges exist in both of them. Have a paper in draft which
has details and next steps.
However, it still leads to a question: if these solutions are spherically symmetric, what of the intrinsic angular momentum of the electron and every other "spin half" particle?
The formula S = (hbar/2)sigma, imparting exactly the same amount of angular momentum to
every spin half particle, is just as striking as the universality of electric charge. One would
not expect it to arise by coincidence, for example, in different axially symmetric solitons
of different boundary conditions and charges.
If it is not just an illusion or emergent effect, what could cause it?
For a couple of days, just as a mathematical exercise, I have asked how one might
modify "HIggs" terms (e.g. velocity-dependent Higgs terms) to try to hardwire
and quantize angular momentum in the tight way we now know how to do with charge.
There are neat ways to do that kind of thing. For example, given a two-dimensional
velocity vector on the surface of a sphere S(R), one could define a Lagrangian on
that surface like ((|v|**2)-1/g(r)**2)**2+ag(r)**2||grad v||**2-k , a sort of Higgsy term;
with the right choice of a, it would yield an "earthy" flow field around an axis. k can be chosen to make sure energy is zero for that flow field, and more for all others, to make it like a real HIggs term.
A neat mathematical exercise. But since angular momentum is not "visible" at S2(r),
it does not overcome the ad hoc problem for ordinary axial solutions. And it would also be a bit of work to infuse this as part of a relativistic system, analogous, say, to BPS (or the EWT type
Langrangians I have recently developed.). A nice exercise, but I don't see it as the most promising use of my very limited time. (A great toy for someone in differential geometry perhaps...)
Note that there is spontaneous symmetry breaking in this toy system -- an emergent choice of axis z of rotation.
So for now, I still believe that the more fundamental aspects of quantization are at work here,
Perhaps a more careful study of quantization and scattering experiments will show that my two
existing alternative Lagrangians are all we need. However, I think right now more
about the simple formula for angular momentum on pages 35-40 of Mandl and Shaw (much clearer for purposes here than Itzykson and Zuber, or Lovelock and Rund). It feels as if the usual r x p terms should be zero somehow, and the "S" terms for Lorentz transformations should be where this emerges. But perhaps that means I do need to couple to a true covariant vector here -- classical, but still.... and while the statistics remain spherically symmetric in vacuo, specific states probably do need to have spontaneous symmetry breaking -- in other words, hard fixing of charges
and angular momentum, but loss of simple spherically symmetric ansatzes. A mess.
So is reality a mess, even at this level? Not SO surprising... we know even the proton is a mess
(an object requiring lots of supercomputer time to simulate at best)... but at least the basics and the fixing may be elegant. In need of a third Lagrangian? Maybe...
================================
12/12/12: Two new Lagrangians developed. Draft in process, which
I suppose I will send to the Russian journal which invited a new paper
on this general topic. Lagrangians are simple enough, in some ways simpler
than what I had before, but axial solutions can't be avoided, and of course scattering PDE simulations would be downright 3D in any case.
Two new subsections on some empirical aspects.. and then a major section on quantization, like the scribd quantization paper...
Wherever it may or may not go, it's nice to have the current full story on paper
(full with citations at least).
But it also seems that people do not yet know about it. Myself, I am overwhelmed