Category: Nonlinear Field Theories

I have just gotten back from an extended business trip to the United States.

This was a lot of fun. I went to the SIAM Boston meeting on Computers in Science & Engineering. Later in the trip I went to PyCon2013 and PyData2013 to catch up with where things are at in the Pythonic universe.

It is probably close to fifteen years since I have been to a scientific conference. I very much enjoyed the developments in computational science since they involved many practical problems.

Regular readers of this blog will appreciate my love-hate relationship with mainstream physics. I tend to get very annoyed when I read physics journals, but I do enjoy the computer and applied mathematics literature.

However, I may yet come to some rapprochement with physics-at-large.

The epiphany for me was the sudden realization that most physicists today do not seem to believe in the Copenhagen interpretation but they do believe in point-like particles.

I don’t know exactly when that dawned on me, but it was somewhere in the middle of listening to some talk on numerical solutions of the Schroedinger equation in the semi-classical regime. None of the correspondents in that session seemed to know anything about my work in this area so I kept quiet.

I did not want to spoil the fun of searching for the elusive classical limit.

Reflecting upon what I already knew to be a grand folly I realized that physicists in general seem very sure that they know what rests within the Schroedinger equation. Look at it this way. They know three things:

1) the Schroedinger equation is right
2) the Classical world seems real enough and sensible
3) therefore, it must fall out of the mathematics somehow

All very sensible, but sadly untrue. The relevant mathematics does exist but is not what they are looking for!

Now to the epiphany. The physics community, in general, believe that we know particles are real. However, they are confused about what a wavefunction might be or mean since a wave is clearly not a particle!

What the community has yet to appreciate is that in a certain limit, that of the Classical Schroedinger Equation, the wavefunctions do bounce around, scatter and behave exactly like particles. However, that is a limit where the wave aspect of the physics is frozen out. You get to the idealised particle from the wave description.

However, that regime of behavior is very special and only ever approximate.

Let us now turn things around. Suppose there are no particles at all. Suppose that is merely an illusion and matter is pure wave. Even further, suppose that the phase of that wave is the gauge field (e.g. the photon). Then there is no light separate from its sources. They are part and parcel of the same phenomenon.

In that case, we have things perfectly upside down. We are using a micro-level mental picture of particles and then wondering why the classical level continuum physics is not easily recovered. Turn that around and employ a micro-level picture of pure waves. How then do you recover familiar billiard ball behavior?

That is easy, it is the decorrelated approximation of mean-field theory for which the Classical Schroedinger Equation is just the simplest example in a family (the next one up is the Hartree-Fock approximation).

What then is the reality? Well, in the alternative picture it is a matter wave in configuration space.

What about measurement theory, you ask? How come we only see fields in 3+1 dimensions not (3+1)N dimensions? How can you possibly shrink a beast in configuration space to a manageable quasi-classical field?

Easy. The very same rule invented by Max Born still works!

If you calculate the one-body density of the matter wave you get the classical level of experience. The mapping from (3+1)N dimensions to (3+1) is trivial and we already know the empirically supported rule.

Once we abandon thinking about particles, then there is no measurement theory at all. There is nothing separate, just a whole. There is no particle being measured by something, and no observer either. You can just junk the lot.

There is only one single wave and a continuum foundation to everything. Since the wave equation does admit eigenfunctions, that wave is quantized. Think of it as Universal Jelly with Natural Modes of Vibration.

Furthermore, the classical limit is merely an idealization. The wave never breaks up into independent particles in the sense that quantum field theory describes. Entanglement is ubiquitous and permanent (eat your qubit socks quantum computing zealots).

Wilder still, if you analyze the causality structure of such a theory it is not determinable.

While the underlying wave evolution may well be deterministic the one-body quantities are not sufficient as initial data. You could know them exactly and still not be able to predict what happens.

There is no need for God to Play Dice. You can have a non-local fully deterministic hidden wave theory. This you cannot do for particles, since local hidden variable theories are excluded!

The epiphany which struck me in Boston was the realization that hidden variable theories are alive, but only when constructed in the wave formulation.

If you insist that the wave describes a point-like particle then you cannot conceive of a hidden variable theory. This is because particles are local entities. Waves, on the other hand, are non-local entities. If you only think of particles you will miss this possibility.

The totality of the wave in configuration space is a non-local hidden variable.

I think it is possible to build a new theory now. However, this journey is not for the faint hearted. Quantum Field Theory simply cannot survive such a revision of physical interpretation in its present form.

QFT is necessarily a theory of particles not waves.

If we do get new physics it will be New Physics. Interpretation changes everything.

Getting the particle back as an idealized concept is easy. That is the Classical Schroedinger Equation.

Sionara QED I. Enter QED II, the sequel with added Gravitation.

The question is what to call this hidden layer of reality we are positing. Since the idea of junking QFT in its entirety is so utterly subversive I favor the term subverse.

To be really perverse, let us call the perceivable layer the PERVERSE and the imperceivable hidden layer the SUBVERSE. As Confucius himself might opine:

Contemplation of perverse perceivables guides us to subversive imperceivables.

We are now on the road to SUBVERSIVE QUANTUM FIELD THEORY (SQFT).

SIAM meetings are a lot of fun.

I’ll be back.

There is a common view, perhaps, that discovery is straightforward and the people involved know what is going on. In the interest of encouraging independent thought, and experimentation, I would like to present some apposite opposite evidence.

In my view, the course of history rarely runs straight, and it is often more confusing to those directly involved. You may think that you are doing one thing, but soon events and knowledge overtake you and you realize that you are actually doing something different. That has been my stark experience several times in scientific research.

I would like to present that experience in summary because I hope it will encourage young researchers to stick at their investigations.

Do not be dissuaded from your course of action by uncertainty, peer resistance or non-comprehension – whether your own, or that of ofter people. We are all human and the wonder of the new is that it can be most perplexing. Nature is a Chinese Puzzle. She has her way to be — your task is to find that out. The true Scientist is compelled to mold his or her mind to Nature and not Public Superstition.

Follow Nature where the evidence leads you, even against the Conventional Wisdom.

So it was for me in early 1991. I had just finished my first publication dealing with my doctoral work on Quantum Inference. This was one of the early works on Quantum Information Theory. It showed that the Heisenberg Uncertainty Principle was more subtle than people first supposed. When you take account of the statistical Law of Large Numbers, then you discover that quantum states can be precisely measured. However, you would need an infinite supply of identically prepared systems to be able to do that. I was looking for applications of Quantum Inference and so thought of experimental tests of the quantum transformation theory.

I reasoned that if you could measure states precisely, which you can given enough of them, then you might make a sensitive test of the so-called Wigner theorem. I started out as an experimentalist in Chemical Physics, so I knew all about ultra-cold atomic and molecular beams. I had done my Bachelor of Science thesis in the Research School at ANU on Vibrational Pre-Dissociation Spectroscopy. Using an atomic-beam setup, to shoot cold atoms through a cavity, you should be able to detect non-linearity via a failure of Wigner’s theorem. This is the simple thought that got me started.

So I started looking at possible generalizations of the Schrödinger equation. I knew there had been some work and that the subject was of considerable mathematical difficulty. Hence I resolved to test my understanding with a simpler problem. Since quantum measurement involves the coupling of a formerly well-isolated quantum system to a classical-level measuring device I wondered if the classical limit might involve some non-linearity. Perhaps when a whole bunch of particles got together, there would be a stronger non-linearity. It made sense to me, given that the Schrödinger Cat Paradox only applies to linear theories.

To test this idea I framed a simple mathematical question:

Can one find a wave-equation which propagates waves so that their position and momentum expectation values follow classical trajectories?

Of course, I knew about the Ehrenfest Theorem, but I also knew it was approximate. The goal was to look for an exact equation. Obviously, that could not possibly be the ordinary linear Schrödinger equation. It took me a whole year of constant reflection. Then the answer suddenly struck me like a bolt from the blue:

Of course you could! Moreover, the answer would be unique.

It all followed from a beautifully simple piece of group theory due to Herman Weyl.

I could see the entire proof laid out before my eyes. It was, in essence, a very simple geometrical construction. It was so simple I could not believe it. However, it was also perplexing because I knew I had done something, but could not figure out what.

It can be that way in mathematics sometimes. People like to pretend they know what they are doing when they do mathematics. This is a fiction. If you know what you are doing it is certainly old mathematics. I knew I had new mathematics. Then things fell into place. I could prove the equation was necessarily non-linear, and was unique.

Aha! I said. This is the nonlinear representation theory of dynamical systems.

Reasoning about my own mathematics, I recalled a paper by Weinberg which I had glanced at when finishing my PhD thesis. It was called Testing Quantum Mechanics. In a frenzy, I searched for his paper and was immediately gratified that the equations looked different. However, I am a good enough mathematician to know that simple appearances can be deceptive. Sure enough, I figured out how to generalize his scheme to wave-functions and find an equivalent result.

The Classical Schrödinger Equation was for real, both ways!

I was elated for a simple reason. Many people might assume that it is discouraging to find prior work on a similar track. This is not so. I was elated because I had found something peculiar in my mathematics.

Operators were secondary to wave-functions.

Try as I might, I could not banish this feature. Once I found the result in the equivalent scheme of Weinberg I could see why. I had discovered something. Schrödinger mechanics was more general than Heisenberg mechanics.

Operators were an artifact.

Furthermore, the Classical Schrödinger Equation remained exact for any value of the Planck constant. This was a great surprise. I checked and re-checked everything but it always came out the same way. Greatly puzzled, I figured out why. It involved a Taylor series style expansion that got you back the ordinary Schrödinger Equation.

Excitedly, I prepared a summary of my result and sent it to Weinberg. He was kind and very generous with his time and I presented a seminar at the University of Texas, at Austin on Jan 30th 1992.

I was very nervous and prepared and rehearsed my talk extensively. At that time, I had very little seminar experience. When you first start out it is difficult to get much practice. I worried greatly that I would make a slip in my math somewhere. This is natural for young people. However, important work is never that way.

If I can offer any advice to young researchers it would be this:

Frame and motivate your question clearly.

So, I gave my talk. The result was silence. I was baffled and perturbed. Weinberg was kind enough to ask me later what I thought my talk was really about. I explained that the scheme of mathematics he had proposed for testing quantum mechanics actually contained a complete and exact copy of classical mechanics.

He looked at me intently and said: Oh!

In that moment, I knew I had a problem. As the saying goes:

You broke it, you fix it!

Later I completed the mathematical proof that Copenhagen Quantum Mechanics is Incomplete. I was grateful nobody noticed.

It is a bad feeling to break a perfectly good theory when you don’t know how to fix it.

However, where there is a will there is a way…

What got me really fired up with my last post was some of the recent literature I have read announcing ever more elaborate philosophical interpretations of the wave-function. It is my opinion that these attempts (too many to cite here) are afflicted with what E.T. Jaynes called the Mind Projection Fallacy.

The essence of this idea is the common affliction to mistake models for reality. There is a popular and equivalent concept of distinguishing carefully between Map and Territory. For those who like big words, the difference between Ontology (what is real) and Epistemology (what is known). It is my contention that contemporary physics is in a deep bind precisely because it has ignored this distinction.

The Shut up and Calculate crowd are complacent about what they think they know, while the philosophers obsessively change the words without changing the content of the theory. Each of these extremes leads to trouble.

So let me now make a bold assertion. I now firmly believe that important sections of the new quantum theory have already been published. Yes, you heard me right. Accepted for publication and in print.

This statement ought to shock people.

If there is a new theory of quantum mechanics already out there, then: How come nobody has ever heard of it? Surely with all these hyper-smart people running around in physics they could not be blind to the existence of a new quantum theory right under their noses?

In the journals they know and read? Surely not!

Let me explain why I think this to be the case and exactly why nobody has noticed.

Firstly, if the assertion is correct then how could people not notice?

Answer: Jaynes was absolutely correct. Physicists as a community suffer from the Mind Projection Fallacy. They fervently believe that their theory is correct and therefore they have stopped noticing difficulties. Now we are in the terminal phase of this illness. People not only do not notice difficulties, they do not notice solutions. How can you notice a solution to a difficulty you do not believe exists? Of course, you won’t.

It all started with a few small issues being swept under the rug. However, as time goes on little problems build up. I will state only two of direct relevance to this post:

1) the ongoing perplexing problems of infinities in treatment of self-energy; and

2) the apparent existence of a non-zero cosmological constant.

Of course, there are answers to both conundrums. The first is dealt with in the Standard Model through the program of renormalization, while the second was catered for early on by Einstein’s “great blunder” – his modification of general relativity to achieve a stationary universe by including a cosmological constant.

What if these two pat answers hide something deeper? Perhaps renormalization is simply a stop-gap. Perhaps the presence of the cosmological constant signals something else completely like a self-energy of geometry?

Secondly, if the assertion is correct, then: Where is the evidence of a new theory?

The best example I know is the Self-Field Quantum Electrodynamics of the late Asim Barut and co-workers. There is a lot more than that, but we are simply establishing prime-facie evidence for the logical possibility of my thesis.

Thirdly, if there already is a new theory: Why don’t people pay attention to it?

Here is an old conundrum: Why does mankind oftentimes fail to take heed of warnings?

A host of very distinguished physicists: Einstein, de Broglie, Schrödinger, Bohm and Bell, foremost amongst many, have been critical of the Copenhagen Interpretation of quantum mechanics. They thought it was missing something: that is was incomplete.

Niels Bohr persuaded physicists that these problems either: 1) had deeply philosophical answers; OR 2) were not to be considered part of the practical program of calculation.

In other words, Bohr persuaded folks that it was okay to blur ontology and epistemology. When one did not work, simply slip effortlessly into the other mode until your audience is mentally exhausted and simply gives up.

Whenever something did not make sense you should either: change the words employed, or try not to think about calculating anything. With this device you could happily sweep any problem under the rug.

However, over time this ruse has caught up with physics. There are now so many different words used to say what a wavefunction means that nobody knows anymore. Further, the different tricks and devices have turned fundamental physics into one giant hairball. It is next to impossible to know what to say, or what to do.

I believe this situation is now self-evident. However, the very many professional academic physicists out there will doubtless dismiss this appraisal as betraying a lack of familiarity with the wondrous advances of recent years.

I say: Hooey to that!

Let me close with a simple observation.

Some years ago I conjectured something I called the hypothesis of restricted observables. For now, I will give the one sentence, just go with the flow, formulation:

The hypothesis of restricted observables posits that matter waves are real, but that reality, as we perceive it and experience it, constitutes only the reduced one-body fields projected from the total entangled multi-body matter wave in configuration space.

In short, Schrödinger was right, with one small proviso of restriction, and Bohr and the rest of the team were dead wrong. And no, Dorothy, this ain’t philosophy. You missed your stop in Kansas. This is physics with a big P. The surf is up and the waves are huge today.

As a corollary to that statement, I would go further and state:

If I am correct then 99.999% of the Academic Physics community are dead wrong and we need a new theory. Since they do not believe a new theory is needed, and supposing I am right, they will simply read about it one day. They are not even in the race.

Let us close with a simple general challenge to the Academic physics community.

Here is a “picture” of the Electron Localization Function (charge density) as used daily by Chemical Physicists. The image is form a recent Nanotechnology article on experimental imaging of the electron density. That is (essentially) a one-body density.

I started out in experimental Chemical Physics so I have awesome respect for these people. They solve truly hard problems and do not have Particle Physics Big Head Syndrome. In short, they are good old-fashioned scientists.

Here is my challenge to academic physicists:

According to the hypothesis of restricted observables the above picture is a description of laboratory reality. Yeah, you read that right. That is it. Done and dusted.

The challenge is to prove this statement incorrect through experiment.

In return, I accept my own challenge – one of theory.

Can I build a self-consistent field theory based on the hypothesis of restricted observables?

I believe yes. The physics community (I warrant) believes no. One challenge speaks to ontology (decisive experiment) the other to epistemology (constructive theory).

People who do not understand why this challenge is real are welcome to simply have their heads explode on the spot. There is no shame in saying:

I won’t play, this physics game is just too perverse for words!

First you say: particle; then wave; then both; then what, exactly?

As a final icing on this cake, let us reprise the de la Vega Confusión de Confusiones theme and an old advertisement from the South Sea Bubble days. In that time, there were many prospectus offerings to raise money for dodgy ventures.

I am a financial guy, so here is the prospectus for my part of this here challenge:

…a challenge for carrying out an undertaking of great advantage, but nobody to know if it has been done already

Let the fun begin!

Today I am happy to report that I have resurrected a good number of my “lost works” on Nonlinear Quantum Theory and its physical interpretation.

Twenty years ago, I was a post-doctoral researcher at University of Queensland. This was the early 1990s, and I was very active at that time in developing some new lines of research in nonlinear dynamics and quantum measurement.

The goal was to rebuild the present quantum theory from an alternative base. The different base was to use the Schroedinger interpretation of wavefunctions and not the Copenhagen interpretation.

That may seem like a small thing but it is a change with big consequences.

In particular, the Schroedinger interpretation is compatible with nonlinear field theories.

Nonlinear field theories occur naturally if you treat self-interaction effects directly rather than via a second-quantized quantum field theory.

There are many issues to overcome in rebuilding quantum theory along these lines. However, back in the early 1990’s I could see how to move forward.

Unfortunately, getting grant funding is a more difficult task than that of making progress in fundamental physics. The majority of physicists I interacted with at that time considered this line of research to be “too risky”, perhaps even “crazy”.

So it is with the sociology of science. It is pretty close to that of religion.

The majority of my “lost works” were submitted to various peer-reviewed scientific journals and rejected. The standard reason for rejection was: “We do not think this is important enough to warrant publication”. An all too common story with discovery.

However, one paper I actually withdrew from publication. This was the first paper I wrote on the Classical Schroedinger Equation. At the time, I felt this had become obsolete. However, even today, twenty one years later, it seems the results are unknown.

In that period, from 1991-1995, I was making progress so rapidly that my understanding of the field of nonlinear field theories changed extremely quickly. It was, so to say, opening out before me in real-time and there were few other active researchers. Indeed, the majority of physicists simply thought that to even try to build an alternative theory of quantum mechanics was a “crazy thing to do”.

So be it. People are quite rightly “crazy” until they succeed, I said. Then, in a single instant, everybody else is “crazy” for never having tried.

In such circumstances, the traditional model of refereed publications does not function very well. The referee cannot know about other works that are already in the pipe, but which cannot be cited since they are not published.

Pretty soon, things fall apart with the scientific process.

I had moved to a different place than most workers in physics and they knew next to nothing about why. I may has well have been speaking Greek.

It is a funny circumstance in hindsight, but was rather serious at the time.

The upshot was that I had to “find a new career”. I boxed up my notes and moved on.

I quit physics in 1996 and sought to make a living in financial markets. It was always a vague plan to return to my grand project one day. However, I needed some financial independence, liberty and freedom from chasing government funding. Aside: The sociology of herding is the biggest drag on human progress IMHO.

Today I am happy to relate two pieces of progess: 1) I am financially independent of the government; 2) I managed to restore my lost papers from obsolete storage formats.

I have found The Lost Papers of Highgate Hill.

Very Indiana Jones. Perhaps the Mayan prediction was right…

Many Worlds do end in 2012!