Most scientists do not enter their chosen fields because the work is easy. They do their science mainly because it is challenging and rewarding when triumphant. Yet few scientists will ever taste the sweet dew drops of triumph — real world-changing success — in their lifetimes. So it is remarkable perhaps that the small delights in science are sustaining enough for the human soul to warrant persistence and hard endeavour in the face of mostly mediocre results and relatively few cutting edge break-throughs.

Still, I like to think that most scientists get a real kick out of re-discovering results that others before them have already uncovered. I do not think there is any diminution for a true scientist in having been late to a discovery and not having publication priority. In fact I believe this to be universally true for people who are drawn into science for aesthetic reasons, people who just want to get good at science for the fun of it and to better appreciate the beauty in this world. If you are of this kind you likely know exactly what I mean. You could tomorrow stumble upon some theorem proven hundreds of years ego by Gauss or Euler or Brahmagupta and still revel in the sweet taste of insight and understanding.

Going even further, I think such moments of true insight are essential in the flowering of scientific aesthetic sensibilities and the instilling of a love for science in young children, or young at heart adults. “So what?” that you make this discovery a few hundred years later than someone else? They had a birth head start on you! The victory is truly still yours. And “so what?” that you have a few extra giants’ shoulders to stand upon? You also saw through the haze and fog of much more information overload and Internet noise and thought-pollution, so you can savour the moment like the genius you are.

Such moments of private discovery go unrecorded and must surely occur many millions of times more frequently than genuinely new discoveries and break-throughs. Nevertheless, every such transient to invisible moment in human history must also be a little boost to the general happiness and welfare of all of humanity. Although only that one person may feel vibrant from their private moment of insight, their radiance surely influences the microcosm of people around them.

I cannot count how many such moments I have had. They are more than I will probably admit, since I cannot easily admit to any! But I think they occur quite a lot, in very small ways. However, back in the mid 1990’s I had, what I thought, was a truly significant glimpse into the infinite. Sadly it had absolutely nothing to do with my PhD research, so I could only write hurriedly rough notes on recycled printout paper during small hours of the morning when sleep eluded my body. To this day I am still dreaming about the ideas I had back then, and still trying to piece something together to publish. But it is not easy. So I will be trying to leak out a bit of what is in my mind in some of these WordPress pages. Likely what will get written will be very sketchy and denuded of technical detail. But I figure if I put the thoughts out into the Web maybe, somehow, some bright young person will catch them via Internet osmosis of a sort, and take them to a higher level.

There are a lot of threads to knit together, and I hardly know where to start. I have already started writing perhaps half a dozen manuscripts, none finished, most very sketchy. And this current writing is yet another forum I have begun.

The latest bit of reading I was doing gave me a little shove to start this topic anew. It happens from time to time that I return to studying Clifford Geometric Algebra (“GA” for short). The round-about way this happened last week was this:

- Weary from reading a Complex Analysis book that promised a lot but started to get tedious: so for a light break YouTube search for a physics talk, and find Twistors and Spinors talks by Sir Roger Penrose. (Twistor Theory is heavily based on Complex Analysis so it was a natural search to do after finishing a few chapters of the mathematics book).
- Find out the Twistor Diagram efforts of Andrew Hodges have influenced Nima Arkani-Hamed and even Ed Witten to obtain new cool results crossing over twistor theory with superstring theory and scattering amplitude calculations (the “Amplituhedron” methods).
- That stuff is ok to dip into, but it does not really advance my pet project of exploring topological geon theory. So I look for some more light reading and rediscover papers from the Cambridge Geometric Algebra Research Group (Lasenby, Doran, Gull). And start re-reading Gull’s paper on electron paths and tunnelling and the Dirac theory inspired by David Hestene’s work
- The Gull paper mentions criticisms of the Dirac theory that I had forgotten. In the geometric algebra it is clear that solving the Dirac equation gives not positively charge anti-electrons, but unphysical negative frequency solutions with negative charge and negative mass. So they are not positrons. It’s provoking that the authors claim this problem is not fully resolved by second quantisation, but rather perhaps just gets glossed over? I’m not sure what to think of this. (If the negative frequencies get banished by second quantisation why not just conclude first quantisation is not nature’s real process?)
- Still, whatever the flaws in Dirac theory, the electron paths paper has tantalising similarities with the Bohm pilot wave theory electron trajectories. And there is also a reference to the Statistical Interpretation of Quantum Mechanics (SIQM) due to Ballentine (and attributed also as Einstein’s preferred interpretation of QM).
- It gets me thinking again of how GA might be helpful in my problems with topological geons. But I shelve this thought for a bit.
- Reading Ballentine’s paper is pretty darn interesting. It dates from 1970, but it is super clear and easy to read. I love that in a paper. The gist of it is that an absolute minimalist interpretation of quantum mechanics would drop Copenhagen ideas and view the wave function as more like a description of what could happen in nature, tat is, the wave functions are descriptions of statistical ensembles of identically prepared experiments or systems in nature. (Sure, no two systems are ever prepared in the exact same initial state, but that hardly matters when you are only doing statistics rather than precise deterministic modelling.)
- So Ballentine was suggesting the wave functions are;
- not a complete description of an individual particle, but rather
- better thought of as a description of an ensemble of identically prepared states.

This is where I ended up, opening my editor to draft a OneOverEpsilon post.

So here’s the thing I like about the ensemble interpretation and how the geometric algebra reworking of Dirac theory adds to a glimmer of clarity about what might be happening with the deep physics of our universe. For a start the ensemble interpretation is transparently not a complete theoretical framework, since it is a statistical theory it does not pretend to be a theory of reality. Whatever is responsible for the statistical behaviour of quantum systems is still an open question in SIQM. The Bohm-like trajectories that the geometric algebra solutions to the Dirac theory are able to compute as streamline plots are illuminating in this respect, since they seem to clearly show that what the Dirac wave equation is modelling is almost certainly not the behaviour a single particle. (One could guess this from Schrödinger theory as well, but I guess physicists were already lured into believing in the literal wave-particle duality meme well before Bohm was able to influence anyone’s thinking.)

Also, it is possible (I do not really know for sure) that the negative frequency solutions in Dirac theory can be viewed as merely an artifact of the statistical ensemble framework. No single particle acts truly in accordance with the Dirac wave equation. So there is no real reason to get ones pants in a twist about the awful appearance of negative frequencies.

(For those in-the-know: the Dirac theory negative frequency solutions turn out to have particle currents in the reverse spatial direction to their momenta, so that’s not a backwards time propagating anti-particle, it is a forwards in time propagating negative mass particle. That’s a particle that’d fall upwards in a gravitational field if the principle of equivalence holds universally. As an aside note: it is a bit funky that this cannot be tested experimentally since no one can yet clump enough anti-matter together to test which way it accelerates in a gravitational field. But I presume the sign of particle inertial mass can be checked in the lab, and, so far, all massive particles known to science at least are known to have positive inertial mass.)

And as a model of reality the Dirac equation has therefore, certain limitations and flaws. It can get some of the statistics correct for particular experiments, but a statistical model always has limits of applicability. This is neither a defense or a critique of Dirac theory. My view is that it would be a bit naïve to regard Dirac theory as *the* theory of electrons, and naïve to think it should have no flaws. At best such wave-function models are merely a window frame for a particular narrow view out into our universe. Maybe I am guilty of a bit of sophistry or rhetoric here, but that’s ok for a WordPress blog I think … just puttin’ some ideas “out there”.

Then another interesting confluence is that one of Penrose’s big projects in Twistor theory was to do away with the negative frequency solutions in 2-Spinor theory. And I think, from recall, he succeeded in this some time ago with the extension of twistor space to include the two off-null halves. Now I do not know how this translates into real-valued geometric algebra, but in the papers of Doran, Lasenby and Gull you can find direct translations of twistor objects into geometric algebra over real numbers. So there has to be in there somewhere a translation of Penrose’s development in eliminating the negative frequencies.

So do you feel a new research paper on Dirac theory in the wind just there? Absolutely you should! Please go and write it for me will you? I have my students and daughters’ educations to deal with and do not have the free time to research off-topic too much. So I hope someone picks up on this stuff. Anyway, this is where maybe the GA reworking of Dirac theory can borrow from twistor theory to add a little bit more insight.

There’s another possible confluence with the main unsolved problem in twistor theory. The Twistor theory programme is held back (stalled?) a tad (for 40 years) by the “googly problem” as Penrose whimsically refers to it. The issue is one of trying to find self-dual solutions of Einstein’s vacuum equations (as far as I can tell, I find it hard to fathom twistor theory so I’m not completely sure what the issue is). The “googly problem” stood for 40 years, and in essence is the problem of “finding right-handed interacting massless fields (positive helicity) using the same twistor conventions that give rise to left-handed fields (negative helicity)”. Penrose maybe has a solution dubbed Palatial Twistor Theory which you might be able to read about here: “On the geometry of palatial twistor theory” by Roger Penrose, and also lighter reading here: “Michael Atiya’s Imaginative Mind” by Siobhan Roberts in Quanta Magazine.

If you do not want to read those articles then the synopsis, I think, is that twistor theory has some problematic issues in gravitation theory when it comes to chirality (handedness), which is indeed a problem since obtaining a closer connection between relativity and quantum theory was a prime motive behind the development of twistor theory. So if twistor theory cannot fully handle left and right-handed solutions to Einstein’s equations it might be said to have failed to fulfill one it’s main animating purposes.

So ok, to my mind there might be something the geometric algebra translation of twistor theory can bring to bear on this problem, because general relativity is solved in fairly standard fashion with geometric algebra (that’s because GA is a mathematical framework for doing real space geometry, and handles Lorentzian metrics as simply as Euclidean, not artificially imposed complex analytic structure is required). So if the issues with twistor theory are reworked in geometric algebra then some bright spark should be able to do the job twistor theory was designed do do.

By the way, the great beauty and advantage Penrose sees in twistor theory is the grounding of twistor theory in complex numbers. The Geometric Algebra Research Group have pointed out that this is largely a delusion. It turns out that complex analysis and holomorphic functions are just a sector of full spacetime algebra. Spacetime algebra, and in fact higher dimensional GA, have a concept of monogenic functions which entirely subsume the holomorphic (analytic) functions of 2D complex analysis. Complex numbers are also completely recast for the better as encodings of even sub-algebras of the full Clifford–Geometric Algebra of real space. In other words, by switching languages to geometric algebra the difficulties that arise in twistor theory should (I think) be overcome, or at least clarified.

If you look at the Geometric Algebra Research Group papers you will see how doing quantum mechanics or twistor theory with complex numbers is really a very obscure way to do physics. Using complex analysis and matrix algebra tends to make everything a lot harder to interpret and more obscure. This is because matrix algebra is a type of encoding of geometric algebra, but it is not a favourable encoding, it hides the clear geometric meanings in the expressions of the theory.

* * *

So far all I have described is a breezy re-awakening of some old ideas floating around in my head. I rarely get time these days to sit down and hack these ideas into a reasonable shape. But there are more ideas I will try to write down later that are part of a patch-work that I think is worth exploring. It is perhaps sad that over the years I had lost the nerve to work on topological geon theory. Using spacetime topology to account for most of the strange features of quantum mechanics is however still my number one long term goal in life. Whether it will meet with success is hard to discern, perhaps that is telling: if I had more confidence I would simply abandon my current job and dive recklessly head-first into geon theory.

Before I finish up this post I want to thus outline very, very breezily and incompletely, the basic idea I had for topological geon theory. It is fairly simplistic in many ways. There is however new impetus from the past couple of years developments in the Black Hole firewall paradox debates: the key idea from this literature has been the “ER=EPR” correspondence hypothesis, which is that quantum entanglement (EPR) might be almost entirely explained in terms of spacetime wormholes (ER: Einstein-Rosen bridges). This ignited my interest because back in 1995/96 I had the idea that Planck scale wormholes in spacetime can allow all sorts of strange and gnarly advance causation effects on the quantum (Planckian) space and time scales. It seemed clear to me that such “acausal” dynamics could account for a lot of the weird correlations and superpositions seen in quantum physics, and yet fairly simply so by using pure geometry and topology. It was also clear that if advanced causation (backwards time travel or closed timelike curves) are admitted into physics, even if only at the Planck scale, then you cannot have a complete theory of predictive physics. Yet physics would be deterministic and basically like general relativity in the 4D block universe picture, but with particle physics phenomenology accounted for in topological properties of localised regions of spacetime (topological 4-geons). The idea, roughly speaking, is that fundamental particles are non-trivial topological regions of spacetime. The idea is that geons are not 3D slices of space, but are (hypothetically) fully 4-dimensional creatures of raw spacetime topology. Particles are not apart from spacetime. Particles are not “fields that live in spacetime”, no! Particles are *part of spacetime*. At least that was the initial idea of Geon Theory.

Wave mechanics, or even quantum field theory, are often perceived to be mysterious because they either have to be interpreted as non-deterministic (when one deals with “wave function collapse”) or as semi-deterministic but incomplete and statistical descriptions of fundamental processes. When physicists trace back where the source of all this mystery lies they are often led to some version of non-locality. And if you take non-locality at face value it does seem rather mysterious given that all the models of fundamental physical processes involve discrete localised particle exchanges (Feynman diagrams or their stringy counterparts). One is forced to use tricks like sums over histories to obtain numerical calculations that agree with experiments. But no one understand why such calculational tricks are needed, and it leads to a plethora of strange interpretations, like Many Worlds Theory, Pilot Waves, and so on. A lot of these mysteries I think dissolve away when the ultimate source of non-locality is found to be deep non-trivial topology in spacetime which admits closed time-like curves (advanced causation, time travel). To most physicists such ideas appear nonsensical and outrageous. With good reason of course, it is very hard to make sense of a model of the world which allows time travel, as decades of scifi movies testify! But geon theory doe snot propose unconstrained advanced causation (information from the future influences events in the past). On the contrary, geon theory is fundamentally limited in outrageousness by the assumption the closed time-like curves are restricted to something like the Planck scale. I should add that this is a wide open field of research. No one has worked out much at all on the limits and applicability of geon theory. For any brilliant young physicists or mathematicians this is a fantastic open playground to explore.

The only active researcher I know in this field is Mark Hadley. It seemed amazing to me that after publishing his thesis (also around 1994/95 independently of my own musings) no one seemed to take up his ideas and run with them. Not even Chris Isham who refereed Hadley’s thesis. The write-up of Hadley’s thesis in NewScientist seemed to barely cause a micro-ripple in the theoretical physics literature. I am sure sociologists of science could explain why, but to me, at the time, having already discovered the same ideas, I was perplexed.

To date no one has explicitly spelt out how all of quantum mechanics can be derived from geon theory. Although Hadley I surmise, completed 90% of this project! The final 10% is incredibly difficult though — it would necessitate deriving something like the Standard Model of particle physics from pure 4D spacetime topology — no easy feat when you consider high dimensional string theory has not really managed the same job despite hundreds of geniuses working on it for over 35 years. My thinking has been that string theory involves a whole lot of ad hockery and “code bloat” to borrow a term from computer science! If string theory was recast in terms of topological geons living as part of spacetime, rather than as separate to spacetime, then I suspect great advances could be made. I really hope someone will see these hints and connections and do something momentous with them. Maybe some maverick like that surfer dude Garett Lisi might be able to weigh in and provide some fire power?

In the mean time geometric algebra has so not been applied to geon theory, but GA blends in with these ideas since it seems, to me, to be the natural language for geometric physics. If particle phenomenology boils down to spacetime topology, then the spacetime algebra techniques should find exciting applications. The obstacle is that so far spacetime algebra has only been developed for physics in spaces with trivial topology.

Another connection is with “combinatorial spacetime” models — the collection of ideas for “building up spacetime” from discrete combinatorial structures (spin foams, or causal networks, causal triangulations, and all that stuff). My thinking is that all these methods are unnecessary, but hint at interesting directions where geometry meets particle physics because (I suspect) such combinatorial structure approaches to quantum gravity are really only gross approximations to the spacetime picture of topological geon theory. It is in the algebra which arises from non-trivial spacetime topology and it’s associated homology that (I suspect) combinatorial spacetime pictures derive their use.

Naturally I think the combinatorial structure approaches are not fundamental. I think topology of spacetime is what is fundamental.

* * *

That probably covers enough of what I wanted to get off my chest for now. There is a lot more to write, but I need time to investigate these things so that I do not get too speculative and vague and vacuously philosophical.

What haunts me most nights when I try to dream up some new ideas to explore for geon theory (and desperately try to find some puzzles I can actually tackle) is not that someone will arrive at the right ideas before me, but simply that I never will get to understand them before I die. I do not want to be first. I just want to get there myself without knowing how anyone else has got to the new revolutionary insights into spacetime physics. I had the thrill of discovering geon theory by myself, independently of Mark Hadley, but now there has been this long hiatus and I am worried no one will forge the bridges from geon theory to particle physics while I am still alive.

I have this plan for what I will do when/if I do hear such news. It is the same method my brother Greg is using with *Game of Thrones*. He is on a GoT television and social media blackout until the books come out. He’s a G.R.R. Martin purest you see. But he still wants to watch the TV adaptation later on for amusement (the books are waaayyy better! So he says.) It is surprisingly easy to enforce such a blackout. Sports fans will know how. Any follower of All Black Rugby who misses an AB test match knows the skill of doing a media blackout until they get to watch their recording or replay. It’s impossible to watch an AB game if you know the result ahead of time. Rugby is darned exciting, but a 15-aside game has too many stops and starts to warrant sitting through it all when you already know the result. But when you do not know the result the build-up and tension are terrific. I think US Americans have something similar in their version of Football, since American Football has even more stop/start, it would be excruciatingly boring to sit through it all if you knew the result. But strangely intense when you do not know!

So knowing the result of a sports contest ahead of time is more catastrophic than a movie or book plot spoiler. It would be like that if there is a revolution in fundamental physics involving geon theory ideas. But I know I can do a physics news blackout fairly easily now that I am not lecturing in a physics department. And I am easily enough of an extreme introvert to be able to isolate my mind from the main ideas, all I need is a sniff, and I will then be able to work it all out for myself. It’s not like any ordinary friend of mine is going to be able to explain it to me!

If geon theory turns out to have any basis in reality I think the ideas that crack it all open to the light of truth will be among the few great ideas of my generation (the post Superstring generation) that could be imagined. If there are greater ideas I would be happy to know them in time, but with the bonus of not needing a physics news blackout! If it’s a result I could never have imagined then it’d be worth just savouring the triumph of others.

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