A Discussion on a Electron vs Positron Mass Paper

I have paper that is heavy in math, that basically no one I know is reading and its in a preprint state; paper. I find it really interesting and believe it has a lot of potential although far from finished. So I would like to explain in simple language what it contains in ideas and results for a broader audience. Also this post is about to spread ideas. My approach might not be the best and better approaches can be done picking up on some of the novel ideas within the paper.

Overview

In Simple Terms:

Particles as Natural Limits: Imagine there's a limit to how much energy a certain space can hold. This limit creates stable particles when it's reached, like a balloon becoming stable when it's full of air.
Challenge with Lorentz Invariance: In physics, there's a rule that says experiments should work the same way regardless of how fast you're moving (as long as it's a constant speed). But when trying to model the energy limits mentioned above, this rule breaks.
Speed of Light as a Solution: If everything is defined based on the speed of light (which is constant and doesn't change), the challenges can be overcome. The approach is to make use of limiting arguments by approaching the speed of light and study how different aspects converges.
Particle Formation through Light Speed Currents: Using some complex math, we can model particles as currents approaching the speed of light. This method simplifies the analysis.
Derive Niels Bohr's assumption of angular momentum: The paper suggests that the angular momentum is a constant and the same formula, that Bohr assumed, that relates Planck's constant to the angular momentum.
Understanding the Mysterious Number 137: This number has puzzled many great minds in physics. In the paper, particles are modeled in a unique way that relates to this number, giving an insight into a long-standing mystery.
Electron vs. Positron: While traditionally thought of as equal in mass, this approach suggests a slight difference between them. This is potentially revolutionary as it goes against conventional wisdom.
Potential Implications: Discoveries like these can change our understanding of the universe and could lead to new technologies or solutions to big mysteries in physics.

Key Takeaway:

The paper explores a novel approach to understanding fundamental particles using the concept of energy limits and making sense of a speed of light reference frame. This approach not only offers insights into the mysterious number 137 in physics but also hints at a potential difference between the masses of the electron and positron that can be resolved with measurements project now in the pipeline.

The Detailed Overview

The basic idea of introducing limits

The basic idea is that particles result from certain limiting features of nature. The paper provides an example, possibly the simplest one, of how to develop such a theory. A theory that passes the special relativity test, that is necessary for fundamental theories. This means, in principle, that physics is the same independent of the Galilean reference system in which you conduct experiments. For the layperson, a Galilean reference system can be explained as a system that moves at a constant speed or not at all. Technically, we cannot determine the location of the rest frame, and the existence of a universal rest frame is typically dismissed. This means that velocity is always defined relative to other Galilean reference systems. In technical terms, experiments in these systems being invariant (or constant) means that we require the theory to be Lorentz invariant.

The natural theory to start with is, in a sense, electromagnetism (a theory that combines electricity and magnetism) as it plays a significant role in the particle world and is, in a sense, foundational. This theory has this property of Lorentz invariance. The next step is to figure out how to introduce a limit for e.g. how much energy density (or similar quantity) space can cope with (an analogy is that you cannot pack an infinite number of persons in an elevator - there is a limit). Mathematically this is a nonlinearity (think about linearity as straight lines, planes, ...) introduced into the equations. The simplest approach to add them is to just use limits and say that below x, we have the usual theory, and and all values above are forbidden.

The attractive thing with this is that particle formations becomes in a sense natural. And this limiting behavior has the potential to fix the electric and magnetic fields into a stable self contained system. A system where the most extreme parts reaches these boundaries. Mathematically speaking we can analyze this by introducing mathematical constraints in addition to the usual equations and solve them. Finally under these constraints energy (related to work) should be minimized and then you should, if all goes well, get your particle. In essence the paper linked above, takes on this project, and derive the mathematical consequences and produces a formula for the total energy of the confined system (particle). Hence also mass through Einsteins famous \(E=mc^2\). However, adding these constants is practically impossible it turns out. It always breaks the Lorentz invariance property. To see this complexity, one just need to note that the electric and magnetic parts changes as you change the observers reference system. E.g. just by letting the observe move, the system will potentially not reach the limit which is nonphysical. Also there is the complexity to put a limit on both of the electric and magnetic parts. It is also really messy and hard to do mathematics on it. One simply gets stuck.

The way out, an intriguing idea of using the speed of light.

However, we do have an invariant (constant); The speed of light. And if we somehow could define physics in a reference system moving at the speed of light, we can hope that these limits could be defined there. And then accomplish that the model satisfy special relativity. This is outside the classical domain and the only possible approach to do this mathematically is to define a unique limiting procedure and use properties at the limit, as a modeling tool. What you practically do is creating an infinite sequence_of reference systems and charge configurations (that are at rest in the limit, in the approach used here). Then show that all interesting quantities indeed have a unique mathematical limit, as these systems goes to infinity. If so; the model is well defined mathematically. Also we would have good hope that the resulting model is Lorentz invariant, as each construct in the sequence has this property.

The paper takes that approach and model all currents as a limiting process of currents reaching the speed of light and with no mass in the limit. The paper suggests that this implies that those non traditional objects satisfy a new quite simplifying and odd model. The mathematics for this model can, as it turns out, be analyzed with pen and paper. For example you only need to consider the electrical field (properties of the electric phenomena, spread out in space) and its constraints in this model. This is actually an interesting observation as the heavily used traditional Dirac equation is defined only through just the electrical part or more precise; the electrical potential.

The resulting odd interaction model.

This odd interaction depends on the fact that orthogonal lines (they cross each other like a plus sign) is preserved in all Galilean reference frames although according to special relativity there is a length contraction when reaching the speed of light. The thing with this new interaction between two different straight currents are that one only consider cases where theses are parallel. Let's call two points, normal points if the connecting lines between two points on both of them form a "plus" at both points, or in math lingo; the line between them is a normal to both points. Now loops can curve in all possible ways generally, as a loop can mean just that you have a deformed rubber band, that connect to it self. The crucial thing is that we associate the lines to be defined as lines that touches the loop curve as smooth as possible. Mathematically these lines are called tangent lines. In this model the interaction is only for pair pair of points on two different loops, and where these points are normal points using the tangent lines. Finally, the limiting procedure used means that in the limiting reference system all charges are at rest, at least if the loop does not move relative the observer. This means that there is no magnetic phenomena in the limiting reference system as that needs the charges to move. This model is hence very attractive, as an example, as it simplify a lot of analysis. It cut the complexity in half figuratively speaking. At the same time this challenges contemporary researchers intuition about how electromagnetism works; to the degree that one could consider it as a completely new kind of model.

The idea is to work with such current loops in different constellations and see what those implies. The first basic setup is to put two such current loops, with different signs of the charges, as concentric circles (circles with the same center) in a plane. In this new model that configuration has really nice properties. Like being a good candidate for a stable in-plane system. And if we also put slightly different limits on the "energy density" depending on the sign of the charges, which is considered as a continuum (no point charges) along the current loop in the observer's reference system, we can get a final candidate of a confined stable system. The charges will cancel each other to a degree and produce the fundamental charge that the electron and positron has but with different signs. There are two ways to set up this system that can be interpreted as an electron and its anti particle; the positron.

The result is then a formula for mass. And some more formulae.

Moving into space from the model in a plane.

The next step is to create an even more complex setup in order to reach a stable 3D configuration. In this step, let the currents of different charge signs spirals in a closed loop on a _torus or doughnut, if you want a more vivid and common image.

If you consider the spiraling currents "on the torus", one variation that is even more complex is to let the whole system move at a constant speed \(v\) by turning the torus like turning a car wheel. This will still keep the underlying current loops at the speed of light according to the mathematical construction. However it will introduce some new features as one "move the energy in a circle" e.g. along the torus. Or using Einstein's famous correspondence between mass and energy; "one move the mass in a circle". This is like spinning a wheel and we know that a wheel that spins want's to continue to spin; a fact used in e.g. flywheels. This phenomena is captured in the physical derived quantity; angular momentum.

Angular Momentum

We know from atom theory that there is a nice relationship between angular momentum and Planck's constant for electrons, positrons, and all other particles. All measurements indicates that this is a constant for each particle and don't change in magnitude from individual to individual of the same kind.

The paper also argues that the final setup in space, also has for each scale a velocity that lead to a constant angular momentum independent of scale and we also deduces a formula for it that coincides with Bohr's famous assumption of the angular momentum in his revolutionary atom theory.

Let's digress and tell a story about Niels Bohr: When asked why he, a scientist, had a horseshoe above his door, a symbol of superstition, Bohr supposedly said, "Of course I don't believe in it, but I understand it brings you luck, whether you believe in it or not". A funny way to introduce the idea of unexpected results or the intersection of belief and science.

We continue with presenting an important further remark that starts with the angular momentum that directly relates to the important Planck's constant via Bohr's assumption. Interestingly this is derived in the paper and one does not need to assume it.

From Planck's constant one can calculate the mysterious and fundamental constant known as the fine structure constant. This constant is denoted by the Greek letter alpha (\(\alpha\)) in physics papers and literature.

The Mysterious Integer 137

The physicist Richard Feynman, one of the most influential figures in quantum electrodynamics, was fascinated with the number 137 (or more precisely, the approximate value of the inverse of the \(\alpha\)). He once said, "It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man". Feynman was so intrigued by this number that he even had it written on his chalkboard at the time of his death.

Also the earlier quantum physicist Francesco Pauli himself searched all his life for an explanation on why this constant was related to the integer 137.

137 is a prime number (it has not factors other than 1 and itself). In this setup it is argued in the paper that one could think of 137 as the number of turns the spiral takes to form a loop. 137, being a prime number, is also quite natural. This is because if it had factors you could enlarge the loop and let it span many turns until it reconnects and still in the end form a finite loop. That system could then, shrink to something less than 137. It is also energetically unlikely that you can form an infinite long loop which will in this case cover the whole torus, as it would then be unstable. In any case, another observation is that the outer part of the torus will not be at the constraint (or limit), as it is less dense there. It's the inner part of the torus that is most dense. Hence one need to do a slight correction which explain why \(\alpha\) is not exactly related to \(1/137\), but almost.

An Entertaining Numerology Result

There is a joke going around internet that goes something like this. Pauli meets God when arriving in heaven. They walk to a blackboard. Pauli asks God about what the mystery of \(\alpha\) really is and then God begin to write on the blackboard and Pauli nods, then nods again, and again, and... (an infinite number of times).

In one section in this paper I write down a hypothetical (assumes) and quite accurate continuing fraction (an infinite mathematical construction) that describes how \(\alpha\) relates to integer 137. This mathematical object is a lovely reference to this story. I chuckled all day when I found that beautiful formula. This formula was discovered through some trial and error with the numerical value of \(\alpha\) and some motivation for parts of it. Of course it has aspects of numerology (rightly frowned at) and I acknowledge this in the section heading.

Although there are some seriousness to this formula as the beginning of it can be motivated by a correction, that one get from noticing that it's the inner part of the torous, where the limits are reached. And this correction improves the value of \(\alpha\) compared to using \(1/137\).

The mass formula

Anyhow the amazing thing is that the mass formula will, when plugging in all known constants about the electron, lead to how the mass of the positron will differ from the electron. As it turns out, this system is quite extreme, as the torus is very thin, hence also why \(1/137\) is a good approximation of \(\alpha\). You have a relation of about \(1/1000\) between the two circles that defines the torus shape. Also the two loops of different charge sign (positive and negative) is very close to each other.

I found in a conversation with ChatGPT 3.5 (See reference at the end) that it knew about some results about a mass difference between the electron and positron, done by an experiment that matches (associated with MIT according to the bot) quite well, the calculation done in the paper. This is intriguing but not something we can assume is true. I don't know why it is so hard to find a basic reference for this result. I mostly found that they are exactly the same when searching for references in google.

One possibility is that this result is controversial, as it breaks our well established and well verified current theories, that thorough confirmation is needed before it can go public. Another possibility is that ChatGPT "hallucinated".

The current best measurement up to today (see reference below) seems to be 130 ppb (parts per billion) in relative error and it confirms as it should the current theory up to the measurement's precision. You need to go slightly around the order of 10 ppb relative inaccuracy according to the paper, we discuss here, to see a difference between the electron and positron mass. Interestingly there is one paper below in the references, that indicates that there are effort to produce measurements that are in the order of 40 times more accurate then the state of the art. This would mean that difference suggested by the paper here indeed could be verified if this estimate materializes.

Conclusion

Let's contemplate that the art of theory is indeed important.

Paul Dirac, famously predicted the existence of antimatter based purely on the mathematics of his equations. He could've removed the solutions that predicted antimatter, but he trusted the math, and the discovery of the positron validated his work.

This is the art of science methodology and theory when it works in concert with each other. The model idea came before the validation. The paper we discuss here also went into the direction it did with no knowledge whatsoever of what the chat bot suggested - that the masses differed. That's why this information is interesting; albeit can be a simple result of "bot hallucination".

If we discovered a confirmed and robust experimental inconsistency in line with what was suggested by ChatGpt3.5 we might have another story that line up similarly. A story that could potentially change the path of our understanding of the fundamental properties of nature. Improved theoretical understanding often leads to intriguing applications, and we can only speculate about how it might enhance our world. The most immediate and evident need is for new, safe, and cost-effective energy production methods. Also there are still a lot of scientific unsolved mysteries out there like dark matter, dark energy, and a unified theory that combines general gravity with our particle physics models. Mysteries that may be solved by a development of our understanding of nature's fundamental properties.

References

An overview of A measurement technology for measuring mass of e.g., the positron

Penning traps as a versatile tool for precise experiments in fundamental physics K Blaum, YN Novikov, G Werth Contemporary Physics, 2010•Taylor & Francis

Current best accuracy of 130 ppb relative measurement error

Ding, Y.; Olewiler, T.D.; Rawnak, M.F. Penning-Trap Searches for Lorentz and CPT Violation. Symmetry 2021, 13, 1703. https://doi.org/10.3390/sym13091703

Discusses new measurements of positron mass that are in the pipeline (40X)

X Fan, TG Myers, BAD Sukra, G Gabrielse - Physical review letters, 2023 - APS

My ChatGpt3.5 conversation

https://chat.openai.com/share/acf77625-9d7f-4aa5-a199-fe62699862b4

Special relativity

Einstein, Albert. Relativity: The Special and the General Theory. Crown, 2016.

Mass equivalence and its connection to symmetry.

Close, Frank. The Infinity Puzzle: Quantum Field Theory and the Hunt for an Orderly Universe. Basic Books, 2013.

About Richard Feynman

Feynman, Richard P., as told to Ralph Leighton. Surely You're Joking, Mr. Feynman! (Adventures of a Curious Character). W. W. Norton & Company, 1997.

About Paul Dirac

Farmelo G. The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom. Basic Books; 2009. ISBN-13: 978-0465018277.

About Niels Bohr

Moore R. Niels Bohr: The Man, His Science, and the World They Changed. Alfred A. Knopf; 1966. ISBN-13: 978-0394445099.

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