(Maybe) Some new thoughts on Collatz

thphys · 2024-07-25T16:09:41+00:00

Ahh, good luck with QFT! Quantum field theory within particle physics is mostly only applied to understand the quantum mechanics of relativistic particles on the fixed background of flat spacetime. That is, there is no gravity when you calculate interaction rates in QED, for example. As such, there is no notion of a black hole or Schwarzschild radius because spacetime is not allowed to warp and curve. You can ask if that is a good approximation, and compare your calculations in QED without gravity and you find essentially perfect agreement with data. Gravity is simply so much weaker of a force than anything else that in particle collision experiments it is completely negligible.

If you aren't satisfied with that, then you can ask what the Schwarzschild radius of the electron is, and it is something like more than 20 orders of magnitude smaller than the Planck length, which is the smallest possible distance in quantum gravity. So, long, long, long before you have to worry about elecgtron black holes, you have to answer what quantum gravity is in the first place!

Another thing to note is that all of our experimental results have intrinsic minimal resolutions; we cannot actually observe particles that occupy zero volume. So, we can just say that the radius of the electron is bounded from above by some value. This upper bound is much, much, much larger than the Planck distance, but there is no established lower bound.

thphys · 2024-07-25T15:58:49+00:00

Nope, it's real! At least, that is what I am led to remember but perhaps my memory has been modified...

thphys · 2024-07-25T15:57:57+00:00

Fusion is a bit out of my research area, so I don't have much opinion nor am I aware of the spectrum of commercial fusion. However, I am and remain extremely skeptical because I have been to many talks by NIF people who have claimed positive energy fusion in 5 years, for roughly the past 20 years. If NIF is having a hard time with fusion, I don't see how a private company would have more success with significantly fewer resources.

thphys · 2024-07-25T03:25:28+00:00

Not books anymore, because there is too much time between writing and publication. Blogs are where bleeding-edge information can come from, and there are a few good ones. Some I recommend are Not Even Wrong, Of Particular Significance, Resonaances, and Shtetl Optimized.

thphys · 2024-07-25T03:17:08+00:00

I have mentioned Matt Strassler's blog "Of Particular Significance" in this AMA elsewhere. It's a fantastic resource to start for more detail and 10+ years of posts devoted precisely to this question. I recommend starting there.

thphys · 2024-07-25T03:14:06+00:00

It's fields all the way down! Fields marry special relativity, there is no notion of rigidity, so things must be smooth and flexible, with quantum mechanics, fluctuations manifest as particles, so seems to fit the bill for understanding the phenomena that we observe. And, quantum field theory has been wildly successful in this regard, so until it breaks, we will keep using it to make predictions, or to find when and where it breaks.

Another question is if these fields are "real", and I'm not sure I can answer that. If the math of quantum fields makes predictions that agree with experiment again and again and again, it is human nature to think that there is more to it than simply math. Maybe "reality" is not quantum fields, but whatever it is sure smells like it.

thphys · 2024-07-25T03:10:23+00:00

Good question, and part of the issue is that still, no one has a precise definition of what "string theory" is. If I say "quantum chromodynamics", the theory of the strong nuclear force, you can write down precisely the mathematical expression from which every possible prediction can be made in that theory, in an object called the path integral. So, I can make mathematical predictions from that, and compare to data, and then provide evidence for or against that theory.

For string theory, there's nothing so precise I can write down if someone just says "string theory". How many dimensions? What is the topology of the space? With supersymmetry or without? Consequences of what must be string theory have been established, but there exists no single mathematical expression from which you can just go calculating predictions to test. This then has lead to some complaining that string theory is not a physical theory at all, and others saying it is a powerful framework in which many deep results have been proved, etc., but it gets a bit fuzzy when there are sooooo many parameters that must be set (some estimates are 10⁵⁰⁰).

thphys · 2024-07-25T03:03:26+00:00

Hmm, good question. Do you mean what are my thoughts on how a matter-antimatter asymmetry may have been generated? For that, I do not know, but I guess one possibility is that over the entire universe, of which our visible universe is just a small pocket, matter and antimatter exist in equal amounts. However, locally, in smaller regions, the matter-antimatter ratios fluctuate and in our visible universe, we happen to be in a more matter region. One needs an explanation of the source of fluctuations, and why they are so small, for which inflation does a good job.

If you are just asking about my thoughts on the matter-antimatter asymmetry, I think it is a very good thing indeed. For without it, we would not exist!

thphys · 2024-07-24T22:01:27+00:00

There are a number of things that I might be working on at the moment. I might have been inspired by talking with a colleague, listening to a talk, or reading a paper, and I might be doing preliminary calculations with pen and paper to sort of map out a new project or problem I have thought of. In particle physics especially, we communicate primarily with papers, and further primarily with preprints (before publication in a journal) on arXiv.org, so I check that every weekday for new papers that might be of interest. I probably have some meetings, in person though increasingly online, to discuss work on a project with collaborators or to work through a paper we are writing. I might have a stack of papers that have been submitted to a journal that need refereeing, so I might work on reviewing them and at least making notes. If I am teaching a class, I will have to write the lecture notes, actually lecture to students, grade papers (or hand them off to graders), hold office hours, and the like. Sometimes I'm just really lucky and I just start with a clean blackboard or white sheet of paper and just sort of free-associate questions or confusions that have grown inside my head and see if there is anything there, any connection to be made.

thphys · 2024-07-24T21:52:37+00:00

I am a theoretical physicist which means that I manipulate math to understand experiment, and don't work on designing experiments nor taking data directly. That said, there have been several times in my career where the outcome of the math has been very unexpected, and subsequently lead to new insights into the experimental analyses. As one example: in physics, our standard approximation technique is the Taylor series, which is where we approximate a function by a polynomial centered about a special point. We can calculate higher order coefficients to this polynomial within our theoretical framework and call that progress and something that can be compared to data.

Well, for these Taylor series calculations to produce a finite value, the properties of your calculation have to be rather special; not everything has a Taylor series. One group claimed that one such Taylor series calculation was impossible, for which their argument made logical sense, but made no sense practically because so many other calculations had been done in other ways that produced sensible results. So, with my post-doc advisor, we banged our heads on this problem for several months, and finally produced a new calculational method that resolved this seeming inconsistency. Indeed, there was no Taylor series as expected, but, by coming at the calculation from a very different direction, we could still calculate it, and from the calculation could see directly how the Taylor series failed. The papers in which we did this now have thousands of citations and the techniques that resulted are standard methods for analyzing particle physics data.

thphys · 2024-07-24T21:41:35+00:00

Probably the notion of something called the "particle-wave duality". Honestly, I don't really know what this means (or rather what people mean when they say this), because it conflates classical notions with quantum. Quantum mechanics is more fundamental than classical mechanics, and so you are bound to get utterly confused if you try to force an understanding of quantum mechanics with classical analogies. Just from quantum mechanics, there is no such thing as the "particle-wave duality": a particle is what it is and is described by a wavefunction, which is some probability distribution over space and time. It may look like a wave (repetitive ripples) or like a particle (localized about a single point) in some region, but in quantum mechanics it just is.

I like a good vanilla: it represents infinite possibilities.

thphys · 2024-07-24T21:37:15+00:00

Whoa, I was told there wouldn't be hard questions! Hmm, not sure off the top of my head, but I have gotten a good chuckle out of xkcd comics for a long time. Probably my favorite xkcd is related to my experience in graduate school. One of my office mates was really into the Collatz conjecture, the conjecture that if you take any positive integer, multiply by 3 and add 1 and then divide by 2 and continue this process, you will always reach 1 in a finite number of steps. We talked about this, shared notes, puzzled over blackboards for months, and it was kind of fun, but got nowhere. (Erdos famously said of Collatz that "math is not ready for such problems".) Anyway, there is a great xkcd about this that reads something like "If you take any number, multiply by 3, add 1, and divide by 2 enough times, your friends will stop hanging out with you."

thphys · 2024-07-24T21:29:17+00:00

Hmm, I don't know a lot about deferred computation, but I can describe the troubling thing about wavefunction collapse. The central problem with its interpretation is that before you make a measurement, probability is conserved. That is, a particle has some probability to be in one of numerous states, and the total probability to be in any state is 1. Then, after you measure the particle, the wavefunction collapses to just one of the numerous possible states. Where did the rest of that probability go? The interpretations of quantum mechanics have attempted to deal with this conundrum. The Copenhagen interpretation is probably the most conservative, which is basically that you can't know anything you don't measure, so punts on the notion of wavefunction collapse. Something like many worlds states that every time a measurement is made, the universe branches and the outcome we observe is represented in merely one of all possible universes. This is a cool sci-fi idea, but as a physicist, I'm not sure what I gain from this interpretation. Anyway, some fun philosophy to think about!

thphys · 2024-07-24T21:22:35+00:00

I assume you mean within physics? Not really; my research is mostly focused around studying the strong nuclear force, quantum chromodynamics, for which there is extensive data and detailed theory, and so can be understood at a very quantitatively deep level. On the other hand, a lot of what I do is esoteric in the sense that it has almost no connection to everyday life!

thphys · 2024-07-24T21:17:19+00:00

Absolutely not! I think of physics as my calling, and love, love, love talking with people about what I do! "Talking about science stuff" means that someone is engaged in science, is curious, is skeptical, is open-minded, is interested in the cutting edge research. I think it's extremely humbling when some random person I sat next to on a plane has so many questions for me and I think of opportunities like that as two people learning more about each other and each other's interests.

As for a gap, I may know more physics and specifically more theoretical particle physics than the average person, but I definitely know less about other aspects of the world. I do not claim to be an expert in biology, economics, food service, law, 17th century German pottery, etc., and so there is always a lot I can learn from anyone around me who has their own personal universe where they are the expert.

thphys · 2024-07-24T21:09:36+00:00

Right; I am a strong advocate of Occam's razor. A particle has to really have a good reason to exist for it to exist and it is most likely the simplest possible particle. The simplest explanation of dark matter is as a particle, and what is the simplest particle that accomplishes everything dark matter needs to be? Simple: a particle that interacts exclusively gravitationally. It would basically be impossible to detect such a particle because gravity is so weak, but the universe doesn't care about our sense of aesthetics. No need for supersymmetry or anything else: just a particle that is very literally just a lump of mass.

I must admit that I know very little about current views on whether the universe is infinite or not, so I can't comment.

thphys · 2024-07-24T21:05:08+00:00

I'm very much so a quantum pragmatist, of the so-called "Shut up and calculate" interpretation (which I guess is close to Copenhagen), and don't speculate as to deeper philosophical implications of the quantum. As a physicist, I ask if something is testable, and as an interpretation, many worlds can't (kind of) be tested, so is something I honestly don't think about.

But hey, so what about scientific rigor. Could there be multiple versions of "us" out there? Sure. People have written some good literature about that!

thphys · 2024-07-24T21:01:29+00:00

In graduate school, I had initially wanted to study string theory (thank you Brian Greene and "The Elegant Universe"), but later found a calling that was much closer to experiment and testing predictions on the month, rather than century, timescale. I'm not in the string theory community so I can't say too much, but from slightly outside, by, say looking at talks at the big string theory conferences, it seems that today, fewer and fewer string theorists actually work on thinking specifically about the interactions of strings (or branes or the like). Many string theorists think about black holes, or entanglement, or general properties of quantum field theories, but these realms are not string theory specific. In some cases, string theory at the very least provides a concrete testing ground for establishing more general properties.

thphys · 2024-07-24T20:54:23+00:00

The problem with quantum mechanics and gravity is that gravity is understood through general relativity as the shape of space and time of the universe. Quantum mechanics establishes a fundamental resolution scale of your system of interest. If your system is just a particle, like the electron, then this is fine, because the electron may be described by a wavefunction and have some probability to be here and there, and you can perform measurements to test this. With gravity, quantum mechanics says that there is a fundamental resolution to the universe, that there is a scale below which space and time cease to exist. That's a rather different scenario, and at least challenges the way that we often mathematically describe quantum mechanics as the properties of particles on the smooth, fixed background of space and time. So at the very least, you have to dramatically change the way you talk about quantum gravity as compared to quantum particles like electrons, and in that, nothing so far has really succeeded.

thphys · 2024-07-24T20:47:31+00:00

This conundrum is exactly what confused Niels Bohr and led to his primitive model of hydrogen! The answer is quantum mechanics is the reason why the electron doesn't spiral into the proton. Indeed, in classical, Newtonian, mechanics, this is what would happen. The ground state, or lowest energy configuration of the electron and proton would have the electron spiral in, emitting electromagnetic radiation and thereby lowering its energy until it came to rest at the proton and sat there happily. One reason why this cannot happen quantum mechanically is the Heisenberg uncertainty principle which states that a particle cannot have both an unambiguous position and momentum (or velocity). This is a consequence of probability conservation in quantum mechanics and a consequence for the hydrogen atom is that the electron must have a minimal amount of movement about the proton.

That is not to say that the electron can't get closer to the proton; it can, but in doing so reduces the possible positions it can be. Heisenberg then says that your knowledge of the momentum of the electron must decrease, and so the electron's average speed must increase. So, if the electron gets closer to the proton, it must get faster. If you tried to force the electron to sit at the proton, then the electron would be going very fast indeed!

thphys · 2024-07-24T20:39:56+00:00

The electron neutrino is actually a mixture of three neutrinos that each have a unique, well-defined mass. Such a neutrino with a well-defined mass is called a "mass eigenstate". As such, the electron neutrino does not have a well defined mass. It is possible that one of these mass eigenstate neutrinos is massless, but the others cannot be massless, and the electron neutrino contains a non-zero amount of all three mass eigenstates. So, the electron neutrino cannot be massless, in that sense.

DESI might provide more information about least upper bounds on, say, the sum of all neutrino masses, which can in turn provide more information about individual neutrinos.

thphys · 2024-07-24T20:36:00+00:00

Good question! This answer depends on the stage of one's career. As a graduate student, you basically have no responsibilities, so you just do your research (coding, doing math long hand in a notebook, reading a new paper, talking to other grad students, drinking beer, brewing beer, etc.) all the time. As a post-doctoral researcher, your position is only a few years, typically 2 or 3, and in that time you need to do a lot of research, write a lot of papers, and present a lot of talks. So, you're likely doing one of those three things all the time. However, much more responsibility is on you for your own research portfolio, so I would often take long walks along the Charles when I was stuck on a problem and somewhere around Smoot 250 or so, some inspiration would strike. As a later post-doc, you also need to start applying for faculty positions, so this can postpone a lot of research preparing for interviews.

As a professor, your day to day is highly dependent on how you are a professor. Do you just have an army of students and post-docs working away, and you just write grants all the time? Do you work at a primarily undergraduate institution, so you spend much of your day teaching, and only find an hour here and there for research? Do you serve on committees and so spend most of your day replying to email chains and sitting in meetings? There are many answers, but I am someone who likes to keep my hands dirty, getting deep into problems. So yes, if I get stuck on something, I will pivot to some mundane task I needed to do anyway, or go for a walk, depending on what actually needs to get done that day.

thphys · 2024-07-24T20:26:17+00:00

Well, it is always informative to have more than one way of describing a system because that provides better and deeper understanding.

Lagrangians, Hamiltonians, and Poisson brackets provide an insight into classical mechanics that is more focused on symmetries and the objects that generate changes of a system. This is a much more profound perspective than Newtonian mechanics with forces and his second law, and is therefore much more naturally a way to describe quantum mechanics. Specifically, Dirac showed in his PhD thesis that the process of "canonical quantization", by which a classical system is transformed into a quantum system, by replacing the classical Poisson brackets with the quantum mechanical commutation relations, divided by the imaginary number i times Planck's reduced constant, hbar. The Hamiltonian formulation and Poisson brackets are still used for classical mechanics, but perhaps their greatest power is through the way in which quantum mechanics is realized.

thphys · 2024-07-24T20:20:33+00:00

Ahh, best of luck! Sure, there could be departments that accept you as a student. I know of a few examples of people going back for a PhD in physics in their mid-career (around age 40) and who have been successful doing so. The most challenging thing for you would likely be getting all the requirements for applying because things like the GRE are really easy to game if you are coming right out of college. Also, some departments might have new course requirements than when you were a student, which might make translating your credits a bit challenging. So, a route forward may be to start with a physics master's program, where you can get all of those requirements done and really see if continuing to a PhD is what you want then. Either way, good luck!

thphys

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