Thinking about giving up physics

[Geometry_dude]

Hi there,

it seems that there are some good physicists and also some good people around here, so I'm just going to try this. This is going to be a bit long, so maybe just browse over it, if you feel like helping. It is the story of my "career" as a physicist so far.

I have a bachelor's degree in physics from a German university and I actually did quite well during my studies there. However, when it came to writing my bachelor's thesis, things went south.

My original goal was to understand, as in really understand, quantum mechanics and while I have gained a lot of insight, I still couldn't explain to my grandma what spin or what a particle actually is without either resorting to half-truths or speculation.

Anyhow, so I went to this one professor who was supposed to be doing research in quantisation and asked him whether I could write a thesis on geometric quantisation. Quantisation seemed to be a fishy thing when we had it in quantum mechanics, so I wanted to take a peek.
He said yes and it actually turned out to be one of the standard topics he was giving out. At that time my knowledge of geometry and topology was rather limited. I didn't know him at all at that point and he said that he would like to have me working mostly on my own, which I took a bit too literal, I must confess.

So I started reading the book he was referring me to and at first I didn't understand a single sentence. After reading up a bit on general manifold theory and symplectic geometry, I was actually able to understand what was going on, even though I sometimes needed half an hour to understand a single sentence. When it came to the part I was supposed to write about, I had a good, though not fundamental, idea of what was going on and I found an error in the book, which was one of the main motivators for something called "metaplectic correction". So I wrote my second supervisor about this and he was saying some things, which turned out to be not very sensible. I then started to question what they were doing and why, and played the skeptic.

Without going too much into detail, what I figured out was that they knew what they were starting out with and what they had to get to, so all they tried was trying to find a sequence of ad-hoc steps to get to the desired result. There was no real physics behind, just mathematical playing around.

It seems to me that we never actually understood what was going on this tiny scale and we were just looking for mathematical structures that seemed to reasonably model what is going on without actually knowing what is going on. I find this incredibly unsatisfactory - as opposed to putting in clear cut principles and then using the math to understand the physics behind, one just uses the math without having any idea what is actually going on.

I know the quantum world is fundamentally different from our macroscopic world, but that doesn't mean it doesn't admit a clear-cut realist interpretation leading the path to an actual understanding what is going on. It seems like physicists have just given up on this.

For this reason, so I believe, we look at this in the wrong direction. We shouldn't start out with a classical description (which is based on principles that directly contradict the few quantum principles we know) and then find one or more corresponding quantum descriptions. We should start with reasonable ("new") principles and see whether we can use these to account for the effects we observe. This was the first time I denounced a scientific concept.

Anyhow, I couldn't support what they were doing there so I just restricted myself to the classical part of the thesis. Obviously, my professor didn't like that at all and when I told him that it is the wavelike-behavior with which one can already do a lot of physics (e.g. derive the Schrödinger equation), he lost his temper and became very patronizing. I remained respectful and polite, but I haven't talked to him since and any attempt in doing so failed.

Anyhow, I then went on to continue my studies in theoretical physics at another university in the Netherlands. So when I started grad school, the first semester we had to dive into quantum field theory. I understood why they were doing what they were doing, where all the equations came from, but a) I was bored by these endless calculations without actually knowing what was going on physically and b) I didn't really believe in what they were doing. Don't get me wrong, I know QED for example is really well tested and stuff, but there's so many ad-hoc assumptions, guessed equations and conceptual problems that I just didn't know what was right and what was wrong anymore. Even worse, they actually quantized a second time! I consider it close to a miracle that they actually made it work and I find it very impressive, but what I was looking for was a coherent, clear theory - like GR.

I wanted to understand on a fundamental, everyday language level what was going on and when I saw what they were doing in renormalization I started to denounce the entire thing. I said to myself "until and here and no further", applied to the mathematics master and didn't even bother to show up to the exam. I talked to the QFT professor and while he was rather helpful and kind, he couldn't help me with my further path. They did admit me to the mathematics master.

Moreover, I didn't really like the GR course at my current university, because they didn't treat stuff in the language of modern differential geometry and also because my mathematical maturity was not developed enough to translate everything into a formal mathematical language.

I prefer to really understand the math first and then dive into the physics, but the physics courses here are designed to make you understand enough math to apply it and then they make you calculate loads and loads of examples just for the sake of making you calculate stuff faster. I felt like an inferior form of a computer algebra system and it seemed like "shut up and calculate" was the main message we were supposed to get. I do not think this is the way physics should be. The lack of mathematical rigor and non-well-definedness of some of the concepts really gave me the rest.

So, currently I study mathematics and while I find what I learn here a lot more interesting and useful, it lacks what I was actually looking for: Trying to understand what this world around us is and how it works.
I believe I could get the master's degree with my abilities, but my situation for this at the moment is far from optimal.

This is because I have trouble making new friends and I have been feeling incredibly lonely and I feel separated from my peers because of the political and scientific views I hold. I have been diagnosed with schizoid personality disorder and related to that I suffer from depressive episodes. I am about to start treatment, but I don't want to do their drugs (they have horrible side-effects!), don't really think they can help me and I am afraid that if I tell them what I think about all day (spacetime geometry) they'll label me totally nuts.

I really like physics and I am very interested in it, but the direction of my own questions is orthogonal to the direction my lectures took. I also believe that I am good at it and that I could actually figure something out, but I know that I need independent guidance and honest discussions with physicists about my ideas to get there.

Do you have any good advice for me?

 

[Vanadium 50]

It sounds like physics is not for you. You're looking for a degree of certainty that it does not provide, along with an emphasis on qualitative over quantitative understanding. A physics education won't provide that.

 

[TheKracken]

Homeomorphic; a user on these forums may have quite a bit to say on this topic. I believe he experienced and believes something similar to what you are saying. I am very interested in what he will have to say.

 

[WannabeNewton]

I can sympathize to an extent because I started learning GR before learning QM and that made it really hard to appreciate QM because as a conceptual framework (if one can even call it that) it leaves a lot to be desired and aesthetically it is unequivocally inferior to GR in every way. The supreme beauty of GR did produce some disillusionment when it came time to learn QM, especially given that QM lacks any and all aesthetic quality. However physics is about calculations and results, and only in a very narrow sense about beauty, so such disillusionment is immaterial.

Mathematical rigor has its merits in physics but it is arguably much more of a burden than a leg up, especially in a theory like QED. It is unfortunate that you were force fed calculations in QFT without being given any connection to the underlying physics but this is not ubiquitous amongst all QFT classes (Sidney Coleman's lectures come to mind as a counter-example). Even then, calculations are the most integral aspect of QFT so it is only pragmatic that one become fluent in QFT calculations before attempting to get down to the conceptual foundations. To stress again, physics is about calculations and so your dislike of your GR course due to a lack of use of modern differential geometry is ill-founded because such a language would be, for the most part, a huge handicap when it comes to actual calculations. There are very few scenarios in GR wherein the language of modern differential geometry would prove more efficient for calculations than the much more graceful abstract index language.

All in all, I think V50 hit the nail on the head. You can't have your cake and eat it too. Have you considered mathematical physics instead?

 

[Geometry_dude]

Thank you guys for the responses, it's good to have an outside perspective.

It sounds like physics is not for you. You're looking for a degree of certainty that it does not provide, along with an emphasis on qualitative over quantitative understanding. A physics education won't provide that.

With all due respect, how can one claim to understand something without knowing or at least having a rough idea of what's behind? What is, for example, the point of calculating trajectories of masspoints without the concept of the center of mass?
Either way, I do agree with you that I started the wrong studies. The program I entered is supposed to prepare students for the (scientific) job market, which implies they have to train them to quickly calculate property $A$ under given conditions. Whether the applied theory is conceptually problematic or even contradictory to other principles of nature is not of importance as long as it is reasonably reliable.
I get it, but I didn't expect it when I applied here. What I expected was a sort of scientific spirit along with the corresponding values and principles - questioning stuff, looking at things from new perspectives, getting rid of one's own misconceptions, connecting to fellow grad students and their views. I was so wrong.

My biggest asset is that (I think) I can get to the bottom of things, understand the concepts behind of whatever topic it might be and then apply this knowledge accordingly. I just happen to like physics, because it's comparibly hard there to get to the bottom of things and I always had a taste for far out stuff. Maybe it is time to let it go and move onto other endeavors...

I can sympathize to an extent because I started learning GR before learning QM and that made it really hard to appreciate QM because as a conceptual framework (if one can even call it that) it leaves a lot to be desired and aesthetically it is unequivocally inferior to GR in every way.

In my mind, GR is a lot more than mere mathematical beauty, it is direct evidence that Einstein was one of the most capable philosophers of nature of the twentieth century. Deep philosophical concepts like causality and the Einstein equivalence principle were melted together with the mathematical superiority of differential geometry. What he did was good old school physics: First think about the principles, then find a suitable mathematical formalism, that is an axiomatic system based on the principles, compare with known laws and constants of nature, try to produce a contradiction (mathematical or physical), go back and change the principles that caused the contradiction, repeat. He was a true master of his subject.

However physics is about calculations and results, and only in a very narrow sense about beauty, so such disillusionment is immaterial.

I totally agree. My problem is that I don't know what I would work on. I'm really interested in what's going on in the quantum realm, because I think that's where nature's biggest mysteries lie. However, I believe, spacetime geometry and spacetime topology has a lot more to do with these strange phenomena than what we want to give those things credit for. Consider this a hypothesis or speculation, whatever you prefer. Fact is that I have not seen any satisfactory theory that is able to explain quantum phenomena in terms of such geometrical notions - and I am convinced that this is the proper framework.
Most of the stuff they're doing in that direction is either completely ad-hoc, fundamentally misguided or so far removed from an actual model that they wouldn't be able to make predictions in decades. Other people's opinion may differ.
Of course, it's always easy to rant when you do not supply a replacement yourself.

All in all, I think V50 hit the nail on the head. You can't have your cake and eat it too. Have you considered mathematical physics instead?

Yes, mathematical physics is pretty much the path I'm going right now. Very often it is said, that mathematical rigor is not that necessary, because the final test for physical theories is experiment.
While the latter is of course true, truth is when you make a mathematical mistake, you make a mathematical mistake (I love tautologies). You cannot apply theorems when they are not applicable (another one!), you cannot apply approximations when they do not hold (that's enough). I have seen how such things have lead to serious problems in well-established theories. Problems that weren't actually there or pseudo-problems like the meaning of the "zero-point-energy", an artifact of a poorly defined "quantization map" that has no predictive value whatsoever.

Math is our tool that we use to understand nature. If we do not apply it properly, then what we get out at the end directly corresponds to this. In the worst case, you started with a very good ansatz, didn't do the math right or applied a method wrong, got nonsense in the end and hence got rid of the ansatz.
So what are you actually testing then? Your inability to do proper math or your intuition that gives you a correct result despite the fact that you logic is seriously flawed? The method I outlined above is how physics works: We are constantly in this circle! We are not trying to defend theories, we are trying to get rid of them, make them break to learn something out of it! How are you supposed to learn something if your chain of arguments isn't coherent or even correct?

I mean, look, I see why people do that. Proving stuff is hard work! But it is necessary work, because every one coming after you has a scientific obligation to question what you are doing - even if he is just a small grad student or a bachelor's student. Questioning things and denouncing concepts is as much part of science as accepting scientific facts when reason demands so.

The mathematicians have understood all these things and they had to - they don't have experiments after all! They have developed standards that (mostly) prevent the occurence of errors. This is important because every error made, can result in propagated errors.
I promise you, there will come a point in theoretical physics, maybe we have reached it already, where the lack of such rigor will seriously risk the science, because you cannot make an experiment and are therefore dependent on precise mathematical arguments.

Also I think that we have hit a point technologically where, as a physicist, you could leave most of the proving and calculating stuff to a computer (e.g. use Mathematica and theorem provers) and focus on the work a computer cannot do. Like understanding concepts, finding mathematical shortcuts, playing around with different ansatzes, questioning your ideas etc. ...


Summa summarum, I think both of you are right when you say that theoretical physics is not the place for me to feel at home. I will try to establish contact with a professor at the local philosophy of physics department and see whether that helps. I still have some time to decide whether to finish my math master's degree or take a jump into the job market. Luckily, I don't have to fight with financial problems like someone in the US would have to worry about in a similar situation. Sorry for the long reply.

 

[micromass]

Well, I'm sad to say it but it's true: mathematical rigor is completely orthogonal to creativity and intuition.

If you look at historical evidence (recent or far in the past), you'll always see the same thing: mathematicians and physicists who don't care about being rigorously correct and go doing stuff anyway. They do stuff which from a rigorous point of view is totally wrong and unmotivated, but eventually they do reach the correct result! This happens all the time in physics and math. If the mathematician/physicist were required to keep himself to the current mathematical rules, then he would get nowhere.

The entire mathematical structure of axiom -> definition -> theorem -> proof -> application is not a effective structure to do research with, certainly not in physics. It is only a way to present the math as clean as possible to the audience. Many students then think this is the only way to do math properly, which is far from the truth. The truth is probably closer to application -> vague calculations -> theorem -> proof -> definition/axioms.
As you see, the axioms come last, not first. I cannot find the exact name, but a mathematician once said that when a mathematical system is ready for axiomatization, then it is ready for burial and the axiom are its obituary. Maybe a bit too extreme, but I think it's rather accurate nevertheless.

Now, in mathematics, I agree that eventually everything should be axiomatized and put in a deductive structure. But that's the nature of mathematics. I don't see why this is necessary for physics. You can't discover new physics by keeping close to some kind of axiomatic method. You need to break the math, abuse the math and get to a good answer anyway. This is how physicists have always done things. And it worked. It is then up to the mathematician to make things formal.

This is why I think physics is way harder than mathematics. In mathematics you have some structure to rely on. In physics, you have to rely on your physical intuition and on the experimental evidence. I find that much harder.

 

[homeomorphic]

Well, first of all, as a topology PhD, my knowledge of physics is a bit superficial, even though I'm not exactly a laymen in it, either. I was a bit like that, especially in terms of wanting to understand nature, but my perspective is slightly different.

As far as approaching quantum mechanics by means of classical mechanics and quantization, it could be that you need to start with non-classical principles, but to me, the Schrodinger equation is suggested by the same optical/mechanical analogy which lead to Hamiltonian mechanics. Some people have argued that that was Schrodinger's motivation. So, I don't know what the right approach is, but I think there is a least something to gain in terms of understanding by approaching it via classical physics.

I appreciate physical understanding, and I am not as pessimistic about it as some physicists, but I also recognize that it can be difficult to understand the quantum world, and physicists have to do what they can.

I can definitely relate to being bored by endless calculations. Calculations are PART of physics, but I can't agree that they are what physics is ABOUT. I have to reason conceptually/visually or else I can't remember anything. To me, there's no point in studying something that is just a bunch of symbols flying around because it isn't going to stick, and I can't use it to think about the big picture. For a while, I considered getting a physics PhD, but after a classical mechanics class that was almost entirely moving meaningless symbols around, I got a little nervous about following that path. It was a very painful experience, even though I did fairly well in the class and impressed the professor. Later, with the help of other books, I proved my initial hunch, which was that that very formal approach that provided no insight was just completely, 100% unnecessary. Something similar is probably true in other subjects. But I don't think it's good to frown so much on calculations that you don't get practice doing them.

Calculation has its place. Sometimes, in math, you get too far into the realms of abstraction, and you don't really understand it until you move some x's and y's around, as one of my topology profs once said. I see calculation as a sort of training ground where I can test my ideas and make sure I know what's going on. What I don't like is when it becomes an end in itself. My end goal is understanding, not calculation.

Oddly enough, I took a general relativity class as a grad student, and I didn't think it was that bad, even though it didn't use modern differential geometry. I don't really see a huge conflict between the mathematician's geometry and that of a physicist. Differential geometers will sometimes do coordinate calculations just like a physicist. I really don't think the extra theory gets in the way, if you understand it well. I like to have a deep understanding to go along with my calculations when I do calculations. That way I'm not just moving symbols around (unless I want to) and there's some soul to the calculations. Of course, the way differential geometry is taught is sometimes too abstract (or, again, too many symbols flying around and not enough intuition) and if you learn it that way, without understanding except on a formal level, it doesn't seem too helpful to me.

I don't blame physicists so much for lack of rigor because it slows you down. I care more about having physical intuition, anyway, which seems lacking, too. However, that doesn't mean I think that rigor has nothing to say. The old argument that experiments remove the need for rigor sounds a little too empirical for me. Empirical is good because reality is the ultimate test, and in the end that's what matters, but ideally, I think you would want to have a theoretical idea of what is going on, rather than just saying we measured it and it came out right.

My take on it is that if you are like me, what you need to do is make a lot of money doing something else first, and then you can just bypass the system and do whatever you want (and until then, you can keep doing it on the side, as time allows). That's really the only option, as far as I'm concerned. I may never make any substantial contributions to math or physics, except perhaps on an expository level, but I will get to pursue my own curiosity, rather than someone else's curiosity, and to me, that is much more important.

 

[WannabeNewton]

Deep philosophical concepts like causality and the Einstein equivalence principle were melted together with the mathematical superiority of differential geometry.

Indeed, without a doubt GR is the most perfect theory of physics in more ways than one. To quote Max Born, "The general theory of relativity seemed and still seems to me at present to be the greatest accomplishment of human thought about nature; it is a most remarkable combination of philosophical depth, physical intuition and mathematical ingenuity. I admire it as a work of art."

What he did was good old school physics...

I would argue that it isn't old school physics. Old school physics consisted primarily of theory engendered by or born out of experiment and was often Ad hoc. Einstein formulated GR purely by theoretical and philosophical considerations (albeit misguided in certain respects) with no experimental vanguard so in that sense it foreshadowed a new way of doing physics as evidenced by the astounding success of gauge theory decades later. However unlike GR, QM was not developed in such an elegant manner and was in fact developed in a completely opposite manner with theory being fit together in an overtly Ad hoc manner in order to match experiment. That's just how QM was developed and given the fact that it is much more counter-intuitive than GR it is understandable that it needed a myriad of experimentation to bring it into fruition. Practically speaking, QM has been much more useful than GR and has been verified to much higher accuracy so the Ad hoc assumptions and formulations don't take away from the end result at all, which is what's really important in physics.

I totally agree. My problem is that I don't know what I would work on.

Have you considered working on TQFT?

Problems that weren't actually there or pseudo-problems like the meaning of the "zero-point-energy", an artifact of a poorly defined "quantization map" that has no predictive value whatsoever.

Well the problem of the zero-point energy is not one of mathematical rigor but rather one of cutoff frequency scales so I'm not sure what you are alluding to.

But it is necessary work...

Not for physics it isn't. You don't need to prove everything rigorously in order to justify a calculation or a framework. It would be a waste of time if physics students had to endure that. The last thing I want, as a physics student, is to be force fed detailed proofs of all the subtle, tricky parts of calculations. I am content with a hand-wavy but convincing argument. Feynman intuitively formulated his rules for calculating scattering cross sections which Dyson later proved rigorously and every QFT book spends a few pages droning out the calculations needed to justify the Feynman rules but Feynman first came up with them using physical intuition-intuition is what drives things forward not the A posteriori rigorous proofs, as micromass elucidated above.

I promise you, there will come a point in theoretical physics, maybe we have reached it already, where the lack of such rigor will seriously risk the science, because you cannot make an experiment and are therefore dependent on precise mathematical arguments.

Well many (rightfully) would argue that in such a case you aren't doing physics.

Like understanding concepts, finding mathematical shortcuts, playing around with different ansatzes, questioning your ideas etc. ...

As of now there are many kinds of calculations that computers are not clever enough to do i.e. those calculations in physics which very much require human ingenuity. Calculations don't necessarily mean rote or mindless algebraic manipulations.

 

[micromass]

I think Feynman formulates it very well:

https://www.youtube.com/watch?v=YaUlqXRPMmY

I don't fully agree with all of his statements about mathematicians in the following video, but it's mostly accurate:

https://www.youtube.com/watch?v=obCjODeoLVw

 

[homeomorphic]

Have you considered working on TQFT?

I worked on TQFT, and I would hesitate to recommend it to someone who's more interested in understanding the ways of the universe. It would probably be similar to what he's doing now. Mathematicians can understand TQFT without really knowing anything about physics, based on Atiyah's axioms. That's the side of it I experienced, and, as someone who wanted to be closer to physics, I didn't find it very satisfying. Perhaps, if you know more physics than I did, it would be different, but it's my sense that such a person may as well be working in string theory or quantum gravity and just using TQFT, rather than working in TQFT specifically.

 

[Geometry_dude]

However unlike GR, QM was not developed in such an elegant manner and was in fact developed in a completely opposite manner with theory being fit together in an overtly Ad hoc manner in order to match experiment.

Yes, absolutely, that's exactly what I'm saying. And it was absolutely useful and necessary! Because this is an area of physics where we don't really understand what is actually going on, while having enough math to do something about it. But we need to be honest about this and tell those who come after us where we made a jump, where we did something that was somewhat wiggly and why we did it. If we would have been clear about this, people wouldn't try to "quantize gravity" today and see the philosophical flaws in this train of thought.
Sometimes just playing with the math doesn't do the trick and one needs to put in new physics. One cannot do that without having at least an intuition of what's actually going on in "the real world".

For example the Dirac equation was a lucky (while educated) guess. It works, but we don't know what's behind, because it was never derived from principles!

It's like thermodynamics before the advent of statistical mechanics. Thermodynamics was necessary and useful from a technological point of view, but the real understanding came when it was tied to mechanics via statistics.

It's exciting, but maybe we're at a point today when it's time to get back to the drawboard and see what we can do. At the time when they developed quantum field theory, Einstein's view of gravity as a result of the underlying geometry of spacetime was still met with scepticism. Noone was willing to take this view to an extreme with the exception of maybe Einstein himself. We have to let go of our Newtonian understanding of the world where we have a fixed stage and particles being actors on that stage separate from it.

Concepts like spacetime being a four-dimensional smooth continuum, causality and scientific realism are useful and they serve as a guide to new mathematical hypotheses. And if that doesn't work, fine, let's figure out where we went wrong without it. It's the fundamental belief of science that this world can be understood, so why aren't we trying?

Well, I'm sad to say it but it's true: mathematical rigor is completely orthogonal to creativity and intuition.

Thank you for your reply. You would probably agree that we need both and they can go hand in hand. Intuition leads, mathematics proves.

If you sacrifice everything for intuition you are bound to end up where the string theorists are now: In the realm of "Not even wrong."
Without intution, you will just focus on mathematical structures without seeing what't going on in the big picture. You can do this in mathematics, but in physics it won't help much. Well, in the worst case one can always work on examples.

If you look at historical evidence (recent or far in the past), you'll always see the same thing: mathematicians and physicists who don't care about being rigorously correct and go doing stuff anyway. They do stuff which from a rigorous point of view is totally wrong and unmotivated, but eventually they do reach the correct result! This happens all the time in physics and math. If the mathematician/physicist were required to keep himself to the current mathematical rules, then he would get nowhere.

The entire mathematical structure of axiom -> definition -> theorem -> proof -> application is not a effective structure to do research with, certainly not in physics. It is only a way to present the math as clean as possible to the audience.

Again, I agree with you. It is also a way to present results very effectively and coherently. That is exactly why I believe physics would benefit from it even more so than mathematics.
Which axiomatic system are you working in, i.e. what are the assumptions of what you are doing?
If you state these relevant things in a theorem, the important result and support it with a proper proof, then people can look at it, say "I believe those things are true" and when they believe your proof, use your result and continue. Else they can look at it, immediately see "I don't agree with this" and move on. It makes things more effective and more coherent.
Of course, when you read a physics paper you immediately see the area where it is placed in. It's the fine branches and the small assumptions that are important. For example, you might have a formula like $E=mc^2$. But this formula is meaningless if you don't connect it to the axiomatic system of special relativity with the additional assumption that the $3$-momentum of the point particle vanishes in the frame. All this can be stated in a short proposition.
Relating to the first video from Feynman, that is exactly what he is saying: You need to categorize what is part of what and what can be deduced from what. All this can be done on a formal level and it shows where the blind spots are. We can formally draw the implications and equivalences, categorize results. But again, you cannot do this when it's not proven.

The truth is probably closer to application -> vague calculations -> theorem -> proof -> definition/axioms.

That is how you work, but this is not how you present your result. I am talking about the point when your findings are ready for presentation and how to put them into a proper form. By all means, interchange limits freely when you are trying to figure something out, put in ad-hoc assumptions to look where it leads to, explore your ideas. But once you have found something, you need to frame it in a brief and proper way. Clearly stating your result in a proposition and supporting it with a separate proof/reference to a proof for the sceptic is an excellent way to do this.

I don't see why this is necessary for physics. You can't discover new physics by keeping close to some kind of axiomatic method. You need to break the math, abuse the math and get to a good answer anyway. This is how physicists have always done things. And it worked. It is then up to the mathematician to make things formal.

Look, there's different kinds of proofs. Even in pure mathematics, some proofs are just sloppy and have subtle problems that don't influence the result. But if you state your result, add a proof, that proof turns out to be wrong or flawed, then the sceptic who reads it and maybe is more of a mathematician (who might not have had the intuition to get there) can take this, remove the flaws and everyone is happy.
If not, then your result is a conjecture, not a proposition. I find making this distinction very important.

Moreover, that this is the way things have always been is not an argument to keep them this way. Mathematics also started out this way, only later it was formalized. I believe that it is time for physics to become formalized as well. Hilbert has already foreshadowed this path for physics with the statement of his famous sixth problem.

Of course, there's different ways to do physics, and that's totally fine. What I'm trying to say is that in the future all these boring annoying steps like proving that an integral converges, etc., can be done by machines. So there is a way for people to do rigorous mathematical science and still focus on ones own intuition and ideas.

It would be a waste of time if physics students had to endure that. The last thing I want, as a physics student, is to be force fed detailed proofs of all the subtle, tricky parts of calculations. I am content with a hand-wavy but convincing argument.

Fine, then skip the proof. What I'm arguing for is that the proof is there for the skeptic.

This is why I think physics is way harder than mathematics. In mathematics you have some structure to rely on. In physics, you have to rely on your physical intuition and on the experimental evidence. I find that much harder.

Yes, we are all inclined more in one way or the other. However, it is when there is a contradiction between our careful logic and experimental results when the really interesting stuff happens. If you logic is flawless, you are able to go directly back to the axioms and principles. Because from a correct statement, you cannot deduce a wrong one, hence you know your original statement is wrong. If you got to your result by guesses and by constantly putting in information "from the outside", you cannot do this.

As far as approaching quantum mechanics by means of classical mechanics and quantization, it could be that you need to start with non-classical principles, but to me, the Schrodinger equation is suggested by the same optical/mechanical analogy which lead to Hamiltonian mechanics. Some people have argued that that was Schrodinger's motivation. So, I don't know what the right approach is, but I think there is a least something to gain in terms of understanding by approaching it via classical physics.

Sure, but QM is not classical and never will be. I actually had the latter point of view before working on this subject. Yet if you are able to find those non-classical principles, this is inherently more valuable in terms of understanding what is going on.

Let's try it. I postulate that free particles are superpositions of matter plane-waves in Minkowski spacetime satisfying the relation $E= \hbar \omega$. Relativistic covariance now implies
$\bar p = \hbar \bar k$. We know then
$$ \psi = \int_{\mathbb{R}^4} d^4p \, A(p) \, e^{\frac{i}{\hbar}(p_\mu x^\mu)} $$
By deriving and using the kinetic energy relation in the non-relativistic limit
you can check then that this directly leads to the Schrödinger equation or the Klein-Gordon equation. You can introduce potentials by first looking at what happens for a constant potential, then a step potential plus smoothness conditions, then many steps, then a smooth potential in the limit. Then you got the SE or KGE and no quantisation was necessary (2nd quantisation is a different story). Of course, this way you replaced you ad-hoc quantisation step by an (ad-hoc?) ontological statement. However, the latter is mathematically flawless, while the path using canonical quantisation is not even well-defined.

Thank you, homeomorphic, for your reply, I share many of your views, especially this one:

I like to have a deep understanding to go along with my calculations when I do calculations. That way I'm not just moving symbols around (unless I want to) and there's some soul to the calculations.

My take on it is that if you are like me, what you need to do is make a lot of money doing something else first, and then you can just bypass the system and do whatever you want (and until then, you can keep doing it on the side, as time allows). That's really the only option, as far as I'm concerned. I may never make any substantial contributions to math or physics, except perhaps on an expository level, but I will get to pursue my own curiosity, rather than someone else's curiosity, and to me, that is much more important.

I find the idea that there are already people out there like me very comforting. I will keep your advice in mind.

I would argue that it isn't old school physics.

You may be right, I might have imposed this interpretation a posteriori.
As you said, the split of GR and quantum theory is more than a split in physics, it is a split in the philosophy of how to do physics. While disparity can be very healthy, it seems dangerous to me that there's such a fundamental disagreement on how to proceed with the science.

The way I would like to do physics is definitely inspired by the former, while the latter is my topic of interest.

Have you considered working on TQFT?

Funny that you ask that, yes I have. I haven't read much into it though as it appeared to me that they're not getting anywhere with this. For this one would have to put in more physical principles.

Well the problem of the zero-point energy is not one of mathematical rigor but rather one of cutoff frequency scales so I'm not sure what you are alluding to.

This was referring to QM. The constant $\hbar/2$ cannot be measured, because only energy differences can be measured. Yes, it leads to a divergence in QFT and using the same argument it can be removed. Mathematical rigor here would have shown that the Hamiltonian is only defined up to an additive constant and avoided confusion.

As of now there are many kinds of calculations that computers are not clever enough to do i.e. those calculations in physics which very much require human ingenuity. Calculations don't necessarily mean rote or mindless algebraic manipulations.

Yes, that's a different story. What I'm saying is that we live in the 21st century and that we can and should let computers do the work when it does boil down to mindless algebraic manipulations. That's one of the reasons why we invented them!

micromass, thanks for the Feynman videos I haven't seen them for a while.

 

[micromass]

That is how you work, but this is not how you present your result. I am talking about the point when your findings are ready for presentation and how to put them into a proper form. By all means, interchange limits freely when you are trying to figure something out, put in ad-hoc assumptions to look where it leads to, explore your ideas. But once you have found something, you need to frame it in a brief and proper way. Clearly stating your result in a proposition and supporting it with a separate proof/reference to a proof for the sceptic is an excellent way to do this.

I don't see how the physicist would benefit from this. Sure, it would all be nice and correct, but I think it's a bit a waste of time. First of all, the physicist who deduced something mathematically would need to spend time figuring out why it works mathematically. This is not a trivial problem and several of the things that people do in QFT are still unsolved mathematically. Even then, the physicist wastes quite valuable time that he could use for something else. Second, the physicist would need to have taken rigorous math courses in order to justify everything (and to even see the problem!). This would go instead of physics courses, which are way more valuable to him. Third, the average reader doesn't really care about the mathematical correctness.

Look, there's different kinds of proofs. Even in pure mathematics, some proofs are just sloppy and have subtle problems that don't influence the result. But if you state your result, add a proof, that proof turns out to be wrong or flawed, then the sceptic who reads it and maybe is more of a mathematician (who might not have had the intuition to get there) can take this, remove the flaws and everyone is happy.
If not, then your result is a conjecture, not a proposition. I find making this distinction very important.

Moreover, that this is the way things have always been is not an argument to keep them this way. Mathematics also started out this way, only later it was formalized. I believe that it is time for physics to become formalized as well. Hilbert has already foreshadowed this path for physics with the statement of his famous sixth problem.

Look, I'm all for the rigorization of mathematics. I think it's a nice thing and an interesting thing. But it didn't add anything mathematically, conceptually or physically. All it really did was to make our work more difficult.
See, the interesting part is that all the mathematics before the rigorization was virtually all correct and yield good results empirically. All the rigorization did was introduce weird functions like the Dirichlet and popcorn functions which never come up in real life ayway. It introduced very artificial constructions like the real numbers that nobody really cares about anyway. Sure, its cool it can be done, but other then that, it's pretty useless.

Euler, for example, did work on series which was horrible from a rigorous point of view. But all his results are useful and correct. Proving them rigorously didn't really do anything. Sure, it convinced the skeptics, but seeing that the results worked in other math and physics should have been enough.

Second, the use of nonrigorous mathematics lead to brilliant work on divergent series. After, the rigorization, only convergent series were allowed. This was a big loss since divergent series really do give a lot of information. Mathematicians afterwards had to create a rigorous theory of divergent series anyway to allow them. So what we should check first is whether the results are physically meaningful. We shouldn't throw out mathematics that gives physically meaningful and good results. The physical observations should be more important than mathematical rigor, even in mathematics. If our mathematical rigor disagrees with the physical observations, then it's the mathematical rigor that is wrong and it is the mathematician who needs to change the mathematics in order to allow the "bad" mathematics anyway. This should be the case and this has been the case in history, with for example divergent series and dirac delta functions.

So from the physics point-of-view, they see that the mathematics they use will always be justified by the mathematicians later on granted that they yield physically meaningful results. So then, what is the point of physicists doing things mathematically rigorous? The rigor and mathematical theories are variable anyway and if you think about, they are also quite arbitrary. So the physicist rightfully sees that rigor has no place in physics and just makes things harder than they are. What matters is the observation.

It's nice that you like mathematical rigor so much. But I doubt mathematical rigor has ever helped physicists or mathematicians much in discovering new math and physics. Mathematics worked fine before the rigorization (and the results were mostly all correct too). Physics worked fine and continues to work fine.

For example, Poincare: "Logic sometimes makes monsters. For half a century, we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times, when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that."

De Morgan: "The history of algebra shows us that nothing is more unsound than the rejection of any method which naturally arises, on account of one or more apparently valid cases in which such method leads to erroneous results. Such cases should indeed teach caution, but not rejection; if the latter had been preferred to the former, negative quantities, and still more their square roots, would have been an effectual bar to the progress of algebra... and those immense fields of analysis over which even the rejectors of divergent series now range without fear, would have been not so much as discovered, much less cultivated and settled... the motto which I should adopt against a course which seems to me calculated to stop the progress of discovery would be contained in a word and a symbol - remember $\sqrt{-1}$"

And besides, what is the use of deducing everything so painstakingly from basic axioms, when Godel has shown that we can never prove a (sufficiently rich) axiom system to be consistent anyway. Thus even our rigorous mathematics might as well yield contradictions. So it is much less certain than what people make it seem.

Fine, then skip the proof. What I'm arguing for is that the proof is there for the skeptic.

I feel that physical observation is always superior to mathematical rigor. So if a certain mathematical operation yields physically meaningful results, then that should be proof enough for the skeptic. Rigorization is all very fun and nice, but adds nothing to the physics. What matters in mathematics are the ideas and the creativity.

 

[homeomorphic]

That is how you work, but this is not how you present your result. I am talking about the point when your findings are ready for presentation and how to put them into a proper form. By all means, interchange limits freely when you are trying to figure something out, put in ad-hoc assumptions to look where it leads to, explore your ideas. But once you have found something, you need to frame it in a brief and proper way. Clearly stating your result in a proposition and supporting it with a separate proof/reference to a proof for the sceptic is an excellent way to do this.

Actually, I think this sort of thing can create problems. Formal proofs are not always fit for human consumption. Things would be a lot easier if we didn't beat around the bush and just talked directly about our ideas instead of encoding them and forcing everyone else to decode them. The formal proof is like a form of excrement that comes out as a byproduct of convincing yourself that there are no holes in your arguments. Mathematicians have substituted this excrement for explanations and, as a consequence, many of their books and papers are unreadable. The desire to produce a perfectly polished logical result often runs contrary to the desire to explain ideas. That becomes clear if you imagine your goal as explanation, rather than logical proof. Probably, only rarely would you think to write a perfectly logical proof if your goal was to really explain something to someone.

It's not that it's useless, though. When I wrote my dissertation, I felt like it was always on the verge of collapsing due to lack of rigor. I thought my results were correct, but they had to be fixed many times. It wasn't until it was all carefully written out that I was confident of my results. Even in its final form, it isn't 100% rigorous, although I am confident in its results and my adviser and some of my committee members appear to have read it fairly carefully and gave it the green light. The important thing was to really convince myself of the correctness of my results, and that I could write it all out if I had to.

Are epsilons and deltas that big of a deal for calculus? Maybe not, especially with all the standard fairly nice functions that you come across in elementary calculus, engineering, and physics. I'm not sure they are insignificant, but their role is fairly minor. They do yield an explanation of why it's okay to differentiate or integrate power series term by term, for example. People like Euler would just do it and get away with it, but if you want an honest (and intuitive) explanation, the only way I know how to explain it is by uniform convergence, along with epsilons and deltas. Note that that is in marked contrast to most of the rest of elementary calculus, which is pretty believable, even without epsilons and deltas.

Also, people today forget that Riemann was trying to deal with uglier functions for number theory purposes (he introduced Riemann sums and his Riemann-Lebesgue lemma in this context). For dealing with stranger situations where our intuition doesn't apply as well, we do need the rigor to keep us on track.

Are epsilons and deltas a big deal for math? Yes. I can't imagine point-set topology without them, or functional analysis, or a lot of things. And how about Hilbert's program to formalize mathematics, with its grand finale being Godel answering that it isn't possible? Without the attempt for rigor, we wouldn't have gotten those results, and all the other logic that came with it. A lot of the "use" of rigor comes when you apply it to other things beyond what you were supposed to be rigorizing.

Another interesting point is that epsilons and deltas might be viewed as just introducing MORE intuition to math, not just replacing it by rigor. It gives you another set of intuitive concepts that can be applied to other situations. When I'm coming up with an analysis proof, I typically don't think in terms of epsilons and deltas, I think in terms of "this is small compared to that" or "this thing shrinks faster than this other thing".

Terence Tao says the purpose of rigor is to elevate good intuition and correct bad intuition.

 

[WannabeNewton]

For example the Dirac equation was a lucky (while educated) guess. It works, but we don't know what's behind, because it was never derived from principles!

Historically yes it was quite an ingenious but lucky guess. Now however we do know that it is a straightforward result of transforming into position space the momentum space relation that one obtains for Dirac spinors under spinorial representations of the Lorentz group so we do have a deeper understanding of it from more fundamental principles.

Funny that you ask that, yes I have. I haven't read much into it though as it appeared to me that they're not getting anywhere with this. For this one would have to put in more physical principles.

Unfortunately there seems to be a rather huge give and take in every which venue available i.e. with TQFT you end up doing something purely academic with only a superficial connection to physics whereas proper physics doesn't have the amount of rigor that you personally seem to desire. I suppose you should, if you want to stick to physics, go with the choice that you feel would ultimately leave you in the least miserable state.

Yes, it leads to a divergence in QFT and using the same argument it can be removed. Mathematical rigor here would have shown that the Hamiltonian is only defined up to an additive constant and avoided confusion.

I don't want to sidetrack the thread topic but it's actually more subtle than that because the gravitational field couples directly to energy (and not just energy differences): http://ticc.mines.edu/csm/wiki/images/7/72/VacuumCatastrophe.pdf

 

[thecage]

Have you considered philosophy of physics? While it's a small field, I still find it very interesting. Last semester I took a course in it and we spent a lot of time talking why quantum mechanics doesn't fit with SR, and about concepts such as non-locality, randomness, flashes, etc. It's one of those rare courses which is not all about calculations but about taking a step back, looking closely at the assumptions and their implications.

 

[micromass]

Have you considered philosophy of physics? While it's a small field, I still find it very interesting. Last semester I took a course in it and we spent a lot of time talking why quantum mechanics doesn't fit with SR, and about concepts such as non-locality, randomness, flashes, etc. It's one of those rare courses which is not all about calculations but about taking a step back, looking closely at the assumptions and their implications.

Good suggestion! And don't think philosophy of physics is some kind of fake thing, they actually do legit physics things there. If you don't believe me, check out https://www.amazon.com/dp/0444515607/?tag=pfamazon01-20