Nobody complains about physicists' math?

[Hurkyl]

They're two different things. To physics, math is a tool, not a holy grail. We use mathematical language because we can express things to each other that way. We have other tools though, like experiment and qualitative analysis.

At the risk of stressing the analogy to its limits -- the problem is that there is a trend to reject the shiny new nailgun sitting on the next shelf in preference to using an ancient, rusty hammer that's barely held together with chewing gum and duct tape. They both do an adequate job of putting a nail into wood, but the hammer is more cumbersome and could fall apart unexpectedly.

You seem to forget the important point here : it works!

No it doesn't; every time a prominent physicist says "nobody understands quantum mechanics", that's a strike against your thesis. Quantum mechanics has been around for over a century, and has very simple toys requiring only very elementary mathematics; there is no excuse for its simplest and most elementary notions to still be considered mysterious and unintuitive by its experts!

(of course -- I hope that my knowledge of prevailing opinion is behind the times, and that things have improved)

 

[ZapperZ]

No it doesn't; every time a prominent physicist says "nobody understands quantum mechanics", that's a strike against your thesis.

I won't repeat my argument against this. You can http://physicsandphysicists.blogspot.com/2007/04/no-one-understands-quantum-mechanics.html" .

This attack against physicists now has taken a very strange turn far from the original post.

Zz.

 

[jostpuur]

"What do you mean, funny? Let me understand this cause, I don't know maybe it's me, I'm a little ****ed up maybe, but I'm funny how? I mean, funny like I'm a clown, I amuse you? I make you laugh... I'm here to ****in' amuse you? What do you mean funny, funny how? How am I funny?"

I would not mess with the Newton.

I think that's way overrated.
"rigor" in math came after the giants who developed modern math in the first place, and they're "giants".

There's no way of knowing what the giants would say in this thread.

As one can guess from my OP, IMO the physicists' way could get a little bit more criticism, than it is receiving now. Even though it is true that the details of rigorous proofs can be complicated, the ideas are usually simple. I have noticed, that when physicists explain mathematical things with their "physical arguments", the ideas can get quite incomprehensible. I'm not criticizing use of approximations. I'm criticizing making illogical conclusions and making people get used to them. IMO the physicists' way is "not working as well as it looks from inside". Well I know that's a student's opinion against huge amount of scientists, but here's one related quote:

http://aleph0.clarku.edu/~djoyce/hilbert/problems.html

Besides it is an error to believe that rigor in the proof is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof. It also frequently leads the way to methods which are more capable of development than the old methods of less rigor.

 

[yasiru89]

I can't really put it any better than Hilbert, and Zz- I don't aim to stereotype the entire landscape of physics, merely remark on the general trend hovering over much of today's physics(as far as Arxiv goes anyway- my readings are also very occassional). This is all the more evident when someone can say (and I quote) "..physics...where every function equals the first few terms of its Taylor expansion.." and people aren't appalled by it.

The 'giants' as you call them were indeed quite 'rigorous' for their times, Euler's foresights into the realms of rigour weren't all that well considered (perhaps because he sometimes picked up a habit of Leibnitz and brought metaphysics to the mix). Even in his doodlings with divergent series he hints at interpreting his results as representations. A basis not only of the mathematical theory but for things as physically 'profound' as Quantum Field Theory.
In fact I'd say Euler paved the way towards the rigour so vehemently stuck to by Cauchy et al. His work hinted at definition instead of thinking about something as 'that' thing, (
(which gave things an aura of mystery and divinity in their very existence and deterred certain individuals from serious considerations)

Also at the risk of over assumption I believe what Hurkyl meant was not in anti thesis of your rather well played out argument ZapperZ. Indeed we fully 'understand' a theory when it can be put in completely mathematical terms (an intuitive grasp may be helpful but it isn't everything- certainly not absolute comprehension) and we actually have that to a rather large extent in quantum theories. What 'nobody understands quantum mechanics' really connotes is a general reluctance to accept the implication of it, something of which one could relate with Einstein. Also quantum theories don't exactly fit perfectly with observations do they(this is taking into account uncertainty and limited measurement ability)? This is because it is an asymptotic theory as are so many in physics. Then there's the very word 'mysterious', this need not imply that you know absolutely nothing of a theory or its application, only that one has trouble coping with implications. Where intuition ceases to be of very great use - The collapse of determinism.

 

[yasiru89]

Oh I refer to 'your rather well played out' argument in the URL you provided.

 

[ZapperZ]

I really don't see this going anywhere, and it certainly isn't productive from my perspective. Why?

1. I still haven't gotten any concrete example on where this really is a huge problem, to the extent that it can induce physical sickness.

2. I also haven't seen any concrete evidence that the physics that was done has been compromised due to such "mathematical practice". All I have seen are hand-waving speculation that such-and-such can happen and shouldn't happen. I've even brought up specific physics for someone to point out where exactly is the fault in the end result. I had no takers.

3. Because of #2, all the arguments so far have been nothing more than a matter of tastes. So you like the color green better while I like purple.

4. This thread now has become an issue on ... er... quantum mechanics? And what it actually means?! All because of some seemingly sloppy differentials?

I can tackle a lot of issues when the issue itself is clearly presented, but not in this thread that seems to meander from an outright attempt at dissing physicists to the validity of quantum mechanics. Up to this point, no one has managed to actually pin-point faulty physics that is a result of the mathematics being used, something that I've asked for repeatedly.

I'm sorry that I actually wasted any effort and intruded into this thread in the first place.

Zz.

 

[Hurkyl]

I found the article I was looking for:

http://www.theorie.physik.uni-muenchen.de/~serge/why_physics_is_hard.pdf


When I think of "not understanding quantum mechanics", I think of opinions like "delayed choice quantum erasure is mysterious". This is certainly a popular opinion, but I don't know how much of it is due to non-physicists, how much is due to physicsts trying to popularize the subject, and how much is due to physicists actually believing it.

Of course, I would like to believe that most physicsts¹ find this, and other sorts of things, to be natural rather than mystifying -- would I be correct to interpret your essay in that fashion?

1: Note that I'm not asking about the "very best quantum physicists", but your typical quantum physicist. I should probably even include good graduate students of the subject.

 

[Mororvia]

Thanks for that article. I can certainly appreciate being confused about mathematics in physics courses heh.

However, I don't know how much this has to do with the mathematics in physics being complainable. Its seems more to be about how it may need some better math instruction which I have some ideas about, but that's for a different thread

 

[Hurkyl]

Thanks for that article. I can certainly appreciate being confused about mathematics in physics courses heh.

However, I don't know how much this has to do with the mathematics in physics being complainable. Its seems more to be about how it may need some better math instruction which I have some ideas about, but that's for a different thread

The opening posters differential example is very much in spirit with the Green's function example in the article I linked (but much more elementary). I think what yasiru89 was saying about replacing rigour with "intuitive obviousness" is the same sort of thing.

 

[Jonny_trigonometry]

AFAIK nearly all mathematicians laugh at the way Physicists do math. Of course, not nearly as hard as they laugh at the way engineers do math.
But this is so unsurprising that it is not discussed outside of Math dept coffee rooms.

oh man, have you seen much of continuum mechanics? It's way cool. It's an engineering dicipline, with amazingly complicated math. This doesn't say much, but I'm quite impressed at the formulation. You can take this subject to a mathematical extreme.

 

[Galileo]

There are things to be said for both sides.

Although physicists surely do not need to be as mathematically rigorous as mathematicians, I believe when a physical theory is well established a clear and logically valid mathematical derivation of the results of the theory to go with the physical idea is beneficial to a quick understanding of the subject.

However, most physicists are in areas where the theory is not yet well or fully developed to an extend where the results are cast in a formally rigorous way. And so we are guided by our intuition and use mathematical symbols to formulate physical ideas and results to bring new understanding, concepts and ideas which can be tested by experiment. It is the faster way to new discoveries.

When a mathematician lays down the axioms for a mathematical theory. All the 'truths' (theorems) are fixed and mathematicians proceed to find them theorems by working -within- the system. Ofcourse, when foundational issues arise, there's a lot of thinking outside of the system, or about the system (what _are_ the right axioms?). And that is much more in line with what physicists do.
Most physicists at the frontiers of discovery do not work -within- a system. That is, a mathematically rigorous decription of the system under research. We think -about- the system and as such we are justified to use any creative means we have to discover 'the right axioms' Nature has chosen for us.

But again, to clear the dust when the results are well established, a comprehensive and mathematically sound formulation will be a boon to anyone new to the subject and wished to learn about it.

I believe that is also the reason for the contrasting remark in the interesting article Hurkyl posted, where the writer says that 'excellent self-contained textbooks often appear even in relatively new branches of mathematics, as soon as the major new achievements are recognized', while it takes a while for a physical theory to get a nice textbook. It's a lot easier to do so when you work within a formal system.

 

[Thrice]

Although physicists surely do not need to be as mathematically rigorous as mathematicians, I believe when a physical theory is well established a clear and logically valid mathematical derivation of the results of the theory to go with the physical idea is beneficial to a quick understanding of the subject.

Is that found outside undergraduate physics? Because I've never seen it except once in a relativity book that I found shelved in the math library. Coming from the math side of things, this is really my main complaint with physics (that I've done). The theories are quite elegant & easy to understand if presented differently, but it's like they're intentionally stitched together in the ugliest way possible.

You seem to forget the important point here : it works!

Given the number of QM interpretations & the passion I see during discussions, I'd say that is obviously not the only consideration even in physics.

 

[Hurkyl]

However, most physicists are in areas where the theory is not yet well or fully developed to an extend where the results are cast in a formally rigorous way. And so we are guided by our intuition and use mathematical symbols to formulate physical ideas and results to bring new understanding, concepts and ideas which can be tested by experiment. It is the faster way to new discoveries.

You don't need a fully developed theory to have rigor; you can crystallize your intuition into a conjecture, axiom, assumption, et cetera. Then, you can rigorously base your subsequent work upon that.

In fact, that's one of the main ways mathematical research is done -- intuition, analogy, and/or experiment suggest some theorem, but it resists both proof and disproof. It then gets stated as a conjecture, and some people march forward to deduce things assuming the conjecture and others "backsolve" to find cases where we can prove the conjecture, and to develop new tools to attempt to prove the conjecture in full.

 

[jostpuur]

In fact I have lot of different examples of how physicists' math can suck pretty badly, but most of the examples are quite complicated, and some, for other reasons, perhaps not very good to make my point. I thought there wouldn't be much people reading my OP if it started with a too long and complicated story, so I put a simple example of differentials, Taylor series and derivatives first. It could be it was on the other hand slightly too simple, because it didn't seem very serious. I'll try to gather some energy to write better examples at some time.

The "Why physics is hard" paper seemed interesting, although when Winitzki talks about how physicists' lectures can be incomprehensible because the lecturers talk only about technical details of the calculations, it slightly looks like he would be whining about everything annoying he has encountered in physics. Unfortunate, that this seems quite lonely paper. There should be more discussion about this topic.

it works!

It works... "it works" can mean so many different things. Consider the chi-energy that some kungfu guys supposedly use. They first do some useless movements with their arms, to better control/gather/direct(whatever) the chi. Then they punch a brick, and the brick breaks. It works! See, it's like theoretical physics: Kungfu guys don't know what they are doing, but it works.

 

[ZapperZ]

It works... "it works" can mean so many different things. Consider the chi-energy that some kungfu guys supposedly use. They first do some useless movements with their arms, to better control/gather/direct(whatever) the chi. Then they punch a brick, and the brick breaks. It works! See, it's like theoretical physics: Kungfu guys don't know what they are doing, but it works.

You don't know the workings of the quasiparticles that are in the semiconductors that made up your transistors that you use to run your modern electronics, even the computer that you just used to typed that. All you care about is that it works! So how come you don't make the same complain to yourself?

Zz.

 

[jostpuur]

You don't know the workings of the quasiparticles that are in the semiconductors that made up your transistors that you use to run your modern electronics, even the computer that you just used to typed that. All you care about is that it works! So how come you don't make the same complain to yourself?

Zz.

I understand that it is impossible to understand everything, and sometimes it is good way to be satisfied by something merely working. The kungfu example was an example of using the "it works" argument badly. Your computer example is an example of the "it works" argument been used appropriately.

I already know what I'm going to write about delta functions, to better explain how physics can start going in the wrong direction, but I need more time for it. I'll return to this later.

 

[Hurkyl]

Your computer example is an example of the "it works" argument been used appropriately.

What makes it appropriate? As far as I can tell, they are exactly the same formal argument.

(And since when is 'appropriateness' a quality applicable to a logical argument?)

 

[Poop-Loops]

It works... "it works" can mean so many different things. Consider the chi-energy that some kungfu guys supposedly use. They first do some useless movements with their arms, to better control/gather/direct(whatever) the chi. Then they punch a brick, and the brick breaks. It works! See, it's like theoretical physics: Kungfu guys don't know what they are doing, but it works.

That's where experimentation comes in. You make the following test: can someone break bricks without all the mumbo jumbo? Yes? Chi is BS.

Haven't found one after a bunch of trials? There might be something to it.

You can do the same thing in math. Do you get the same answer doing it the correct way and then doing it the short-hand way? Yes? Then you might as well use it as long as it suits your needs. No? Don't use it.

I understand that you can't cancel infinities or zeros. But if I ever stumble upon a problem where doing it would give me the correct answer, I'd do it.

This is basically due to CONTEXT, like Zz keeps telling you all. If you expand something in a series, and the terms go like x^n and x is already small (i.e. <1, or even <<1), you will likely stop after the first or 2nd term. Why? Because it's pointless to go on. Even if you calculate it exactly, you couldn't test it that precisely with an experiment. Secondly, you just stated that in reality it's an infinite series, but since you don't want to bother with it, you're just going to approximate it.

I saw an example saying what you could do is say f(x) ~a +bx = g(x) right? Well what we chose to do instead is say that "~" = "=".

I mean, if you wanted to denote all approximations as "~" instead of "=", you would almost NEVER use an equals sign in physics, because even classical mechanics is an approximation of QM. If you're going to be that anal about the details, then the equals sign would only be valid in pure math.

 

[Poop-Loops]

(And since when is 'appropriateness' a quality applicable to a logical argument?)

Funny you should say that. I am currently learning Perturbation Theory in QM. One of the examples stated you use an "appropriate" small parameter to approximate your perturbation.

Other places say the same thing, that you can only use PT when appropriate, when your perturbation is small compared to your unperturbed state.

Physicists use "appropriateness" as a measure of an argument all the time.

 

[Hurkyl]

I saw an example saying what you could do is say f(x) ~a +bx = g(x) right? Well what we chose to do instead is say that "~" = "=".

I mean, if you wanted to denote all approximations as "~" instead of "=", you would almost NEVER use an equals sign in physics, because even classical mechanics is an approximation of QM. If you're going to be that anal about the details, then the equals sign would only be valid in pure math.

There's a difference between scratch work and technical writing. Making true and accurate statements can mean the difference between an accessible paper/textbook and an impenetrable document readable only by those with secret insider knowledge. (This flaw isn't unique to physics, of course)

There are lots of actual equalities in physics. For example, in classical mechanics, the kinetic energy of a point particle is given by K = m v^2 -- this is an actual equality, not a mere approximation.

That classical mechanics only approximates reality is an entirely separate issue.

 

[Pythagorean]

"it works" may be an oversimplified phrase for what's really going in most physics people's minds (or at least in mine).

When I say "it works" I mean "It's the only thing that works to make reliable predictions thus far, and it's really, truly all I got as a means of answering the questions I'm asking, and I'm sure a better way can and will be discovered, but I'm doing this other thing, and I'm using this tool; my job is not to make tools, but to make things with these tools."

The mathematicians are our tool makers in this sense.

 

[jostpuur]

fourier transform and physical boundary conditions

This story is about the identity

<br /> \frac{1}{2\pi}\int\limits_{-\infty}^{\infty} e^{ikx} dk = \delta(x).<br />

All theoretical physicists students must be familiar with the example. We first calculate

<br /> \int\limits_{-L}^{L} e^{ikx} dk = \frac{2\sin(Lx)}{x},<br />

and then, for a<0<b,

<br /> \lim_{L\to\infty}\int\limits_{a}^{b} f(x)\frac{2\sin(Lx)}{x} dx = \lim_{L\to\infty} 2\int\limits_{La}^{Lb} f(\frac{u}{L})\frac{\sin(u)}{u} du = 2\int\limits_{-\infty}^{\infty} f(0)\frac{\sin(u)}{u}du = 2\pi f(0).<br />

At this point the delta function identity makes sense.

Frankly, I don't know how the step where the limit is taken, would be carried out rigorously. But that's not the point. Let us assume that it can be done when something is assumed of the f.

The Fourier transform is defined as

<br /> \tilde{\psi}(k) = \int\limits_{-\infty}^{\infty} \psi(x) e^{ikx} dx.<br />

Now we want to understand why

<br /> \psi(x) = \frac{1}{2\pi}\int\limits_{-\infty}^{\infty} \tilde{\psi}(k) e^{-ikx} dk<br />

gives the inverse transform. This is how it goes. First substitute the Fourier transform into the inverse transform

<br /> \frac{1}{2\pi}\int\limits_{-\infty}^{\infty} \Big(\int\limits_{-\infty}^{\infty}\psi(x&#039;) e^{ikx&#039;} dx&#039;\Big) e^{-ikx} dk<br />

change the order of integration

<br /> = \int\limits_{-\infty}^{\infty}\Big(\frac{1}{2\pi}\int\limits_{-\infty}^{\infty} e^{ik(x&#039;-x)} dk\Big) \psi(x&#039;) dx&#039;<br />

notice that we have the delta function, and complete the calculation

<br /> = \int\limits_{-\infty}^{\infty} \delta(x&#039;-x) \psi(x&#039;) dx&#039; = \psi(x).<br />

Next we want to understand why this calculation works. It doesn't seem very clear. How does the divergent integral become a delta function? Divergent integrals at least don't become distributions in the strict sense, because distributions are something else than divergent integrals. This is how this was explained to me by physicists, when I wondered about this:

"Mathematicians could not do a calculation like this, because the integrals don't really exist. However we can use physical arguments. By changing the integrand by some arbitrary \epsilon&gt;0 like this

<br /> \int\limits_{-\infty}^{\infty} e^{-\epsilon k^2 + ik(x&#039;-x)} dk<br />

we impose physical boundary conditions at infinities \pm\infty. Thus the integral can be made convergent and the calculation is justified."

One explanation was given, but I have now, after lot of thinking, come up with a different explanation. It goes like this:

We first write the expression we want to deal with carefully

<br /> \frac{1}{2\pi}\lim_{L\to\infty}\int\limits_{-L}^L \Big(\lim_{L&#039;\to\infty}\int\limits_{-L&#039;}^{L&#039;} \psi(x&#039;) e^{ikx&#039;} dx&#039;\Big) e^{-ikx} dk = \cdots<br />

We can write it in a form

<br /> \cdots =\lim_{L\to\infty}\int\limits_{-L}^L \big(\lim_{L&#039;\to\infty}\phi(L&#039;,k) \big) dk = \cdots<br />

where

<br /> \phi(L&#039;,k) = \frac{1}{2\pi}\int\limits_{-L&#039;}^{L&#039;} \psi(x&#039;) e^{ik(x&#039;-x)} dx&#039;.<br />

Let us assume that \|\psi\|_1 &lt;\infty. We then get

<br /> |\phi(L&#039;,k)| \leq \frac{1}{2\pi}\int\limits_{-L&#039;}^{L&#039;} |\psi(x&#039;)| dx&#039; \leq \frac{1}{2\pi} \|\psi\|_1<br />

The constant function

<br /> g(k) = \frac{1}{2\pi} \|\psi\|_1<br />

is integrable over the interval [-L,L], and dominates |\phi(L&#039;,k)| for all L', so according to the Lebesgue's dominated convergence, we get

<br /> \cdots =\lim_{L\to\infty} \lim_{L&#039;\to\infty} \int\limits_{-L}^L \phi(L&#039;,k) dk = \cdots<br />

The integration over [-L,L]\times [-L&#039;,L&#039;] can be carried out in either order, so we get

<br /> \cdots =\lim_{L\to\infty} \lim_{L&#039;\to\infty} \int\limits_{-L&#039;}^{L&#039;}\Big(\int\limits_{-L}^L \frac{1}{2\pi} \psi(x&#039;) e^{ik(x&#039;-x)} dk\Big) dx&#039; = \lim_{L\to\infty} \int\limits_{-\infty}^{\infty} \psi(x&#039;) \Big(\frac{1}{2\pi}\int\limits_{-L}^{L} e^{ik(x&#039;-x)} dk\Big) dx&#039; = \cdots<br />

and now, inside the integral over x', we have a delta function representation that works so that if you integrate over it with fixed L, and after integration take the limit L\to\infty, it works as a delta function. So the calculation is complete

<br /> \cdots = \psi(x).<br />

I see that the technical steps in this calculation were complicated, but the ideas were simple. If you ask me that how does this delta function identity work, I think it works according to the principle "integrate first, then take the limit last", which could not be simpler. Indeed, compare these ideas:

Idea 1: Integrate first, then take the limit.

Idea 2: The integral does not exist in mathematical sense, but in physics we can impose physical boundary conditions in infinity, to make the calculation justified.

Which one of these ideas is simpler? Is there anyone who wants to defend the idea 2 as a simpler one? To me it doesn't seem simple at all. It is completely incomprehensible: Nobody even knows what the physical boundary conditions are supposed to mean, not to mention how they would make the calculation justified. That is an example of how exact natural science can become exact natural poetry. It might seem a good explanation only as long as you don't know about anything better.

 

[jostpuur]

I'm taking a huge risk in writing this, because I've come up with this on my own, and it could be full of mistakes In fact I of course know that I don't know the complete proof of the Fourier inverse transform: There was at least one gap in the beginning of the proof, of which I'm aware of myself. It is not my intention to appear as an expert how knows all math better than physicists. My point should become clear despite of this.

 

[jostpuur]

The idea that it is sufficient to tell physics students precisely what is needed to calculate, and nothing more, does not work. It does not work for this reason: Even though the student would have already accepted the principle that mathematics is merely a tool, physicists don't need epsilon-delta-approach and blablabla... when the student is shown how to use the delta function in calculations, inside his head, the students asks himself "why does that work?". Because he is curious! That is why he is a student of physics! He wants to understand how things work! And if the student is not been given proper answer to this, he will come up with his own answer. The student is not necessarily capable of coming up with anything close to a correct answer, and that is why we start getting these explanations with physical boundary conditions.

It is not like that I have merely thought about this inside my head, and then concluded that the physicists' way wouldn't be working. When I'm being told that for physicists, the mathematics is a tool, it sounds like a good idea. I've come to the conclusion that the physicist's way isn't working, because I have observed, that it is not working. With their physical arguments, physicists destroy their capability to think logically.

This is of course only my observation. One person's experience isn't convincing scientific evidence, but it is sufficient for me to make my mind.

btw. Let me ask converse. Has there been studies carried out, that would prove, that physicists who no nothing about mathematical argumentation, are still capable of making progress on their own field equally well as physicists who know mathematics very well? My point, in this, is that how do you really know that the "physicists' way" is working as everybody seems to instist?

 

[jostpuur]

What makes it appropriate? As far as I can tell, they are exactly the same formal argument.

(And since when is 'appropriateness' a quality applicable to a logical argument?)

My point with the kungfu example was that the kungfu guys are doing things in unnecessarily complicated manner, when they first do some movements with their arms to gather chi. If I believed that my computer would not work unless I did some rituals before using it, and then justified these rituals by noting that my computer has always worked after them, that would be analogous to the kungfu chi.

In fact the western martial artists don't seem to understand brick breaking very well either, since they often explain that the great velocity of the fist would result in the greater force being exerted on the brick than on the fist, in contradiction with the Newton's laws. But at least they are not doing the punch in unnecessarily complicated manner. That is "when it works, it is fine"-thinking at its best.

The reason why I connected this to the theoretical physics, is because also in theoretical physics, I can see that the ideas behind the calculations can become unnecessarily complicated, even though simpler and more rigorous ideas would be available.

 

[Pythagorean]

It is not like that I have merely thought about this inside my head, and then concluded that the physicists' way wouldn't be working. When I'm being told that for physicists, the mathematics is a tool, it sounds like a good idea. I've come to the conclusion that the physicist's way isn't working, because I have observed, that it is not working. With their physical arguments, physicists destroy their capability to think logically.

This is of course only my observation. One person's experience isn't convincing scientific evidence, but it is sufficient for me to make my mind.

Discussion:

From your words before, I don't know if you're saying that it doesn't work in education, or it doesn't work for everybody, or it doesn't work for it's intended purpose.

It's intended purpose is to understand the phenomena we experience in the world around us. Consistent phenomena that we can somehow measure, emulate the conditions of, and record things about.

We're desperate to understand these things more than qualitatively. It's our relationship with reality that is important to me as a physics student.

Retort

"With their physical arguments, physicists destroy their capability to think logically."

Following from my discussion above, it depends on how you define "logically". If you speak of pure logic, then that may very well be so, but pure logic isn't always helpful to our relationship with reality... in fact, at some point pure logic is completely detached from reality.

The important side of physics is experiment. We can interact with our world over and over with consistent results as long as we have sufficient control over the conditions. There is error in approximation but if the error is insignificant to the physical question you're asking (like it's pointless to factor the force of gravity into your calculations of a circuit).

The best way to keep track of the conditions is with numbers, and as we start to track the relationship between the numbers, we find complicated relationships and are thus forced to use the math... and eventually, we see certain mathematical tools popping up a lot within physics and this becomes part of our toolbag (mathematical physics) and this is what's taught to physics students.

your Delta function and Fourier Transform discussion

Is interesting. I actually don't remember seeing the integral form of the delta function...

Looking through my Griffith's E&M book, it's not present in the introduction of the "dirac delta function"

I've always boggled over exactly how the Fourier Transform works too. It's a very awesome "tool" as it were and I'm still reading over your post trying to understand some of it.

 

[jostpuur]

lagrange's undetermined multipliers

I'll show how I encountered the Lagrange's undetermined multipliers for the first time. This is from lecture notes of one "mathematical methods for physicists" course:

We want to find extrema of function f(x,y) under the constraint \phi(x,y)=0. In the extremal point we have df(x,y)=0 and d\phi(x,y)=0, so we also have df(x,y) - \lambda d\phi(x,y)=0, where \lambda is the Lagrange's undetermined multiplier. When we substitute

<br /> df(x,y) = \frac{\partial f(x,y)}{\partial x} dx + \frac{\partial f(x,y)}{\partial y} dy<br />
<br /> d\phi(x,y) = \frac{\partial \phi(x,y)}{\partial x} dx + \frac{\partial \phi(x,y)}{\partial y} dy<br />

we get

<br /> \Big(\frac{\partial f(x,y)}{\partial x} - \lambda \frac{\partial \phi(x,y)}{\partial x}\Big) dx + \Big(\frac{\partial f(x,y)}{\partial y} - \lambda \frac{\partial\phi(x,y)}{\partial y}\Big) dy = 0<br />

Hence we must have

<br /> \frac{\partial f(x,y)}{\partial x} - \lambda \frac{\partial \phi(x,y)}{\partial x} = 0<br />
<br /> \frac{\partial f(x,y)}{\partial y} - \lambda \frac{\partial\phi(x,y)}{\partial y} = 0<br />

Now I have few questions about this derivation. Isn't the conclusion

<br /> df = 0\quad\textrm{and}\quad d\phi=0\quad\implies\quad df - \lambda d\phi = 0<br />

true for arbitrary lambda. How is the lambda something that must be solved, in the end? The last step seems to be

<br /> A dx + B dy = 0\quad\implies\quad A=0\quad\textrm{and}\quad B=0<br />

Why like this? Why not for example like this

<br /> \implies\quad \frac{dy}{dx} = -\frac{A}{B}?<br />

(Ok, if these are rigor differentials, then they are indeed linearly independent vectors, and the conclusion A=B=0 is correct, however I know that with this particular lecturer, they were not rigor differentials, instead they were infinitesimal quantities.)

I believe the answer to these questions is that this "proof" is 100% pseudo proof. It has nearly nothing to do with mathematics. This is not necessarily a very good example of why physicists' way of dealing with math would be bad, since I believe most competent theoretical physicists agree with me in that this proof was complete nonsense. So it is merely evidence of how dumb some individual physicists can be.

However, this is still evidence of how harmful the physicists' way can be. Otherwise it would not be possible, that proofs like this were lectured in a university, but the popular thinking that physicists' proofs are allowed to be incorrect, now make this possible.

 

[jostpuur]

Also I must ask, that if the mathematics is supposed to be a tool to physicists, then why not simply take some already proven theorem about Lagrange's multipliers, and then use it as a tool? Why invent incorrect proofs, and then justify them by the argument "math is a only a tool for us"? I believe the answer to this is what I already mentioned in my previous post. It is because in reality physicists want to understand what they are doing, and if mathematical proofs are not given to them, they invent their own proofs instead.

 

[ZapperZ]

btw. Let me ask converse. Has there been studies carried out, that would prove, that physicists who no nothing about mathematical argumentation, are still capable of making progress on their own field equally well as physicists who know mathematics very well? My point, in this, is that how do you really know that the "physicists' way" is working as everybody seems to instist?

I will turn this around and ask you this, which I believe I had asked you way in the beginning of this thread. Can you show, from any of the published works that have been produced, where the WRONG conclusion to such work came out of the result of a faulty understanding and application of mathematics in the SAME vein that you are describing?

This is VERY important, because if you can't, or if it doesn't exist, then your concern doesn't occur and your point is completely moot!

To make this clearer, I can give an example where someone MIGHT use as an example, but it isn't in the same vein as what you are describing. This will be the use of the mean-field approximation in many-body theory (which, btw, is full of nothing more than modeling a valid approximation for complex systems). Such an approximation is made because it is convenient, and it WORKS, most of the time. However, there are many instances where it fails, and we KNOW this. This is why we call it an approximation in the first place! This is not a faulty application of mathematics, but a convenient representation of a physical system that is not solvable analytically.

And speaking of delta function, remember that "quasiparticles" that I brought up earlier? Guess what? If I want to correctly model its behavior in an ordinary metal (y'know, the one that you are using in your electronics and in your house) and derive all the basic equations that you know and love such as Ohm's Law and such that comes out of the Drude model, I will have to model the spectral function of the quasiparticles (i.e. electrons) in those conductors as delta functions! Considering that electrical engineers accept such basic equations as being valid, I would consider this as an example that it works!

So I'll ask this again. So me a clear example where the faulty understanding of mathematics has actually produced a wrong conclusion in the workings of physics.

Zz.

 

[jostpuur]

I will turn this around and ask you this, which I believe I had asked you way in the beginning of this thread. Can you show, from any of the published works that have been produced, where the WRONG conclusion to such work came out of the result of a faulty understanding and application of mathematics in the SAME vein that you are describing?

This is VERY important, because if you can't, or if it doesn't exist, then your concern doesn't occur and your point is completely moot!

I don't have any specific example.

I believe that when student's understanding of mathematics and capability to think logically is destroyed with the education policy that encourages in the "calculate without thinking" style, it has an effect on the quality of his publications later, but naturally I cannot prove this by an example.

To make this clearer, I can give an example where someone MIGHT use as an example, but it isn't in the same vein as what you are describing. This will be the use of the mean-field approximation in many-body theory (which, btw, is full of nothing more than modeling a valid approximation for complex systems). Such an approximation is made because it is convenient, and it WORKS, most of the time. However, there are many instances where it fails, and we KNOW this. This is why we call it an approximation in the first place! This is not a faulty application of mathematics, but a convenient representation of a physical system that is not solvable analytically.

And speaking of delta function, remember that "quasiparticles" that I brought up earlier? Guess what? If I want to correctly model its behavior in an ordinary metal (y'know, the one that you are using in your electronics and in your house) and derive all the basic equations that you know and love such as Ohm's Law and such that comes out of the Drude model, I will have to model the spectral function of the quasiparticles (i.e. electrons) in those conductors as delta functions! Considering that electrical engineers accept such basic equations as being valid, I would consider this as an example that it works!

I did explain (in post #53) that I have no problems with making approximations. These accusations that mathematicians cannot stand approximations stem from some stereotypes.

I even understand that people can take some results as given, and use them as tools since they work, even if they don't understand the underlying proofs or mechanisms. What I don't understand is that when physicists are not satisfied by taking something as a tool, but instead come up with an intentionally incorrect proof, and then say "this proof is wrong, but it is ok because the result is only a tool for us, so the proof is right in a sense". When that happens, it is a clear sign of cargo cult science to me. A scientist should have a more realistic picture of what he knows and what he doesn't.

So I'll ask this again. So me a clear example where the faulty understanding of mathematics has actually produced a wrong conclusion in the workings of physics.

Zz.

Talking about cargo cult science... has it produced incorrect results ever either? I don't think it happens. If science goes cargo cult, the research goes meaningless, but not wrong.

 

[ZapperZ]

I don't have any specific example.

Then don't you see that this has rendered your complain absolutely moot?

Remember, you are dealing with many physicists here. This is similar to my asking for physical evidence and you telling me that you have none or that it is there, but we can't measure the effect.

If what you are complaining about has no manifestation in the workings of physics that leads to false or incorrect outcome, then this is a waste of time.

Zz.

 

[jostpuur]

relativistic mass situation

In mathematics there is no general agreement about which of the definitions of the natural numbers should be considered correct, \mathbb{N}=\{0,1,2,3,\ldots\} or \mathbb{N}=\{1,2,3,\ldots\}. Sometimes the zero is there, and sometimes it is not. Consider a following hypothetical claim.

"The definition \mathbb{N}=\{0,1,2,3,\ldots\} is in contradiction with mathematics. In reality zero does not belong to the \mathbb{N}. If zero was in the \mathbb{N}, a contradiction would arise, because on the other hand 0\notin\mathbb{N}=\{1,2,3,\ldots\}."

This is ridiculous circular conclusion of course, but fortunately I have never heard anyone explaining anything like this. Why have I not heard this? Well, because mathematicians are not so stupid that they would claim anything like this Mathematicians understand the difference between definitions and theorems.

In physics not everybody agrees whether mass m should mean the rest mass m_0, or the relativistic mass m_{\textrm{rel}}. Consider the following claim.

"The relativistic mass is in contradiction with relativity. If, in reality, mass grew as a function of speed, then a contradiction would arise, because this could be used to measure absolute speed."

This is equally ridiculous circular conclusion, but for some reason, this example is not hypothetical, this is reality. I have encountered this argument numerous times on the internet and in universities. Why? I have no other explanation for this, that the physicists, who speak like this, don't understand the difference between definitions and theorems. In other words, the stupidity stems from poor understanding of mathematical thinking. These physicists don't consider symbols m,m_0,m_{\textrm{rel}} as something that have defined meanings, but instead as something that represent some kind of... mhmhmh... physical truths?

This probably doesn't qualify as an answer to ZapperZ's question, but don't you still find this example somewhat disturbing? I have difficulty believing, that a physicists who have this bad conceptual problems with the mass concept, could ever come up with anything relevant on other matters either.

 

[ZapperZ]

In physics not everybody agrees whether mass m should mean the rest mass m_0, or the relativistic mass m_{\textrm{rel}}. Consider the following claim.

"The relativistic mass is in contradiction with relativity. If, in reality, mass grew as a function of speed, then a contradiction would arise, because this could be used to measure absolute speed."

This is equally ridiculous circular conclusion, but for some reason, this example is not hypothetical, this is reality. I have encountered this argument numerous times on the internet and in universities. Why? I have no other explanation for this, that the physicists, who speak like this, don't understand the difference between definitions and theorems. In other words, the stupidity stems from poor understanding of mathematical thinking. These physicists don't consider symbols m,m_0,m_{\textrm{rel}} as something that have defined meanings, but instead as something that represent some kind of... mhmhmh... physical truths?

This probably doesn't qualify as an answer to ZapperZ's question, but don't you still find this example somewhat disturbing? I have difficulty believing, that a physicists who have this bad conceptual problems with the mass concept, could ever come up with anything relevant on other matters either.

This is a poor attempt at reflecting how we work in physics. Where did you get all of this from? Some webpage that is trying to explain to the GENERAL PUBLIC on what mass is?

You'll note that in the particle data book, there is ZERO AMBIGUITY of the mass that is defined in there. Why do you think that is when you seem to be convinced that ... In physics not everybody agrees whether mass m should mean the rest mass m_0, or the relativistic mass m_{\textrm{rel}}." So what "physics" are you dealing here when the rest of physics really has no issues with what "mass" we are dealing with.

Wait till I tell you about the "mass" of the electrons in your semiconductor.

If you are so concerned about a poor misrepresentation of mathematics by physicists, you shouldn't do the same crime by making a poor representation of physicist.

Zz.

 

[jostpuur]

Where did you get all of this from?

From university teaching. Professional physicists have explained me that relativistic mass is in contradiction with relativity. One graduate student has accused me of attempting to overthrow relativity, when I said that their relativistic mass arguments are stupid.

 

[ZapperZ]

From university teaching. Professional physicists have explained me that relativistic mass is in contradiction with relativity. One graduate student has accused me of attempting to overthrow relativity, when I said that their relativistic mass arguments are stupid.

So you are equating what is going on at your university as part of teaching as representative of what is going on in physics in general? Do you seriously think this is a valid observation when I had just cited to you what professionally is going on?

Look in physics papers. Is there EVER such a confusion that you just described? Look in condensed matter physics where in heavy fermionic systems, the electron being measured can have an effective mass as high as 200 times the bare mass? Are there EVER any "confusion" in the mass of anything based on the context that it is being presented?

I can deal with trying to argue and discuss legitimate problems in the workings of physics. However, I have almost no patience when I have to deal with non-existent problems that is being brought up based simply on a very narrow observations. If you see a physics professor abusing female colleagues at your university, do you also draw a conclusion that physics is hostile towards women in the field?

You need to be aware of how you are drawing up generalized conclusion about something based on flimsy, or in this case by your own admission, non-existent evidence. If you are not concerned about this pattern that you are exhibiting, then there's nothing I can say here to convince you of otherwise.

Zz.

 

[Marco_84]

Ok... after half an hour or more I've read al the thread.

I study theorethical physics in Milano Italy, i'll tell you my story:

The first two year of UNI Physicist and mathematicians have all the same course (CalculusI,II,Lin Alg, EM, CM, Diff geom...) during the examinations, written and orals teachers wants from us a perfect distinction of what a def or a theorem is OBVIOUSLY.
Maybe during an oral examination the teacher prefers that a physicist can explain much better how was done the experiment and the setup... but wants that the mathematicians solve the maxwell equation with proper BC and stuff without esitations...

What I am trying to say (its not so easy in another language) is that I've seen mathematician that don't even understand the difference between hp and th. And physicist (me) that cannot plug a capacitor!

I like to be rigourous also, but what I've learned is that, in science generally not only physics, we need to try and guess... and maybe, later, when everything is more clear, we can fix all the details...

All the differences everybody is making are just in his mind because the important is:

KNOWLEDGE AND SHARE IT

Never forget this.

Galileo, Newton, Huygens... they were physicst or mathematicians??
For me Nothing of this... They were just liberal thinkers...


Who can tell which method is better??

Ill tell the story of the Galielo and its cycloide; this is to justify the sentenceof ZZ: IT WORKS.

He was working in his lab and during that period he was trying to quantify the area underneath this curve.
Unfortunately he didn't know the calcus that we "bearly understand", integrals or diff equations...
But He had scissors, steel,pencils,libras and wheels.
He mesaured the total weight of the steel, then he drew the curve and cutted out the area desired.
After that he mesaured the weight of the cycolid and with a proportion he found out that
the ration of that area upon the genereting circle was 3:1.
Well he didint believe at the first that the formula could be so easy:

A=3 \pi r^{2}

In any case, i don't know elsewhere, but in italy we have a course at third year on functional analisys and other stuff (mathematical method..) where they theach us what a generalized function is and how to get the proper def of delta dirac distribution.
Obviously in QM course or QFT course we don't always have to take the Rudin book in our hand and remember the dominate convergence theorem or every time i solve an ODE should i re-deomnstrate its excistence (difficult) and uniqness (easy)?!?

regards
Marco

 

[jostpuur]

If you see a physics professor abusing female colleagues at your university, do you also draw a conclusion that physics is hostile towards women in the field?

I don't draw conclusions like this. One of my original points in this thread was that why is there not discussion or debate about physicists' way of dealing with math. For example, there is discussion about sexual harassment. The complete lack of discussion (with the exception of Winitzki's "Why physics is hard" paper) is what keeps puzzling me.

I also can't help thinking about some kind of politics of fear in this thing. It is always dangerous to complain about somebody's math, because one's own math could get more attention as well. Complaining about some math being incomprehensible could be interpreted as a sign of weakness.

I have seen enough bad math, that I'm not going to interpret the lack of discussion as lack of problem anymore. The lack of discussion is evidence of quite opposite.

 

[ZapperZ]

I don't draw conclusions like this. One of my original points in this thread was that why is there not discussion or debate about physicists' way of dealing with math. For example, there is discussion about sexual harassment. The complete lack of discussion (with the exception of Winitzki's "Why physics is hard" paper) is what keeps puzzling me.

I also can't help thinking about some kind of politics of fear in this thing. It is always dangerous to complain about somebody's math, because one's own math could get more attention as well. Complaining about some math being incomprehensible could be interpreted as a sign of weakness.

I have seen enough bad math, that I'm not going to interpret the lack of discussion as lack of problem anymore. The lack of discussion is evidence of quite opposite.

If there is sexual harassment in physics, it will be due to a well-documented evidence, not simply due to one person doing it at one location.

I've asked to document such "mistakes" in mathematics that translates into faulty physics. You offered none, and could not find any. What do you expect me to do? You started a thread to complain about something that does not register in the workings of physics or the validity of the results. You complained about things that appears to be based more on your lack of knowledge on how physics work rather than what actually transpired. How exactly do you expect me to debate with you on things that you can't even show to exist?

Even for mathematicians, that is really a rather illogical task to do!

The fact remains that you cannot find any evidence that such faulty mathematics is being practiced unknowingly in all of these physics papers. If you can, you would have written your "complain" to the journals already as a rebuttal. Instead, you made up a series of faulty mathematics that you claim physicists practice (with no evidence, mind you) without knowing anything of what they are doing, and you want physicists to defend or response to such a thing. I'm surprised you didn't ask us to defend religious beliefs on angels!

Zz.

 

[ZapperZ]

Mary Boas is a stickler for presenting mathematics accurately, but also making sure that students can actually USE the mathematics quickly. In her book "Mathematical Methods in the Physical Sciences", she even had this to say:

There is no merit in spending hours producing a many-page solution to a problem that can be done by a better method in a few lines. Please ignore anyone who disparages problem-solving techniques as "tricks" or "shortcuts". You will find that the more able you are to choose effective methods of solving problems in your science courses, the easier it will be for you to master new material.

I find absolutely nothing wrong with that statement. I have said this many times already, that we tend to know the mathematics that we are using. Unless someone can find fault with the mathematics being presented in Boas's text, I can't find a single problem with the way it has been presented for USE by physicists. The same can be said about Arfken text on Mathematical Physics.

So that is why I want to be shown evidence that "bad mathematics" has crept into the practice of physics, and this has somehow caused faulty conclusion. If there is such a thing, I want to write a rebuttal to it so that I can pad my publication list, since obviously no one here wants to do that.

Zz.

 

[Hurkyl]

I'm taking a huge risk in writing this, because I've come up with this on my own, and it could be full of mistakes

It turns out that it's fairly close to at least one rigorous method. The broad strokes for the univariate case are thus:

Choose a space \Phi of "test functions". Then, we can turn any function f into a linear functional (which I will denote as \hat{f}) on \Phi by convolution:

\hat{f}[ g ] := \int_{-\infty}^{+\infty} f(x) g(x) \, dx

The upshot is that \Phi^* (the "continuous dual space" of \Phi, which contains all linear functionals) contains some functionals, such as:

\delta[ g ] := g(0)

which are not given by functions. But in \Phi^*, they are limits of functions. Limits are computed "pointwise", so:

\left( \lim_n \hat{f}_n \right)[g]<br /> = \lim_n \left( \hat{f}_n [g] \right)<br /> = \lim_n \int_{-\infty}^{+\infty} f_n(x) g(x) \, dx<br />

which is precisely what you suggested:

Idea 1: Integrate first, then take the limit.

Going back to your expression, if we fix x as a constant, then your family of functions was

f_L(x&#039;) = \frac{1}{2\pi} \int_{-L}^{+L} e^{i k (x&#039; - x)} \, dk

and the relevant assertion is that

\lim_{L \rightarrow +\infty} \hat{f}_L = \delta.



Physicists seem to use the integral notation for all distributions, and not just for convolutions of ordinary functions; one thing that would please me greatly would be to see a specification for the grammar they use for that. It's not always clear to me what is a function, what is a functional, and so forth. It doesn't help that everybody likes to pretend that it's nothing more than elementary calculus.

 

[Marco_84]

Why everybody thinks that physicist don't know what you have just wrote down about delta function?
I think this is very stupid...

regards
marco

 

[explain]

Somebody asked whether there are actual examples when insufficient knowledge of mathematics gave wrong results in physics.

There certainly exist many examples where physicists have arrived to incorrect conclusions due to wrong understanding of some mathematical issues. However, these examples are merely isolated articles because other physicists do the calculations more carefully or by different methods and eventually get the correct results, discovering and pointing out out the errors.

I can give two actual examples:

1) Calculation of Hawking radiation from the so called "tunneling formalism" (Parikh-Wilczek). Some people used a similar calculation and got twice the Hawking temperature and for a while there was a discussion about why that was so (e.g. Padmanabhan's review in Phys.Reports has this). The main problem with the calculation is that one has some integral that diverged but introduces an ad hoc "physical" prescription about how to go around the pole in the complex plane. The result is bogus. Later the error was explained in another paper, I forgot where (I think by E. Vagenas).

2) There is a paper in Nature (a "short communication") and also in some J. Physics A (see ref. below), where the authors "prove" by some "physical" arguments that all calculus textbooks can't compute derivatives of cotangent. It was claimed that

\frac{d}{dx}\frac{\cosh x}{\sinh x} = -\frac{1}{\sinh^2 x} + 2 \delta(x).

Of course their arguments are total \textrm{bulls}\textrm{hit}; there cannot be a delta function here. But the authors were solid state physicists and didn't really seem to understand such mathematical subtleties. They needed to get some answer out of some sloppily calculated integral; actually they already knew the correct answer but they could not find it unless they inserted an extra delta function as above. In other words, they wanted to fix a wrong calculation by inventing the wrong formula for d/dx (cosh x / sinh x) and arguing that the new formula is more correct than the standard formula. See arXiv:0705.1512 for references and explanations of why this is wrong (no, I'm not the author of that publication :).

3) When I was a student I passed a math exam for physicists where I was given a problem similar to the following.

Compute the definite integral

F(a,b)= \int _0 ^\infty \left( \frac{\sin(ax)}{ax^2} - \frac{\sin^2(bx)}{b^2x^3} \right) \textrm{d}x .

My solution was (unnecessarily) long and complicated and produced a complicated formula. The examiner's solution was short and brilliant. It ran something like this. Let's change variable as u=ax in the first term and as u=bx in the second term; this produces

F(a,b) = \int _0 ^\infty \left( \frac{\sin u}{u^2} - \frac{\sin^2 u}{u^3} \right) \textrm{d}u .

It follows that F(a,b) is actually independent of the parameters a,b. Now integrate the second term by parts and obtain

F(a,b) = \int _0 ^\infty \left( \frac{\sin u}{u^2} - \frac{\sin 2u}{2u^2} \right) \textrm{d}u + \left. \frac{\sin^2 u}{2u^2}\right|_0^\infty.

Again change variable to 2u in the second term in the integral. Then the integral vanishes, and so the result is just -1/2.

Of course this "brilliant" solution is wrong. The limit at x\to 0 needs to be handled more carefully. I don't remember the answer for F(a,b) but it's definitely a nontrivial function of the parameters. (I was not given the full score on this problem because the examiner believed in his solution.)

Also it's an interesting remark made by someone before that physicists invent their pseudo-explanations because of lack of real explanations in physics textbooks, and lack of real mathematical culture. Indeed the mathematics used in such pseudo-explanations is most of the time totally confusing. The answer is correct (especially if it is known beforehand), but of course there are infinitely many ways of obtaining a correct answer by incorrect calculations. This does not help students who try to understand some new material, and this does not help researchers to push science forward if they never learned correct explanations for Lagrange multipliers.

As an example: \textrm{Landau and Lifsh}\textrm{itz} vol.2, ch. 9 or 10 "prove" that the covariant derivative of the metric is always zero "from first principles", i.e. merely from the transformation properties of the covariant derivative. In fact their proof silently assumes that the covariant derivative of the metric is zero at some step where they appear to be just "juggling the indices."

 

[ZapperZ]

I don't think this is what I am looking for. I mean, people makes mistakes all the time, both in mathematics and in physics. So these corrections are not unexpected.

What I was looking for was an accepted standard practice of mathematics in physics. If you look at the first post, it isn't about someone making a math mistake, but rather the whole accepted and common practice of using math that is "at fault".

Zz.

 

[explain]

Then I would give the example of renormalization in quantum field theory. It is an accepted standard practice to perform renormalization using totally unjustified mathematical steps (Feynman integral, perturbative expansion, cutoffs, counterterms, etc.). These unjustified mathematical steps are presented as "black magic that works." For example, one computes the cross-section of scattering of photons on electrons as a perturbative asymptotic expansion in the fine structure constant \alpha. This would make sense if the cross-section were a well-defined function of that constant, say \sigma(\alpha); then we can certainly compute the asymptotic expansion of \sigma(\alpha) at small \alpha. However, the cross-section is actually undefined as a function of the coupling constant; only the asymptotic expansion is defined by ad hoc tricks (each term of the perturbative expansion is a divergent integral that needs to be replaced by a convergent integral in some way). This is how things have been for the last 50 years in mainstream high energy physics. There is by now a growing body of mathematically rigorous QFT work that will probably soon culminate in a book (say in 20 years or so) showing how to compute "renormalized" quantities without fraudulent mathematical operations. Maybe. But in any case, the "black magic" of renormalization has been accepted and remains in the curricula everywhere.

Now the catch is that we actually do not know whether the "black magic" works and why. All we know is experimental agreement in certain theories (perturbative electroweak + some QCD calculations). But we do not have experiments in quantum gravity, and we do not have a good handle on the nonperturbative QFT regimes. If there are several different methods of performing renormalization, we do not know which method is "more correct". For example, there is Pauli-Villars regularization, dimensional regularization, zeta-function regularization, momentum cutoff regularization, and maybe other methods. There are some cases (some theories of gravity + matter fields) when different methods of renormalization give different results. Since all these methods are "black magic", we have no idea which result is correct and why the results are different. This means that we have no way of finding predictions of a given quantum field theory unless by some happy coincidence all our regularization methods give the same result. But even in that case we still don't know whether the result is correct. (What if the "black magic" is wrong, but the theory is also wrong, in such a way that some calculations by accident give correct results that agree with some limited set of experiments?) This is a very unsatisfactory state of affairs.

 

[ZapperZ]

Then I would give the example of renormalization in quantum field theory. It is an accepted standard practice to perform renormalization using totally unjustified mathematical steps (Feynman integral, perturbative expansion, cutoffs, counterterms, etc.). These unjustified mathematical steps are presented as "black magic that works." For example, one computes the cross-section of scattering of photons on electrons as a perturbative asymptotic expansion in the coupling constant. This would make sense if the cross-section were a well-defined function of that constant of which we compute the asymptotic expansion. However, the cross-section is actually undefined as a function of the coupling constant; only the asymptotic expansion is defined by ad hoc tricks (each term of the perturbative expansion is a divergent integral that needs to be replaced by a convergent integral in some way). This is how things have been for the last 50 years in mainstream high energy physics. There is by now a growing body of mathematically rigorous QFT work that will probably soon culminate in a book (say in 20 years or so) showing how to compute "renormalized" quantities without fraudulent mathematical operations. Maybe. But in any case, the "black magic" of renormalization has been accepted and remains in the curricula everywhere.

This occurs in condensed matter physics as well, and this is rather a well-known "problem". But is this done unknowingly, though? Or is this done rather out of necessity? And in physics, there are many "ad hoc" introduction. One would say plugging in the fundamental constants into anything is an ad hoc process. And the whole field of phenomenology might even be loosely termed as that. I don't see this as being wrong, because the "guide" here has always been the physics and the empirical results.

I can even bring one example very easily in which I am intimately connected to right now since that's what I'm looking into. The field-emission process has been very much described by the Fowler-Nordheim equation. In fact, with very little variation, it is used in many application, from SEM, to the new flat-panel displays. Yet, if you look carefully at the FN-model, it makes a number of important simplification, the first of which is that the working temperature is very close to T=0K! It also makes a very important assumption that the applied field or the effective field at the field emitter is very much smaller than the work function of the metal. These and a number of other assumptions that are not as crucial, allowed for an analytical equation to be derived.

Yet, even with such assumptions, it is used in a wide range of conditions in which those two assumptions are not satisfied. Many applications use field emitters that are at hundreds of degrees celsius, while others are in field up to the scale of MV/m. Yet, the FN model is STILL useful in extracting the important aspects of the system, so much so that devices can be designed using it! For that they are used for, the FN model is as good as any.

But yet, for many, this is a "mathematical error" being propagated. I don't. I see it as a simplification of the physical description of a phenomena to make it usable, and usable accurately enough.

Zz.

 

[explain]

What you are saying is that an approximation is used beyond the range of its validity, just because a more precise computation is too complicated. This is a different story. What I meant to say is that the "black magic" of renormalization is not even presented as anything different from the usual "physical" nonrigorous calculation.

Many physicists do not think that renormalization is ill-defined but say after Feynman, "shut up and calculate!" These physicists have been taught hundreds of useful facts using either wrong or almost wrong mathematics, and they have been made insensitive to it. Other physicists try to invent "physical" arguments that "show" that the Feynman integral or the "black magic" of renormalization make sense somehow in terms of "physical interpretation". For example, one can "calculate" 1-1+1-1+... by writing

1-1+1-1+...=\left.\frac{1}{1-x}\right|_{x=-1}=\frac{1}{2}.

This "calculation" is quite meaningless, but I am sure there are some cases where the "result" 1/2 can be useful. Imagine a physicist confronted with this "calculation." The physicist might say something like, "Well, in this case we need the analytic continuation of this series to x=-1, which takes care of the singularity, and we do not expect a singularity for physical reasons, so we need to remove it... let's check this against the experimental result..." Many physicists are in denial or insensitive to the fact that their routinely done calculations are mathematically quite meaningless.

 

[jostpuur]

It is a different thing to face a problem to which no rigor mathematical theory exists yet, than to distort already existing mathematics and to do logical mistakes. I was complaining about the latter. (edit: I mean that I don't see a point in complaining about renormalization)

 

[jostpuur]

hello, explain. Your post #92 looks interesting! I'll have to take a closer look at those things.

 

[explain]

hello, jostpuur. You seem to be interested in the lack of rigorous mathematics in physics. I am trying to show examples where physicists proceed regardless of nonrigorous mathematics. The examples I showed are not simply mistakes in calculations but mistakes that originated from lack of understanding of certain mathematical issues.

My post about renormalization was a response to ZapperZ's question about something that is "accepted standard practice" in physics but is mathematically incorrect.

 

[Poop-Loops]

The way I see it, physicists knowingly use "bad math", so they can tell you "it works here because we're making an approximation" or some such. For example, when instead of an sum (big sigma, forget what it's called), you use an integral, because the number of particles you are integrating over is so huge it roughly works out.

They're not doing it because they don't know any better, they are doing it because it makes life a lot easier, and they acknowledge the limitations.