Rigorousness of mathematics in physics classes

  • #1
372
0
Does anybody else here find that their physics classes - while providing an acceptable working understanding of the maths involved, don't treat the mathematics with enough rigorousness for their students to be able to understand other discussions on the subject?

I ask because I'm quite often baffled by the mathematical discussions that go on here; and also struggle to understand many textbooks on areas that I nevertheless do quite well in. As an example, I am very happy with wave quantum mechanics and can get by with matrix mechanics, but I don't know what a Hilbert space is and don't seem to be able to find any sources to explain what one is in terms I understand.

In general this phenomenon means I'm stuck with only my lecturer's interpretations of the subject and nothing else, because the amount of mathematical terms in most mathematical physics books I've found goes way over my head. Am I alone here?
 

Answers and Replies

  • #2
ZapperZ
Staff Emeritus
Science Advisor
Education Advisor
Insights Author
35,847
4,663
Wait... you're contradicting yourself.

First you said that the mathematics do not have enough "rigorousness". But then, you said that most mathematical physics books goes way over your head. Have you ever considered that maybe those things that you don't understand ARE the "rigoriousness" that you were looking for. Just because you do not understand it, doesn't mean such a thing doesn't exist.

I know that Arfken deals with this in his book. How "rigorous" you want it to be depends on you, because these books, and the physics classes, are not meant to teach you the mathematics, but rather how to use the mathematics as a means to a specific end, which is the physics. You can always pursue the mathematics itself, either on your own, or by taking extra classes from the math department. That is where they do everything rigorously, to death, if that is what you want.

Zz.
 
  • #3
372
0
What I'm saying is that I wish my classes were more rigorous so that I could understand the level of mathematics involved in textbooks on the subject. As it stands, the style of teaching I receive makes it difficult to read around the subject because, unlike the class teaching, most authors treat the mathematics much more technically and thus beyond my understanding. To go back to dedicated mathematics textbooks below that level to get a grounding in the general principles first would take me far, far too long.
 
  • #4
664
3
What is stopping you from trying to learn the mathematics you do not understand? There are mathematics textbooks available in print and there are several online resources as well. Or you could just discuss it with someone in real life. Don't see problems, see solutions.
 
  • #5
372
0
What is stopping you from trying to learn the mathematics you do not understand?
In short, time. I could give up what free time I have to study mathematics on top of my course, but...well, I didn't come here to do mathematics. I care about it right now merely as a means to an end - understanding physics better.

I get the impression that many british universities are actually similar in this respect. Physics lecturers seem to frame their material in a functional style, rather than delving into technicalities. This would be fine, if not for the different approach taken by authors. The two approaches don't mesh terribly well. Pragmatism, or laziness?
 
  • #6
990
58
You might try to find Arno Bohm's Quantum Mechanics book at the library. He takes more time with the mathematics of Hilbert space at the beginning.

The reason physics books aren't as rigorous as mathematical physics books is that that level of detail is not needed to get ones work done, and there really isn't time to cover it. Take a look at some functional analysis books that cover Hilbert space: they are quite thick and dense.
 
  • #7
664
3
In short, time. I could give up what free time I have to study mathematics on top of my course, but...well, I didn't come here to do mathematics. I care about it right now merely as a means to an end - understanding physics better.

I get the impression that many british universities are actually similar in this respect. Physics lecturers seem to frame their material in a functional style, rather than delving into technicalities. This would be fine, if not for the different approach taken by authors. The two approaches don't mesh terribly well. Pragmatism, or laziness?
Trying to figure out how little effort one need to spend to barely reach ones goal is a very bad idea. There is nothing that is keeping you from using the time you do have more efficiently. You do not have to give up anything. Still, if you do, learn to sort out your priorities.

Mathematics is a tool for problem solving and a vital tool for understanding physics. Mathematics is a part of physics, whether you like it or not.
 
  • #8
359
3
Should a physicist study math proofs? is found here:

https://www.physicsforums.com/showthread.php?t=149605

Generally, the physicists said that math is just a tool for them to get the physics job done. There are many, many mathematical tools that a theoretical physicist needs to know, and they basically said that they don't have the time nor the inclination to study the foundation behind these mathematical tools.

I believe that a theoretical physicist probably knows more mathematical tools (relating to physics) than a mathematician knows, and that is because a physicist will just take up marginal time to learn the tools to know how to use them, whereas a mathematician will devote a great deal of time to work out the properties, theorems, abstractions, generalizations, etc... for each tool. A theoretical physicis probably knows about 5 times (rough number) as many mathematical tools (relating to physics) than a mathematician, but a mathematician probably understands each tool 10 times (again rough number) better than the physicist.
 
Last edited:
  • #9
372
0
Trying to figure out how little effort one need to spend to barely reach ones goal is a very bad idea.
Far from it. I'm more than happy with my current performance already - it's just that I get this sinking feeling over not having the deep understanding of theoretical physics that I'd like to. I'm just saying that I'd like to be able to feel more confident that I understand the formalisms of the theoretical stuff, but I'm not willing to devote half my life to doing so. All I want from lecturers is a bit more attention to detail, and "read this, this and this" occasionally. Not too much to ask.
 
  • #10
664
3
andytoh, this has nothing to do with studying proofs. At all. It is about understanding the mathematics behind it in order to understand the physics. See post #3 for a better definition of what the OP wants.

Sojourner01, you cannot except that the world will change according to your needs, just because you want to learn the underlying mathematics in physics courses. If instructors started wasting time going over every single concept in mathematics that is in use in that particular course, they would waste hours and hours. Instead, you should adapt to the world.

It is all about discipline. If there is something you do not understand to the extent that it is disturbing your physics studies, find information about it for yourself instead of blaming the instructor for your lack of understanding. That is not the way the world works. Just look up the underlying information about that particular section of mathematics. How hard can it be?
 
  • #11
mathwonk
Science Advisor
Homework Helper
10,996
1,187
I also struggled as a student to understand the mathematical parts of my physics classes. This was in my case because the mathematics clases had evolved to presenting the material in a sanitized way that ignored its physical beginnings.

The physics use of math was actually linked to the rigorous math but no one took time to tell me how. for example in feymans lectures he gives a nice little approximation of the length of a planetary orbit, just by adding up polygonal terms. This blew my mind because i thought an integral was an antiderivative and had forgotten it was about an idealized way of adding things up.

another time i was tempted in a problem to use integerals in a suggestive but imprecise way prompted by the leibniz notation. I spent an hour or so justifying that use to myself before using it in my homework, and was praised by the physics grader for being the first person in over a hundred homeworks to do so (this was at harvard).

Again the rigor was achievable but it took time no one wanted to spend. I thionk this is aprtly because the correctness of the use of math in physics is not based on the rigor of the math but on the track record of its success in physics and on the intuition of physicists, which is actually more reliable than mathematical rigor.

It is a challenge and an instructive one to bring the two into accord, but in some cases this is still an open challenge. In any case, when studying with physicists, one should try to acquire the physicists ability to know his use of math is correct without an epsilon and delta proof.

However, this is a math guys point of view, and Zapper is more qualified on the use of math in physics than I.
 
  • #12
372
0
And mathwonk comes up with the most balanced and insightful response I've yet received to this problem, on the forum and elsewhere.
 
  • #13
1,306
19
One example is people just think that integration is just a way to calculate areas underneath curves...its scope is much broader than that, and if you don't understand the true meaning of integration and how it works, then you are screwed in most of calculus. Same goes for differentiation.

Too much mathematical rigor can also be bad for you and your studies of physics. If you get too formal you lose sight of the intuitive nature of the mathematics. While a mathemeticians can take 5 to 6 lines of derivation to show a particular fact, you often can see that the result is quite obvious by thinking with less rigor and with a little more intuition and abstraction. Sure, physicists need to have a certain element of mathematical rigor in their work, but intuition is just as important.

Another example that makes me cringe (and I hear it all the time) is when people refer to a differential as a 'rate of change'. They don't see the distinction between dy and dy/dx, for instance.

Quit looking at derivations as just mathematical manipulations and whatnot, and start looking at derivations with more of an abstract mindset. Open your mind.
 
Last edited:
  • #14
mathwonk
Science Advisor
Homework Helper
10,996
1,187
crash course in hilbert space

thank you for the kind response. i will try to remind you also that you probably do know what a hilbert space is.

R^n has many important properties. the game of axiomatizing consists in taking some of those properties, assigning a name to them, and calling any other object sharing those properties by that name.

For hilbert space we take quite a few of the properties of R^n, namely, ability to add vectors and multiply by scalars, ability to take dot products, and hence to define lengths using them, and finally the validity of the cauchy criterion for convergence. that is all.

i.e. a hilbert space is any vector space having a positive definite dot product, such that all cauchy sequences converge in the distance |V| defined as the square root of V.V.

Do not feel bad if you did not remember this definition of hilbert space, as one of david hilbert's most famous attributed quotes is something like "and what is this 'hilbert space'?".

a consequence of the axioms choaen is that one can use infinite series, i.e. infinite linear combinations of vectors, which converge if and only if their absolute values do so.

moreover one has a concept of "hilbert basis" which is an infinite orthonormal set, such that every vector in the space is either a finite or an infinite linear combination of those.

this allows one to deal with a space of uncountable linear dimension, with a countable "hilbert basis" of vectors.

then the usefulnes of this concept is that most of the basic theorems of fourier analysis go thropugh in this context. I.e. the canonical example of a hilbert space is the space of functions expresible as infinite series of sines and cosines, with the usual dot product, namely integral of the product over some convenient interval, like -pi to pi.

abstractly, this is isomorphic, as are all hilbert spaces having a countable hilbert basis, to the space of "square summable sequences" of real or complex numbers.

these are the smallest infinite dimensional hilbert spaces. they are called "separable". thus we have taken most properties of R^n except finite dimensionality.:tongue:
 
  • #15
mathwonk
Science Advisor
Homework Helper
10,996
1,187
in general try never to be intimidated by a big name attached to a concept, like hilbert or banach space, A HEISENBERG GROUP, OR A HAAR MEASURE, or galois cohomology, or a normal separable proper scheme of finite type over an artin ring. it is merely shorthand for some ist of proeperties which someone is too lazy to give you, and also a mask for frightening the uninitiated.

grothendieck always tried to choose terminology for his ideas which was suggestive of their meaning (a separated scheme behaves like a topological space where one can separate distinct points by disjoint open sets), but too often concepts are merely named for someone a student or colleague wishes to honor, but who often had no role in their discovery.

The "Kodaira dimension" of an algebraic variety e.g. is apparently a concept introduced by moishezon in the Steklov Institute seminar on surfaces and named later by others. This of course is not kodaira's fault.
 
Last edited:
  • #16
George Jones
Staff Emeritus
Science Advisor
Gold Member
7,384
1,001
Sojourner01, you might find https://www.physicsforums.com/showpost.php?p=1008436&postcount=60" of mine interesting.

This type of stuff is important only to a small minority of phsyicists, but is becomng increasingly useful in some area of string theory, high energy physics, relativity, etc.

As ZapperZ says, if you are interested in mathematics as mathematics, you can fill some of your options with pure math courses. This is what I did, both as an undergrad, and as a grad student.
 
Last edited by a moderator:
  • #17
George Jones
Staff Emeritus
Science Advisor
Gold Member
7,384
1,001
The "Kodaira dimension" of an algebraic variety e.g. is apparently a concept introduced by moishezon in the Steklov Institute seminar on surfaces and named later by others. This of course is not kodaira's fault.
This reminds me of story about John Wheeler (of MTW) and the popularization of Kruskal-Szekeres coordiantes that are so well known, and so important for the study of black holes.

From

http://www.princeton.edu/~paw/archive_old/PAW98-99/09-0210/0210irtx.html

"In one amusing incident, Wheeler tells the story of an "elegant discovery" concerning spacetime and relativity made by Martin Kruskal, a mathematical physicist and colleague of his. At the time Kruskal made the discovery, he related it to Wheeler. When Wheeler found that, for whatever reason, Kruskal had not published the results, he wrote a paper up himself with Kruskal's name, and sent it to Physical Review. He forgot, however, to tell Kruskal he had done so. Kruskal only learned of "his" article upon receiving mysterious galley proofs from the journal; when he figured out what had happened, he suggested coauthorship, which Wheeler politely rejected."
 
  • #18
mathwonk
Science Advisor
Homework Helper
10,996
1,187
One of my collagues was also surprized to see a joint article with his own name on it in the bibliography of another department member who had added it out of homage to the contribution made by the other.

I know of other cases but will not list them since the parties are still active and I could get it wrong.
 
  • #19
mathwonk
Science Advisor
Homework Helper
10,996
1,187
Sojourner what think you of the crash course in hilbert space?
 
  • #20
D H
Staff Emeritus
Science Advisor
Insights Author
15,393
684
Too much mathematical rigor can also be bad for you and your studies of physics. If you get too formal you lose sight of the intuitive nature of the mathematics.
Too much mathematical rigor can even be bad for the study of even applied mathematics. I took a graduate-level ODE class many years ago. The instructor was very rigorous, which I liked because it was appropriate to the subject at hand. I had the same instructor the following semester for optimal control theory, an applied math class.

The first 50 or so pages of the text provided the context for the subject: The nature of the subject, necessary and sufficient conditions for the existence of an optimal controller, stability of the controller. The instructor got so hung up on the lack of rigor in this introductory section that we only made it through the first 60 pages of the book.:eek: We never got into the meat of the subject. I had to learn optimal control theory on my own.

In reality, the book is one of the most rigorous applied math texts in my library. My advice: Beware a pure mathematician teaching an applied math course. You might not learn anything.
 
  • #21
372
0
Thanks, mathwonk. It's ever so slightly clearer now, perhaps only in that I now know that the definition of a Hilbert space doesn't really do anything of great importance and that it's the only kind of space that makes sense to use with quantum mechanics anyway. The definitions still went a little over my head, believe it or not;
cauchy sequences,
basis,
uncountable linear dimension,
isomorphic,
countable hilbert basis,
square summable sequences


Are a few I'm not familiar with. This is how rudimentary my mathematical teaching is.
 
  • #22
mathwonk
Science Advisor
Homework Helper
10,996
1,187
a cauchy sequence is a sequence {a(n)} whose entries cluster together as n gets larger. technically, for every e>0 there is an N such that whenever n,m>N then |a(n)-a(m)| < e.

this means that ideally they should have a limit, and the only thing that can make this fail is that the place where they are clustering is a hole in your space.

R^n does not have any holes in it, because the real numbers are defined not to.

but infinite dimensional spaces do have holes, and where the holes are can depend on how you measure distances. for instance with the integral distnace i mentioned above, you can have have continuous functions converging closer and closer to discontinuous ones, so if you leave out the discontinuous functions from your space then you have holes.

A basis is a subset of vector in your space that can be used to represent other vectors. a linear basis is a set {v(j)} indexed by some index set, such that for every vector v in the space, there is a unique finite set of elements of the basis, say v1,...,vk, and unique corresponding numbers a1,...,ak such that v = a1v1+...+akvk.


the number of vectors in the basis is the dimension of the space. so the simplest spaces are finite dimensional. the next more complex are the "countable dimensional" ones, i.e. ones with a sequence of vectors as a basis, i.e. one basis vector for each positive integer.

but we also want a hilbert space to have no holes in it, and it is a theorem that an infinite dimensional inner product space with no holes in it can never have a basis which is a sequence, you need more basis vectors than there are integers, like maybe one for each real number. thats a lot of vectors.


but in a separable hilbert space you can get away with a sequence of vectors in your "basis" if you allow infinite linear combinations instead of finite ones. i.e. you broaden the definition of "basis".


think of fourier series. have you had that? a functon is a vector in your space, and even if your function is discontinuous, it can often be represented as an infinite fourier series in terms of sins and cosins.

whereas any finite linear combination of sins and cosins would be differentiable.

so allowing infinite sums gives us the ability to represent more functions, and worse behaving ones, using a smaller set of really nice ones.

i seem to recall that quantum mechanics HAS A SPEcial vocabulary speaking of "states" and representing them as linear transformnations of hilbert spaces, not always separable either, but this is not my field. I could understand all the math in a book on quantum mechanics, like that by mackey, but i do not know what the physics is about.

it sounds like you have a lot of math to learn to get up to speed in this area, mostly linear analysis, i.e. convergence in linear spaces, inner products, bounded and unbounded linear transformations, and then spectral theory, which can also benefit from some complex analysis.

a nice little book on spectral theory we used at harvard in my youth was by edgar lorch, fairly self contained with intro to hilbert space, and yet going quite far.
 
Last edited:
  • #23
mathwonk
Science Advisor
Homework Helper
10,996
1,187
a square summable sequence is a sequence whose squares have a finite sum.

this refers to a vector in infinite dimensional space, with a sequence of corrdinates which lies at a finite distnace from the origin in the usual infinite euclidean distance, i.e. the square root of the sum of the coordinates.
 
  • #24
mathwonk
Science Advisor
Homework Helper
10,996
1,187
"isomorphic" is a very basic idea in math and should be learned as soon as possible. it means roughly "having the same structure".

of course this only refers to the list of properties under consideration being the same.

isomorphism allows us to stop viewing every object in thw qorld as different from every other, and realize the commonalities that some thigns share. it is the mnost important idea in all of abstract mathematics.

in afct it is the defining concept of all abstract mathematics.

the simpelkste xample is congruence in geometryge
 
  • #25
D H
Staff Emeritus
Science Advisor
Insights Author
15,393
684
the simpelkste xample is congruence in geometryge
Which is isomorphic to

the simplest example is congruence in geometry
:smile:
 

Related Threads on Rigorousness of mathematics in physics classes

Replies
14
Views
3K
Replies
0
Views
5K
Replies
4
Views
871
  • Last Post
Replies
10
Views
6K
  • Last Post
Replies
6
Views
3K
Replies
12
Views
2K
Replies
9
Views
1K
  • Last Post
Replies
1
Views
2K
Replies
12
Views
438
Top