The usefulness of proofs to a physicist: eg The Schwarz Inequality

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I'm trying to really solidify my maths knowledge so that I'm completely comfortable understanding why and how certain branches of mathematics are introduced in physics and inevitably that leads me to a study of proofs. I usually skip proofs as I found them annoying, unintuitive and just seemingly tedious guess work.

I started a course on analysis at uni for my physics degree but switched option because I hated it so much.

For example The Schwarz Inequality proof in a book I'm using:

For any pair of vectors [itex]|a\rangle[/itex],[itex]|b\rangle[/itex] in an inner product space V, the schwarz inequality holds:
[itex]\langle a|a\rangle \langle b|b\rangle \geq |\langle a|b\rangle |^2[/itex]. Equality holds when [itex]|a\rangle[/itex] is proportional to [itex]|b\rangle[/itex]

The proof then starts:
Let [itex]|c\rangle = |b\rangle - (\langle a|b\rangle / \langle a|a\rangle)|a\rangle[/itex]

Therefore [itex]|b\rangle = (\langle a|b\rangle / \langle a|a\rangle)|a\rangle + |c\rangle[/itex]
Then take inner product of [itex]|b\rangle[/itex] with itself:

[itex]\langle b|b\rangle = \left|\dfrac{\langle a|b\rangle}{\langle a|a\rangle}\right|^2 \langle a|a\rangle + \langle c|c\rangle[/itex]
From which point it is clear the inequality can be proven.
For me however, I immediately get stuck on where [itex]|c\rangle[/itex] came from.

I tried to prove this on my own multiple times and probably spent about 6 hours total on it on and off and end up messing about writing [itex]|a\rangle[/itex] and [itex]|b\rangle[/itex] in terms of orthornormal vectors before feeling like an idiot and ultimately getting nowhere and just giving up.

If left to work on this problem for a year i can't even imagine what on earth would cause me to think of introducing a third vector [itex]|c\rangle[/itex] into the mix.
Then, not only that, but assigning it the very specific value:
[itex]|c\rangle = |b\rangle - (\langle a|b\rangle / \langle a|a\rangle)|a\rangle[/itex]

I'd have a better chance of winning the lotto than happening to come up with this.

Clearly I'm lacking something fundamental here, I'm guessing the choice of the vector c might seem obvious to someone good at this, but in my mind the great majority of non trivial proofs proceed in the following way:
-> 1) Algebra that I can follow clearly
-> 2) Insert step that seems to have been conjured out of thin air (ie lets introduce the vector [itex]|c\rangle = |b\rangle - (\langle a|b\rangle / \langle a|a\rangle)|a\rangle[/itex])
-> 3) Algebra that I can follow clearly
-> 4) QED.

I then sit there for ages trying to justify how I would come up with step 2 and, i sort of can see why, but it is just 99.9% hindsight because once it comes time for me to do practise proofs on my own, I still don't even know where to start, and when i look for the proofs online, the above scenario plays out again. This repeats until I just eventually get fed up!

This isn't my first foray into trying to learn these things, but I never get further than this and end up just deciding to just accept proofs and move on because its just a massive waste of my time and has seemingly little if any benefit.
 

Answers and Replies

  • #2
UltrafastPED
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|c> is an auxilliary variable which is defined as it is to speed up the proof ... you will often see such "tricks" in well-polished proofs. The most common tricks are multiplying by one or adding zero.

Making something concise may make it easy to remember, but does not make it intuitive.

If you want to write proofs you must first develop an "intuition" for the subject matter - sometimes this will be an analogy, or a mapping to something that works the same, like a geometric visualization. In physics there are some implicit assumptions which speed demonstrations along: everything is very smooth, etc.

The mathematician Terrance Tao speaks to the mathematical side in his blog:
http://terrytao.wordpress.com/career-advice/there’s-more-to-mathematics-than-rigour-and-proofs/
 
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I guess these 'tricks' become obvious once you have a firm grasp of the material, and it is this reason I just can't shake that by not learning proofs I'm somehow not really grasping the material.

Thanks for that great article from Tao
As a physics student it is safe to say my mathematical education falls into the "pre-rigorous" stage as Tao mentioned. I became aware that in working within vector calculus that I was constantly making errors as I didn't really understand how the underlying mathematics worked, I was simply applying rules and often I wasn't sure if certain things I was doing made sense.
I found when I actually paid closer attention to the precise and formal definitions, I made fewer of these mistakes.

This is what has led me to have another go at trying to patch up my less than rigorous mathematical knowledge as I can't help but feel it will benefit me greatly if I can. But I cannot get past the proof stage.

Are there any books that teach when and how and more importantly why these these methods and tricks work.
What would aid my understanding would be for example if the book gave a long winded proof followed by an explanation on how the proof could be speed up 'retrofitting' various tricks back into the original working.
That way I could get an intuitive idea as to where and why they were used in the first place and how to spot opportunities to use them in the future.

If the trick is just thrown into the solution right off the bat, it just leaves me confused and feeling like I've learnt nothing. Unfortunately all the books I've seen on this topic seem to go the latter method and I'm yet to find one that I could make use off.
 
  • #4
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In the case of this proof, the specific vector [itex]|c\rangle[/itex] you named has very meaningful interpretation. The reason we like inner-product spaces is that they give us a meaningful notion of angles. The vector [tex]|s\rangle=\dfrac{\langle a|b \rangle}{\langle a|a\rangle}|a\rangle[/tex] is what we call the orthogonal projection of [itex]b[/itex] onto [itex]a[/itex]. It basically means that if [itex]|a\rangle[/itex] is the ground, and it's noon on a sunny day, then the vector [itex]|s\rangle[/itex] (which is just a multiple of [itex]|a\rangle[/itex]) is the shadow cast by [itex]|b\rangle[/itex]. So what your book is doing is breaking [itex]|b\rangle[/itex] into two pieces, [tex]|b\rangle = |s\rangle+|c\rangle,[/tex] where [itex]|s\rangle[/itex] is the shadow on [itex]|a\rangle[/itex], and [itex]|c\rangle[/itex] is everything else.

Two things to say here.
1) A good textbook would point out that [itex]|c\rangle[/itex] came from this decomposition.
2) Why would we think to break one of the vectors up like this? Well, that's the intuition that UltrafastPED was describing. Whenever we deal with a given structure, we get used to a few tricks that exploit that structure. In an inner-product space, we often break things into two perpendicular pieces that have a meaningful interpretation. As a different (but related) example, in a metric space, we often bound the distance between [itex]x[/itex] and [itex]z[/itex] by making a clever choice of [itex]y[/itex] and bounding the distances between [itex]x[/itex] and [itex]y[/itex] and between [itex]y[/itex] and [itex]z[/itex].
 
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  • #5
AlephZero
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You need to realize that math often gets invented in the reverse order to what is published as a formal proof.

The Schwarz inequality is a generalization of elementary ideas: the triangle equality (the sum of two sides of a triangle is longer than the third side), or the fact that ##| \cos \theta| <= 1## for real values of ##\theta##.

If you come to something like this "cold" in a math book, playing around with the simplest special cases you can think of is often a good way to start.
 
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All this makes sense now.

I did have the 'intuition' of splitting the vectors up into orthogonal components, however i decided to do it using a standard basis. This worked in [itex]\Re^2[/itex] and I could prove the equality by expanding and factorising, but couldn't do it in the general case.

The idea of splitting one the vectors up into its component parallel and perpendicular to the other components didn't occur to me, but I can perfectly see why this is a good idea, why it works, and ultimately how they arrived at [itex]|c\rangle[/itex] being what it is. The book just quoting and expression for [itex]|c\rangle[/itex] right out of nowhere seems almost certain to confuse any newcomer to the topic!

Many thanks, this has frustrated me all day, but I finally get it!
 
  • #7
UltrafastPED
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Since you are in physics I would suggest an upper-level linear algebra course - this course will be proof-based and will really give you a good feel for linear vector spaces ... which are really useful in physics!
 

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