## Mathematical rigor in theoretical physics

We have seen in the last post that mathematical sloppiness
can easily lead you astray. But is this always the case?

This is not a new problem. John von Neumann sorted out the
mathematical foundation of quantum mechanics in Mathematical Foundations of Quantum Mechanics.
The first time I read this book I found it incredibly boring: this is what you
learn in school. Then I learned to appreciate its sheer brilliance. The reason
it looks so boring is because it was so good it became the standard. At the
same time, competing with von Neumann was Dirac who introduced the well known
“Dirac functions” – an invaluable tool for any quantum mechanics computation. Here
is what von Neumann had to say about it:

*“The method of Dirac [...] in no way satisfies the requirements of mathematical rigor – not even if these are reduced in a natural and proper fashion to the extent common elsewhere in theoretical physics”*– OUCH!!!

Now I am not a historian and I don’t know the year von
Neumann wrote the book (I have only the translation to English year), but it
was probably in the 1930s well before the theory of distributions put Dirac’s
delta function on a solid mathematical foundation.

Fast forward to present, I have found a series of outstanding
lectures by Carl Bender which shook me to the core regarding to what it means to be a theoretical
physicist. Towards the end of the series, the fog clears and I came back to my
original beliefs about mathematical rigor along the lines of von Neumann, but Mr.
Bender managed to give me a scare with some mathematical voodoo.

To give you a taste of the lectures, let me ask a question
from Lecture 4: How much is:

1 - 1 + 1 - 1 + … = ?

This is stupid you may say: it is clearly a divergent
series. Worse you can make it converge to any number. You pick one, say 26, Add
the first 26 ones and then cancel the rest of the series. Does Hilbert hotel ring a bell?

Now how about this series:

1 + 0 – 1 + 1 + 0 – 1 + 1 + 0 – 1 + … = ?

Would it surprise you if I can prove that this gives a
different answer than the first series? And all that we have extras are an
infinite numbers of zeros!!!

Let’s proceed.

First we can introduce the Euler summation machine which
takes a divergent series and spits out a number E:

So let our sum: Sum (a_n) be not convergent. Construct the
following function:

f(x) = Sum(a_n x^n) for x < 1 where x is such that the
sum converges

Define E=lim_{x->1} f(x)

Let’s apply it to: 1 – 1 + 1 – 1 + 1 – 1 …

f(x) = 1-x+x^2-x^3+… = 1/(1+x)

Therefore E = 1/2

Can we make other machines in this spirit?

Yes, and here is another one, the Borel summation:

Again Sum (a_n) is not convergent.

We know that: Integral dt exp^(-t) t^n = n! which means that

1 = Integral dt e^{-t} t^n / n!

Replace Sum (a_n) -> Sum (a_n)*1 = Sum (a_n)* Integral dt
e^{-t} t^n / n!

Then flip the sum with the integral:

B = Integral dt exp^(-t) Sum (t^n a_ /n!)

Do E=B? Yes they do and here is why:

E and B are machines obeying two rules:

Rule 1: summation property

S(a0 + a1 + a2+ …) = a0 + S(a1 + a2+ …)

Rule 2: linearity

S( Sum(alpha a_n + beta b_n)) = alpha S( Sum (alpha a_n)) +
beta S( Sum(b_n))

Let’s apply it to our two divergent series:

sum(1 -1 + 1 -1 + …) = S

S=1+ sum(-1 + 1 -1 + …) (by Rule 1)

S = 1 – sum(1 - 1 + 1 - 1 + …) (by Rule 2)

S = 1-S

2*S= 1

S= 1/2 BINGO!

Now the second series

S= sum( 1 +
0 -1 +1 +0 -1 +1+…) =

S = 1+ sum( 0 – 1
+1+0 -1 +1 +0+…)=

S = 1+0+ sum(-1+ 1 +0 -1 +1
+0 -1+…)

3S = 1+1+0 +nothing(cancel term by term, no commutation of
the order of the numbers in the series)

S = 2/3

Let’s double check with Euler:

f(x) = 1 – x^2 + x^3 – x^5 + x^6 –x^8 +…

= (1+x^3+x^6+…) – (x^2 + x^5 +x^8+…)

=1/(1-x^3) – x^2/(a-x^3) = (1-x^2)/(1-x^3)

lim x-> 1 f(1) = 2/3

Mr. Bender is also making provocative (but true) statements
like:

*“If you are given a series and you have to add it up the dumbest thing that you can possibly do is add it up […] and if the series diverges it’s not only a stupid idea, it doesn't work.”*

Here is the complete series on You Tube:

Lecture 1:

Lecture 2:

Lecture 3:

Lecture 4:

Lecture 5:

Lecture 6:

Lecture 7:

Lecture 8:

Lecture 9:

Lecture 10:

Lecture 11:

Lecture 12:

Lecture 13:

Lecture 14:

Lecture 15:

"Fast forward to present, I have found a series of outstanding lectures by Carl Bender which shook me to the core regarding to what it means to be a theoretical physicist."

ReplyDeleteCan you elaborate on exactly what about the lectures was so perturbing? How did this shake the core so to speak?

Dear Anonymous,

ReplyDeleteIf you watched the lectures, in the overview lesson 1, Professor Bender starts by saying something like:

"There are different styles to do mathematical physics. One style is a very rigorous approach ... where is all theorem and proof and I personally find that ... less interesting (to be polite)...

A lot of what I'm going to tell you is not rigorous (at least not rigorous yet) and because of that it is more powerful. The more rigorous you get in fact the less powerful you are."

OK I said, this looks a bit like a nice spin to engage and entice the audience, but the series builds a crescendo of powerful techniques:

lecture 3-Shanks, lecture 4-Richardson, Borel, Euler, lecture 5- divergent series are not bad (converge much faster than convergent series) easier to extract info than convergent series, lecture 6-Pade.

At this point I told myself: OMG, maybe I am doing research the wrong way, just look how powerful those "voodoo techniques" are. But the fog cleared in lesson 7 when he talked about Stieltjes function. The point was that Taylor series converge in a circle, which is only up to first singularity, but the domain of convergence for Stieltjes functions has a different shape.

It is OK to explore a new domain before the foundations of it are settled, but in my area of research (of finding the answer to why questions) you can very easily go astray chasing red herrings if you do not intimately understand the math. Here is one example. People now talk about EPR=ER, and only yesterday I found a paper on PRL on this. This is the translation of Bohmian mechanics idea into field theory, and has a kernel of truth, but to say that entanglement is explained by wormholes is completely crackpot IMHO.