## 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: