Archive for the ‘Quantum Field Theory’ Category

Choosing Zeidler on QFT

Some weeks back just before the Singapore trip, it was suggested that I should do reading seminars on quantum field theory (QFT) and/or particle physics. A decision to be made is which book to be chosen. My first exposure to quantum field theory is through Herbert Green‘s course on “Elementary Field Theory” in Adelaide and as usual, his treatment is unique and usually has no text-book reference similar to what he is doing. The next course that had field theory in it was particle physics for which I referred to is T.D. Lee‘s “Particle Physics and Introduction to Field Theory“, which is clear enough for me though his space-time signature is (+1,+1,+1,+i). It was only later (after graduating), that I was introduced to Mandl & Shaw by my eldest brother, when he gave me the book as a present. Later, I used this as one of my main reference during my Part III course on Quantum Field Theory in Cambridge. At the time, however, I heard praises of the book by Itzykson & Zuber, which is of course harder to read but my guess contained a lot of gems (never finished reading it).

For our group, I thought I will have something different, tailored for a more mathematical-inclined audience. Even here, there are several choices. I settled for the thickest book I can find. Saw Zeidler with his intended six-volume book. Apparently, he did not quite finish the six volumes since he passed away in November 2016. Three volumes of “Quantum Field Theory – A Bridge Between Mathematicians and Physicists” were published. So we begin with Volume 1: Basics in Mathematics and Physics.

A Prologue was written to describe the style and scope of  the book with an outline of the 6 volumes. The book is filled with quotations and historical anecdotes, making it untiring to read. It opens with five different quotations of famous physicists  and mathematicians leading to the five golden rules of the book:

  1. Write with an open landscape with depths of perspectives.
  2. Teach the content with a battery of problems.
  3. A technical book which is readable beyond the first several pages.
  4. Delves into deep questions of physics interlinking with mathematics.
  5. Teach for the appreciation of the physics.

The prologue also uses the often-quoted example of how successful QFT is in calculating the anomalous magnetic moment of the electron. Let us recall that any current-carrying loop produces a magnetic moment \underline{M} =  \frac{1}{2}\underline{r} \times\underline{j} where \underline{j} is the current density of the loop. The current may be carried by a charged particle orbiting in a loop. From the formula given, the magnetic moment is then proportional to its angular momentum.

If one let the particle be an electron carrying only the spin angular momentum \underline{S}, then its magnetic moment is

\underline{M}_e = \cfrac{g_e\mu_B\underline{S}}{\hbar}\ ,

where \mu_B=e\hbar/(2m_e) is the Bohr magneton and g_e is the gyromagnetic factor (dimensionless magnetic moment). Now according to Dirac’s theory, g_e=2 but this quantity receives corrections from QFT which can be written as

g_e=2(1+a)\ .

The correction a can be written as a (divergent) power series of the fine structure constant \alpha. Up to fourth power of \alpha (using 891 Feynman diagrams), the correction is

a_{th}=0.001 159 652 164 \pm 0.000 000 000 108\ .

Experimentally it is

a_{expt}=0.001 159 652 188 4\pm 0.000 000 000 004 3\ .

This is indeed excellent agreement. The calculation does not stop there. The next fifth power of \alpha correction involves 12,672 Feynman diagrams giving a correction which is accurate to more than 1 part in a billion. See the 2006 PRL paper here. The question now that this series that gave the correction is said to be divergent.

QFT is a practical theory which goes on to compute useful quantities for physicists:

  • cross-sections of scattering processes of particles
  • masses of stable elementary particles
  • lifetime of unstable elementary particles

Thus one ultimate goal of QFT is to bring it a rigorous mathematical setting. Perhaps it is worth mentioning here the work of Kreimer that has connections to noncommutative differential geometry.

In the search for a mathematical framework, some guiding principles from mathematics are

  1. Going from concrete structures to abstract ones
  2. Combining abstract structures
  3. Functor between abstract structures
  4. Statistics of abstract structures

In the last one, much help was given by physics (Zeidler used the term physical mathematics which I will not adopt) through Feynman functional integrals which we will go into the coming readings.

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