Monday, August 28, 2006

Review of 'Deep Down Things' by Bruce Schumm

Review posted at here.

Fed up with useless metaphors which equate the Higgs particle with hangers-on at a party slowing a celebrity’s passage? Exasperated at continual references to Lie algebras and gauge theories, which are never explained?

In Peter Woit’s recent book ‘Not Even Wrong’, he comments (p. 205) that relativistic quantum field theory is not even studied until the second or third year of graduate school. For the rest of us, there is ‘Deep Down Things’.

Schumm’s objective is to take us on a conceptual tour of the Standard Model of quantum mechanics, without requiring a mastery of the technical apparatus. The first half of the book introduces the four fundamental forces, wave-particle duality and the wave function itself. The approach is historical and visual - plenty of Feynman diagrams - and Schumm assumes the reader is happy with complex exponentials. By chapter 5 we are deep in the eightfold way, and the classification of quarks, leptons (electrons, muons, neutrinos) and bosons (the force quanta).

Chapter 6 begins the process of diving deeper with a discussion of Lie groups and Lie Algebra, motivated by plenty of examples. A Lie group is defined via: (i) a continuous set (i.e. a real or complex manifold such as R^n or C^n) with (ii) operators which are continuous functions over the manifold. Chapter 7 introduces Noether’s theorem: ‘to every differentiable symmetry generated by local actions, there corresponds a conserved quality’ and this is linked with symmetries under transformations by the Lie group operators (such as rotations in isospin space which interchange protons and neutrons).

Introductory quantum mechanics courses talk about the physical irrelevance of the phase of the wave function when it comes to the calculation of probabilities of observables. We thus have the concept of global phase invariance. However, this is unphysical - we cannot have the universe adjusting phase by the same amount everywhere at the same time. Yang and Mills in the mid-50s proposed to force the wave function to be invariant under local changes of phase: it turns out the only way to achieve this is to add a new term of the form gA(x)psi(x) where g is a charge parameter associated with the particle, psi(x) is the wave function and A(x) is a new term which turns out to be the field potential function for the relevant force field (electromagnetic in chapter 8). The freedom of choice in choosing the function A is called a gauge freedom, hence gauge theory.

Choose a fundamental particle. Write down its wave function. Identify the spaces in which the particle participates (space-time, isospin, ...). Identify the Lie group which rotates the wave function (state vector) in each of these spaces - U(1), SU(2), SU(3). By the principle of local phase invariance, adjust the original wave function with gauge terms gA(x)psi(x) as above. From making this work mathematically, out pop the corresponding force quanta (= the number of generators of the corresponding Lie algebra above). As the chapter heading puts it: ‘Physics by Pure Thought’!

Chapter 9 explains how the standard model assigns a mass of zero to all force-field quanta. Any attempt to add mass destroys the local phase invariance that we just discussed. The only way to retrieve the situation is to assume the existence of a new field (the Higgs field) which somehow pervades the universe and which interacts with non-zero-mass force quanta (via the weak force) in a ‘screening’ way which gives them mass. The Higgs field is also responsible for the masses of quarks and leptons. If this is true, there should be a Higgs particle within reach of CERN’s Large Hadron Collider in 2007.

This is a really excellent book. If you dimly recall how to solve a differential equation, and are unfazed by the notion of an abelian group, then this book is accessible. By book-end you have the sense that you ‘get’ the big picture of the standard model and its remaining conceptual weaknesses. I would say that if you were an undergraduate interested in theoretical physics and wanted a tour d’horizon, this is the one book which will give it (Penrose’s ‘The Road To Reality’ is still too difficult for this purpose).