[Dr. Chris Oakley's Home Page] [The search for a quantum field theory] [The book so far (PDF)]
There are a lot of introductory sources for relativistic quantum mechanics, but I find none of them very satisfying. I much prefer good mathematical argument to the woolly semi-historical accounts and plausibility arguments that seem to be the norm. Not only that, I believe that we already have all the ingredients for a compact and compelling development of the subject. They just need to be assembled in the right way. The important departure I have made from the "standard" treatment (if there is such a thing) is to switch round the roles of quantum field theory and Wigner's irreducible representations of the Poincaré group. Instead of making quantizing the field the most important thing and Wigner's arguments an interesting curiosity, I have used Wigner's work to drive the whole process. One advantage of doing this is that since I am not expecting the field quantization program to be the last word, I need not be too disappointed when I find that it does not work as I may want it to.
Here is an anecdotal account of my discovery of the Wigner work.
The book is to some extent a re-working of portions of my doctoral thesis. Here I tried to build the theory up from a set of axioms, but of these, I was least happy with spacelike commutativity/anticommutativity of quantum fields, as I prefer to take the view that quantum fields are not fundamental. I therefore delay mention of quantum fields here as long as possible.
Understanding of special relativity, classical mechanics, quantum mechanics and the theory of continuous groups is assumed here.
3. The Poincaré group
This is standard stuff. My arguments are based on my Cambridge Part 3 mathematics lecture notes (Ian Drummond was the lecturer, I think).
3.1 Finite-dimensional representations of the Lorentz group. The group SL(2,C)
Based on my Cambridge notes and notes from Paul Tod's lectures in Oxford in 1981. I have changed or extended some of the arguments in places.
3.2 Parity, time reversal and spacetime reversal
The role of these operations is not always spelled out clearly, so I am spelling it out here. And by the way, I do not believe in an anti-unitary time reversal operator (I cannot fully justify that statement just yet, but I will).
3.3 The unitary irreducible representations of the Poincaré group
Wigner's main contribution to particle physics. Lifted from my Part 3 notes, although I worked out the tachyonic case myself for completeness. However, I do not like the Wigner rotation. In my view, one should always prefer covariant to non-covariant notation. I developed this alternative originally in my D.Phil. thesis, but recently reworked the whole thing.
3.4 Normalisations of irreducible representations
4. The position representation
The lack of a Hermitian four-position operator. Group velocity of Klein-Gordon wave disturbances and the motion of a relativistic particle.
5. Fock space
Exchange symmetries. Bose-Einstein and Fermi-Dirac statistics. Creation and annihilation operators defined from the states. The argument I use is based on that in Lowell S. Brown, Quantum Field Theory (Cambridge University Press, 1992), section 2.1. Steven Weinberg, The Quantum Theory of Fields (Cambridge University Press, 1995), vol. 1, ch. 4 also does this, but note that there is no need to postulate a creation operator as it can just be defined from the states.
6. Quantum field theory; the spin-statistics theorem
If we assume that quantum field operators in the context of relativistic quantum mechanics (i) exist and (ii) commute or anticommute for spacelike separations then the spin-statistics theorem follows. Less clear to me is why spacelike (anti)commutation should be a requirement. One argument is that quantum field theory would not be possible if this was not the case. Another is that this is to do with causality, i.e. the commutativity of currents formed as local products of field operators expresses the need for events that cannot be connected by light rays to be independent. The latter argument I find more convincing than the former, but - for reasons that I will explain at greater length when I have time - it is still not satisfactory.
7.1 Quantization of classical electrodynamics; Haag's theorem
A simple proof of Haag's theorem.
7.2 Local field equations
This is work in progress. So far there is just an introduction and a simple proof of Haag's theorem.
Still to do: sums of tensor products of free fields. Matrix elements and the correspondence with time-dependent perturbation theory in quantum mechanics. The argument for the latter is given here, section 6.
A number of issues arising I have, to some extent, at least, investigated, but I do not have enough by way of substantial results to write up in "text book" format or a scientific paper.
In no particular order, these are:
(i) The Haag expansion (i.e. the expansion of the interacting field as a sum of tensor products of free fields) makes no assumptions about the fields other than the axiomatic ones and is therefore to be preferred to the expansion in powers of the coupling constant. So the question arises: can we drop the latter assumption in solving the spacelike commutators? If so, greater ingenuity would be required as, unlike the power series expansion, each term in the Haag expansion of the commutator contains combinations of the two input field expansions up to infinite order.
(ii) Section 2 of my QED paper here would seem to indicate that any local, non-derivative, Lorentz invariant set of equations derived from an action principle will solve the spacelike (anti)commutators. The vector bosons, though, must be massive, however small this mass is, and a cursory examination of the extensions of the theory to include multiple vector fields and chiral fermions seemed to indicate a need for mutual interactions among bosons. It would be interesting to discover whether these interactions follow the pattern of a (Higgsless) spontaneously-broken gauge theory. Also, can one generalise the theory to spins other than ½ and 1?
(iii) Bound states. The expansion in powers of the coupling works well for scattering processes, but for bound states is problematic. A two-particle bound state needs to have an invariant mass which is less than the sum of its components, something that cannot happen at any particular order in the perturbation expansion when the raw material is free field states (as used in my work). However, one may identify a bound state with the required property in, for example, a time dependence of exp[it(m 1+m2-mre4/(2n2 ))], which gives the energy levels of a single-electron atom, even though this may not be so easy to recognise when expressed in powers of the coupling constant, i.e. exp[it(m1+m2)](1-itmre4/(2n2)-t2mr2e8/(8n4)+...). To be sure that we are looking at a bound state, the perturbation series needs to be taken to infinite order, which is most readily done when there is a recurrence relation for classes of graphs on the lines of the Bethe-Salpeter equation. But what exactly comprises the Bethe-Salpeter equation in this circumstance is far from clear. Examination of the order e2 single-photon-exchange graph merely shows that it does not contradict the required result. But without an understanding of bound states, there is - apart from the usual hand waving - no calculation of the Lamb shift. Actually, there may be an alternative to expanding to infinite order in the coupling. The wavefunction of the H atom in the language used here has poles at the invariant masses of each energy level, and this could translate into a delta function enabling penetration below the invariant mass of the components. One is still required to develop a dynamical theory but this might be a first step. Here are the associated notes.
I find it depressing the reflect that, in today's research climate, it would be career suicide for a young theoretical HEP researcher to investigate any of these. In fact, the only advice I can usefully proffer to a young researcher who values mathematical consistency is to finish their Ph.D. and then get out (contrary to popular belief, having a Ph.D. does help you in the job market). Anything goes now, except high standards, and no person who takes pride in their work should stay a day longer than is needed to obtain their doctoral degree.
Update (9 November 2012): Though I am not necessarily repudiating the last paragraph, there may be progress on relativistic bound states here: http://arxiv.org/abs/1211.1619. Greenberg's "N-Quantum Approach" is another variation on the Stückelberg-Källén theme. Equation 24 is a fully-relativistic two-body wave equation that, the authors claim, replicates the gross and fine structure of a Hydrogen atom. I will post more when I have understood the arguments better.
Update (4 April 2013): I have still been too busy trying to earn a living, and have not yet managed to spend significant amounts of time on the Cowen-Greenberg paper. However, their equation 24 is - I think - also just a self-consistency equation for the appropriate term in the Haag expansion of the two-fermion state. One interesting thing about it is that if you try to express the solution - which is perfectly well-behaved - as a power series in the coupling every term will be a divergent integral! So maybe the pathological infinities that arise in quantum field theory are simply a result of the expanding in the coupling being an illegal action, and the solution is just to re-express the theory in terms of self-consistency relations like the Cowen-Greenberg one. One proposal I made before was to require annihilation operators to be pre-commuted to the right at each order in perturbation theory. This may solve the divergence problem, but it also seems to require one to give up any hope of modelling bound states. So self-consistency equations of terms in Haag expansions look like the way to go.