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Relativistic Quantum Mechanics

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.

This is what I have so far:

1. Introduction

2. Definitions

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

Again, something not really touched on in any text book, but in my opinion, very important. When you determine the eigenvalues of a matrix you normally go on to determine the eigenvectors. This is the continuous equivalent.

4. 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.

5. 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. A physical interpretation.

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. Interactions

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. 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. Although nimble footwork is needed to dodge pathological infinities, the expansion in powers of the coupling works well for scattering processes. For bound states, though, it is problematic. A two-particle bound state needs to have an invariant mass less than the sum of its components, something that cannot happen in my scheme, which requires free particles as a basis.

Here are some notes demonstrating the connection between bound states and poles in the momentum-space wave function of the simplest bound state - interesting (well I think it is interesting, at least), but probably irrelevant.

Update (9 November 2012): 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 30 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.

Update (3 August 2013): further to the above, here are notes deriving my counterpart to the Cowen-Greenberg equation (up to equation 14).

Update (15 January 2014). Here (same link) the gross structure of the Hydrogen atom is obtained in the case of non-relativistic motion.