CHAPTER 6. THE PARTICLE INTERPRETATION OF QFT 77 The next section deals with the best-known scientific context in which elementary particles are investigated. The point of concern will be the tension between this experimental basis of QFT on the one side and the conceptual investigations about the corresponding theory on the other side.
6.2 Theory and Experiment in Elementary
CHAPTER 6. THE PARTICLE INTERPRETATION OF QFT 78 like tajectories which are allowed (provided their boundaries are unsharp8) when only the constraints of Heisenberg’s uncertainty relations are taken into account.9 From this theoretical point of view it thus appears to be impossible that our world is composed of particles when we assume that localizability in any region of space-time is a necessary ingredient of the particle concept. Surprisingly, the very theory which excludes localizabil-ity is remarkably good in predicting experiments which apparently involve localizable particles.
For the working physicist this contradiction is not an important issue because it does not cause any problems, neither for the theoretical nor the experimental physicist, as long as conceptual questions as such are not at stake. In the last few decades Rudolf Haag and his colleagues, a group of theoretical physicists which puts much emphasis on conceptual clear-ness in their often pioneering work, have tried to fill this longstanding gap between theory and experiment. Within the framework of AQFT they pro-posed a mathematical model for ‘almost localized’ particles as they appear in scattering experiments10. The main ideas are firstly to describe scat-tered particles in terms of measurement results of a certain arrangement of particle detectors and secondly to assume only approximate localizability.
Before coming to more details of this model the next few paragraphs will give some necessary background information about experiments in High Energy Physics and the sense in which there is a gap between the achieved results and their theoretical description.
All the experimental information which was used to test QFT comes from particle detectors. These are employed in target regions of particle ac-celerators where very fast and therefore very energetic elementary particles can hit each other and sometimes give rise to new particles which emerge
8The requirement that the tubes are to have unsharp walls is due to Jauch’s theorem.
For details consult Jauch (1974).
9The blurredness of real particle tracks, however, is much larger than the minimum which is required by Heisenberg’s uncertainty relations.
10Haag (1996), pp. 75-94, 271-289, is an introduction in two steps with an increasing degree of complexity.
CHAPTER 6. THE PARTICLE INTERPRETATION OF QFT 79 from the scattering process. Since the aim of these scattering processes is to break the involved particles apart, one has to use very high energies and therefore very high velocities in order to exceed the binding energies. Since the velocities can be of the order of the velocity of light (though of course smaller) one has to take relativity theory into account for an appropriate theory of these processes, whereas ordinary quantum mechanics is non-relativistic. The founders of QFT proceeded in a somewhat conservative fashion: They used the formalism and the methods of classical Lagrangian mechanics and only modified it where necessary which is in particular due to the fact that some formerly scalar- or vector-valued functions had to be replaced by operators (see physics glossary). The result of this procedure were operator-valued quantum fields corresponding to different kinds of el-ementary particles and certain quantum states which these quantum fields can be in.
A surprising result was that particles themselves no longer appear in this theory. Although there are entities like “N-particle states” (see physics glossary) among the possible states of QFT it is not clear how these states relate to N particles. This is not only due to problems with the lack of individuality in systems with superposed identical particles. There is another essential characteristic of the particle concept which gets lost in QFT, viz. localizability. Since this topic is mathematically involved and not easy to display separately I wish to demonstrate the problem together with the way it is addressed in advanced QFT by using a mathematical model which is directly linked to modern detection devices for elementary particles.
The well-known cloud chamber photographs show particle tracks which are e. g. split after collisions or after a creation of new particles or which are curved due to a magnetic field. This detection method from the early days is visually compelling but has disadvantages in the numerical analysis because the only data it is based upon are graphical. Today one uses much more elaborate detection devices which directly supply the elementary par-ticle physicist with electrical signals that can be processed by computers.
This procedure allows of various possibilities to improve the exactness and
CHAPTER 6. THE PARTICLE INTERPRETATION OF QFT 80 value of the experimental data. A serious problem for the selective detec-tion of elementary processes consists in the fact that there is always a large proportion of signals which are irrelevant for the process in which one is in-terested. A method to suppress background signals lies in the use of energy thresholds which have to be exceeded before the detector responds. The more intricate task is the discrimination of different elementary processes which occur in the same region at a similar time. A very successful way of achieving this aim is a coincidence arrangement of detectors: Only those signals are assumed to originate from one and the same process which were detected at exactly the same time.
In AQFT this detection method has been employed to tackle a con-ceptual question by modelling the described experimental situation in a mathematical way. The detector model is meant to demonstrate that a relativistic N-particle state after a scattering process can be understood as a state of N “singly localized” particles at least in the asymptotic limit, i. e.
when time goes to infinity. The coincidence arrangement of N detectors is described by a product of N operators. Due to the Reeh-Schlieder theorem (see section 6.3.1) these operators can only be “almost localized”, since strict localization is incompatible with the condition that the detectors must not respond to the vacuum. Operators are said to be almost local-ized when they are smeared out with test functions which vanish quickly when their arguments go to infinity.
The significance of the described detector model and in particular of the notion of almost localized operators for the tenability of a particle interpretation will be discussed in subsection 10.4. The reason why I do not consider this question here already is that I will have arguments why an answer depends crucially on certain philosophical presuppositions which are investigated somewhat better in the context of chapter 10.