5.2 World lines
5.2.3 Lessons from experiments
The sole possibility to theoretically construct effective localised states is not enough to take them seriously. What ultimately grounds localised states must be physical practice, that is, experiments and their theoretical description. Effectively localised states have to be shown as actually occurring and not only as a possibility. Even though it will not suffice to only point towards measurement outcomes, since effective localisation is an assumption about states and not phenomena, a clear picture of how experimental phenomena come about will help to make inferences to the states that come before them. So what can we learn from experiments?
The primary use of QFT is the description of scattering events. Scattering experiments start with the production of particles.4 Since these particles are supposed
to be accelerated by magnets, they have to be charged. Electrons and ions are the particles that generally are produced first. There are a large number of different kinds of devices that can be used to produce them, but the fundamental processes are always more or less the same. Electrons are usually emitted from surfaces by either heating up the surface, putting a high voltage on it or bombarding it with photons. Protons on the other hand are extracted out of a plasma, e.g., by bombarding it with high energy electrons. All of these methods have in common that they can be designed such that the energy of the produced particles can be controlled extremely well (cf. Hill 1994). Any other particles can subsequently be produced from collisions of electrons or ions (cf. Grupen & Schwartz 2008, p. 85). These particles are then accelerated in a magnetic guiding field to very high energies, before being directed on a target.
4In the present context the term ‘particle’ is merely a practical placeholder for whatever actually
After the collision the resulting particles are led into detectors.5 The evolution of
detectors started with scintillation materials and counting light flashes with the naked eye; but has now reached enormous dimensions with detectors such as the ATLAS at CERN, where signals are not anymore read out optically, but only electronically. Even though the size and technical complexity of detectors changed, the (sub-)atomic processes that are used in particle detection have not. Detection of particles is only possible if the particles interact with the detector in a way that changes the state of the detector and can be recorded, that is, some amount of energy has to go from the particle to the detector (cf. Maxwell 1988, p. 20). In principle, each of the four known forces of nature is capable in many different reactions of serving this role (cf. Grupen & Schwartz 2008, p. 1). However, in practice mostly only two processes are used, namely, ionisation and emission of radiation. Both processes are based on the electromagnetic force, and therefore, neutral particles have to be converted through scattering within the detector into charged particles before they can be measured. Generally, measurements can be divided into non-destructive and destructive measurements. The former do not change the position and energy- momentum of the measured particle considerably, so that the particle can be used for further measurements, while in the latter the particle is destroyed and its energy is more or less completely absorbed in the detector or transferred to other particles. Non-destructive methods are typically used to measure time, velocity and mo- mentum. Time can be measured, e.g., by letting a charged particle excite atoms in a scintillation material, which then emits light while de-exciting. The emitted light can be led into a photomultiplier where it is transformed into an electric signal by using the photoelectric effect. Velocity can be measured in time-of-flight measurements by doing two time measurements on one particle at different places. Finally, momentum can be measured, e.g., in drift chambers set into an electric field. A charged particle will ionise atoms of the gas that lie on its track within the chamber. By measuring the ionisation current the track of the particle can be monitored and from the curvature of the track, due to the external electric field, the momentum can be calculated using the Lorentz force law.
On the contrary, the total energy of a particle can only be measured destructively. By using a solid, such as silicon, instead of gas, as detector material, charged particles will very quickly lose all their energies due to ionisation at low, and due to bremsstrahlung at high energies. While for low energy photons the photoelectric effect dominates, for high energy photons, electron-positron pair creation dominates. Strongly interacting hadrons on the other hand first have to be transformed through scattering into a shower of other particles of which a fraction that is proportional to the total energy is charged and can be measured by the above-mentioned methods.
Additionally, every detector measures the location of a particle just because the detector has a finite size. By separating detectors into small parts, e.g., small strip-
5Standard textbooks about measurement techniques in particle physics are, e.g., Grupen &
Schwartz (2008) and Green (2000). Most of my presentation is derived from them, but see also Falkenburg (2007, ch. 3, 4) for a concise presentation.
like electrodes, very accurate location measurements can be made. Furthermore, types of particles can be identified by combining several different measurement methods. The ATLAS detector for example consists of eleven different detectors, one after the other (cf. LHC-Experiments-Committee 1999).
The theoretical means to describe measurement processes and outcomes used in textbooks and by experimental physicists are usually either taken from classical physics, like the Lorentz force law, or empirically justified approximations or simpli- fications of the fundamental quantum field theoretical description. As Falkenburg (2007, p. 96) notes: “Today, the theory of position measurement is interlocked with many dynamic laws which form a complicated piecemeal assemblage of classical and quantum theoretical assumptions.” For example the Bethe-Bloch formula, which is commonly used to describe the energy loss due to ionisation, or the Bethe-Heitler formula for bremsstrahlung do not work with operators and states in Fock space, as one might expect in QFT, but merely with more or less classical notions (even though it has to be kept in mind that properties like energy are not continuous, but quantised). Another example is Cerenkov radiation that is commonly described by using the classical Maxwell equations.
Both the Bethe-Bloch and the Bethe-Heitler formula, are based on the Born approximation for scattering events. Since the latter is only valid if the interaction is small, it is clear that what really goes on can only be described by QFT. The total ionisation energy loss along the track of a charged particle, which can be calculated at once with the Bethe-Bloch formula, is actually a series of distinct interactions of the particle with bound electrons on its way. For an exact quantum field theoretical calculation, each of these interactions has to be taken as a scattering process between the incoming charged particle and a bound electron. Radiation due to bremsstrahlung or deexcitation, for which the Maxwell equations can give useful results, have to be described as the interaction between a charged particle and the Coulomb field of an atom by which photons are created. For all of these processes a certain transition probability and a cross section can be calculated, using the methods of QFT.6
Even though part of classical physics are still in use to analyse the data, captured in detectors, the initial scattering event and its outcome, that is, the cross section can only be described and predicted by QFT. I will go into the details of interactions in section 5.3, at this point it is only important to emphasise the connection between localised states and localised measurement outcomes. The most important results of the previous discussion are that measurement outcomes are always fairly well localised events, which can be evaluated partly with notions from the mechanics of classical point particles. However, this by itself does not mean that the states that are measured have to be localised, because measurement outcomes, for example tracks in cloud chambers, are not properties of the state, but “properties of the interaction process” (Haag 1996, p. 309) of the state with the detector. The link to
6For a quantum field theoretical treatment of interactions used in measurements see Nagashima
QFT is made by the fact that localised states are sufficient for localised measurement outcomes (cf. Haag 1996, p. 301; Wallace 2001, p. 12). The reason is that one property of interactions in QFT is that they happen locally (I will treat locality in sect. 5.3.4). The fact that measurement outcomes are localised can therefore be explained by the assumption that the states are localised.
This locality is also in accordance with the theoretical description of scattering events. Here the initial and final states are constructed as wave packets, that is, a superposition of plane waves with well defined momentum. In practice often simpler plane waves are used without culminating them into packets, but then it has to be assured by boundary conditions that initial states behave as wave packets, so that they do not interfere before and after the scattering. The result is then the same as when working with packets (cf. Knight 1961, p. 459; Peskin & Schroeder 1995, p. 102; Greiner & Reinhardt 2009, p. 2). The relation between plane waves, wave packets and effectively localised states is summarised by Knight (1961, p. 462) when he writes:
In the theoretical treatment of the scattering of fields, the initial and final states are usually idealized to one-particle states of definite momentum. Such states are not localized, and cannot be produced by apparatus confined to a bounded region of space-time. The localization associated with production and detection is accounted for in single-particle scattering theory by using a wave-packet description of initial and final states. It is in this sense that the localized states defined here provide a certain field theoretic analog of the wave-packet description.
In conclusion, the reliance on effectively localised states in order to define world lines in QFT is justified on the one hand by the use of wave packets in the theoretical description of interactions, and on the other hand by the fact that measurement outcomes always show well localised events. Both these aspects are furthermore connected by the principle of locality.
However, it is important to stress that even though the above justifies us in thinking of entities in QFT as localised in certain conditions, there is certainly no constraint for them from QFT to always be localised in that way. It will therefore be necessary to have a closer look into the theoretical description of interaction that QFT provides, in order to see whether there is something more suitable and convincing than local world lines. This I will be doing in the next sections.