On the one side, the adoption of a particle interpretation of QFT would make the importance of particle experiments and the predominance of speaking in terms of particles comprehensible. It could explain why charge only exists in discrete amounts which is a typical feature of particles and not continuously which is characteristic for field quantity
On the other side, we saw that there are various problems for a par-ticle interpretation. Some results indicate that parpar-ticle states cannot be localized in any finite region of space-time no matter how large it is. Other results show that the particle number might not be an objective feature.
Nevertheless, it turned out that most arguments need to be seen in relative terms. At this stage of research it can only be recorded that there are various potential threats for the tenability of a particle interpreta-tion. However, before one can take these arguments as conclusive evidence against a particle interpretation of QFT alternative explanations have to ruled out at first. A more comprehensive and detailed evaluation of possi-ble arguments against a particle interpretation will be carried through in subsection 12.2.1 of the conclusion when we have a better background for this discussion.
Chapter 7
Field Interpretations of QFT
Many textbooks on QFT include in their introduction some remarks about the term ‘quantumfieldtheory’ and the entities about which it is a theory.
Some textbooks stress that QFT is just as much a particle theory as it is a field theory. Others stress that it is even more a particle than a field theory and that the term ‘quantum field theory’ is somewhat misleading.
Still other textbooks say that the term is fully justified since the incorpora-tion of relativity theory into quantum physics leads to the inevitable field character.
One thing one can learn from this is that there is obviously no agree-ment among physicists and that the situation is by no means clear. Another thing one can learn is that these two possibilities, particles or fields, are the standard options for the kinds of entities to which QFT refers. Ac-cordingly, particle and field ontology are the first two approaches which are under investigation in this study. I considered the particle ontology first because it is the most immediate option for that theory which is the theoretical basis for electrons, quarks and protons after all. Nevertheless, the field interpretation of QFT is arguably the kind of ontology to which most physicists would subscribe if pressed for a decision. In this chapter I will investigate how well-founded and viable this choice actually is.
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7.1 The Field Concept
Classical Newtonian mechanics is formulated as a theory about bodies and forces with pure “action at a distance”. It is only stated which force bodies exert on each other. Nothing is said about how these effects are mediated since it is assumed that there is an instantaneous interaction between two massive bodies. It turned out that the electromagnetic interaction between charged bodies cannot be described within this framework. A mediating field, the electromagnetic field, had to be introduced which accounts for the local transmission of electromagnetic forces.
The systematic and efficient formulation of the theory of electromag-netism with Maxwell’s equations at its core revealed another famous fea-ture. There is a limiting velocity for the transmission of signals, namely the velocity of light. In classical electromagnetism the existence of a limiting velocity for the transmission of signals simply emerged from this theory which rests on observed electromagnetic phenomena. It was Einstein who established this feature as a requirement for any physical theory. Hence the term ‘Einstein causality’ which was introduced in section 3.4 already.
Before decscribing how this principle was put to use in the formation of QFT I will say a little more about the notion of a field in general.
While the introduction of fields as mediators for the transmission of forces is a good starting point for getting an intuitive idea, the standard definition of a field is somewhat different. A field is generally defined as a system with an infinite number of degrees of freedom for which certain field equations must hold. A comparison of the specification of a field to the one of a point particle makes it clear what this definition means. A point particle can be described by its position x(t) which changes as the time t progresses. In a three-dimensional space there are three degrees of freedom for the motion of a point particle corresponding to the three coordinates x1 - x3 of the particle’s position. In the case of a field the description is more complex. The field is represented by the specification of a field value φfor each pointxin space where this specification can change as the timet progresses. A field is therefore specified byφ(x, t), i. e. a (time-dependent)
CHAPTER 7. FIELD INTERPRETATIONS OF QFT 98 mapping from each point of space to a field value.
As I indicated already the formal specificationφ(x, t) is not enough for something to be a field. Certain field equation need to be fulfilled. Without giving any further details I wish to point out just one extreme case why the formal specificationφ(x, t) cannot be sufficient. Considerφ(x, t) where φ(x, t) = 0∀x 6= ˜x with ˜x being a particular point in space. In this case φ(x, t) would just describe an ordinary point particle instead of a proper field.
One further information about fields should be supplied in order to make it understandable how one can come across the idea to think of fields as being the basic entities in the world. The intuitive notion of a field is that it is something transient and fundamentally different from matter.
However, in physics it is perfectly normal to ascribe energy and even mo-mentum to a pure field where no particles are present. This surprising feature shows how gradual the distinction between fields and matter can be.