Chapter 4
(A)QFT as Objects of Philosophy
Before beginning with my main study I will address two questions which might occupy the thoughts of some readers by now. Why do I look at QFT while the much less complicated quantum mechanics (QM) already displays the same problems in a far clearer way? Why does algebraic quantum field theory (AQFT) play such a prominent role in my investigation? I will try to answer these questions in the following three sections.
CHAPTER 4. (A)QFT AS OBJECTS OF PHILOSOPHY 49 clarification about the location and interconnection of various problems.
The use of no-go-theorems was particularly valuable and these results are arguably the highest achievements in this area of research. Outstanding examples are the Gleason and the Kochen-Specker theorems1 , EPR/Bell inequalities, non-objectification and probability theorems.2 On the other hand none of the proposed solutions has led to a proper solution which is satisfactory in all respects. The most prominent proposals are the decoherence approach, consistent histories, modal interpretations, many worlds, many minds, Bohmian mechanics and nonlinear alternatives of the Schr¨odinger equation. Each particular proposal solves certain problems at the cost of having new problems at a different place.3 To use a picture by David Armstrong in another context, each solution can only flatten the bulge in the carpet to the effect that the bulge appears somewhere else again. The impression is getting ever stronger that there will not be sweeping new results on the foundations of QM in the foreseeable future.
On first sight it might be surprising that the discussion on the concep-tual foundations of the quantum domain has always been primarily con-cerned with QM and not with QFT. After all QFT is, in a certain sense, more comprehensive than QM and in particular relativistically invariant in contrast to QM. There have been at least two reasons for neglecting QFT in favor of QM for conceptual reflections. First, for a long time the attitude was dominating that the decisive philosophical problems show up in QM already. Accordingly a conceptual analysis of QFT appeared not to be necessary. It even seemed that looking at QFT would only blur the view on the central features since QFT is much more complex and mathemat-ically advanced than standard QM. A second reason for neglecting QFT
1Redhead (1987), section 1.5 and chapter 5, gives a very systematic exposition.
2Mittelstaedt (1998), chapters 3 and 4, is a very recent state of the art source for details.
3Barrett (1999) is a very instructive and comprehensive current exposition of prob-lems faced by new interpretations of or alternatives to QM. An impressively short and persuasive analysis of the stalemate reached in the interpretation of QM can be found in Peres and Zurek (1982) where only the most general features of different approaches are used. Barrett (1996) reflects on certain aspects of some formulations of QM.
CHAPTER 4. (A)QFT AS OBJECTS OF PHILOSOPHY 50 was the fact that QFT has not yet reached the status of a consistent and complete theory. The lack of a quantum field theory of gravitation is felt as a pressing need. Since it cannot be excluded that the incorporation of the fourth force might lead to deep changes of QFT as a whole, the current version of QFT must be seen as a preliminary theory.
There have been various studies on the historical development of QFT and in particular Quantum Electro Dynamics (QED).4This is partly due to some charismatic figures involved, especially Richard Feynman, and some spectacular successes of methods like renormalization, Feynman diagrams and the extensive use of symmetry groops. For the preference of historical studies on QFT over philosophical ones it might have been more impor-tant, however, that history does not change afterwards like theories do.
QFT as an object of philosophical reflection only began to receive more thorough attention around 1990. Apart from a certain saturation in the research on the conceptual foundations of QM the two above-mentioned arguments against QFT as a philosophical topic were weakened for the fol-lowing reasons. First, a careful analysis of the specifically relativistic traits of QFT led to results which at least aggrevate the conceptual problems of QM severely. Possibly they even surmount those problems qualitatively.
Second, due to the development of QFT and of the theory of super strings in the last decade the initial hope is fading away that QFT is near to its final completion. This hope flourished in the aftermath of the elec-troweak unification which seduced some people to euphorically anticipate the achievement of the final theory of everything. Today, string theorists like Dieter L¨ust cautiously estimate some 15 years before there will be any contact between string theory and experimental testing of it.
There are not only indirect or negative arguments in favour of QFT as an object of philosophical research. Some further arguments support the hope that a conceptual analysis of QFT will deliver results which finally enable us to tackle problems which seemed insoluble when looking at QM.
However, looking at QFT the fundamental difficulty to find and to
under-4Darrigol (1986), Schweber (1994) and Cao (1997) are arguably the most famous ones.
CHAPTER 4. (A)QFT AS OBJECTS OF PHILOSOPHY 51 stand the nature of the basic entities of the quantum regime might lead to a solution which only makes it necessary to explain why we have the impressionof “elementary particles”. In that case there would be no need to take “elementary particles” ontologically serious.
One often encounters the opinion that quantum field theory is just as well a particle as a field theory despite of its supposedly misleading name. Some newer results, however, seem to make it almost impossible to maintain that view. In particular the Reeh-Schlieder theorem and the Unruh effect seem to display features which show that QFT is essentially a field theory. Further results like a no-go theorem by David Malament and e. g. Robert Wald’s research on QFT in curved space-time strongly support such a view.
One problem for investigating the conceptual aspects of QFT consists in the fact that many results with conceptually important consequences can only be stated within a formalism which is mathematically involved, namely algebraic QFT (AQFT), even if it is physically clearer than the standard formalism of QFT in at least some respects. Philosophical aspects cannot always be seen immediately, however. It is instructive to realize that e. g. the Reeh-Schlieder theorem is already nearly 40 years old without having received any notice from philosophy of physics for most of that time.
At least until recently one (if not the) longest and most intensive track of discussions on the philosophy of QFT was an investigation by Paul Teller and Michael Redhead on different formal descriptions of many particle sys-tems containing identical particles.5 This discussion which was initiated in the late eighties finally became a central part in Teller’s An Interpretive Introduction to Quantum Field Theory(1995) which is the first systematic monograph on the philosophy of QFT.6 Teller’s book displays two deficien-cies, however. First, a major part of his studies can already be performed
5Redhead (1975), Redhead (1980), Redhead (1983), Teller (1983), Redhead (1988), Redhead and Teller (1991) and finally Teller (1995) is a selection of publications dealing at least partly with this question.
6The anthology Philosophical Foundations of Quantum Field Theory Brown and Harr´e (1988) was the first book to appear about this field of research.
CHAPTER 4. (A)QFT AS OBJECTS OF PHILOSOPHY 52 with respect to non-relativistic standard QM. Second, the formalism upon which Teller reflects is on the one hand somewhat out-of-date (about from the fifties) and on the other hand too restricted in its scope of application since only free field theory is considered. It can be seen as partly Teller’s merit, however, that a broad discussion on the conceptual foundations of QFT has begun in the recent years.7