The motivation for AQFT:
LetAbe an observable andF(A) a real valued function of one real variable.
It can be shown that F(A) is just another way to label the same measure-ment outcomes. It is thus reasonable to consider the algebra generated by A as physically more basic than A itself or any other way to refer to pos-sible measurement outcomes. Segalgeneralized this insight by saying that the C*-algebra generated by all bounded19 operators (together with the norm topology) should be the basic entity in the mathematical description of physics. While in standard QM the Hilbert space representation and the C*-algebra formalism are equivalent this is no longer the case in QFT.
Quantum fields itself are not in general observable, but rather the local observables which are built from quantum fields. Two quantum fields are therefore physically equivalent when they generate the same algebras of local observables. It is thus reasonable to take these algebras as the starting point.
19See footnote 3 on page 169.
CHAPTER 3. FUNDAMENTALS OF QUANTUM PHYSICS 46 Basic Ideas of AQFT
One of the main traits and possibly the most unusual one of AQFT is the idea that all the physical information is contained in the mapping
O 7→ A(O)
from (bounded) space-time regions to algebras of local observables. This is to say that it is not necessary to specify observables explicitly in order to get physically meaningful quantities.20 The very way how algebras of local observables are linked to spacetime regions is completely sufficient to supply observables with physical significance. To express this idea in a slightly different way let us introduce the notion of the algebra Aloc of all local observables in the sense of set theoretic union, i. e.
Aloc=∪OA(O).
We can now say that it is this partition of Aloc into subalgebras which contains all the physical information about the observables. In this sense the primary concern of AQFT are the local algebras as wholes and the partition of Aloc into such algebras.21 At this stage nothing is said which would physically discriminate observables within any one algebra. This is not to say that there are no differences between observables of one algebra.
The claim is that the allocation itself of observable algebras to finite space-time regions suffices to account for the physical meaning of observables. It is not necessary to start with any such information explicitly.
20The knowledge of this correspondence allows one to calculate e. g. collision cross sections.
21The so-called quasilocal C*-algebra A generated by all local algebrasA(O) is the C*-inductive limit of the system{A(O)}which is the completion ofAin norm topology:
A=Aloc.
This means that A is the smallest C*-algebra containing all local algebras. Aloc is an algebra as well but not a C*-algebra. It is for this reason that A and not Aloc is used in other contexts when reference to the set of all possible observables is made.
In the concrete approach using von Neumann algebras the equivalent of thequasilocal (abstract) C*-algebraAis the so-called globalalgebraR.
CHAPTER 3. FUNDAMENTALS OF QUANTUM PHYSICS 47 The physical justification for this approach consists in the recognition that the experimental data for QFT are exclusively space-time localization properties of microobjects from which other properties are inferred. The Stern-Gerlach experiment may serve as an illuminating example: All one gets in this experiment are certain space-time distributions of dots from detected particles originating from a particle source and hitting a photo-graphic plate. Only in a second step one recognices particles with certain spin directions after having passed an inhomogeneous magnetic field. This example might help to imagine that space-time localisation can be the basis for all other physical properties.
Physically the most important notion of AQFT is the principle of local-itywhich has its effect both on the external as well as the internal structure of AQFT.22 The external aspect is the fact that AQFT considers only ob-servables connected with finite regions of space-time and no global observ-ables like total charge or the total energy momentum vector which refer to infinite space-time regions. The physical idea behind this is that QFT is a statistical theory and that the experimental information comes from measurements in certain always finite space-time regions. Accordingly ev-erything is expressed in terms oflocal algebrasof observables. The internal aspect is that there is a constraint on the observables of such local alge-bras: All observables of a local algebra connected with a space-time region O are required to commute with all observables of a second algebra which is associated with a space-time region O0 that is spacelike separated from O. This principle of (Einstein)causality is the main relativistic ingredient of AQFT.
TheAssumptions of AQFTare explicitly stated in appendix B together with various remarks about their contents and significance. It is instructive to have a look at theisotonycondition which looks like a truism but which is arguably one of the two most important assumptions.
22Slightly different expositions of the twofold meaning oflocalitycan be found e. g.
in Haag and Kastler (1964), p.848 and Horuzhy (1990), p. 3.
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.