Since no-go theorems are pivotal for some steps in the ensuing investigation I will conclude this chapter with some general remarks about no-go theo-rems, their role in quantum physics and their significance for ontological studies.
There are different ways to find out something about the ontology of a scientific theory. For QM and QFT a very precise and successful method is to look for no-go theorems like the famous one by John Bell which will be summarized later6 on hidden variable theories or a more recent one found
6See “On the problem of hidden variables in quantum mechanics” Bell (1966) and
“On the Einstein-Podolsky-Rosen paradox” Bell (1964). Interestingly, Bell himself seems to have been annoyed by this business and in “On the impossible pilot wave”
Bell (1982), he wondered “[...] why did people go on producing ‘impossibility’ proofs [...]”, p. 160 in Bell (1987), mentioning famous names like J. M. Jauch, C. Piron, B.
Misra, S. Kochen, E. P. Specker, S. P. Gudder and, last but not least, himself!? Probably
CHAPTER 2. ONTOLOGY AND PHYSICS 33 by David Malament on the impossibility of a certain particle interpreta-tion for relativistic QM.7 The advantage of such no-go theorems is a very high degree of precision. Most no-go theorems suffer from a very limited scope, however. Malament’s no-go theorem for instance only shows that non-relativistic QM of a fixed number of localizable particles cannot be reconciled with relativity theory. It thus does not rule out a particle in-terpretation for QFT because here the precondition of a fixed number of particles is not met. The relevance of this no-go theorem for the interpre-tation of QFT is not immediately clear therefore. Further thought is thus necessary for an understanding of its ontological significance with regard to QFT.
The origin of no-go theorems in quantum physics consists in the fact that there is no undebated correct way to understand various entities ap-pearing in the formalism of QM. More on this can be found for instance in chapter 3. So far, the most successful way to handle this situation is the construction or discovery of proofs which demonstrate that certain sets of assumptions (e. g. locality, separability, determinism, value definiteness of all possible physical quantities etc.) lead to contradictions. Assuming that an interpretation of a (piece of) formalism can sometimes be condensed into a set of assumptions we can thus at least exclude some interpreta-tions. Since we can by this exclusion procedure show that one or the other interpretative option is not an admissable way to go these results are called
‘no-go theorems’.
There are three particularly famous examples for no-go theorems. The first one isJohn von Neumann’salleged proof of the impossiblity ofhidden variable theories. Later it turned out that von Neumann’s proof rested on implicit assumptions which narrow the applicability of his result consid-erably.8 For instance, von Neumann’s proof is not a legitimate argument
the reason for this stance are almost ideological feelings concerning Bohmian Mechanics.
7See “In defense of dogma: Why there cannot be a relativistic quantum mechanics of (localizable) particles” Malament (1996).
8Jammer (1974) is the authoritative account of the historical background as well as of the change in the evaluation of Neumann’s alleged proof.
CHAPTER 2. ONTOLOGY AND PHYSICS 34 against Bohm’s version of quantum mechanics which is explicitly holistic or non-local.
The second famous example of a no-go theorem is Bell’s theorem.
(Derivation of Bell inequalities under certain conditions, proof of viola-tion in QM).9 The third example are non-objectification theorems against ignorance interpretation of QM (nonvanishing ’interference terms’.)
9Redhead (1987)
Chapter 3
Fundamentals of Quantum Physics
In this chapter I will deal with some salient features of quantum mechanics (QM), quantum field theory (QFT) and algebraic quantum field theory (AQFT), an axiomatic reformulation of QFT. Instead of aiming at any kind of completeness I will emphasize some issues which are of particular significance for ontological considerations. Moreover, I will introduce some pieces of formalism which prepare the ground for investigations in some later chapters. Nevertheless, the most important points will be taken up again in those chapters themselves. For general introductions to QM, QFT or AQFT the ‘References’ contain various suggestions.
3.1 The Legacy of Quantum Mechanics
As regards ontological considerations about QFT the legacy of QM is mostly a negative one. Most of the notorious obstacles for an ontologi-cal understanding of QM are equally troublesome in QFT. In section 4.1 I will reflect upon the question whether it is appropriate to start ontolog-ical investigations about QFT before corresponding matters with respect to QM are settled. For now, I put that concern aside.
Problems concerning the individuation and reidentifiability of particles,
35
CHAPTER 3. FUNDAMENTALS OF QUANTUM PHYSICS 36 are the most fundamental ones for our context especially when considered together. They are concerned with the distinguishability of particles in its transtemporal and in its instantaneous aspect respectively. Both as-pects have notoriously caused trouble for the idea of individual traceable particles. These problems arise in QM already and lose nothing of their importance in QFT.
The Problem of Individuation
The problem of individuation results from the study of systems with many quantum mechanical particles. The starting point for the statistics of such systems is the fact that the Maxwell-Boltzmann statistics (i. e. the energy distribution law) which is valid in classical statistical mechanics leads to false predictions for systems of ‘identical’ quantum mechanical particles.
In QM, particles are called ‘identical’ when they have all their permanent properties (e. g. rest mass, charge, spin) in common. A set of permanent properties fixes a class of particles (e. g. electrons) rather than a particular particle.1 It turned out that one gets the experimentally correct statistics when the possible micro states which lead to the same macro state are counted differently for systems of identical particles: Micro states which differ merely by the ‘exchange of two particles’ must be counted as just one state. This fact is referred to as ‘non-occurrence of degeneracy of exchange’2or ‘indistinguishability of identical particles.’
What are the consequences of the indistinguishability of identical parti-cles for our main issue, the basic entities of QFT? The emerging problems become clearer after a short look at the symmetrisation postulate which follows from the indistinguishability of identical particles given some ad-ditional assumptions.3 Depending on the spin of the respective particles the wavefunction of a many-particle system has to be symmetric or
anti-1Cf. Mittelstaedt (1986) (Sprache und Realit¨at in der modernen Physik), chapter viii.
2In German: “Nichtauftreten von Austauschentartung”.
3The logical connection of the indistinguishability of identical particles and the sym-metrisation postulate is discussed in detail in St¨ockler (1988), p. 12.
CHAPTER 3. FUNDAMENTALS OF QUANTUM PHYSICS 37 symmetric under the ‘exchange of two particles,’ or, to be more careful, under the exchange of two particle labels.4 ‘Non-symmetric’ wavefunctions (which are neither symmetric nor antisymmetric) are excluded.
The symmetrisation postulate is only necessary5 in the Schr¨odinger many-particle formalism which, despite the problems with indistinguisha-bility, uses labels that are obviously meant to refer to individual particles of the overall system. Because of the symmetrisation postulate, however, not all wavefunctions of the ’overall’ or ’compound’ system, which could be con-structed by the standard way of forming the tensor product of one-particle states, are allowed any more. Inside the Schr¨odinger many-particle for-malism non-symmetric wavefunctions can be formulated but get excluded.
The formalism is therefore richer than the experimental reality which it is designed to encompass.6 This fact could be taken as an indication that the theory is built upon inadmissible assumptions which lead to a piece of structure that has to be excluded artificially.
A closer look at a symmetrisized wavefunction of a compound system hints at a reason for these difficulties: An anti-symmetric wavefunction of a system of fermions is a superposition of product wavefunctions, i. e. a sum of tensor products of one-particle states. A sufficient example is the wavefunction of a system of two identical fermions (e. g. electrons):
Ψ(x1, x2) = 1
√2
ψα(x1)ψβ(x2)−ψβ(x1)ψα(x2)
, (3.1)
where ψα(x1) and ψβ(x2) are energy eigenfunctions of one-particle Hamil-tonians and α and β represent sets of quantum numbers characterizing one-particle states. Sincex1 andx2 are variables of the “single particles” 1
4Pauli’s well-known ‘exclusion principle’ is thereby fullfilled automatically: The wave-function of two fermions in the same single state, i. e. with the same quantum numbers, vanishes as can be seen in equation (3.1) on p. 37. In other words there is no compound state where two fermions have the same quantum numbers.
5The symmetrisation postulate is true but trivial in the so-called ‘occupation number representation’ which I am going to discuss later.
6Especially M. Redhead worked on the so-called “surplus structure” of scientific theories: Redhead (1975), Redhead (1980).
CHAPTER 3. FUNDAMENTALS OF QUANTUM PHYSICS 38 and 2 it is natural to ask what the states of these “single particles” are. It turns out that it is impossible to give a satisfactory answer to this question if one holds on to the conception of individual particles. Each “single par-ticle” is in thesamestate as a part of the compound system even though in the wavefunction of the compound system different one-particle wavefunc-tions are used. The location of a “particle exchange” has obviously become problematic. The usage of labels for individual particles in the usual sense might lead one astray. We thus have an ontological problem since on the one hand we can successfully use labels which seem to number something but on the other hand we are not dealing with particles in the usual sense any more for whom the labels were introduced originally.7 In section 10.2 I will elaborate on this issue in more detail.
The Problem of Reidentifiability
The problem of individuation exerts its full force only in connection with the second one, the problem of reidentifiability: If in certain classes of mi-croparticles we cannot distinguish individual particles by permanent prop-erties why do we not simply look where they are and keep track of their location while time elapses? The following argumentation shows that even this way is obstructed.
On first sight the claim that we cannot follow a particle in space-time is astonishing since we seem to have exactly these looked-for tracks of ticles in cloud-chamber photographs, showing, for example, charged par-ticles on curved trajectories. A closer look reveals, however, that these
‘particle tracks’ have very little in common with sharp trajectories of clas-sical physics. On the micro level we have smeared tube-like objects. Each of these tube-like trajectories is the result of a vast amount of unsharp quantum mechanical position measurements8 in close succession. The
de-7Problems with particle labels are discussed extensively in Teller’s recently published book Teller (1995) on some philosophical problems of free QFT, with main emphasis on QED.
8In our context a measurement is every interaction of a quantum object and a macro-scopic system with a definite result, e. g. a dot on a photographic plate.
CHAPTER 3. FUNDAMENTALS OF QUANTUM PHYSICS 39 gree of unsharpness or “smearedness” is even far bigger than the theoretical minimum which is given by Heisenberg’s uncertainty relation for position and momentum.
With a particle track being the result of many successive measurements, the immediate suggestion is that one and the same particle gave rise to the track because numerous measurements were performed on this particle.
Unfortunately we have difficulties with this assumption in QM: Even if the time interval between two quantum mechanical position measurements that contribute to one particle track is extremely small we cannot be sure to have measured the same particle. The reason for this is the fact that even in the case of a sharp position measurement after an arbitrarily small time interval the measured object can be detected infinitely far away from the first point of detection.9 Between the results of two quantum mechan-ical measurements of a continuous observable - like position - there is in principle no deterministic connection.10
The problems discussed above partly reflect a general difference between classical and quantum mechanics. In classical mechanics when position and momentum of a free particle are given both is fixed for any later point of time, we have a so-called ‘path law.’ In quantum mechanics, however, we do not have a deterministic law of this kind. The first reason is that a quantum object cannot have a sharp position and a sharp momentum at the same time. The second reason is more fundamental: There is, in general, no deterministic connection between single (or groups of) measurement outcomes. All we have in quantum mechanics is a law for the evolution of the statistics of measurement outcomes: The stastitics is given by the
9See for example Hegerfeldt and Ruijsenaars (1980).
10In contrast to that for discrete observables (i. e. where the measurement outcomes are discrete numbers) ‘repeatable measurements’ are possible if we are dealing with
‘state preparation measurements.’ In this case after the measurement the measured object can be found in that eigenstate which corresponds to the measured eigenvalue.
A repeated measurement leads with certainty to the same measurement outcome (this is the defining property of a ‘repeatable measurement’). For a proof of the impossibility of repeatable measurements of continuous observables see the classical paper Ozawa (1984).
CHAPTER 3. FUNDAMENTALS OF QUANTUM PHYSICS 40
‘state function’ and the deterministic law according to which it evolves in time by the Schr¨odinger equation. We know, therefore, how the statistics of possible measurements are connected, but we do not know in general how single measurements are connected. This means that a particle track cannot be interpreted as a succession of connected measurements on one object. The possibility to identify an object by tracing it through space-time is excluded.