The meaning of the gauge invariance constraint

MODELING IN QUANTUM FIELD THEORY

2.1. The meaning of the gauge invariance constraint

In Symmetries and Reflections Wigner (1970) places great weight on the physical irrelevance of the absolute value of potential energy. Thus he introduces the idea of a global symmetry, general gauge invariance. Laws which involve differences in or differentials of potential energy are unchanged in form under the transformation

V(x') = V(x) + C

where V(x) is the potential energy term, and C is a constant, representing a change in bench- mark, common to all observers. But could a local change in bench-mark be tolerated, one which would represent a different change in measuring conventions for each observer? Suppose C were to be a function of position C(x). The observer Oi would recalibrate his or her measures of potential energy by adding C(xi), but the observer Oj, would recalibrate his or her measures by C(xj). Could there be laws which were invariant in form under the transformation

V(x') = V(x) + C(x)?

It seems that certain laws can be reformulated to be locally gauge-invariant, including Lagrangeans and wave equations.

For example in the wave equations of quantum mechanics the absolute value of the phase of the wave front is not relevant to the form taken by these equations as laws of nature. It is only phase difference that counts in quantum mechanics. This is a global phase invariance. But the SchriSdinger equation can be modified so that it is locally gauge-invariant. A change of phase representation by each observer can be accommodated by making the phase a function of x and t.

The overall form of the Schr/Sdinger equation is preserved if, when we change ~ t o e icu hi[/, we change A to A + 17x(x, t). One could say that the requirement of local phase invariance has provided a reason for introducing the vector potential.

Quantum field theory is that branch of high-energy physics that is concerned with the creation of hypotheses about the exchange processes that stand for forces in the subatomic realm.

As I understand it the current research programme in this sector of scientific research is to make the Lagrangeans describing the electromagnetic, the weak and the strong interactions, locally

A Metaphysics for Physics

151

gauge-invariant. This has been achieved for quantum electrodynamics by adding the photon as a 'gauge particle' to compensate for local changes in the electric field. Though it would not be historically accurate to say that there was a coherent programme of extending this treatment analogously to all interactions, this is roughly what seems to have happened. The theory of the weak interaction was achieved by introducing the W +, the W and the Zo particles to 'carry' the interaction (the conditions for this step will be tackled below), and to be the gauge particles which make the Lagrangean description of the weak interaction locally gauge-invariant, under some well-chosen symmetry group. The further generalization of the programme to the strong interaction depended on a number of further considerations, which amount to the need for the theory to entail the existence of very weak 'forces' at short distances. I understand that only non- abelian gauge theories behave like this. It will emerge as the argument unfolds that this empirical-cum-mathematical feature chimes in well with the metaphysical constraints that lead to the gauge theory programme in general.

2. 2. The basic physics of interactions

The essential step is the replacement of any explanatory apparatus that refers to forces acting between corpuscles with a scheme in which all interactional phenomena are mediated by the exchange of particles. Collisions between particles and decays of particles, as well as the forces that bind together the components of the atomic nuclei, are comprehended under the same scheme. This class of phenomena are represented in Feynrnan diagrams as in Figure 1. Such a diagram represents the simplest mode of exchange commensurate with the demands of the conservation laws relevant in each case. The contributions of more complex modes of exchange to the interaction are dealt with by a technical device, 'renormalization.' I shall discuss the relation between renormalization and gauge invariance below. The expressions which refer to intermediate vector particles are theoretical terms, whose place in theory is justified by their role in maintaining the 'book-keeping' conservation principles germane to the kind of interaction being described.

y

Figure 1

Electromagnetic interaction by photon exchange

152

Quantum Field Theory

2.3. The properties of virtual particles

The properties of the intermediate vector particles can be deduced directly from the conservation principles of the 'good' symmetries which are supposed to hold overall in each class of interactions. But in general it turns out that the rule

which links energy, momentum and mass for real observable particles is not obeyed by virtual particles. For example E 2 - p2 is not zero for the virtual photon of electrodynamics.

So far I have discussed the use of conservation principles as bookkeeping rules in quite general terms. However, the methodologically prescriptive force of the adoption of a general principle of conservation does not come out fully until one looks in some detail at the physics of a particular class of interactions. The electromagnetic field appears in the form of an exchange of virtual particles in any coupling mediated by that field.

The naYve picture treats the coupling as the conjunction of the emission of a photon at one vertex and its absorption at the other, as in Figure 2. Suppose the energy of the initial state is E^

and of the final state EB. Then

EA = EB.

But if Es is the energy of the intermediate state of the system it can be shown that neither E^ nor EB is equal to EN. Book-keeping rules germane to the deduction of the properties of photons as IVPs of the electromagnetic field break down.

that is

E~+ E2- [E~'+ E~+ 1~] ~ 0

E^- EB~0.

Figure 2

The naive emission/absorption picture

A Metaphysics for Physics

¹⁵³

9 p~

El, '

EI'Y ~ P 2

Figure 3 The alternative process

To get a more sophisticated picture we must incorporate the Feynman 'covariance'. The above picture includes the process direction A -~ B. But from the point of view of equivalent reference frames we must also consider the equally likely picture displayed in Figure 3. The total description will be a superposition of the two equally likely descriptions. From the analysis of the joint picture we get an expression for the amplitude which is Lorentz covariant. And in the combined picture both energy and momentum are conserved. According to the substantialist metaphysics to which we seem committed we are once again at least in touch with reality.

The properties of the IVP can now be calculated by using the restored conservation laws.

'Its' energy is (El - El') = (E2 + E2'), and 'its' momentum is (pl - pl') = (p2 - P2'). And for the virtual photon as IVP

q --- (El- E l ' ) 2 - ( P l - PI') 2 = (Pl- Pl') 2 which is an invariant.

For the ordinary photon p2 _ E 2 _ c2p2 = 0, but for the virtual photon (pro - p,)2 ;e 0, that is the virtual photon has mass. From a phil osophical point of view the 'anomalous' property of the virtual photon seems to arise directly out of the Feynman restoration of the full bouquet of conservation principles typical of the substantialist metaphysics.

The choice of symmetry group and the achievement of gauge invariance relative to that group and the ascription of properties to IVPs through the use of strict conservation book- keeping rules on the basis of a conservative 'grammar' both seem to be mutually motivated pairs of operations. The empirical basis of this elaborate, internally supported structure is simple, namely the preservation of the observed initial and final states of an interaction or decay. Despite the complexity of the structure of quantum field theory the IVPs are apparently typical hypothetical entities of an advanced corpuscularian physical theory. The diagrams are, then, despite Feynman's disclaimers, treated as quasi-pictorial representations of the mechanisms of processes. We have already seen how the diagrams are

both

corpuscularian metaphors and guides to research designed to find the free species of being whose genus is defined by their 'hidden' constituents, that is those which do not have or cannot be assigned tracks in the real world on the basis of photography, computer reconstructions, and soon, but which can be associated in some way with particles which do leave tracks.

But should we call the electromagnetic IVPs 'photons'? After all, they have one startlingly anomalous property, 'mass'. The justification for the assimilation that the use of the common term implies lies, I think, in the natural history of quantum electrodynamics. Once the decision to

154 Quantum Field Theory

separate the Hamiltonian for matter from that for radiation is taken, the theory of exchanges must develop from the conjunction of two 'pictures', one representing the emission of a free photon and the other its absorption. Thus the basic physics invokes standard photons as elementary quanta of excitation of the electromagnetic field considered as an infinite array of elementary oscillators. In this way the genus of the electromagnetic IVPs is fixed. It is only in the further development of the theory that we come to see that the IVPs of quantum electrodynamics are a different species from the photons of tradition. (Some would say that all photons are virtual.

Those which are 'free' link distant vertices and are of near-zero mass.) Thus far the existence of virtual photons remains open. The framework of thought is still exactly that of the policy realism of part four. A free 'X' need not have exactly the same properties as a captive 'X' provided they belong to the same kind.

Suppose theory tells us of a necessary condition that must obtain for a certain, quite definite phenomenon to be possible. When the phenomenon turns up, physicists make a confident claim for the existence of the necessary condition, in just the same sense as the phenomenon exists. A cause, after all, must be at least as real as its effect. This reasoning is apparent in the following quotation:

The researchers do not actually 'see' gluons in their apparatus [for instance they do not manage to photograph their trails]. At the highest energy at which [the apparatus] runs they find that a small fraction of electron-positron collisions produce three sprays or 'jets' of particles ... [of] three pronged appearance which all lie in the same p l a n e . . , the physicists believe that the jets originate in a gluon, quark and anti-quark that materialize from the electron-positron annihilation (Quoted by Picketing 1981).

But the imperative to accept such particles is stronger than the force of a necessary physical condition. The 'intermediate vector boson" comes not only from the 'book-keeping' rules of the relevant conservation principles, but also from the conditions for the gauge invariance of the relevant Lagrangean. IVPs are also gauge particles, GPs.

A particular case of the use of this kind of reasoning is the Gargamelle experiment. On the basis of the properties assigned to the Z particle of the weak interaction a phenomenon called the 'neutral-current event' would be expected (see Figure 4). But neutrinos are involved in this episode. Since they are uncharged, they leave no ionization tracks, so a real-world neutral- current event would look something like the track in Figure 5. It is not hard to visualize the photograph of real tracks which would count as a picture of a neutral-current event.

V V

Figure 4 The neutral-current event

A Metaphysics for Physics 155

Figure 5

Schematic track of a neutral-current event

At first sight this looks like a simple case of hypothetico-deductive reasoning. It seems to make gauge invariance under whatever group of transformations is popular (in this case SU (3)) as ontologically pregnant as was covariance under the Lorentz group. But the matter is not so simple. In the relativity case the choice of the Lorentz group as the relevant 'base-line' is based upon considerations that are independent of the particular form that covariance takes in the harmonized laws. The 'contraction' rule can be deduced directly from the Michelson-Morley experiment. But, so far as I understand the reasoning that leads to the choice of symmetry groups in quantum field theory, it involves choosing a symmetry group just so that the favoured Lagrangean will turn out to be gauge-invariant. This makes the gauge invariance condition more potent than the group of transformations. However, there is another way of reasoning which leads to the community accepting the IVP in question.

The photograph which we read as a picture of a neutral-current event appears in the discussion as interpreted. I have already pointed out the importance of the balance between plausibility and revisability of theories. When the neutral-current event does turn up, that balance shifts slightly towards the left. But in a recent study of the history of the development of this approach to subatomic physics, Picketing (1984, pp. 188-95; 300-2) points out an interesting feature of the control of the experimentation. The 'observability' of neutral-current events is relative to the measures taken to eliminate from the tables of results another class of events, neutral background events. The latter are interactions between neutrons, which like neutrinos leave no tracks, and the circumambient environment, apparatus, etc. Picketing shows that the conventions of interpreting were so set in the 1960s as to strike out all events initiated by trackless particles as neutral-background events. Only when the convention was reset in the 1970s did neutral-current events, initiated by neutrinos, become observable. He argues, no doubt with justice, that the resetting of the convention had to do with growing confidence in the theory that said that neutral-current events should exist. Of course confidence in the theory will not make neutral current events come into existence. Neutral-current events might not have been detected within the reset convention. Resetting is a necessary but not a sufficient condition for their discovery. This point hardly seems worth emphasizing, but one must remember that there is a reading of the programme to sociologize all explanations of scientific decision-making which seems to suggest that conventions make existents, rather than making possible the setting up of experimental programmes for their detection.

But the history of the study of the weak interaction shows that it is actually much closer to the kind of example that I used to ground the case for policy realism for the beings of Realm 2.

Experiments at CERN have produced sprays of particles, the immediate antecedent of which, though not itself leaving a trail, seems to have the physical properties assigned to the W particles by quantum field theory. If by 'free' particle is meant a being which leaves an ionization track that can be photographed or otherwise recorded, and which is the only particle required to explain the length, curvature and density of the track, then the CERN experiments have not quite identified a free W particle. But at least it is the next best thing. Quantum field theory then looks

156 Quantum Field Theory

very like a theory-family in the sense outlined in Chapter Six. The difference lies in the mode of manifestation of the hypothesized beings when they are 'free'. Realm 2 beings appear to ordinary observation within the world of perception. Any problem about their existence is reducible to a philosophical problem about the status of Realm 1 beings. But even the most robust 'free' particles in high-energy physics are distanced from the nai've observer by the causal processes that intervene between the moving particle and the ionization track it leaves. These processes are epistemically ineliminable. It follows that when we, the scientific community, are using observational predicates in our description of these beings, they must appear in dispositional attributions. These are the characteristic attributions of Realm 3 discourse.

3. A S U M M A R Y O F T H E M E T H O D O L O G Y

From the point of view of quantum field theory there are two interlocking steps.

1. The mechanism of the fields (forces) is described in terms of exchanges of virtual particles. They are introduced as intermediate vector particles, IVPs, and their properties are determined by the 'bookkeeping' requirements of the conservation principles that control the attribution of quantum numbers for such properties as 'charge', 'spin', 'colour', etc. The properties that are conserved differ between the three interactions, strong, weak and electromagnetic. This aspect of the theorizing goes on within the framework of a corpuscularian metaphysical model.

2. Virtual particles also appear as gauge particles, GPs. Their properties are determined by the requirement that they compensate local field changes so that the Lagrangean for the whole interaction is locally gauge-invariant under some appropriate symmetry group. The choice of symmetry group is not independently motivated as is the choice of the Lorentz group for relativity. The symmetry groups which 'discipline' the invariances are derived from the rules for the conservation of quantum number properties, rules found necessary to keep the books in the descriptions of the results of experiment. Properties and symmetry groups 'grow up together' so to speak.

2 A defence of a shadowy reality for IVPs as phenomenal "would be's" is attractive-the counterfactual might be explicated through an ontology based on vacuum physics. Dispositions, in general, can be grounded in some state, structure or condition of the substance to which they are ascribed. Photons can be treated as non-fundamental beings when they are taken as elementary quanta of excitation of the oscillator plenum of which the ground state is the vacuum. In the vacuum state only the average energy of the plenum is zero. Has the oscillator plenum any claim to be a representational model of reality?

Can both 'real' and 'virtual' photons be interpreted within a common ontology based on the idea of the oscillator plenum? The metaphysics of the oscillator story is tricky, since the oscillators so invoked are mathematical abstractions of 'something else' (cf. Aitchison 1985, pp. 335-6).

Nevertheless the idea deserves exploration. The Lamb shift can be described in photonic talk, in terms of the emission and reabsorption of a virtual photon. It can also be thought of as the effect of an interaction between an electron and a random fluctuation in the vacuum field. Perhaps virtual photon talk is a photonic way of describing flutuations in the vacuum field, while real photon talk describes the behaviour of quanta when the average energy is non-zero. Approached in this way the apparently naive claim 'Virtual particles might be real' is a photonic way of claiming that the dispositions (affordances) of the basic stuff (the glub) can be notionally grounded in the oscillator picture. Since oscillator talk is also metaphorical the oscillator plenum is another icon, a Cartwright model of the glub.

A Metaphysics for Physics

¹⁵⁷

If, by each route, we get concepts such that I V P = GP

the community seems to be satisfied with the theory. It is plausible in just the sense of that term I developed for the assessment of theorizing for Realm 2.

The story of the W and Z particles also illustrates this procedure. By the use of the methodology sketched above the weak interaction is pictured as a process mediated by particle exchange, and the properties of that class of corpuscles worked out in the standard way. The Gargamelle experiment shares the weakness of all attempts to prove the plausibility of a theory conceived hypothetico-deductively, by the test of a prediction of phenomena of the same natural kind as the lowest-level 'facts' encompassed by the theory.

But if the parallel with Realm 2 methodology is taken more seriously the policy realist treatment of quantum field theory suggests a different kind of experiment, an existential search for examples of the beings referred to in the theory.

The quasi-pictorial reading of the Feynman diagrams, or if you like the taking of W

particle

talk fairly literally, motivates the 'search for the free W particle'. This amounts to the conceiving of and realizing experimentally a physical situation in which the vertices of the W process are well separated and the 'particle itself' can become manifest. 3 In fact the W particle does not manifest itself,

in propria persona,

but rather the phenomena are such that any alternative explanation of the 'cross-section' of the process is ruled out. Nevertheless the argument hinges on the general point about the difference between the status of 'virtual' and 'real' particles-4hat it is a matter of the genus encompassing two species, one of which can be tracked. But we are still in Realm 3. The W particle is not manifested by virtue of generating its own unique track. It has to be inferred from the tracks of a shower of particles to which it is antecedent. But that last and crucial condition can be satisfied at all only through the use of the appropriate bookkeeping rules--even when free, Ws are, in some sense, virtual!

The case against subsuming the virtual status as a species of physical reality turns on the unclarity of the particle concept in the context of exchanges. The analysis above, in which WPs seem to have most particulate a representation, is the lowest order of exchange. The 'true picture', so to speak, is a superposition of the higher-order modes of exchange as well-for instance, Figure 6. In the 'true picture' the number of virtual photons is not sharp, since the system fluctuates over an infinity of superposed states. On the reasonable principle that ontologically photons are entity-like and have some of the 'grammar' of individuals, the unsharpness of the number of virtual cousins counts against their reality. But the upset of the metaphysical clarity of the virtual particle picture goes deeper.

The metaphysics and methodology of quantum field theory in terms of particles meshes only with the simplest kinds of exchange. The more complex modes of exchange are conceivable within the discipline of the conservation rules which are represented by higher-order terms in the mathematical description of the interaction or decay. Integrating over these possibilities should

3 In the reasoning that leads to a policy-realist interpretation of the W-particle there are two analogies.

The photonic concept is fhst legitimated in the electromagnetic case through its traditional application to luminiferous phenomena (in Bohr's sense). Its success through explanatory power in 'parsing the amplitude' justifies the concept of the 'virtual' species. The whole scheme is applied by analogy to the weak interaction. But the analogy between real and virtual species is reversed. It is the concept of the virtual species that is legitimated via explanatory power in the tidy accounting of the weak interaction, and the search for the real species is dependent on an analogy in which the virtual species is the source, and the real species the subject.