How Is Quantum Field Theory Possible?
This book provides a sustained attempt to give an interpretation of quantum field theory along broadly Kantian lines. Kant is sometimes represented as trying to bolster Euclidean geometry and Newtonian physics as necessary to articulating intelligible phenomenal experience, but Auyang attempts to abstract a categorical framework at a level of generality which will accommodate everyday experience, classical physics, quantum physics, and ultimately quantum field theory. I am sure this is the right way to approach Kant, but Auyang does not attempt to apply the dense jungle of Kantian distinctions and terminology, or to reproduce a genuinely transcendental argument to justify the synthetic a priori as Kant himself does. Rather Auyang's project is in the tradition of Strawsonian descriptive metaphysics, but attempting at a deep level of abstraction to capture the general concepts of object, property, quantity, and relation that she sees as underpinning all intelligible discourse about the physical world. This is the Kantian turn.
Let us start with the general concept of object. For Auyang this is at a higher level of abstraction than the substantial concept of 'thing', but nevertheless, in accordance with her programme, an everyday thing such as a table falls under the general concept of object, which incorporates for Auyang two essential features. Firstly there is the variety of representations corresponding crudely, for example, to viewing the table from different directions or perspectives which are linked by definite transformation rules, and secondly there is the crucial ingredient of binding together or transcending these representations in a way that objectifies the experiences as being of something, the object, over and above the phenomenal representations. Auyang argues that the general concept of object applies equally in quantum mechanics. Quantum objects display a variety of properties corresponding to differing measurement contexts. The objective feature is the state vector which can be represented by expanding in terms of various complete sets of eigenstates of different possible 'observables' (in the simple case of a discrete spectrum of eigenvalues). The transformation theory of quantum mechanics shows how these different representations are related by unitary transformations. Again we have the basic structure of representations linked by appropriate transformations and the objective state vector uniting and transcending and hence, says Auyang, objectifying the state of the quantum objects.
In full generality we have a state space M which encompasses all possible states of an object. What Auyang calls 'representative rules' [f.sub.[Alpha]] and [f.sub.[Beta]] assign definite predicates in the representations [f.sub.[Alpha]](M) and [f.sub.[Beta]](M) for the objective state x. As Auyang puts it (p. 96), 'the transformation [Mathematical Expression Omitted] does not merely relate the predicates in different representations, it is a composite map that points to the objective state x. The objectivity of x is guaranteed by its invariance under all transformations, and thus abstracts from them all.'
For Auyang it is this representation-transformation-invariance structure that underpins the general concept of object. There is no privileged representation either in observation, such as the sense datum, or in reality, providing the unique God's Eye point of view. Thus Auyang rejects both phenomenalism and metaphysical realism in the sense in which Putnam uses that term. If we were given only the representations this, says Auyang, would lead to relativism, which she also rejects. Every part of the structure explained above is necessary, according to Auyang, for the general concept of object.
When discussing the example of the table we explained the representations as viewing the table from different perspectives, but in discussion of Macbeth's hallucinatory dagger Auyang makes it plain that representation must involve all the relevant modalities of sight, touch, and so on, in the case of an everyday 'thing'. It is not perhaps totally clear what the relevant transformation rules would be, but presumably would comprise the coherent linkage of visual and tactile impressions, etc. At all events Auyang wants her account to give necessary, not just sufficient, conditions for objectivity. It should also be stressed that the distinction between the physical object and its representations is not the distinction between reality and appearance. The topic of knowledge, which Auyang refers to as the 'empirical object', involves the fusion of the whole integrated structure, of which the physical object is just one conceptual element.
Auyang sometimes refers to the representations of an object as conventions. All she means by this is the choice available to us, which representation to use, but again the concept of object involves all the possible choices, taken at one gulp, so to speak.
It is clear that Auyang arrives at her analysis of object from the corresponding definitions of a manifold in differential geometry, where x is a point in the manifold and [f.sub.[Alpha]](x), [f.sub.[Beta]](x) are different coordinate representations of the point. The composite map [Mathematical Expression Omitted] is then just a passive coordinate transformation on the manifold. There are also active point transformations of the manifold specified by the map [Mathematical Expression Omitted]. Auyang is perfectly clear on the mathematical distinction between coordinate transformations and point transformations, but we shall see later that it may pose problems for her subsequent discussion of field theories, in particular general relativity.
So let us turn to Auyang's discussion of field theories. Objectification is now applied not to things but to events. Basically the events consist of the fields at a given spacetime point being in a certain state. So attached to each spacetime point there is a 'quality space' comprising possible states of the field at that point. These states may be represented in various ways, the representations being connected by a local symmetry group, local because it is indexed by the spacetime point at which it acts. But the spacetime points are also variously represented by different coordinatizations of the spacetime manifold, which in turn are related by general coordinate transformations. So there are two symmetry groups in play, one relating different representations of the local quality space, which Auyang refers to as the local symmetry group, the other relating representations of the 'spacetime' points, which Auyang refers to as the spatio-temporal group.
We have been careful to introduce the two symmetry groups in their passive versions, because that ties in with the ideal of different representations of objective entities. Auyang, however, tends to slide between the passive and active interpretations, which could make for some confusion, as we shall explain later.
But for the moment we shall continue with the exposition of Auyang's own ideas. The invariant object in the quality space tells us what kind of event we are dealing with; the invariant location in the spacetime manifold tells us which particular event of that kind we are referring to.
So for Auyang the role of spacetime is to individuate events. At this stage in the discussion the individual events are 'loose', there is no causal connection between events. That will come later. But first, I want to explain Auyang's rather interesting approach to the nature of spacetime. It derives essentially from interpreting the structures which mathematicians call fibre bundles. Speaking very crudely, a fibre bundle can be thought of in two ways. Firstly, it can be thought of as constructed by attaching one sort of space, the fibre, to each point of a second sort of space, the base space, so that locally the structure is just the familiar Cartesian product. But there is a structure group imposed which transforms each fibre into itself, but in such a way that the transformations of different fibres are independent of each other.
In the field-theory case, the fibres are replicas of the quality space and the base space is the spacetime manifold. The local symmetry group is essentially just the structure group of the bundle.
On this reading one can imagine the base space as existing, ontologically speaking, in its own right quite independently of the fields which are then 'attached' to the points of spacetime. This would give a 'substantival' interpretation of spacetime as an independently existing entity.
But there is a second way of thinking of a fibre bundle. Take the whole bundle as primary and introduce an equivalence relation, effectively of points that lie on the same fibre. Then define the base space as the quotient of the bundle by this equivalence relation. Applied to field theory the whole array of particular field events is what we start with, and spacetime is a derivative notion constructed as the quotient space, but making no sense independently of the fields. As Auyang puts it, spacetime is absolute, but not substantival. It arises as a structural aspect of the field which is itself the primary ontological entity. In particular, on this second reading, it makes no sense to talk of 'emptying' spacetime of the fields - if there are no fields then, for Auyang, there is no spacetime.
It should be noted that the mathematics does not dictate which reading to adopt - that is essentially a philosophical decision on Auyang's part to avoid the chimera of a substantival spacetime. But it is also important that the mathematics can be glossed that way.
Auyang also rejects a relational theory of spacetime as imposed 'externally' on events. The role of spacetime is to confer identity on events, to account for their diversity. As Auyang puts it (p. 139), 'space [is] ... neither a container nor a relator but a kind of divider.'
I now want to turn to Auyang's discussion of how to introduce interactions in field theory, i.e. causal relations between events. To be specific, consider a charged matter field [Psi](x). We treat the field classically to start with, so one can think of [Psi] as a (first-quantized) Schrodinger field, or if we want to be relativistic a Klein-Gordon or Dirac field. The Lagrangian for the matter field is invariant under global phase transformations [Psi](x) [approaches] [Psi](x) [e.sup.i[Theta]], which leads via Noether's theorem to the conservation of charge. But suppose we want to impose invariance under local phase transformations, i.e. [Psi](x) [approaches] [Psi](x) [e.sup.i[Theta](x)], where [Theta](x) is now an arbitrary function of the spacetime location x.
The presence of derivatives [[Delta].sub.[Mu]][Psi] in the Lagrangian spoils the local invariance, but it can be restored by 'correcting' the derivative with a correction term that specifies what is to count as 'the same phase' at different spacetime locations.
So [[Delta].sub.[Mu][Psi] in the Lagrangian is replaced by ([[Delta].sub.[Mu]] - ie[A.sub.[Mu]])[Psi] where [A.sub.[Mu]] specifies what mathematicians call the connection, which ties adjacent phases together and itself transforms as [A.sub.[Mu]] [approaches] [A.sub.[Mu]] + [e.sup.-1][[Delta].sub.[Mu]][Theta](x) under the local symmetry transformations. It is then easily checked that the corrected derivative is indeed invariant under the local symmetry transformations. But the [A.sub.[Mu]] field is formally identical to what we would get by coupling the matter field [Psi](x) to the electromagnetic potential [A.sub.[Mu]], and the constant e just measures the strength of the coupling, namely the electric charge.
So, to summarize, we have derived the electromagnetic coupling by imposing invariance under the local symmetry group. For historical reasons the local symmetry group is known as a gauge group (strictly of the so-called second kind - the corresponding global symmetry is a gauge symmetry of the first kind) and the hope is widely shared that all interactions, including electromagnetic, strong, weak, and even gravitational, can be derived by imposing the appropriate local gauge symmetry.
As we have seen, gauge symmetries are guaranteed by introducing connections on the relevant fibre bundles, which serve to spread the conventional choice of representation of the matter field across spacetime, so tying together the local symmetry transformations, at different spacetime points. The connection is the additional structure that binds the independent events at different spacetime points into an interactive unity, and underlies the fundamental concept of causation, the final element, says Auyang, required for the intelligibility of the physical word over and above the objectivity and identity of the fundamental events.
Having expounded Auyang's own ideas, I will now turn to some possible criticisms.
Let us start with the reasons for imposing local gauge symmetries. Auyang sees this as arising from, and indeed in some sense solving, the general philosophical problem of consistent predication, how to relate the blue of this cup to the blue of that cup. Why should not we call the second cup yellow? After all, it is a free world! But this trivial semantic conventionalism is not the metaphysical problem of what it is about the word that makes it true to say that different objects exhibit the same colour property (whatever we choose to call it). However, it is not clear that Auyang's exposition of local gauge symmetry really deals with that problem at all. Firstly, for Auyang the gauge group is interpreted passively. The connection relates representations of objective states of affairs at different spacetime points, but it does not relate the states of affairs themselves. It is a bit like the trivial semantic conventionalism we discussed a moment ago. Secondly, it is not at all clear that conventions should be treated as propagating locally (indeed according to some authors with subluminal speed). Conventions just aren't the sort of thing that relativistic constraints apply to. Thirdly, even if the first two points were met, physics does not admit local gauge invariance for all predicates, but only for a very limited number, so it could not be the solution to the general philosophical problem.
Indeed, more generally, how can symmetry under a mere choice of conventional representation dictate any genuinely physical principle at all?
I do not think Auyang really deals with any of these points, so let me hazard my own views, which are related to Auyang's, but nevertheless are, I believe, distinctively different.
There are two ways of dealing with the gauge freedom in physics which come immediately to mind:
1. Remove the gauge freedom by just choosing a single representation in Auyang's sense - that is, do what physicists call 'fixing the gauge'. But there are two problems here: (a) it may not be possible to fix the gauge consistently across the whole bundle. This goes in the trade under the name of the Gribov obstruction. And (b) when we quantize a gauge theory, gauge fixing leads to a new sort of gauge freedom associated with the requirement of preserving unitarity. This is a rigid fermionic symmetry involving so-called ghost fields, known as the BRST symmetry after its discoverers, Becchi, Rouet, Stora, and Tyutin. This symmetry is fundamental to the general proof of the renormalizability of gauge theories.
2 Since the objective physical quantities are gauge-invariant, formulate gauge theories in terms of these invariants. For example, the gauge potentials specified by the connection depend on the gauge but the gauge fields themselves, defined geometrically in terms of the curvature of the connection, are gauge-invariant. So why not just use the fields as opposed to the potentials in formulating gauge theories? The difficulty here is a rather subtle one, that prevents the formulation of a gauge theory as a local theory at all. This is best seen by considering the Aharonov-Bohm effect, which effectively measures the line integral of the connection (i.e. the potential) around a closed curve enclosing a flux of 'curvature', that is, a gauge field. This 'loop integral' is a gauge-invariant quantity, but depends on the fields in regions in general remote from the loop in question. So if the fields are all that is real their effect on the nonvanishing of the loop integral is in general highly non-local. Another way of expressing this situation is that the general gauge-invariant quantities are defined, not over a space of points in spacetime but over a space of loops in spacetime, again showing that they cannot in general be specified locally.
The reaction of most physicists to this situation is that the gauge potentials are in some sense 'real', rather than being conventional, i.e. that the gauge group relates not just representations of events but events themselves. So the gauge group is now being given an active interpretation, but with the proviso that events linked by gauge transformations are observationally indistinguishable. So, in a slogan, the real transcends the observable, but that perhaps is not too high a price to pay for restoring a truly local physics of the real.
So far we have discussed the first-quantized version of gauge theories. When we turn to the second-quantized version, the full quantum field theory, new difficulties of interpretation arise.
Firstly, Auyang claims that the fields are operator-valued functions when quantization is employed. Technically that is not quite right: fields are operator-valued distributions defined over a space of test functions which are themselves defined over spacetime. Crudely, the fields are not indexed by points of spacetime, as Auyang opines, but are 'smeared' over regions of spacetime. This may seem a technicality but requires some revision of Auyang's account of spacetime points as conferring identity on field events.
But, more importantly, let me stress that in general the fields themselves, for example a charged matter field, are not quantum-mechanical observables, that is, represented by self-adjoint operators. The observables are quantities like charge densities or energy densities which are self-adjoint constructions from the fields. The fields themselves serve as intertwining operators connecting different superselection sectors of the theory. So the [Psi](x) in the above discussion of the first-quantized gauge theory is far removed from the notion of event in the second-quantized theory, even at the level of representation of the state in a basis provided by 'observable' operators, as recommended by Auyang in her general discussion of objectification in quantum mechanics. Put simply, the second-quantized charged fields are components of purely mathematical structure. For example, the Dirac fields do not commute at spacelike separation. Far from it, they anti-commute. Moreover, any attempt to construct the fields out of local observables is highly non-unique, so in a sense the fields may be thought of as 'coordinatizing' the local observables. (This ambiguity in specifying fields in observable terms was first formalized by Borchers , although the basic point. really goes back to Dyson .)
If we are to take the fields seriously in an ontological sense, then we may need to invest the purely mathematical ingredients in a theory with a kind of reality. That, after all, was our conclusion when discussing the unobservable gauge freedom in the first-quantized version of the theory. But all this leads us far from Auyang's Kantian analysis.
Finally I want to say something about Auyang's treatment of general relativity (GR) as a gauge theory. There is considerable confusion in the literature as to what is meant by the gauge group of GR. Considered as a constrained Hamiltonian system the gauge group is the group of general coordinate transformations. Noether's so-called second theorem results, not in conservation laws, but in identities, indeed the contracted Bianchi identities, which reduce from ten to six the number of independent field equations, and so allow a gauge freedom associated with four arbitrary functions, corresponding to the arbitrary choice of coordinate system. The general coordinate transformations do not in general constitute a group from the global point of view, since in general they cannot be defined globally. But there is a globally defined symmetry group, which is an invariance group of GR, namely the diffeomorphism group, diff, which, from the local point of view, is the active version of local coordinate transformations. It is this group that leads to the so-called hole argument in GR. Briefly, a hole diffeomorphism is a diffeomorphism that acts only inside a patch of spacetime, but reduces to the identity outside the patch. From the invariance under diff it follows that the dragged fields, metric, etc. inside the patch cannot be specified uniquely in terms of the fields outside the patch, so determinism fails in the strongest possible sense. Now this state of affairs is often used as an argument against substantivalism, but it clearly also hits Auyang's absolutist conception of spacetime points. A possible response is again to recognise that models of GR related by hole diffeomorphisms are observationally equivalent, so the failure of determinism applies only to a reality which outstrips the observational, a similar lesson to the one we already seem to learn from trying to interpret gauge theories generally, as in our discussion above.
Auyang signally fails to distinguish diff from coordinate transformations, and seems unaware of the extensive literature on the hole argument. It is perhaps the most serious lacuna in the whole book.
But there is another sense in which one can understand GR as a gauge theory, namely by considering the tangent bundle of the spacetime manifold, or more appositely the related principal bundle, namely the frame bundle. The local gauge group is GL(4, R) which reduces to SO(1, 3) if consideration is restricted to Lorentzian frames (or one might want to consider SL(2, C), the covering group of SO(1, 3) if spinor fields are to be introduced). There are now two ways to go. Stick with the Lorentz group SO(1, 3) or introduce an affine structure in the fibres (to be sharply distinguished from an affine connection on the bundle), so the local symmetry group becomes the inhomogeneous Lorentz group, that is, the Poincare group. Auyang follows some influential authors in claiming that the Poincare group is needed if one wants to allow for torsion in the spacetime manifold. I do not think this is right and refer the reader to Invanenko and Sardanashvily  or Gockeler and Schucker , who support, in my view correctly, a contrary view. We do not need an affine bundle at all in order to extend GR to the Einstein-Cartan [U.sub.4] theory incorporating spin and torsion.
So far we have only discussed GR as a classical gauge theory. This, indeed, is all that Auyang does. But beyond this beckon the philosophically uncharted waters of the quantum gravity programme. This potentially raises new difficulties for Auyang's interpretation of the role of absolute spacetime points, if in some sense spacetime is itself being quantized, perhaps even at the level of the local manifold structure.
So, in summary, I have some disagreements with Auyang, both of a general nature and also at the level of some of the technical detail. But I want to end this review on a positive note. Auyang's book is beautifully written. She has produced the work, as far as I can tell, in intellectual isolation both from the philosophical community and indeed from the quantum field theory community. It is a noble effort to investigate the philosophical underpinnings of field theories and in particular of gauge theories. I have learnt much from reading this book, not least in formulating my disagreements with some of the arguments. At all events I thoroughly agree with her final quotation from Einstein: 'Das Wirkliche is uns nicht gegeben sondern aufgegeben (nach Art eines Ratsel).' As Auyang translates Einstein: 'The real is not given to us but is set us as a task (by way of a riddle).'
Botchers, H. J. : 'Uber die Mannigfaltigkeit der interpolierenden Felder zu einer kausalen S-Matrix', Nuovo Cimento, 15, pp. 784-94.
Dyson, F. J. : 'The Interactions of Nucleons with Meson Fields', Physical Review, 73, pp. 929-30.
Gockeler, M. and Schucker, T. : Differential Geometry, Gauge Theories, and Gravity, Cambridge, Cambridge University Press.
Ivanenko, D. and Sardanashvily, G. : 'The Gauge Treatment of Gravity', Physics Reports, 94, pp. 1-45.
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|Publication:||The British Journal for the Philosophy of Science|
|Article Type:||Book Review|
|Date:||Sep 1, 1998|
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