Quantum Gravity and the Structure

DISKUSSION

Quantum Gravity and the Structure of Scientific Revolutions

J U R G E N A U D R E T S C H

Summary

In a case study Kuhn's morphology of scientific revolutions is put to the test in confronting it with the contemporary developments in physics. It is shown in detail, that Kuhn's scheme is not compatible with the situation in physics today.

1. INTRODUCTION

Scientists are human beings. They are within their scientific activities subject to influences which can only properly be described by psychological and sociological methods. There are not many attempts today to do this adequately. The number of people working out how scientists should behave, is still larger than the number of people undertaking the difficult task to study how scientists really do behave. It is the great merit of Kuhn to have pointed out psychological, sociological as well as logical aspects of this question in his essay "The Structure of Scientific Revolutions" (Kuhn, 1962). Being a historian, Kuhn restricted to a historical method. He tried to extract some general traits from a collection of historical examples. Those philosophers of science trained in logical-empiricist analysis, in the construction of axiomatics and the like, turned his historical analysis into the logical reconstruction of the dynamics of theories, thus obtaining a certai~ general scheme for the development of science. In the following we will give a critical discussion of this scheme.

To extract a scheme and claim general validity for it, is in complete accordance with Kuhn's intentions: "I am not less concerned with rational reconstruction, with the discovery of essentials, than are philosophers of science. My objective, too, is an understanding of science, of the reasons for its special efficacy, of the cognitive status of its theories. But unlike most philosophers of science, I began as a historian of science, examining closely the facts of scientific life" (Kuhn 1970 b, p. 236). It is this generality of Kuhn's scheme which, taken seriously, causes severe problems when it is confronted with historical examples not used by Kuhn, or even with what is happening in physics today, as we will demonstrate below.

A second motivation for this essay originates in the following observation: Usually, analytical philosophy of science is concerned with what has been

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Quantum Gravity and the Structure of Scientific Revolutions 323

called in a slightly exaggerated way the "autopsy of dead theories ''l. Similarly, history of science prefers geocentric astronomy, phlogiston theory and case studies like this. Examples are only classified post hoc. What is needed instead is the logical, psychological and sociological analysis of physics today, especially of those parts which are still "under construction". It is necessary to finally reach a "contemporary history" of science, and to examine closely the facts of contemporary scientific life.

This essay tries to make such an attempt. It confronts Kuhn's morphology of scientific revolution with actual developments of modern physics. The result will give an answer to the question, whether Kuhn's scheme can adequately be related to the situation in physics today, and consequently, whether it can be taken as a basis for a generally valid dynamics of physical theories.

The reader who is familiar with modern physics may immediately turn to chapters 2c, 4 and 5. For readers who are not, the other chapters may be helpful.

2. POINTS OF DEPARTURE

Before classifying metric theory of gravitation and quantum theory with regard to Kuhn's concepts, we briefly recall to the reader's mind some characteristic physical traits of both schemes without trying to give a complete or systematic description.

2a. Metric Theory of Gravitation

Maxwell's theory of electromagnetism is incompatible with a non-relativistic geometry of space and time. This was the starting point for the introduction of special relativity. An appropriate description of electromagnetism must incorporate the special relativistic theory of the behaviour of measuring rods and clocks, which in turn makes essentially use of light rays. Special relativity represents a framework for the construction of theories and a prescription for the extraction of observer-dependent physical predictions. It consists of i) a flat Riemann space, representing the unified concept of space-time, ii) a system of worldlines, representing in a cinematical (!) way a field of classical observers 2, iii) the physical substratum, e. g. a Maxwell field or the worldtine of a point particle, which is mathematically described for example by tensor fields embedded in the space-time.

Measuring results are obtained by interaction between the observer and the observer-independently introduced physical system. This measurement is a

Shrader-Frechette (1977). 2 Usually considerations are based on canonical systems of observers representing inertial

systems, but this is not a fundamental point. Observer fields representing accelerated frames of reference can be introduced in a corresponding way.

324 Jiirgen Audretsch

classical one in the sense that the result is obtained without disturbing the system. It is assumed that the influence of the observer can in principle be eliminated. Furthermore it is assumed that the measurement can be performed pointwise. Mathematically, the measuring results are obtained as scalars by essentially "projecting" (taking scalar products) the geometrical quantities, describing the physical system in a point, on quantities describing the observer in the same point.

Accordingly, the framework of special relativity consists out a flat space time which in no way can be influenced but in turn influences everything else, because it represents the geometry which governs the kinematics of observers and of physical objects. This is completed by a theory of classical measurements and by an all comprising demand • all parts of physics are to be described in such a way that the description fits into the scheme sketched above (all physical theories are to be described in a special-relativistic way). It is this all- claim which we will rediscover again in general relativity. Theories of space and time are the basis of other physical theories.

Because Newton's theory of gravitation is based on a different space-time, it is the immediate consequence of the all-claim above to ask for the construction of a special relativistic theory of gravitation. In fact this step can be omitted. In Newton's theory it is a property of matter called mass which reacts to gravity. As experiments show, it is again exactly this property which is responsible for the inertial forces in accelerated frames of references. On this equality Einstein based his procedure of unifying gravity and inertia in eliminating the concept of gravitational force and inertial force all together. This is done by replacing the flat space-time of the scheme above by an appropriately curved one. Curvature takes on the role of gravity, the other ingredients of the scheme above are taken over. The result is called metric theory of gravitation. Frames of reference are still to be represented by the congruence of worldlines of the respective observers. And again all non-gravitational fields are to be embedded in this now curved space-time. Measurement remains a local process.

Another concept of special relativity is also taken over. Rest-mass is only a special form of energy. And accordingly all sorts of energy, pressure and so on are taken as source of the curvature representing gravity. Consequently, and this is reflected by the non-linearity of the respective equations, even gravity can act as source of gravity. Remembering that every measurement and that the way a physical system develops is influenced by the underlying space-time geometry, we have: all physical processes influence geometry (are to be taken as source of curvature), and on the other hand all physical processes are influenced by geometry (develop within the respective space-time). Note that the all-claim of special relativity has now even been strengthened, because to determine the geometry of space-time no single non-geometrized part of a physical system may be omitted.

Essential consequences of the theory can be demonstrated experimentally. The main domains of application of the theory up to now are those showing strong gravitational fields: cosmology of the early stages of the universe, exterior and interior of very compact stars.


2b. Quantum Theory

An essential characteristic of quantum theory is that it exactly predicts the amount of our possible knowledge or ignorance of a physical state. Uncer- tainty relations prevent that particular physical quantities, for example position and momentum, can be measured simultaneously with any desired accuracy. It is this non-classical measurement which plays a central role in the theory. Measurement always contains an interaction of the system with the classical measuring device. This interaction itself is usually not treated within quantum mechanical dynamics. Instead, its outcome is made a part of the underlying axiomatic structure of the theory. Complete knowledge about a physical state can only be obtained by measuring a complete set of commutating observables. This state then develops according to the respective dynamics in a mathematically well defined way. But future measurements can be predicted only in the sense of a probability, accordingly future knowledge of the system will be restricted by uncertainties. Quantum theory takes seriously the point that we only know something for certain about a system immediately after having performed an appropriate measurement.

The standard formulation of the quantum mechanics of a system with a finite number of degrees of freedom makes use of a Hilbert space description, with the physical state described by a Hitbert vector and the physical variables attributed to selfadjoint operators defined by commutator relations. They contain Planck's constant ~as a characteristic indicator because the basic commutator between position and momentum, on which a Hamiltonian approach is based, is of the dimension of an action.

The quantum theory of systems with an infinite number of degrees of freedom like fields (as for example the electromagnetic field) is based on essentially the same ideas and formulated as a quantum field theory. For special relativistic fields the whole quantum mechanical apparatus can be formulated in accordance with the demands of special relativity. As part of quantum field theory, interacting many-particle systems with the possibility of creation and annihilation of particles can appropriately be described.

With regard to the formalism there is no principle distinction between "matter" and "interaction". For electromagnetically interacting electrons for example both the Dirac field and the Maxwell field have to be quantized. Nowhere within the quantum theoretical scheme are there any interactions, substances or physical systems in general which are excluded from quantization. This is again an all-claim, of different nature but of equal totality and universality. Roughly speaking, the fact that there is no physics without measurement implies that there is no physics without quantization, demanding the quantum mechanical form of measurements and of dynamics.

Traditionally the domain of application of quantum theory is microphysics, but there are also important macroscopic quantum phenomena of laboratory scale like superconductivity and superfluidity.

326 J/irgen Audretsch

2c. The Structure of the Two Basic Schemes As it is common in theoretical physics, we have used the word "theory" for

the metric theory of gravitation and for quantum theory. But as we have pointed out above, both of them are not a physical theory like for example electrodynamics or the theory of the Dirac field or strong interaction.

Both are instead: - Collections of basic terms and procedures on which a physical explanation

has to be based, i. e. to which physical theories must submit. -Col lect ions of this sort which govern the work of two groups of

practitioners (sometimes not very precisely called "relativists" and "field- theorists").

- Frameworks or schemes for the cognition, comprising a mathematical and a conceptual apparatus.

- Schemes describing the structure of physical knowledge. - Basic sets of beliefs and committments of the scientific community. For good reasons both schemes must therefore be called a paradigm in the sense of Kuhn (1962).

These two paradigms exist simultaneously. In Kuhn's scheme too, there is a situation, when there are two simultaneous paradigms. To enable a compari- son, we will characterize the contemporary situation in physics in more detail.

Kuhn's paradigm is first of all sociologically defined and in this sense prior to theory. He says (Kuhn 1970a, p. 10): "Normal science means research firmly based upon one or more past scientific achievements, achievements that some particular scientific community acknowledges for a time as supplying the foundations for its further practice." These achievements "are sufficiently unprecedented to attract an enduring group of adherents away from competing modes of scientific activity '~ and they are simultaneously "sufficiently open- ended to leave all sorts of problems for the redefined group of practitioners to resolve". "Men whose research is based on shared paradigms are committed to the same rules and standards for scientific practice" (p.11).

Are there today such habits which can be attributed to our two paradigms ? The sociological situation is characterized by the fact that there are indeed, as far as the research interests are concerned, the "'family" of "relativists (general- relativists, geometers)" and the much larger "family" of "field-theorists", which base their activity on quantum theory. But for about fifty years, both groups coexisted without much interaction, both convinced that their work has nothing to do with the work of the other group. Furthermore this living together was and still is essentially a peaceful one, and accordingly at least.at the better universities, students are educated in both paradigms only chosing afterwards to which one their research will be committed.

In addition it has to be stressed, that the two contemporary paradigms have a particular trait in common which most of the historical paradigms (as quoted by Kuhn) do not have: their claim for totality, universality, i. e. their all-claim. From the point of view of quantum theory this means : Because the role of the observer and the corresponding statistical interpretation is only appropriately described within the scheme of quantum theory, all physical theories have to


be formulated in a quantum mechanical manner. According to the usual procedure, certain physical properties of finite or infinite degrees of freedom have to be represented by self-adjoint operators obeying commutator or anti- commutator relation and so on. The all-claim of the paradigm "metric theory of gravitation" on the other hand is : The gravitational interaction is a universal one, it is to be represented by curved space-time, i. e. by geometry. Therefore alltheories describing physical interactions are to be formulated in such a way tfiat the influence of the omni-present gravity is encorporated in a geometrical way, which means that the influence is to be formulated as the influence of a nontrivial space-time geometry. And furthermore, that for all processes it is to be taken into account, that they influence geometry (back-reaction). In the standard procedure, space-time is represented as a manifold and its geometrical properties are formulated in every point in the sense of a classical field theory. Accordingly, to incorporate the interaction with geometry in this way, all physics is to be described in a localized way.

As a third important fact it should be noted that the two paradigms described above are incompatible. This should be distinguished from Kuhn's opinion that paradigms are incomparable or incommensurable. One possibility to demonstrate their incompatibility is to make use of the fact, that an essential part of the metric gravitation paradigm is the concept of the classical observer which enables pointlike measurements without disturbances. This is opposed to the quantum mechanical process of measurement according to which between two measurements only the probability for the future outcomes of the subsequent measurement is "propagating". But of course the physical system itself does not disappear in the meantime. Therefore, following the first paradigm, the system is affected by curvature and it gravitates itself. Now simply take this influence on the geometry and measure the respective change of the geometry according to the rules of the metric gravitation paradigm, to obtain in every point a rather complete information (no probability) about the system between the two quantum mechanical measurements. A corresponding physical situation, where this contradiction can be demonstrated, is for example an electron passing a double-slit. Therefore, applying both paradigms successively in the sense of a thought experiment, may lead to contradictions. What we have briefly sketched above is, that there can be no submission of one paradigm to the other.

Why then do the two groups of supporters of the respective paradigms live nevertheless peacefully together ? Simply because despite of the all-claim and despite of the incompatibility the two paradigms are traditionally applied to different domains, namely macrophysics and microphysics. But today this situations has changed considerably. There is an elaborate scheme to treat at least certain aspects of the influence of metrically described gravity on quantum field-theoretically described systems 3. This can be done in a tolerably consistent way only in the sense of approximations.

3 There is also an experiment demonstrating the influence of gravity on macroscopic quantum systems. But gravity is thereby treated on the Newtonian level, so that this experiment is of no importance for our discussion.


3. THE PRESENT SITUATION 4

3a. Exterior Field Approximation

In the following we briefly discuss such an approximation in order to show firstly that restricted physical questions can be answered already within the approximation scheme, secondly that these approximations have only a limited domain of application and finally that the importance of the intended physical domain of application makes it highly desirable to have a complete theory (with regard to which the approximation mentioned above will become at all proper approximations). This last point then will in turn make understandable the psychological situation of those working hard to find the new over-all scheme.

It seems heuristically plausible that strong gravitational fields like for example strong electric fields can create particles. We restrict our discussion to cosmological particle creation. A reader not familiar with theoretical physics may find the Appendix A helpful, which contains some arguments based essentially on a dimensional analysis.

In the exterior field approximation the particle creating field enters the calculation as an unquantized background field. It influences the quantized field, but is not influenced itself. Accordingly, in our case the cosmological space-time acts as a classical background. Geometry remains unquantized as it is in a metric theory of gravitation. Its particle creating influence is represented by the fact that quantum field theory is embedded in the given curved space- time. Feasible as an in-out-approach to the process by which the incoming physical state defined in an in-region in the distant past (preparating quantum mechanical measurement) is compared with the outgoing state registered in the out-region in the distant future (registrating quantum mechanical measurement). For the interval in between it is assumed that the interaction of the state with the curved background develops without additional external disturbances. This implies that during this time no measurements of what sort so ever may happen, and correspondingly nothing is known about the state in between. Correlated to this is the fact that there exists no particle concept for the region of interaction where, according to the naive picture, the particles are produced. Only in the in-and-out-region, provided that the interaction disappears appropriately (i. e. the space-time becomes flat), it is possible to introduce a meaningful particle concept.

Referring to the dynamics in the Heisenberg picture, particle creation means that if the system starts as empty space (i. e. as a vacuum state) in the in-region then a measurement in the out-region will register particles. The mean-value of the particle number operator attributed to the definition of particles in the out- region and taken with regard to the invacuum state is unequal to zero

i < vac,in T a+utaout I vac,in > 12 =k 0

4 For a more detailed study of the topics mentioned in this chapter, see the review articles in: (Hawking and Israel, 1979), (Held, 1980), (Isham et al., 1975), (L6vy and Deser, 1979).


To work this out mathematically the creation operators in the out-region a+o,,t have to be expressed by the creation and annihilation operators in the in- region. It can be done by solving the particle field equation (e. g. the Dirac equation for electrons and positrons) embedded into the respective space-time. This is the way geometry comes in. Along these lines it has been worked out for several cosmological space-times how the cosmological curvature can create particles, thus filling the universe with matter.

What are the deficiencies of the procedure? Although feasible, it is of a limited applicability: i) Because space-time itself remained unquantized, although quantization seems to be necessary for the high curvature near the big bang. ii) Because at the beginning of the universe there is a big bang and not a flat space-time region, therefore the procedure cannot be applied for universe with big bang. iii) Because the in-out-approach allows no statement about what happens in between. This would be necessary for treating the back- reaction of the created particles on the curvature. The general opinion is, that these are deficiencies of an approximation which can be overcome in the complete framework which encorporates geometry in a quantized way.

The approximation indicates already that there is a quantum mechanical interaction between non-gravitational fields (matter fields) and geometry, and that this interaction will be of a decisive importance for example during the early stages of our universe. There are further examples for overlapping domains of application, which cannot be treated appropriately with today's methods.

Already at this stage of the development of the theoretical physical procedure it is worthwhile to analyse the logical, conceptual and systematical structure of the theory using the tools of modern philosophy of science to obtain new answers to old philosophical questions concerning space, time, matter and the way they are related. In order not to disturb our case study of the structure of a contemporary scientific revolution, some comments are given in Appendix B.

3b. Today's Attempts

It is characteristic for today's situation that those working in the field are aware that something new is necessary, in fact that a new paradigm 5 in the sense described in chapter 2 must be built up, and that this can only be done in a paradigm unification absorbing the two momentary paradigms. To prepare the appearance of a new paradigm and to prepare oneself for the time after its appearance when it is to be applied, there is only one reasonable strategy which can systematically be persued by a larger group of people: push as far as possible one paradigm into the prevailing domain of application of the other in integrating as many of its characteristics as possible, and at the same time try to learn as much as possible about the "interaction" between the two paradigms. This will at least enable you to appreciate the new paradigm when it comes up.

5 Of course a physicist would not cai1 it a paradigm.


According to this strategy, the problems are attacked today along several lines. We mention some very briefly. For details we refer the reader to the literature cited above. The whole area of research is usually called Quantum Gravity.

Quantum field theory in a given curved space-time (external field approximation) as sketched above is an attempt to incorporate the unquantized gravity described as curved space-time into quantum field theory. Another attempt is the so-called covariant approach to quantize geometry in quantizing the metric. It is based on a separation of the metric according to

g~t~ = ~ht~ + h~¢

into a classical background vla¢ (usually the one of the flat space-time of special relativity) and a disturbance h~ which is to be quantized. In a metric theory of gravitation the metric represents the gravitational dynamics in the form of a non-trivial kinematics thus abolishing both concepts. The covariant approach on the other hand artificially reintroduces both concepts in splitting the metric into a kinematical part ~1,~ and a dynamical gravitational potential h~f3. This is against the basic concept of the curved geometry paradigm, but has the advantage that everything is reduced to field theory in Minkowski space. Now the elaborated methods of the particle physics paradigm, learned in connection with Yang-Mills theories respectively non-abelian gauge theories, can tenta- tively be applied. Thereby gravity comes out as the theory of gravitons and gravitinos.

Another attempt is the canonical approach. In this case one takes gravity as geometry seriously and tries to extract from the metric tensor the true dynamical degrees of freedom, which then are taken as canonically conjugate variables on which commutation relations are imposed.

4. QUANTUM GRAVITY AND THE ROLE OF THE CRISIS

Having discussed the characteristic traits of today's situation, we are now prepared to compare it with Kuhn's view of the structure of scientific revolutions. To do so, we briefly recapitulate Kuhn's scheme and then contrast it with what is happening instead.

4a. Kuhn's Scheme

According to Kuhn's paradigm view of science the development of science does not go by analytic continuation but shows the following steps (all quotations refer to Kuhn, 1970a)

- normal science ("paradigm-based research" (p. 25), "puzzle-solving activity" (p. 52), "a highly cummulative enterprise" (p. 52))

- awareness of anomalies (p. 52) ("opens a period in which conceptual categories are adjusted until the initially anomalous has become the anticipated" (p. 64))


- crisis ( " w h e n . . . an anomaly comes to seem more than just another puzzle of normal science, the transition to crisis and to extraordinary science has begun" (p. 82))

- e m e r g e n c e of a new paradigm as the response to crisis ("the new pa rad igm. . , emerges all at once, sometimes in the middle of the night, in the mind of a man deeply immersed in crisis" (p. 89), "a relatively sudden and unstructured event like the gestalt switch" (p. 122) "scales are falling from the eyes" (p. 122))

- competition between the old and the new paradigm, succession of the new one, ("that decision must be based less on past achievement than on future promise" (p. 157), "a decision of that kind can only be made on faith" (p. 158), "a new scientific trfith does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it" (quoted from Max Plan&) (p. 151) "The transfer of allegiance from paradigm to paradigm is a conversion experience that cannot be forced" (p. 151))

- normal science ruled by the new paradigm. Scientific revolutions are thereby called "those non-cummulative developmen- tal episodes in which an older paradigm is replaced in whole or in part by in imcompatible one" (p. 92). Extraordinary science as opposed to normal science appears when normal science is faced with a crisis.

For further use we have to describe in more detail the two basic concepts "anomaly" and "crisis" which play a central role in Kuhn's scheme: According to Kuhn, scientific revolutions are initiated by anomalies and because a paradigm will not too easily surrender, only those anomalies which "penetrate existing knowledge to the core" will lead to paradigm change (p. 65). Anomalies are thereby seen predominantly as "anomalies in relation of an existing theory to nature" (p. 97) or "anomaly in the fit between theory and nature" (p. 81). "Anomaly appears only against the background provided by the paradigm" (p. 65). Being engaged in a discussion with falsificationists, for Kuhn anomalies were related to experiments which cause a failure of existing rules. Technical breakdown is the core of the crisis (p. 69). Proliferation of competing versions of a theory is a very usual symptom of a crisis (p. 71). There is a state of a crisis when it is recognized that the paradigm is "failing in application to its own traditional problems" (p. 69).

Studying a particular collection of historical examples, Kuhn based his morphology of scientific revolutions implicitly on the following fundamental assumption: For a domain of mature science there is always only one paradigm A which is then replaced in a revolutionary process by a new paradigm B, which again will be replaced in the same way by a third paradigm C and so on.

A --~ B --~ C --~

When Kuhn speaks of competing paradigms, he has successive paradigms in mind. During extraordinary science a paradigm and its predecessor are both struggling for the mastery. Correspondingly during the period of normal


science, mature science is seen as a single paradigm science (Masterman 1970). It is this assumption of a "one-dimensional" scheme for the paradigm change which makes the crucial role of anomalies and crises plausible at all. Also the sociological and psychological aspects of extraordinary science go back to this. Furthermore, this assumption is responsible for the "rationality gaps", which are sometimes attributed to Kuhn's scheme (compare for example Stegmiiller, 1976). This will become apparent when the assumption is given up.

4b. What is Happening Instead

Physics is an advanced mature science. Nevertheless it is not single- paradigm science, as the situation today demonstrates most drastically. Kuhn seems not to be aware of the fact that a multi-paradigm status may occur in mature science 6. This will become a central point, when Kuhn's scheme is now confronted with contemporary research in physics.

As we have described above in chapt. 2 and 3, physics today is characterized by the following facts: a) Contemporary physics is multi-paradigm science 7 with two paradigms

having an all-claim and overlapping domains of application. But there is no competition between the two paradigms. It is not assumed that one paradigm will ever replace the other. There are neither anomalies nor is there a crisis.

b) Nevertheless, apart from the elaboration of the twoparadigms in the sense of normal science, there are evergrowing attempts to unify the two paradigms

A..,~ / , C

B

There are psychological, sociological and logical reasons for these attempts. Physicists are thereby aware of the fact that the task of paradigm unification necessarily means abolition of the two oM paradigms A and B in favour of a new one C. A rational strategy exists to do systematically the first step towards the new paradigm.

In the following we will describe these points in more detail. Both paradigms do not show any anomalies. There is not even one single

experiment related to the intended domain of application of the unified paradigms. Accordingly there cannot be any experimental anomalies or

6 The idea that in normal science there is always only one paradigm in each field of science has as well been criticized by Popper (1970, p. 55): "Although I find Kuhn's discovery of what he calls 'normal science most important, I do not agree that the history of science supports his doctrine (essential for his theory of rational communication) that 'normally' we have one dominant theory - a 'paradigm' in each scientific domain . . . "

7 Of course there are within each paradigm rivalling theories, for example based on different field equations (Brans-Dicke eqs. versus Einstein eqs.). But this is of no importance for our arguments.


falsifications. There are only within each paradigm the usual difficulties, in connection with the standard puzzle solving of normal science. None of these experimental "anomalies" is penetrating existing knowledge to the core. There is no technical breakdown. Neither are there within each paradigm theoretical anomalies. Infinities occur within quantum field theory, but there is as part of the theory a canonical procedure to remove them (renormalization, regularization) 8. In General Relativity singularities occur. But all those which have been found up to today are hidden behind horizons 9.

Although there are no anomalies, is there perhaps nevertheless a state of crisis ? The contrary is the case; both paradigms are extremely efficient and both are highly respected today. The quantum field theoretical paradigm has just shown to be very successful. Within its framework one has been able to unify electromagnetic and weak interaction. The scientific community has recognized this in awarding the 1979 Nobel Prizes in physics. Furthermore people are hopeful to include strong interaction in a similar way (grand unification), because the only plausible theory of strong interaction is also a gauge theory. Among physicists there is no widespread belief that one is approaching a critical point or, even more dramatic, that one is pushed into a CFISIS.

A third trait which characterizes today's multi-paradigm status in physics is that there is no competition between the two paradigms. One does not find two groups of supporters of different paradigms, but instead only scientists temporarily engaged in normal science activities within one paradigm. The two

8 Shrader-Frechette (1977) has argued that high energy physics is passing to a crisis because the elementary particle paradigm is to be replaced. From our point of view the elementary particle concept is only one of the more or less elaborated concepts within the quantum field theory paradigm. It is not a paradigm itself. The arguments given by Shrader-Freehette (1977) like "absense of clear criteria", "particles versus excited states" and so on could be taken to demonstrate this.

9 Kanitscheider (1977) has discussed the philosophical implications of the singularity theorem of Hawking and Penrose in detail. Singularities (in black holes as big bang and so on), horizons and similar concepts form an important part of today's metric theory of gravitation. Physics as we know it from elsewhere breaks down when space-time horizons and singularkies come in. Not only "usual" physics, but also philosophical concepts introduced in relation to this physics can break down (e. g. the traditional causal structure in connection with horizons). Furthermore, it is a characteristic trait that philosophical arguments can strongly interfere a discussion about for example the relevance of singularities inside black holes. Based on this, Kanitscheider has concluded that physics has reached a state of crisis in the sense of Kuhn.

The objection to this conclusion is the following: If a physical situation (e. g. occurrence of horizons) can be called anomalous with regard to certain philosophical concepts or if certain philosophical positions are assumed, this may well have little or no influence on physics. From a singularity which is hidden behind a horizon, we can on principle never obtain informations. It therefore depends on the respective philosophical position, if the occurrence of a singularity which is hidden empirically (not theoretically) should be called a crisis. There is in fact no technical breakdown as far as the mathematical treatment of space-time singularities is concerned. Accordingly, it is the general opinion among physicists, that as long as the theory does not lead to naked singularities (where the singularity would be empirically visible), there is no crisis. This is important, because crisis in the sense of Kuhn has a large sociological component.

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paradigms are well established. Accordingly many physicists are educated parallely in both paradigms; and an ever growing number is changing paradigm from time to time.

Although there are neither anomalies nor the sense of a crisis, a group of physicists is engaged in what Kuhn would call extraordinary science. Why are physicists trying to unify the two paradigms hoping to initiate the uprise of a new paradigm? There are the following psychological, sociological and, last not least, logical reasons for this:

There are some physicists, who, although trained this way, have an aversion to paradigm pluralism. For them mature science should be single-paradigm science. In many cases, this may be nothing more but an unreflected preconception absorbed from introductory pages of physics textbooks or by reading memoirs of famous physicists. But, on the other hand, the history of physics shows that unification is a very successful method, what may justify the preconception. In any case, aversion to pluralism is one of the driving psychological motives for paradigm unification among physicists.

A second psychological motive is, as in Kuhn's case, the hope that the new paradigm will show surplus meaning. There is in the first place the hope that unification may remove infinities in both initial paradigms, so that they do not even appear mathematically. Perhaps space-time singularities may be avoided when the source becomes genuine quantum mechanical matter. Perhaps inclusion of gravity may render regularization unnecessary. Another hope is that incompleteness will be abolished, that for example the cosmic initial conditions may be fixed or may become unnecessary. Inclusion of gravity may fix the mass spectrum of the elementary particles. Again it is not a sense of crisis which pushes the physicist, but the promisses of the forthcoming unified paradigm attract them.

A very significant motive, which can only sociologically be explained, is that unification has become fashionable again. From the point of view of interactions we have four interactions: electromagnetic, weak, strong and gravitational. The electroweak synthesis has been successful. The inclusion of the third interaction looks promising. Why not attack gravity along the same lines (for example make use of the gauge structure)? We have briefly mentioned in chapt. 3b how this is done today.

Let us now turn to the logical reasons for paradigm unification. In this connexion it is important to note that there is in fact a rational approach of the sort which philosophers of science, trying to close "rationality gaps", would like to find - and had difficulties to find when Kuhn's scheme was taken as a basis. If instead the structure is multi-paradigm science demanding for paradigm unification, a rational approach emerges quite naturally.

In contrast to an accumulation of anomalies, which according to Kuhn raises the sense of a crisis, the situation today is characterized by the fact that existing paradigms can properly be declared inadequate: The intended domains of application of the two paradigms overlap. Because one agrees with the all- claims of both of them, it is the generally accepted opinion that situations, as for example the early stage of the universe, can impossibly be treated using one


paradigm only. On the other hand, paradigms are incompatible, what can be elucidated by thought experiments (while proper experiments or observations play no role at all in our case !). From this situation inevitably follows the need to reconcile the all-claims in unifying the paradigms, what will lead to one all- embracing paradigm. Therefore, in contradiction to Kuhn's scheme, today's normal science can find its own limitations, not because there is a falsificating experiment, but because normal science is multi-paradigmatic science and the paradigms have an all-claim. The regions where the two paradigms overlap are the sources of tension which leads to innovation.

As a second aspect of rationality we have shown in chapt. 3, that there is a strategy to do the first step towards unification, which is again completely rational • Try to absorb as many elements of one paradigm into the other and take the resulting theory as one particular limiting case of the yet unknown new paradigm (compare the example of the external field approximation sketched above). In doing so, there is a rational discussion between the two groups of practitioners which are trained in the respective paradigms.

As a last comment we mention an aspect, which has also a sociological component. If a new paradigm originating from paradigm unification appears, it will have no difficulties to be accepted as the replacement of the two old ones. Being a unification it comprises at least the two domains of application where the previous ones have in fact been successful. From the very beginning therefore this new paradigm is obviously superior to each of its two predecessors. As compared with Kuhn's scheme of two succeeding paradigms. there is a great deal of rationality also in this part of extraordinary science.

5. CONCLUSION

Our intention was not to criticize Kuhn's ideas from the point of view of other elaborated programs and concepts of philosophy of science. This has vastly been done already (compare for example the review in Stegmiiller, 1976). Neither have-we been interested in proposing an additional alternative scheme and claiming general validity for it. Our aim was instead to put Kuhn's scheme to the test in applying it to the developments in physics today.

We have shown above in detail that this contemporary situation in physics is not compatible with Kuhn's scheme for the structure of scientific revolutions. Accordingly, the minimal coclusion which has to be made is, that Kuhn's scheme is at least incomplete and has to be adjusted in one way or the other. Because there are indications that today's situation is not a singular one, that there have been similar situations in the past as well l°, our critique is in fact potentially more far-reaching.

I0 It would be worthwhile to study in detail the following conjecture: Non-relativistic quantum mechanics and special relativity were both coexisting paradigms with an all-claim. Finally these all-claims were convincingly conciliated in relativistic quantum field theory. As in the case discussed above, there has been a phase in the meantime, when the deficiencies of certain "approximations" to the new theory, which was still to be created, were discussed (Dirac equation and Klein-Gordon equation in the framework of first quantization).

22*


Let us, at the end of this essay, prevent a misunderstanding. It has not been written to contribute to the debate between Kuhn, Popper, Lakatos, Feyer- abend and the respective followers which dominate today's discussion on rationality and progress in science. We have criticized the work of Kuhn in particular, because in this case the underlying structure of a "one-dimensional" dynamics of paradigms (a paradigm has only one predecessor) and its consequences for the over-all scheme is most evident. But our objections apply mutatis mutandis also to other approaches to theory dynamics, to the extent as the are based on "one-dimensional" schemes which neglect the importance or the necessity of unifications.

APPENDIX A: HEURISTIC APPROACH TO COSMOLOGICAL PARTICLE CREATION

A rather simple curved cosmological space-time is the 3-flat Robertson- Walker universe. It describes galaxies moving away from each other (expansion of the universe) and is characterized by the fact that the 3-space between the galaxies is 3-flat. Attributing fixed coordinate values x, y, z to particular galaxies, the measured distance between them is given by the line-element

(1) ds 2 = c2dt 2 - R2(t) {dx 2 + dy 2 + dz 2}

where the common factor R(t) determines the time dependence of the spacelike distances and c is the velocity of light. For a radiation filled universe R(t) is characteristically of the form

(2) R ~ t 1/2

and for a matter filled universe

(3) R ~ t 2/3

both showing a big bang at t = 0. The gravitational interaction between the galaxies causes attraction. Accor-

dingly, the velocity with which the galaxies move away from each other is permanently diminished (decelerated expansion). There is a relative acceleration of the galaxies towards each other

In this universe not only two galaxies but any two test particles separated at a small distance A1 experience an acceleration b towards each other of the form

(4) b = Q-AI

where for the expansion laws above we have

(s) Q - 1/÷.


In a very heuristic way the field theoretical vacuum may be described as containing virtual pairs of particles and antiparticles of mass m, which are continuously created and immediately afterwards annihilated. This creation "out of nothing" violates the conservation of energy by an amount of

(6) AE = 2mc 2.

Such a violation can be tolerated, but only if it does not last very long. The respective duration At is related to the energy uncertainty AE according to the uncertainty relation

(7) AE " At ~ h

The maximum distance a pair of particles can separate from each other during At is

(8) A1 -< c a t

where c is the velocity of light. This gives with (6) and (7)

(9) AI~<h/mc = Compton wavelength = 4 -10 -11cm (for electrons, positrons).

The energy E the particle can accumulate because of the gravitational acceleration when travelling the distance 1, is because of (4) and (5)

(10) E = force • distance ~ m • b - A1 ~ ~ (A1) 2.

On the other hand, the energy, a virtual particle has to accumulate to be able to become real, is at least its rest energy E = mc 2. This implies with (10) taking into account (9) the following condition.

m h 2 (11) E ~ - --Z-c 2 > mc 2

which is a condition for the age t of the universe. According to (11) it is only during the time

(12) t ~<~h/mc? = 1,3 • lO-21sec (for electrons, positrons)

after the big bang, that the gravitational forces are strong enough to produce particles.

What can be seen from the very simple picture is, that there is a cosmological particle creation and that it is relevant only for the very early stages of the universe. H o w efficient the process is, i. e. how much matter is produced, can only be worked out by a more serious calculation.


APPENDIX B: SOME NEW COMMENTS ON OLD PHILOSOPHICAL QUESTIONS

The study of the exterior field approximation already leads to some new comments on the old philosophical discussion of the relation between space, time and matter. We add the following comments here in order to demonstrate that contemporary physics may be of interest as well for those philosophers, who are not predominantly interested in schemes for the dynamics of scientific theories.

Geometry on one hand and matter/energy on the other, are they dependent in a causal sense ? Are they dependent in an ontological sense (one is of a prior reality, the other is derived)? Is there a predominance of one over the other in the structure of the theory? (Is "Mach's Principle" incorporated in Einstein's theory?)

Before the inclusion of quantum theory the arguments have been: Accor- ding to Einstein's field equations

G=~ = T=~

the existence of matter/energy causes curvature. But solutions of

G ~ = 0

for empty space-time exist. Because curvature without matter/energy is possible, the conclusion has been, that geometry is the more basic concept. The geometry is more primary because it may exist independently.

The external field approximation shows that in general space-time curvature creates matter. Does this now imply a total symmetry between geometry and matter content because every empty space-time immediately fills itself with particles ? There is at least one counterexample that this is not the case. For massive particles without spin (Klein-Gordon equation) it has been shown in the exterior field approximation, that a plane sandwich gravitational wave leads exactly to no creation of particles (Gibbons, 1975) Accordingly there are still empty space-time solutions possible which remain empty. Nevertheless it is obvious that the relation between matter/energy and curvature has become more symmetric because each one may create the other.

As a second comment we would like to draw attention to the fact that empty space-time has got a new quality. Instead of being describable as what is relating objects or events, empty space-time is the place of gravitational forces, and now as well of the quantum field theoretical vacuum which may decay.

REFERENCES

Gibbons, G. W. 1975: "Quantized Fields Propagating in Plane-Wave space-times" Comm. math. Phys. 45, 191.

Hawking, S. W. and Israel, W. (eds.) 1979: "General Relativity", Cambridge. Held, A. (ed.) t980: "General Relativity and Gravitation", VoL I, New York. Isham, C. J., Penrose, R., and D. W. Sciama (eds.) 1975: "Quantum Gravity", Oxford.


Kanitscheider, B. : "Singularit~iten, Horizonte und das Ende der Zeit", Philosophia naturalis 16, 480.

Kuhn, T. S. 1962: "The Structure of Scientific Revolutions", Chicago. Kuhn, T. S., 1970a: "The Structure of Scientific Revolutions" (second edition, enlarged)

International Encyclopedia of Unified Science Vol. 2, No. 2, Chicago. Kuhn, T. S., 1970b: "Reflections on my Critics" in Criticism and the Growth of Knowledge,

I. Lakatos, A. Musgrave (eds.) Cambridge. L~vy, M. and Deser, S., 1979: "Recent Developments in Gravitation", New York. Masterman, M., 1970: "The Nature of Paradigm" in Criticism and the Growth of Knowledge,

I. Lakatos, A. Musgrave (eds.), Cambridge. Popper, K., 1970: "Normal Science and its Dangers" in Criticism and the Growth of Knowledge,

I. Lakatos, A. Musgrave (eds.), Cambridge. Shrader-Frechette, K, 1977: "Atomism in Crisis: An Analysis of the Current High Energy

Paradigm", Philosophy of Science 44, 409. Stegmiiller, W., 1976: "The Structure and Dynamics of Theories", New York 1976. Translation of

"Theorienstrukturen und Theoriendynamik" originally published as V.2, pt. 2 of "Probleme und Resultate der Wissenschaftstheorie und analytischen Philosophic".

Adresse des Autors:

Prof. Dr. Jiirgen Audretsch, Universit~it Konstanz, Postfach 5560, D-7750 Konstanz

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Quantum Gravity and the Structure