EDGE 10 — March 20, 1997


Stuart Kauffman & Lee Smolin

"We argue that in classical and quantum theories of gravity the configuration space and Hilbert space may not be constructible through any finite procedure. If this is the case then the "problem of time" in quantum cosmology may be a pseudoproblem, because the argument that time disappears from the theory depends on constructions that cannot be realized by any finite beings that live in the universe. We propose an alternative formulation of quantum cosmological theories in which it is only necessary to predict the amplitudes for any given state to evolve to a finite number of possible successor states. The space of accessible states of the system is then constructed as the universe evolves from any initial state. In this kind of formulation of quantum cosmology time and causality are built in at the fundamental level."


Murray Gell-Mann

"It may well be that Hartle's approach makes it unnecessary to invent all this ponderous machinery. But then, of course, your communication refers to Quantum Cosmology, and who knows what is needed there?"

Julian Barbour

"It is very good that Stu Kauffman and Lee are making this serious attempt to save a notion of time, since I think the issue of timelessness is central to the unification of general relativity with quantum mechanics. The notion of time capsules is still certainly only a conjecture. However, as Lee admits, it has proven very hard to show that the idea is definitely wrong. Moreover, the history of physics has shown that it is often worth taking disconcerting ideas seriously, and I think timelessness is such a one. At the moment, I do not find Lee and Stu's arguments for time threaten my position too strongly."

(11,833 words)

John Brockman, Editor and Publisher | Kip Parent, Webmaster


Stuart Kauffman & Lee Smolin

In The Third Culture, I noted that physicists had come to the wrong book. They had little to say about the other scientists, and, vice versa. This may have to do with the fact that the language of physics is mathematics; it may also be that ideas about complexity and evolution have not had the same relevance for cosmology and physics as they have for biology and computer science. Astronomers have studied the spectra of light emitted by distant stars billions of years ago, and have so far found no indication that the laws of physics have changed over this epoch.

Cosmology, which came into its own as a science only about thirty years ago, is concerned in part with pinning down the parameters of the universe: its expansion rate, the amount of its mass, the nature of its "dark matter." Cosmologists today are also speculating on more far-reac hing questions, such as how the universe was created and how its structure was determined. While some cosmologists are speculating that the laws of physics might explain the origin of the universe, the origin of the laws themselves is a problem so unfathomable that it is rarely discussed. Might the principles of adaptive complexity be at work? Is there a way in which the universe may have organized itself? Does the "anthropic principle"‹the notion that the existence of intelligent observers like us is in some sense a factor in the universe's existence‹have any useful part to play in cosmology?

The theoretical physicist Lee Smolin is interested in the problem of quantum gravity‹of reconciling quantum theory with Einstein's gravitational theory, the theory of general relativity, to produce a correct picture of spacetime. He also thinks about creating what he calls a theory of the whole universe, which would explain its evolution, and he has invented a method by which natural selection might operate on the cosmic scale.

The cosmologist Sir Martin Rees noted in The Third Culture that "one of the key issues in physics is to reconcile gravity with the quantum principle and the microphysical forces. There are various schools of thought; the Stephen Hawking School, the Roger Penrose School, and a number of others. My view is that we're a long way away from a consensus in that field, but Smolin and Ashtekar have injected important new ideas into that debate."

"Quantum gravity was one of the subjects beyond the fringe, when John Wheeler talked about it in the 1950s. Now it's something where serious approaches are being adopted. But we're still a long way from experimental test. Lee Smolin's most important insight was to suggest a new way of looking at space and time in terms of a lattice structure on a tiny scale. It relates in a way to Wheeler's very farsighted ideas of spacetime foam: the idea that if you look at space and time on a very tiny scale, there are no longer three dimensions of space and one of time but the dimensions all get screwed up in a complicated way.

"The other idea with which Smolin is associated is "natural selection" of universes. He's saying that in some sense the universes that allow complexity and evolution reproduce themselves more efficiently than other universes. The ensemble itself is thus evolving in some complicated way. When stars die, they sometimes form black holes. (This is something which I wear my astrophysical hat to study.) Smolin speculates‹as others, like Alan Guth, have also don e‹that inside a black hole it's possible for a small region to, as it were, sprout into a new universe. We don't see it, but it inflates into some new dimension. Smolin takes that idea on board, but then introduces another conjecture, which is that the laws of nature in the new universe are related to those in the previous universe. This differs from Andrei Linde's idea of a random ensemble, because Smolin supposes that the new universe retains physical laws not too different from its parent universe. What that would mean is that universes big and complex enough to allow stars to form, evolve, and die, and which can therefore produce lots of black holes, would have more progeny, because each black hole can then lead to a new universe; whereas a universe that didn't allow stars and black holes to form would have no progeny. Therefore Smolin claims that the ensemble of universes may evolve not randomly but by some Darwinian selection, in favor of the potentially complex universes."

The physicist Alan Guth points out that "a possible reason that Discover magazine dubbed Lee "The New Einstein" on a recent cover is that his work is motivated by the same goal‹to construct a unified theory of physics‹and his approach is to keep Einstein's original theory as the fundamental basis of it. Superstring theory basically puts Einstein's theory in the background. The belief is that Einstein's theory will reemerge as a low-energy limit, but it's not the fundamental ingredient of the theory. The fundamental ingredient of the superstring theory is this microscopic string. In Smolin's formulation, the fundamental ingredient remains the gravitational field, and the goal is to treat it quantum mechanically. What he hopes to do that's different from the failed approach‹the approach that successfully quantizes electromagnetism but fails for gravity‹ is to exploit the fact that the theory of gravity is fundamentally nonlinear."

"The relativity physicists belong to a small club. It's a club that has yet to convince the majority of the community that the approach they're pursuing is the right one. Certainly Smolin is welcome to come and give seminars, and at major conferences he and his colleagues are invited to speak. The physics community is interested in hearing what they have to say. But the majority looks to the superstring approach to answer essentially the same questions."

The physicist Murray Gell-Mann noted "Smolin? Oh, is he that young guy with those crazy ideas? He may not be wrong!"

The synthetic path to investigating the world is the logical space occupied by Gell-Mann, the biologist Stuart Kauffman, the computer scientist Christopher G. Langton, and the physicist J. Doyne Farmer, and their colleagues in and around Los Alamos and the Santa Fe Institute.

The Santa Fe Institute was founded in 1984 by a group that included Gell-Mann, then at the California Institute of Technology, and the Los Alamos chemist George Cowan. Some say it came into being as a haven for bored physicists. Indeed, the end of the reductionist program in physics may well be an epistemological demise, in which the ultimate question is neither asked nor answered but instead the terms of the inquiry are transformed. This is what is happening in Santa Fe.

Stuart Kauffman is a theoretical biologist who studies the origin of life and the origins of molecular organization. Twenty-five years ago, he developed the Kauffman models, which are random networks exhibiting a kind of self-organization that he terms "order for free." Kauffman is not easy. His models are rigorous, mathematical, and, to many of his colleagues, somewhat difficult to understand. A key to his worldview is the notion that convergent rather than divergent flow plays the deciding role in the evolution of life. With his colleague Christopher G. Langton, he believes that the complex systems best able to adapt are those poised on the border between chaos and disorder.
Kauffman asks a question that goes beyond those asked by other evolutionary theorists: if selection is operating all the time, how do we build a theory that combines self-organization (order for free) and selection? The answer lies in a "new" biology, somewhat similar to that proposed by Brian Goodwin, in which natural selection is married to structuralism.

The evolutionary biologist Stephen Jay Gould noted in The Third Culture that "He's following in the structuralist tradition, which should not be seen as contrary to Darwin but as helpful to Darwin. Structural principles set constraints, and natural selection must work within them. His "order for free" is an outcome of sets of constraints; it shows that a great deal of order can be produced just from the physical attributes of matter and the structural principles of organization. You don't need a special Darwinian argument; that's what he means by "order for free." It's a very good phrase, because a strict Darwinian thinks that all sensible order has to come from natural selection. That's not true."

According to the computer scientist Danny (W. Daniel) Hillis: "Stuart Kauffman is a strange creature, because he's a theoretical biologist, which is almost an oxymoron. In physics, there are the theoretical types and the experimental types, and there's a good understanding of what the relationship is between them. There's a tremendous respect for the theoreticians. In physics, the theory is almost the real stuff, and the experiments are just an approximation to test the theory. If you get something a little bit wrong, then it's probably an experimental error. The theory is the thing of perfection, unless you find an experiment that shows that you need to shift to another theory. When Eddington went off during a solar eclipse to measure the bending of starlight by the sun and thus to test Einstein's general-relativity theory, somebody asked Einstein what he would think if Eddington's measurements failed to support his theory, and Einstein's comment was, "Then I would have felt sorry for the dear Lord. The theory is correct."

"In biology, however, this is reversed. The experimental is on top, and the theory is considered poor stuff. Everything in biology is data. The way to acquire respect is to spend hours in the lab, and have your students and postdocs spend hours in the lab, getting data. In some sense, you're not licensed to theorize unless you get the data. And you're allowed to theorize only about your own data‹or at the very least you need to have collected data before you get the right to theorize about other data."

"Stuart is of the rare breed that generates theories without being an experimentalist. He takes the trouble to understand things, such as dynamical-systems theory, and tries to connect those into biology, so he becomes a conduit of ideas that are coming out of physics, from the theorists in physics, into biology."

Kauffman and Smolin began working together a year ago and a result of this collaboration is a paper entitled "A Possible Solution For The Problem Of Time In Quantum Cosmology" which I plan to publish in next week's edition of EDGE (#12). What follows is an introductory letter from Smolin with initial comments from the theoretical physicist Julian Barbour and Murray Gell-Mann.

While some of this material, particularly in the paper, is mathematical, most of it is readable by non-scientists. A few weeks ago I received an email from the novelist Bruce Sterling, who wrote: "This is truly a remarkably interesting mailing list; despite its recherche topics it seduces me into reading it almost every time." It is in this spirit I present Stu Kauffman and Lee Smolin's paper, "A Possible Solution For The Problem Of Time In Quantum Cosmology."


Stuart Kauffman & Lee Smolin

STUART KAUFFMAN is a biologist; professor of biochemistry at the University of Pennsylvania and a professor at the Santa Fe Institute; author of Origins of Order: Self-Organization and Selection in Evolution (Oxford), and At Home in the Universe (Oxford).

LEE SMOLIN is a theoretical physicist; professor of physics and member of the Center for Gravitational Physics and Geometry at Pennsylvania State University; author of The Life of The Cosmos (Oxford, forthcoming).

From: Lee Smolin
To: John Brockman

March 20, 1997

Dear John,

As you know, Stu Kauffman and I met each other a year ago. After an exchange of emails I invited him to Penn State to give a talk. From the moment he got off the plane he was talking, and what he wanted to talk about was how we might invent rules for knots and networks to evolve in time in such a way that they may replicate themselves and hence create autocatalytic systems. It is a true fact about Stu that he is always talking, but it is also a very interesting fact that this idea of networks replicating themselves is relevant for both the origin of life and quantum gravity. The first, because living cells consist of a myriad of chemical reactions that are collectively autocatalytic ‹ that is each molecule is synthesized in reactions that are catalyzed by other molecules.

The question is interesting for quantum gravity because, as Carlo Rovelli (now professor at the University of Pittsburgh) and I discovered a few years ago, the quantum states of the gravitational field-or equivalently of the geometry of space-are described by networks of a certain kind. These are called spin networks, and they were originally invented by Roger Penrose in the 1960's, but for a completely different reason. We rediscovered them as part of our search for a quantum description of space and time. These networks are graphs with integers on the edges, representing quantum mechanical spins.

Carlo and I, and then some others‹especially two very good postdocs Thomas Thiemann and Rouman Borissov‹figured out how these networks should evolve in time if Einstein's equations hold at the quantum mechanical level. But there is a problem, which is they seem unlikely to do something nice. The networks are very very tiny, a typical volume occupied by a node is 10 to the power -99 centimeters cubed (that there is a discrete scale to quantum geometry-even if very tiny, is one of the things Carlo and I discovered.) But the world seems regular, and its geometry seems Euclidean on scales much much larger than these tiny quanta of geometry. Thus, to describe the real world these networks must grow slowly and smoothly, so that they describe the geometry of the space we see. This is unlikely, just as it is unlikely that atoms organize themselves into a regular crystal such as in a metal. And this is the essence of the problem-to understand why the world is big (compared to 10 to the -33 centimeters) and the geometry of space is almost Euclidean requires that these networks that describe the quantum geometry of space organize themselves into a very regular arrangement.

I don't know what Stu knew about quantum gravity when we met‹I think he had only heard something about our knots and networks. But somehow he had intuited right away that a key problem in quantum gravity and cosmology was going to be a problem of how these networks can self-organize themselves into a regular, slowly changing state. For him this was analogous to problems of self organization as they appear in biology and other areas such as non-equilibrium statistical physics. And this is what he was talking about when he got off the plane to meet me.

I had also thought about this issue, and seen the possibility of an analogy
between problems of self-organization in quantum gravity and cosmology and problems in biology. This was one reason I wanted to meet Stu. But of course I'm a physicist, this is what I do. How it is that Stu understood this I don't know, but this shared feeling of the importance of analogies between problems in fundamental physics and biology has been the basis of what quickly became a great friendship and a collaboration.

In the past year I've taught Stu a bit of quantum mechanics and in return he has taught me something about biology. So after a while it seems we have fallen into working together on both kinds of problems. I am part of a collaboration he organized to make a try at understanding the origin of life and I have also tried to formulate mathematically his notion of an autonomous agent. I also described to him a problem in astronomy I was working on‹which is to understand how spiral structure forms in galaxies. He told me about models of pattern formation biologists and chemists use, due originally to Turing, called reaction diffusion equations, which I then applied to the interactions among the stars, gas and dust in the disk of a spiral galaxy. And Stu has also gotten involved in our work on quantum gravity. This paper is one outcome of this work.

For me, this paper came from the coming together of three lines of thought. The first was a long conversation Stu and I were having about how to formalize theoretical biology. Stu is very worried that you cannot specify all the possible tasks or niches or properties that might give an organism selective advantage all at once, in advance of watching a system evolve. He is not sure that the standard model of natural selection in which there is a fitness landscape, labeled in advance by fixed properties is really useful. Instead, he has been trying to develop a notion of the adjacent possible states: that a system evolves by moving into a space of configurations that are just one step‹one new reaction or one exaptation‹away from the present state. At some point he mentioned that he was also worried about whether you could describe a complex system quantum mechanically, because it might not be possible to specify a basis for the possible quantum states of the system in advance.

I had dismissed this last point as his taking things to far‹as he sometimes does. But then all of a sudden I realized that this might in fact apply to the space of states of quantum gravity. The reason is that the mathematical structures involved, which are the networks I mentioned, might not be classifiable. This means that one might not be able to invent any test that could be guaranteed to tell any two apart in a finite amount of time.

I realized this because I was caught in an argument between two views of time. On the one side Julian Barbour, whose ideas on understanding of space and time in relativity theory have been very influential. Nonetheless, I had been unable to agree with his thinking of the last few years, in which he has come to the conclusion that in quantum cosmology time cannot be fundamental. Time, according to him, should play no fundamental role in nature.

But although I instinctively disagree with this, I have been unable to defeat the argument that leads Julian to it. Nor has anyone else. The result is a famous problem in quantum cosmology called the problem of time‹time is nowhere to be found in the fundamental equations of the theory.

On the other side I have been worrying about the meaning of some work I have been doing with Fotini Markopoulou, of the University of London. Fotini has her own strong views about time, which are very different from Julian's. In particular, she wants the notion of causality to be present at a fundamental level in quantum gravity. She is not the first to think this, Roger Penrose also argued for it a long time ago, and it was a main motivation of his twistor theory. But what is new is that Fotini proposed that it should be possible to construct a version of quantum gravity based on the spin networks, in which there is a notion of causality at the fundamental level. And in fact it is possible, and we have constructed a version of quantum gravity that works this way.

This seems right to me, but I had been bothered that I saw no way to refute Julian's argument that time disappears from the basic equations of quantum cosmology. But at some point I realized that what I had been discussing with Stu might be relevant. For if Stu is right and the space of quantum states of quantum gravity cannot be specified in advance, Julian cannot run the argument that leads to the disappearance of time from the basic equations. Instead, if the space of possible quantum states must be constructed as the system evolves, then a notion of causality, and hence of time, cannot be avoided. This is the basic idea in the paper.

I should say right away that I am not sure that this is the right solution to the problem of time. It is just the first one I've heard of that seems plausible to me and I have been thinking and writing about this problem for many years. I should say also that its not known that the states of quantum gravity can't be classified‹certain classes of knots, which are only a bit simpler, can be. Furthermore, the actual networks that Fotini and I use in our version of quantum gravity are much simpler than the ones that Carlo and I found to be the states of quantum gravit y‹and these simpler set can be classified. The idea of using only these simpler states was Fotini's‹and she argued for it for an entirely different reason (in technical language these are simpler because they are described intrinsically‹they are not embedded in any prior three dimensional space). But I think Julian would be happy to say maybe she is right if in the end it allows him to keep his argument that there is no time at a fundamental level. So the only thing I would say definitely for the proposal in the paper is that it is new, it is making us think, and we are having fun with it. And I guess I would add that if it's right, it's important, but again that's what we often say about ideas in this field.

I am pleased you are emailing the paper to the Third Culture Mail List and also posting it to the Website. I hope this will be the beginning of an interesting discussion and Stuart and I look forward to comments.

Lee Smolin


by Stuart Kauffman and Lee Smolin

We argue that in classical and quantum theories of gravity the configuration space and Hilbert space may not be constructible through any finite procedure. If this is the case then the "problem of time" in quantum cosmology may be a pseudoproblem, because the argument that time disappears from the theory depends on constructions that cannot be realized by any finite beings that live in the universe. We propose an alternative formulation of quantum cosmological theories in which it is only necessary to predict the amplitudes for any given state to evolve to a finite number of possible successor states. The space of accessible states of the system is then constructed as the universe evolves from any initial state. In this kind of formulation of quantum cosmology time and causality are built in at the fundamental level. An example of such a theory is the recent path integral formulation of quantum gravity of Markopoulou and Smolin, but there are a wide class of theories of this type.


The problem of time in quantum cosmology is one of the key conceptual problems faced by theoretical physics at the present time. Although it was first raised during the 1950's, it has resisted solution, despite many different kinds of attempts[1,2,3,4,5]. Here we would like to propose a new kind of approach to the problem. Basically, we will argue that the problem is not with time, but with some of the assumptions that lead to the conclusion that there is a problem. These are assumptions that are quite satisfactory in ordinary quantum mechanics, but that are problematic in quantum gravity, because they may not be realizable with any constructive procedure. In a quantum theory of cosmology this is a serious problem, because one wants any theoretical construction that we use to describe the universe to be something that can be realized in a finite time, by beings like ourselves that live in that universe. If the quantum theory of cosmology requires a non-constructible procedure to define its formal setting, it is something that could only be of use to a mythical creature of infinite capability. One of the things we would like to demand of a quantum theory of cosmology is that it not make any reference to anything at all that might be posited or imagined to exist outside the closed system which is the universe itself.

We believe that this requirement has a number of consequences for the problem of constructing quantum a good quantum theory of cosmology. These have been discussed in detail elsewhere [4,6,7]. Here we would like to describe one more implication of the requirement, which appears to bear on the problem of time.

We begin by summarizing briefly the argument that time is not present in a quantum theory of cosmology. In section 3 we introduce a worry that one of the assumptions of the argument may not be realizable by any finite procedure. (Whether this is actually the case is not known presently.) We explain how the argument for the disappearance of time would be affected by this circumstance. Then we explain how a quantum theory of cosmology might be made which overcomes the problem, but at the cost of introducing a notion of time and causality at a fundamental level. As an example we refer to recent work on the path integral for quantum gravity [causal], but the form of the theory we propose is more general, and may apply to a wide class of theories beyond quantum general relativity.


The argument that time is not a fundamental aspect of the world goes like this (For more details and discussion see [1,2,3,4,5]). In classical mechanics one begins with a space of configurations C of a system S. Usually the system S is assumed to be a subsystem of the universe. In this case there is a clock outside the system, which is carried by some inertial observer. This clock is used to label the trajectory of the system in the configuration space C. The classical trajectories are then extrema of some action principle.
Were it not for the external clock, one could already say that time has disappeared, as each trajectory exists all at once as a curve g on C. Once the trajectory is chosen, the whole history of the system is determined. In this sense there is nothing in the description that corresponds to what we are used to thinking of as a flow or progression of time. Indeed, just as the whole of any one trajectory exists when any point and velocity are specified, the whole set of trajectories may be said to exist as well, as a timeless set of possibilities.

Time is in fact represented in the description, but it is not in any sense a time that is associated with the system itself. Instead, the t in ordinary classical mechanics refers to a clock carried by an inertial observer, which is not part of the dynamical system being modeled. This external clock is represented in the configuration space description as a special parameterization of each trajectory, according to which the equations of motion are satisfied. Thus, it may be said that there is no sense in which time as something physical is represented in classical mechanics, instead the problem is postponed, as what is represented is time as marked by a clock that exists outside of the physical system which is modeled by the trajectories in the configuration space C.

In quantum mechanics the situation is rather similar. There is a t in the quantum state and the Schroedinger equation, but it is time as measured by an external clock, which is not part of the system being modeled. Thus, when we write,

ih d/dt f = H f (t)

the Hamiltonian refers to evolution, as it would be measured by an external observer, who refers to the external clock whose reading is t.

The quantum state can be represented as a function f over the configuration space, which is normalizable in some inner product. The inner product is another a priori structure, it refers also to the external clock, as it is the structure that allows us to represent the conservation of probability as measured by that clock.

When we turn to the problem of constructing a cosmological theory we face a key problem, which is that there is no external clock. There is by definition nothing outside of the system, which means that the interpretation of the theory must be made without reference to anything that is not part of the system which is modeled. In classical cosmological theories, such as general relativity applied to spatially compact universes, or models such as the Bianchi cosmologies or the Barbour-Bertotti model[12,BB-Royal], this is expressed by the dynamics having a gauge invariance, which includes arbitrary reparameterizations of the classical trajectories. (In general relativity this is part of the diffeomorphism invariance of the theory.) As a result, the classical theory is expressed in a way that makes no reference to any particular parameterization of the trajectories. Any parameterization is as good as any other, none has any physical meaning. The solutions are then labeled by a trajectory, g, period, there is no reference to a parameterization.

This is the sense in which time may be said to disappear from classical cosmological theories. There is nothing in the theory that refers to any time at all. At least without a good deal more work, the theory speaks only in terms of the whole history or trajectory, it seems to have nothing to say about what the world is like at a particular moment.

There is one apparently straightforward way out of this, which is to try to define an intrinsic notion of time, in terms of physical observables. One may construct parameter independent observables that describe what is happening at a point on the trajectory if that point can be labeled intrinsically by some physical property. For example, one might consider some particular degree of freedom to be an intrinsic, physical clock, and label the points on the trajectory by its value. This works in some model systems, but in interesting cases such as general relativity it is not known if such an intrinsic notion of time exists which is well defined over the whole of the configuration space.

In the quantum theory there is a corresponding phenomenon. As there is no external t with which to measure evolution of the quantum state one has instead of the Schroedinger equation the quantum constraint equation

Hf =0

where f is now just a function on the configuration space. Rather than describing evolution, generates arbitrary parameterizations of the trajectories. The wavefunction must be normalizable under an inner product, given by some density on the configuration space. The space of physical states is then given by this constraint equation subject to a condition that the state is normalizable.

We see that, at least naively time has completely disappeared from the formalism. This has led to what is called the "problem of time in quantum cosmology", which is how to either A) find an interpretation of the theory that restores a role for time or B) provide an interpretation according to which time is not part of a fundamental description of the world, but only reappears in an appropriate classical limit.

There have been various attempts at either direction. We will not describe them here, except to say that, in our opinion, so far none has proved completely satisfactory (For good critical reviews that deflate most known proposals, see [1,2].). There are a number of attempts at A) which succeed when applied to either models or the semiclassical limit, but it is not clear whether any of them overcome technical obstacles of various kinds when applied to the full theory. The most well formulated attempt of type B), which is that of Barbour[5], may very well be logically consistent. But it forces one to swallow quite a radical point of view about the relationship between time and our experience.

Given this situation, we would like to propose that the problem may be not with time, but with the assumptions of the argument that leads to time being absent. Given the number of attempts that have been made to resolve the problem, which have not so far led to a good solution, perhaps it might be better to try to dissolve the issue by questioning one of the assumptions of the argument that leads to the statement of the problem. This is what we would like to do in the following.


Both the classical and quantum mechanical versions of the argument for the disappearance of time begin with the specification of the classical configuration space C . This seems an innocent enough assumption. For a system of N particles in d dimensional Euclidean space, it is simply R^(Nd) . One can then find the corresponding basis of the Hilbert space by simply enumerating the Fourier modes. Thus, for cases such as this, it is certainly the case that the configuration space and the Hilbert space structure can be specified a priori.

However, there are good reasons to suspect that for cosmological theories it may not be so easy to specify the whole of the configuration or Hilbert space. For example, it is known that the configuration spaces of theories that implement relational notions of space are quite complicated. One example is the Barbour-Bertotti model[12,13], whose configuration space consists of the relative distances between N particles in d dimensional Euclidean space. While it is presumably specifiable in closed form, this configuration space is rather complicated, as it is the quotient of R^(Nd) by the Euclidean group in d dimensions[5].

The configuration space of compact three geometries is even more complicated, as it is the quotient of the space of metrics by the diffeomorphism group. It is known not to be a manifold everywhere. Furthermore, it has a preferred end, where the volume of the universe vanishes.

These examples serve to show that the configuration spaces of cosmological theories are not simple spaces like R^(Nd) , but may be considerably complicated. This raises a question: could there be a theory so complicated that its space of configurations is not constructible through any finite procedure? For example, is it possible that the topology of an infinite dimensional configuration space were not finitely specifiable? And were this the case, what would be the implications for how we understand dynamics.

(There is an analogous issue in theoretical biology. The problem is that it does not appear that a pre-specifiable set of "functionalities" exists in biology, where pre-specifiable means a compact description of an effective procedure to characterize ahead of time, each member of the set[8,7]. This problem seems to limit the possibilities of a formal framework for biology in which there is a pre-specified space of states which describe the functionalities of elements of a biological system. Similarly, one may question whether it is in principle possible in economic theory to give in advance an a priori list of all the possible kinds of jobs, or goods or services[8].)

We do not know whether in fact the configuration space of general relativity is finitely specifiable. The problem is hard because the physical configuration space is not the space of three metrics. It is instead the space of equivalence classes of three metrics (or connections, in some formalisms) under diffeomorphisms. The problem is that it is not known if there is any effective procedure which will label the equivalence classes.

One can in fact see this issue in one approach to describing the configuration space, due to Newman and Rovelli[9]. There the physical configuration space consists of the diffeomorphism equivalence classes of a set of three flows on a three manifold. (These come from the intersections of the level surfaces of three functions.) These classes are partially characterized by the topologies of the flow lines of the vector fields. We may note that these flow lines may knot and link, thus a part of the problem of specifying the configuration space involves classifying the knotting and linking among the flow lines.

Thus, the configuration space of general relativity cannot be completely described unless the possible ways that flow lines may knot and link in three dimensions are finitely specifiable. It may be noted that there is a decision procedure, due to Hacken, for knots, although it is very cumbersome[15]. However, it is not obvious that this is sufficient to give a decision procedure for configurations in general relativity, because there we are concerned with smooth data. In the smooth category the flow lines may knot and link an infinite number of times in any bounded region. The resulting knots may not be classifiable. All that is known is that knots with a finite number of crossings are classifiable. If these is no decision procedure to classify the knotting and linking of smooth flow lines then the points of the configuration space of general relativity may not be distinguished by any decision procedure. This means that the configuration space is not constructible by any finite procedure.

When we turn from the classical to the quantum theory the same issue arises. First of all, if the configuration space is not constructible through any finite procedure, then there is no finite procedure to define normalizable wave functions on that space. One might still wonder whether there is some constructible basis for the theory. Given the progress of the last few years in quantum gravity we can investigate this question directly, as we know more about the space of quantum states of general relativity than we do about the configuration space of the theory. This is because it has been shown that the space of spatially diffeomorphism invariant states of the quantum gravitational field has a basis which is in one to one correspondence with the diffeomorphism classes of a certain set of embedded, labeled graphs W , in a given three manifold. [10,11]. These are arbitrary graphs, whose edges are labeled by spins and whose vertices are labeled by the distinct ways to combine the spins in the edges that meet there quantum mechanically. These graphs are called spin networks, they were invented originally by Roger Penrose[16], and then discovered to play this role in quantum gravity(For a review of these developments see [17]. These results have also more recently been formulated as theorems in a rigorous formulation of diffeomorphism invariant quantum field theories[19,18].).

Thus, we cannot label all the basis elements of quantum general relativity unless the diffeomorphism classes of the embeddings of spin networks in a three manifold may be classified. But it is not known whether this is the case. The same procedure that classifies the knots is not, at least as far as is known, extendible to the case of embeddings of graphs.

What if it is the case that the diffeomorphism classes of the embeddings of spin networks cannot be classified? While it may be possible to give a finite procedure that generates all the embeddings of spin networks, if they are not classifiable there will be no finite procedure to tell if a given one produced is or is not the same as a previous network in the list. In this case there will be no finite procedure to write the completeness relation or expand a given state in terms of the basis. There will consequently be no finite procedure to test whether an operator is unitary or not. Without being able to do any of these things, we cannot really say that we have a conventional quantum mechanical description. If spin networks are not classifiable, then we cannot construct the Hilbert space of quantum general relativity.

In this case then the whole set up of the problem of time fails. If the Hilbert space of spatially diffeomorphism invariant states is not constructible, then we cannot formulate a quantum theory of cosmology in these terms. There may be something that corresponds to a "wavefunction of the universe" but it cannot be a vector in a constructible Hilbert space. Similarly, if the configuration space C of the theory is not constructible, then we cannot describe the quantum state of the universe in terms of a normalizable function on C .

We may note that a similar argument arises for the path integral formulations of quantum gravity. It is definitely known that four manifolds are not classifiable; this means that path integral formulations of quantum gravity that include sums over topologies are not constructible through a finite procedure[19].

Someone may object that these arguments have to do with quantum general relativity, which is in any case unlikely to exist. One might even like to use this problem as an argument against quantum general relativity. However, the argument only uses the kinematics of the theory, which is that the configuration space includes diffeomorphism and gauge invariant classes of some metric or connection. It uses nothing about the actual dynamics of the theory, nor does it assume anything about which matter fields are included. Thus, the argument applies to a large class of theories, including supergravity.


What if it is the case that the Hilbert space of quantum gravity is not constructible because embedded graphs in three space are not classifiable? How do we do physics? We would like to argue now that there is a straightforward answer to this question. But it is one that necessarily involves the introduction of notions of time and causality.

One model for how to do physics in the absence of a constructible Hilbert space is seen in a recent formulation of the path integral for quantum gravity in terms of spin networks by Markopoulou and Smolin[14](This followed the development of a Euclidean path integral by Reisenberger[20] and by Reisenberger and Rovelli[21]. Very interesting related work has also been done by John Baez[22]. We may note that the theory described in [14] involves non-embedded spin networks, which probably are classifiable, but it can be extended to give a theory of the evolution of embedded spin networks.). In this case one may begin with an initial spin network W_0 with a finite number of edges and nodes (This corresponds to the volume of space being finite.) One then has a finite procedure that constructs a finite set of possible successor spin networks W_1^a , where a labels the different possibilities. To each of these the theory associates a quantum amplitude A(W_0 -> W_1^a) .

The procedure may then be applied to each of these, producing a new set W_2 [a, b] . Here W_2 [a , b] labels the possible successors to each of the W_1^a . The procedure may be iterated any finite number of times N , producing a set of spin networks S^N [W_0] that grow out of the initial spin network W_0 after N steps. S^N [W_0] is itself a directed graph, where two spin networks are joined if one is a successor of the other. There may be more than one path in S^N [W_0] between W_0 and some spin network W_f . The amplitude for W_0 to evolve to W_f is then the sum over the paths that join them in S^N [W_0] , in the limit in which N is taken to infinity, of the products of the amplitudes for each step along the way.

For any finite N , S^N [W_0] has a finite number of elements and the procedure is finitely specifiable. There may be issues about taking the limit N goes to infinity, but there is no reason to think that they are worse than similar problems in quantum mechanics or quantum field theory. In any case, there is a sense in which each step takes a certain amount of time, in the limit N goes to infinity we will be picking up the probability amplitude for the transition to happen in infinite time.

Each step represents a finite time evolution because it corresponds to certain causal processes by which information is propagated in the spin network. The rule by which the amplitude is specified satisfies a principle of causality, by which information about an element of a successor network only depends on a small region of the its predecessor. There are then discrete analogues of light cones and causal structures in the theory. Because the geometry associated to the spin networks is discrete[10], the process by which information at two nearby nodes or edges may propagate to jointly influence the successor network is finite, not infinitesimal.

In ordinary quantum systems it is usually the case that there is a non-vanishing probability for a state to evolve to an infinite number of elements of a basis after a finite amount of time. The procedure we've just described then differs from ordinary quantum mechanics, in that there are a finite number of possible successors for each basis state after a finite evolution. The reason is again causality and discreteness: since the spin networks represent discrete quantum geometries, and since information must only flow to neighboring sites of the graph in a finite series of steps, at each elementary step there are only a finite number of things that can happen.

We may note that if the Hilbert space is not constructible, we cannot ask if this procedure is unitary. But we can still normalize the amplitudes so that the sum of the absolute squares of the amplitudes to evolve from any spin network to its successors is unity. This gives us something weaker than unitarity, but strong enough to guarantee that probability is conserved locally in the space of configurations.

To summarize, in such an approach, the amplitude to evolve from the initial spin network W_0 to any element of S^N [W_0] , for large finite N is computable, even if it is the case that the spin networks cannot be classified so that the basis itself is not finitely specifiable. Thus, such a procedure gives a way to do quantum physics even for cases in which the Hilbert space is not constructible.

We may make two comments about this form of resolution of the problem. First, it necessarily involves an element of time and causality. The way in which the amplitudes are constructed in the absence of a specifiable basis or Hilbert structure requires a notion of successor states. The theory never has to ask about the whole space of states, it only explores a finite set of successor states at each step. Thus, a notion of time is necessarily introduced.

Second, we might ask how we might formalize such a theory. The role of the space of all states is replaced by the notion of the successor states of a given network. The immediate successors to a graph Gamma_0 may be called the adjacent possible[7]. They are finite in number and constructible. They replace the idealization of all possible states that is used in ordinary quantum mechanics. We may note a similar notion of an adjacent possible set of configurations, reachable from a given configuration in one step, plays a role in formalizations of the self-organization of biological and other complex systems[7].

In such a formulation there is no need to construct the state space a priori, or equip it with a structure such as an inner product. One has simply a set of rules by which a set of possible configurations and histories of the universe is constructed by a finite procedure, given any initial state. In a sense it may be said that the system is constructing the space of its possible states and histories as it evolves.

Of course, were we to do this for all initial states, we would have constructed the entire state space of the theory. But there are an infinite number of possible initial states and, as we have been arguing, they may not be classifiable. In this case it is the evolution itself that constructs the subspace of the space of states that is needed to describe the possible futures of any given state. And by doing so the construction gives us an intrinsic notion of time.


We must emphasize first of all that these comments are meant to be preliminary. Their ultimate relevance rests partly on the issue of whether there is a decision procedure for spin networks (or perhaps for some extension of them that turns out to be relevant for real quantum gravity [17].). But more importantly, it suggests an alternative type of framework for constructing quantum theories of cosmology, in which there is no a priori configuration space or Hilbert space structure, but in which the theory is defined entirely in terms of the sets of adjacent possible configurations, accessible from any given configuration. Whether such formulations turn out to be successful at resolving all the problems of quantum gravity and cosmology is a question that must be left for the future. (We may note that the notion of an evolving Hilbert space structure may be considered apart from the issues discussed here[23].).

There are further implications for theories of cosmology, if it turns out to be the case that their configuration space or state space is not finitely constructible. One is to the problem of whether the second law of thermodynamics applies at a cosmological scale. If the configuration space or state space is not constructible, then it is not clear that the ergodic hypothesis is well defined or useful. Neither may the standard formulations of statistical mechanics be applied. What is then needed is a new approach to statistical physics based only on the evolving set of possibilities generated by the evolution from a given initial state. It is possible to speculate whether there may in such a context be a ``fourth law" of thermodynamics in which the evolution extremizes the dimension of the adjacent possible, which is the set of states accessible to the system at any stage in its evolution[7].

Finally, we may note that there are other reasons to suppose that a quantum cosmological theory must incorporate some mechanisms analogous to the self-organization of complex systems[6]. For example, these may be necessary to tune the system to the critical behavior necessary for the existence of the classical limit[24,14]. This may also be necessary if the universe is to have sufficient complexity that a four manifolds worth of spacetime events are completely distinguished by purely relational observables[4,6]. The arguments given here are complementary to those, and provide yet another way in which notions of self-organization may play a role in a fundamental cosmological theory.


We are indebted to Julian Barbour and Fotini Markopoulou for conversations which were very helpful in formulating these ideas. We would also like to thank John Baez, Louis Crane, Lou Kauffman and Adrian Ocneanu for discussions and help concerning the mathematical questions about classifiability. This work was supported by NSF grant PHY-9514240 to The Pennsylvania State University and a NASA grant to The Santa Fe Institute. Finally, we are grateful to the organizers of the conference on Fundamental Sources of Unpredictability for providing the opportunity of beginning discussions that led to this paper.


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[5]J. B. Barbour,to appear in the {\it Procedings of the NATO Meeting on the Physical Origins of Time Asymmetry] eds. J. J. Halliwell, J. Perez-Mercader and W. H. Zurek (CAmbridge University Press, Cambridge, 1992); {\it On the origin of structure in the universe] in {\it Proc. of the Third Workshop on Physical and Philosophical Aspects of our Understanding of Space and Time] ed. I. O. Stamatescu (Klett Cotta); {\it Time and the interpretation of quantum gravity] Syracuse University Preprint, April 1992.

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From: Murray Gell-Mann
Submitted: 3/20/1997

Jim Hartle, in papers published during the last few years and based in part on our collaborative work on quantum mechanics, has greatly clarified the issue of how time is to be treated in quantum cosmology. The authors whose work you are distributing seem to be unaware of this clarification, although they do refer to some other papers by Jim. It may well be that Hartle's approach makes it unnecessary to invent all this ponderous machinery. But then, of course, your communication refers to Quantum Cosmology, and who knows what is needed there?

Best regards.

MURRAY GELL-MANN is a theoretical physicist; Robert Andrews Millikan Professor Emeritus of Theoretical Physics at the California Institute of Technology; winner of the 1969 Nobel Prize in physics; a cofounder of the Santa Fe Institute, where he is a professor and cochairman of the science board; a director of the J.D. and C.T. MacArthur Foundation; one of the Global Five Hundred honored by the U.N. Environment Program; a member of the President's Committee of Advisors on Science and Technology; author of The Quark and the Jaguar: Adventures in the Simple and the Complex (1994).

From: Julian Barbour
Submitted: 3/20/1997

Lee finds my idea that time flow does not exist "scary." This was his comment after a sleepless night following my explanation to him of the notion of time capsules: that each experienced instant of time is really a completely self contained entity, a time capsule, that gives the impression it is embedded in a flow if time solely because it is structured in a very special way. The instant is not in time. Time is in the instant. There are just lots of different nows, and the ones we experience happen to be structured in such a way that they lead us to believe in an external flow of time.

I was led to this radical solution to the problem of time in quantum gravity, which was first revealed in its full depth in 1967 by Bryce DeWitt, because all other attempts to reconcile our strong impression of the flow of time flow with the static nature of DeWitt's wave function of the universe seemed to me to smuggle in time through one illicit tacit assumption or another. The basic idea of time capsules is explained in qualitative terms in my paper in the CUP paperback Physical Origins of Time Asymmetry (edited by Halliwell, Perez-Mercader, and Zurek) and in more detail in Classical and Quantum Gravity, 11, 2853-2897 (1994).

It is very good that Stu Kauffman and Lee are making this serious attempt to save a notion of time, since I think the issue of timelessness is central to the unification of general relativity with quantum mechanics. The notion of time capsules is still certainly only a conjecture. However, as Lee admits, it has proven very hard to show that the idea is definitely wrong. Moreover, the history of physics has shown that it is often worth taking disconcerting ideas seriously, and I think timelessness is such a one.

At the moment, I do not find Lee and Stu's arguments for time threaten my position too strongly. Starting with Stu's interesting point that it seems effectively impossible to specify all at once and in advance "all the possible tasks or niches or properties that might give an organism selective advantage," I do not think a similar difficulty applies to my basic notion, which is that of the space of all possible relative configurations the universe might have. The set of possible configurations stands in correspondence, not with a set of selectively advantageous tasks, but with the set of all possible living organisms. There is a very beautiful description of The corresponding space of possible organisms is Dawkins's The Blind Watchmaker, which reminded me strongly of the configuration space of general relativity (and hence of quantum gravity in the so-called metric representation) when I read that book a year or two ago. Since living organisms are essentially defined by their DNA, I do not think that the difficulties in defining a space of tasks applies to the definition of the space of possible living organisms or the space of possible relative configurations of the universe.

Turning to the difficulties Lee raises, two points can be made: first, he suspects there may be difficulties in specifying in advance the space of quantum states of quantum gravity. However, this presupposes that the quantum mechanics of the universe as a whole has the same basic structure as the quantum mechanics of the subsystems of the universe, in particular a Hilbert space of quantum states. I doubt whether this is the case, and I think Lee must be sympathetic to my doubt, since a central argument of his forthcoming The Life of the Cosmos is our shared conviction that the physics of the complete universe will almost certainly be very different from the physics of its subsystems. My timeless proposal for quantum gravity only relies on a configuration space (which is like the space of possible vector potentials modulo gauge transformations in QED), not necessarily a quantum state space of the kind Lee and his collaborators are seeking (which is somewhat analogous to the Fock space of quantum field theory). I do not think the difficulty of specifying vector-potential configurations in a finite time (as opposed to the finite time needed to specify elements of Fock space) in any way undermines that way of looking at QED.

Second, it is not entirely clear to me that spin-networks, which (as Lee is careful to point out) have to be embedded in a prior three-dimensional space (a bare manifold), so that an uncomfortable amalgam of the definitely concrete (the network) and an intangible (the manifold) arises, represent the final definitive framework of quantum gravity. Personally, I would be much happier with the entirely intrinsically defined structures that Fotini Markopoulou advocates, which seem to me completely in the spirit of Riemann's conjecture that the metrical properties of space arise from a discrete foundation. But then there is no problem with specification, and a timeless arena for quantum physics of the universe will be available. Of course, one might still try for a timelike causal evolution within such an arena, as Fotini advocates. I shall be interested to see how that idea develops, since my strong conviction is that the deep structure of general relativity considered as a dynamical theory indicates a totally different timeless structure for the quantum theory. My guess is that Fotini and Lee will seek in vain, but no doubt there will be fun and interest along the way. I think a theory of space is more likely to arise from such work than a theory of causal evolution in time.

Julian Barbour

JULIAN BARBOUR, a theoretical physicist, is the author of Absolute or Relative Motion? The Discovery of Dynamics and The Frame of Mind (Cambridge) and the editor of Mach's Principle: From Newton's Bucket to Quantum Gravity (Birkhauser).

Copyright ©1997 by Edge Foundation, Inc.


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