Stuart A. Kauffman
Chapter 10: A Coconstructing Cosmos?
Oxford University Press, 2000, 243-266
FROM BIOSPHERES to the cosmos? Yes, because they may share general themes. The major enquiry of Investigations has concerned autonomous agents and their coconstruction of biospheres and econospheres whose configuration spaces cannot be finitely prestated. These themes find echoes in thinking about the cosmos as a whole. But abundant caution: I am not a physicist, the problems are profound, and we risk nonsense.
Whatever the risk, two facts are true. First, since the big bang our universe has become enormously complex. Second, we do not have a theory for why the universe is complex. Equally unarguably, the biosphere has increased in molecular diversity over the past four billion years, just as the standing diversity of species has increased. And equally unarguably, the econosphere has become more complex over the past few million years of hominid evolution. We know this with confidence. If we lack a theory, it is not because the staggering facts of increasing diversity and complexity that stare us in the face do not deserve a theory to account for them.
But we have seen hints of such a theory in the coconstruction of biospheres and econospheres by the self-consistent search of autonomous agents for ways to make a living, the resulting exapting novel ways of making livings, the fact that new adjacent niches for yet further new species grow in diversity faster than the species whose generation creates those new adjacent possible niches, and the search mechanisms to master those modes of being. We have seen a glimmer of something like
a fourth law, a tendency for self-constructing biospheres to enlarge their workspace, the dimensionality of their adjacent possible, perhaps as fast, on average, as is possible - glimmers only, not yet well-founded theory nor well-established fact. But glimmers often precede later science.
Consider again how the chemical diversity of the biosphere has become more diverse in the past four billion years, urged into its adjacent possible by the genuine chemical potential from the chemical actual into the adjacent possible, where the actual substrates exist and the adjacent possible products do not yet exist. Each time the molecular diversity of the biosphere expands, the set of adjacent possible reactions expands even faster. Recall our simple calculation that for modestly complex organic molecules, any pair of molecules could undergo at least one two substrate-two product reaction. But then the diversity of possible reactions is the square of the diversity of chemicals in the system. As the diversity of molecular species increases, there are always proportionally more novel reactions into the adjacent possible. If we take the formation of a chemical species that has never existed in the biosphere, or perhaps the universe, as a breaking symmetry, then the more such symmetries are broken, the more ways come into existence by which yet further symmetries may be broken.
And the chemical case makes clear the linking of the flows of matter and energy in this sprawling chemical diversity explosion. Many such reactions will link exergonic and endergonic processes. As this occurs, energy is pumped from the exergonic partner into the products requiring endergonic synthesis. These products - the chemical diversity in the bark of a redwood tree, for example - take their place in the chemical actual, poising the biosphere, and thus the universe, for its next plunge into the chemical adjacent possible.
Consider again equilibrium statistical mechanics. At its core, statistical mechanics relies on the same kind of statistical argument as does the flipping of a fair coin 10,000 times. We all understand that distributions of roughly 5,000 heads and 5,000 tails are far more probable macrostates than distributions with all heads or all tails. Now consider the general argument I have made that as molecular diversity increases, the diversity of reactions increases even faster, and that there is a genuine chemical potential from the actual into the adjacent possible. And consider again the general argument made just above that the greater the diversity of molecular species and reactions, the more likely the coupling of exergonic and endergonic reaction pairs driving the endergonic synthesis of new adjacent possible molecules that poise the system to advance again into the next adjacent possible. While the detailed statistical form of these chemical reaction graphs are not yet known, they too smell of “law.” As in the case of fair coin flips and equilibrium statistical mechanics, it is as if here again the mathematical structure compels the consequent behavior of matter and energy. In the case of the nonergodic and non-
equilibrium chemical flux into the adjacent possible, the universe is busy diversifying itself into myriad complexity.
The universe is enormously complex, and we don’t really yet know why. May there be new ways of thinking of the cosmos itself? If a mere glimmer can be acceptable as potentially useful early science, then the burden of this chapter is to suggest perhaps, yes.
It is not obvious, in fact, that the universe should be complex. One can imagine universes governed by general relativity that burst briefly into big bang being, then recollapse in a rapid big crunch within parts of a second or a century. Alternatively, one can imagine universes governed by general relativity that burst into big bang being and expanded forever with no further complexity than hydrogen and helium and smaller particles in an open and ever-expanding dark, cold vastness.
Here we are, poised, it seems (but see below) between a universe that will expand forever and a universe that will eventually ease into gentle contraction, then rush to a big crunch.
Our fundamental theories in physics, and just one level up, biology, remain un-united. Einstein’s austere general relativity, our theory of space, time, and geometry on the macroscale, floats untethered to quantum mechanics, our theory of the microscale, seventy-five years after quantum mechanics emerged in Schrodinger’s equation form for wave mechanics. Theoretically apart, general relativity and quantum mechanics are both verified to eleven decimal places by appropriate tests. But it remains true that general relativity and quantum mechanics remain fitfully fit, fitfully un-united. And Darwin’s view of persistent coevolution remains by and large unconnected with our fundamental physics, even though the evolution of the biosphere is manifestly a physical process in the universe. Physicists cannot escape this problem by saying, “Oh, that’s biology.”
Why the universe is complex rather than simple is, in fact, beginning to emerge as a legitimate and deep question. In the past several years, I have had the pleasure to come to know Lee Smolin, whose career is devoted to quantum gravity and cosmology. Most of what I shall write in this chapter reflects my conversations and work with Lee and his colleagues, who have been wonderfully welcoming to this biologist. Sometimes outsiders can make serious contributions. Sometimes outsiders just make damned fools of themselves.
Caveat lector, but I will continue.
In Smolin’s book, The Life of the Cosmos, he raises directly the question of why the universe is complex. Current particle physics has united three of the four fundamental forces - the electromagnetic, weak, and strong forces called the “standard
model.” With general relativity, which deals with the remaining force, gravity; this provides a consistent framework. Particle physics plus general relativity have altogether some twenty “constants” of nature, which are parameters of the standard model and general relativity, such as the value of Planck’s constant, h; the fine structure constant, that is, the ratio of the electron rest mass to proton rest mass; the gravitational constant, g; and so forth. Smolin puts approximate maximum and minimum bounds on these twenty constants and asks a straightforward question: In a twenty-dimensional parameter space ranging over the plausible values of these twenty constants, what volume of that parameter space is consistent with values of the constants that would yield a complex universe with stars, chemistry, and potentially, life?
Smolin’s rough answer is that the volume of parameter space for the constants of nature that would yield a complex universe are something like 10 raised to the minus 27th power. That is, a tiny fraction of the possible combinations of the values of the constants are consistent with the existence of chemistry and stars, as well as life. For the universe to be complex, the constants must be sharply tuned.
Smolin’s argument could be off by many orders of magnitude without destroying his central point: The fact that our universe is complex, based on our current theories of the standard model and general relativity, is surprising, even astonishingly surprising.
Many physicists have remarked upon this fine tuning of the constants.
There have been several responses to this issue, some raised prior to Smolin’s work. One is based on a view of multiple universes and the “weak anthropic principle”. This principle states that there exist multiple universes, but only those universes that were complex would sport life forms with the wit to wonder why their universe was complex. So the very fact that we humans are here to wonder about this issue merely means that we happen to be in one of the complex universes among vastly many universes. The argument is at least coherent. But it’s hard to be thrilled by this answer.
The “strong anthropic principle” goes further - indeed, too far - and posits that, for mysterious reasons, the universe is contrived such that life must arise to observe it and wonder at it. Few think the strong anthropic principle counts as science in any guise.
Smolin points out that there are two possible answers to the puzzle of the complexity of the universe. Either we will find a parameter-free description - a kind of supertheory - that yields something like our complex universe with its constants, or some historical process must pick those constants. Smolin proposes the possibility of “cosmic natural selection?’ Here, daughter universes are born of black holes. Universes with more black holes will have more daughter universes. Given minor heritable variation in the constants of the laws of the daughter universes, cosmic natural selection will select for universes whose constants support the for-
mation of a near-maximum number of black holes. He then argues that on very crude calculations most alterations in the known constants would be expected to lower the number of black holes. Lee points out that his theory is testable, for example, by deducing that our constants correspond to near-maximum black hole production, and that his theory has not been ruled out yet.
I confess I am fond of and admire Lee Smolin a great deal, but I don’t like his hypothesis. Why? Well, preferably, one would like a theory that had the consequence that any universe would be complex like ours and roughly poised between expansion and contraction. We have no such theory at present, of course. The remainder of this chapter discusses ideas and a research program that just might point in this direction.
As a start, we can begin with the most current view of the large-scale structure and dynamics of the universe. The most recent evidence suggests that on a large enough scale the universe is flat, the matter distribution is isotropic, and - a recent surprise - the universe may be expanding at an accelerating rate. This latest result, if it holds true, contravenes the accepted view of the past several decades that the rate of expansion of the universe has been gradually slowing since the big bang. The hypothesis that the universe is exactly poised between persistent expansion and eventual collapse has held that the rate of expansion of the universe will gradually slow, but never stop.
One way to explain a persistent accelerating expansion of a flat universe is to reintroduce Einstein’s “cosmological constant” into the field equations of general relativity. A positive cosmological constant expresses itself as a repulsive force between masses that increases with the distance between those masses. Some physicists think that a positive cosmological constant must be associated with some new source of energy in free space. The source of such an energy is currently unknown.
Before turning to the huge difficulties of quantum gravity, we should review the fundamental mystery of quantum mechanics. Most readers are familiar with the famous two-slit experiment, which exhibits the fundamental oddness of quantum interference. Feynman, in his famous three-volume lectures on physics, gives the mystery as simply as possible: We begin with a gun shooting bullets. The bullets pass through one of two holes in a metal plate and fly further on, landing on a flat layer of sand in a box. Bullets passing through either hole may be deflected slightly by hitting the walls of the hole. Thus, in the sandbox behind the metal plate, we would expect, and actually would find, two mounds of bullets. Each mound would be centered on the line of flight of the bullet from the gun through the corresponding hole to the sandbox, with a “Gaussian” or normal bell-shaped distribution of bullet densities falling away from the peak of each mound.
When we use monochromatic light rather than bullets, we note the following: If the light hits the sandbox, changed into a photon-counter surface, we find that the size of the energetic impact is the same each time a photon hits the surface. Photons of a given wavelength have a fixed energy. A photon either is recorded at a point on the surface or not. Whenever one is recorded, the full parcel of energy has been detected at the surface. Now if only one hole is open, one gets the Gaussian mound result. Most photons pass through the hole unscathed and arrive in a straight line at the photon-counter surface. A Gaussian distribution peaked at that center is present because some photons are deflected slightly by the edges of the hole.
But if two holes are open, then one gets the famous interference pattern of light and dark interfering circles spreading from the centers on the photon-counter surface that were the peaks of the mounds seen when hole 1 or hole 2 was open. Of course, as Feynman points out, there is no way to account for this oddness in classical physics.
Quantum mechanics was built to account for the phenomenon. The Schrodinger equation is a wave equation. The wave that propagates from the photon gun is an undulating spherically spreading wave of probability “amplitude?’ The amplitude at any point in space and time is the square root of the probability that the photon will be found located at that point. To obtain the actual probability, the amplitude must be squared.
A central feature of Schrodinger’s equation is its linearity. If two waves are propagating, the sum and differences of those waves are also propagating. It is the essential linearity of quantum mechanics that makes the next puzzle, the link from quantum to classical worlds, so difficult. For a central puzzle of quantum mechanics becomes the relation between this odd quantum world of possible events, where the possibilities can propagate, but never become actual, and the classical world of actual events.
A variety of approaches to the liaison between the quantum and classical realms exist. The first is the “Copenhagen interpretation,” which speaks of the “measurement event;’ when the quantum object interacts with a macroscopic classical object, the measuring device, and a single one of the propagating possibilities becomes actual in the measurement event, thereby “collapsing” the wave function. A second approach is the Everett multiworld hypothesis, which asserts that every time a quantum choice happens, the universe splits into two parallel universes. No one seems too happy with the Everett interpretation. And few seem very sure what the Copenhagen interpretation’s collapse of the wave function might really mean.
Meanwhile, there are two other long-standing approaches to the link between the quantum and classical worlds. The first is Feynman’s famous sum over all possible trajectories, or histories, approach. In quantum mechanics, we are to imagine a given possible pathway of the photon from the photon gun through the screen with the two slits to the photon-counting surface. For each pathway, there is a well-
defined procedure to assign an “action.” This action can be thought of as having an amplitude and a phase, and the phase rotates through a full circle, 2 pi, many times along the pathway. According to the Feynman scheme, classical trajectories correspond to quantum pathways possessing minimal action.
Consider, says Feynman, all the pathways that start at the photon gun and end up at the same point on the photon-counting surface. Nearly parallel, nearly straight-line pathways, have nearly the same action. So when those pathways interact, they have nearly the same phase, and their interaction yields constructive interference, which tends to build up amplitude. Thus, pathways that are near the classical pathway interact constructively to build up amplitude. By contrast, quirky crooked pathways between the photon gun and the same point on the counter screen have very different actions, hence very different phases, and interact destructively, so their amplitudes tend to cancel. The classical pathway, therefore, is simultaneously the most probable pathway over the sum of histories of all possible pathways, and the pathway that requires the least action.
The result is beautiful, but has two problems. First, Feynman assumes a continuous background space and time in his theory. Quantum gravity, as we will see, cannot make that assumption in the first place . Rather space, or geometry, is a discrete, self-constructing object on its own. Thus, achieving a smooth space and time is supposed to be a consequence of an adequate theory of quantum gravity. If Feynman’s sum over histories must assume a smooth background space and time, then it cannot as such be taken as primitive in quantum gravity. Second, granting a continuous background space and time, Feynman’s sum over all histories still only gives a maximum of the amplitude for the photon to travel the classical pathway, it never gives an actual photon arriving at the counting surface. No more than any other does Feynman overcome the fundamental linearity of quantum mechanics. We still have to collapse the wave function. Despite these problems, Feynman’s results are brilliant, and at least we see a link between the classical and quantum worlds, if not yet actual photons striking counters.
But there is an alternative approach to the link between the quantum and the classical worlds. This possible approach is based on the well-established phenomenon of “decoherence.” Decoherence arises when a quantum-coherent Schrodinger wave is propagating and the quantum system interacts with another quantum system having many coupled variables, or degrees of freedom. The consequence can be that the Schrodinger wave function of the first quantum system becomes entangled in very complex ways with the other complex quantum system, which may be thought of as the environment. Rather like water waves swirling into tiny granular nooks and crannies along a rugged fractal beach, the initial coherent Schrodinger equation representing the initial quantum system swirls into tiny and highly diverse patterns of interaction with the quantum system representing the environment. The consequence of this intermixing is decoherence.
To understand the core of decoherence, one must understand that the exhibition of interference phenomena, the hallmark of quantum mechanics noted in the double-slit photon experiment, requires that literally all the propagating possible pathways in Feynman’s sum over histories that are to arrive at each point on the photon-counter surface, do in fact arrive at that point. If some fail to arrive, the sum over all histories fails. In effect, if some of the phase information, the core of constructive and destructive interference, has been lost in the maze of interactions of the quantum system with its environment, then that phase information cannot come to be reassembled to give rise to quantum interference.
Decoherence is accepted by most physicists. For example, in attempts to build quantum computers that can carry out more than one calculation simultaneously due to the linear features of quantum mechanics, actual decoherence is currently a technical hurdle in obtaining complex quantum calculations.
Decoherence, then, affords a way that phase information can be lost, thereby collapsing the wave function in a nonmysterious fashion. Thus, some physicists hope that decoherence provides a natural link between the quantum and classical realms. Notable among these physicists are James Hartle and Murray Gell-Mann, whose views can be found in Gell-Mann’s The Quark and the Jaguar. In essence, Hartle and Gell-Mann ask us to consider “the quantum state of the universe” and all possible quantum histories of the universe from its initial state. Some of these histories of the universe may happen to decohere. Hartle and Gell-Mann argue that the decoherent histories of the universe, where true probabilities can be assigned, rather than mere amplitudes, correspond to the classical realm. Others have argued that decoherence itself can be insufficient for classical behavior.
It is striking that there appear to be two such separate accounts of the relation between the quantum and classical worlds, Feynman’s sum over histories in a smooth background space-time and decoherence. For an outsider, it is hard to believe that both can be correct unless there is some way to derive one from the other. I will explore one such possibility below. In particular, I will explore the possibility that decoherence of quantum geometries is primary and might yield a smooth space-time in which Feynman’s account is secondarily correct.
I turn next to an outsider’s grounds for doubts about some of the core propositions of quantum mechanics. Roland Omnes, in The Interpretation of Quantum Mechanics, is at pains to argue that decoherence is the plausible route to classicity. In his discussion, two major points leap to attention. The first concerns the concept of an elementary predicate in quantum mechanics. Quantum mechanics is stated in the framework of Hilbert spaces, which are finite or infinite-dimensional complex spaces, that is, spaces comprised of finite or infinite vectors of complex numbers. In
effect, an elementary predicate is a measurement about an “observable” that returns a value drawn from some set of possible values. And so the first striking point is Omnes’ claim that all possible observables can be stated in Hilbert space. The second striking point is Omnes’ claim that some observables cannot be observed.
The first point is striking because it is not at all clear that all possible observables can be finitely stated in Hilbert space. My issue here is precisely the same as my issue with whether or not the configuration space of the biosphere is finitely prestatable. As I argued above, there does not seem to be a finite prestatement of all possible causal consequences of parts of organisms that may turn out to be useful adaptations in our or any biosphere, which arise by exaptation and are incorporated in the ongoing unfolding exploration of the adjacent possible by a biosphere.
In quantum mechanics, an observable corresponds to a mathematical operator that “projects out” the subspace of Hilbert space corresponding to the desired observable in a classical measurement context that allows detection of the presence or absence of the observable. But the biosphere is part of the physical universe, and the exapted wings of Gertrude the flying squirrel are manifestly observables, albeit classical observables. If we cannot finitely prestate the observable, “Gertrude’s wings,” then we cannot finitely prestate an operator on Hilbert space to detect the presence or absence of Gertrude’s wings. In short, there seems to be no way to pre-specify either the quantum or classical variables that will become relevant to the physical evolution of the universe.
It turns out that the above issues may bear on the problem of time in general relativity, as Lee Smolin realized from our conversations and as I return to shortly.
Now, the second point. Omnes follows up on it. An observable requires a measuring device. There are some conceivable observables for which the measuring device would be so massive that it would, of itself, cause the formation of a black hole. Thus, no information resulting from the measurement could be communicated to the outside world beyond the black hole.
A strange situation; even if we could finitely prestate all possible observables, only some observables can manage to get themselves observed in the physical universe.
What shall we make of conceivable observables that cannot, in principle, be observed? More important, it seems to this outsider, is the following: If observation happens by coupling a quantum system to some other system, quantum or classical, whereby decoherence occurs and, in turn, classicity arises because loss of phase information precludes later reassembly of all the phase information to yield quantum interference, then there appears to be a relation between an observable being observed and the very decoherence by which something actual arises from the quantum amplitude haze.
If that is correct, then only those observables that can get themselves observed can, in principle, become actual. More, it begins to seem imperative to consider the
specific possible pairs of quantum systems that can couple and decohere, for only thereby can such pairs become classical via decoherence. This begins to suggest preferred histories of the universe concerning such comeasuring pairs of quantum systems. Preferentially, those comeasuring pairs of quantum systems that decohere and become classical will tend to accumulate, due to the irreversibility of classicity. Thereafter quantum-classical pairs of systems that cause decoherence of the quantum system will preferentially accumulate into classicity.
If comeasuring yields classicity, and classicity is irreversible, the classical universe begins to appear to coconstruct itself. In particular, it is generally accepted that bigger systems, that is, systems with more coupled degrees of freedom, deco-here more rapidly when they interact than smaller systems. If so, this begins to refine the suggestion of preferred histories of the universe concerning comeasuring pairs of quantum systems toward a preference for the emergence of classical diversity and complexity: If quantum systems with more coupled degrees of freedom irreversibly decohere more rapidly into classical behavior when they interact than smaller, simpler systems, then the kinetics of decoherence should persistently favor the irreversible accumulation of bigger, more complex quantum systems, rather than of smaller, simpler, quantum systems.
Chemistry should be an example. Molecules are quantum objects, yet flow into the chemical adjacent possible. The adjacent possible explodes ever more rapidly as molecular diversity, and hence molecular complexity, increases. Reactions of complex molecules are precise examples of the couplings of quantum systems whereby decoherence can happen. Decoherence presumably happens more rapidly among complex reacting molecules than among very simple molecules or the same total number of mere atoms, nucleons, and electrons in the same total volume. This hypothesis ought to be open to experimental test. If confirmed, the flow of possible quantum events into the chemical adjacent possible should, in part, be made irreversible by the decoherence of complex molecular species as they couple and react with one another.
If the general property obtains that complex quantum entities can couple to and interact with other complex quantum entities in more ways than can simple systems and that the number of ways of coupling explodes faster than the diversity of entities, and thus faster than the complexity of those quantum objects, then decoherence should tend to lead to favored pathways toward the accumulation of complex classical entities and processes. I return to these themes below.
To my delight, I soon found myself coauthor on a paper with Lee Smolin concerning the problem of time in general relativity. Lee had done the majority of the
work, but had taken very seriously my concern that one cannot finitely prestate the configuration space of a biosphere.
In general relativity, space-time replaces space plus time. A history becomes a “world-line” in space-time. But that world-line is a geometrical object in space-time. Time itself seems to disappear in general relativity, to be replaced by the geometrical world-line object in space-time.
But argued Lee, with my name appended, general relativity assumes that one can prestate the configuration space of a universe. In that prestated configuration space, a world-line is, indeed, merely a geometrical object. What if one cannot prestate the configuration space of the universe? If so, one cannot get started on Einstein’s enterprise, even if general relativity is otherwise correct. As concrete examples, Lee pointed out that four-dimensional manifolds are not classifiable.
How might one do physics without prestating the configuration space of the universe? Lee postulated use of spin networks, as described below, with the universe constructing itself from some initial spin network. In this picture, time and its passage is real. If there can be a framework in which time enters naturally, and possibly there is a natural flow of time, or an arrow of time preferentially from past to future, then, among other possible consequences, we may be able to break the matter-antimatter symmetry, for antimatter can be stated as the corresponding matter flowing backward in time. Break the symmetry of time in fundamental physics and you may buy for free the breaking of the symmetry between matter and antimatter. If time flows preferentially from past to future, matter dominates antimatter. That would be convenient since matter does dominate antimatter, and no one knows just why.
We will head in this direction.
For the sixty years following 1926 and the emergence of matrix mechanics and the Schrodinger formulation of quantum mechanics, scant progress was made on quantum gravity. Now, in the past decade or so, there are two alternative approaches, string theory and spin networks. Of the two, string theory has captured the greatest attention. I discuss it briefly below.
Spin networks were invented by Roger Penrose three decades ago as a framework to think about a quantized geometry. Quite astonishingly, spin networks appear to have emerged from a direct attempt to quantize general relativity by Carlo Rovelli and Lee Smolin. In outline, part of the tension between quantum mechanics and general relativity lies in the very linearity of quantum mechanics and the deep nonlinearity of general relativity.
Building on work of Astekar and his colleagues, Rovelli and Smolin proceeded directly from general relativity along somewhat familiar pathways of canonical
quantization. In outline, general relativity is based on a metric tensor concerning space-time. The metric tensor is a 4 x 4 symmetric tensor. It turns out that this tensor yields seven constraint equations. The solutions of six of the seven have turned out to be spin networks. The solution of the seventh equation would yield the Hamiltonian function, hence the temporal unfolding, of spin networks in a space x time quantum gravity.
Spin network theories can be constructed in different dimensions. The two most familiar are for two spatial and one temporal or three spatial and one temporal dimension. We will concern ourselves with three plus one spin networks for concreteness. The minimal objects in a spin network are discrete combinatorial objects that constitute first a tetrahedron, with four vertices and four triangular faces. A tetrahedron represents a primitive discrete unit of geometry, or space. Integer-valued labels are present on the edges and vertices of these tetrahedra. The labels on the edges represent spin states. The labels on the vertices represent “intertwinors” and concern how edges entering a vertex are connected to one another into and out of the vertex.
Analytic work has associated an area with a face of a tetrahedron and a volume with its volume. There is, at present, no way to represent the length of an edge connecting vertices. On the other hand, one can think of the integer values on the edges around a face of a tetrahedron as associated with the area of the tetrahedron, such that larger integers correspond to larger areas.
A geometry is built up by minimal moves, called “Pachner moves,” in which a given tetrahedron can give rise to daughter tetrahedra off each face. In addition, several tetrahedra can collapse to a single tetrahedron.
Thus we may picture an initial spin network, say, a single tetrahedron. In analogy with chemistry and combinatorial objects, the founder set of a chemical reaction graph, and the adjacent possible in the chemical reaction graph, we may consider the single initial tetrahedron as a founder set, gamma 0. Consider next all possible adjacent spin networks constructible in any single Pachner move. Let these first adjacent possible spin networks lie in an adjacent ring, gamma 1. In turn, consider all the spin networks constructible for the first time from the founder set in two Pachner moves, hence constructible for the first time in one Pachner move from the gamma-1 set of spin networks. Let this new set be the gamma-2 set of spin networks.
By iteration, we can construct a graph connecting the founder spin network with its i-Pachner move “descendants,” 2-Pachner move descendants,... N-Pachner move descendents.
Each spin network in each gamma ring represents a specific geometry, subject to the constraint that two spin network tetrahedra that share one triangular face must assign the same spin labels to the common edges, hence, the same area to the common face.
Changes in the values of spins on the edges that change the areas and volumes of the tetrahedra can be thought of as deforming the geometry so that it warps in different ways. However, it should be stressed that there is no continuous background space or space-time in this discrete picture. Geometry is nothing but a spin network, and a change in geometry is nothing but a change in the tetrahedral structure of the spin network by adding or deleting tetrahedra or by changing the spin values on the edges of tetrahedra.
Within quantum mechanics, there is an appropriate way to consider the discrete analogue of Schrodinger’s equation, namely a means over time of evolving amplitudes from an initial distribution. In particular, the appropriate means of evolving amplitudes concern what are called “fundamental amplitudes,” which specify initial and final values of the integer values on edges before and after Pachner moves.
Consider a given graph linking spin networks from an initial tetrahedron in gamma 0, outward as in a mandala, to all daughter networks in gamma 1, gamma 2,... gamma N, where N can grow large without limit.
I now describe one approach to thinking about quantum gravity and the emergence of a smooth large-scale geometry based on this mandala and on Feynman’s idea of a sum over all histories. Endow the spin networks throughout with the same fundamental amplitudes, thus, the same law propagating amplitudes applies everywhere in the spin network mandala. Begin with all amplitude concentrated in the initial spin network tetrahedron in gamma 0. In this vision, a unit of time elapsing is associated with a Pachner move, such as a move from gamma 0 to a point in gamma 1. With analogy to Feynman’s sum over all possible histories, consider the set of all pathways that begin at the initial tetrahedron in gamma 0 and end on a given specific spin network N time steps later, for N = 1000. That final spin network might lie in the gamma-0 ring, the gamma-1 ring, the gamma-2 ring, or any ring out to the gamma-N ring.
Here is a hopeful intuition that may prove true. If we consider the family of all histories beginning on gamma 0 and ending in a specific spin network in the gamma N = 1000 ring, those pathways must be very similar and few in number. By contrast, if we consider all pathways length 1000 that begin on the gamma-0 tetrahedron and end, 1000 steps later, on a specific spin network in the gamma-23 ring after wandering all over the spin network mandala graph, there may be many such pathways, and they can be very dissimilar. Now, during the amplitude propagation along any pathway, an action can be associated with each Pachner move, hence, we can, with Feynman, think about the constructive or destructive interference among the family of pathways 1000 steps long that begin on the gamma-0 tetrahedron and
end on any specific spin network. Then the hopeful intuition is that those pathways that begin on gamma 0 and end on a spin network member of the gamma N = 1000 ring in 1000 Pachner moves will have very nearly the same action, hence, show strong constructive interference. By contrast, those pathways that begin on the gamma-0 tetrahedron and end, 1000 Pachner moves later, on a specific spin network in the gamma-23 ring will have very different actions, hence, show strongly destructive interference.
If the constructive interference among the few pathways to ring N overwhelms any residual constructive interference in the inner rings - such as ring 23, due to the larger number of pathways from gamma 0 to gamma 23 - then the hopeful concept is that amplitude will tend to accumulate in the gamma-N ring. Then (goes the hope shared with Smolin) the neighboring spin networks in the gamma-N shell constitute nearly the same geometry and nearly the same action in the sum of histories reaching them, which begins to suggest that a smooth large-scale geometry might emerge.
For this line of theory to succeed, it is not actually necessary that amplitude preferentially accumulate in the outermost, gamma-N, ring. Rather it is necessary as N increases that there be some ring, M, where M is less than N but increases monotonically with N, such that a sufficiently large number of alternative pathways with sufficiently similar phase end on members of the M ring that constructive interference is maximum for members of the M ring. Further, it is necessary that as N increases and M increases, amplitude continue to accumulate on the Mth ring.
In short, the concept is that, via constructive and destructive interference as amplitudes propagate in the mandala, some large-scale smooth geometry will pile up amplitude, hence probability, and a smooth classical geometry will emerge. Here is at least one image of how a large-scale smooth geometry might emerge from spin networks and constructive interference.
At least three major caveats are required. First, no calculation has yet been carried out for such a model, so such a theory may not work. Second, Feynman’s sum over histories assumes a classical continuous space and time. It may be entirely invalid to attempt to use a sum over histories argument in this quantum geometry setting. Third, assuming we can use Feynman’s sum over histories, we still have possible quantum geometries, not an actual geometry.
Recall the puzzle, nay, the deep mystery, about what processes, if any, might have “chosen” the twenty constants in the standard model such that the universe happens, improbably, to be complex. To answer this deep mystery we have, at present, the anthropic principle, Lee Smolin’s concept of cosmic natural selection for black
hole density, and the hope to find the ultimate parameter-free theory that would not require multiple universes or a historical process.
With caveats, I now briefly describe a way that may be useful to begin to think about the emergence of the constants such that any universe would have a given set of constants.
Tuning the constants corresponds to tuning the laws of physics. Is there a way to imagine a self-tuning of a universe to pick the appropriate values of its constants, to tune its own laws? I think the answer may be yes. And if the following is wrong in detail, the pattern of thought may prove useful.
In the spin network mandala picture, a 15J symbol, present throughout the spin networks in the mandala, generates an analogue of Schrodinger’s equation, hence, the means to propagate amplitudes in the graph of spin networks. Thus, a change in a 15J symbol would correspond to changing the laws of physics about how amplitudes propagate.
Importantly, the fundamental amplitudes are an ordered listing of 15 integers, hence, there is a family of all possible fundamental amplitudes. Since each fundamental amplitude can be thought of as the “law” about propagating amplitudes among spin networks, Louis Crane pointed out that there is an infinite family of all possible laws to propagate amplitude among spin networks.
Thus, imagine an infinite stack of our spin network mandalas, in which each manadala is a graph from gamma 0, the tetrahedron, outward to gamma N, for N allowed to be arbitrarily large, of spin networks reachable in N steps by Pachner moves. The mandala members of the infinite stack of mandalas differ from one another only in the fundamental amplitudes, hence laws, that apply to each mandala. (I concede it may be necessary to have a means to encode in each manadala the given fundamental amplitudes that apply to that manadala.)
Now consider how amplitudes propagate in each mandala from an initial state with all the amplitudes concentrated in the gamma-0 tetrahedron. And consider any two mandalas whose fundamental amplitudes are minimally different. For some such adjacent mandalas with adjacent laws, the small change in the law may lead to a large change in how amplitudes propagate in the mandalas. For other pairs with minimal changes in the fundamental amplitudes or law, the way the amplitude propagates throughout the mandala may be very slight. Assuming this is true, one intuitively imagines that the total system is spontaneously drawn to those tuned values of the fundamental amplitude laws, where small changes in the laws make minimal changes in how amplitudes propagate.
A simple possible mechanism might accomplish this. Imagine a sum of histories from an initial gamma-0 tetrahedron in a mandala with some given fundamental amplitude laws (thereby the initial and boundary conditions are specified), where the pathways in that set of histories pass up and down the stack of mandalas such that the fundamental amplitude laws change, as does the spin network, and then
consider the bundle of all such histories that end on a given spin network in a given gamma ring with given, perhaps new, fundamental amplitude laws. In effect, this conceptual move allows there to be quantum uncertainty not only with respect to spin networks along histories, but also quantum uncertainty with respect to the law by which amplitude propagates.
Then one can imagine a sum over all histories that, by constructive interference alone, picks those pathways, hence fundamental amplitude laws, that minimize the change in the ways amplitudes propagate. Such pathways would have similar phase, hence, accumulate amplitude by constructive interference. Then, by mere constructive interference, one can hope that such a process would pick out not only the history, but also tune the law to the well-chosen fundamental amplitudes laws that maximized constructive interference. Hopefully, that constructive interference would pick out smooth large-scale geometries like classical flat or near flat space. In such a large-scale classical-like space and time, Feynman’s familiar sum over histories that minimizes a least action along classical trajectories would emerge as a consequence.
Smolin and I discuss this possibility in a second paper. I find the idea attractive as a research program because it offers a way in which a known process, constructive interference, modified to act over a space of geometries and laws simultaneously, chooses the law. It is, of course, rather radical to suppose that there is quantum uncertainty in the law, but it does not seem obviously impossible.
On an even grander scale, particle physicists build the standard model from an abstract algebra called SU(3) x SU(2) x U(i). One can imagine a similar research program that by constructive interference alone picks out the particles, constants, and laws of the standard model. Presumably, particles governed by sufficiently “nearby” laws would be able to interact, hence undergo constructive or destructive interference, thus picking the particles and the laws simultaneously.
There is a further interesting feature, for we appear to have in our mandala, or mandalas, a new arrow of time. Allow that at any step, any Pachner move can happen. Some moves add tetrahedra. A equal number delete tetrahedra. Yet the number of spin networks in ring N + i is larger than the number of spin networks in ring N. Statistically, there are more ways to advance into ring N + 1 than to retreat from ring N into ring N - 1. Other things equal, amplitude should tend to propagate outward from the gamma-0 tetrahedron. There is an analogy in chemical reaction graphs to the adjacent possible and the real chemical potential across the frontier from the actual to the adjacent possible.
But if so, time enters asymmetrically due to the graph structure of the spin network mandala. Then, statistically, time tends to flow in one direction, from simpler toward more complex spin networks into the ever-expanding adjacent possible.
String theory has gained very substantial attention as a potential “theory of everything;’ namely, a theory that might link all four forces and all the particles of the standard model into a single coherent framework. I do not write with even modest expertise on the subject. Nevertheless, it is possible that the concept of the law selecting itself via maximum constructive interference in a sum over all possible histories in a space of both spin networks and laws might possibly have relevance to string theory. The description of string theory that I give draws heavily on Brian Greene’s The Elegant Universe.
As is known qualitatively by many outside the confines of the physics community, string theory began by giving up the picture of fundamental particles as zero-dimensional, point particles. In its initial version, in the place of point particles, string theory posited one-dimensional strings that might be open, with two ends, or closed loops, with no free ends. Among the fundamental ideas of string theory is the idea that the different particles and the different forces can all be thought of as different modes of vibration of such strings. Because strings have finite length, string theory can hope to overcome the infinities that emerge when attempts are made to marry point particle quantum theories with general relativity in a continuous space-time. In effect, the finite length of the strings prevents consideration of space becoming infinitely curved at a point. Thus, string theory can dream of uniting quantum mechanics and general relativity, and it has, in fact, produced the entity carrying the gravitational force, the graviton, in a natural way.
Current string theory has gone beyond single-dimensional strings, and now considers two-or-higher-dimensional entities called M-branes. The rough present state of the art has shown that there are at least five one-dimensional string theories and M-brane theory. All of these theories appear to be linked as cousins of one another via various dualities among the theories.
String theories posit either eleven-or-fewer-dimensional space and time, with three of the spatial dimensions unfurled and large scale, corresponding to our familiar three-dimensional space. The remaining dimensions are imagined as curled up on the Planck length scale in what are called “Calabi-Yau” spaces, or more generally, compactified moduli. Compactification of an eleven-dimensional space and time can be thought of as a large-scale three-dimensional space and time, but with the additional dimensions curled up at each point in the large-scale three-dimensional space.
Calabi-Yau spaces can have different topologies. Consider as an analogy a long thin tube with two ends and a one-hole torus, like a donut. These two are topologically different. As a consequence, closed one-dimensional string loops can “live” on these surfaces in different ways. Thus, if you think of a string as a closed loop,
that loop might live on the long tube in two ways, either wrapped around the tube one or more times or not wrapped around the tube, but lying on the tube’s surface like a rubber band lying on a surface. By contrast, consider the torus. The closed string might wrap around the torus in either of two ways, through the hole or around the torus. In addition, the string loop might live on the surface of the torus without wrapping either dimension. Each of these different ways of being on the tube or torus and the corresponding modes of vibration constitute different particles and forces. Calabi-Yau spaces are more complex than the tube or torus, but the basic consequences are the same. Different Calabi-Yau spaces, or more generally, different compactified moduli, with different kinds of holes around which strings can wrap zero, one, or more times correspond to different laws of physics with different particles and forces.
Physicists have shown, furthermore, that one Calabi-Yau space can smoothly deform into another with a “gentle” tearing of space and time. Hence, the laws, forces, and particles can deform into one another in a space of laws, forces, and particles. Within current string theory, it appears that it is still not certain that there exists a Calabi-Yau space whose string or M-brane inhabitants would actually correspond to the known particles and forces, but hopes are high.
However, even if there is a Calabi-Yau space whose strings and M-branes do correspond to our known particles and forces, string theorists have the difficulty that it is not clear how the current universe happens to choose the correct Calabi-Yau space. The familiar ideas on this subject include the existence of a multiverse and the weak anthropic principle. For example, one could imagine Lee Smolin’s arguments for choices of Calabi-Yau spaces that lead to fecund universes with a near maximum of black holes, which are the birthplaces of still further universes.
The parallel between spin networks with different fundamental amplitude laws and the family of string and M-brane theories that can deform into one another is that in both theories we confront a family of theories having the property that different members of the family correspond to different particles, forces, and laws. In both cases, physicists do not at present have a theory to account for how, in this embarrassment of riches, our universe happens to pick the correct laws. I therefore make the suggestion that the same pattern of reasoning that I described above, a sum over histories of trajectories that vary both in configurations and in the laws, which maximizes constructive interference, might prove a useful approach. In the string theory context, one would consider a hyperspace of Calabi-Yau spaces, in which neighboring Calabi-Yau spaces would propagate amplitudes from the same initial condition in different ways. Presumably, somewhere in the hyperspace of Calabi-Yau spaces, small changes in Calabi-Yau spaces would yield small changes in how amplitudes propagate. For other locations in the hyperspace of Calabi-Yau spaces, small changes in the Calabi-Yau space would yield large
differences in how amplitudes propagate. In the hyperspace of Calabi-Yau spaces, where one Calabi-Yau space can deform into its neighbors, it should be possible to construct a sum over all histories of trajectories between an initial and final state in the same or different Calabi-Yau space, then seek such sums over histories that maximize constructive interference. The hope is that maximizing constructive interferences would pick out the Calabi-Yau space corresponding to our particles and forces. Presumably, this would occur in the region of the hyperspace of Calabi-Yau spaces, where small changes in the Calabi-Yau space yield the smallest changes in how amplitudes propagate. In short, maximization of constructive interference may be a useful principle to consider to understand how the universe chooses its laws.
String theorists recognize the need to confront further major problems. Most notably, string theory posits a background space-time in which strings and M-branes vibrate. But if string theory is to be the theory of everything, including space and time, then space and time cannot be assumed as a backdrop. Thus, a virtue of spin networks is that it affords the hope of a quantized geometry from the outset. On the other hand, particles and the three nongravitational forces have yet to be incorporated into a spin network picture.
However the universe picks its presumably quantum laws, somehow the classical realm emerges. I noted above that current theory sees two approaches to linking the quantum and classical realms. The first is based on Feynman’s sum over histories, but as a perturbative theory assumes a continuous background space-time and does not get rid of the linear superposition of possibilities that is the core of quantum mechanics and interference.
What of the second approach, decoherence? The reality of decoherence is established. If one is to take decoherence seriously, and also to consider geometry constructing itself, then presumably decoherence can apply to geometry as it constructs itself. What would it mean to apply decoherence to quantum gravity itself, to the vacuum, to geometry itself?
Well, to an outsider, the following seems possible. If we are to conceive of an initial spin network, say a tetrahedron, and all possible daughter spin networks, as well as all their possible daughter spin networks, propagating amplitudes on the mandala, then at any moment N steps away from moment 0, more than one geometry is possible - namely all those reachable in N Pachner moves.
We seem to confront the same problem we confront with quantum systems coupling to quantum systems, such as electrons coupling to organic molecules, or to classical systems, such as rocks. These quantum systems can decohere. Can
quantum geometries become coupled with one another or different parts of one quantum geometry become coupled, so to speak, and decohere?
Why not try the idea?
I now discuss one possible approach to this issue. The approach posits a quantum of action, h, to the generation of a tetrahedron, hence a Planck energy and thus a Planck mass to a tetrahedron, and decoherence setting in at a sufficient mass and size scale.
By use of an equation suggested by Zurek relating the decoherence timescale, Td, to the relaxation timescale, Tr, of the system, in which increasing mass and area increase the rate of decoherence in proportion to their product, it can be qualitatively shown (via sufficiently rough arguments) that geometry may well be thought of as decohering, and doing so on a length scale of about 10-15 cm, which is smaller than the Compton radius of the electron and even smaller than the radius of a nucleus.
Now, there are some interesting features of this rough calculation. First, if we begin with an initial tetrahedron of geometry, it can have four daughter tetrahedra. In turn, each daughter tetrahedron can have two or more daughter tetrahedra, hence, the initial spin network can grow exponentially in the number of tetrahedra before decoherence sets in. This is a clue that a purely quantum account might be given of an initial exponential expansion of a universe starting with a single tetrahedron. Thus, it might be possible to do without the “inflationary hypothesis” of exponential expansion of a classical space in the early moments after the big bang.
Second, an initial exponential expansion of geometry might overcome, as the inflationary hypothesis does, the particle-horizon problem in cosmology, in which we confront the puzzle of why parts of the universe that have been out of apparent causal contact since the big bang can be so similar. If the initial expansion is exponential, then slows to linear, as in the inflationary hypothesis or perhaps in this purely quantum approach, then the particle-horizon problem may disappear.
Third, a purely quantum exponential expansion over many orders of magnitude should presumably yield a flat power spectrum over many size scales for quantum fluctuations in the earliest small fraction of a second of the life of the universe prior to the end of this exponential expansion when decoherence of geometry occurs.
Fourth, we must consider when geometries decohere whether there may be geometries that are the slowest to decohere. If different parts of a single spin network geometry can become coupled, it is natural to assume that flat parts might decohere more slowly than distorted parts of that geometry. Intuitively, phase information can get lost more readily when two lumpy parts of a geometry couple
than when two flat parts of a geometry couple. Of course, an explicit model exploring this is badly needed and entirely missing, but in its absence, let’s make the assumption. Given that, then an initial exponential explosion of flat and warped geometry occurs until decoherence sets in on a length scale of something like 10-15 cm. At this point, flat geometry “wins” because it decoheres most slowly. Hence, as soon as decoherence of geometry sets in, space tends to be flat in the absence of matter.
But even after decoherence sets in, geometry is busy all the time trying to build geometry exponentially and everywhere, while simultaneously decohering. Now an interesting feature of the Td/Tr equation alluded to above is that whatever the exponential rate of expansion of geometry may be per Planck time unit, the exponential rate of decoherence, Td, which grows as the mass times the size scale squared of the geometry, increases, until eventually the exponential rate of formation and exponential rate of decoherence of geometry must balance. The exponential expansion of the universe is over. However, linear expansion by construction of geometry can continue. The fastest linear construction of geometry from any tetrahedron would be at the speed of light.
When the rate of geometry formation and decoherence balance, geometry keeps building tetrahedra as fast as possible everywhere, but flat geometry, by hypothesis, decoheres most slowly. In the limit, perhaps flat geometries do not decohere at all. Then, the geometry of the universe tends to be flat in the absence of matter, as Einstein requires in general relativity. And, once again, the flatness after exponential expansion may overcome a need for the inflationary scenario and solve the particle-horizon problem.
It may be of interest that the assumption of an action, h, in the generation of each tetrahedron implies an expanding total energy in geometry itself, the vacuum itself, as geometry constructs itself. Indeed, one expects that the assumption of an action, h, per tetrahedron would lead to a uniform energy density, a constant scalar quantity even as geometry grows. Such an energy could be related to the cosmological constant.
It may also be of interest that string theory posits “extra dimensions” that “curl up” on themselves to yield four-dimensional space-time. Could the decoherence of geometry afford a parallel way that extra dimensions can curl up? And could the ever-generating possible geometries, as they generate exponentially even as they decohere, yield sufficient extra degrees of freedom to correspond to the modes of oscillation of strings or M-branes in six or seven extra dimensions?
It may be interesting that the energy content of geometry could be enormous compared to that of familiar particles of the same size scale. That might allow the familiar particles with rest mass to borrow a tiny bit of the vacuum energy for their mass. Such a possibility could hint that matter, energy, and geometry might be able
to interconvert. Perhaps different particles would be different kinds of topological “knots” in the spin network structure of geometry with interconvertible particles being nearby knot topologies.
We are obviously far from anything like a coherent theory that implements any of the intuitions above. They remain at best mere suggestions for a research program.
I began this chapter wondering why the universe is complex. In place of the anthropic principle or Lee Smolin’s cosmic selection, I have suggested one possible approach to the choices of the constants of nature by maximizing constructive interference over a sum of all histories through a space of both configurations and laws. Even if that program were to succeed, it does not necessarily yield a complex universe, let alone one poised roughly between expansion and contraction.
Might we see ways to understand why the universe is complex? Perhaps, but merely perhaps. I return to the thoughts earlier in this chapter that decoherence requires coupling systems and the loss of phase information. If, in general, complex and high-diversity quantum systems with many coupled degrees of freedom lose phase information when they interact more rapidly than an equal number of simple, low-diversity quantum systems with the same total number of interacting parts, then the comeasuring of entangled quantum systems should tend toward higher complexity, diversity, and classicity. In short, complexity and diversity would beget classicity irreversibly. In turn, this would lead to a preferred tendency toward a lock-in of complexity and diversity. There is a sense in which classical objects are like the constraints on the release of energy that permits work to be done. Classical objects, interacting with quantum objects, lead to decoherence and more classicity. Complex pairs of quantum objects that decohere readily, or classical objects and those quantum objects that are caused to decohere readily when interacting with the classical object, form preferred pairs that tend to decohere, hence become frozen into classicity. We begin to have an image of preferred pairs of quantum systems coupling and decohering, hence, an image of a complex and diverse universe constructing itself as it nonergodically invades the adjacent possible, rather as a biosphere constructs itself. And if more complexity and diversity means more comeasurement and faster decoherence of a wider variety of complex quantum systems, in analogy with the concept that extracting work from increasingly subtle nonequiibrium systems requires increasingly subtle measuring and coupling devices, the universe as a whole may persistently break symmetries as new entities come into existence, and hence expand its diversity, complexity, and classicity as fast as possible.
Loose arguments? Yes. Testable? Here and there. Wrong? Probably. Deeply wrong? Maybe not. Does this get the universe to the edge of expansion versus contraction or
to a flat universe expanding forever more rapidly? I would love it to be so. Indeed, would love a view in which matter, energy, and geometry can all interconvert. After all, if geometry, the vacuum, has energy, such interconversion does not seem impossible. Do the considerations of this chapter require detailed models and supporting calculations to be taken as more than the merest suggestions? Absolutely. This chapter, like much of Investigations, is protoscience. But science grows from serious pro toscience, and I take Investigations to be serious protoscience.
We enter a new millennium. There will be time for new science to grow.