CAN WE DO PHYSICS WITHOUT A CONSTRUCTIBLE STATE SPACE?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 (This followed the development of a Euclidean path integral by Reisenberger and by Reisenberger and Rovelli. Very interesting related work has also been done by John Baez. We may note that the theory described in  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, 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. 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.
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.
CONCLUSIONSWe 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). 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).
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.
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. 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.
ACKNOWLEDGEMENTSWe 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.