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A Possible Solution For The Problem Of Time In Quantum Cosmology
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.

INTRODUCTION

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.


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