James B. Hartle

Department of Physics, University of California, Santa Barbara

If the Universe is a quantum mechanical system, then it has a quantum state. This state provides the initial condition for cosmology. A theory of this state is an essential part of any final theory summarizing the regularities exhibited universally by all physical systems and is the objective of the subject of quantum cosmology. This chapter is concerned with the role that the state of the Universe plays in anthropic reasoning - the process of explaining features of the Universe from our existence in it [1]. The thesis will be that anthropic reasoning in a quantum mechanical context depends crucially on assumptions about the Universe's quantum state.

Every prediction in a quantum mechanical Universe depends on its state, if only very weakly. Quantum mechanics predicts probabilities for alternative possibilities, most generally the probabilities for alternative histories of the Universe. The computation of these probabilities requires both a theory of the quantum state as well as the theory of the dynamics specifying its evolution.

To make this idea concrete while keeping the discussion manageable, we consider a model quantum Universe. The details of this model are not essential to the subsequent discussion of anthropic reasoning but help to fix the notation for probabilities and provide a specific example of what they mean. Particles and fields move in a large - perhaps expanding - box, say, presently 20 000 Mpc on a side. Quantum gravity is neglected - an excellent

1 This is a reworking of an article, reproduced with permission, from The New Cosmology: Proceedings of the Conference on Strings and Cosmology, eds. R. Allen, D. Nanopoulos and C. Pope, AIP Conference Proceedings 743 (New York: American Institute of Physics, 2004).

Universe or Multiverse?, ed. Bernard Carr. Published by Cambridge University Press. © Cambridge University Press 2007.

approximation for accessible alternatives in our Universe later than 10-43 s after the big bang. Spacetime geometry is thus fixed with a well defined notion of time, and the usual quantum apparatus of Hilbert space states and their unitary evolution governed by a Hamiltonian can be applied.2

The Hamiltonian H and the state |tf) in the Heisenberg picture are the assumed theoretical inputs to the prediction of quantum mechanical probabilities. Alternative possibilities at one moment of time t can be reduced to yes/no alternatives represented by an exhaustive set of orthogonal projection operators {Pa(t)} (a = 1,2,...) in this Heisenberg picture. The operators representing the same alternatives at different times are connected by:

For instance, the Ps could be projections onto an exhaustive set of exclusive ranges of the centre-of-mass position of the Earth, labelled by a. The probabilities p(a) that the Earth is located in one or another of these regions at time t is given by p (alH, tf) = \\Pa(t) |tf)\\2 . (18.2)

The probabilities for the Earth's location at a different time is given by the same formula with different Ps computed from the Hamiltonian by Eq. (18.1). The notation p (alH, tf) departs from usual conventions (e.g. ref. [2]) to indicate explicitly that all probabilities are conditioned on the theory of the Hamiltonian H and quantum state |tf).

Most generally, quantum theory predicts the probabilities of sequences of alternatives at a series of times - that is histories. An example is a sequence of ranges of centre-of-mass position of the Earth at a series of times giving a coarse-grained description of its orbit. Sequences of sets of alternatives {Pkk(tk)} at a series of times tk (k = 1,...,n) specify a set of alternative histories of the model. An individual history a in the set corresponds to a particular sequence of alternatives a = (a1,a2,... ,an) and is represented by the corresponding chain of projection operators Ca:

C a = Pnn (tn) •••Pi, (t1), a = (a1,...,an). (18.3)

The probabilities of the histories in the set are given by p (alH, tf) = p (an, ...,a1lH, tf) = \\Ca |tf )\\2 (18.4)

2 For a more detailed discussion of this model in the notation used here, see ref. [2]. For a quantum framework when spacetime geometry is not fixed, see, e.g., ref. [3].

provided the set decoheres, i.e. provided the branch state vectors Ca\^) are mutually orthogonal. Decoherence ensures the consistency of the probabilities given by Eq. (18.4) with the usual rules of probability theory.3

To use either Eq. (18.2) or (18.4) to make predictions, a theory of both H and is needed. No state means no predictions.

'If you know the wave-function of the Universe, why aren't you rich?' This question was once put to me by my colleague Murray Gell-Mann. The answer is that there are unlikely to be any alternatives relevant to making money that are predicted as sure bets, conditioned just on the Hamiltonian and quantum state alone. A probability p(rise\H, for the stock market to rise tomorrow could be predicted from H and through Eq. (18.2) in principle. But it seems likely that the result would be a useless p(rise\H, w 1/2, conditioned just on the 'no boundary' wave-function [7] and M-theory.

It is plausible that this is the generic situation. To be manageable and discoverable, the theories of dynamics and the quantum state must be short -describable in terms of a few fundamental equations and the explanations of the symbols they contain. It is therefore unlikely that H and contain enough information to determine most of the interesting complexity of the present Universe with significant probability [8,9]. We hope that the Hamiltonian and quantum state are sufficient conditions to predict certain large-scale features of the Universe with significant probability. Approximately classical spacetime, the number of large spatial dimensions, the approximate homogeneity and isotropy on scales above several hundred Mpc, and the spectrum of density fluctuations that were the input to inflation are some examples of these. But even a simple feature, such as the time the Sun will rise tomorrow, will not be usefully predicted by our present theories of dynamics and the quantum state alone.

The time of sunrise does become predictable with high probability if a few previous positions and orientations of the Earth in its orbit are supplied in addition to H and That is a particular case of a conditional probability of the form

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