Every time a new paper by Stephen Hawking appears it sparks inevitably a lot of commentary, and the one that came out last Thursday is no exception. No, sorry to say this is not the one in which he finally gives due credit to H. Simpson (1999)
for the toroidal universe theory. This one is written with Thomas Hertog and is called:Populating the Landscape: A Top Down ApproachPeter
already have comments, but they are both pretty non-committal. I wouldn't dare at all to make a comment were it not because the most part of the paper is rich in conceptual discussion and scarce in technical math; this makes me feel
as if I understood it quite well, which surely isn't the case. I'll try to make a summary of what I felt
What Hawking and Hertog are proposing is that when you try to apply quantum mechanics in cosmology, that is to regard the whole universe as a quantum system, you have to change the rules for extracting predictions from QM. The standard rules are: you start with a quantum system in state A, and you want to know the probability of finding it some time T later in state B. Assign to each possible way the system could get from A to B in a time T (these ways are called "paths" or "histories") a complex number called the "amplitude" of the history; the rules for assigning amplitudes to histories need not concern us here. Now sum your amplitudes over all possible histories (this is called "the path integral") to get a total amplitude for the A-B transition. Now a simple mathematical operation (multiplying the amplitude by its complex conjugate) gives you the probability of the "A to B in T" transition.
According to H&H, this is not the way you should apply QM to the whole universe. In the universe we don't know and can't know its initial state; we only know the final state, the one we are in now. We are not preparing the system in A and wondering if it will appear later in B; we are now in B, and we are wondering how we got here. So their idea is that we should write a path integral for all the possible histories of the universe that end up in a universe consistent with present data. They expect such a path integral would be dominated by a particular subset of histories (that means, almost all the probability for the present data comes from a restricted set of possible pasts) and then assuming that our history is in this subset we can derive further testable predictions.
This description seems to be lacking something: if normally we need both an initial and a final state to evaluate a path integral, how can we do it with only a final state? The answer is that the notion of a fixed, initial state disappears from Hawking's framework and is replaced by the "no-boundary condition". This means (roughly and so far as I understand it) that there is no initial state for the universe, because the integral sums over universe-geometries in which time is imaginary, and in this case time becomes indistinguishable from a space coordinate. Thus an "initial state in time" is replaced by "all possible states of universes with 4 spatial dimensions" (assuming we have fixed as part of our data that the "really space" dimensions are 3). However I don't really understand completely how this so-called "Euclidean path integral" is supposed to be related to the actual universe in which time is distinct from the other dimensions.
Anyway, I should make clear that H&H do not apply this framework to any calculation in the real universe. (They do a calculation with an ultra-simplified toy model to illustrate how it is supposed to work). What they are doing is proposing a way to frame questions about the state of the universe and the values of its most general parameters (such as the number of dimensions or the masses of elementary particles) which according to the string theory are determined, perhaps, by accidental events in the history of the universe instead of being included or predicted from the ultimate theory of nature. The article is therefore a participant in the enduring "String theory - Landscape - Anthropic principle" controversy, of which I could say a lot more... but I have no time now, so that will be for another day. I have to leave for a delicious event you will learn about in the next post. So long!