Once Before Time - Martin Bojowald [141]
Vilenkin’s tunneling condition relies on another effect of quantum mechanics, again a consequence of properties of the wave function. A wave function can often penetrate barriers with its tails, even if those would be too high for a corresponding classical particle. Many a physical phenomenon is based on this possibility, such as some kinds of radioactive decay and several technological developments including new transistors in microelectronics and the scanning tunneling microscope. Boldly stretching the analogy, Vilenkin proposed in 1983 that the universe itself might have emerged by such a tunneling process. Our universe would be the tail of a pioneering wave function that had once penetrated the barrier of the big bang and its singularity. But from where did the universe tunnel, and from where came the bulk of the wave function, whose tail our universe is supposed to be, before the tunneling process? Vilenkin’s answer, obvious only at first sight: From nothing.
One should perhaps, in general and especially in this case, not read too much into the notions of physical theories and grasp them as merely what they are: names that paraphrase a mathematical fact by analogies. Nothing cannot be thought; or, formulated more strongly, “Who thinks of nothing does not think at all”:
SOCRATES: He then who sees some one thing, sees something which is?
THEAETETUS: Clearly.
SOCRATES: And he who hears anything, hears some one thing, and hears that which is?
THEAETETUS: Yes.
SOCRATES: And he who touches anything, touches something which is one and therefore is?
THEAETETUS: That again is true.
SOCRATES: And does not he who thinks, think some one thing?
THEAETETUS: Certainly.
SOCRATES: And does not he who thinks some one thing, think something which is?
THEAETETUS: I agree.
SOCRATES: Then he who thinks of that which is not, thinks of nothing?
THEAETETUS: Clearly.
SOCRATES: And he who thinks of nothing, does not think at all?
THEAETETUS: Obviously.
SOCRATES: Then no one can think that which is not, either as a self-existent substance or as a predicate of something else?
—PLATO, Theaetetus
One can hardly attribute physical meaning to tunneling from nothing in a literal sense. Regardless, Vilenkin’s postulate does have sense with regard to the wave function of the universe, endowed by the tunneling condition with certain initial values at vanishing volume. There, the wave function is no longer zero, and it is not doomed to imply a wave function that vanishes everywhere. Instead, it prescribes the rate of change of the wave function at this place, a sufficient specification to determine it completely. One can try to draw at least rough predictions for the further course of an expanding universe at larger volume, and compare them with observations.
The no-boundary condition of Hartle and Hawking from the year 1984 sets up an initial condition in a similar way, motivated by a physical picture. Here, the picture is that of a universe closed and rounded off in the past, where classically the singularity would lie; consequently there is no boundary to the past—hence the condition’s name. Also here, one can imagine that the classical singularity is replaced by this rounded-off space-time; the proposal thus combines the singularity problem with the question of uniqueness. It may at first appear problematic that general relativity does not allow any space-time closed off at its bottom in the past, like some kind of chalice. This is the crucial point where Hartle and Hawking postulate new equations deviating from general relativity. Since deviations occur only for a small universe near the singularity, they might well incorporate implications of quantum gravity. Like all initial conditions mentioned so far, however, this one does not arise as