These last few proposals by Vilenkin, Hartle-Hawking and others are solutions to the Wheeler-DeWitt equation and exist in a category of proposals called "quantum gravity cosmologies" which make cosmic applications of an approach to quantum gravity called "closed dynamic triangulation" or CDT (also known as Euclidean quantum gravity). I've noticed the philosopher and theologian, William Lane Craig makes a series of false claims about the Hartle-Hawking model and was not being seriously challenged on those claims, so I wanted to correct those statements and to at least in part, explain the scheme proposed by James Hartle and Stephen Hawking.
1. The Hartle-Hawking Model requires Ontological commitments to Imaginary time
William Lane Craig takes the uncharitable interpretation that Stephen Hawking supposes the Euclideanization of time, literally rather than a mathematical trick deployed in the model. He states
Now in his model this imaginary time and Euclidean spacetime are suddenly supposed to be, not merely conceptual devices, but actual representation ... of physical reality in the very early history of the universe.Craig's interpretation becomes even more bizarre when he not only falsely attributes this position to Stephen Hawking but he asserts that the Hartle-Hawking model requires this interpretation in order to work. Neither of these statements are true, the Hartle-Hawking proposal deploys a mathematical technique called a wick rotation, whereby the universe's evolution is describable using a minisuperspace consisting of two regions. A Euclidean region where space has the metric signature (+,+,+,+) and a Lorentzian region where space time has the metric (-,+,+,+).
A separable component of the wave function corresponds to each region of the minisuperspace, an exponential component to the Euclidean region and an oscillating component to the Lorentzian region.
In an article titled 'The Quantum State of the Universe' pg. 272 Hawking clearly alludes to himself returning to real time, contrary to what Craig states: "We live in a Lorentzian geometry and therefore we are interested really only in the oscillating part of the wave function". James Hartle and Stephen Hawking appear interested in the euclidean region as a mathematical technique, in so far as they integrate over all initial spacetimes of a Euclidean 4-geometry with a given 3-space (the surface of that 4-geometry) as its only boundary. Such that the space that constitutes its boundary has a certain metric on a manifold with a specified matter field.
This gives the unconditional probability for that initial 3-space S to arise from a zero-dimensional point, not dependent on some prior 3-space S'.
2. Hartle-Hawking presupposes the Everett Many Worlds Interpretation of quantum mechanics
When Craig turns to ordinary quantum mechanics he presumes that James Hartle and Stephen Hawking commit themselves to the Many World's Interpretation.
In short, Hawking's wave functional analysis of the universe requires the Many Worlds Interpretation of quantum physics.Even if Hartle-Hawking believed the MWI was correct, this wouldn't justify the claim that their model requires the MWI in order to work. Taking the lesser claim even this is false, James Hartle is one of the original architects of the Consistent Histories approach, under this interpretation there is no bloated ontology of fantastically unimaginable, parallel universes branching off every time there's a measurement or sufficiently decohering state. Only one universe exists under the Consistent Histories interpretation, the one mapped out by the relevant solution to the Wheeler-DeWitt equation.
I tend to think of the Consistent Histories approach is a lot like the ordinary Copenhagen interpretation, corrected so that its consistent with decoherence experiments and generalized so that it describes all closed systems. Including the entire Universe but I'm aware there are several different avenues one can take this interpretation down.
3. Stephen Hawking believes the Universe had a beginning
William Lane Craig believes that Stephen Hawking is on his side about whether or not the universe is eternal. He writes
In the Hartle-Hawking model ... there is no breakdown of the laws nor a singular beginning, but there is a beginning nonetheless, and the universe is finite to the past.Craig is correct to say there is no singular beginning in the Hartle-Hawking model and one cannot trace the trajectory of a particle arbitrarily far into the past in classical spacetime but he's wrong to say there is "a beginning nonetheless". Classical concepts of space and time break down when
Where a is the scale factor, I is the Euclidean Einstein action (don't worry too much about that), and phi is the value of a massive scalar field.
But there is still what Hawking calls a "purely quantum mechanical" region of the model. I spoke earlier of the wick rotation Hartle-Hawking use and the oscillating component of the wave function, the Wheeler-DeWitt equation describing it is further reduced using the WKB approximation, to the Hamilton-Jacobi equation. What Hartle-Hawking get are (or what looks like) two non-interfering WKB (approximately Friedman) solutions. These solutions produce a universe which is eternal into the past and collapses from large radii and "bounces" back into our FRW-type universe. Alexander Vilenkin on pg. 11 describes the Hartle-Hawking model "The simplest interpretation of the Hartle-Hawking wave function describes a contracting and re-expanding universe with an AG boundary at the bounce".
Its more correct to say a contracting and re-expanding universe is most likely what results from the Hartle-Hawking wave function, its assigned an extraordinarily high probability but small non-zero probabilities are assigned to other states.
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