Bianca Dittrich takes questions from Butterfield and Pitts.
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We can apply all this to spin foams / spin nets!
The key question is this. Is there a phase which leads to smooth space times?
Some promising results! Dittrich & collaborators have found fix points from which one could extract different systems of embedding maps. Those would correspond to different (non-) realisations of ’emergence of space-time).
Task for the future: Expand the theory around this vacuum!
Coming back to the initial question: What is (quantum) space-time? Dittrich suggests the time scales for constructing an answer are ~2500ys. Challenge accepted!
An example: The Ising model.
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Now moving from 1+1D to 2D!
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How do we get a vacuum of a statistical system? Dittrich has the answer.
More boxes. Lots more boxes!
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We can embed a discrete state into a contiuum theory and describe those states as equivalence classes of states in that theory.
Cylindrical consistency ensures that the result of a computation (e.g. an inner product, an expectation value) does not depend on which representative one chooses. This gives us a contiuum theory!
Huggett: where does the cylinder come in? Artefact from QFT!
How do we embed? embeddings should be determined by the dynamics of the theory; and fine- and coarse graining should play a role!
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ad 2). What is vacuum?
It surely isn’t a simple object. Maybe one of the following?
– lowest-energy state
– invariant under symmetries of the background space-time
– encode the geometry of Minkowski space
– cyclic state which generates the Fock space.
In LQG, we have a kinematical vacuum. (Simple: it’s a box!)
In all seriousness: It’s a peculiar state. In LQG, it’s maximally squeezed state, in the sense that all spatial geometry operators have zero expectation value on it, and zero fluctuations! It’s a state of no spatial geometry. Truly nothing!
As in non-gravitational quantum theory, we can build a highly excited state from the vacuum by applying loads of ‘creation operators’, based on graphs.
The LQG representation of the kinematical observable algebra is unique! I.e., it is
– cyclic
– irreducible, and it
– carries a representation of spatial diffeos.
Thus:
– relational observables solve the problem of time in many situations;
– are useful, especially for quantum cosmology;
BUT, Dittrich asks: What do we measure if there is no space-time?
We have aether theories on the one side, and “emergence of spacetime” approaches on the other. No clear winner at this point.
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More difficult example (due to Dittrich & Tambornino): Two-point function of scalar field relative to (four) clock scalars.
where G is the Green’s function on fixed background.
The resolution limit for degrees of freedom depends on the energy of clocks: the higher the energy, the higher the resolution.
There is general amusement at the proposal for “fashionable” clocks: Just change clocks whenever necessary!
But what is aether, really?
Some special properties include
– not being detectable
– not interacting with other matter
– has a peculiar non-relativistic kinetic term (linear and not quadratic dependence on momentum)
An example is the time-of-arrival operator, à la Aharanov & Unruh. With additional uncertainty relation, this can be quantised!
Beyond Spacetime 