# Freedman, Universe from a Single Particle, Metric Crystallization

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Fun fact: Freedman did work on topological quantum computing.
Interesting fun example: cone on $S^3\times S^3 \embeds \RR^7$.
Idea for today: break metric symmetry on a Lie group.
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Setup toy model for the beginning of the universe: finite dimensional Hilbert space $X\cong \CC^n$ with symmetry group $G\da \liesu_n$, and a metric $g_{ij}$ on $\lieg\da \liesu_n$.
This gives $G$ a left-invariant metric.
Now add a probability distribution on $\lieg$, which are basically Hermitian matrices, whose draws are random Hamiltonians.

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Several layers of randomness: choose metric in a Boltzmann manner, then choose a Hamiltonian using a Gaussian based on the metric.
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Natural metric: Killing form, $\ad\dash$invariant and a local (global) extremum in the space of metrics.
Other metrics yield unit ellipsoids instead of unit spheres.
Write an energy functional on this space, essentially scalar curvature (i.e. Ricci scalar curvature).
Can extract an explicit formula from a paper of Milnor from the 70s on left-invariant metrics on Lie groups.
Need these invariant metrics to sort out how to actually compile a quantum circuit.
Nielsen-Brown-Susskind define a word metric $g_{ij} = \delta_{ij} e^{? w(i)}$ where $w(\wait)$ counts the number of letters, e.g. 
\[
w(1\tensor x\tensor 1 \tensor 1 \tensor y \tensor z \tensor 1\tensor 1 \tensor 1) = 3
.\]
This makes travelling along long words or words with capitals exponentially more difficult, rethinks the combinatorial problem of building a circuit as a differential geometric problem.
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Qubit structure: $J: (C^z)\tensorpower{}{n}\iso C^{z^N}$.
Define $g_{ij}$ to be KAQ ("knows about qubits") if it has a basis of principle axes $\ts{H_a}$ which admit a tensor product decomposition.
Forms about a $\sqrt{d}$ dimension subvariety in a $d\dash$dimensional space -- codimension 1 is already very thin, so these are exceedingly rare.
However, about 30% of critical points for the action functional satisfy this property.
Try to do perturb a Gaussian and expand/truncate.
Slight issue: get an integral like $\int e^{g_{ij}\cdots + c_{ij} x^i x^j}$, but the $x^i, x^j$ are commuting variables and the structure constants $c_{ij}$ anticommute, so this term vanishes after integrating.
Trying to integrate fermionically cancels the first term.
See Majorana operators.
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How they studied various metrics: gradient flow with respect to the action functional.
I wonder how one actually does this in practice..?
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Conclusions: nature abhors naked Hilbert spaces and attempts to equip them with tensor or Majorana structures.
In $D=2^n$, we see Majorana degree groupings and Brown-Susskind exponential penalty factors.
For random initial dimensions, conjecture all but log many dimensions develop such a structure.
A few exceptions: leaky universe scenario. Not good!
Setup is well-suited to studying pairs of an initial Hamiltonian and initial state to model near-beginning universe scenarios.
Consider the configuration space of these pairs and see how entropy evolves.
Outstanding problem: where does "space" come from?
Finding "space" in a model means finding the Leech lattice, wild!
But checking for this by brute force requires $10^{12}$ qubits.
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Feynman diagrams: if no thickness, kind of old school.
Modern treatments involve ribbon diagrams.
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