Copenhagen May 25, 2015
Michael Freedman
Microsoft Research—Station Q
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Quantum Mathematics and the
Relationship Between Math and Physics
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Eugene Wigner
“The Unreasonable Effectiveness of Mathematics in the Natural Sciences” 1960
I’d like to propose a “dual” aphorism:
“. . . the unreasonable effectiveness of physics in mathematics . . .”
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𝐖𝐢𝐠𝐧𝐞𝐫
Dualities in Mathematics
Poincaré Duality (in topology)
Fourier Duality (in analyis)
p
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Physics
• Particle ↔ wave • ADS/CFT
• Donaldson ↔ Seiberg/Witten
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Field Theory
Effective Infrared Limit Effective Ultraviolet Limit
Both and Wigner are supported by the work of Ed Witten and Vaughan Jones others in the past 30 years.
Story of: Jones polynomial (topology)
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Wigner
History Quantum Mechanics
↓ Operator algbebras
↓ von Neumann algebras
↓ braid representa?on Link invariants
‖
Jones Polynomial ↓
“Topological Quantum Field Theory” ↓
Categorifica?on ↓
Khovanov homology
Topological quantum computa?on
Five branes, equa?ons in 4 & 5D
3-‐manifold topology
More
Robert Langlands
↓ Algebraic number theory / Galois groups
⇕ Automorphic forms / rep theory
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𝐖𝐢𝐠𝐧𝐞𝐫:
Langlands Program
N=4 supersymmetric Yang-Mills → 4D families of TQFTs …
Energy of loops in non-‐linear Σ-‐model
Raoul BoX
Alexei Kitaev
↓
Controlled k-‐theory
↓
Kitaev’s classifica?on of free fermions according
to dimension and symmetry
↓
BoX periodicity
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The central player in string theory and perhaps mathematics as a whole is the complex curve.
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Things sounding highly specialized to mathematicians: super-symmetric string field theories, turn out to be fundamental. They enumerate basic algebraic-geometric objects, as shown by Candelas et. al. using a duality between Calabi-Yau manifolds.
Shing-Tung Yau Candelas, de la Ossa, Green, Parkes 10
• “Wigner” says that the universe is regular.
• suggests that the universe is not a realization of an arbitrary consistent system but rather a system that is maximal or even unique.
• Otherwise math would far outreach physics.
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Wigner
Leibniz spoke of “the best of all possible worlds.”
• Maybe there is only one possible world.
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Gottfried Leibniz
Could the universe have been different?
• Could there be a world where NP-complete problems can be solved efficiently?
• What about a world where Grover search runs in cube root rather than square root time?
• Many (Aaronson) think not – that just like perpetual motion, such worlds cannot be consistent.
Free Will! Indeed.
Descarte Aaronson
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I’d like to expand the discussion to include Information and Computation as promising, younger colleagues of Physics and Mathematics.
• The Godel, Turing and Shannon’s theories of proof,
computation and communication evolved in the 1960’s into the theory of computational complexity
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Good computational models are rare. Modern Church-Turing (MCT) Thesis:
• There are only two maximal physically realistic models of computation: – One based on Classical Physics P – One based on Quantum Physics BQP
Alonzo Church Alan Turing 15
We believe BQP is stronger than P
(evidence Shor’s factoring Algorithm ) • Why is the quantum world superior?
• Our Classical world emerges through neglect, that is failure to observe an entire system but merely a piece of it.
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• Mathematically this neglect is called partial trace and
averages the unobserved degrees of Freedom. Physically the process is called decoherence.
• Planck’s contstant ћ ≤ ∆ p ∆ x is the quantum of phase space volume and neglecting a portion of phase space large with respect to ћ produces “classical outcomes.”
Quantum Classical
Corollary of MCT: All we will ever know (or at least compute) will lie in BQP
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Despite quandaries involving unitarity and black holes I am happy in believing
Quantum Mechanics governs the universe.
Schrödinger
(Amplitudes NOT probablities) 18
• Amplitudes are square roots of probabilities • Square roots of probabilities are not intuitive. • Nothing in our large scale classical world, nothing in
our evolutionary experience, prepares our mind for superposition of amplitudes within a Hilbert space.
• Superposition was born amid mystery and paradox in the period 1900-1927.
Born Bohr Heisenberg Schrodinger Planck
Radiation, Diffraction, Scattering, Atomic Spectra
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• We need something like the double slit experiment to see amplitudes at work
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Source
Blind Screen
Observed paXern
|α + β|2 = |α|2 + |β|2
All closed 1 open 2 open
0
-|α|2
-|β|2
+|α + β|2
It is amplitudes not probabili?es which add
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This has lead us to a new type of numeral : Let us “hash” Mankind’s history into a
Brief History of Numbers • -‐13,000 years: Coun?ng in unary
• -‐3000 years: Place nota?on • Hindu-‐Arab, Chinese
• 1982: Configura?on numbers as basis of a Hilbert space of states
Possible futures contract for sheep in Anatolia
7,123,973,713
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• Quantum computers manipulate numbers in superposi?on – essen?ally crea?ng a new kind of numeral.
• We believe that quantum computers will do amazing things.
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• But we’re not sure exactly what. • Prominent possibili?es:
• Quantum Chemistry
• Drug Design
• High Tc
• Machine Learning
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• Quantum mechanics does not permit copying of informa?on (no cloning theorem). Thus
• Long quantum mechanical computa?on requires either – painful error correc?on: For big problems 99.9%-‐99.99% of resources go to s?fle error, even given physical gates with 99.9% fidelity, or
– extreme accuracy (topology)
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Topology
• Charlie Marcus of KU and NBI is making breathtaking advances in the topological direc?on:
• Within our life?mes a new tool will lie within our or collec?ve toolbox.
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Stanescu, Lutchyn, Das Sarma, PRB’11 Sankar Das Sarma Charlie Marcus
6.23 × 109
decimal
number
computer nuclear
biology
α |0> + β |1> quantum number/ quantum computer
fire
machines
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But a skep?c might ask: • If topological quantum systems are so great at processing informa?on,
why don’t natural biological systems exploit them? • Quantum effects are most pronounced in cold environments, T« gap. • Maybe biological systems will, but we’ll have to wait 1011 years for the
cosmic background temperature to drop low enough.
• Our joint endeavor with Marcus and others is designed short-‐cut this
tedious hundred billion-‐year wait.
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Let me finish by revisi?ng two old ideas with “quantum thinking.” The first is modest and sober, the second is not. Max Flow = Min Cut (1956 Shannon, and Ford-‐Fulkerson) Classical Max Flow = Min Cut = 2 Quantum (2015 Cui, F., Stong, R.)
Max (rank (network)) = 7 ˂ 8 T, T’, T”
in out
T 3
2
2
2
2
2
3
T’
T”
Now a less sober idea
• Taking seriously, let’s reverse a fifty-year effort to construct a mathematical foundation for field theory and instead seek a field theoretic foundation for mathematics.
• A Feynman diagram (let’s take cubic interactions) has the same structure as a proof in a formal system X: two things come together and a third thing gets “spit out.”
Physics (field theory) Logic (modus ponens)
A
BA⇒
B
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Wigner
Wigner
• We should attempt to “reverse engineer” the “field theory” whose perturbative expansions are the deductions of some fixed formal system—X.
• For such a theory, the partition function Zi, f would (perturbatively) be a weighted sum of all possible proofs from the initial conditions i, the axioms, to the final condition f, the statement in question.
i f
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But a field contains more than perturbative information (think of kinks, instantons, phase transition, etc.), so one would expect situations in which
Zi, f > 0 even in the absence of a formal proof, in system X, of statement f, from the axioms i.
32 David Thouless Gerard t’Hooft
• In a field based system, more things would be “provable.” Corresponding to nonperturbative effects.
• Perhaps, Gödel’s incompleteness theorems disappear: there seems no enumeration scheme for our more general “proofs.”
• “Proofs” would no longer be something you can “write down”, but merely accumulate evidence about.
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• The early 20th century paradoxes within set theory might eventually be interpreted as “pushing a low energy effective theory beyond its limits.”
• With luck, we might undo all the good work of the twentieth century on logic and set theory and return to the world of Hilbert and Bohr.
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David Hilbert Niels Bohr
• There are two types of numbers (integers) in our experience that are effec?vely non-‐overlapping:
0, 1, 2, …, 1070 10(10(22))
Number of things Number of configurations or system states
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