Copenhagen May 25, 2015 Michael Freedman Microsoft Research—Station Q 1 Quantum Mathematics and the Relationship Between Math and Physics 2 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 . .” 3 Dualities in Mathematics Poincaré Duality (in topology) Fourier Duality (in analyis) p 4 Physics • Particle ↔ wave • ADS/CFT • Donaldson ↔ Seiberg/Witten Field Theory Effective Infrared Limit Effective Ultraviolet Limit 5 Both ​ and Wigner are supported by the work of Ed WittenWign and Vaughan Jones others in the past 30 years.er History Quantum Mechanics ↓ Story of: Operator algbebras Jones polynomial (topology) ↓ von Neumann algebras ↓ braid representaon Link invariants ‖ Jones Polynomial ↓ “Topological Quantum Field Theory” ↓ Topological quantum Categorificaon 3-manifold computaon ↓ topology Khovanov homology Five branes, equaons in 4 & 5D 6 More​ ������ : Langlands Program N=4 supersymmetric Yang-Mills → 4D families of TQFTs … ↓ Algebraic number theory / Galois groups ⇕ Automorphic forms / rep theory Robert Langlands 7 Energy of loops in non-linear Σ-model ↓ Bo periodicity Raoul Bo ↓ Kitaev’s classificaon of free fermions according Alexei Kitaev to dimension and symmetry ↓ Controlled k-theory 8 The central player in string theory and perhaps mathematics as a whole is the complex curve. 9 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. ​ • Wign suggests that the universe is not a errealization of an arbitrary consistent system but rather a system that is maximal or even unique. • Otherwise math would far outreach physics. 11 Leibniz spoke of “the best of all possible worlds.” • Maybe there is only one possible world. Gottfried Leibniz Could the universe have been different? 12 • 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 13 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 14 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. 16 • 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 17 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. Planck Born Bohr Heisenberg Schrodinger Radiation, Diffraction, Scattering, Atomic Spectra 19 • We need something like the double slit experiment to see amplitudes at work 20 It is amplitudes not probabili?es which add All closed 1 open 2 open Blind Screen -|α|2 0 +|α + β|2 -|β|2 Source |α + β|2 = |α|2 + |β|2 Observed paern 21 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 Possible futures contract for sheep in Anatolia • -3000 years: Place notaon • Hindu-Arab, Chinese 7,123,973,713 • 1982: Configuraon numbers as basis of a Hilbert space of states 22 • Quantum computers manipulate numbers in superposi?on – essen?ally creang a new kind of numeral. • We believe that quantum computers will do amazing things. 23 • But we’re not sure exactly what. • Prominent possibili?es: • Quantum Chemistry • Drug Design • High Tc • Machine Learning 24 • Quantum mechanics does not permit copying of informaon (no cloning theorem). Thus • Long quantum mechanical computaon 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) 25 Topology • Charlie Marcus of KU and NBI is making breathtaking advances in the topological direc?on: Stanescu, Lutchyn, Das Sarma, PRB’11 Sankar Das Sarma Charlie Marcus • Within our life?mes a new tool will lie within our or collec?ve toolbox. 26 6.23 × 109 decimal computer nuclear machines biology number α |0> + β |1> quantum number/ quantum computer fire 27 But a skep?c might ask: • If topological quantum systems are so great at processing informaon, 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. 28 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 in out T’ 2 3 T Quantum 2 (2015 Cui, F., Stong, R.) 2 2 T” 2 3 Max (rank (network)) = 7 ˂ 8 T, T’, T” 29 Now a less sober idea • Taking​Wigner 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.” A ⇒ B ​Wigner B A Physics (field theory) Logic (modus ponens) 30 • 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 31 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. David Thouless Gerard t’Hooft 32 • 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. 33 • 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. David Hilbert Niels Bohr 34 • 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 configurations Number of things or system states 35 .