Copenhagen May 25, 2015
Michael Freedman Microsoft Research—Station Q
1 Quantum Mathematics and the Relationship Between Math and Physics
“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 representa on Link invariants ‖ Jones Polynomial ↓ “Topological Quantum Field Theory” ↓ Topological quantum Categorifica on 3-manifold computa on ↓ topology Khovanov homology Five branes, equa ons 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 classifica on 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 pa ern 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 nota on • Hindu-Arab, Chinese 7,123,973,713
• 1982: Configura on numbers as basis of a Hilbert space of states
22
• 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.
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 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)
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 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.
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
• TakingWigner 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