Copenhagen May 25, 2015

Michael Freedman

Microsoft Research—Station Q

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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  

𝐖𝐢𝐠𝐧𝐞𝐫     

Dualities in Mathematics

Poincaré Duality (in topology)

Fourier Duality (in analyis)

p  

4  

Physics

•  Particle ↔ wave •  ADS/CFT

•  Donaldson ↔ Seiberg/Witten

5  

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)

6  

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

   

7  

 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  

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.

•  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.

11  

Wigner 

Leibniz spoke of “the best of all possible worlds.”

•  Maybe there is only one possible world.

12  

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  

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.

Born   Bohr   Heisenberg   Schrodinger  Planck  

Radiation, Diffraction, Scattering, Atomic Spectra

19  

•  We need something like the double slit experiment to see amplitudes at work

20  

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  

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  

•  -­‐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

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:              

•  Within  our  life?mes  a  new  tool  will  lie  within  our  or  collec?ve  toolbox.  

26  

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  

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.  

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29  

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

30  

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

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.

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.

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.

34  

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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