Rosetta Slides Mine

7/30/2019 Rosetta Slides Mine

http://slidepdf.com/reader/full/rosetta-slides-mine 1/23

Physics, Topology, Logic, and Computation:A Rosetta Stone

John C. Baez

UC Riverside

Mike Stay

Google, U. of Auckland

1

presented by

Wilson de Oliveira



Hieroglyphic

Rosetta Stone 196 BC

Demotic

Greek



The Rosetta Stone (pocket version)

Category Theory Physics Topology Logic Computation

object system manifold proposition data type

morphism process cobordism proof program

2



Objects

• String diagrams have ‘strings’ or ‘wires’:

X

• Quantum mechanics has Hilbert spaces: X Cn

• Topology has manifolds: X

• Linear logic has propositions:

X = “I have an item of type X .”

• Computation has datatypes: interface X;

• SET has sets: X

3

worldline of a

particle of type X

you can only use any

given proposition once

smooth spaces, like a circle, torus,

hypersphere; no corners or edges



Morphisms

• String diagrams have vertices:

f

X

Y

• Quantum mechanics has linear transformations:

f : X → Y f : Cn → Cm

(An m × n matrix with complex entries)

4

string's label changes from

X to Y



Morphisms

• Topology has cobordisms: X

Y

• Linear logic has constructive proofs:

X Y

• Computation has programs: Y f(X);

• SET has functions: f : X → Y

5

the "borders" of a cobordism

are the source and target manifold

a process that consumes an item of

kind X and produces one of kind Y

f(X) is of type Y



Morphisms compose associatively• String diagrams:

f

g

X

Y

Z

• Quantum mechanics: matrix multiplication

• Topology:

6



Morphisms compose associatively

• Linear logic: Y Z X Y X Z

(◦)

• Computation:

Y f(X x);

Z g(Y y);

...

z = g(f(x));

• SET: (g ◦ f ) : X → Z

7

X x

means: x is of type X



Identity morphisms

• String diagrams:X

• Quantum mechanics: identity matrix

1 0

0 1

• Topology:

• Linear logic: X X (i)

• Computation: X id(X x) { return x; }

• SET: 1 X : X → X

8



Monoidal categories

• String diagrams:

f

X

Y

f

X

Y

= f ⊗ f

X

Y

X

Y

= f ⊗ f

X ⊗ X

Y ⊗ Y

• Quantum mechanics: tensor product

a b

c d ⊗ e f g

h j k =

a e a f a g b e b f b g

a h a j a k b h b j b k

c e c f c g d e d f d gc h c j c k d h d j d k

9

you can take two objects

and put them together to form a

new object

you can also put together two

morphisms so they act in parallel on thepair of objects



Monoidal categories

• Topology:

• Linear logic: AND X Y X Y

X ⊗ X Y ⊗ Y (⊗)

• Computation: parallel programming

Pair<X, X’> pair;

• SET: f × f : X × X → Y × Y

10

Disjoint union of manifolds and cobordisms

"parallel data type"

Note that while we can use projection maps to "take apart" a pair of functions, we can't do that

with matrices: there's no way to write the list (1, 0, 0, 1) as the tensor product of two lists.

This is called entanglement in quantum mechanics. Similarly, we can't split an arbitrary cobordismfrom two circles to two circles into two pieces (go back four pages to the composition picture)



Monoidal unit

• String diagram:

• Quantum mechanics: I = C, the phase of a photon

• Topology:

• Linear logic: I , trivial proposition

• Computation: I = void or I = unit type

• SET: one-element set I

11

denoted I, such that (X tensor I) and (I tensor X) are both isomorphic to X

acts a bit like T



Braided monoidal categories

• String diagrams:

X Y

• Quantum mechanics: swap the particles. Bosons

commute, fermions anticommute; quantized mag-

netic flux tubes in thin films, or “anyons”, can have

arbitrary phase multiplier.

12

careful: who is over who?

the wave functions stays the

the wave function changes sign

But if you restrict particles to 2 dimensions, then other possibilities arise. If you take a thin conductive film,

put it in a strong magnetic field, and drop the temperature down to near absolute zero, then the flux tubes

get quantized and you can braid them: the change in the phase of the wave function depends on the

handedness of the swap and the strength of the field. It turns out that you can change the phase of the

wavefunction by any complex number instead of just +/-1, so they call these things "anyons"



Braided monoidal categories

• Topology:

• Linear logic: W X ⊗ Y W Y ⊗ X

(b)

• Computation: pair.swap();

• SET: b( x, y) = y, x

13

this makes the parallel processors trade their data

you can permute elements of a

pair, but you can't braid them:

SET is a symmetric monoidal



Braided monoidal closed categories• String diagrams:

f

X Y

Z

→ f

X

Y

Z

• Quantum mechanics: antiparticles

1 X : X → X pair : I → X ∗ ⊗ X

e−

→ e+

e−

14

"Closed" means that for any pair of objects X, Y, there's an object Z such that the set of morphisms from X to Y is

isomorphic to the set of morphisms from I toZ. Z is usually denoted X -o Y



Braided monoidal closed categories

• Topology:→

• Linear logic: IMPLIES X ⊗ Y Z

Y X Z (c)

• Computation: Currying

z = f(x, y);

or

z = f(y)(x);

• SET: f : X × Y → Z f : Y → Z X

15

here we see an identity morphism on the circle being bent

into a morphismfrom the empty manifold to a pair of circles.

If we can orient the manifold at the top (say, put an arrow on

this circle), then when we bend over the morphism,

the orientation has swapped, so it's a "backwards output"



Model Theory

X Y

f

g

16

a small category with two objects and two nontrivial morphisms

At first glance, it's not a particularly interestingcategory; but if I change the names here ...



Model Theory

E V

s

t

Th(Graph)

17

"the theory of graphs": if you map the objects of this category to finite sets and the morphisms of the category to

functions between those sets, then we get the description of a particular graph. It's pretty easy to see that any such

assignment will give a graph, and all graphs arise this way



Model Theory

Th(Graph) SET

object V set of vertices

object E set of edges

morphism s : E → V function that picks out the source of each edge

morphism t : E → V function that picks out the target of each edge

18

This is a simple example of the notion of syntax and semantics. On the left, we have the syntax, the API, theabstract description of the mathematical gadget we're trying to describe. Its objects are simply placeholders,

without any implicit meaning. On the right, we have the semantics; here is where the symbols acquire

meaning; in this instance, V now means a set of vertices.

Functor



Model Theory (in Java)interface ThGraph {

// Internal interfaces

interface V;

interface E;

// Methods

V s(E);

V t(E);

}

A functor is a structure-preserving map. In Java terms, a func-tor picks out a class that implements the interface.

19

Python !!!!



Model Theory (Classical)

Syntax [Programming language] Semantics [CE-SET]

data type computably enumerableset of values

method partially recursive function

20



Model Theory (Quantum)

Syntax [Programming language] Semantics [QM]

data type Hilbert space of values

method linear transformation

21



Model Theory (Quantum)

Syntax [Topology] Semantics [QM]

manifold Hilbert space of statescobordism linear transformation

Topological Quantum Field Theory

22

Documents

Rosetta Slides Mine