Mathematicians look at particle

physics

Matilde Marcolli

“Year of Mathematics” talk – July 2008

We do not do these things

because they are easy.

We do them because they are hard.

(J.F.Kennedy – Sept. 12, 1962)

1

Elementary particle physics

Constituents of all known matters and forces

(except gravity)

• Is there new physics beyond?

(massive neutrinos; supersymmetry? dark matter?

dark energy?)

• Unification with gravity?

(loops? strings? branes? noncommutative spaces?)

2

Parameters of the Standard Model

from experiments (particle accelerators)

3

Particle accelerators are giant microscopes

Higher energies = smaller scales

Theory: perform calculations that predict results of

events that can be seen in accelerators

Formula: Standard Model Lagrangian

4

LSM = −1

2∂νg

aµ∂νg

aµ − gsf

abc∂µgaνg

bµg

cν −

1

4g2

sfabcfadegb

µgcνg

dµge

ν − ∂νW+µ ∂νW

−µ −

M2W+µ W−

µ − 1

2∂νZ

0µ∂νZ

0µ − 1

2c2wM2Z0

µZ0µ − 1

2∂µAν∂µAν − igcw(∂νZ

0µ(W+

µ W−ν −

W+ν W−

µ ) − Z0ν (W+

µ ∂νW−µ − W−

µ ∂νW+µ ) + Z0

µ(W+ν ∂νW

−µ − W−

ν ∂νW+µ )) −

igsw(∂νAµ(W+µ W−

ν − W+ν W−

µ ) − Aν(W+µ ∂νW

−µ − W−

µ ∂νW+µ ) + Aµ(W

+ν ∂νW

−µ −

W−ν ∂νW

+µ )) − 1

2g2W+

µ W−µ W+

ν W−ν + 1

2g2W+

µ W−ν W+

µ W−ν + g2c2

w(Z0µW

+µ Z0

νW−ν −

Z0µZ0

µW+ν W−

ν ) + g2s2w(AµW

+µ AνW

−ν − AµAµW

+ν W−

ν ) + g2swcw(AµZ0ν (W+

µ W−ν −

W+ν W−

µ ) − 2AµZ0µW

+ν W−

ν ) − 1

2∂µH∂µH − 2M2αhH

2 − ∂µφ+∂µφ− − 1

2∂µφ

0∂µφ0 −

βh

(

2M2

g2 + 2Mg

H + 1

2(H2 + φ0φ0 + 2φ+φ−)

)

+ 2M4

g2 αh −

gαhM (H3 + Hφ0φ0 + 2Hφ+φ−) −1

8g2αh (H4 + (φ0)4 + 4(φ+φ−)2 + 4(φ0)2φ+φ− + 4H2φ+φ− + 2(φ0)2H2) −

gMW+µ W−

µ H − 1

2g M

c2wZ0

µZ0µH −

1

2ig

(

W+µ (φ0∂µφ− − φ−∂µφ0) − W−

µ (φ0∂µφ+ − φ+∂µφ

0))

+1

2g

(

W+µ (H∂µφ

− − φ−∂µH) + W−µ (H∂µφ+ − φ+∂µH)

)

+ 1

2g 1

cw(Z0

µ(H∂µφ0 − φ0∂µH) +

M ( 1

cwZ0

µ∂µφ0+W+µ ∂µφ

−+W−µ ∂µφ+)−ig

s2w

cwMZ0

µ(W+µ φ−−W−

µ φ+)+igswMAµ(W+µ φ−−

W−µ φ+) − ig

1−2c2w2cw

Z0µ(φ

+∂µφ− − φ−∂µφ+) + igswAµ(φ+∂µφ

− − φ−∂µφ+) −

1

4g2W+

µ W−µ (H2 + (φ0)2 + 2φ+φ−) − 1

8g2 1

c2wZ0

µZ0µ (H2 + (φ0)2 + 2(2s2

w − 1)2φ+φ−) −1

2g2 s2

w

cwZ0

µφ0(W+µ φ− + W−

µ φ+) − 1

2ig2 s2

w

cwZ0

µH(W+µ φ− − W−

µ φ+) + 1

2g2swAµφ

0(W+µ φ− +

W−µ φ+) + 1

2ig2swAµH(W+

µ φ− − W−µ φ+) − g2 sw

cw(2c2

w − 1)Z0µAµφ

+φ− −

g2s2wAµAµφ+φ− + 1

2igs λa

ij(qσi γµqσ

j )gaµ − eλ(γ∂ + mλ

e )eλ − νλ(γ∂ + mλ

ν)νλ − uλ

j (γ∂ +

mλu)u

λj − dλ

j (γ∂ + mλd)d

λj + igswAµ

(

−(eλγµeλ) + 2

3(uλ

j γµuλ

j ) −1

3(dλ

j γµdλ

j ))

+ig

4cwZ0

µ{(νλγµ(1 + γ5)νλ) + (eλγµ(4s2

w − 1 − γ5)eλ) + (dλj γ

µ(4

3s2

w − 1 − γ5)dλj ) +

(uλj γ

µ(1 − 8

3s2

w + γ5)uλj )} + ig

2√

2W+

µ

(

(νλγµ(1 + γ5)U lepλκe

κ) + (uλj γ

µ(1 + γ5)Cλκdκj )

)

+

ig

2√

2W−

µ

(

(eκU lep†

κλγµ(1 + γ5)νλ) + (dκ

j C†κλγ

µ(1 + γ5)uλj )

)

+ig

2M√

2φ+

(

−mκe (ν

λU lepλκ(1 − γ5)eκ) + mλ

ν(νλU lep

λκ(1 + γ5)eκ)

+

ig

2M√

2φ−

(

mλe (e

λU lep†

λκ(1 + γ5)νκ) − mκν(e

λU lep†

λκ(1 − γ5)νκ)

− g

2

mλν

MH(νλνλ) −

g

2

mλe

MH(eλeλ) + ig

2

mλν

Mφ0(νλγ5νλ) − ig

2

mλe

Mφ0(eλγ5eλ) − 1

4νλ MR

λκ (1 − γ5)νκ −1

4νλ MR

λκ (1 − γ5)νκ + ig

2M√

2φ+

(

−mκd(u

λj Cλκ(1 − γ5)dκ

j ) + mλu(u

λj Cλκ(1 + γ5)dκ

j

)

+

ig

2M√

2φ−

(

mλd(d

λj C

†λκ(1 + γ5)uκ

j ) − mκu(d

λj C

†λκ(1 − γ5)uκ

j

)

− g

2

mλu

MH(uλ

j uλj ) −

g

2

mλ

d

MH(dλ

j dλj ) + ig

2

mλu

Mφ0(uλ

j γ5uλ

j ) −ig

2

mλ

d

Mφ0(dλ

j γ5dλ

j ) + Ga∂2Ga + gsfabc∂µG

aGbgcµ +

X+(∂2 −M2)X+ + X−(∂2 −M2)X−+X0(∂2 − M2

c2w)X0 + Y ∂2Y + igcwW+

µ (∂µX0X− −

∂µX+X0)+igswW+

µ (∂µY X− − ∂µX+Y ) + igcwW−µ (∂µX

−X0 −

∂µX0X+)+igswW−

µ (∂µX−Y − ∂µY X+) + igcwZ0µ(∂µX+X+ −

∂µX−X−)+igswAµ(∂µX+X+ −

∂µX−X−)−1

2gM

(

X+X+H + X−X−H + 1

c2wX0X0H

)

+1−2c2w2cw

igM(

X+X0φ+ − X−X0φ−)

+1

2cwigM

(

X0X−φ+ − X0X+φ−)

+igMsw

(

X0X−φ+ − X0X+φ−)

+1

2igM

(

X+X+φ0 − X−X−φ0)

.

1

We have a formula: does it mean we understand?

The task of mathematics:• Is there a simple principle behind?• Does the formula follow?• What does it mean?

Geometry: guiding principle for tackling complexity

“Collision II” Dawn N. Meson, San Francisco artist

5

Geometrization of physics

Kaluza–Klein theory

General Relativity:

gravity = metric on 4-dim spacetime

Electromagnetism: 5-dimensions

Circle bundle over spacetime

⇒ Gauge theories

6

Evolution of the Kaluza Klein idea, I

Gauge theories: vector bundles

connections and curvatures (gauge potentials, force fields)

sections (matter particles/fields)

bundle symmetries (gauge symmetries)

7

Evolution of the Kaluza Klein idea, II

String theory: fibration of Calabi-Yau

manifolds over 4-dim spacetime

“Kaluza-Klein (Invisible Architecture III)” Dawn N.Meson

8

Evolution of the Kaluza Klein idea, III

Noncommutative geometry (Connes, 1980s)

Product of spacetime by a

“noncommutative space”

Jackson Pollock “Untitled N.3”

9

What is a noncommutative space?

Example: composition law for spectral lines

Compose when target of one is source of next:

G = groupoid

10

First instance of noncommutive variablesin Quantum Mechanics

(

a b

c d

) (

u v

x y

)

=

(

au + bx av + by

cu + dx cv + dy

)

6=

(

au + cv bu + dv

ax + cy bx + dy

)

=

(

u v

x y

) (

a b

c d

)

Observables in quantum mechanics usually don’tcommute ⇒ “Uncertainty principle”NCG: Geometry of Quantum Mechanics

11

The main idea: There are more dimensions

than the 4-dimensions of space and time

The extra dimensions account for forces and particles

and their interactions (internal symmetries)

12

But when is a mathematical model a good

model of the physical world?

• Simplicity: difficult computations follow

from simple principles

• Predictive power: new insight on physics,

new testable calculations

• Elegance: “entia non sunt multiplicanda

praeter necessitatem” (Ockham’s razor)

More than one mathematical model may be

needed to explain different aspects of the same

physical phenomenon

13

Example: Composite particles (baryons)

Classification in terms of elementary particles(quarks)

Mathematics: Lie group SU(3)• Linear representations of Lie groups

14

Example: Noncommutative Geometry:

Standard Model Lagrangian computed

from simple input

Matrix algebras and quaternions

A = C ⊕ H ⊕ M3(C)

Predictions: Higgs mass, mass relation

Mathematics: Spectral triples, spectral action

15

Mathematics and reality

The test of experiments

(different models predict different Higgs masses)

Large Hadron Collider (CERN)

September 2008 (?)

16