Particle Physics Quiz EPPOG Hands on Particle Physics Masterclasses 2011.
Mathematicians look at particle physics - math.fsu.edumarcolli/PopTalkSlidesFinal.pdf · Elementary...
Transcript of Mathematicians look at particle physics - math.fsu.edumarcolli/PopTalkSlidesFinal.pdf · Elementary...
We do not do these things
because they are easy.
We do them because they are hard.
(J.F.Kennedy – Sept. 12, 1962)
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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?)
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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
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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)
.
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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
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Geometrization of physics
Kaluza–Klein theory
General Relativity:
gravity = metric on 4-dim spacetime
Electromagnetism: 5-dimensions
Circle bundle over spacetime
⇒ Gauge theories
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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)
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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
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Evolution of the Kaluza Klein idea, III
Noncommutative geometry (Connes, 1980s)
Product of spacetime by a
“noncommutative space”
Jackson Pollock “Untitled N.3”
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What is a noncommutative space?
Example: composition law for spectral lines
Compose when target of one is source of next:
G = groupoid
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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
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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)
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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
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Example: Composite particles (baryons)
Classification in terms of elementary particles(quarks)
Mathematics: Lie group SU(3)• Linear representations of Lie groups
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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
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