2006 high energy lecture 1 -...
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IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Introduction to gauge theory2006 High energy lecture 1
�©� �©� �&³
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September 22, 2006
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Table of Contents
Introduction
Dirac equation
Quantization of Fields
Gauge Symmetry
Spontaneous Gauge Symmetry Breaking
Standard Model
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
ReferencesThe basic
References for quantum field theory
“Quark and Leptons” Halzen and Martin“Quantum Field Theory” Ryder“Quantum Field Theory” Mandl and Show“Gauge Theory of Elementary Particle Physics” Cheng and Li“Quantum Field Theory in a Nutshell” Zee“An Introduction to Quantum Field Theory” Peskin and Schroederand many more. . .
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
ReferencesThe basic
Introduction
The Standard Model (SM) is The basis of High Energy Physics.
SM is a local quantum gauge field theory with Spontaneous gaugesymmetry breaking mechanism a.k.a Higgs Mechanism.
Object of this lecture is to learn the basic concept of the gaugesymmetries and their breaking mechanism to understand SM.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
ReferencesThe basic
Introduction
The Standard Model (SM) is The basis of High Energy Physics.
SM is a local quantum gauge field theory with Spontaneous gaugesymmetry breaking mechanism a.k.a Higgs Mechanism.
Object of this lecture is to learn the basic concept of the gaugesymmetries and their breaking mechanism to understand SM.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
ReferencesThe basic
The lecture will be a short introduction course to Quantum fieldtheory and gauge theory.
A modern approach to this subject is to use path integral andpropagator theory.
However, we will follow traditional Lagrangian approach. For thepath integral method, look for the references.
In elementary particle physics, we use the unit where ~ = c = 1.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Dirac equation
The classical field theory which describes EM field is consistentwith Special theory of relativity
but not with Quantum mechanics.
The Schrodinger equation describes low energy electrons in atombut it is not consistent with relativity.
Non-relativistic quantum mechanics cannot describe High energyparticle interactions.Need to combine quantum mechanics with special relativity.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Dirac equation
The classical field theory which describes EM field is consistentwith Special theory of relativitybut not with Quantum mechanics.
The Schrodinger equation describes low energy electrons in atombut it is not consistent with relativity.
Non-relativistic quantum mechanics cannot describe High energyparticle interactions.Need to combine quantum mechanics with special relativity.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Dirac equation
The classical field theory which describes EM field is consistentwith Special theory of relativitybut not with Quantum mechanics.
The Schrodinger equation describes low energy electrons in atom
but it is not consistent with relativity.
Non-relativistic quantum mechanics cannot describe High energyparticle interactions.Need to combine quantum mechanics with special relativity.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Dirac equation
The classical field theory which describes EM field is consistentwith Special theory of relativitybut not with Quantum mechanics.
The Schrodinger equation describes low energy electrons in atombut it is not consistent with relativity.
Non-relativistic quantum mechanics cannot describe High energyparticle interactions.Need to combine quantum mechanics with special relativity.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Dirac equation
The classical field theory which describes EM field is consistentwith Special theory of relativitybut not with Quantum mechanics.
The Schrodinger equation describes low energy electrons in atombut it is not consistent with relativity.
Non-relativistic quantum mechanics cannot describe High energyparticle interactions.
Need to combine quantum mechanics with special relativity.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Dirac equation
The classical field theory which describes EM field is consistentwith Special theory of relativitybut not with Quantum mechanics.
The Schrodinger equation describes low energy electrons in atombut it is not consistent with relativity.
Non-relativistic quantum mechanics cannot describe High energyparticle interactions.Need to combine quantum mechanics with special relativity.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
In 1928, Dirac realized that the wave equation can be linear to thespace time derivative ∂µ ≡ ∂/∂xµ.
(iγµ∂µ −m)ψ = 0 (1)
Applying (iγµ∂µ −m) to (1) leads(1
2{γµ, γν}∂µ∂ν + m2
)ψ = 0 (2)
where {A,B} = AB + BA is anticommutator. If
{γµ, γν} = 2ηµν (3)
ηµν is the Minkowski metric η00 = 1, ηjj = −1 and otherwise zero.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Then the Dirac equation becomes
(∂2 + m2)ψ = 0
This is the same form as Klein-Gordon equation for the scalarfields.
(∂2 + m2)φ = 0
γµ satisfies Clifford algebra (3) can be written as 4× 4 matrices.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
One representation of γµ satisfies (3) is
γ0 =
(I 00 −I
)γi =
(0 σi
−σi 0
)(4)
I is 2× 2 identity matrix and σi (i = 1, 2, 3) are Pauli matrices.
It is called Dirac basis.
Some useful notations:γµ ≡ ηµνγ
µ
6p ≡ γµpµ
e.g. Dirac equation(i 6∂ −m)ψ = 0
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
The matrixγ5 ≡ iγ0γ1γ2γ3
has the form in Dirac basis
γ5 =
(0 II 0
)(5)
and anticommute with γµ
{γ5, γµ} = 0
(γ5)† = γ5, (γ5)2 = 1
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
With the 6 matrices
σµν ≡ i
2[γµ, γν ]
{1, γµ, σµν , γµγ5, γ5} form a complete basis of 16 elements.
All 4× 4 matrices can be written as a linear combination of above16 matrices.
γµ can have different basis with the same physics.e.g. Weyl basis,
γ0 =
(0 II 0
), γi =
(0 σi
−σi 0
), γ5 =
(−I 00 I
)
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
With the 6 matrices
σµν ≡ i
2[γµ, γν ]
{1, γµ, σµν , γµγ5, γ5} form a complete basis of 16 elements.
All 4× 4 matrices can be written as a linear combination of above16 matrices.
γµ can have different basis with the same physics.e.g. Weyl basis,
γ0 =
(0 II 0
), γi =
(0 σi
−σi 0
), γ5 =
(−I 00 I
)
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
If we transform the spinors to momentum space
ψ(x) =
∫d4p
(2π)4e−ipxψ(p)
The Dirac equation becomes
(γµpµ −m)ψ(p) = 0 (6)
Dirac spinor ψ can be divided into two 2-component spinors,
ψ =
(φχ
)
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
In Dirac basis, (γ0 − 1)ψ(p) = 0 in the rest frame pµ = (m,~0).
Only φ describes electron, which has two component.
For slowly moving electron, χ(p) is very small.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
In Dirac basis, (γ0 − 1)ψ(p) = 0 in the rest frame pµ = (m,~0).
Only φ describes electron, which has two component.
For slowly moving electron, χ(p) is very small.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Lorentz transformation is defined as
Λ = e−12ωµνJµν
anti-symmetric ωµν = −ωνµ are 3 rotation and 3 boost parameters.
J ij are rotation generators and J0i are boost generators.
The coordinate xα transforms
x ′α = Λαβxβ
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Quantization of FieldsGauge Symmetry
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Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Spinors transforms under Lorentz transformation is
ψ′(x ′) = S(Λ)ψ(x)
whereS(Λ) = e−
i4ωµνσµν
AlsoSγνS−1 = Λν
µγµ
andS(Λ)† = γ0e
i4ωµνσµν
γ0
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IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Define ψ = ψ†γ0 then
ψ(x)ψ(x) is invariant under Lorentz transformation (scalar).
ψ(x)γµψ(x) transform as Lorentz vector.
ψ(x)γ5ψ(x) transform as a pseudoscalar.
ψ(x)γ5γµψ(x) transform as Lorentz pseudovector.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Define ψ = ψ†γ0 then
ψ(x)ψ(x) is invariant under Lorentz transformation (scalar).
ψ(x)γµψ(x) transform as Lorentz vector.
ψ(x)γ5ψ(x) transform as a pseudoscalar.
ψ(x)γ5γµψ(x) transform as Lorentz pseudovector.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Lagrangian L and Lagrangian density L is defined from the action
S =
∫dtL =
∫d4xL
In High energy physics(HEP) Lagrangian means Lagrangian densityL.
If the L is a function of field φ(x), the Euler-Lagrange eq. ofmotion should satisfied.
∂µ
(∂L
∂(∂µφ)
)− ∂L∂φ
= 0 (7)
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IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Lagrangian of free Dirac field
L = ψ(iγµ∂µ −m)ψ (8)
From the eq. of motion
∂µ
(∂L
∂(∂µψ)
)− ∂L∂ψ
= 0
Dirac equation can be obtained
(iγµ∂µ −m)ψ = 0
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Define chiral projection,
ψL(x) = PLψ(x) , ψR(x) = PRψ(x) .
The projection operators
PL =1− γ5
2, PR =
1 + γ5
2.
Then the Dirac spinor is sum of two chiral components
ψ(x) = ψL(x) + ψR(x)
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
In Weyl basis
γ5 =
(−I 00 I
).
Thus
ψ(x) =
(ψL
ψR
).
Some properties to notice
P2L = PL, P2
R = PR , PLPR = 0
γ5ψL = −ψL, γ5ψR = +ψR .
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IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Dirac Lagrangian can be written in chiral components
L = ψ(iγµ∂µ −m)ψ
= ψLiγµ∂µψL + ψR iγµ∂µψR −m(ψLψR + ψRψL) (9)
If m = 0, ψL and ψR are independent and have additionalsymmetry
ψL −→ e iθLψL, ψR −→ e iθRψR ,
Weak interaction is called ‘chiral’ because it only interacts withleft-handed leptons.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Dirac Lagrangian can be written in chiral components
L = ψ(iγµ∂µ −m)ψ
= ψLiγµ∂µψL + ψR iγµ∂µψR −m(ψLψR + ψRψL) (9)
If m = 0, ψL and ψR are independent and have additionalsymmetry
ψL −→ e iθLψL, ψR −→ e iθRψR ,
Weak interaction is called ‘chiral’ because it only interacts withleft-handed leptons.
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Dirac equationDirac gamma matricesSpinor and Lorentz transformationDirac LagrangianChirality
Further readings
Check the references for Charge conjugation, Paritytransformation, CP and CPT .
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Quantization of FieldsGauge Symmetry
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Noether’s theoremScalar field quantizationDirac field quantization
Noether’s theorem
If a Lagrangian L with a field φa is invariant of a continuoustransformation φa −→ φa + δφa
0 = δL =δLδφa
δφa +δL
δ(∂µφa)δ(∂µφa) (10)
use the eq. of motion
δLδφa
= ∂µ
(δL
δ(∂µφa)
)(10) becomes
0 = ∂µ
(δL
δ(∂µφa)δφa
)�©� �©� �&³ Introduction to gauge theory
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We define a current
Jµ ≡ δLδ(∂µφa)
δφa
Then∂µJµ = 0
We have a conserved current Jµ.
Noether’s Theorem
A conserved current is associated with a continuous symmetry ofthe Lagrangian.
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Charge is defined as
Q =
∫d3xJ0 =
∫d3x
δLδ(∂0φa)
δφa. (11)
Since dQ/dt = 0, the charge is conserved.
π(x) ≡ δLδ(∂0φa)
is canonical momentum (density) corresponding to φa.
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Free scalar field Lagrangian
L = |∂µφ|2 −m2|φ|2 (12)
of Klein-Gordon equation of motion
(∂2 + m2)φ = 0.
The Noether current
Jµ = i [(∂µφ∗)φ− φ∗(∂µφ)], (13)
corresponds the symmetry φ→ e iθφ.
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Quantization of FieldsGauge Symmetry
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For free fermion
L = ψ(iγµ∂µ −m)ψ (14)
The Noether current
Jµ = ψγµψ (15)
corresponds the symmetry ψ → e iθψ.
∂µJµ = (∂µψ)γµψ + ψγµ∂µψ = (imψ)γµψ + ψγµ(−imψ) = 0
This symmetry is called U(1) global symmetry, since θ is the samefor any space-time x .
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Quantization of scalar field
For free scalar field, the canonical momentum is π = φ.
Being a “quantum” field theory requires:
1. φ(x) and π(x) becomes operator
2. and they satisfy canonical commutator relation.
[φ(~x , t), π(~y , t)] = iδ(3)(~x − ~y)
[φ(x), φ(y)] = [π(x), π(y)] = 0
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If E~p =√|~p|2 + m2,
φ(x) =
∫d3p
(2π)31√2E~p
(a(~p)e−ip·x + a(~p)†e ip·x
)∣∣∣p0=E~p
π(x) = ∂0φ(x) (16)
[a(~p), a(~p′)†] = (2π)3δ(3)(~p − ~p′), (17)
a(~p)† creates one particle state from vacuum |0〉
|~p〉 =√
2E~pa(~p)†|0〉
a(~p) destroys vacuum a(~p)|0〉 = 0.
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Quantum field is a harmonic oscillator with continuous degree offreedom.
φ(x) acting on vacuum, create a particle at x .
φ(x)|0〉 =
∫d3p
(2π)31
2E~pe−ip·x |~p〉
〈0|φ(x)|~p〉 = e ip·x is free particle wave function.
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Dirac field quantization
For free Dirac field, canonical momentum is
π =δL
δ(∂0ψ)= iψ†.
Not like the scalar case, fermion field should satisfyanticommutation relation
{ψa(~x , t), ψ†b(~y , t)} = δ(3)(~x − ~y)δab
{ψa(~x , t), ψb(~y , t)} = {ψ†a(~x , t), ψ†b(~y , t)} = 0
a, b are spinor components
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We can write the field operators
ψ(x) =
∫d3p
(2π)31√2E~p
∑s
(b(p, s)u(p, s)e−ip·x + d†(p, s)v(p, s)e ip·x
)ψ(x) =
∫d3p
(2π)31√2E~p
∑s
(d(p, s)v(p, s)e−ip·x + b†(p, s)u(p, s)e ip·x
)s = 1, 2 is spin index.
{b(p, s), b†(p′, s ′)} = {d(p, s), d†(p′, s ′)} = (2π)3δ(3)(~p − ~p′)δss′
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Both b(p, s) and d(p, s) annihilate vacuum
b(p, s)|0〉 = d(p, s)|0〉 = 0
b†(p, s) and d†(p, s) creates particle with energy momentum p
but they are charge conjugated state with each other.
We define b†(p, s) creates a fermion and
d†(p, s) creates an anti-fermion.
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Quantization of FieldsGauge Symmetry
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From the Noether current Jµ = ψγµψ, there is a conserved charge
Q =
∫d3xψ†(x)ψ(x) =
∫d3p
(2π)3
∑s
(b†(p, s)b(p, s)− d†(p, s)d(p, s)
)b†(p, s) creates a fermion with +1 charge and d†(p, s) creates afermion with −1 charge.
For instance Qe is the electric charge of electrons.
U(1) symmetry must be related with electric charge conservation!
�©� �©� �&³ Introduction to gauge theory
IntroductionDirac equation
Quantization of FieldsGauge Symmetry
Spontaneous Gauge Symmetry BreakingStandard Model
Noether’s theoremScalar field quantizationDirac field quantization
From the Noether current Jµ = ψγµψ, there is a conserved charge
Q =
∫d3xψ†(x)ψ(x) =
∫d3p
(2π)3
∑s
(b†(p, s)b(p, s)− d†(p, s)d(p, s)
)b†(p, s) creates a fermion with +1 charge and d†(p, s) creates afermion with −1 charge.
For instance Qe is the electric charge of electrons.
U(1) symmetry must be related with electric charge conservation!
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Maxwell equationGauge invarianceComplex Scalar FieldGauge field quantization
Maxwell equation
The Maxwell equation is eq. of motion for photon field Aµ(x)
∂µFµν = 0 or ∂2Aν − ∂ν∂µAµ = 0 (18)
whereFµν = ∂µAν − ∂νAµ
The Lagrangian for photon is
LMax = −1
4FµνF
µν (19)
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Lint = eAµψγµψ is a covariant interaction term between Dirac and
Maxwell field where e is a coupling constant.
Then the combination of electro and fermion Lagrangian is
L = LDirac + LMax + Lint
= ψ(iγµDµ −m)ψ − 1
4FµνF
µν (20)
Dµ = ∂µ − ieAµ is a covariant derivative.
Then the Dirac equation with electromagnetic interaction is
[iγµ(∂µ − ieAµ)−m]ψ = 0 (21)
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Gauge invarianceThe most significant property of the Lagrangian (20) is that it isinvariant under gauge transformation.
LMax = −14FµνF
µν is invariant under the transformation
Aµ(x) → Aµ(x) +1
e∂µΛ(x)
for any scalar function Λ(x)
While LDirac is invariant under
ψ(x) → e iΛ(x)ψ(x)
only if Λ(x) = constant.
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However the total Lagrangian with interaction term (20)
L = ψ(iγµDµ −m)ψ − 1
4FµνF
µν
and covariant Dirac equation (21)
[iγµ(∂µ − ieAµ)−m]ψ = 0
is invariant under local U(1) gauge symmetry.
ψ(x) → e iΛ(x)ψ(x)
Aµ(x) → Aµ(x) +1
e∂µΛ(x)
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The gauge boson mass term M2AAµAµ is not invariant
under the gauge transformation
Aµ(x) → Aµ(x) +1
e∂µΛ(x).
Thus, the gauge invariant field Aµ should be massless.
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Complex Scalar Field
φ = (φ1 + iφ2)√
2 , φ∗ = (φ1 − iφ2)√
2
The Lagrangian of a free complex scalar field
L = (∂µφ)(∂µφ∗)−m2φ∗φ (22)
is invariant under global gauge transformation
φ→ e iΛφ , φ∗ → e−iΛφ∗ ,
where Λ is a real constant.
However, it is not invariant for local gauge Λ(x)
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Complex Scalar Field
φ = (φ1 + iφ2)√
2 , φ∗ = (φ1 − iφ2)√
2
The Lagrangian of a free complex scalar field
L = (∂µφ)(∂µφ∗)−m2φ∗φ (22)
is invariant under global gauge transformation
φ→ e iΛφ , φ∗ → e−iΛφ∗ ,
where Λ is a real constant.
However, it is not invariant for local gauge Λ(x)
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For small Λ(x), the local gauge transformation can be written as
φ→ φ+ iΛ(x)φ ,
∂µφ→ ∂µφ+ iΛ(x)(∂µφ) + i(∂µΛ(x))φ .
Then Euler-Lagrange equation leads (10)
δL = Jµ∂µΛ(x) (23)
where the conserved current is
Jµ = i [(∂µφ∗)φ− φ∗(∂µφ)].
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As the case of fermions, we can add the interaction therm
Lint = −eJµAµ (24)
between scalar field and gauge field. Then
δLint = −e(δJµ)Aµ − Jµ∂µΛ, (25)
whereδJµ = 2|φ|2∂µΛ
for
Aµ(x) → Aµ(x) +1
e∂µΛ(x).
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To cancel the extra term −e(δJµ)Aµ, we must add
Lext = e2AµAµ|φ|2
The total Lagrangian
Lscalar = (Dµφ)(Dµφ)∗ −m2φ∗φ− 1
4FµνFµν (26)
is local U(1) gauge symmetric, where
Dµφ = (∂µ − ieAµ)φ
is a covariant derivative.
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With the gauge invariant Lagrangian (20), the total Lagrangian
L = ψ(iγµDµ −mf )ψ + (Dµφ)(Dµφ)∗ −m2sφ
∗φ− 1
4FµνFµν (27)
gives a complete description of the world with
1. a local U(1) symmetric charged scalar with mass ms ,
2. a local U(1) symmetric charged fermion with mass mf ,
3. a local U(1) symmetric neutral massless gauge boson(photon).
Or, in one sentence, Quantum electrodynamics (QED).
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With the gauge invariant Lagrangian (20), the total Lagrangian
L = ψ(iγµDµ −mf )ψ + (Dµφ)(Dµφ)∗ −m2sφ
∗φ− 1
4FµνFµν (27)
gives a complete description of the world with
1. a local U(1) symmetric charged scalar with mass ms ,
2. a local U(1) symmetric charged fermion with mass mf ,
3. a local U(1) symmetric neutral massless gauge boson(photon).
Or, in one sentence, Quantum electrodynamics (QED).
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With the gauge invariant Lagrangian (20), the total Lagrangian
L = ψ(iγµDµ −mf )ψ + (Dµφ)(Dµφ)∗ −m2sφ
∗φ− 1
4FµνFµν (27)
gives a complete description of the world with
1. a local U(1) symmetric charged scalar with mass ms ,
2. a local U(1) symmetric charged fermion with mass mf ,
3. a local U(1) symmetric neutral massless gauge boson(photon).
Or, in one sentence, Quantum electrodynamics (QED).
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With the gauge invariant Lagrangian (20), the total Lagrangian
L = ψ(iγµDµ −mf )ψ + (Dµφ)(Dµφ)∗ −m2sφ
∗φ− 1
4FµνFµν (27)
gives a complete description of the world with
1. a local U(1) symmetric charged scalar with mass ms ,
2. a local U(1) symmetric charged fermion with mass mf ,
3. a local U(1) symmetric neutral massless gauge boson(photon).
Or, in one sentence, Quantum electrodynamics (QED).
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With the gauge invariant Lagrangian (20), the total Lagrangian
L = ψ(iγµDµ −mf )ψ + (Dµφ)(Dµφ)∗ −m2sφ
∗φ− 1
4FµνFµν (27)
gives a complete description of the world with
1. a local U(1) symmetric charged scalar with mass ms ,
2. a local U(1) symmetric charged fermion with mass mf ,
3. a local U(1) symmetric neutral massless gauge boson(photon).
Or, in one sentence, Quantum electrodynamics (QED).
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For a free (non-interacting) gauge field ( E~p =√|~p|2 + m2 )
Aµ(x)=
∫d3p
(2π)31√2E~p
3∑r=0
(ar (~p)εµr e−ip·x + ar (~p)†εµ∗r e ip·x
)(28)
εµ: polarization vector, r : indices of polarization.
[ar (~p), as(~p′)†] = (2π)3δrsδ
(3)(~p − ~p′), (29)
is a quantization condition for a photon field,
where one photon state is |p〉 ∝ ar (~p)†|0〉 and ar (~p)|0〉 = 0.
Photon is neutral since it is a real field
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Renormalization
Only a rough description of renormalization will be presented.
Read the Field Theory references for the details of this subject.
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If a system of particles |{ki}〉in scatters to |{pf }〉out
{ki}, {pf } are set of initial and final momenta of particles.
The matrix element M is defined as
out〈{pf }|{ki}〉in = (2π)(4)δ(∑
ki −∑
pf ) · iM
Without going through the details, the scattering cross section is
dσ ∝M2.
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M can be expanded with time-evolution, i.e. the Hamiltonian.
out〈{pf }|{ki}〉in = limT→∞
〈{pf }|e−iH(2T )|{ki}〉 (30)
Amplitude of scattering can be expanded with interaction terms.
Interaction term of fields are proportional to coupling constant G .
e.g. (mf ,m2s , e, . . . )
The leading order term of (30) is proportional to G and higherorder term will be G 2, G 3, etc.
This is a basic idea of perturbation expansion.
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M can be expanded with time-evolution, i.e. the Hamiltonian.
out〈{pf }|{ki}〉in = limT→∞
〈{pf }|e−iH(2T )|{ki}〉 (30)
Amplitude of scattering can be expanded with interaction terms.
Interaction term of fields are proportional to coupling constant G .
e.g. (mf ,m2s , e, . . . )
The leading order term of (30) is proportional to G and higherorder term will be G 2, G 3, etc.
This is a basic idea of perturbation expansion.
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Higher order term in M contains momentum integrals.
And the integral diverges with p →∞.
It could be fatal problem of the field theory itself.
The solution that the theorist found(?) is very simple.
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Higher order term in M contains momentum integrals.
And the integral diverges with p →∞.
It could be fatal problem of the field theory itself.
The solution that the theorist found(?) is very simple.
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Cut it off!
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Integrate momentum p to finite cut-off Λ to make the theory finite.
By summing up the perturbation series and re-normalizing it,
we can obtain the physical values.
The final result can depend on at most log(Λ).
The physical parameters (couplings, masses) varies with the energyat the log scale.
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Renormalizability
The leading order term is proportional to G .
If dimension [G ] = x , roughly
the perturbation give G 2Λ−2x contribution.
If x < 0, the sum depends strongly on Λ and renormalization fails.
[G ] > 0 is a condition for renormalizable interaction.
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Renormalizability
The leading order term is proportional to G .
If dimension [G ] = x , roughly
the perturbation give G 2Λ−2x contribution.
If x < 0, the sum depends strongly on Λ and renormalization fails.
[G ] > 0 is a condition for renormalizable interaction.
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From [L] = 4 and [x ] = −1, [∂µ] = [m] = 1,
We obtain for scalar fields
[φ] = [Aµ] = 1,
and for fermion fields
[ψ] =3
2.
Therefore, [e] = 0.
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Any term with dimension more than 4 need a coupling [G ] < 0
and is not renormalizable.
Only gauge invariant dimension 4 term is |φ|4.
The renormalizable Lagrangian with U(1) gauge symmetry is
L = ψ(iγµDµ −mf )ψ + |Dµφ|2 − V (|φ|2) +1
4FµνFµν , (31)
where
V (|φ|2) = µ2|φ|2 + λ|φ|4. (32)
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The minimum of potential (32) is at
φ = 0 and |φ|2 = −µ2
2λ
For λ > 0 and real mass µ, φ = φ∗ = 0 is the absolute minimum.
For µ2 < 0, the potential has more than one minimum.
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If q = |φ|, there is two minimum in the potential.‘
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Since φ = (φ1 + iφ2)√
2, V (φ1, φ2) has a shape of Mexican hat
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Spontaneous Symmetry Breaking
What will happen, if there are more than one ground state?
Like coin flipping, system can choose each ground state with equalprobability.
Even after the system select a specific ground state,
the Lagrangian has the symmetry.
However, the solution itself does not have a symmetry, anymore.
The symmetry is broken spontaneously.
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Spontaneous Symmetry Breaking
What will happen, if there are more than one ground state?
Like coin flipping, system can choose each ground state with equalprobability.
Even after the system select a specific ground state,
the Lagrangian has the symmetry.
However, the solution itself does not have a symmetry, anymore.
The symmetry is broken spontaneously.
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Spontaneous Symmetry Breaking
What will happen, if there are more than one ground state?
Like coin flipping, system can choose each ground state with equalprobability.
Even after the system select a specific ground state,
the Lagrangian has the symmetry.
However, the solution itself does not have a symmetry, anymore.
The symmetry is broken spontaneously.
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The spontaneous symmetry breaking of U(1)
If the scalar field has imaginary mass,
it can have continuous (Mexican hat shape) ground state. Choosea vacuum,
〈φ〉 = v =
√−µ2
2λ
and parametrize it as
φ(x) = ρ(x) exp[iθ(x)]
then〈ρ〉 = v , 〈θ〉 = 0.
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Insert φ(x) = (v + χ(x))e iθ(x) to Lagrangian
L = |∂φ|2 − µ2|φ|2 − λ|φ|4, (33)
= (∂χ)2 − λv4 − 4λv2χ2 − 4λvχ3 − λχ4 + (v + χ)2(∂θ)2.
χ has a real mass and θ is massless.
θ is called Nambu-Goldstone boson.
(33) does not have global U(1) symmetry.
It is broken spontaneously.
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Insert φ(x) = (v + χ(x))e iθ(x) to Lagrangian
L = |∂φ|2 − µ2|φ|2 − λ|φ|4, (33)
= (∂χ)2 − λv4 − 4λv2χ2 − 4λvχ3 − λχ4 + (v + χ)2(∂θ)2.
χ has a real mass and θ is massless.
θ is called Nambu-Goldstone boson.
(33) does not have global U(1) symmetry.
It is broken spontaneously.
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Goldstone theorem
Whenever a continuous symmetry is spontaneously broken,
a massless (Nambu-Goldstone) boson emerge.
Any degree of freedom moves along with the flat direction
does not have a mass.
No mass term m2φ2
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Goldstone theorem
Whenever a continuous symmetry is spontaneously broken,
a massless (Nambu-Goldstone) boson emerge.
Any degree of freedom moves along with the flat direction
does not have a mass.
No mass term m2φ2
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Anderson-Higgs MechanismIf we add gauge field
L = −1
4FµνFµν + |Dµφ|2 − µ2|φ|2 − λ|φ|4, (34)
= −1
4FµνFµν + e2ρ2(Bµ)2 + (∂ρ)2 − µ2ρ2 − λρ4.
where Bµ = Aµ − (1/e)∂µθ and
Fµν = ∂µAν − ∂νAµ = ∂µBν − ∂νBµ
are invariant under U(1) transformation,
φ→ e iΛφ (θ → θ + Λ), Aµ → Aµ +1
e∂µΛ
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If the symmetry is broken spontaneously,
〈φ〉 = v =
√−µ2
2λ
and insert ρ = v + χ to (35),
L = −1
4FµνFµν + (ev)2(Bµ)2 + e2(2vχ+ χ2)(Bµ)2
+(∂χ)2 − 4λv2χ2 − 4λvχ3 − λχ4 − λv4 (35)
There is no Goldstone boson θ, while Bµ gains a mass M =√
2ev .
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If the symmetry is broken spontaneously,
〈φ〉 = v =
√−µ2
2λ
and insert ρ = v + χ to (35),
L = −1
4FµνFµν + (ev)2(Bµ)2 + e2(2vχ+ χ2)(Bµ)2
+(∂χ)2 − 4λv2χ2 − 4λvχ3 − λχ4 − λv4 (35)
There is no Goldstone boson θ, while Bµ gains a mass M =√
2ev .
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In the Anderson-Higgs (also, Landau-Ginzburg-Kibble) mechanism,
the massless degree of freedom is eaten by gauge field.
As a consequence, the gauge field becomes massive.
Since the massive photon has an extra degree of freedom inaddition to two polarizations,
the total degree of freedom does not change.
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Ferromagnetism
The Hamiltonian of Ferromagnet has rotational symmetry of spin,
A spin can point any direction which is global SO(3) symmetry.
If spin aligns one direction, SO(3) is spontaneously broken to
SO(2): a symmetry of rotation around spin direction.
Since a continuous symmetry is spontaneously broken,
there exists Goldstone mode called spin wave.
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Superconductivity
When temperature went down SO(2) which is equivalent to U(1)is also broken spontaneously.
At low temperature a pair of electron in superconducting materialact like a boson (Higgs scalar).
This is the same case as we discussed, local U(1) gauge symmetry.
There is no Goldstone mode, but the photon becomes massive.
Which explains why electric force becomes short-ranged andmagnetic field cannot penetrate in superconducting material.
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The Standard Model
I will close this lecture with a comment about the Standard Model.
The Standard Model(SM) is a SU(2)L × U(1)Y local gauge theory
with 6 quarks and leptons as a basis.
SU(2)L is non-Abelian gauge symmetry, which is rather
complicated than Abelian gauge group U(1)
But the basic concept is the same.
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In SM, instead of U(1), SU(2) is broken by Higgs mechanism,
As a result, there exist three massive gauge bosons W±,Z .
Also the quarks and leptons which are chiral field to SU(2)L
and originally massless, obtain the masses.
SM is extremely successful, both in theory and experiments.
Except, we have not seen Higgs scalar, yet.
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In SM, instead of U(1), SU(2) is broken by Higgs mechanism,
As a result, there exist three massive gauge bosons W±,Z .
Also the quarks and leptons which are chiral field to SU(2)L
and originally massless, obtain the masses.
SM is extremely successful, both in theory and experiments.
Except, we have not seen Higgs scalar, yet.
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Read the references on Quantum field theory!
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