Redshift Academy


Gauge Theories (Yang-Mills)
---------------------------

Abelian Gauge Theory
--------------------

A gauge theory is a field theory in which some GLOBAL continuous 
symmetry of the theory is replaced by a stricter LOCAL continuous
symmetry requirement.  The imposition of a local symmetry requires
the introduction of new fields that make the Lagrangian invariant 
under the local transformation.  For each group generator there 
necessarily arises a corresponding field (usually a vector field) 
called the gauge field.  Consider the following Lagrangian for a 
simple complex field:

L = ∂_μφ∂^μφ* - m²φφ*

It is easy to see that the global transformation φ' = φexp(iθ) leaves
the Lagrangian invariant. Now consider if one wished to not only 
make global changes of phase but also local transformations of the 
form φ' = φexp(iθ(x)).  Now the situation is not so straightforward 
since θ is now a function of x and the kinetic term picks up a 
derivative of θ(x).  As a result the action is no longer invariant 
under this type of change.

In order to make it invariant and enforce such a symmetry, one must 
rewrite the transformation law so that there is a new type of 
derivative D_μφ, which under the change of phase on φ transforms in 
the same fashion, D_μφ -> exp(iθ(x))D_μφ.  D_μ is called the GAUGE 
COVARIANT DERIVATIVE and is defined as:

D_μ = ∂_μ + igA_μ

Where A_μ is a new quantity called the GAUGE FIELD and g is a 
coupling constant.  Therefore, gauge fields are included in the 
Lagrangian to ensure its invariance under the local transformations.  

Proof:

φ' = φexp(iθ(x))

φ*' = φ*exp(-iθ(x))

D_μ'φD^μ'φ* = (∂_μφ' + igA_μ'φ')(∂^μφ*' - igA^μ'φ*')

        = (∂_μφ + i∂_μθφ + igA_μ'φ)exp(iθφ)
                          (∂^μφ* - i∂^μθφ* - igA^μ'φ*)exp(-iθφ)

        = (∂_μφ + i∂_μθφ + igA_μ'φ)
                          (∂^μφ* - i∂^μθφ* - igA^μ'φ*)

To get back to the original form we need A_μ' to transform as 
follows:

A_μ' = A_μ - (1/g)∂_μθ and A^μ' = A^μ - (1/g)∂^μθ

Therefore,

D_μ'φD^μ'φ* = (∂_μφ + i∂_μθφ + ig(A_μ - (1/g)∂_μθ)φ)
                          (∂^μφ* - i∂^μθφ* - ig(A^μ - (1/g)∂^μθ)φ*)

        = (∂_μφ + igA_μφ)(∂_μφ* - igA_μφ*)

        = D_μφD^μφ* 

Also,
                    
m²φφ^* -> m²exp(iθ)φexp(-iθ)ψ^*
        
      = m²φφ^*

Aside:  The above also applies to the Dirac Lagrangian.
    _
L = ψ(iγ^μ∂_μ - m)ψ

ψ -> exp(iθ)ψ
_            _
ψ -> exp(-iθ)ψ

Now,

∂_μ(ψexp(1θ)) = exp(iθ)∂_μψ + ψi∂_μθexp(iθ)

The covariant derivative is D_μ = ∂_μ + igA_μ 

Where,

A_μ -> A_μ - (1/g)∂_μθ

Therefore,

D_μ = ∂_μ + ig(A_μ - (1/g)∂_μθ)

D_μ(ψexp(iθ)) = ∂_μ(ψexp(iθ)) + iψexp(iθ)gA_μ - iψexp(iθ)∂_μθ

            = exp(1θ)∂_μψ + iψexp(iθ)∂_μθ + iψexp(iθ)gA_μ - iψexp(iθ)∂_μθ

            = exp(1θ)∂_μψ + iψexp(iθ)gA_μ

            = exp(1θ)[∂_μψ + iψgA_μ]

            = exp(1θ)[∂_μ + igA_μ]ψ

            = exp(1θ)D_μψ

Therefore,
_
ψiγ^μD_μψ 
   _
=> ψexp(-iθ)iγ^μD_μ(ψexp(iθ))
   _
=> ψexp(-iθ)exp(iθ)iγ^μD_μψ
   _
=> ψiγ^μD_μψ

Also,
  _                     _
mψψ => mexp(iθ)ψexp(-iθ)ψ
        _
    => mψψ

Invariance of A_μ by Itself
-------------------------

We have looked at how the introduction of the gauge field makes
φ invariant under a local transformation. However, the gauge
field itself must also have its own gauge invariant dynamic term 
in the Lagrangian.  Consider the construction, F_μν, defined as:

F_μν = ∂_μA_ν - ∂_νA_μ

Intuitively, this is a good choice since it contains the first 
derivatives, ∂_μ, of the field that is characterisitic of all 
Lagrangians.  

If we plug in A_μ' = A_μ - (1/q)∂_μθ into the above we get:

F_μν = ∂_μA_ν + ∂_μ∂_νθ - (∂_νA_μ + ∂_ν∂_μθ)

  = ∂_μA_ν - ∂_νA_μ 

Thus, under the gauge transformation, F_μν, remains unchanged.  To make 
this both gauge and Lorentz invariant and also match the quadratic
nature of the other terms in the Lagrangian, it is necessary to form 
the product:

F_μνF^μν

We can now rewite our a Lagrangian for the gauge field as:

L = -F_μνF^μν

The is the Lagrangian that describes that electromagnetic field in 
the absence of any charges (complex fields are charged fields so in
the absence of charge φ = 0).

F_μν is referred to as the FIELD STRENGTH TENSOR.  In the Abelian 
case this is equivalent to the ELECTROMAGNETIC TENSOR.

Note:  By convention L is normally wriiten as L = -(1/4)F_μνF^μν

In the abelian U(1) theory there is 1 gauge fields whose quanta 
is the photon.

Non-Abelian Gauge Theory (Yang-Mills)
-------------------------------------

Yang and Mills extended the above Abelian theory to Non-Abelian 
groups.  Therefore, it extends the U(1) gauge theory to a gauge 
theory based on the SU(N) group.  Yang–Mills theory seeks to 
describe the behavior of elementary particles using these non-
Abelian Lie groups and forms the basis of our understanding of 
the Standard Model of particle physics.  The Lagrangian for the
gauge fields is:

L = -(1/4)F^aμνF^a_μν

Aside:  This is the kinetic term of the PROCA ACTION that 
describes a massive spin-1 field of mass, m, in Minkowski 
spacetime (i.e. the W and Z bosons).  The Proca action is 
given by:

L = -(1/4)F^μνF_μν + m²A_μA^μ

The mass term of the Proca Lagrangian is not invariant under 
a gauge transformation since (A_μ + ∂_μθ)(A^μ + ∂^μθ) ≠ m²A_μA^μ.  The 
consequence is that m = 0 unless the symmetry is spontaneously
broken.  This discussed in detail in the section on the Higgs 
mechanism.

Continuing, the gauge covariant derivative becomes:

D_μ = ∂_μ - igT_aA^a_μ

Where T_a are the group generators that satisfy the Lie algebra:

[T_a,T_b] = if^abcT_c

[D_μ,D_ν] = -igT_aF^a_μν

Proof:

[D_μ,D_ν] = igT(∂_μA_ν - ∂_νA_μ + gT[A_μ,A_ν]

      ≡ igT_a(∂_μA^a_ν - ∂_νA^a_μ + gf^abcA^b_μA^c_ν)

      = igT_aF^a_μν 

Note that in the abelian case F^a_μν reduces to the electromanetic 
tensor and [D_μ,D_ν] = igF_μν.

SU(2)
-----

φ' = φexp(iθ_iσ_i/2)

φ*' = φ*exp(-iθ_iσ_i/2)

[σ_i/2,σ_j/2] = iε^ijkσ_k/2

D_μ = ∂_μ - igW_μⁱσ_i/2

G_μνⁱ = ∂_μW_νⁱ - ∂_νW_μⁱ + f^ijkW_μ^jW_ν^k

In the non-abelian SU(2) theory there are 3 gauge fields whose 
quanta are the W⁺, W^- and Z bosons.

SU(3)
-----

φ' = φexp(iθ_iλ_i/2)

φ*' = φ*exp(-iθ_iλ_i/2)

[λ_i/2,λ_j/2] = if_ijkλ_k/2

D_μ = ∂_μ - ig''G_μⁱλ_i 

G_μν = ∂_μG_νⁱ - ∂_νG_μⁱ + f^ijkG_μ^jG_ν^k

In the non-abelian SU(3) theory there are 8 gauge fields whose 
quanta are the gluons.

U(1) ⊗ SU(2) ⊗ SU(3)
---------------------

D_μ = ∂_μ - ig''G_μⁱλ_i/2 - igW_μⁱσ_i/2 - ig'YB_μ

Footnote:

Gauge symmetry is a powerfull symmetry particularly in the context
of General Relativity where coordinates vary from place to place
based on the curvature of spacetime.

QED is described by a U(1) group that represents electric charge.  
The unified electroweak interaction is described by SU(2) ⊗ U(1) 
group where the U(1) group represents the weak hypercharge, Y_W, 
rather than the electric charge.  QCD is an SU(3) Yang–Mills theory. 
The massless bosons from the SU(2) ⊗ U(1) theory mix after 
spontaneous symmetry breaking (Higgs mechanism) to produce the 3 
massive weak bosons, and the photon field.  The Standard Model 
combines the strong interaction with the unified electroweak 
interaction (unifying the weak and electromagnetic interaction) 
through the symmetry group U(1) ⊗ SU(2) ⊗ SU(3).  At the current 
time, however, the strong interaction has not been unified with the 
electroweak interaction, but from the observed running of the 
coupling constants it is believed they all converge to a single 
value at very high energies.