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Units, Constants and Useful Formulas
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 = ∂_{μ}φ∂^{μ}φ* - m^{2}φφ*
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_{μ}φ, which under the change of phase on φ transforms in
the same fashion, D_{μ}φ -> exp(iθ(x))D_{μ}φ. 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* = (∂_{μ}φ' + igA_{μ}'φ')(∂^{μ}φ*' - igA^{μ}'φ*')
_{ } = (∂_{μ}φ + i∂_{μ}θφ + igA_{μ}'φ)exp(iθφ)
(∂^{μ}φ* - i∂^{μ}θφ* - igA^{μ}'φ*)exp(-iθφ)
_{ } = (∂_{μ}φ + i∂_{μ}θφ + igA_{μ}'φ)
(∂^{μ}φ* - i∂^{μ}θφ* - igA^{μ}'φ*)
To get back to the original form we need A_{μ}' to transform as
follows:
A_{μ}' = A_{μ} - (1/g)∂_{μ}θ and A^{μ}' = A^{μ} - (1/g)∂^{μ}θ
Therefore,
D_{μ}'φD^{μ}'φ* = (∂_{μ}φ + i∂_{μ}θφ + ig(A_{μ} - (1/g)∂_{μ}θ)φ)
(∂^{μ}φ* - i∂^{μ}θφ* - ig(A^{μ} - (1/g)∂^{μ}θ)φ*)
_{ } = (∂_{μ}φ + igA_{μ}φ)(∂_{μ}φ* - igA_{μ}φ*)
_{ } = D_{μ}φD^{μ}φ*
Also,
m^{2}φφ^{*} -> m^{2}exp(iθ)φexp(-iθ)ψ^{*}
= m^{2}φφ^{*}
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)∂_{μ}ψ + ψi∂_{μ}θexp(iθ)
The covariant derivative is D_{μ} = ∂_{μ} + igA_{μ}
Where,
A_{μ} -> A_{μ} - (1/g)∂_{μ}θ
Therefore,
D_{μ} = ∂_{μ} + ig(A_{μ} - (1/g)∂_{μ}θ)
D_{μ}(ψexp(iθ)) = ∂_{μ}(ψexp(iθ)) + iψexp(iθ)gA_{μ} - iψexp(iθ)∂_{μ}θ
_{ } = exp(1θ)∂_{μ}ψ + iψexp(iθ)∂_{μ}θ + iψexp(iθ)gA_{μ} - iψexp(iθ)∂_{μ}θ
_{ } = exp(1θ)∂_{μ}ψ + iψexp(iθ)gA_{μ}
_{ } = exp(1θ)[∂_{μ}ψ + iψgA_{μ}]
_{ } = exp(1θ)[∂_{μ} + igA_{μ}]ψ
_{ } = exp(1θ)D_{μ}ψ
Therefore,
_
ψiγ^{μ}D_{μ}ψ
_
=> ψexp(-iθ)iγ^{μ}D_{μ}(ψexp(iθ))
_
=> ψexp(-iθ)exp(iθ)iγ^{μ}D_{μ}ψ
_
=> ψiγ^{μ}D_{μ}ψ
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} = ∂_{μ}A_{ν} - ∂_{ν}A_{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)∂_{μ}θ into the above we get:
F_{μν} = ∂_{μ}A_{ν} + ∂_{μ}∂_{ν}θ - (∂_{ν}A_{μ} + ∂_{ν}∂_{μ}θ)
^{ }= ∂_{μ}A_{ν} - ∂_{ν}A_{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_{μν}F^{μν}
We can now rewite our a Lagrangian for the gauge field as:
L = -F_{μν}F^{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_{μν}F^{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^{aμν}F^{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^{μν}F_{μν} + m^{2}A_{μ}A^{μ}
The mass term of the Proca Lagrangian is not invariant under
a gauge transformation since (A_{μ} + ∂_{μ}θ)(A^{μ} + ∂^{μ}θ) ≠ m^{2}A_{μ}A^{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_{a}A^{a}_{μ}
Where T_{a} are the group generators that satisfy the Lie algebra:
[T_{a},T_{b}] = if^{abc}T_{c}
[D_{μ},D_{ν}] = -igT_{a}F^{a}_{μν}
Proof:
[D_{μ},D_{ν}] = igT(∂_{μ}A_{ν} - ∂_{ν}A_{μ} + gT[A_{μ},A_{ν}]
_{ } ≡ igT_{a}(∂_{μ}A^{a}_{ν} - ∂_{ν}A^{a}_{μ} + gf^{abc}A^{b}_{μ}A^{c}_{ν})
_{ } = igT_{a}F^{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_{i}σ_{i}/2)
φ*' = φ*exp(-iθ_{i}σ_{i}/2)
[σ_{i}/2,σ_{j}/2] = iε^{ijk}σ_{k}/2
D_{μ} = ∂_{μ} - igW_{μ}^{i}σ_{i}/2
G_{μν}^{i} = ∂_{μ}W_{ν}^{i} - ∂_{ν}W_{μ}^{i} + f^{ijk}W_{μ}^{j}W_{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_{i}λ_{i}/2)
φ*' = φ*exp(-iθ_{i}λ_{i}/2)
[λ_{i}/2,λ_{j}/2] = if_{ijk}λ_{k}/2
D_{μ} = ∂_{μ} - ig''G_{μ}^{i}λ_{i}
G_{μν} = ∂_{μ}G_{ν}^{i} - ∂_{ν}G_{μ}^{i} + f^{ijk}G_{μ}^{j}G_{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_{μ}^{i}λ_{i}/2 - igW_{μ}^{i}σ_{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.