2. Brief Outline
1. Gauge invariance in classical electrodynamics
2. Local gauge invariance in quantum mechanics
3. Yang-Mills theory
3. Gauge Invariance and Classical Electrodynamics
• In classical electrodynamics, the electric and magnetic fields can
be written in terms of the scalar and vector potentials
B = × A E = − φ −
∂A
∂t
• However, these potentials are not unique for a given physical
field. There is a certain freedom in choosing the potentials.
• The potentials can be transformed as
Aµ
(x) → A µ
(x) = Aµ
(x) + ∂µ
Λ(x)
without affecting the physical electric and magnetic fields.
4. Local Gauge Invariance in Quantum Mechanics
Charged Particle in the Electromagnetic Field
• Hamiltonian of a charged particle moving in the presence of the
electromagnetic field is given by
H =
1
2m
(p − qA)2
+ qφ
• Quantum mechanically, the charged particle is described by the
Schr¨odinger equation,
−
1
2m
− iqA
2
ψ(x, t) = i
∂
∂t
+ iqφ ψ(x, t)
5. Local Gauge Invariance in Quantum Mechanics
Gauge Invariance and Quantum Mechanics
• Classically, the potentials φ and A are not unique for a given
physical electromagnetic field.
• We can transform the potentials locally without affecting the
physical fields (and hence the behaviour of the charged particle
moving in the field).
• We want to investigate whether an analogous situation exists in
quantum mechanics (i.e. whether quantum mechanics respects
the gauge invariance property of electromagnetic fields)
6. Local Gauge Invariance in Quantum Mechanics
• The gauge transformation of the potentials does not leave the
Schr¨odinger equation invariant.
• However, it is possible to restore the form invariance of the
Schr¨odinger equation, provided the transformation of the
potentials
Aµ
→ A µ
= Aµ
+ ∂µ
Λ(x)
is accompanied by a transformation of the wave function
ψ → ψ = e−iqΛ(x)
ψ
• With these two transformations together, the form invariance of
the Schr¨odinger equation is assured (i.e. A µ and ψ satisfy the
same equation as Aµ and ψ.)
7. Local Gauge Invariance in Quantum Mechanics
Summary
Quantum mechanics respects the gauge invariance property of the
electromagnetic field. It gives the freedom to change the
electromagnetic potentials but at the cost of a simultaneous change in
the phase of the wave function.
8. Local Gauge Invariance in Quantum Mechanics
Reversing the Argument
(Demanding Local Gauge Invariance)
• Instead of starting with the charged particle Schr¨odinger
equation, we start with the free particle Schr¨odinger equation
−
1
2m
2
ψ(x, t) = i
∂
∂t
ψ(x, t)
• We demand that this equation remains invariant under the local
phase transformation of the wave function
ψ(x) → ψ (x) = e−iqΛ(x)
ψ(x)
• However, the new wave function ψ (x) does not satisfy the free
particle Schr¨odinger equation.
9. Local Gauge Invariance in Quantum Mechanics
• We conclude that the local gauge invariance is not possible with
the free particle Schr¨odinger equation.
• However, the demand of local gauge invariance can be satisfied
by modifying the free particle Schr¨odinger equation.
• It turns out that by modifying the derivative operators in the free
particle Schr¨odinger equation as
∂µ → Dµ = ∂µ + iqAµ
we can achieve the required goal, provided the vector field Aµ
also transforms under the phase transformation of the wave
function ψ.
10. Local Gauge Invariance in Quantum Mechanics
Summary
• Local gauge freedom in the wave function in quantum mechanics
is not possible with the free particle Schr¨odinger equation.
• The insistence on the local gauge freedom forces us to introduce
in the equation a new field which interacts with the particle.
11. Yang-Mills Theory
• We now turn to extend the concept of local gauge invariance to
field theories.
• In field theory, the quantity of fundamental interest is the
Lagrangian density of the fields and accordingly, we demand the
local gauge invariance of the Lagrangian density.
12. Yang-Mills Theory
Lagrangian Density
• We consider a Lagrangian density which depends upon the scalar
field φ and its first derivative ∂µφ
L ≡ L(φ(x), ∂µφ(x))
• We also assume that the Lagrangian density is constructed out of
the inner products (φ, φ) and (∂µφ(x), ∂µφ(x)) (where bracket
denotes the inner product in field space), e.g.
L = (∂µφ)†
(∂µφ) − m2
φ†
φ − λ(φ†
φ)2
where the field φ, in general, is a multi component field.
13. Yang-Mills Theory
(Infinitesimal group theory)
• We are mainly interested in the compact Lie groups such as
SU(N) and SO(N).
• One basic property of the compact groups is that their finite
dimensional representations are equivalent to the unitary
representation.
• The advantage of unitary transformations is that they preserve
the inner products
φ†
φ → (Uφ)†
(Uφ) = φ†
(U†
U)φ = φ†
φ
14. Yang-Mills Theory
(Infinitesimal group theory)
• Associated with each Lie group is a Lie algebra. The elements ω
of the group can be written as
T(ω) = eiλaT(ta)
where ta are the generators of the group and T is some
representation.
• One important representation is the adjoint representation for
which the Lie algebra space coincides with the vector space on
which the group elements act. The action is given by
Ad(ω)A = ωAω−1
where, A is an element of the Lie algebra space.
15. Yang-Mills Theory
(Global symmetry transformation)
• We now assume that the Lagrangian density remains invariant
under a global symmetry transformation
φ(x) → φ (x) = T(ω)φ(x)
where ω is an element of the symmetry group and T(ω) is some
unitary representation under which the fields φ transform.
• For example, the field φ may be a two component object
transforming under the fundamental representation of the SU(2)
group, i.e.
φ(x) ≡
φ1(x)
φ2(x)
→
φ1(x)
φ2(x)
= eiΛaσa/2 φ1(x)
φ2(x)
16. Yang-Mills Theory
(Local symmetry transformation)
• We now generalize the global transformation to a local
transformation
φ(x) → φ (x) = T(ω(x))φ(x)
• Under a local transformation, the inner product φ†φ remains
invariant. However, the inner product involving the derivative of
the fields (∂µφ)†(∂µφ) does not remain invariant, since
∂µφ(x) → ∂µφ (x) = T(ω(x))∂µφ(x) + ∂µT(ω(x))φ(x)
(The second term in the right hand side prevents the invariance
of the inner product involving the derivatives)
17. Yang-Mills Theory
(Introducing gauge fields)
• To ensure the invariance of the Lagrangian density, the same
procedure, as in the case of quantum mechanics, is followed.
• We replace the ordinary derivative by a covariant derivative
∂µφ(x) → Dµφ(x) = (∂µ − igT(Aµ))φ(x)
introducing a field Aµ known as the gauge field.
• The field Aµ is constructed in such a way that the covariant
derivative transforms exactly as the field φ, namely
Dµφ(x) → (Dµφ(x)) = T(ω(x))Dµφ(x)
18. Yang-Mills Theory
(Gauge fields belong to the lie algebra)
• The last demand leads to the following transformation property
for the gauge fields Aµ
T(Aµ) = T(ωAµω−1
) +
i
g
T(ω∂µω−1
)
• Both the terms in the right hand side belong to the lie algebra of
the corresponding symmetry group.
• The first term is a result of the action of adjoint representation.
For the second term, we look at the group elements near identity
ω(x) = 1 + iλa(x)ta + o(λ2
)
This gives
ω∂µω−1
= −i(∂µλa)ta
19. Yang-Mills Theory
(Gauge fields belong to the Lie algebra)
• Since gauge fields Aµ belong to the Lie algebra space, it follows
that we can write them as a linear combination of the generators
ta
Aµ
= Aµ
a ta
• From this, it also follows that the number of independent gauge
fields is equal to the number of generators of the group. Thus,
e.g., if the symmetry group is SU(N), the number of gauge fields
will be (N2 − 1).
• Thus, the number of gauge fields depends only upon the
underlying symmetry group and is independent of the number of
matter fields present in the system (of course, the number of
matter fields should match with the dimension of some
representation of the symmetry group)
20. Yang-Mills Theory
(Lagrangian density for the gauge fields)
• Since we have introduced the gauge fields Aµ in our system, we
need to have a term in the Lagrangian density which describes
their dynamical behavior.
• Moreover, this term should also be gauge invariant to preserve
the gauge invariance of the Lagrangian density.
• We recall that the electromagnetic Lagrangian density is given by
L = −
1
4
FµνFµν
where, Fµν = ∂µAν − ∂νAµ is the field strength tensor.
21. Yang-Mills Theory
(Lagrangian density for the gauge fields)
• To construct the field strength tensor for the gauge fields, we take
guidance from the following theorem (Rubakov, chapter 3)
“A Lie algebra is compact if and only if it has a
(positive-definite) scalar product, which is invariant under the
action of the adjoint representation of the group ”
• Since our aim is also to have a gauge invariant term, we demand
that the field strength tensor for the gauge fields should also
transform according to the adjoint representation, i.e.
Fµν → Fµν = Ad(ω)Fµν = ωFµνω−1
and we construct the gauge invariant Lagrangian density using
this field tensor.
22. Yang-Mills Theory
(Lagrangian density for the gauge fields)
• This demand leads to the field strength tensor
Fµν = ∂µAν − ∂νAµ − ig[Aµ, Aν]
• Since Fµν belongs to the Lie algebra, we can write it as a linear
combination of the generators
Fµν
= Fµν
a ta
• In terms of the components Aµ
a and Fµν
a , we have
Fµν
a = ∂µ
Aν
a − ∂ν
Aµ
a + gfabcAµ
b Aν
c
• This differs from the electromagnetic case by the presence of a
non linear term.
23. Yang-Mills Theory
(Lagrangian density for the gauge fields)
• The Lagrangian density for the gauge fields is postulated to be
the inner product
Lgaugefield = −
1
2
Tr(Fµν
Fµν) = −
1
4
Fµν
a Faµν
• Since Fµν transforms as the adjoint representation, this inner
product is invariant (basically due to cyclic property of trace).
24. Yang-Mills Theory
(Full Lagrangian density)
• For the example given earlier, the complete Lagrangian density
thus becomes
L = (Dµφ)†
(Dµ
φ) − m2
(φ†
φ) − λ(φ†
φ)2
−
1
4
Fµν
a Faµν
where,
Dµφ = (∂µ − igT(Aµ))φ
25. Yang-Mills Theory
(Energy-momentum tensor)
• The energy momentum tensor can be obtained using the
definition
δS = −
1
2
d4
x
√
−g Tµν
δgµν
• This gives
Tµν
=
1
4
ηµν
Fλρ
a Faλρ − Fµλ
a Fν
a λ + 2(Dµ
φ)†
Dν
φ − ηµν
Lφ
• Energy is given by integrating the (00)th component of this
tensor over the spatial volume and is positive definite.
E = d3
x (D0
φ)†
D0
φ + (Di
φ)†
Di
φ + m2
(φ†
φ) + λ(φ†
φ)2
+
1
2
F0i
a F0i
a +
1
4
Fij
a Fij
a
26. Yang-Mills Theory
Summary
The interaction between the scalar fields and the gauge fields can be
obtained by invoking the local gauge invariance principle. This
principle also dictates the kind of terms which can be present in the
Lagrangian.
27. Why Yang-Mills Theory
• Every physical phenomenon is believed to be governed by four
interactions. Two of these, namely, Gravity and
Electromagnetism are felt in day to day life.
• Due to this, it is possible to formulate a classical version of these
interactions.
• The formulation of the universal Gravitational force law by Isaac
Newton from the observation of the motion of an apple and the
moon is an excellent example of this.
• Similarly, the laws of Electrodynamics were discovered by
observing the behavior of magnets, current carrying wires and so
on. James Clark Maxwell gave the exact mathematical form of
these laws using these observation (and his excellent insight).
30. Why Yang-Mills Theory
• The quantum version of the Electrodynamics (Quantum
Electrodynamics) was constructed with the help of its known
classical version.
• However, there is no guidance in the form of classical laws for
the strong and weak interactions. We have to directly deal with
the quantum version of these interactions.
• The Gauge invariance principle comes to rescue. The
mathematical form of the strong and weak interactions has been
constructed by using this principle.
31. References
1. Valery Rubakov, Classical Theory of Gauge Field, Princeton
University Press, Princeton, New Jersey (2002)
2. Aitchison and Hey, Gauge Theories in Particle Physics: Volume 1,
3rd Ed., IOP (2004)
