February 2016
PX430: Gauge Theories for Particle Physics
Tim Gershon
(T.J.Gershon@warwick.ac.uk)
Handout 1: Revision, Notation and The Gauge Principle
Relativistic quantum mechanics is a prerequisite for this module, and therefore we will assume
knowledge of the material covered in PX408, available from
http://www2.warwick.ac.uk/fac/sci/physics/teach/module home/px408/.
Nonetheless, let’s start with some very brief revision.
Relativity
Einstein’s principle of special relativity can be formulated thus
“The laws of physics are the same in all inertial frames of reference.”
This has several far reaching consequences, including the fact that space and time, rather than
being distinct entities, become mixed together as components of the 4-dimensional “space-time” – any
relativistically invariant theory has to treat space and time on the same footing. This can be made
manifest using four-vector notation, since the product of a covariant 4-vector and a contravariant
4-vector is invariant.
Quantum Mechanics
Most of the important consequences of quantum physics will be relevant to this module, including
the concept of spin, Heisenberg’s uncertainty principle (especially when stated in the form that the
commutator of position and momentum operators gives
–i in natural units – the imaginary number:
h
˙
[x̂, p̂] = i), Heisenberg’s equation of motion (Â = −i Â, Ĥ , so that any operator that commutes with
the Hamiltonian represents a constant of motion), the Pauli exclusion principle, and so on. For our
purposes one particularly important outcome of quantum theory is the wavefunction interpretation:
the state of a system is described by a complex valued wavefunction (let’s call it ψ), with the square
of the magnitude of the wavefunction (|ψ|2 ) giving the probability density (this is sometimes called
the Born interpretation).
Relativistic Quantum Mechanics
We have previously encountered two relativistic wave equations: the Klein-Gordon equation
(∂µ ∂ µ + m2 )φ = 0 ,
(1)
(iγ µ ∂µ − m)ψ = 0 ,
(2)
and the Dirac equation
where the gamma matrices are defined by their anticommutation relations {γ µ , γ ν } = 2g µν .
We have seen that the solutions to the Dirac equation are four-component spinors, and that consequently the quantum property of spin and the existence of antiparticles emerge as natural outcomes
of the theory. We have also discussed the limitations of relativistic quantum mechanics, specifically
the need for a theory that naturally handles particle-antiparticle pair creation and annihilation, and
the experimental evidence that such a theory is necessary.
Hamiltonian & Lagrangian Mechanics
It can be difficult to avoid getting Hamiltonian & Lagrangian mechanics mixed up, since, for
example, the starting point for Lagrangian mechanics is Hamilton’s principle of least action. Without
worrying too much about which is which, we define the action
Z
S = Ldt
(3)
where L is the Lagrangian, and the integral is between initial and final points. In classical physics, the
Lagrangian is usually defined as the difference between the kinetic and potential energies: L = T −V .
Considering a particle travelling between initial and final points, the principle of least action states
that the path taken will be the one that minimises the action. Writing the Lagrangian L = L(qi , q̇i )
in terms of generalised coordinates qi (t) and their derivatives with respect to time q̇i (t), allows us to
obtain the Euler-Lagrange equations. The derivation goes as follows:
Z
B
S =
L(qi , q̇i )dt ,
(4)
δL(qi , q̇i )dt ,
(5)
A
Z B
δS = 0 =
A
Z B
0 =
A
Z
B
0 =
A
Z
B
0 =
A
!
∂L
δqi +
δ q̇i dt ,
∂qi
∂ q̇i
i
!
X ∂L dqi ∂L dq̇i
+
dt ,
∂qi dt
∂ q̇i dt
i
!
X ∂L d ∂L
q̇i −
q̇i dt ,
∂qi
dt ∂ q̇i
X ∂L (6)
(7)
(8)
i
RB 0
RB
where in the last stage we have used integration by parts, A uv 0 = [uv]B
A − A u v, and used
boundary conditions to identify that the constant term goes to zero. Since each term of q̇i is, in
general, non-zero, this identity is only satisfied when:
d ∂L
∂L
−
= 0.
(9)
dt ∂ q̇i
∂qi
These are the Euler-Lagrange equations.
Q1 Work carefully through the steps of the derivation of the Euler-Lagrange equations (Eq. (9).
Q2 For a classical particle in a potential V = V (qi ), write down the kinetic energy T , and show
that the solution of the Euler-Lagrange equations gives Newton’s law, F = ma.
Q3 Consider further a particle in an electromagnetic field, with kinetic energy given by T =
2
1
2m (p − qA) . Solve the Euler-Lagrange equations for this case.
It can be preferable to replace the generalised velocities q̇i by the corresponding conjugate momenta, defined as
∂L
pi =
(10)
∂ q̇i
Q4 For the classical Lagrangian discussed above, what are pi ?
The Hamiltonian is then defined as (here including explicitly a possible dependence on t)
X
H(qi , pi , t) =
q̇i pi − L(qj , q̇j , t)
(11)
i
Q5 For the classical Lagrangian discussed above, what is H?
Perhaps surprisingly, this Lagrangian-Hamiltonian formulation of mechanics, which appears initially to be little more than an interesting digression in classical physics, turns out to be extremely
useful in quantum physics, and indeed in quantum field theories. In such cases, it is more common to
consider the Lagrangian density, L, such that the action is obtained from the integral over all space
and time dimensions
Z
S = L d4 x
(12)
Writing the Lagrangian density as a function of a field ψ, which itself is a function of (generalised)
coordinates xµ , the quantum field theoretic version of Eq. (9) is
∂L
∂L
∂µ
−
= 0.
(13)
∂(∂µ ψ)
∂ψ
(Actually, in quantum field theory we should really replace both the field and the Lagrangian density
with operators: ψ −→ ψ̂, ∂µ ψ −→ ∂µ ψ̂ and L −→ L̂.)
Q6 Write Eq. (13) in full, expanding the Lorentz invariant term into
∂
∂t ,
∇, etc.
Essence of the Gauge Principle
As mentioned above, the wavefunction interpretation of QM tells us that physical systems are
defined by a complex wavefunction. Such a complex wavefunction has two degrees of freedom, which
we can write as the real and imaginary parts, or as the magnitude and the phase. Since all physically
observable quantities depend only on the magnitude of the wavefunction (or rather, the magnitude
squared, which gives the probability density), the phase is not an observable quantity.
Consider a complex wavefunction ψ = |ψ|ei arg(ψ) . Suppose this is made up of two interfering
terms, then the observable quantity is
|ψ|2 = |ψ1 + ψ2 |2 = |ψ1 |2 + |ψ2 |2 + 2 Re(ψ1 ψ2∗ ) = |ψ1 |2 + |ψ2 |2 + 2|ψ1 ||ψ2 | cos(arg(ψ1 ) − arg(ψ2 )) (14)
Clearly this depends on the relative phase between the two interfering components. There are very
many examples of such effects, the most famous being Young’s double slit experiment. So relative
phases (ie. phase differences) are observable, but absolute phases are not.
Consequently, there is a symmetry inherent in any quantum mechanical theory: if the wavefunction is changed by a (global) phase factor, the physics remains unaffected. Such a change of phase is
known as a (global) gauge symmetry.
Since Noether’s theorem states that for every symmetry there is an associated conserved quantity,
it is natural to ask what is the conserved quantity associated with the above global gauge symmetry?
This is difficult to answer, since the question has been posed in very general terms. However, bearing
in mind that we will be principally concerned with the electromagnetic interaction and associated
wavefunctions, it might be possible to guess that the answer is the electric charge.
Q7 Familiarise yourself with Noether’s theorem. How does it apply to a) continuous symmetries
and b) discrete symmetries?
Q8 What is the difference between a “gauge” symmetry, and, for example, the symmetry with
respect to translation of spatial coordinates that is related to conservation of momentum?
The “gauge principle” states that such symmetries of the Lagrangian with respect to rotations
in some internal space are of profound importance. We shall be concerned not only with global
gauge transformations, but also with local transformations, where the change of phase can vary in
space and time. As we shall see, the requirement that our theory be invariant under such gauge
transformations can be used to determine the entire dynamics of interacting (quantum) gauge fields.
Gauge Invariance of Maxwell’s Equations
Before turning to quantum fields, we can get an idea of the principle from classical physics. We
start with Maxwell’s equations (in Heaviside-Lorentz units):
∇.E = ρem
∇ × E = − ∂B
∂t
∇.B = 0
∇ × B = j em + ∂E
∂t
Gauss
Faraday − Lenz
no magnetic monopoles
Ampère − Maxwell
(15)
together with the continuity equation:
∂ρem
+ ∇.j em = 0 .
∂t
(16)
We can write Eq. (15) in terms of the vector potential Aµ = (V, A), where B = ∇ × A and
E = −∇V − ∂A
∂t . This is achieved by defining the field strength tensor
F µν = ∂ µ Aν − ∂ ν Aµ ,
(17)
from which Maxwell’s equations can be written
ν
∂µ F µν = jem
,
(18)
ν = (ρ , j
where jem
em em ).
Q9 Show that Eq. (18) implies all of Eq. (15) as well as Eq. (16).
Q10 Show that F µν is antisymmetric. What are the elements of F µν ?
We now consider transformations of the form
Aµ 7→ A0µ = Aµ − ∂ µ χ
(19)
where χ is an arbitrary real-valued scalar function, i.e. it depends on 4-position, but its value is a
real number. It should be clear that this transformation of Aµ has no effect on physics, since both
B and E are unaffected; this is a local gauge transformation.
Q11 Satisfy yourself that both B and E are unaffected by the gauge transformation of Eq. (19).
Q12 Show the the field strength tensor F µν is unaffected by the gauge transformation of Eq. (19).
The fact that we can change Aµ without affecting physics allows us to choose a particular form
with which it is convenient to work. This is called choosing a gauge. One popular choice is called the
Lorenz gauge, ∂µ Aµ = 0. [n.b. Lorenz 6= Lorentz!] With this choice, and after combining Eqs. 17
and 18 to get
ν
Aν − ∂ ν (∂µ Aµ ) = jem
(20)
we find
ν
Aν = jem
.
(21)
From here we can see that each component of Aµ satisfies the massless Klein-Gordon equation in
ν = 0).
free space (jem
Q13 Suppose that ∂µ Aµ = f 6= 0. Show that a gauge transformation can be applied to change to
A0 where ∂µ A0µ = 0. What (in terms of f ) is the form of the transformation?
Q14 In fact, the Lorenz gauge condition does not specify Aµ uniquely. Show that further gauge
transformations of the form Aµ 7→ A0µ = Aµ − ∂ µ Λ, where Λ is any scalar function that
satisfies Λ = 0 will (i) leave F µν (and hence the observable physics) unaffected, and (ii)
preserve the Lorenz condition.
Q15 The Coulomb gauge specifies ∇.A = 0. If the Coulomb gauge is satisfied, find an expression
for the Lorenz condition. How many independent components does Aµ have?
The fact that expressing Maxwell’s equations in the manifestly gauge invariant form of Eq. (18),
automatically embodies the continuity equation, suggests that there is a close relation between gauge
invariance and charge conservation.
Gauge Invariance in Quantum Mechanics
Starting with the Schrödinger equation for a free particle
Hψ =
∂ψ
1
(−i∇)2 ψ = i
2m
∂t
(22)
we can introduce the electromagnetic interaction via the minimal substitution
∇ 7→ D = ∇ − iqA ,
∂
∂
7→ D0 =
+ iqV
∂t
∂t
(∂ µ 7→ Dµ = ∂ µ + iqAµ )
(23)
where the contravariant form is in parentheses since we do not expect to be able to apply a Lorentz
invariant treatment to the non-relativistic Schrödinger equation. We then find the Schrödinger
equation for a particle in an electromagnetic field
1
∂ψ
2
Hψ =
(−i∇ − qA) + qV ψ = i
.
(24)
2m
∂t
Q16 Show that the classical expression for the force on a charged particle in an electromagnetic
field, F = qE + qv × B, where v is the particle’s velocity, can be derived from Eq. (24), using
Lagrange-Hamilton mechanics.
We now apply a gauge transformation. Both ψ and Aµ are transformed. At first, let us not
specify the form of the transformation, but simply replace these variables by their primed versions:
1
∂ψ 0
0 0
0 2
0
Hψ =
(−i∇ − qA ) + qV ψ 0 = i
.
(25)
2m
∂t
We require that physics should be unaffected under this transformation, which means that we
need to obtain an equation of the same form (it does not have to be exactly the same equation, just
to take the same form). If the transformation of Aµ has the form of Eq. (19), it is clear that we need
to make some transformation ψ 7→ ψ 0 in order to keep the physics unaffected. What is the form of
this transformation? Since |ψ|2 is an observable quantity, we require |ψ 0 |2 = |ψ|2 , and so we are only
allowed to multiply by a phase factor. In fact, the answer is ψ 7→ ψ 0 = eiqχ ψ. To see this, consider
(−i∇ − qA0 )ψ 0 = (−i∇ − qA − q(∇χ))(eiqχ ψ)
= eiqχ (−i∇ − qA)ψ
(26)
and
∂
∂
∂χ
0
0
i − qV ψ =
i − qV + q
(eiqχ ψ)
∂t
∂t
∂t
∂
iqχ
i − qV ψ
= e
∂t
(27)
i.e.
− iD 0 ψ 0 = eiqχ (−iDψ)
&
iD00 ψ 0 = eiqχ (iD0 ψ)
(28)
and so we find
1
1
(−iD 0 )2 ψ 0 = eiqχ
(−iD)2 ψ = eiqχ iD0 ψ = iD00 ψ 0 .
2m
2m
We have recovered the Schrödinger equation for a particle in an electromagnetic field.
(29)
Q17 Prove Eqs. (26) and (27).
Q18 Prove Eq. (29).
Therefore, the gauge invariance of Maxwell’s equations remains an invariance in quantum mechanics, provided we make the combined transformation
Aµ 7→ A0µ = Aµ − ∂ µ χ
& ψ 7→ ψ 0 = eiqχ ψ .
(30)
The Gauge Principle
What we have shown above is that if we make the gauge transformation Aµ 7→ A0µ = Aµ − ∂ µ χ,
then we require a transformation of the wavefunction of the form ψ 7→ ψ 0 = eiqχ ψ in order to
recover the same physics. The gauge principle, which is ultimately more powerful, makes the reverse
argument: we require physics to be invariant under the local gauge transformation ψ 7→ ψ 0 = eiqχ ψ,
and consider the effect on derivative terms:
∂ µ ψ 7→ ∂ µ ψ 0 = eiqχ ∂ µ ψ + iq (∂ µ χ) eiqχ ψ .
(31)
We can then see that Dµ ψ = (∂ µ + iqAµ ) 7→ D0µ ψ 0 = eiqχ Dµ ψ if Aµ 7→ A0µ = Aµ − ∂ µ χ. Thus we
can determine both the transformation required of Aµ and the form of the minimal substitution (and
the reason for calling it the minimal substitution becomes clear – to have gauge invariance we need
to introduce a field Aµ and this is the simplest way to do it). In other words starting from a free
particle equation, by making a local gauge transformation, we can determine the form of Aµ and of
the minimal substitution itself.
Q19 Start from the free particle Schrödinger equation, and by requiring physics to be invariant under
the local gauge transformation ψ 7→ ψ 0 = eiqχ ψ, determine the form of the electromagnetic
gauge field Aµ .
It might be reasonable at this stage to ask if there are any other gauge transformations we could
make to our wavefunction ψ, which might determine interactions other than the electromagnetic
one. We will address this question (though rather briefly) later. For now, we simply note that the
transformation ψ 7→ ψ 0 = eiqχ ψ is a phase change, and can be thought of as a rotation in the internal
(real, imaginary) space of the wavefunction. This is called a U(1) gauge transformation. Successive
transformations of this form can be applied without regard to the order – i.e. U(1) rotations commute.
This is called an Abelian gauge transformation (we will encounter the non-Abelian (non-commuting)
case later).
[Aside: U(1) is a symmetry group, sometimes called the circle group. It is the multiplicative
group of all complex numbers with unit magnitude. The mathematical language of group theory is
quite helpful to discuss the more complicated gauge transformations that we will encounter at the
end of the module. Unfortunately (or perhaps fortunately, depending on your perspective), time
limitations prevent any proper discussion of this aspect.
The interested student is encouraged to read further into this topic, for example Appendix M of
Volume 2 of “Gauge Theories in Particle Physics”, by Aitchison & Hey, 3rd edition, or Chapter 4
of “Mathematical Methods for Physicists” by Arfken and Webber (6th edition). There is also some
material available on the web that is associated to relevant modules offered in the Mathematics
department: MA249 – Groups and Rings, MA3E1 – Groups and Representations and MA4E0 – Lie
Groups, though parts of these modules may not be at the appropriate level.]
The Gauge Principle in Classical Electromagnetism
As an application of the gauge principle, let’s go back and consider electromagnetism. We know
that we can form a theory around a Lagrangian density that is both (i) gauge invariant and (ii)
Lorentz invariant. We already have an object which satisfies (i), namely F µν , and to satisfy (ii) we
will need to contract the indices. So the only possibility is
1
L = − Fµν F µν ,
4
(32)
where the factor of 14 is conventional, but the minus sign is not (remember, we need to minimise the
action).
So, simply requiring that the theory should be invariant under local gauge transformations, as
well as Lorentz invariant, is enough to derive electromagnetism, as well as the form of the interaction
between electrons and photons (electrodynamics). Clearly this is a very powerful principle.
Gauge Invariance in Relativistic Quantum Mechanics
Naturally, we would like to extend the above discussion to a relativistic quantum mechanical
treatment, to see how the particles described by the Klein-Gordon and Dirac equations (i.e. the
solutions to those equations) interact with the associated gauge fields. Indeed, there is nothing to
stop us from trying to do so: we can either apply to minimal substitution ∂ µ 7→ Dµ = ∂ µ + iqAµ
to introduce electromagnetic interactions, then gauge transform Aµ and see what happens, or alternatively we could attempt to apply the gauge principle, and assert that physics should be invariant
under U(1) transformations. However, the effects are much clearer when we use the Lagrangian
formulation within quantum field theory.
Q20 Apply the minimal substitution to (i) the free particle Klein-Gordon equation (∂µ ∂ µ +m2 )φ = 0,
(ii) the free particle Dirac equation (iγ µ ∂µ −m)ψ = 0. Obtain conserved currents in both cases.