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1.4
Particle physics
It is an empirical fact that subatomic particles are also governed by a Lagrangian, L and
its corresponding action, S. The Lagrangian that describes fundamental particles is not as
simple as the Lagrangian of Eq. (1.15), its form and meaning is shaped by Lorentz symmetry
and quantum mechanics, and the generalized coordinates do not represent the position of a
particle, but rather fields in spacetime. But at the end of the day the classical dynamics of
fields is still based on the principle of least action and the quantum mechanics of fields, even
relativistic fields, is still based on path integrals. The Lagrangian is still just a function of
some generalized coordinates with a form not too di↵erent from Eq. (1.15) with kinetic terms
minus potentials. In a sense, the kinetic terms are describing what are the dynamical degrees
of freedom (what are the particles) whereas the potential terms describe how they interact.
Below I will discuss symmetries and conservation laws associated with Lagrangians with
finite number of degrees of freedom, but much of the intuition and insight carries forward to
the case of fundamental particles, which are described by Lagrangian with many (possibly
infinite) number of degrees of freedom.
How to build Lagrangians for relativistic quantum systems and how to calculate the resulting
quantum dynamics is what you would learn in an advanced course on quantum field theory.
It is certainly beyond the scope of this course. Instead, I would like to expose you to what I
think are some of the core ideas and modes of thought in particle physics and along the way
describe those phenomena that do not require the full machinery of quantum field theory.
Some of the key concepts that we will encounter in this course, and which I think underlie
much of particle physics are Lagrangians, symmetries, and scattering experiments, as shown
in Fig. (2). Lagrangians are the theoretical constructs by which we define our theories
and describe what particles are present and how they interact. By performing scattering
experiments we can learn about the existence of particles, deduce their interactions, and
construct the appropriate Lagrangian. Conversely, we can use the Lagrangian to make
predictions to be tested through scattering experiments3 . Finally symmetries are powerful
constraints that restrict the form of the Lagrangian and the resulting dynamics. Symmetries
result in very strong predictions for experiments and thus also form very sensitive tests of
our theories of Nature.
3
Scattering experiments are of course not the only type of experiments that teach us about the subatomic
world, but they are by far the most common. Also important are precision experiments involving bound
states (for example atomic spectroscopy) and experiments monitoring the decay of some long-lived metastable
state (for example measuring the lifetime of a free neutron), which are often produced through scattering,
but not always.
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Scattering
experiments
Lagrangians
Symmetries
Figure 2: Some of the key concepts underlying particle physics.
1.5
Symmetries
The description of a classical system with a particular set of generalized coordinates, {q1 , . . . , qN },
is implicitly assuming the vantage point of a particular observer. Another observer may use
a di↵erent set of coordinates to describe the system. Two di↵erent observers that relate to
each other in some well-defined way correspond to two di↵erent coordinate systems that can
be transformed into each other in a well-defined manner. Such a transformation is known
as a symmetry of the system if the Lagrangian written in the transformed coordinates is
identical to the original Lagrangian4 . In that case the equations of motion, and hence the
physics, would look the same for the two observers. Let me show you a few simple examples.
As the first, and most straightforward example, consider a free theory with N non-interacting
particles. The Lagrangian is made up of only the kinetic terms,
L=
N
X
1
m q̇ 2
2 i i
(1.21)
i=1
Another observer situated some distance d away from the first observer would use coordinates,
qi ! qi0 = qi + d
(1.22)
The arrow here is common notation indicating a coordinate transformation, in this case from
0
{q1 , . . . , qN } to the shifted coordinates {q10 , . . . , qN
}. Then the Lagrangian in the transformed
4
Actually, the more general condition is that the Lagrangian does not change by more than a total time
derivative. Since it is the minimization of the action that determines the equations of motion, and hence
the physics, a change of the Lagrangian by a total time derivative only results in the action changing by a
constant. This subtle distinction will not be important for what follows, but it is worth keeping in mind.
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coordinates is
L(qi0 , q̇i0 )
=
N
X
1
m (q̇ 0 )2
2 i i
i=1
=
N
X
1
m
2 i
i=1
✓
d
(qi + d)
dt
◆2
=
N
X
1
m q̇ 2
2 i i
.
(1.23)
i=1
where in the last step I used the fact that d is constant in time. Thus, the shifted Lagrangian
is identical to the one we started with,
L ! L0 = L
(1.24)
(by the primed Lagrangian I just mean the same functional form, but in the primed coordinates) Hence, observers that relate to each other by a constant translation of the coordinates
see the same physics. A constant translation is a symmetry of the free Lagrangian. In fact,
you can easily convince yourself that the Lagrangian is even invariant under independent
di↵erent constant translations of each of the coordinates.
As a second example let’s consider a system in two dimensions that is described by two
degrees of freedom q1 and q2 . I will assume that the Lagrangian is given as usual by the
di↵erence between the kinetic terms and the potential,
L = 12 mq̇12 + 12 mq̇22
V (q12 + q22 ) .
(1.25)
Note the special form of the potential which is assumed to depend only on the combination
q12 + q22 (but is otherwise an arbitrary function of this combination). For example, if we were
describing a two dimensional harmonic oscillator then q1 and q2 would just be the x and y
coordinates in a cartesian system and the potential would be V = 12 k(x2 + y 2 ), with k some
spring constant. Let us now consider another observer which is viewing the same system but
from an angle rotated counter clockwise with respect to the first observer. The appropriate
coordinate transformation relating the two observers is
✓
◆
✓ 0 ◆ ✓
◆✓
◆
q1
q1
cos
sin
q1
!
=
(1.26)
q2
q20
sin
cos
q2
The Lagrangian in the rotated coordinates is then,
L(qi0 , q̇i0 ) =
=
cos + q̇2 sin )2 + 12 m(q̇1 sin
q̇2 cos )2
V (q1 cos + q2 sin )2 + (q1 sin
q2 cos )2
1
m(q̇1
2
1
mq̇12
2
+ 12 mq̇22
In the last step I used the relation cos2
(1.27)
V q12 + q22
+ sin2
= 1 twice. Once again we have that,
L ! L0 = L ,
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(1.28)
and the Lagrangian seen by the two observers is the same. We conclude that the physics of
such a system would look identical between any two observers related by a rotation of the
plane.
The last two examples we have seen are known as continuous symmetries, as they involve a
continuous parameter that parametrize the transformation and can be dialed to zero (in the
first case it was the shift, d, and in the second case it was the rotation angle, ). There are
also examples of discrete symmetries associated with transformations that only take specific
values. For example, consider once again the Lagrangian of Eq. (1.25). It remains unchanged
under the transformation,
q1 ! q10 =
q1
q2 ! q20 = q2
(1.29)
(1.30)
This is known as a parity transformation on q1 . Notice that it is not a rotation, there is no
value of in Eq. (1.26) that would yield the same transformation. This is a reflection of the
coordinates about the q2 axis rather than a rotation and correspond to an observer looking
at the mirror image of the system.
The path integral formulation of quantum mechanics makes it manifest that classical symmetries are also important when going over to quantum mechanics. Classically, a symmetry
of the Lagrangian, i.e. a transformation of the generalized coordinates that leaves the Lagrangian unchanged, results in di↵erent observers seeing the same equations of motion. In
quantum mechanics we see through the path integral that a symmetry of the Lagrangian results in di↵erent observers seeing the same transition amplitude (up to an irrelevant phase),
since the path integral would be the same. There are rare instances where this conclusion
is invalid and those occur when the measure associated with the path integral, Dqi itself
changes under the transformation. When this happens we speak of the classical symmetries
as being anomalous symmetries. The decay of the neutral pion ⇡ 0 into two photons, ⇡ 0 !
is one of the most important manifestations of this phenomenon in particle physics and is
the result of what is known as the chiral anomaly.
1.6
Conservation laws
I want to now show you a beautiful connection between continuous symmetries of the Lagrangian and conservation laws. Before I state the general theorem, we can already see a
particular example for this connection through the equations of motion, Eq. (1.9). If the
Lagrangian does not depend on the coordinate qi , then the corresponding conjugate momentum, pi , is conserved,
dpi
@L
=
=0
dt
@qi
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(1.31)
If the Lagrangian is indeed independent of qi it is also easy to show that it is symmetric
under a translation of the coordinate qi . Consider the infinitesimal transformation,
qi ! qi0 = qi + ✏i
(1.32)
where ✏i is an infinitesimal translation (since it is a continuous symmetry we can be as close
to the identity transformation as we like). Using Taylor’s theorem we see that the Lagrangian
changes as,
@L
= L(qi )
(1.33)
@qi
where in the last step I used the fact that the Lagrangian does not depend on qi . This
is a sufficient and necessary condition — the Lagrangian is invariant under infinitesimal
translations of the coordinate qi if and only if it is independent of this coordinate. Thus we
establish the first example where the existence of a continuous symmetry of the Lagrangian
implies the existence of a conserved quantity and vice versa.
L ! L(qi0 ) = L(qi + ✏i ) = L(qi ) + ✏i
Let me now state for you the general theorem. Suppose we have a general Lagrangian,
L, of the generalized coordinates {q1 , . . . , qN } and their generalized velocities {q̇1 , . . . , q̇N }.
Consider now the most general infinitesimal transformation of the coordinates given by
qi (t) ! qi (t) + ✏fi (t)
and
q̇i (t) ! q̇i (t) + ✏f˙i (t) ,
(1.34)
where ✏ reminds us that these are infinitesimal transformations, and fi (t) are general functions of time and even of the of the coordinates themselves. If this transformation is a
symmetry of the Lagrangian, then the following quantity is conserved,
I=
N
X
@L
i=1
@ q̇i
fi . That is to say
dI
=0
dt
(1.35)
Let’s prove this assertion. Under the general coordinate transformation Eq. (1.34) we have,
0
L!L =L+
N
X
@L
i=1
@qi
(✏fi ) +
N
X
@L
i=1
@ q̇i
(✏f˙i ) .
(1.36)
where I have used Taylor’s theorem to expand the transformed Lagrangian. If the transformation is a symmetry then we must have,
◆
✓ ◆
◆
N ✓
N ✓
X
X
@L
@L ˙
d @L
@L ˙
0=✏
fi +
fi = ✏
fi +
fi
(1.37)
@q
@
q̇
dt
@
q̇
@
q̇
i
i
i
i
i=1
i=1
Where in the last step I used the Euler-Lagrange equations, Eq. (1.7), to write @L/@qi =
d(@L/@ q̇i )/dt. This last term can finally be written as a total time derivative and we conclude
that,
!
N
d X @L
0=✏
fi
(1.38)
dt i=1 @ q̇i
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Since the parameter ✏ is arbitrary this equation implies that the term in brackets is conserved.
The reverse is also evidently true, if the quantity in brackets is conserved then the general
coordinate transformation we considered is a symmetry of the system and L0 = L through
Eq. (1.36). Hence we established what is known as Noether’s theorem,
dI
=0
dt
1.7
()
The transformation (1.34) is a symmetry of L
(1.39)
Approximate symmetries
Consider a system described by two generalized coordinates, q1 and q2 with both coordinates
experiencing an harmonic potential,
✓
◆
2
2
2
2
1
1
1
1
L = 2 mq̇1 + 2 mq̇2
k q + 2 k 2 q2 .
(1.40)
2 1 1
This for example could be describing a particle moving in two dimension. It is convenient to
canonically
normalize
p
p the kinetic terms to include only a factor of two, and use the variables
Q1 = m q1 , Q2 = m q2 ,
✓
◆
2
1 2
1 2
1 2 2
1
L = 2 Q̇1 + 2 Q̇2
! Q + 2 !2 Q2 ,
(1.41)
2 1 1
with the angular frequencies given as usual by !12 = k1 /m and !22 = k2 /m. This is of course
a trivial system, just two uncoupled harmonic oscillators and you can solve it completely:
the first coordinate undergoes harmonic motion with period T1 = 2⇡/!1 while the second
coordinate also undergoes harmonic motion, but with period T2 = 2⇡/!2 . Instead, I want
to draw your attention to a more qualitative aspect of the system, one that carries a more
general lesson with it. I want to analyze the system when !1
!2 , that is when the
oscillation time scale for Q1 is much much shorter than the oscillation time scale for Q2 . The
equations of motion associated with this Lagrangian are simply,
d
Q̇1 = !12 Q1
dt
(1.42)
d
Q̇2 = !22 Q2
(1.43)
dt
Now, an experimentalist measuring this system over short time scales T1 (⇠ 1/!1 ) would
naturally use units of time with T1 , and define her clock with t0 = !1 t. With this change the
equations of motion take the form,
d
Q̇1 = Q1 ,
dt0
(1.44)
d
!22
Q̇
=
Q2 ⇡ 0 ,
2
dt0
!12
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because !1
!2 .
(1.45)
Here I have used the chain-rule twice with d/dt = (dt0 /dt)(d/dt0 ) = !1 d/dt0 and a dot
now represents di↵erentiation with respect to the rescaled time coordinate, t0 . This is very
interesting. On these short time scales the experimentalist monitoring the particle would
conclude that there exists a conservation law,
d
Q̇2 ⇡ 0 .
dt0
(1.46)
That is to say, the conjugate momentum associated with the coordinate Q2 is approximately
conserved. This conservation law is only violated by the small dimensionless ratio !22 /!12 .
If the experimentalist has a sufficiently sensitive measuring device, or she has the luxury of
observing the system over time scales comparable or larger than T2 then she would be able
to detect violations in the conservation of Q̇2 . She would then reach the following important
conclusion:
There are two separate forces acting on the system, one associated with a time scale T1
and another, weaker force, associated with a time scale T2 . The first of these respects the
conservation of the quantity Q̇2 while the second violates it!
This precise situation arises very frequently in particle physics. For example, particles carry
a quantity called strangeness that is exactly conserved by the strong force. However, it is
violated by the weak force as seen in the decay of Kaons (which carry one unit of strangeness)
into pions or light leptons neither of which carry strangeness. Another example, which we
will encounter later in the course, is isospin symmetry which is conserved by the strong
force but is weakly violated by electromagnetism. Yet another, more subtle example, is the
separate conservation of the number of electrons, the number of muons, and the number of
tau-leptons by all known forces. We now know that these separate conservation laws are
only approximate and are in fact violated by the existence of neutrino mass terms.
I could give you many more examples, but that would again carry us into the realm of storytelling. I hope the basic message is clear. The Lagrangian describing the particles contains
di↵erent interactions (forces) that come with di↵erent time scales (di↵erent strengths) and
respect di↵erent conservation laws (di↵erent symmetries). It is sometimes the case that
the stronger interactions result in some conservations laws that is violated by the weaker
interactions, but violated only weakly. We can then speak of approximate conservation
laws or approximate symmetries of the Lagrangian. Conversely, if we observe experimental
evidence for the violation of a well-established conservation law, it would immediately imply
the existence of a new weak interaction (new force) that had been hitherto unobserved. This
is one of the best ways to look for and discover new weak interactions/forces!
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