Download Symmetry

6
Symmetry∗
Warning: This chapter is a bit technical. Have some humour with it.
In this lecture, I will tell you about one of the most important themes of modern theoretical
physics: symmetry. By this I mean
the symmetry of the physical laws themselves,
rather than the symmetry of objects governed by these laws. The fundamental nature of symmetries in physics wasn’t appreciated until recently, but today symmetry shapes how we think
about physics. Most of this material is not taught until graduate classes in physics. But it is
such beautiful and important stuff that I want to try to give you the gist of it. It will be a bit
tougher than the average material in this course, so please bear with me ...
Take a piece of paper. Now draw a closed loop on it. You might have drawn something like
this:
But this is only one of many possible closed loops you could have drawn and many of you will
have drawn different loops. Now draw a loop with rotational symmetry. This time, you all had
to draw a circle:1
You will have drawn circles of different sizes, but circles nonetheless. This is the power of
symmetry: it cuts down the number of possibilities. In this lecture, we will see how similar
reasoning applies to the laws of physics.
1
It fact, you might have drawn a circle to begin with. This is because we intrinsically prefer symmetry. Even
if you drew two circles, I hope you still get the point.
67
6. Symmetry∗
68
6.1
Symmetry in Physical Laws
What is symmetry? How can a physical law be “symmetrical”? We need a definition. I will use
the following:
a “thing” is symmetrical if there is “something” we can do to it,
so that after we have done it, it “looks” the same as it did before.
For example, you could rotate your piece of paper by any angle and the circle you drew on it
would look the same. Here is a list of symmetry operations that are interesting in physics:
• Translation in space
• Translation in time
• Rotation by a fixed angle
• Uniform velocity in a straight line
• Reversal of time
• Reflection of space
• Exchange of identical particles
• Exchange of matter and anti-matter
• Phase rotation of the wavefunction
As we will see, not all of these are actually symmetries of the laws of physics.
6.2
6.2.1
Global Symmetries
Symmetries of Space and Time
Let us start with the symmetries of space and time. In short, we believe that
the laws of physics are the same everywhere and at all times.
Of course, this is a question that can be addressed experimentally. Consider measuring the
energy levels of a gas of hydrogen atoms. We do the experiment in a certain region of space.
Now imagine that the whole experimental apparatus (and everything in its surroundings that
could possibly affect it) is translated in space to a different location. We expect the outcome of
the experiment to be the same. In fact, we see this in Nature: we look at the light from distant
stars and measure the energy levels of the hydrogen in their atmospheres. We find the same
spectral lines as in our labs on Earth. The laws of atomic physics don’t seem to be different at
the locations of the stars.
6.2 Global Symmetries 69
How is this symmetry reflected in our equations? As an example, consider Newton’s law of gravity
between the Sun and the Earth located at positions �xS and �xE . Newton’s F� = m�a for the motion
of the Earth is
d2 �xE
�xE − �xS
m✟
= −G ✟
m✟
.
(6.2.1)
E
E MS
✟
dt2
|�xE − �xS |3
Translation in space corresponds to a shift of all spatial coordinates
�x → �x � = �x + �a ,
(6.2.2)
where �a is a constant vector. For this to be a symmetry it should not change the dynamics.
Applying (6.2.2), we see that both sides of (6.2.1) don’t change
d2 �xE
d2 (�xE + �a)
d2 �xE
→
=
,
2
2
dt
dt
dt2
�xE − �xS
�xE + �a − (�xS + �a)
�xE − �xS
−GMS
→ −GMS
= −GMS
.
|�xE − �xS |3
|�xE + �a − (�xS + �a)|3
|�xE − �xS |3
(6.2.3)
(6.2.4)
The law of gravity therefore only depends on the relative positions of two objects, not on their
absolute positions in space. (This means that we are free to set �xS ≡ 0, which we will do from
now on.) The same is true for all laws of physics. If you ever find an equation in physics that
doesn’t satisfy this, it is wrong.
We also believe that translation in time has no effect on physical laws. Whether we measure
the energy levels of hydrogen today, tomorrow or in ten years makes no difference. Moreover,
when we look at distant stars we are really looking back in time, so the fact that the energy
levels of hydrogen are the same in all stars we ever looked at tells us about symmetry both in
space and in time.
Let’s see how time translation symmetry is manifested in our Newtonian example. It is easy to
see that eq. (6.2.1) doesn’t change under a constant time shift
t → t� = t + a .
(6.2.5)
All correct equations in physics have this property.
Similarly, rotation in space leaves the physical laws invariant. If we turn our hydrogen experiment at an angle it gives the same energy levels, provided we turn everything else that is
relevant along with it. Indeed, we have never observed that the orientation of an experiment
affects the outcome.
What happens to eq. (6.2.1) if we rotate all spatial coordinates through a constant angle θ,
�x → �x � = R(θ)�x .
(6.2.6)
This time the equation does change. However, it changes in exactly the same way on the lefthand-side and the right-hand-side
� 2 �
d2 �xE
d2 (R�xE )
d �xE
→
=
R
,
(6.2.7)
2
2
dt
dt
dt2
�
�
�xE
R�xE
�xE
−GMS
→
−GM
=
R
−GM
.
(6.2.8)
S
S
|�xE |3
|R�xE |3
|�xE |3
Hence, the dynamics doesn’t change after rotation. Constant rotations are a symmetry of Newtonian gravity.
70
6. Symmetry∗
To avoid giving the impression that you could do practically anything without changing the
phenomena, let me give an example to the contrary—just to see the difference. Suppose that
we ask: “Are the physical laws symmetrical under a change of scale?” Again, our hydrogen
example illustrates the point. Imagine we make the apparatus ten times bigger in every part—
will it work exactly the same way? The answer, in this case, is no! The wavelength of light
emitted by the gas of hydrogen atoms in the new experiment in not ten times longer than in
the original, smaller experiment. In fact, it is exactly the same. So the ratio of the wavelength
to the size of the emitter will change. The laws of atomic physics are not scale-invariant.
Special relativity rests on the belief that the laws of physics for someone standing still are the
same as for someone moving at a uniform speed in a straight line. We called this fact boost (or
Lorentz) symmetry. This means that we don’t expect to find a different outcome if we perform
our hydrogen experiment on a train in uniform motion.
What happens to the Newtonian gravity example if both the Sun and the Earth are boosted
ˆ? (Here, we have rotated the
with velocity v along the axis defined by the direction �xE − �xS ≡ x �x
coordinates such that the x-axis lies along the boost direction). The relevant boost transformations
are
t − vx/c2
t → t� = �
1 − v 2 /c2
x → x� = �
and
x − vt
1 − v 2 /c2
.
(6.2.9)
I leave it to you as an exercise to show that the left-hand-side and the right-hand-side of eq. (6.2.1)
now transform differently
d2 x�
d2 x
GMS
= γ 3 2 �= − �2 .
(6.2.10)
�2
dt
dt
x
Evidently, boosts are not a symmetry of eq. (6.2.1). But, we claimed that boosts are a symmetry
of the laws of physics? What is going on? The resolution is that Newton’s equations are only
valid for small speeds, v � c. Both F = ma and Fg = GmM/r2 are wrong at speeds close to the
speed of light. Einstein’s theory of relativity (see Lecture 7) fixes this and leads to equations that
are symmetric under boosts.
In Lecture 4 we uncovered a hidden symmetry of Maxwell’s equations. Since this will be
important for the rest of the lecture, let us recall what we had found: We showed that the four
Maxwell equations can be written as a single equation
✷Aµ = µ0 J µ ,
(6.2.11)
if we define the four-vectors
�
Aµ = (φ/c, A)
where
� =∇×A
�
B
J µ = (cρ, J� ) ,
(6.2.12)
� = −∇φ − A
�˙ .
E
(6.2.13)
and
and
It is easy to see that Lorentz transformations—Aµ� = Λµ ν Aν and J µ� = Λµ ν J ν —are a symmetry
of the Maxwell equation (6.2.11),
✷Aµ → ✷� Aµ� = ✷(Λµ ν Aν ) = Λµ ν ✷Aν ,
J
µ
→ J
µ�
µ
ν
= Λ νJ ,
(6.2.14)
(6.2.15)
i.e. ✷� Aµ� = µ0 J µ� if ✷Aµ = µ0 J µ , and vice versa. This is just like spatial rotations are a
symmetry of Newton’s equations.
6.2 Global Symmetries 71
We also discussed that the choice of Aµ is not unique. The electric and magnetic fields in
(6.2.13) don’t change if we make the following transformations
�→A
�� = A
� + ∇α ,
A
(6.2.16)
Aµ → A�µ = Aµ + ∂µ α .
(6.2.18)
φ → φ� = φ − α̇ ,
(6.2.17)
or
This is called a gauge symmetry of Maxwell’s equations.
6.2.2
Noether’s Theorem
One of the most profound results in all of physics is Noether’s theorem:
For every continuous symmetry of the laws of physics,
there must exist a conservation law.
This theorem provides a direct connection between dynamics and the abstract world of symmetry. The most important examples are:
Symmetry under translation in space
Symmetry under rotation in space
⇔
Symmetry under translation in time
⇔
Conservation of momentum
Conservation of angular momentum
⇔
Conservation of energy
The general proof of Noether’s theorem is too boring to reproduce here.
Let me remark that the reverse of Noether’s theorem is also true:
For every conservation law, there must exist
a continuous symmetry of the laws of physics.
For example, you have all heard about the conservation of electric charge. But, what is the
corresponding symmetry? It turns out that the conservation of charge is a consequence of
the laws of quantum mechanics being invariant under a phase rotation of the wavefunction,
Ψ → eiα Ψ, i.e.
Conservation of charge
⇔
Symmetry under phase rotation of wavefunction
Modern physics is full of these abstract symmetries and the corresponding new conservation
laws. But we are getting a bit ahead of ourselves ...
6. Symmetry∗
72
6.3
Local Symmetries
So far, we have only considered global symmetries—i.e. transformations that act the same at
every point in space and time. Next, we want to look at local symmetries—i.e. transformations
that can be different at different points in space and time.
6.3.1
Let There Be Light!
In Lecture 5, I told you that modern particle physics associates a field with every matter particle
Ψ(t, �x) .
(6.3.19)
These fields are usually complex-valued, i.e. they have an amplitude and a phase at every point
in spacetime. We can represent the phase information by little arrows:
spacetime
(Here, I have reduced four-dimensional spacetime to a one-dimensional line.)
The dynamics of the field is described by its Lagrangian
L[Ψ] = ∂µ Ψ∂ µ Ψ∗ + · · · .
(6.3.20)
We won’t need to know the specific form of the Lagrangian, only that it is a function of the field
configuration Ψ(t, �x).
Now, consider the following transformation of the field
Ψ → Ψ� = eiα Ψ ,
α = const.
(6.3.21)
This corresponds to rotating all phase arrows by the same angle α:
spacetime
This operation is a symmetry; i.e. nothing changes after rotating the arrows. The black arrows
and the grey arrows represent exactly the same physical situation.2
In particular, the Lagrangian for the field Ψ before and after the transformation (6.3.21) is the
same:
L[Ψ] → L[Ψ� ] = L[Ψ] .
(6.3.22)
By Noether’s theorem, the symmetry Ψ → eiα Ψ should correspond to a conservation law—
the global conservation of charge. (Unfortunately, I don’t know a simple way to show this.)
The word ‘global’ is important: it allows for a charge to be destroyed on the Earth as long
2
Global phase symmetry isn’t anything deep. What we call the zero phase angle is really up to us. All that
the global phase symmetry is saying is that the physics is independent of the arbitrary choice we made. It would
be more shocking if this was not true.
6.3 Local Symmetries
73
as another charge is created simultaneously on the Moon. (Obviously, this violates relativity
because it would require information to travel faster than light.) In the real world, we would
like something better than that: we want local conservation of charge—meaning that (isolated)
charges can’t disappear spontaneously, they can only move from one point to the next. Moreover,
particles and anti-particles can only annihilate if they meet at a point.
The symmetry associated with local charge conservation is local phase rotation
Ψ → Ψ� = eiα(t,�x) Ψ ,
α(t, �x) �= const.
(6.3.23)
This corresponds to rotating the little phase arrows by different amounts at different points in
space and different moments in time:
spacetime
Unfortunately, this transformation does not seem to leave the physics unchanged.
In particular, the Lagrangian before and after the transformation (6.3.23) is not the same:
L → L� = L + J µ ∂ µ α ,
(6.3.24)
where J µ [Ψ] is a function of Ψ (called the ‘current’). The detailed form of J µ won’t be important,
but could be obtained from eq. (6.3.20). Note that the extra term in eq. (6.3.24) has to be
proportional to ∂µ α, since it should vanish for α = const.
Can we fix this? What do we have to do to make local phase rotation a symmetry? It turns out
that we need to introduce another field—a four-vector field Aµ (t, �x). We represent the field Aµ
by a line sticking out of each spacetime point (mathematicians call this a ‘fibre’). The length of
the line represents the magnitude of the field:
spacetime
We then introduce the following strange rule: when we rotate the phase arrows of Ψ by α(t, �x)
we simultaneously change the length of the lines representing Aµ by ∂µ α(t, �x):
spacetime
This rule corresponds to the transformation
Aµ → A�µ = Aµ + ∂µ α .
(6.3.25)
74
6. Symmetry∗
Importantly, the α’s in eqs. (6.3.23) and (6.3.25) are the same. The field Aµ therefore “knows”
about the relative phases of Ψ between different points in spacetime.
The mathematical reason that this works is the following: We start by defining a new Lagrangian
� ≡ L − J µ Aµ ,
L
(6.3.26)
i.e. we add to the old matter Lagrangian a coupling between Aµ and the current J µ . It is easy to
see that this Lagrangian doesn’t change under the transformations (6.3.23) and (6.3.25):
� = L − J µ Aµ → L� − J µ A� = (L + ✘
� . (6.3.27)
L
J µ✘
∂µ✘
α) − (J µ Aµ + ✘
J µ✘
∂µ✘
α) = L − J µ Aµ = L
µ
The term that spoiled the symmetry in eq. (6.3.24) has been cancelled exactly. Introducing the
field Aµ , with the transformation property (6.3.25), has fixed the problem.
But, what is Aµ ? In fact, you have seen it before! It is precisely the vector potential of
electromagnetism (see Lecture 4). In particular, the transformation (6.3.25) is exactly the same
as in (6.2.18). If we want local phase rotation symmetry of the matter fields, we have to introduce
the electromagnetic force. If you had never seen electromagnetism before, this is how it would
arise from pure thought!
Phase rotations are described by the group U (1). What about other symmetries of the matter
fields, described by other groups? Indeed, as we have seen in Lecture 5, the Standard Model
uses two more groups: SU (2) and SU (3). Asking these symmetries to be local requires new
forces: the weak and the strong nuclear force.
6.3.2
Gravity from Symmetry
Above we stated that all laws of physics are invariant under constant (or global) translations in
space and time
�x → �x � = �x + �a ,
t → t� = t + a .
(6.3.28)
(6.3.29)
Imagine that we want these symmetries to be local
�x → �x � = �x + �a(t, �x) ,
�
t → t = t + a(t, �x) .
(6.3.30)
(6.3.31)
For this transformation to be a symmetry we again have to introduce a force: gravity! We will
have much more to say about gravity in the next lecture. However, even if we had never heard
about gravity, even if we have never seen an apple fall, considerations of symmetry lead us to
the concept of gravity by pure thought. It is a powerful way of doing theoretical physics.
6.3.3
From Symmetries to Forces
Let me summarize the key insight of this rather technical section:
The forces of Nature are dictated by symmetries, they could not be other than what they are!
Once we require the laws of particle physics to be symmetric under (local) transformations by
certain symmetry groups, we have to add forces. Symmetries are input, forces are output. The
6.4 Discrete Symmetries∗
75
symmetry groups associated with the four fundamental forces are:
U (1) → electromagnetic force
SU (2) → weak force
SU (3) → strong force
SO(1, 3) → gravity
6.4
Discrete Symmetries∗
So far, we have discussed only continuous symmetries, in the sense of rotations, translations and
boosts by any arbitrary amount. For example, a circle comes back to itself after being rotated
by any angle:
There are also important discrete symmetry operations in physics, meaning that the symmetry
only acts by a certain amount. An equilateral triangle only comes back to itself if rotated by
multiples of 120◦ :
Of particular importance in particle physics are symmetries where one flips the sign of an
object that can take two values. It turns out that this will be intimately linked to the concept
of anti-matter.
The three most important discrete symmetries are parity (P), charge conjugation (C), and
time reversal (T). Rather surprisingly, physics chooses not to obey these symmetries. And it is
a good thing. As we will see, this act of rebellion allowed the universe to form interesting things
such as galaxies and life.
Let me explain what these discrete symmetry operations are and how they act on elementary
particles. We represent an elementary particle by a sphere:
The particle can move:
It could spin about some axis. For massless particles, quantum mechanics dictates that the spin
can only be left-handed or right-handed around the axis defined by the direction of motion (but
nothing in between):
left-handed
right-handed
76
6. Symmetry∗
If the particle has negative charge it could be the electron. If it has positive charge it could be
the positron (the anti-particle of the electron):
6.4.1
-
+
electron
positron
C, P, and T
Parity (P) flips the sign of every spatial coordinate, �x → −�x. This is equivalent to reflection in a
mirror. Note that the mirror image of a left-handed spinning particle is a right-handed spinning
particle:
P
left-handed
right-handed
Charge conjugation (C) exchanges every particle with its corresponding anti-particle:
-
+
C
particle
anti-particle
Performing both C and P together leads to a CP transformation. This exchanges a left-handed
particle with a right-handed anti-particle:
-
+
CP
left-handed
particle
right-handed
anti-particle
Time reversal (T) flips the sign of the time coordinate, t → −t. A left-handed particle moving
in the forward direction becomes a left-handed particle moving in the backward direction. Alternatively, we can interpret this as a right-handed anti-particle moving in the forward direction.
left-handed
particle
-
-
T
+
=
right-handed
anti-particle
left-handed
particle
Finally, we can perform C, P and T together and call it CPT. A forward moving left-handed
particle stays a forward moving left-handed particle:
left-handed
particle
CPT
left-handed
particle
6.4 Discrete Symmetries∗
77
For a long time, physicists believed all of these transformations to be symmetries of the laws
of physics. Shockingly, experiments of the weak force showed that, except CPT, none of these
transformations are actually symmetries of physics!
6.4.2
CP Violation!
For a long time, it seemed obvious that nothing would change if we swapped left-handed and
right-handed particles. It therefore came as quite a surprise when parity violation was detected
in the weak decay of unstable pions3 (π − ) in muons4 (µ− )) and anti-neutrinos ν̄µ ,
π − → µ− + ν̄µ .
(6.4.32)
The pions don’t have any spin, so angular momentum conservation requires the produced muon
and anti-neutrino to have opposite spins. Still, there are two possibilities: left-handed muons
and right-handed muons. Both experimental outcomes are related by parity symmetry. It was
absolutely shocking when muons were found to be always left-handed, and never right-handed:
left-handed
always
P
P
never
right-handed
The weak force, which governs the pion decay, violates P!
Pions are composite particles made out of pairs of (anti)quarks, e.g. π − is made out of a down quark and a
up anti-quark.
4
A muons is like an electron, just more massive.
3
78
6. Symmetry∗
Now, if charge conjugation is a good symmetry, then the process
π + → µ+ + ν µ ,
(6.4.33)
should always produce left-handed anti-muons. Instead, experiments found that this decay never
produces left-handed anti-muons, only right-handed ones:
left-handed
always
C
never
left-handed
The weak force also violates C!
Maybe there is a saving grace and CP is conserved? Indeed this is the case in the pion decay:
left-handed
always
CP
always
right-handed
However, within a couple of years is was discovered that decays of neutral K mesons5 violate
CP symmetry. Not even CP survived as a good symmetry of Nature. This is a good thing! It
explains why the universe is filled with matter and not anti-matter (see Lecture 8).
6.4.3
CPT Invariance?
What about CPT? We have strong reasons to believe that this is, in fact, an exact symmetry of
Nature. It is possible to show that, without CPT symmetry, quantum mechanics would make
no sense. However, experimenters should never trust theorists. There are significant efforts to
find CPT violation. Maybe we will be surprised again.
5
A neutral K meson is made out of a down quark and a strange antiquark.