Symmetries and conservation laws:

1

• Symmetries and conservation laws
• What do we mean by a symmetry and a conservation
  law?
• What is the relationship between a symmetry and a
  conserved quantity?
• Continuous symmetries and constants of motion
• Time and space translation symmetry
• Rotational symmetry
• Symmetry with respect to moving observer
• Gauge symmetries and conserved additive quantum
  numbers
• Electric charge
• Baryon (quark) number and quark flavor
• Lepton number and lepton flavor
• Discrete symmetries of charge conjugation, parity
  and time reversal

2

• What do we mean by a symmetry?
• A symmetry is a change of something that leaves
  the physical description of the system unchanged.
• Physical objects have certain symmetries people
  are approximately bilaterally symmetrical, a
  sphere is symmetrical with respect to rotation
  about any axis through its center. I will not
  talk about this kind of symmetry.
• The laws of nature (the mathematical way in which
  we describe objects and their interactions) are
  unchanged with respect to changes in some things.
• We need to be careful that everything appropriate
  is changed. For example, if I move horizontally,
  the laws of nature arent different, but if I
  alone move, my motion may change. The laws are
  the same, but their application is different if I
  move outside the building.

3

• Translations in time and space and rotations
• Physical laws are unchanged if time or any of the
  space coordinates is shifted by a constant
  amount.
• This is plausible forces generally depend on
  differences in coordinates, unaffected by the
  origin of the coordinate system. Velocity and
  acceleration have to do with time derivatives of
  positions, also unaffected by changing the origin
  of the coordinate system (3 symmetries for 3
  orthogonal directions of translation).
• Physical laws are also unaffected by changing the
  origin of the time coordinate things work the
  same if we come back tomorrow and do the same
  experiment ( 1 symmetry).
• Physical laws also unaffected by rotating the
  coordinate system (3 symmetries for 3 axes of
  rotation.)

4

• Associated with each continuous symmetry
  operation is a conserved quantity
• This fact can be derived from the laws of
  dynamics in a way that is straightforward but
  beyond the scope of this talk (ref. Landau and
  Lifshitz Mechanics).
• This is the first link between symmetries and
  conserved quantities. It is true even in
  classical mechanics, and also true in quantum
  mechanics and field theories.
• No evidence for violation of energy, momentum, or
  angular momentum conservation is seen.

Symmetry Space translation Time translation Rotation
Conserved quantity Linear momentum Energy Angular momentum
5

• A second type of symmetry has to do with a
  reference frame moving with respect to one in
  which the laws of physics are valid. A reference
  frame in which Newtons laws work is called an
  inertial reference frame.
• Physical laws are unchanged when viewed in any
  reference frame moving at constant velocity with
  respect to one in which the laws are valid.
• It is not true that all measured quantities are
  unchanged for example, energy and momentum will
  have different values when calculated in
  different frames.
• The fact that the laws of motion are unchanged
  plus the principle that the speed of light is a
  quantity that has the same value in any reference
  frame is the essence of the theory of special
  relativity.

6

• Special relativity has certain consequences
• Two events that are simultaneous in one reference
  frame are not simultaneous in a reference frame
  moving with respect to it.
• There are some quantities (called Lorentz
  scalars) that have values independent of the
  reference frame in which their value is
  calculated. One example is the rest mass,
  defined by
• m02c4 E2 p2c2

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• Is a reference frame that is rotating at constant
  angular velocity with respect to an inertial
  frame also an inertial frame?
• Newtons laws do not work in such a frame, in the
  sense that particles will not continue to move in
  a straight line in the absence of an applied
  force.
• Machs principle says that the preferred
  rotational frame is one that is not moving with
  respect to the large mass of the universe. This
  is a conjecture that is difficult to test.

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• A different kind of conserved quantity is the
  electric charge. So far as we know, the total
  electric charge is conserved. Physical processes
  can move charge from one particle to another, but
  only in ways that keep the total charge, gotten
  by summing all the charges, constant.
• The conservation of electric charge also follows
  from a symmetry of nature, this a bit more
  abstract. The laws of electricity and magnetism
  are described by what is called a field theory,
  where particles are represented by fields. Fields
  can be represented by complex functions
  (including real and imaginary parts) that have a
  value at all points in space. The field theory
  describing electricity and magnetism is extremely
  successful.
• A gauge transformation is one in which the field
  is changed by multiplying it by a complex number
  with magnitude one.
• F FeiQwhere F is the field
  representing the particles and Q is an arbitrary
  number. If Q depends (does not depend) on the
  space coordinate, this is known as a local
  (global) gauge transformation.
• Since the phase Q is not observable, the laws of
  physics should not depend on the value of Q.
  Invariance of the laws of physics under local
  gauge transformations requires the existence of a
  conserved charge.

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• The electric and magnetic forces act on particles
  that carry electric charge. Similarly, the strong
  force acts on particles that carry color charge
  quarks and gluons. Color charge is also
  conserved, for a reason very similar to that for
  electric charge.
• Strong forces are described by a field theory
  (quantum chromo dynamics or QCD), and invariance
  with respect to local gauge transformations in
  QCD requires the existence of color charges that
  are conserved.
• QCD describes very well the strong interactions.
  A property of the theory is that only
  color-neutral objects can propagate long
  distances hence it is not possible to directly
  test the conservation of color charge.
• Weak interactions are similarly described by a
  field theory that is unified with that of
  electricity and magnetism. Again, invariance with
  respect to local gauge transformations implies
  the existence of a conserved weak charge.

10

• Other conserved quantities that are similar to
  electric charge in the sense that the total value
  is (approximately) conserved and that the
  conserved quantity takes on integer values. These
  are quark (baryon) number and lepton number.
• Protons, neutrons and other baryons each have
  three quarks, so conservation of quark number
  also implies conservation of baryon numberB
  Q/3.
• There is no field theory that would imply the
  existence of a conserved quantity such as lepton
  number and baryon number. For that reason, it is
  believed that baryon and lepton number are only
  approximately conserved. No evidence is yet seen
  for baryon or lepton number violation.
• There are also approximately conserved numbers
  associated with each separate type of lepton and
  with each type of quark.

Particle Quark number Lepton number
Each quark 1 0
Each anti-quark -1 0
Each lepton 0 1
Each anti-lepton 0 -1
11

• Finally, there are three discrete symmetries
  associated with reversing the direction of some
  quantity. These are
• Charge conjugation changing particles into
  anti-particles.
• Parity inversion reversing the direction of
  each of the three spatial coordinates.
• Time reversal changing the direction of time.
• These are interesting because it is not obvious
  whether the laws of nature should look the same
  for any of these changes, and the answer was
  surprising when these symmetries were first
  tested. I will use the example of a neutron and
  its decay to illustrate each of the three
  symmetries. Neutrons have spin angular momentum
  of ½ and decay in a process called b decay
• n?p e n

12
Charge conjugation (C) simply means to change
each particle into its anti-particle. This
changes the sign of each of the charge-like
numbers. The neutron is neutral, nonetheless it
has charge-like quantum numbers. It is made of
three quarks, and charge conjugation change them
into three anti-quarks. Charge conjugation leaves
spin and momentum unchanged. The interesting
question is, does a world composed completely of
anti-matter have the same behavior. For example,
in neutron decay, there is a correlation between
the spin of the neutron and the direction of the
electron that is emitted when the neutron decays.
The electron spin is also directed opposite to
its direction of motion.
? ?
momentum direction
n ? p e
? ? ? ?
spin direction Charge conjugated
? ? momentum direction
n
? p e
? ? ? ? spin
direction This is not what an anti-neutron decay
looks like! The laws of physics responsible for
neutron decay are not invariant with respect to
charge conjugation. This feature is restricted to
the weak interaction.
13
The parity operation (P) changes the direction
(sign) of each of the spatial coordinates. Hence,
it changes the sign of momentum. Since spin is
like angular momentum (the cross product of a
vector direction and a vector momentum, both of
which change sign under the parity operation),
spin does not change direction under the parity
operation.
? ? momentum
direction
n ? p e
? ? ? ? spin
direction Parity operation
? ?
momentum direction
n ? p e
? ? ? ?
spin direction The world would look
different under the parity operation, since now
the electrons spin would be in the same
direction as its momentum. The world is not
symmetric under the parity operation! Parity
violation occurs only in the weak interaction.
14
The lack of symmetry under the parity operation
was discovered in the fifties following the
suggestion of Lee and Yang that this symmetry was
not well tested experimentally. It is now known
that parity is violated in the weak interaction,
but not in strong and electromagnetic
interactions. The situation with charge
conjugation symmetry is similar the lack of
symmetry under charge conjugation exists only in
the weak interaction. The Standard Model
incorporates parity violation and charge
conjugation symmetry violation in the structure
of the weak interaction properties of the quarks
and leptons and in the form of the weak
interaction itself.
15
Now lets consider what happens when we apply
both the charge conjugation operation and the
parity operation.
? ? momentum
direction
n ? p e
? ? ? ? spin
direction Parity operation
? ?
momentum direction
n ? p e
? ? ? ?
spin direction Parity operation plus
charge conjugation
? ?
momentum direction
n ? p e
? ? ? ?
spin direction This is in fact what an
anti-neutron decay looks like! The world appears
to be symmetric under the CP operation (at least
for neutron decay). CP is in fact weakly
broken, which I will come to later.
16
Time reversal means to reverse the direction of
time. Here we need to be a bit more careful.
There are a number of ways in which we can
consider time reversal. For example, if we look
at collisions on a billiard table when the cue
ball strikes the colored balls on the break, it
would clearly violate our sense of how things
work if time were reversed. It is very unlikely
that we would have a set of billiard balls moving
in just the directions and speeds necessary for
them to collect and form a perfect triangle at
rest, with the cue ball moving away. However, if
we look at any individual collision, reversing
time results in a perfectly normal looking
collision (if we ignore the small loss in kinetic
energy due to inelasticity in the collision). The
former lack of time reversal invariance has to do
with the laws of thermodynamics we here are
interested in individual processes for which the
laws of thermodynamics are not important. Time
reversal reverses momenta and also spin, since
the latter is the cross product of a momentum
(which changes sign) and a coordinate, which does
not.
17
Now lets consider what happens when we apply
time reversal (T) to the case of the neutron
decay.
? ? momentum direction
n
? p e
? ? ? ? spin
direction Time reversal
? ?
momentum direction
n ? p e
? ? ? ?
spin direction This looks just fine, the
electron spin is opposite to its momentum and the
electron direction is opposite to the neutrons
spin. So, at least for neutron decay, the laws
of physics appear to be symmetric under time
reversal invariance.
18
Now lets consider what happens when we apply all
three symmetry operations to the case of neutron
decay.
? ? momentum direction
n
? p e
? ? ? ? spin
direction Time reversal
? ? momentum direction
n ?
p e
? ? ? ? spin
direction Parity plus time reversal
? ? momentum direction
n ? p
e ?
? ? ? spin direction T, P and
Charge conjugation ? ?
momentum direction
n ? p e
? ? ? ?
spin direction
19
The result that applying C, P, and T leaves the
physical laws unchanged is not surprising. Since
CP leaves things unchanged (for neutron decay)
and T also does, applying all three should also
work fine. In fact, there is a theorem that
says that under rather general conditions, any
set of physical laws that can be described by a
field theory will be unchanged under the CPT
operation. There are many consequences to this
theorem, for example that the total lifetime and
mass of a particle is identical to that of its
anti-particle. There are some considerations of
conditions under which physical laws are not
invariant under CPT, but sensitive experimental
tests of CPT invariance have not shown any
evidence for its breakdown.
20

Could there be evidence of violation of one or
more of these symmetries in neutrons? Consider
the case if neutrons had an electric dipole
moment (edm). The neutron has no charge, but it
does have charged quarks inside it. If the charge
is distributed such that the negative and
positive charge is separated by some distance
(within the neutron), then it would have a dipole
moment. The value of the dipole moment is the
value of the positive charge times the distance
between the positive and negative charges. The
direction of the dipole moment must be aligned
with the spin. Assume the neutron dipole moment
points in the same direction as the spin .
T P C
CP CPT ? ? ?
? ? ? dipole moment direction
n n n n
n n ? ?
? ? ? ? spin direction Charge
conjugation correctly turns a neutron into an
anti-neutron, with the spin and electric dipole
moment in opposite directions. CPT does the same.
However, both T and CP produce non-physical
particles, with the relative direction of spin
and edm incorrect. The existence of a neutron edm
explicitly violates CP and T symmetries. No such
evidence for a neutron edm is seen.
21
There is evidence for violations of CP symmetry
and hence of T symmetry. Until recently, that
evidence existed solely in the decays of neutral
kaons. A neutral kaon is a meson consisting of a
strange quark and a down quark. The physical
particles with definite mass and lifetime are
combinations of a kaon and an anti-kaon, much the
way that circularly polarized light is a
combination of vertically and horizontally
polarized light. Now, one combination of K0 and
K0 that makes a physical particle with definite
mass and lifetime is mostly a CP eigenstate with
eigenvalue 1 (K0S) and another combination is a
CP eigenstate with eigenvalue 1 (K0L) . It was a
surprising result found in 1964 that the K0L
decayed into a pair of pions that were in a CP
eigenstate with eigenvalue 1. This implied
violation of CP symmetry in kaon decays. The
manifestation of CP violation is restricted to
the weak interaction. Hence processes that
involve only the electromagnetic and strong
interactions appear to be CP conserving. Only
very recently has other evidence of CP violation
been found, and that is the subject of the next
lecture.
22

Symmetry C P CP T CPT Q L Li B
Weak interaction no no weakly broken weakly broken yes yes yes no yes
Electromagnetic interaction yes yes yes yes yes yes yes yes yes
Strong interaction yes yes yes yes yes yes yes yes yes