Download Symmetries and Conservation Laws

Symmetries and conservation laws:
1. What do we mean by a symmetry and a conservation
law?
2. What is the relationship between a symmetry and a
conserved quantity?
3. Continuous symmetries and constants of motion
a. Time and space translation symmetry
b. Rotational symmetry
c. Symmetry with respect to moving observer
4. Gauge symmetries and conserved additive quantum
numbers
a. Electric charge
b. Baryon (quark) number and quark flavor
c. Lepton number and lepton flavor
5. Discrete symmetries of charge conjugation, parity and
time reversal
What do we mean by a symmetry?
1. A symmetry is a change of something that leaves the
physical description of the system unchanged.
a. 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.
b. The laws of nature (the mathematical way in which
we describe objects and their interactions) are
unchanged with respect to changes in some things.
2. We need to be careful that everything appropriate is
changed. For example, if I move horizontally, the laws of
nature aren’t 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.
Translations in time and space and rotations:
1. Physical laws are unchanged if time or any of the space
coordinates is shifted by a constant amount.
2. 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).
3. 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).
4. Physical laws also unaffected by rotating the coordinate
system (3 symmetries for 3 axes of rotation.)
Associated with each continuous symmetry operation is a
conserved quantity:
1. 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).
Symmetry
Space translation Time translation
Rotation
Conserved
quantity
Linear
momentum
Angular
momentum
Energy
2. 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.
3. No evidence for violation of energy, momentum, or
angular momentum conservation is seen.
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 Newton’s 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.
1. It is not true that all measured quantities are unchanged;
for example, energy and momentum will have different
values when calculated in different frames.
2. 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.
Special relativity has certain consequences:
1. Two events that are simultaneous in one reference
frame are not simultaneous in a reference frame moving
with respect to it.
2. 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
Is a reference frame that is rotating at constant angular
velocity with respect to an inertial frame also an inertial frame?
1. Newton’s 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.
2. Mach’s 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.
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.
1. 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.
2. A gauge transformation is one in which the field is changed by
multiplying it by a complex number with magnitude one.
F` = FeiQ
where 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.
3. 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.
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.
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.
Quark Lepton
Particle
number number
Each quark
1
0
Each anti-quark
-1
0
Each lepton
0
1
Each anti-lepton 0
-1
Protons, neutrons and other baryons each have three quarks, so
conservation of quark number also implies conservation of baryon number:
B = Q/3.
1. 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.
2. There are also approximately conserved numbers associated with
each separate type of lepton and with each type of quark.
Finally, there are three discrete symmetries associated with reversing the
direction of some quantity. These are:
1. Charge conjugation – changing particles into anti-particles.
2. Parity inversion – reversing the direction of each of the three spatial
coordinates.
3. 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
Charge conjugation (C) simply means to change each particle into its antiparticle. 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 antiquarks. 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
n p e

  
spin direction
Charge conjugated:
n



n p e
  
momentum direction
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.
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



n p e
  
momentum direction
n



n p e
  
Parity operation:
spin direction
spin direction
The world would look different under the parity operation, since now the
electron’s 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.
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.
Now let’s consider what happens when we apply both the charge
conjugation operation and the parity operation.


momentum direction
n
n p e

  
spin direction
Parity operation:
n



n p e
  
Parity operation plus charge conjugation:


n
n p e

  
momentum direction
spin direction
momentum direction
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.
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.
Now let’s consider what happens when we apply time reversal (T) to the
case of the neutron decay.


momentum direction
n
n p e

  
spin direction
Time reversal:
n



n p e
  
momentum direction
spin direction
This looks just fine, the electron spin is opposite to its momentum and the
electron direction is opposite to the neutron’s spin.
So, at least for neutron decay, the laws of physics appear to be symmetric
under time reversal invariance.
Now let’s consider what happens when we apply all three symmetry
operations to the case of neutron decay.

n

Time reversal:
n

p
momentum direction
e
  
spin direction

momentum direction

n
n

  
spin direction

momentum direction
Parity plus time reversal:
p
e

n
n

  
spin direction

momentum direction
T, P and Charge conjugation:
p
e

n
n

  
p
e
spin direction
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.
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.
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.
Symmetry
C
P
Weak
interaction
no
no
Electromagnetic
yes yes
interaction
Strong
interaction
yes yes
CP
T
weakly weakly
broken broken
CPT
Q
L
Li
B
yes yes yes no yes
yes
yes
yes yes yes yes yes
yes
yes
yes yes yes yes yes