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 symmetryb. Rotational symmetryc. Symmetry with respect to moving observer

4. Gauge symmetries and conserved additive quantum numbers

a. Electric chargeb. Baryon (quark) number and quark flavorc. 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).

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.

Symmetry Space translation Time translation Rotation

Conserved quantity

Linear momentum Energy Angular

momentum

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.

`ei

where is the field representing the particles and is an arbitrary number. If depends (does not depend) on the space coordinate, this is known as a local (global) gauge transformation.

3. Since the phase is not observable, the laws of physics should not depend on the value of 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.

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.

Particle Quark number

Lepton number

Each quark 1 0Each anti-quark -1 0Each lepton 0 1Each anti-lepton 0 -1

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 decay:

np e

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.

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 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 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.

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 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 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.

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

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 CP T CPT Q L Li B

Weak interaction no no weakly

brokenweakly 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