Dirac Equation

0),(

0),(

00

trmii

trmi

jj

under parity exchange where r r

0),(00 trmii j

j +

Looks like this satisfies a new equation

No longerlooks likeDirac’s eq

That assumed P (r,t)→ (r,t)and simply neglects the 4-momentum-dependent spinor components!

tr ,Whereas satisfies the Dirac equation (in the original coordinate system)

It is which satisfies Dirac in the inverted system. tr ,0

0),(

0),(

0),(

0),(

0),(

0

00

0

00

0

00

trmi

trmii

trmii

trmii

trmi

jj

jj

jj

0 0

applying r r

so

P (r,t)= 0 (r,t)

P (r,t)= 0 (r,t)

0 =

1 0 0 00 1 0 00 0 1 00 0 0 1

The parity of a fermion is the opposite of its antifermion.

You just showed in homework:

LbLaweak

J bLaL

PP 0†

ba

52105

21 11

††

.

.

.

baba 5

21

21

ba )1( 5

21

55

5005

55

†

VECTOR AXIAL VECTOR

This V-A coupling clearly identifies a parity-VIOLATING interactionThe relative sign between these two terms changes under PARITY!

The full HAMILTONIAN cannot conserve parity!

As we have seenELECTROMAGNETIC and STRONG INTERACTION terms

in addition to free particle terms in the lagrangian

CONSERVE PARITY

Up until 1956 all the laws of physics were assumedto be “ambidextrous” (invariant under parity).

The biological handedness on earth was assumed to be an evolutionary accident.

• heart on left side• intestines winding in same sense• many chemicals synthesized by plants and animals definite handedness• DNA

(no underlying physical constraint)

1956

Chen Ning Yang Tsung Dao Lee

T.D. Lee andC.N. Yangchallenged theassumption thatthe laws of physicsare ambidextrous…

…pressing for more experimentalevidence for theconservation orviolation of parity

Recall: not ALL conservation principles survive decay!

Between Christmas of 1956 and New Year's Day, physicists at the National Bureau of Standards

operated the low temperature equipmentbeing assembled in this photograph.

The vertical tube (upper right)contained 60Co, a beta-ray counter,

and basic thermometry.

A vacuum flask (lower left) is being placed around the tubeto provide insulation to maintain low temperatures.

Chien-Shiung WuColumbia University

clock-wise: Ernest Ambler, Raymond W. Hayward, Dale D. Hoppes, Ralph P. Hudson

National Bureau of Standards

A crystal specimen ofcerium magnesium nitratecontaining a thin surfacelayer of of radioactive 60Co was supported in acerium magnesium nitratehousing within anevacuated glass vessel.

2 cm above an anthracenecrystal scintillator transmitted its light flashes through a lucite rod out to a photomultiplier.

The magnets on eitherside were used to coolby adiabatic demagnetization.

Inductance coils arepart of the thermometry.

Page 90, Ernest Ambler's Notebookfor December 27, 1956.

The first of two successful runs began at 12:04 (middle of page). Hudson's

notation “Field on” refers to the magnetic field produced by the solenoid.

The crystal was cooled and the 60Co nuclei were polarized in one direction.

Hudson later added “PARITY NOT CONSERVED!”

(see top of the page).

After again cooling the crystal andpolarizing the 60Co nuclei in the

opposite direction, the physicists observed the opposite behavior of

the -particle counts with time.

An initially high counting rate of -particles was observed to decrease as the crystal warmed and the 60Co nuclei became randomly polarized.

“  counts decrease”

For 60Co, at 1 K radiation pattern is uniform in all directions.

At lower temperatures the radiation pattern becomes distorted most easily detected by a -detector aligned with the sample axis though frequently an azimuthal detector is also used.

→ →

→

→

The distribution of emitted directions PEAK in the direction of nuclear spin

→ →

→

→

The distribution of emitted directions PEAK in the direction of nuclear spin

Jnucleus · pemittedelectron

→ →

identifying anobservable

quantity that isNOT invariant

to parity!

Here’s a physical processes whose mirror image does NOT occur in nature!

Of course we have already described experiments that determined

π → ν

alwaysleft-handed

alwaysleft-handed

Which strengthens the PARITY-VIOLATING observations:

(π → ν )P L L(π → ν )R R

doesn’t exist!

Weak interactions do NOT conserve parity!

Charge Conjugation InvarianceCharge conjugation reverses the sign of electric charge

Maxwell’s equations remain invariant!(charge/current density and the things derived from them: E and H

just change sign)

In Relativistic Quantum Mechanics this is generalized toparticle antiparticle exchange

Proton p Antiproton p Electron e- Positron e+ Photon

Q +e e e +e 0 B +1 1 0 0 0 L 0 0 +1 0

+2.79 2.79 0

½ћ ½ћ ½ћ ½ћ ћ

eћ2mc

eћ2mc

eћ2mc

+eћ2mc

Charge Conjugate Operator

Obviously C 2| p > = | p > on a proton state

i.e. plus the mesons at the centers of all our multiplet plots:, , , ,

Particles that are their own antiparticle are eigenstates of C

Although C | > = |

>

or even C | > = | > |

>

but applied singly C | p > = | p > | p

>

E and B fields change sign under charge conjugation

C | > = | >

it makes perfect sense to assign the photon c = 1

Then the dominant decay mode of tells us c (0 ) = (-1)(-1) = +1

like parity, a multiplicative quantum number

This seems to explain why

or why

While strong and electromagnetic interactions (productions or decays) are invariantunder CHARGE CONJUGATION

weak interactions:

C : ( + + ) +

both left-handed both left-handed???

Recalling that Dirac particle/antiparticle states have opposite paritymaybe the appropriate invariance is to a simultaneous change of particle antiparticle with parity flipped!

CP : ( L+ + L) R

+ R

restores the invariance!

considering the observed neutrino states

momentum, p

spin, Neutrino,

momentum, p

spin, antiNeutrino,

P

C CPmomentum, p

spin, Neutrino,

momentum, p

spin, antiNeutrino,

pseudovector

K0 = ds K0 = sd

C | K0 > = | K0 > C | K0 > = | K0 >antiparticles of

one another

CP | K0 > = | K0 > CP | K0 > = | K0 >K0s are pseudo-scalars

same pseudo-scalernonet as s and s

So the normalized eigenstates of C P (the states that serve as solutions to the equations of motion)

must be

001 2

1 KKK 002 2

1 KKK and

CP | K1 > = | K1 > CP | K2 > = | K2 >

CP | K1 > = | K1 > CP | K2 > = | K2 >

K1 CP = +1 final states :

K2 CP = 1 final states :

+ or 00

+0 or 000

Here K1 and K2 are NOT anti-particles of one anotherbut each (up to a phase) is its own anti-particle!

K1 “long”-lifetime 10-8 sec(travel ~3 km before decaying!

K2 “short”-lifetime 10-11 sec(travel ~5½ m before decaying!

Different CP states must decay differently, if the weak interaction satisfies CP invariance!

and, in fact kaon beams are observed to decay differently along different points of their path!

1955 Gellmann & PaisNoticed the Cabibbo mechanism, where was the weak eigenstate,

allowed a 2nd order (~rare) weak interaction that could potentially induce the strangeness-violating transition of

cc sds cossin

K o K

o

a particle becoming its own antiparticle!

u u

s

d s

d

Ko

Ko

W

u

u

s

d s

d

Ko

Ko

W W