Conservation Laws - I

[Secs 2.2, 2.3, 16.4, 16.5 Dunlap]

Ever wondered why we have conservation laws?

Well, there was once a mathematician/physicist – Emmy Noether who wondered this and came up with an answer. From 1915 = 1918 working with the laws of special relativity she realized that the conservation of momentum derived from the invarience of physics laws in space, and the conservation of Energy to the invarience of physics laws with respect to time. This is now known as Noether’s theorem – derived from classical dynamics.

Symmetries and Conservation

While the maths behind Noether’s theorem is difficult the general idea can be understood with respect to some imaginary universes:

(a) In this universe laws are not constant in time. As time progresses electric charges increase with time. The spring shown compresses – building potential energy (not conserved) – this could be released to drive an engine.

(b) An electric field is placed across this universe – making space anisotropic. The top (+) charge will be accelerated to the left and (-) to the right. Angular momentum will not be conserved

(c) The electric field is not uniform – the (-) will have more attraction. There will be a net force to the right. Momentum is not conserved.

From the Quantum View

This equation derives (see Sakurai p83) from the Time Dependent Shrodinger Equation. [Although it is called by the name of Heisenberg it was first derived by P.A.M Dirac! ] It basically tells us that if an operator (for some observable) commutes with the Hamiltonian of the system then that observable will not change with time.

]ˆˆˆˆ[1

AHHAidt

dA

ˆiH

ˆiA

2

Introducing the Heisenberg Equation of Motion:

One way to see this is the diagram shown in Hibert space {x} – where we see vectors

{x}1 i

ˆWe see vector ( ) ( ) pointing in the same direction

ˆas vector ( ). If also vector ( ) ( ) then clearly

ˆ ˆˆ ˆ( ) ( ) ( )

i i

i i i

i i i

x E x

x x A x

x x AH x

H

A

HA AH

A synopsis of conservation lawsConservation of BASIC SYMMETRY- Quant. no Interaction violated in

Energy TRANSLATIONS in TIME none

Momentum TRANSLATIONS in SPACE none

Ang. momentum DIRECTIONS in space J none

Parity REFLECTIONS in space (or P) *Weak interaction

Charge conjugation parity Particle - Antiparticle C *Weak interaction

Charge Changes in EM gauge Q none

Lepton number (electron) Changes in Weak charge Lenone

Lepton number (muon) Changes in Weak charge Lnone

Lepton number (tauon) Changes in Weak charge Lnone

Baryon number Quark number invarience B none

Isospin ud quark interchange I for non leptonic)+ *EM

Strangeness u (d)s quark interchange S *Weak interaction (S=1)

Charm qc quark interchange c *Weak interaction (c=1)

Bottomness qb quark interchange b *Weak interaction (b=1)

Topness qt quark interchange T *Weak interaction (T=1)

Energy and MomentumThese two very basic laws stem from the uniformity of space and time. So often we take these laws for granted – While to us these laws seem obvious it is perhaps interesting to note that even as late as 1932 when the missing energy in decay was still a mystery to physicists most physicists still believed that either the law of conservation of energy or the law of conservation of momentum was breaking down on the quantum level! It was W. Pauli who postulated that there was a missing particle – the neutrino – (and had to wait a very long time to hear of its discovery in 1958!).

Wolfgang Pauli (1900 – 1958) Nobel Laureate

WARNING: Some particle decays you see in this course – may be “trick” decays that cannot work because they violate energy conservation

Angular Momentum - J

Angular momentum – resulting from the isotropy of space – is always conserved. Take a look at the gamma decay of the positive pion +

31 12 2 2

0 1 i.e. ( or )J

This reaction is strictly not allowed because the J=0 for a pion and J=1 for the photon so that the RHS must have a half odd integer spin. Later we shall see that this reaction also breaks another conservation law too.

Parity π (P)

State of EVEN parity

State of ODD parity

State of MIXED parity

( ) ( )x x

( ) ( )x x

Inversion of a particle intrinsic wave will map ˆ( ) ( ) ( ) r r P r

Clearly another parity operation will will return the wavefunction to the original one2ˆ ( ) ( ) P r r

Hence the parity operator P has two eigenvalues only:

1P Mixed states are not allowed

Parity π (P)Up until the mid 1950s most physicists strongly believed that conservation of parity was as fundamental as the conservation of momentum and energy. Why?

One reason was that EM processes were clearly parity conserving. Take a look at the EM wave equation:

2 2 2 2

2 2 2 2 2

10

x y z c t

Clearly carrying out the operations x→ -x, y→ -y , z→ -z leaves the equation unchanged. There had been no observation of parity violation in the strong interaction (nuclear reactions etc).

The problem The first K meson even was discovered in cosmic ray tracks in 1947, then in 1953 accelerators that made K- mesons came on-line. Two different decays were found for charged strange mesons

Since the two final states had different parity (π has (-) parity) it was then thought that the initial states should also have different parities, and hence be two distinct particles. However, with increasingly precise measurements, there were found to be no difference between their masses and lifetimes, indicating that they are the same particle. This was known as the τ-θ puzzle. particle— now called the K+.

0 + and

Parity non-conservation

Two theoretical physicists Yang Chen Ning and Lee Tsung-Dao however had become convinced that from a theoretical perspective parity violation could indeed be expected in the weak interaction that causes decay. weak interaction. – They showed that no experiments had yet been performed to demonstrate parity conservation in beta processes.

Yang Chen Ning

(1922)

1957 Nobel Laureates

Lee Tsung Dao

(1926)

Alice in the looking glass

From Alice through the looking glass – Lewis Carroll – 1875. Well is Alice going to see things – or feel things – any different in her world than Alice before she went into the reflected world. No - because most of the world is determined by the electric and strong interactions.

Alice

Right handed coordinate system

Left handed coordinate system

Parity non-conservation

Now if parity is conserved – it must be the case that oriented nuclei in Alice’s right handed world – emit beta radiation isotropically (same in all directions). To consider why – think about the converse – preferential emission. Alice will see the electrons emitted along positive z. Alianna who is Alices friend will see electrons coming off along negative z. But if parity is really conserved Alianna should see electrons in the positive z. Thus Parity conservation → isotropic emission

Alice Aliana

Parity non-conservation

Wu Chien-Shiung

(1912 – 1997)

Prof. Wu graduated from Nanjing 1936, Obtained doctorate in 1940 from UC Berkeley. Worked on Manhatten project 43-45.

In 1956 – 1957 she performed the most difficult of experiments that made her famous. She observed that there is a preferred direction of emission in beta decay, and that therefore, parity was not conserved for the weak interaction. In other words, Dr. Wu was able to prove that identical nuclear particles do not always act alike, and, thereby, disprove, what was then, a widely accepted "law" of nature!

To the disapointment of many she was never awarded the nobel prize.

Mdm. Wu’s difficult experiment

The sample had to be cooled to 10mK using adiabatic de-magnetization. The NaI detectors made sure the nuclei were oriented. The anthracene crystal observed the betas.

“I do not believe in a Left – Handed God”Wolfgang Pauli - 1955

Pauli and Wu – c 1940

Two years latter Wu Chien Shiung experimentally demonstrated that God had planned things slightly differently!

Charge conjugation parityParticles and Antiparticles.

Hole state= e+

Sea of negativeEnergy states

Free electron state e-

2mc2

Dirac Picture Feynman Picture

+Ze

e+

e-

0 energy

Time

Free Electron State

Sea of bound negative electron states

Hole state e+

e-

e+

e-

DIRAC PICTURE FEYNMAN PICTURE

Ze+ TIME

the whole of space consists of “negative energy” states – as obtained from

If a photon comes along with E>2mc2 then it can excite an electron into a real positive energy state – leaving behind a hole – which has the SAME MASS as the electron – the positron – that behaves with opposite dynamics in an electric field.

2 2 2( ) ( )E pc mc

The positron state is not the absence of an electron, but an electron wave oscillating in the opposite sense. – a “time reversed” electron wave. This is why on a Feynman diagram, antiparticles are shown as the particle but with an arrow indicating they are traveling back-wards in time.

Charge conjugation parityThe charge conjugation operation reverses the sign of the charge and magnetic moment of a particle (leaving all other coordinates unchanged).

Maxwell’s equations are invarient under charge conjugation because while charge Q and current density J change so do the electric fields E and magnetic fields B.

In quantum physics, however, charge conjugation means interchange of particle to anti-particle and vice-versa.

Proton Anti-proton

Charge +1e -1e

Baryon number +1 -1

Mag. moment +μ -μ

Spin J J

C

ˆ ˆ and e e p p C C So that electron and positron are not eigenstates of the charge conj. operator

CP conservation

While C and P are not separately conserved in the WEAK interaction – The CP operation should lead to a conserved value of CP. This can be seen from this diagram. Aliana sees positrons going around her loop of wire. The nuclei will thus spin in opposite direction – and give rise to beta particles that Aliana will see as positrons (not electrons).

AliceAliana

μ