Download Conservation Laws I - Department of Physics, HKU

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
Introducing the Heisenberg Equation of Motion:
dA 1 ˆ ˆ ˆ ˆ
 [ AH  HA]
dt i
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.
One way to see this is the diagram shown in
Hibert space {x} – where we see vectors
Aˆ i
Hˆ i
2
1
i
{x}
ˆ  ( x)  E ( x) pointing in the same direction
We see vector H
i
i
ˆ  ( x)  A ( x) then clearly
as vector  ( x). If also vector A
i
ˆ ˆ  ( x)  AH ( x)
ˆ ˆ  ( x)  AH
HA
i
i
i
i
i
A synopsis of conservation laws
Conservation 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
Le
none
Lepton number (muon)
Changes in Weak charge
L
none
Lepton number (tauon)
Changes in Weak charge
L
none
Baryon number
Quark number invarience
B
none
Isospin
ud quark interchange
I
for non leptonic)+ *EM
Strangeness
u (d)s quark interchange
S
*Weak interaction (S=1)
Charm
qc quark interchange
c
*Weak interaction (c=1)
Bottomness
qb quark interchange
b
*Weak interaction (b=1)
Topness
qt quark interchange
T
*Weak interaction (T=1)
Energy and Momentum
Wolfgang Pauli
(1900 – 1958)
Nobel Laureate
These 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!).
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 +

J 0

 
1
2




1
i.e. ( 12 or 23 )
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
 ( x)   (  x)
State of ODD parity
 ( x)   ( x)
State of MIXED parity
ˆ (r )
Inversion of a particle intrinsic wave will map  (r )   ( r )  P
Clearly another parity operation will will return the wavefunction to the original one
ˆ 2 (r )   (r )
P
Hence the parity operator P has two eigenvalues only:
P  1
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
1  2


 2
0
2
2
2
2
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
     0
and
 +      
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+.
Parity non-conservation
1957 Nobel
Laureates
Yang Chen Ning
Lee Tsung Dao
(1922)
(1926)
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.
Alice in the looking glass
Left handed
coordinate
system
Right
handed
coordinate
system
Alice
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.
Parity non-conservation
Alice
Aliana
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
Parity non-conservation
Prof. Wu graduated from Nanjing 1936,
Obtained doctorate in 1940 from UC
Berkeley. Worked on Manhatten project 4345.
Wu Chien-Shiung
(1912 – 1997)
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.
Pauli and Wu – c 1940
“I do not believe in a Left – Handed God”
Wolfgang Pauli - 1955
Two years latter Wu Chien Shiung experimentally demonstrated that God
had planned things slightly differently!
Charge conjugation parity
Particles and Antiparticles.
e+
e+
Free electron
state
Free
Electron
State ee-
2mc2
Hole
state
e+
Hole state=
e+
Sea
Sea ofof
negative
Energy states
bound
negative
electron
states
e-
e-
0 energy
+
Ze+Ze
Dirac Picture
DIRAC
PICTURE
the whole of space consists of “negative
energy” states – as obtained from
E   ( pc) 2  (mc 2 ) 2
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.
TIME
Time
Feynman Picture
FEYNMAN
PICTURE
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 parity
The 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
C
Anti-proton
Charge
+1e
-1e
Baryon number
+1
-1
Mag. moment
+μ
-μ
J
J
Spin
ˆ e  e
C
ˆ p  p
and C
So that electron and positron are not eigenstates of the charge conj. operator
CP conservation
Aliana
Alice
μ
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).