Download Chapter 3, Lecture 1

Chapter 3: Symmetries
Symmetry is very important in physics and especially
particle physics.
Symmetries are connected to conservation laws
(rotational invarianceangular momentum
conservation; translational invariancemomentum
conservation)
Transformations can be continuous or discrete
e.g. translations, rotations, Lorentz transformations are
continuous
Spatial reflection through the origin (parity) is discrete.
Associated conservation laws are additive (cont symmetries)
Associated conservation laws are multiplicative (discrete
symmetries)
Two types of symmetry breaking:
Explicit breaking (when the interactions do not respect
the symmetry. The corresponding quantum numbers are
not conserved in the interaction.)
Spontaneous breaking (when the interaction does
respect the symmetry; mathematically the Lagrangian is
invariant under the corresponding group but its states
are not).
Next example: Parity or mirror reflection is a
discrete transformation
(x, y, z) ® (-x,-y,-z)
P = +1 "even parity"
P= -1 "odd parity"
y = cos x
Þ Py = cos(-x) = cos(x) = +y
Þ P = +1
y = sin x
Þ Py = sin(-x) = - sin(x) = -y
Þ P = -1
y = cos x + sin x
Þ Py = cos x - sin(x)
Þ No definite parity
Question: What does the parity do to the
following quantities
Spatial position r
Time t
Momenta p
Angular momentum r x p
Intrinsic spin
Parity of hydrogenic wave functions
Y(r,q , f ) = c (r)Yl m (q , f )
(2l +1)(l - m)! m
=c (r)
Pl (cosq )eimf
4p (l + m)!
Let’s focus on the angular part
imf
e
im(p +f )
®e
imp imf
=e e
m imf
= (-1) e
Pl m (cosq ) = Pl m (cos(p - q )) = (-1)l+m Pl m (cosq )
The overall angular wave function Y_lm is the product of
these two pieces.
m
l+m
(-1) (-1)
= (-1) (-1)
l
2m
Question: What is the parity of the hydrogenic
wave functions ?
Answer: Parity = (-1)l ; we will use this result
in the future.
Parity is a multiplicative quantum number.
Parity is found to be conserved in electromagnetic and
strong interactions.
Question: What about the weak interaction ?
Elementary particles have intrinsic parity
Pp = -1 PN = +1 (nucleon: n or p)
Pp = -1 PN = +1 (nucleon: n or p)
p+ p ®p+ + p + n
Is this a strong, weak or electromagnetic interaction ?
What is the product of intrinsic parities in the
initial and final states ?
Ans: +1 and -1.
How can parity be conserved ?
Ans: orbital angular momentum between a pair of
particles in the final state.