Pictures Paths Particles Processes

November 7, 2013 297
1∑
m=−1
|1, m〉 µν 〈1, m| αβ
= 1 2 ∆µ α ∆ ν β − 1 2 ∆µ β ∆ ν α ,
|0, 0〉 µν 〈0, 0| αβ
= 1 3 ∆µν ∆ αβ , (10.214)
so that there is a completeness relation of the form
2∑
s∑
s=0 m=−s
This confirms that no states have been overlooked.
10.11.4 Rank-3 tensors
|s, m〉 µν 〈s, m| αβ
= ∆ µ α ∆ ν β . (10.215)
For the sake of illustration we also give the complete set of rank-3 tensorial
states. These fall apart in one spin-3, two spin-2, three spin-1 and a single spin-
0 sector, giving the correct total of 27 possible orthonormal states, listed below.
For reasons of typography I have left out the normalizing denominators ; these
can of course be trivially recovered.
spin-3 :
|3, 3〉 = |+ + +〉
|3, 2〉 = |+ + 0〉 + |+0+〉 + |0 + +〉
|3, 1〉 = 2 |+00〉 + 2 |0 + 0〉 + 2 |00+〉
− |+ + −〉 − |+ − +〉 − |− + +〉
|3, 0〉 = 2 |000〉 − |+0−〉 − |0 − +〉 − |− + 0〉
− |−0+〉 − |+ − 0〉 − |0 + −〉
|3, −1〉 = |+ − −〉 + |− + −〉 + |− − +〉
−2 |−00〉 − 2 |0 − 0〉 − 2 |00−〉
|3, −2〉 = |− − 0〉 + |−0−〉 + |0 − −〉
|3, −3〉 = − |− − −〉
spin-2(1) :
|2, 2〉 = |+0+〉 + |0 + +〉 − 2 |+ + 0〉
|2, 1〉 = 2 |00+〉 − |+ − +〉 − |− + +〉
− |+00〉 − |0 + 0〉 + 2 |+ + −〉
|2, 0〉 = |+0−〉 + |0 + −〉 − |−0+〉 − |0 − +〉
|2, −1〉 = 2 |00−〉 − |+ − −〉 − |− + −〉
− |0 − 0〉 − |−00〉 + 2 |− − +〉
|2, −2〉 = 2 |− − 0〉 − |0 − −〉 − |−0−〉
spin-2(2) :
|2, 2〉 = |+0+〉 − |0 + +〉