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November 7, 2013 293
10.11 States of higher integer spin
10.11.1 The spin algebra for integer spins
In this Appendix we shall consider systems of spinning particles with arbitrary
integer spin. Such particles states can be represented, in the Feynman rules, as
tensors of some rank r :
|s, m〉 µ1µ2µ3···µr
where s stands for the total spin of the particle, and m denotes the spin along
some quantization axis, for which we shal take the z direction here. That is,
once we have found the correct operators of the spin algebra
Then we have, by definition,
(S x,y,z ) µ1µ2···µr ν 1ν 2···ν r
and (S 2 ) µ1µ2···µr ν 1ν 2···ν r
(S 2 ) µ1µ2···µr ν 1ν 2···ν r
|s, m〉 ν1ν2···νr = ¯h 2 s(s + 1) |s, m〉 µ1µ2···µr ,
(S z ) µ1µ2···µr ν 1ν 2···ν r
|s, m〉 ν1ν2···νr = ¯h m |s, m〉 µ1µ2···µr . (10.187)
It is easy to see that the spin algebra is correctly constructed once we have
raising and lowering operators
with
(S ± ) µ1µ2···µr ν 1ν 2···ν r
, S − = (S + ) † ,
We can then find the other algebra elements via
S x = 1 2
[[S + , S − ], S + ] = 2¯h 2 S + . (10.188)
(
S+ + S −
)
, Sy = 1 2i
(
S+ − S −
)
, Sz = 1
2¯h [S +, S − ] ,
S 2 = 1 2
{
S+ , S −
}
+ (Sz ) 2 . (10.189)
We will start with particles in their rest frame 28 . The spin representations are
built using four unit vectors, with obvious notation, as t µ , x µ , y µ and z µ , which
obey
t·t = 1 , x·x = y ·y = z ·z = −1 , t·x = t·y = t·z = x·y = x·z = y ·z = 0 .
(10.190)
Things will become easier if we also define
so that
x ± µ = 1 √
2
(
x µ ± i y µ) (10.191)
x ± · x ± = 0 , x + · x − = −1 , x ± · z = x ± · t = 0 . (10.192)
28 This implies that the particles are massive. For massless partices, see later on.