Pictures Paths Particles Processes

300 November 7, 2013
− 1 4
(∆ µρ ∆ ν α∆ βγ − ∆ µρ ∆ ν β∆ αγ − ∆ νρ ∆ µ α∆ βγ + ∆ νρ ∆ µ β∆ αγ
)
spin-1(3) :
1∑
m=−1
|1, m〉 µνρ 〈1, m| αβγ
=
1
3 ∆µν ∆ ρ γ∆ αβ
spin-0 : |0, 0〉 µνρ 〈0, 0| αβγ
=
1
6
(
∆ µ α∆ ν β∆ ρ γ + ∆ µ β∆ ν γ∆ ρ α + ∆ µ γ∆ ν α∆ ρ β
−∆ ν α∆ µ β∆ ρ γ − ∆ ν β∆ µ γ∆ ρ α − ∆ ν γ∆ µ α∆ ρ β
)
. (10.217)
The total completeness relations is also valid :
3∑
s∑
s=0 m=−s
provided we sum over all sectors with the same s.
|s, m〉 µνρ 〈s, m| αβγ
= ∆ µ α ∆ ν β ∆ ρ γ , (10.218)
10.11.5 Massless particles : surviving states
So far, we have taken our particles to be at rest, with a momentum p for which
p µ = m t µ .
For moving particles, we can obtain the correct states by simply performing the
appropriate Lorentz boost. As already indicated, we shall take the motion of
the particles to be along the z axis ; our states have been prepared for this by
taking z as the spin quantization axis. The momentum of the particle will then
be
p µ = m t µ → p µ = p 0 t µ + |⃗p| z µ , (10.219)
and the vector z µ becomes, under the same boost
( ) ( )
|⃗p| p
z µ → t µ 0
+ z µ . (10.220)
m m
The vectors x ± are not affected by the boost. It is therefore sufficient to replace,
in Eqns.(10.194),(10.211), (10.212),(10.213), and (10.216), z by its boosted form.
Let us now consider the extreme case : that of a massless particle. We can
view this as the limit p 0 /m → ∞ of a massive particle. In that limit, z µ diverges
badly, and we must again adopt the point of view presented in chapter 6 : the
theory wil only be viable if those tensors that diverge in the massless
limit decouple completely. That is, the only observable states must be those