Quantum Field Theory I

110 CHAPTER 3. INTERACTING QUANTUM FIELDS
the LSZ reduction formula
The Lehman–Symanczik–Zimmerman reduction formula relates the S-matrix
elements to the Green functions. It is derived by means of a notoriously known
trick — inserting the unit operator in a proper form at a proper place.
Let us consider the n-point Green function G(x 1 ,...,x n ) with ⃗x i being far
apart from each other. In this region the Hamiltonian can be replaced by the
effective one, i.e. by H0 eff . The basis of the effective Hamiltonian eigenstates
can be taken in the standard form 44
|Ω〉 |⃗p〉 =
√
2E eff
⃗p a+eff ⃗p
|Ω〉 |⃗p,⃗p ′ 〉 =
√
2E eff
⃗p a+eff ⃗p
|⃗p ′ 〉 ...
Our starting point will be an insertion of the unit operator in the form
∫ d 3 ∫
p 1 1 d 3 p 1 d 3 p 2 1
1 = |Ω〉〈Ω|+
(2π) 3 2E⃗p eff |⃗p 1 〉〈⃗p 1 |+
1
(2π) 6 2E⃗p eff
1
2E⃗p eff |⃗p 1 ,⃗p 2 〉〈⃗p 2 ,⃗p 1 |+...
2
between the first two fields. For sake of notational convenience (and without a
loss of generality) we will assume x 0 1 ≤ x0 2 ≤ ... ≤ x0 n and write
G(x 1 ,...,x n ) = 〈Ω|ϕ H (x 1 )|Ω〉〈Ω|ϕ H (x 2 )...ϕ H (x n )|Ω〉
∫ d 3 p 1 1
+
(2π) 3 2E⃗p eff 〈Ω|ϕ H (x 1 )|⃗p 1 〉〈⃗p 1 |ϕ H (x 2 )...ϕ H (x n )|Ω〉+...
1
The main trick which makes this insertion useful, is to express the x-dependence
of the fields via the space-time translations generators P µ
ϕ H (x) = e iPx ϕ H (0)e −iPx
and then to utilize that |Ω〉, |⃗p〉, |⃗p,⃗p ′ 〉, ... are eigenstates of P µ , i.e. that
e −iPx |Ω〉 = |Ω〉 (where we have set E Ω = 0), e −iPx |⃗p〉 = e −ipx |⃗p〉 (where
p 0 = E⃗p eff),
e−iPx |⃗p,⃗p ′ 〉 = e −i(p+p′ )x |⃗p,⃗p ′ 〉, etc. In this way one obtains
∫ d 3 p 1 e −ip1x1
G(x 1 ,...,x n ) =
(2π) 3 2E⃗p eff 〈Ω|ϕ H (0)|⃗p 1 〉〈⃗p 1 |ϕ H (x 2 )...ϕ H (x n )|Ω〉+...
1
wheretheellipsescontainthevacuumandthemultiparticle-statescontributions.
At the first sight the matrix element 〈Ω|ϕ H (0)|⃗p〉 depends on ⃗p, but in
fact it is a constatnt independent of ⃗p. To see this one just need to insert
another unit operator, now written as U −1
⃗p U ⃗p, where U ⃗p is the representation of
the Lorentz boost which transfors the 3-momentum ⃗p to ⃗0. Making use of the
Lorentz invariance of both the nonperturbative vacuum 〈Ω|U −1
⃗p
= 〈Ω| and the
scalar field U ⃗p ϕ H (0)U −1
⃗p
= ϕ H (0), one can get rid of the explicit ⃗p-dependence
in the matrix element 45
〈Ω|ϕ H (0)|⃗p〉 = 〈Ω|U −1
⃗p U ⃗pϕ H (0)U −1
⃗p U ⃗p|⃗p〉 = 〈Ω|ϕ H (0)|⃗0〉
44 Note that we are implicitly assuming only one type of particles here, otherwise we should
make a difference between 1-particle states with the same momentum, but different masses
(i.e.to write e.g.|m,⃗p〉 and |m ′ ,⃗p〉). It is, however, easy to reinstall other 1-particle states, if
there are any.
45 For higher spins one should take into account nontrivial transformation properties of
the field components. But the main achievement, which is that one gets rid of the explicit
3-momentum dependence, remains unchanged.