159
6.3. Applications to the ‘ABC’ theory
and
∫
∫
′
H ˆ = g d
3
x φ ˆ A φ ˆ B φ ˆ C ≡ d
3
x H ˆ ′ .
(6.45)
Each field φ ˆ i , (i = A, B, C) has a mode expansion of the form (5.143), and
associated creation and annihilation operators a ˆ
† and ˆ
a i which obey the comi
mutation relations
†
[ˆ a i (k), a ˆ j (k
′ )] = (2π)
3 δ
3 (k − k
′ )δ ij
i, j = A, B, C.
(6.46)
The new feature in (6.46) is that operators associated with distinct particles
† †
commute. In a similar way, we also have [ˆ a i , a ˆ j ] = [ˆ a i , a ˆ ] = 0.
j
6.3.1 The decay C → A + B
As our first application of (6.42), we shall calculate the decay rate (or resonance width) for the decay C → A+B, to lowest order in g. Admittedly this is
not yet a realistic, physical, example; even so, the basic steps in the calculation
are common to more complicated physical examples, such as W
−
→ e
− + ¯
ν e .
We suppose that the initial state |i> consists of one C particle with 4momentum p C , and that the final state in which we are interested is that with
one A and one B particle present, with 4-momenta p A and p B respectively.
We want to calculate the matrix element
S fi =
(6.48)
fi
To proceed we need to decide on the normalization of our states |p i >. We will
define (for i = A, B, C)
√
†
|p i > = 2E i a ˆ (p i )|0>
(6.49)
i
√
2
2
where E i = m i + p , so that (using (6.46))
i
′
′
(2π) 3 2E i
where the ‘1’ on the right-hand side means the identity in the subspace of
6.3. Applications to the ‘ABC’ theory
and
∫
∫
′
H ˆ = g d
3
x φ ˆ A φ ˆ B φ ˆ C ≡ d
3
x H ˆ ′ .
(6.45)
Each field φ ˆ i , (i = A, B, C) has a mode expansion of the form (5.143), and
associated creation and annihilation operators a ˆ
† and ˆ
a i which obey the comi
mutation relations
†
[ˆ a i (k), a ˆ j (k
′ )] = (2π)
3 δ
3 (k − k
′ )δ ij
i, j = A, B, C.
(6.46)
The new feature in (6.46) is that operators associated with distinct particles
† †
commute. In a similar way, we also have [ˆ a i , a ˆ j ] = [ˆ a i , a ˆ ] = 0.
j
6.3.1 The decay C → A + B
As our first application of (6.42), we shall calculate the decay rate (or resonance width) for the decay C → A+B, to lowest order in g. Admittedly this is
not yet a realistic, physical, example; even so, the basic steps in the calculation
are common to more complicated physical examples, such as W
−
→ e
− + ¯
ν e .
We suppose that the initial state |i> consists of one C particle with 4momentum p C , and that the final state in which we are interested is that with
one A and one B particle present, with 4-momenta p A and p B respectively.
We want to calculate the matrix element
S fi =
(6.47)
to lowest order in g. (Note that the ‘1’ term in (6.36) cannot contribute here
because the initial and final states are plainly orthogonal.) This means that
we need to evaluate the amplitude
∫
(1)
d
4 ˆ
A = −ig
(6.48)
fi
To proceed we need to decide on the normalization of our states |p i >. We will
define (for i = A, B, C)
√
†
|p i > = 2E i a ˆ (p i )|0>
(6.49)
i
√
2
2
where E i = m i + p , so that (using (6.46))
i
′
′
= 2E i (2π)
3 δ
3 (p i − p i ).
(6.50)
′
The quantity E i δ
3 (p i − p i ) is Lorentz invariant. Note that the completeness
relation for such states reads
∫ d
3
p i 1 |p i >
(2π) 3 2E i
where the ‘1’ on the right-hand side means the identity in the subspace of