Master Dissertation

We site Corollary 1.3 of [6] here 2 .
Corollary 4.3. Any (bounded) operator A ∈ F can be expressed as
A = A (−) + A (+)
where A (−) contain annihilation and A (+) creation operators, only.
Definition 4.4. We say that an operator A ∈ F is normally ordered if all
annihilation operators are placed to the right of the creation operators.
The normal ordering of A is expressed by : A :.
In the light of Corollary 4.3 we may define.
Definition 4.5. A contraction between two field operators A and B is
defined by
| AB | := [A (−) , B (+) ]±,
where [·, ·]± is the commutator in the case of Bose fields and
anti-commutator in the case of Fermi fields.
The following theorem tells us how to order field operators normally. We
state it without its proof. 3
Theorem 4.6 (Theorem of Wick). A product of n field operators can be
normally ordered as follows:
A1A2 · · · An =: A1A2 · · · An : + | A1A1 | · · · An + permutations
+ : | A1 | A2 · · · Aj | . . .An : + · · ·
+ : | A1A2 | | A3A4 | · · · : +permutations,
where the sum contains all normal products with all possible pairings of
contractions.
FIXME: think about proving it
We assume the contractions are complex numbers, which therefore can be
taken out of the normal products. Though we have to remember to take
the sign into account in the case of Fermi operators.
In QED there exists only three contractions.
| ψa(x)ψb(y) | := {ψ (−)
a (x), ψ (+)
b (y)} =: 1
i S(+)
ab
| ψa(x)ψb(y) | := {ψ (−)
a (x), ψ (+)
b (y)} =: 1
i S(−)
ba
(x − y) (4.1)
(y − x) (4.2)
| Aµ(x)Aν(y) | := [A (−)
µ (x), A (+)
ν (y)] = gµνiD (+)
0 (x − y), (4.3)
2 I have taken the liberty of reformulating the corollary such that its meaning will be
more apparent for my application. I have not altered the meaning of it, though.
3 See [6]
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