From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 35

Thus:

TrH[A]=

∫

dμ(z)A(z∗,z). (1.160)

This formula is very useful in some applications we will encounter in the following.

Example 1.3.1.

(i) The “delta”-operator:

δ(z∗−z∗ 0 ):ψ(z∗)→ψ(z∗ 0 ) (1.161)

can be written as:

ψ(z 0 ∗)=〈z 0 |ψ〉=〈z 0 |

(∫

dμ(z)|z〉〈z|

)

|ψ〉=

∫

dμ(z)〈z 0 |z〉ψ(z∗)

=

∫

dμ(z)ez

∗ 0 z

ψ(z∗), (1.162)

showing that its kernel is given by:ez

0 ∗z

.

(ii) The kernel of the annihilation operatorais given by:

〈z|a|z′〉=z′〈z|z′〉=z′ez

∗z′

. (1.163)

(iii) The kernel of the creation operatora†is given by:

〈z|a†|z′〉=z∗〈z|z′〉=z∗ez

∗z′

. (1.164)

(iv) The kernel of the number operatora†ais given by:

〈z|a†a|z′〉=z∗z′〈z|z′〉=z∗z′ez

∗z′

. (1.165)

(v) More generally, the kernel of any operator of the form^19 :

K=

∑

pq

kpq(a†)paq (1.166)

is simply given by the expression:

〈z|K|z〉=

∑

pq

kpq〈z|(a†)paq|z〉=

∑

pq

kpq(z∗)p(z′)qez

∗z′

. (1.167)

(^19) A polynomial ina, a†in which the creation operators are all on the left of the annihilation
ones is said to be in its normal form.