From Classical Mechanics to Quantum Field Theory

38 From Classical Mechanics to Quantum Field Theory. A Tutorial

The solutions of these equations can be easily found to be given by:

|φ〉=

∏

j

eφja

†j

| 0 〉=e

∑

jφja†j| 0 〉, (1.180)

whereaj| 0 〉= 0, for allj. It is also not difficult to prove the following relations:

• non-orthogonality condition
  〈φ|φ′〉=e

∑

αφ∗αφ′α; (1.181)

• resolution of identity

I=

∫ (∏

α

(dReφα)(dImφα)

π

)

e

∑

α|φα|^2 |φ〉〈φ|. (1.182)

A similar construction is less simple if we aim at discussing a set of finite or
infinite number of creation/annihilation operators offermionic type, i.e. a set of
operators{(ai,a†i)}isatisfying the canonical commutation relations:

{ai,aj}={a†i,a†j}=0, (1.183)

{ai,a†j}=δijI, (1.184)

acting on the fermionic Fock Hilbert spaceHFwhich is generated by the orthonor-
mal basis|n 1 ···nk···〉=(a† 1 )n^1 ···(a†k)nk···| 0 〉, withnk∈{ 0 , 1 }.
If we insist to define coherent states as common eigenvectors|ξ〉≡|ξ 1 ξ 2 ···〉
of all annihilation operators:

aj|ξ〉=ξj|ξ〉, (1.185)

we see that the commutations relations (1.183) now imply:

ξiξj+ξjξi=0. (1.186)

This condition admits non-trivial solutions only if we allow the “numbers”ξjto
be not inCbut in a Grassmann algebraG.
If we allow so, then coherent states are given by:

|ξ〉≡|ξ 1 ξ 2 ···〉=e

∑

jξja†j| 0 〉=

∏

j

(1−ξja†j)| 0 〉, (1.187)

which are vectors in the generalized Fock space:

H ̃F=

{

|ψ〉=

∑

J

χJ|φJ〉: χJ∈G,|φJ〉∈HF

}

. (1.188)

The set of states{|ξ〉}satisfy again the relationships: