The free real quantum field, satisfying [ˆϕ(x),ˆϕ(y)]=iΔ(x−y), ˆϕ(x)†=ˆϕ(x), with the conventional vacuum state, which has a moment generating function
ω(eiˆϕ(f))=e−(f∗,f)/2 , where
(f,g) is the inner product (f,g)=∫f∗(x)C(x−y)g(y)d4xd4y, C(x−y)−C(y−x)=iΔ(x−y),
has a representation as
ˆϕr(x)†=ˆϕr(x)=δδα(x)+πα∗(x)+∫C(z−x)α(z)d4z,
in terms of multiplication by
α(x) and functional differentiation
δδα(x).
Because
[δδα(x)+πα∗(x),∫C(z−y)α(z)d4z]=C(x−y),
it is straightforward to show that
ˆϕr(x) verifies the commutation relation of the free real quantum field.
For the Gaussian functional integral
ω(ˆAr)=∫ˆAre−π∫α∗(x)α(x)d4x∏zd4zdα∗(z)dα(z),
we find, as required,
ω(eiˆϕr(f))=∫eiˆϕr(f)e−π∫α∗(x)α(x)d4x∏zd4zdα∗(z)dα(z)=∫exp[i∫C(z−x)α(z)d4z]exp[i(δδα(x)+πα∗(x))]×e−(f∗,f)/2e−π∫α∗(x)α(x)d4x∏zd4zdα∗(z)dα(z)=e−(f∗,f)/2.
The last equality is a result of
δδα(x)+πα∗(x) annihilating
e−π∫α∗(x)α(x)d4x, and the Gaussian integral annihilates powers of
∫C(z−x)α(z)d4z.
This is a largely elementary transposition of the usual representation of the SHO in terms of differential operators, with a not very sophisticated organization of the relationships between points in space-time and operators, so I imagine something quite like this might be found in the literature. References please, if anyone knows of any?
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