What effect does multiplying $\mathscr{L}$ by $-1$ have on the propagator?
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Here you have two real scalar fields ϕ1ϕ1 and ϕ2ϕ2 with Lagrangian density
L0[ϕ1,ϕ2]=−12(∂μϕ1)(∂μϕ1)−12m2ϕ21+12(∂μϕ2)(∂μϕ2)+12m2ϕ22L0[ϕ1,ϕ2]=−12(∂μϕ1)(∂μϕ1)−12m2ϕ12+12(∂μϕ2)(∂μϕ2)+12m2ϕ22
Note the relative minus sign between the two pieces.
In this treatment, ϕ1ϕ1 is the physical field and ϕ2ϕ2 is an artificial field that will represent the presence of a heat bath.
He forms a doublet Φ=[ϕ1ϕ∗2]Φ=[ϕ1ϕ2∗] of the fields. But the fields are real, so this is really just Φ=[ϕ1ϕ2]Φ=[ϕ1ϕ2]and then the overall time-ordered propagator of the system is the following 2×22×2 matrix:
−iG(x;y)=⟨0102|T(Φ(x)Φ(y))|0102⟩−iG(x;y)=⟨0102|T(Φ(x)Φ(y))|0102⟩
He then writes the answer in momentum-space with ∫d4p(2π)4Δ(p)eip⋅(x−y)∫d4p(2π)4Δ(p)eip⋅(x−y), it is:
−iΔ(p)=⎡⎣−ip2+m2−iϵ00ip2+m2+iϵ⎤⎦−iΔ(p)=[−ip2+m2−iϵ00ip2+m2+iϵ]
I am confused about the (22)-propagator −iΔ22(p)=ip2+m2+iϵ−iΔ22(p)=ip2+m2+iϵ →→ this is the anti-time ordered propagator. How does this come about?
I know that the source of this is that he defines the doublet ΦΦ first in terms of ϕ∗2ϕ2∗. He in fact does say that ''the Feynman boundary condition is equivalent to adding an infinitesimal imaginary term to the quadratic part of the Lagrangian density'' →→ so complex conjugating the field will supposedly swap the sign of this infinitesimal −iϵ→+i
L0[ϕ1,ϕ2]=−12(∂μϕ1)(∂μϕ1)−12m2ϕ21+12(∂μϕ2)(∂μϕ2)+12m2ϕ22L0[ϕ1,ϕ2]=−12(∂μϕ1)(∂μϕ1)−12m2ϕ12+12(∂μϕ2)(∂μϕ2)+12m2ϕ22
Note the relative minus sign between the two pieces.
In this treatment, ϕ1ϕ1 is the physical field and ϕ2ϕ2 is an artificial field that will represent the presence of a heat bath.
He forms a doublet Φ=[ϕ1ϕ∗2]Φ=[ϕ1ϕ2∗] of the fields. But the fields are real, so this is really just Φ=[ϕ1ϕ2]Φ=[ϕ1ϕ2]and then the overall time-ordered propagator of the system is the following 2×22×2 matrix:
−iG(x;y)=⟨0102|T(Φ(x)Φ(y))|0102⟩−iG(x;y)=⟨0102|T(Φ(x)Φ(y))|0102⟩
He then writes the answer in momentum-space with ∫d4p(2π)4Δ(p)eip⋅(x−y)∫d4p(2π)4Δ(p)eip⋅(x−y), it is:
−iΔ(p)=⎡⎣−ip2+m2−iϵ00ip2+m2+iϵ⎤⎦−iΔ(p)=[−ip2+m2−iϵ00ip2+m2+iϵ]
I am confused about the (22)-propagator −iΔ22(p)=ip2+m2+iϵ−iΔ22(p)=ip2+m2+iϵ →→ this is the anti-time ordered propagator. How does this come about?
I know that the source of this is that he defines the doublet ΦΦ first in terms of ϕ∗2ϕ2∗. He in fact does say that ''the Feynman boundary condition is equivalent to adding an infinitesimal imaginary term to the quadratic part of the Lagrangian density'' →→ so complex conjugating the field will supposedly swap the sign of this infinitesimal −iϵ→+i
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