Given a nonperturbative quantum field, how do you…
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Given a nonperturbative quantum field, how do you determine which theory it is?
I've been reading about how we've constructed nonperturbative ϕ^(4) in all dimensions at this point, and it ends up free in d≥4.
That got be wondering how we know that the solutions are even ϕ^(4) theory. I mean, they're free, so they're clearly solutions to the KG equation without a ϕ^(4) term. How do we know they're actually also ϕ^(4), and we didn't just accidentally construct some *other* theory. Why couldn't there secretly be an interacting nonperturbative Wightman field out there that *actually* describes ϕ^(4), while the field we constructed actually just failed to converge to that solution?
Is it just based on how coarse-graining of the field behaves? I could see that working for d<4, but presumably coarse-graining a free theory doesn't somehow magically produce an extra interaction for the scaling.
Does the triviality in higher dimensions just mean that if you want a Wightman field with a coarse-graining that behaves like it has lagrangian ϕ(☐ - m^(2))ϕ + gϕ^(4), the only possible solutions have g=0? Maybe related to how we expect the ϕ^(4) term to blow up under RG flow to the UV?
#science
I've been reading about how we've constructed nonperturbative ϕ^(4) in all dimensions at this point, and it ends up free in d≥4.
That got be wondering how we know that the solutions are even ϕ^(4) theory. I mean, they're free, so they're clearly solutions to the KG equation without a ϕ^(4) term. How do we know they're actually also ϕ^(4), and we didn't just accidentally construct some *other* theory. Why couldn't there secretly be an interacting nonperturbative Wightman field out there that *actually* describes ϕ^(4), while the field we constructed actually just failed to converge to that solution?
Is it just based on how coarse-graining of the field behaves? I could see that working for d<4, but presumably coarse-graining a free theory doesn't somehow magically produce an extra interaction for the scaling.
Does the triviality in higher dimensions just mean that if you want a Wightman field with a coarse-graining that behaves like it has lagrangian ϕ(☐ - m^(2))ϕ + gϕ^(4), the only possible solutions have g=0? Maybe related to how we expect the ϕ^(4) term to blow up under RG flow to the UV?
#science
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