path integral quantization of EM field derived from canonical quantization?
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Consider the real scalar field ϕ(x,t)ϕ(x,t) on 1+1 dimensional space-time with some action, for instance
S[ϕ]=14πν∫dxdt(v(∂xϕ)2−∂xϕ∂tϕ),S[ϕ]=14πν∫dxdt(v(∂xϕ)2−∂xϕ∂tϕ),
where vv is some constant and 1/ν∈Z1/ν∈Z. (This example describes massless edge excitations in the fractional quantum Hall effect.)
To obtain the quantum mechanics of this field, there are two possibilities:
Perform canonical quantization, i.e. promote the field ϕϕ to an operator and set [ϕ,Π]=iℏ[ϕ,Π]=iℏ where ΠΠ is the canonically conjugate momentum from the Lagrangian.
Use the Feynman path integral to calculate all expectation values of interest, like ⟨ϕ(x,t)ϕ(0,0)⟩⟨ϕ(x,t)ϕ(0,0)⟩, and forget about operators altogether.
S[ϕ]=14πν∫dxdt(v(∂xϕ)2−∂xϕ∂tϕ),S[ϕ]=14πν∫dxdt(v(∂xϕ)2−∂xϕ∂tϕ),
where vv is some constant and 1/ν∈Z1/ν∈Z. (This example describes massless edge excitations in the fractional quantum Hall effect.)
To obtain the quantum mechanics of this field, there are two possibilities:
Perform canonical quantization, i.e. promote the field ϕϕ to an operator and set [ϕ,Π]=iℏ[ϕ,Π]=iℏ where ΠΠ is the canonically conjugate momentum from the Lagrangian.
Use the Feynman path integral to calculate all expectation values of interest, like ⟨ϕ(x,t)ϕ(0,0)⟩⟨ϕ(x,t)ϕ(0,0)⟩, and forget about operators altogether.
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