unchained1978
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In ordinary QFT, everything is formulated in terms of a Fock basis so when we write [itex]|\psi\rangle[/itex] we mean that this is a product of single particle states covering every momentum mode. This leads to a Hamiltonian that's typically of the form [itex]\hat H=\int \frac{d^{3}k}{(2\pi)^{3}} [\omega_{k}(\hat a^{\dagger}_{k}\hat a_{k}+\frac{1}{2}\delta^{3}(0))][/itex].
Is this Fock basis different from the field basis such that [itex]\langle \phi |\psi\rangle=\Psi[\phi][/itex] where [itex]\Psi[\phi][/itex] is the wavefunctional of the field? (I hope that makes sense the way I've asked it). The Hamiltonian seems to have a different form, in the case of a scalar field
[itex]\hat H=\int d^{3}x (-\delta^{2}/\delta \phi^{2}+(\vec\nabla \phi)^{2}+m^{2}\phi^{2})[/itex]
I don't understand very well whether or not these Hamiltonians and quantum states are really describing the same thing, or what the different forms represent. If anyone could enlighten me I would be extremely grateful.
Is this Fock basis different from the field basis such that [itex]\langle \phi |\psi\rangle=\Psi[\phi][/itex] where [itex]\Psi[\phi][/itex] is the wavefunctional of the field? (I hope that makes sense the way I've asked it). The Hamiltonian seems to have a different form, in the case of a scalar field
[itex]\hat H=\int d^{3}x (-\delta^{2}/\delta \phi^{2}+(\vec\nabla \phi)^{2}+m^{2}\phi^{2})[/itex]
I don't understand very well whether or not these Hamiltonians and quantum states are really describing the same thing, or what the different forms represent. If anyone could enlighten me I would be extremely grateful.