In mathematical physics, some approaches to quantum field theory are more popular than others. For historical reasons, the Schrödinger representation is less favored than Fock space methods. In the early days of quantum field theory, maintaining symmetries such as Lorentz invariance, displaying them manifestly, and proving renormalisation were of paramount importance. The Schrödinger representation is not manifestly Lorentz invariant and its renormalisability was only shown as recently as the 1980s by Kurt Symanzik (1981). The Schrödinger functional is, in its most basic form, the time translation generator of state wavefunctionals. In layman's terms, it defines how a system of quantum particles evolves through time and what the subsequent systems look like.
Background Quantum mechanics is defined over the spatial coordinates x {\displaystyle \mathbf {x} } upon which the Galilean group acts, and the corresponding operators act on its state as x ^ ψ ( x ) = x ψ ( x ) {\displaystyle {\hat {\mathbf {x} }}\psi (\mathbf {x} )=\mathbf {x} \psi (\mathbf {x} )} . The state is characterized by a wave function ψ ( x ) = ⟨ x | ψ ⟩ {\displaystyle \psi (\mathbf {x} )=\langle \mathbf {x} |\psi \rangle } obtained by projecting it onto the coordinate eigenstates defined by x ^ | x ⟩ = x | x ⟩ {\displaystyle {\hat {\mathbf {x} }}\left|\mathbf {x} \right\rangle =\mathbf {x} \left|\mathbf {x} \right\rangle } . These eigenstates are not stationary. Time evolution is generated by the Hamiltonian, yielding the Schrödinger equation i ∂ 0 | ψ ( t ) ⟩ = H ^ | ψ ( t ) ⟩ {\displaystyle i\partial _{0}\left|\psi (t)\right\rangle ={\hat {H}}\left|\psi (t)\right\rangle } . However, in quantum field theory, the coordinate is the field operator ϕ ^ x = ϕ ^ ( x ) {\displaystyle {\hat {\phi }}_{\mathbf {x} }={\hat {\phi }}(\mathbf {x} )} , which acts on the state's wave functional as
ϕ ^ ( x ) Ψ [ ϕ ( ⋅ ) ] = ϕ ( x ) Ψ [ ϕ ( ⋅ ) ] , {\displaystyle {\hat {\phi }}(\mathbf {x} )\Psi \left[\phi (\cdot )\right]=\operatorname {\phi } \left(\mathbf {x} \right)\Psi \left[\phi (\cdot )\right],}
where "⋅" indicates an unbound spatial argument. This wave functional
Ψ [ ϕ ( ⋅ ) ] = ⟨ ϕ ( ⋅ ) | Ψ ⟩ {\displaystyle \Psi \left[\phi (\cdot )\right]=\left\langle \phi (\cdot )|\Psi \right\rangle }
is obtained by means of the field eigenstates
ϕ ^ ( x ) | Φ ( ⋅ ) ⟩ = Φ ( x ) | Φ ( ⋅ ) ⟩ , {\displaystyle {\hat {\phi }}(\mathbf {x} )\left|\Phi (\cdot )\right\rangle =\Phi (\mathbf {x} )\left|\Phi (\cdot )\right\rangle ,}
which are indexed by unapplied "classical field" configurations Φ ( ⋅ ) {\displaystyle \Phi (\cdot )} . These eigenstates, like the position eigenstates above, are not stationary. Time evolution is generated by the Hamiltonian, yielding the Schrödinger equation,
i ∂ 0 | Ψ ( t ) ⟩ = H ^ | Ψ ( t ) ⟩ . {\displaystyle i\partial _{0}\left|\Psi (t)\right\rangle ={\hat {H}}\left|\Psi (t)\right\rangle .}
Thus the state in quantum field theory is conceptually a functional superposition of field configurations.
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