Schrödinger functional

Some approaches to quantum field theory are more popular than others. For historical reasons the Schrödinger representation is less favoured than Fock space methods. In the early days of quantum field theory maintaining symmetries such as Lorentz invariance 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).

Within the Schrödinger representation the Schrödinger functional stands out as perhaps the most useful and versatile functional tool, though interest in it is shown only by a few researchers.

The Schrödinger functional is not itself physical. It 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 later systems may look like.

Example: Scalar Field

The basic mathematical definition is as follows. In the quantum field theory of (as example) a scalar field $\phi$ with a time independent Hamiltonian $H$ the Schrödinger functional is defined as

{\mathcal {S}}[\phi _{2},t_{2};\phi _{1},t_{1}]=\langle \,\phi _{2}\,|e^{{-iH(t_{2}-t_{1})/\hbar }}|\,\phi _{1}\,\rangle .

In the Schrödinger representation this functional generates time translations of state wave functionals, via

\Psi [\phi _{2},t_{2}]=\int \!{\mathcal {D}}\phi _{1}\,\,{\mathcal {S}}[\phi _{2},t_{2};\phi _{1},t_{1}]\Psi [\phi _{1},t_{1}]

Schrödinger functional

Example: Scalar Field

Further reading