nLab state-field correspondence

Idea

The following needs clarification.

A field operator $\phi$ in a quantum field theory with a distinguished vacuum vector $|0\rangle$ defines its incoming state as the $|\phi_{in}\rangle \coloneqq U(0,-\infty)\phi |0\rangle$ i.e. as the limit when time goes to infinity of the state $\phi|0\rangle$, here $U(t_1,t_2)$ is the evolution operator from $t_1$ to $t_2$ (which may be written as $U(t_2-t_1)$ when the Hamiltonian is time-independent), which is by definition the inverse of $U(t_2,t_1)$ for $t_2\gt t_1$. The assignment $\phi\mapsto |\phi_{in}\rangle$ is a bijection for conformal field theories.

Related concepts

• state-operator correspondence

References

Discussion in conformal field theory:

• Paul Ginsparg, Section 3.4 in: Applied Conformal Field Theory, lectures at: Fields, strings, critical phenomena, Les Houche Summer School 1988(arXiv:hep-th/9108028)

• Martin Schottenloher, Rem. 9. 11 in: Foundations of Two-Dimensional Conformal Quantum Field Theory (pdf), chapter 9 in: A Mathematical Introduction to Conformal Field Theory, Lecture Notes in Physics, Springer 2008 (doi:10.1007/978-3-540-68628-6, web)