Towards Hamiltonian formalism for string field theory and nonlocality

A bstract String field theories exhibit exponential suppression of interactions among the component fields at high energies due to infinite-derivative factors such as $$ {e}^{\ell^2\square /2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>□</mml:mo> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> in the vertices. This nonlocality has hindered the development of a consistent Hamiltonian formalism, leading some to question whether such a formalism is even viable. To address this challenge, we introduce a toy model inspired by string field theory and construct its Hamiltonian formalism by demanding that it reproduce all correlation functions derived from the path-integral formalism. Within this framework, we demonstrate for this toy model that physical-state constraints can be imposed to eliminate negative-norm states, while zero-norm states decouple from the physical state space. This approach provides a novel perspective on the nonlocality inherent in string field theories.