A relational Hamiltonian for group field theory

Understand

Using a massless scalar field as a clock variable, the Legendre transform of the group field theory Lagrangian gives a relational Hamiltonian.

• In the classical theory, it is natural to define 'equal relational time' Poisson brackets, where 'equal time' corresponds to equal values of the scalar field clock.
• The quantum theory can then be defined by imposing 'equal relational time' commutation relations for the fundamental operators of the theory, with the states being elements of a Fock space with their evolution determined by the relational Hamiltonian operator.
• A particularly interesting family of states are condensates, as they are expected to correspond to the cosmological sector of group field theory.