The Hilbert space of a Quantum Field Theory is constructed as a Fock Space, defined as the direct sum of subspaces containing states with $N$ particles ($0 \le N < \infty$), where identical bosons are described by symmetric wavefunctionals and creation operators commute. An alternative representation exists via field configurations in position space known as the Schrödinger Representation, characterized by wave functionals that assign probability amplitudes to entire spatial fields $\phi(\mathbf{x})$ at a fixed time slice. This structure establishes Quantum Field Theory as an extension of non-relativistic quantum mechanics where degrees of freedom are continuous field values rather than discrete particle coordinates, necessitating functional derivatives for momentum operators and Gaussian wave functionals for the vacuum state in free theory contexts.
The Hilbert space of a Quantum Field Theory is constructed as a Fock Space, defined as the direct sum of subspaces containing states with $N$ particles ($0 \le N < \infty$), where identical bosons ar…