States as Functionals in QFT

Functionals depend on the ENTIRE space configuration for a system. This implies an inherent non-locality for the state. This should be true for all 3 representations – the Path Integral, the Matrix (Heisenberg) and the Schrodinger representation of states.

Path Integral Formulation:
- In the path integral approach, the transition amplitude between an initial state <katex> $\phi_i \rangle$ at time $t_i$ and a final state $\phi_f \rangle$ at time $t_f$ is given by: $eiS[ϕ]\langle \phi_f | e^{-iH(t_f – t_i)} | \phi_i \rangle = \int \mathcal{D}\phi \, e^{iS[\phi]}$ where $S[ϕ]S[\phi]$ is the action functional, and $Dϕ\mathcal{D}\phi$ </katex>denotes integration over all field configurations.
Schrödinger Representation:
- In the Schrödinger representation, states are represented as functionals of the field configuration at a given time slice. For a field configuration <katex> $ϕ(x)\phi(\mathbf{x})$ , a state $\Psi \rangle$ is represented by a wavefunctional $Ψ[ϕ]\Psi[\phi]$ : $Ψ[ϕ]=⟨ϕ∣Ψ⟩\Psi[\phi] = \langle \phi | \Psi \rangle$
- Here, <katex> $ϕ(x)\phi(\mathbf{x})$ denotes the field value at each point $x\mathbf{x}$ in space.</katex>
Non-locality of States:
- The wavefunctional $Ψ[ϕ]\Psi[\phi]$ depends on the entire field configuration $ϕ(x)\phi(\mathbf{x})$ over space, not just at a single point. This non-local dependency reflects the fact that the state of the field is determined by its values across the whole spatial domain.
- For example, in a free scalar field theory, the ground state wavefunctional can be expressed as: $ϕ(x)K(x−y)ϕ(y))\Psi_0[\phi] \propto \exp \left( – \frac{1}{2} \int d^3x \, d^3y \, \phi(\mathbf{x}) K(\mathbf{x} – \mathbf{y}) \phi(\mathbf{y}) \right)$ where $K(x−y)K(\mathbf{x} – \mathbf{y})$ is a kernel that encapsulates the spatial correlations of the field.

Non-local Interactions

Field Correlations:
- The functional form of $Ψ[ϕ]\Psi[\phi]$ indicates that the value of the wavefunctional at a given field configuration depends on correlations between field values at different points in space. This is a direct manifestation of non-locality.
- Even in the vacuum state, fields at different spatial points are entangled, leading to non-local correlations.
Implications for Interactions:
- When interactions are included (e.g., in an interacting scalar field theory with a $ϕ4\phi^4$ term), the wavefunctional becomes even more complex, with non-local dependencies reflecting the interaction terms in the Hamiltonian.
- The path integral formulation sums over all possible field configurations, accounting for all possible interactions, which are inherently non-local because the action $S[ϕ]S[\phi]$ typically includes spatial integrals of interaction terms.
Example: Instantaneous Propagation:
- In the Schrödinger picture, the evolution of states can be influenced by the entire field configuration instantaneously, implying non-local effects. For example, the potential energy in a scalar field theory involves integrals over the entire space, indicating that the interaction energy at a point depends on the field values everywhere.

Summary

In QFT, states represented as functionals of field configurations exhibit non-locality because these functionals depend on the entire spatial field configuration. This non-locality is inherent in the path integral formulation, where the action includes integrals over space, and interactions are accounted for by summing over all possible field configurations. Thus, the non-local nature of states in QFT reflects the fundamental structure of the theory, where interactions and correlations extend across the entire spatial domain.