States as Functionals in QFT

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.

1. Path Integral Formulation:
   • In the path integral approach, the transition amplitude between an initial state <katex> ∣ϕi⟩| \phi_i \rangle∣ϕi​⟩ at time tit_iti​ and a final state ∣ϕf⟩| \phi_f \rangle∣ϕf​⟩ at time tft_ftf​ is given by: ⟨ϕf∣e−iH(tf−ti)∣ϕi⟩=∫Dϕ eiS[ϕ]\langle \phi_f | e^{-iH(t_f – t_i)} | \phi_i \rangle = \int \mathcal{D}\phi \, e^{iS[\phi]}⟨ϕf​∣e−iH(tf​−ti​)∣ϕi​⟩=∫DϕeiS[ϕ] where $S[\varphi ]$ is the action functional, and $\mathcal{D}\varphi$ </katex>denotes integration over all field configurations.
2. 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> $\varphi (\mathbf{x})$, a state $|\Psi ⟩$ is represented by a wavefunctional $\Psi [\varphi ]$: $$\Psi [\varphi ]=⟨\varphi |\Psi ⟩\; </katex>$$
   • Here, <katex>$\varphi (\mathbf{x})$ denotes the field value at each point $\mathbf{x}$ in space.</katex>
3. Non-locality of States:
   • The wavefunctional $\Psi [\varphi ]$ depends on the entire field configuration $\varphi (\mathbf{x})$ 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: Ψ0[ϕ]∝exp⁡(−12∫d3x d3y 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)Ψ0​[ϕ]∝exp(−21​∫d3xd3yϕ(x)K(x−y)ϕ(y)) where $K(\mathbf{x}-\mathbf{y})$ is a kernel that encapsulates the spatial correlations of the field.

Non-local Interactions

1. Field Correlations:
   • The functional form of $\Psi [\varphi ]$ 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.
2. Implications for Interactions:
   • When interactions are included (e.g., in an interacting scalar field theory with a ϕ4\phi^4ϕ4 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 $S[\varphi ]$ typically includes spatial integrals of interaction terms.
3. 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.