Non-locality in the Heisenberg Representation

Non-locality is a fundamental aspect of quantum field theory (QFT) and it arises in both the Schrödinger and Heisenberg representations. Here’s an explanation of how non-locality manifests in each representation:

In the Heisenberg representation, operators evolve with time while states remain fixed. Non-locality in this context is evident through the behavior of field operators and their commutation relations. Here’s how it is manifested:

  1. Field Operators:
    • Field operators ϕ(x,t)\phi(\mathbf{x}, t) and π(x,t)\pi(\mathbf{x}, t) (canonical conjugate momenta) evolve with time.
    • The fields at different spatial points are typically entangled, meaning that an operation or measurement at one point can instantaneously influence the field operators at another point.
  2. Commutation Relations:
    • The fundamental commutation relations for field operators, such as [ϕ(x,t),ϕ(y,t)][ \phi(\mathbf{x}, t), \phi(\mathbf{y}, t) ] and [ϕ(x,t),π(y,t)][ \phi(\mathbf{x}, t), \pi(\mathbf{y}, t) ], reflect the non-local structure of the theory.
    • For spacelike separations, these commutators vanish, preserving causality, but for timelike or lightlike separations, they do not, indicating non-local interactions.
  3. Propagators:
    • The propagators, such as the Feynman propagator D(x−y)D(x – y), show how field correlations extend over spacetime.
    • Even though commutators vanish for spacelike separations, the non-zero value of propagators over spacelike intervals still implies an underlying non-locality in terms of correlations and influence across spacetime.

Summary

Non-locality is an intrinsic aspect of QFT that arises in both the Schrödinger and Heisenberg representations. In the Schrödinger representation, it is reflected in the dependence of the wave functional on the entire field configuration across space. In the Heisenberg representation, it is evident in the evolution and commutation relations of field operators, which encode non-local correlations and interactions. Both representations capture the non-local nature of quantum fields, demonstrating that non-locality is a fundamental feature of QFT rather than an artifact of a particular representation.