The dipole approximation assumes that the wavelength of the electromagnetic field is much larger than the spatial extent of the electron wavefunction. In this case, the interaction Hamiltonian simplifies significantly.

Interaction Hamiltonian

In the dipole approximation, the interaction term becomes:

Hint = -d·E(t),

where:

• d = -er is the electric dipole moment of the electron,
• E(t) is the electric field of the photon.

Simplified Schrödinger Equation

The time-dependent Schrödinger equation becomes:

iℏ∂ψ/∂t = [H0 - d·E(t)]ψ,

where H0 is the unperturbed Hamiltonian of the electron.

Solving for Energy States

Under the dipole approximation, solutions can be obtained using:

1. Time-Dependent Perturbation Theory: To calculate transition probabilities between energy levels.
2. Rabi Oscillations: For resonant interactions between two levels.
3. Floquet Theory: For periodic electric fields (e.g., in laser interactions).

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The system includes:

• An electron with wavefunction ψe(r, t),
• A photon field described by the vector potential A(r, t).

The total Hamiltonian includes:

1. The electron’s kinetic energy, -ℏ2/2me∇2,
2. The coupling between the electron and photon field through minimal coupling,
3. The photon’s energy.

The Hamiltonian in SI units is:

H = (1/2me) [ -iℏ∇ - eA(r, t) ]2
    + eφ(r, t) 
    + (1/2)ε0 ∫ |E(r, t)|2 + (1/2μ0)|B(r, t)|2 d3r,

where:

• φ(r, t) is the scalar potential,
• E = -∂A/∂t - ∇φ is the electric field,
• B = ∇×A is the magnetic field.

Simplifying for interaction only, the Schrödinger equation is:

iℏ∂ψe/∂t = Hψe.

Solving for Energy States

Solving the energy states requires quantizing the photon field. Using second quantization:

• Represent the photon field as a superposition of modes:

      A(r, t) = Σk,λ sqrt(ℏ/2ε0ωk) 
      [ ak,λeik·r + ak,λ†e-ik·r ].
  

Approach to Energy Levels:

1. Electron in an Electromagnetic Field (Perturbation Theory): For weak coupling, perturbation theory gives corrections to the electron’s energy levels.
2. Jaynes-Cummings Model: For resonant interactions (electron treated as a two-level system), one can use this model to calculate Rabi oscillations and energy splitting.
3. Numerical Methods: For more general cases, computational methods are necessary.

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Abstract:

• The paper builds on the author’s previous work on the superluminal Schrödinger equation, which addresses kinetic energy forms that do not become singular at the speed of light.
• It re-derives this equation using quantum field theory approaches, including constructing Lagrangian and Hamiltonian densities.
• The paper solves the superluminal Schrödinger equation for eigenfunctions and iteratively finds the superluminal potential for the deuteron.
• The iterative method shows convergence, yielding a potential similar to subluminal potentials, supporting the validity of the superluminal theories.
• The study implies that particles within the nucleus may exceed the speed of light, challenging traditional physics boundaries.

Key Points:

1. Introduction:
   • The nuclear force is treated analogously to the electrostatic force in Yukawa’s theory, focusing on spherical symmetry to find superluminal eigenfunctions and potentials.
   • The study proposes a boson, with mass equal to the deuteron’s reduced mass, is exchanged between nucleons, leading to the nuclear force and potential.
2. Superluminal Schrödinger Equation:
   • The superluminal form of kinetic energy does not become singular at $v=c$ and increases indefinitely as velocity increases.
   • The paper shows that Lagrangian and Hamiltonian densities can re-derive this superluminal Schrödinger equation via quantum field theory.
3. Eigenfunctions and Potentials:
   • The spherically symmetric superluminal Schrödinger equation is solved for eigenfunctions.
   • An iterative Laplace transform method finds the superluminal potential for the deuteron, with convergence observed after the fourth iteration.
4. Comparison with Subluminal Potentials:
   • The superluminal potentials closely resemble those found by subluminal approaches, like the Reid potential.
   • The potentials have a “hard” core nature, indicating nucleons are not point particles but have a finite size.
5. Implications for Nuclear Physics:
   • The superluminal approach suggests the existence of particles moving faster than light within the nucleus.
   • If experimentally verified, this challenges the speed-of-light limitation in other physics branches.

Conclusion:

• The paper demonstrates the feasibility of constructing a superluminal Schrödinger equation through quantum field theory.
• The derived potentials support the concept of superluminal interactions in nuclear physics.
• This work opens the possibility for future studies and experimental verification of faster-than-light particles in the nucleus.

Overall, Betinis’ work extends quantum mechanics into the superluminal regime, providing theoretical tools and results that could reshape our understanding of nuclear forces and particle physics.

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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 $\varphi (\mathbf{x},t)$ and $\pi (\mathbf{x},t)$ (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 $[\varphi (\mathbf{x},t),\varphi (\mathbf{y},t)]$ and $[\varphi (\mathbf{x},t),\pi (\mathbf{y},t)]$, 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 $D(x-y)$, 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.

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