Numerical methods

This page summarizes the numerical machinery that appears repeatedly across QPE. Exact implementation details remain in the documented classes and functions.

Cartesian sampled fields

A scalar wave function is held by ComplexField2D. Physical grid spacing is stored with the field so numerical operators can validate that cached phase factors belong to the same sampling geometry.

Split-step Fourier propagation

For a scalar potential, QPE uses second-order Strang splitting:

\[ \psi(t+\Delta t) \approx e^{-iV\Delta t/(2\hbar)} e^{-iT\Delta t/\hbar} e^{-iV\Delta t/(2\hbar)}\psi(t), \]

with

\[ T=\frac{p_x^2+p_y^2}{2m}. \]

The kinetic factor is diagonal in Fourier space. FFT2D transforms the field, SplitStepFourierSolver2D multiplies each spectral bin by its precomputed kinetic phase and then transforms back. Potential half-step phases are cached against PotentialField2D::revision().

Gauge-covariant Cayley/ADI step

For U(1) link fields the kinetic term is not propagated by pretending the vector potential is a scalar correction. QPE constructs gauge-covariant nearest-neighbor couplings and applies symmetric Cayley factors along X and Y. The Cayley transform of a Hermitian discrete operator is norm preserving in the absence of the external absorber and finite-precision roundoff.

Two-component Dirac split step

For every momentum bin,

\[ H_0=c(\sigma_x p_x+\sigma_y p_y)+mc^2\sigma_z \]

has a closed two-by-two exponential. DiracSplitStepSolver2D precomputes this matrix on the FFT grid. The special massless zero-momentum bin is handled explicitly so the positive-energy preparation and propagator do not introduce a 0/0 singularity.

Absorbing boundary layer

AbsorbingBoundary2D damps the outer part of an otherwise finite sampled domain. It is an engineering device that approximates an open numerical boundary; it is not a physical dissipative potential unless an experiment explicitly interprets it that way. Norm loss caused by the CAP is therefore tracked separately from unitary propagation expectations.

Normalization and finite-value checks

Regression tests check normalization where the model is expected to be unitary, finite values after propagation, conservation/reference identities where applicable, and known qualitative/quantitative signatures for each specialized module.

Display post-processing

Bright pass, blur, bloom, display transfer and tone mapping act after the physics is computed. They may alter pixels but must not alter psi, the Dirac spinor, the potential, detector measurements or observables.