neuralqx.operators.computational.Euclidean3d.numba.vertex_TRC module

class Euclidean3dVertexTRCOperator(lqx, vertex, *, apply_lapse=False)

Bases: ComputationalOperator

Single-vertex Thiemann-regularised Euclidean Hamiltonian constraint H_v for the 2+1 U(1)^3 weak-coupling model, implemented as a ComputationalOperator.

For each triangulation pairing at vertex v, and for each U(1) copy c in {0,1,2}, the operator produces a loop-shifted configuration:

|σ’> = h_{α_{ij}}^{(c)} |σ>

with matrix element

<σ’|H_v|σ> = eps_term * Δ_k^{(c)}(σ) * Δ_l^{(c)}(σ),

where

Δ_s^{(c)}(σ) = sqrt(V_v(σ)) - sqrt(V_v(h_{s}^{(c)-1} σ)).

The orientation of the loop holonomy is encoded by eps(ij) via an adjoint flip of the loop.

Notes

  • This is the “non-adjoint” (ket-action) ordering

property is_hermitian: bool

This function must return either True or False based on whether your operator is Hermitian or not. Note that unlike for the case of LocalOperator types, you must specify by-hand whether this operator is Hermitian or not, there is no implementation to deduce that information for you as there is no matrices stored in this operator type.

This property plays a role in determining the computational path to be taken when computing gradients. If you specify that the operator is Hermitian while in reality it is not, the computed gradients will be incorrect.

Returns:

True if this operator is Hermitian, False otherwise

property dtype

Specify a JAX NumPy or NumPy dtype for this operator (that is, what is the dtype of the matrix elements returned by this operator)

property max_conn_size: int

The maximum number of non zero ⟨x|O|x’⟩ for every x.

class Euclidean3dVertexTRCAdjointOperator(lqx, vertex, *, apply_lapse=False)

Bases: Euclidean3dVertexTRCOperator

Adjoint of Euclidean3dVertexTRCOperator in the same computational convention.

Differences vs non-adjoint:
  • segment difference uses h_s (forward) instead of h_s^{-1}:

    Δ_s^†(σ) = sqrtV(σ) - sqrtV(h_s σ)

  • loop holonomy is daggered: loop shifts are negated.

ThiemannRegularisedVertexConstraint3d(lqx, vertex, *, apply_lapse=False, adjoint=False)