neuralqx.operators.computational.Euclidean4d.numba.area module

An implementation for the area operator of the Euclidean WCL 4-d model using a ComputationalOperator rather than a LocalOperator

class AreaOperator(H, edges, *, squared=False)

Bases: ComputationalOperator

Returns the area operator for a surface through which edges pass through. The specified surface should be a list of edges of the graph which are assumed to pucnture through the desired surface.

The area for a given graph is given by

\[\hat{A}(S) |\vec{m}\rangle = \sum_{e \cap S\neq\varnothing} \sqrt{ \big(m_e^{(1)}\big)^2 + \big(m_e^{(2)}\big)^2 + \big(m_e^{(3)}\big)^2 } |\vec{m}\rangle\]

i.e. for each edge intersecting (S) take its three U(1) charges, square them, sum, and take the square root. (In the generic normalization one would multiply by 8πℓₚ² here that prefactor is set to 1 by convention, see Notes.)

Orientation of an edge does not affect the result since the operator uses the charge norm. If an edge punctures the surface multiple times, it should be included with several times.

For non-planar graphs, these edges should be supplied in accordance to the edges appearing from the neuralqx.graph.Graph.edges (that is, not in the non-planar representation).

Notes: here it is assumed the following

  • unit Planck length

  • Immirzi parameter of value 1.0

  • Hence the 8πℓₚ² factor is absorbed into units

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)