Redheffer star product
In mathematics, the Redheffer star product is a binary operation on linear operators that arises in connection to solving coupled systems of linear equations.
It was introduced by Raymond Redheffer in 1959,[1] and has subsequently been widely adopted in computational methods for scattering matrices.
This can be rewritten several ways making use of the so-called push-through identity
Redheffer's definition extends beyond matrices to linear operators on a Hilbert space
However, the star product still makes sense as long as the transformations are compatible, which is possible when
[2] The star product is associative, provided all of the relevant matrices are defined.
Provided either side exists, the adjoint of a Redheffer star product is
The star product arises from solving multiple linear systems of equations that share variables in common.
Often, each linear system models the behavior of one subsystem in a physical process and by connecting the multiple subsystems into a whole, one can eliminate variables shared across subsystems in order to obtain the overall linear system.
results in the Redheffer star product being the matrix such that: [1]
Many scattering processes take on a form that motivates a different convention for the block structure of the linear system of a scattering matrix.
Typically a physical device that performs a linear transformation on inputs, such as linear dielectric media on electromagnetic waves or in quantum mechanical scattering, can be encapsulated as a system which interacts with the environment through various ports, each of which accepts inputs and returns outputs.
It is conventional to use a different notation for the Hilbert space,
, has an additional superscript labeling the direction of travel (where + indicates moving from port i to i+1 and - indicates the reverse).
, is defined with an additional flip compared to Redheffer's definition:[5]
Note that for in order for the off-diagonal identity matrices to be defined, we require
(The subscript does not imply any difference, but is just a label for bookkeeping.)
, the exchange operator, is also the S-matrix star identity defined below.
Here the subscripts relate the different directions of propagation at each port.
As a result, the star product of scattering matrices
, is analogous to the following matrix multiplication of transfer matrices [7]
Redheffer generalized the star product in several ways: If there is a bijection
The particular star product defined by Redheffer above is obtained from:
[8] In physics, the Redheffer star product appears when constructing a total scattering matrix from two or more subsystems.
[5] Many physical processes, including radiative transfer, neutron diffusion, circuit theory, and others are described by scattering processes whose formulation depends on the dimension of the process and the representation of the operators.
The Redheffer star product can be used to solve for the propagation of electromagnetic fields in stratified, multilayered media.
[9] Each layer in the structure has its own scattering matrix and the total structure's scattering matrix can be described as the star product between all of the layers.
[10] A free software program that simulates electromagnetism in layered media is the Stanford Stratified Structure Solver.
Kinetic models of consecutive semiconductor interfaces can use a scattering matrix formulation to model the motion of electrons between the semiconductors.
[11] In the analysis of Schrödinger operators on graphs, the scattering matrix of a graph can be obtained as a generalized star product of the scattering matrices corresponding to its subgraphs.
