A Generalized Method for Spatial Operations on Physical Properties of Matter

TL;DR

The paper introduces a generalized ICO method for transforming diverse physical property matrices under spatial operations, simplifying analysis across various physical systems.

Contribution

It provides a unified, intuitive formalism for spatial transformations of coefficient matrices, reducing computational complexity and enhancing interpretability.

Findings

Applicable to high-order nonlinear optics, elastic mechanics, and electromagnetism.

Enables intuitive reasoning about spatial transformations.

Delegates complex calculations to computational tools.

Abstract

The physical properties of matter are typically described by coefficient matrices governed by crystal symmetry. Applying spatial operations, such as rotation, inversion, and mirror, to these matrices provides an effective approach for investigating material properties. However, the diversity of coefficient matrix types complicates their transformation via simple matrix multiplication, and existing methods suffer from cumbersome notation, high computational cost, and lack of intuitive interpretation. Moreover, as coefficient matrices grow in size, conventional approaches become increasingly inadequate. We present a generalized ``input-coefficient-output (ICO)" approach for constructing spatial operation matrices applicable to coefficient matrices across diverse physical systems, including but not limited to high-order nonlinear optics, elastic mechanics, electricity and magnetism. Our approach offers a concise formalism that enables intuitive reasoning about spatial transformations while delegating intensive computations to computational tools, which is analogous to the role of Feynman diagrams in facilitating understanding in physics. This method also offers valuable insights for future theoretical and experimental research.