This project centers on defining and manipulating finitely generated algebras in a differentiable manner, particularly within the context of quantum and mathematical physics. Key features:
- Custom algebra definition: Define algebras via commutation relations, specify traces;
FreeAlgebracan infer the algebra's multiplication rules automatically. - Automorphism-based algebra construction: Construct new finitely-generated algebras by specifying automorphisms, with automatic inference of new commutation relations and multiplication rules.
- Tensor products and expansion: Expand algebraic systems by constructing tensor products or finite powers, useful for modeling multi-particle or iterative systems.
- Differentiability: all operations are built on top of Pytorch with support for automatic differentiation.
This codebase is well-suited for advanced research in quantum information, algebraic geometry, and the computational modeling of algebraic structures in physics. It was originally developed to analyze Grassmann integration and the Clifford-Grassmann Fourier transform, and is not being actively maintained.
- Complex Numbers and Differentiability: Implements complex numbers, including arithmetic and conjugation operations. It integrates with PyTorch to provide differentiable computations, enabling optimization and learning in quantum contexts.
- Algebraic Element Representation: Defines the
AlgebraElementclass for representing elements of an algebra. - Core Operations: Implements basic arithmetic (addition, multiplication) and algebraic properties (conjugation, scalar multiplication).
- Validation and Filtering: Ensures elements maintain consistency with the algebra’s defined relations.
- Commutation Relations: Allows custom algebras to be defined by specifying commutation relations between generators.
- Multiplication Rules: The multiplication rules are inferred based on these commutation relations.
- Trace Specification: Optionally defines the trace operation for algebraic elements, which is crucial for analyzing the algebra’s properties.
- Tensor Product of Algebras: Defines tensor products of finitely generated algebras. This is used to model multi-particle quantum states or higher-dimensional algebraic operations.
- Expansion of Algebras: Allows construction of more complex algebras by combining simpler ones through tensor product operations.
- Automorphisms for Defining New Algebras: Enables the specification of a new algebra by providing an automorphism of an existing algebra.
- Inference of Commutation Relations: New commutation relations are inferred based on the automorphism and the original algebra's structure, allowing efficient definition of related algebras.
- Fermionic Systems: Implements algebraic representations specific to fermions, focusing on anti-commutation relations.
- Fermionic Algebra Constructs: Provides abstractions for defining and manipulating fermionic algebras, which are critical in quantum field theory and many-body quantum physics.
- Finite Power Algebras: Implements functionality for constructing finite power algebras, which involve repeated multiplication of algebraic elements.
- Iterative Constructs: Useful for iterating algebraic structures, such as powers of generators or interaction terms in algebraic expansions.
- Utility Functions: Provides auxiliary functions for matrix operations, enumerations, and algebraic validation.
- Matrix Conversions: Supports conversion of algebraic elements into matrices, particularly for compatibility with numerical libraries like Eigen.
- Enumeration and Validation: Implements enumeration of algebraic terms and validation of element consistency, supporting algebraic structure exploration.