MPP

• • Mħπ Algebraic Structures

    This document formalizes the algebraic structures in the Mħπ system, providing a rigorous mathematical foundation for symbolic operations.

    Overview

    The Mħπ system defines several algebraic structures. A key feature of the system is that these structures remain valid and consistent even under the core postulate of Informational Mass, where [M] = [Ω][T]⁻¹. The test suite verifies the axioms described below hold true within this new physical framework.

    The primary domains are:

    1. Commutative Ring: The basic operations (Add, Mul, Pow) form a commutative ring when applied to symbolic expressions that don’t involve quantum or tensor operations.
    2. Non-commutative Algebra: Quantum mechanical operations, particularly those involving operators like γ^μ, form a non-commutative algebra where the order of operations matters.
    3. Tensor Algebra: Tensor operations form a graded algebra with specific transformation properties under coordinate changes.

    Commutative Ring

    Definition

    A commutative ring is an algebraic structure consisting of a set equipped with two binary operations (addition and multiplication) that satisfy certain axioms.

    In Mħπ, the commutative ring consists of symbolic expressions that don’t involve non-commuting quantum or tensor operators.

    Axioms

    For symbolic expressions a, b, c in the commutative domain:

    • Associativity of Addition: (a + b) + c = a + (b + c)
    • Commutativity of Addition: a + b = b + a
    • Additive Identity: a + 0 = a
    • Additive Inverse: For every a, there exists -a such that a + (-a) = 0
    • Associativity of Multiplication: (a _ b) _ c = a _ (b _ c)
    • Commutativity of Multiplication: a _ b = b _ a
    • Multiplicative Identity: a * 1 = a
    • Distributivity: a _ (b + c) = (a _ b) + (a * c)

    Non-commutative Algebra

    Definition

    A non-commutative algebra is an algebraic structure where the multiplication operation is not commutative. In Mħπ, this primarily applies to quantum mechanical operators and Dirac gamma matrices.

    Axioms

    For quantum operators A, B, C:

    • Associativity of Multiplication: (A _ B) _ C = A _ (B _ C)
    • Non-commutativity of Multiplication: A _ B ≠ B _ A (in general)
    • Distributivity: A _ (B + C) = (A _ B) + (A _ C) and (A + B) _ C = (A _ C) + (B _ C)

    Commutation & Anti-commutation Relations

    The Mħπ simplification engine recognizes and applies specific commutation and anti-commutation relations that define the algebra, such as:

    • [x, p] = iħ
    • {γ^μ, γ^ν} = 2g^μν

    Domain Separation and Implementation

    Mħπ carefully separates the different algebraic domains. The simplify.rs engine respects these separations, applying the appropriate rules based on the domain of the expression. For example, it will reorder factors in a commutative product but preserve the order in a non-commutative one. The tests/algebra_structure.rs file contains property-based tests that verify these axioms hold by comparing the canonical forms of equivalent expressions.

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