MPP

MPP Algebraic Structures

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

Overview

The MPP system defines several algebraic structures:

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 form a non-commutative algebra, where the order of operations matters (e.g., AB ≠ BA for operators A and B).

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 MPP, the commutative ring consists of symbolic expressions that don’t involve quantum or tensor operations, with the operations of addition and multiplication.

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)

Examples

The following expressions belong to the commutative ring:

• Constants: Pi, Planck, Const(42), Rational(3, 4)
• Variables: Var("x")
• Basic operations: Add(Const(1), Const(2)), Mul(Pi, Var("r")), Pow(Var("r"), Const(2))
• Calculus operations: Derive(Var("x"), "x"), Integral(Var("x"), "x")

Non-commutative Algebra

Definition

A non-commutative algebra is an algebraic structure where the multiplication operation is not commutative, meaning that the order of operands matters.

In MPP, the non-commutative algebra consists of quantum mechanical operations, where operators don’t generally commute.

Axioms

For quantum operators A, B, C:

• 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)
• Non-commutativity of Multiplication: A * B ≠ B * A (in general)
• Multiplicative Identity: A * I = A (where I is the identity operator)
• Distributivity: A * (B + C) = (A * B) + (A * C) and (A + B) * C = (A * C) + (B * C)

Examples

The following expressions belong to the non-commutative algebra:

• Quantum states: Ket(Var("psi")), Bra(Var("phi"))
• Inner products: BraKet(Var("psi"), Var("phi"))
• Density matrices: DensityMatrix(Var("psi"))

Commutation Relations

In quantum mechanics, operators often satisfy specific commutation relations. For example, the position and momentum operators satisfy:

[x, p] = i * ħ

where [x, p] = x * p - p * x is the commutator.

MPP supports these commutation relations through the simplifier, which recognizes and applies them during simplification.

Tensor Algebra

Definition

A tensor algebra is a graded algebra that provides a framework for working with tensors of different ranks.

In MPP, the tensor algebra consists of tensor operations, including 4-vectors, tensors of arbitrary rank, and contractions.

Axioms

For tensors T, S of appropriate ranks:

• Linearity: (aT + bS)^{μν…} = aT^{μν…} + bS^{μν…} for scalars a, b
• Tensor Product: (T ⊗ S)^{μν…αβ…} = T^{μν…} S^{αβ…}
• Contraction: Contracting indices follows the Einstein summation convention
• Transformation Law: Tensors transform according to specific rules under coordinate changes

Examples

The following expressions belong to the tensor algebra:

• 4-vectors: Four([t, x, y, z])
• Tensors: Tensor([components], rank)
• Contractions: Contract(tensor, i, j)

Metric Tensor

The metric tensor g_μν plays a special role in tensor algebra, as it defines the inner product between vectors and allows raising and lowering indices.

In MPP, the metric tensor is used for contractions and for computing invariants like the spacetime interval.

Domain Separation

MPP carefully separates the different algebraic domains to ensure that operations are applied correctly:

1. Commutative Domain: Expressions that don’t involve quantum or tensor operations.
2. Quantum Domain: Expressions that involve quantum operations.
3. Tensor Domain: Expressions that involve tensor operations.

The simplifier respects these domain separations, applying the appropriate rules based on the domain of the expression.

Simplification Rules

The simplifier applies different rules based on the algebraic domain of the expression:

1. Commutative Ring Rules:
   • Combine like terms: a + a = 2a
   • Distribute multiplication: a * (b + c) = a * b + a * c
   • Apply power rules: a^0 = 1, a^1 = a
2. Non-commutative Algebra Rules:
   • Preserve operator order: A * B ≠ B * A
   • Apply commutation relations: [x, p] = i * ħ

   • Simplify inner products: ⟨ψ

     ψ⟩ = 1 for normalized states

3. Tensor Algebra Rules:
   • Contract indices: T^{μν} g_{νρ} = T^{μ}_{ρ}
   • Compute invariants: g_{μν} x^μ x^ν = invariant
   • Apply tensor transformation laws

Implementation

The algebraic structures are implemented in the algebra_structure.rs module, which provides functions to:

1. Check if an expression belongs to a specific domain.
2. Verify that an expression satisfies the axioms of its domain.
3. Apply domain-specific simplification rules.

The implementation ensures that the algebraic properties are preserved throughout all operations, maintaining the mathematical rigor of the system.

Future Extensions

Future extensions to the algebraic structures include:

1. Lie Algebras: For symmetry groups and conservation laws.
2. Clifford Algebras: For spinors and relativistic quantum mechanics.
3. Hopf Algebras: For quantum groups and non-commutative geometry.

These extensions will further enhance the expressive power of MPP, allowing it to model more complex physical systems.

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