This document outlines the physical extensions to the Mħπ symbolic mathematics system. The framework is defined by a core postulate: mass is a derived dimension, representing a rate of information flow.
MPP-Physics rejects mass [M] as a fundamental dimension. Instead, it is derived from Information [Ω] and Time [T].
[M] = [Ω][T]⁻¹All mass-dependent physical dimensions are re-derived from this principle.
| Dimension | Standard Derivation | Mħπ Derivation ([M] = [Ω T⁻¹]) | Final Mħπ Dimensions |
|---|---|---|---|
| Action (ħ) | [E]·[T] |
([M L² T⁻²])·[T] = [M L² T⁻¹] |
[Ω L² T⁻²] |
| Momentum (p) | [M]·[v] |
[M]·[L T⁻¹] |
[Ω L T⁻²] |
| Force (F) | [M]·[a] |
[M]·[L T⁻²] |
[Ω L T⁻³] |
| Energy (E) | [M]·[v²] |
[M]·[L² T⁻²] |
[Ω L² T⁻³] |
| Pressure (P) | [F]/[Area] |
[Ω L T⁻³] / [L²] |
[Ω L⁻¹ T⁻³] |
| G | [F r² / M²] |
[Ω L T⁻³]·[L²] / [M²] |
[Ω⁻¹ L³ T⁻¹] |
The self-consistency of this framework has been validated by deriving the Planck Mass mₚ = √(ħc/G) within the system, which correctly yields the dimension [Ω T⁻¹].
The system is built upon these constants, with their dimensions derived from the informational mass postulate.
| Constant | Symbol | Dimensions |
|---|---|---|
| Pi | π |
Dimensionless |
| Planck Length | P |
[L] |
| Planck Time | τ, Δ |
[T] |
| Speed of Light | c |
[L T⁻¹] |
| Reduced Planck Constant | ħ |
[Ω L² T⁻²] (Action) |
| Gravitational Constant | G |
[Ω⁻¹ L³ T⁻¹] |
| Boltzmann Constant | k |
[Ω L² T⁻³ Θ⁻¹] (Energy/Temp) |
| Elementary Charge | e |
[I T] (Charge) |
The tensor module provides a powerful engine for General Relativity. The system can symbolically and automatically derive key geometric objects from any given metric tensor.
Γ^λ_μν): Calculated automatically via christoffel_symbols_from_metric.R^ρ_σμν): Derived from the Christoffel symbols via riemann_tensor_from_christoffel.R_μν) and Scalar (R): Derived by contracting the Riemann tensor.R_μν simplifies to 0, validating Einstein’s vacuum field equations.The quantum module provides tools for working with quantum mechanical concepts. The engine’s algebraic capabilities are particularly suited for non-commutative operator algebra.
|ψ⟩ (Ket), ⟨ψ| (Bra), and ⟨ψ|φ⟩ (BraKet) notation.[A, B]) and anti-commutators ({A, B}). The simplification engine correctly applies canonical commutation relations.γ^μ), including canonical reordering and simplification based on the metric tensor (γ^μγ^ν + γ^νγ^μ = 2g^μν).(iħγ^μ∂_μ ± mc), a key validation of its QFT capabilities.The engine has recently been extended with foundational support for gauge theory, a cornerstone of modern particle physics. This is implemented through new symbolic constructs and simplification rules.
D_μ): The CovariantDerivative expression (D_μ = ∂_μ - igA_μ) is now a first-class citizen in the AST, allowing for its direct manipulation. The ExpandCovariantDerivativeRule can expand this into its partial derivative and connection components.A_μ): The gauge field (or potential) is represented as a four-vector within a Connection expression, which also specifies the associated algebra (e.g., u(1) for QED).gauge.rs module provides functions to apply gauge transformations to fields (ψ → Uψ) and connections (A_μ → A_μ + ∂_μ α), allowing for proofs of invariance.L_QED, the system correctly derives (iħγ^μD_μ - mc)ψ = 0, as validated in the tests/qed.rs suite.