This document outlines the physical extensions to the MPP symbolic mathematics system, providing a framework for expressing physical concepts in terms of the fundamental constants π and P (Planck length).
MPP-Physics builds upon the core MPP constants and adds several physical constants:
π (pi): The geometry constantP: The Planck length (smallest physical unit)τ: Planck time — the time it takes light to travel 1 Planck lengthc: Speed of light constant — defined as P / τΔ: Planck Time Unit (PTU) — defined as P / c, the time required for light to traverse one Planck lengthℏ: Reduced Planck constant — fundamental quantum constantG: Gravitational constantk: Boltzmann constante: Elementary chargeIn the MPP system, constants are categorized as either primitive or derived:
These are fundamental constants that are not defined in terms of other constants:
π (pi): The geometry constantP: The Planck lengthτ: Planck timeℏ: Reduced Planck constantG: Gravitational constantk: Boltzmann constante: Elementary chargeThese are defined in terms of primitive constants:
c: Speed of light, defined as P / τΔ: Planck Time Unit (PTU), defined as P / cThe following relationships between constants are enforced by the simplifier:
c · τ = P
This relationship defines the speed of light as the Planck length divided by the Planck time.
c · Δ = P
This relationship ensures consistency between the Planck Time Unit and the Planck length.
Δ = τ
In the current implementation, τ and Δ are distinct symbols that represent the same physical quantity. This allows for flexibility in notation while maintaining physical consistency.
Each constant has specific physical dimensions:
| Constant | Symbol | Dimensions |
|---|---|---|
| Pi | π | Dimensionless |
| Planck Length | P | Length (L) |
| Planck Time | τ | Time (T) |
| Speed of Light | c | Velocity (L/T) |
| Planck Time Unit | Δ | Time (T) |
| Reduced Planck Constant | ℏ | Energy·Time (ML²/T) |
| Gravitational Constant | G | L³/(M·T²) |
| Boltzmann Constant | k | Energy/Temperature (ML²/T²Θ) |
| Elementary Charge | e | Electric Charge (Q) |
MPP supports multiple natural units systems:
When natural units are active, the simplifier replaces explicit constants with their natural unit values, while maintaining dimensional consistency.
The spacetime module provides tools for working with 4-dimensional spacetime in the MPP system.
Four-vectors represent points in 4D spacetime, with components expressed in terms of Planck units:
(t, x, y, z)
where:
t is expressed in Planck Time Units (Δ)x, y, and z are expressed in Planck lengths (P)The Minkowski metric defines the inner product in spacetime. MPP-Physics supports both signature conventions:
(+, -, -, -): Time-positive convention(-, +, +, +): Space-positive conventionThe spacetime interval between two events is invariant under Lorentz transformations:
Δs² = Δt² - Δx² - Δy² - Δz²
In MPP, this is expressed purely in terms of Planck units.
Lorentz transformations preserve the spacetime interval. The Lorentz factor γ is defined as:
γ(v) = (1 - v²)^(-1/2)
where v is expressed as a fraction of the speed of light.
The proper time along a worldline is the time experienced by an observer following that worldline:
Δτ = √(Δt² - Δx² - Δy² - Δz²)
The quantum module provides tools for working with quantum mechanical concepts in the MPP system.
MPP-Physics implements Dirac notation for quantum states:
|ψ⟩: Ket vector representing a quantum state⟨ψ|: Bra vector, the dual of a ket vector⟨ψ|φ⟩: Inner product between two quantum statesFundamental quantum operators:
xpa†aN = a†aThe commutator of two operators A and B is defined as:
[A, B] = AB - BA
The canonical commutation relation between position and momentum is:
[x, p] = iℏ
The Hamiltonian for a quantum harmonic oscillator is:
H = ℏω(a†a + 1/2)
The Heisenberg uncertainty principle is expressed as:
ΔxΔp ≥ ℏ/2
The tensor module provides tools for working with tensors in general relativity.
Key tensors in general relativity:
The Einstein field equations relate the geometry of spacetime to the distribution of matter and energy:
G_μν = (8πG/c⁴)T_μν
where G_μν is the Einstein tensor and T_μν is the stress-energy tensor.
The dimensional analysis module provides tools for tracking physical dimensions throughout calculations.
The module ensures that operations maintain dimensional consistency:
MPP-TeX has been extended to support physical notation:
\let \c := \P / \tau
\let \Delta := \P / \c
\let \gamma := (1 - v^2)^{-1/2}
\interval{\Delta t}{\Delta x}{\Delta y}{\Delta z}
\vec{A} = (t, x, y, z)
\ket{\psi}
\bra{\psi}
\braket{\psi}{\phi}
MPP-Physics maintains the core MPP philosophy:
By expressing physical concepts in terms of fundamental constants, MPP-Physics provides a framework for symbolic physical calculations that is both elegant and powerful.
In MPP, tensors explicitly track whether each index is contravariant (upper) or covariant (lower). This distinction is crucial in general relativity and other field theories.
The metric tensor g_μν defines the inner product in spacetime and allows:
MPP supports both common metric signatures:
The invariant spacetime interval is calculated as:
Δs² = g_μν Δx^μ Δx^ν
In the mostly-minus convention, this becomes: Δs² = (Δt)² - (Δx)² - (Δy)² - (Δz)²
This quantity remains invariant under all Lorentz transformations.