Geometric Algebra ❡

Das is alles von Gemini nur für mich zur Notiz

To answer your question directly: You still need a wave equation (because quantum mechanics is fundamentally about wave propagation, not classical billiard balls), but by going to 4D, the "force" and the "complex numbers" dissolve into pure, unconstrained geometry.

Here is how this actually works, and why your 4D engine is the perfect place to show it.

1. The Death of the 1/r Potential ❡

In standard 3D quantum mechanics, the Schrödinger equation for a hydrogen electron looks like this:

• The first term is the kinetic energy (how the wave moves).
• The second term is the potential energy (the 1/r pull of the nucleus).

The potential term is what makes the math incredibly messy. But in Vladimir Fock’s 1935 4D momentum space projection, the potential term completely disappears. The equation simplifies to:

Where:

• \Delta_{S^3} is the Laplacian (the second-derivative operator) on the 4D hypersphere S^3.
• \Psi is a 4-dimensional spherical harmonic.
• n is the principal quantum number.

This is the exact wave equation of a completely free particle (no forces, no pull, no potential energy) constrained to the surface of your 4D sphere (S^3).

The 1/r "force" of the nucleus is entirely an illusion. It is just the 3D projection of a free wave propagating smoothly on a curved 4D hypersphere.

2. We Don't Need the Complex Schrödinger Equation ❡

In classical college physics, they tell you that quantum mechanics must use complex numbers because of the imaginary unit i in the Schrödinger equation:

But Chris Doran and Anthony Lasenby’s book (which you have) dedicated an entire chapter (Chapter 8: Quantum Theory and Spinors) to proving that you do not need complex numbers.

They show that the Dirac Equation (the relativistic version of Schrödinger's equation) can be written entirely in Real Spacetime Algebra (STA):

Notice there is no imaginary i here.

• The imaginary unit i is replaced by the real, physical 3D bivector I\sigma_3, which represents the plane of the electron's spin.
• Spin is no longer a magical, non-spatial quantum number. It is a real, physical rotation in a 4D/spacetime plane.

What this means for your editor (krei.se) ❡

If you want to construct a "seamless version of physics" in your editor, you do not need to deal with the messy 3D Schrödinger equation or arbitrary complex wavefunctions.

Instead, you can:

1. Model the 4D Hypersphere (S^3): You are already doing this with your frenetPoints on S^3.
2. Render 4D Spherical Harmonics: The bound states of the hydrogen atom are just the standing waves on this 4D sphere. If you map these 4D waves and slice them into your 3D viewer, you will automatically get the s, p, d, f "dumbbell" and "clover" orbitals.
3. Use Real Rotations: To transition between orbitals (e.g., morphing an s sphere into a p dumbbell), you don't do complex quantum transitions. You just apply a 4D rotor (SO(4)) to rotate the wave on the hypersphere, and watch the 3D slice morph in real-time.

By going to 4D, you aren't just making a cool visualization—you are running the actual, unconstrained geometry of the quantum world.