Physics for Mathematicians – Introduction • Hacker News Board

Re: Physics for Mathematicians – Introduction

#61

Post by scrubs » Fri, Feb 09, 2024, 11:52 PM UTC

nyrikki wrote: post #57

eigenket wrote: post #30

Earlier quoted context omitted.

Could you explain what you think Gibbs style vectors have to do with Banach-Tarski or the axiom of choice? As an aside, I'd like to emphasise that geometric algebra gives exactly the same physics outcomes as doing the maths with vectors or tensors or whatever else you like. The difference is essentially just notation. Some things look prettier.

Specific to building _intuitions_ for why the Banach-Tarski arises in ZF+AC. GA gets rid of the external conventions for coordinate and chirality and also uses SU(2) which is simply connected vs SO(3) which is not. Rotors in GA can be used as elements of the algebra like any number avoiding the complexity of Euler angles, gimbal lock, etc.... GA's rotors are geometrically intuitive and can do rotations around an arbi…

Was there an answer in there somewhere?

Re: Physics for Mathematicians – Introduction

#62

Post by throwaway81523 » Sat, Feb 10, 2024, 9:08 AM UTC

tippytippytango wrote: post #3

This is really cool. Can also serve as a Rosetta Stone for physicists wanting to better understand the language of mathematicians.

I didn't get that impression and it's not really a natural idea. A book about German for English speakers would be at best marginally useful for German speakers trying to learn English. Sachs and Wu's book about GR for mathematicians (which assumes a background in differential geometry) also cautions against expecting the book to work backwards like that.

Re: Physics for Mathematicians – Introduction

#63

Post by eigenket » Sat, Feb 10, 2024, 4:38 PM UTC

nyrikki wrote: post #57

eigenket wrote: post #30

Earlier quoted context omitted.

Could you explain what you think Gibbs style vectors have to do with Banach-Tarski or the axiom of choice? As an aside, I'd like to emphasise that geometric algebra gives exactly the same physics outcomes as doing the maths with vectors or tensors or whatever else you like. The difference is essentially just notation. Some things look prettier.

Specific to building _intuitions_ for why the Banach-Tarski arises in ZF+AC. GA gets rid of the external conventions for coordinate and chirality and also uses SU(2) which is simply connected vs SO(3) which is not. Rotors in GA can be used as elements of the algebra like any number avoiding the complexity of Euler angles, gimbal lock, etc.... GA's rotors are geometrically intuitive and can do rotations around an arbi…

I don't want this to come across as insulting, but this message sounds more like someone trying to sell me something than an objective scientist. You can just say no, there is no link (intuitive or not) between geometric algebra and Banach-Tarski.

To answer some of the other points

1. No physicist is particuarly confused by Euler angles, and gimbal lock is not a problem in physics.

2. I'm not sure I agree that physics even uses computable reals or rationals. I would say in reality we use fuzzy confidence intervals mostly and not exact numbers.

3. Maxwell's equations look different when written in geometric algebra style, you get one neat looking equation rather than the traditional 4, but its just a difference of notation, the same stuff is happening just written in a slightly different way.

4. A tensor can represent a spinor if you let it transform under the correct transformation rule. A basic spinor just looks like (https://en.wikipedia.org/wiki/Dirac_spinor) a complex vector which you let the Clifford algebra act on.

More generally everything I've seen from Geometric Algebra enthusiasts is just a weird way of doing fairly standard stuff in special cases of Clifford algebras in slightly weird old-fashioned notation. Pretty much everyone I've seen doing real work just does stuff in the Clifford algebra.