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> maths degree Then you want differential forms for EM, differential geometry more broadly for GR, and a bit of functional analysis for QM. The hype around ge

by nyssos 3y ago

> maths degree Then you want differential forms for EM, differential geometry more broadly for GR, and a bit of functional analysis for QM. The hype around geometric algebras (Clifford algebras over R) just comes from the fact that it's not the plug'n'chug explicit numbers and coordinates approach, which is all most people ever see. They do not do a good job of tracking the physical structure of electromagnetism, and in fact end up baking in a lot of assumptions about the setting that fail to generalize.

guardingit 3y ago
Thanks for your comment. > They do not do a good job of tracking the physical structure of electromagnetism What do you mean by tracking the physical structure? By this, do you mean the typical EM formulation of Maxwell's laws produces Gauss's law and Faraday's law (which are instructive) where as the Geometric Algebra formula produces ∇F = J (less instructive?). > and in fact end up baking in a lot of assumptions about the setting that fail to generalize. Can you explain a bit more what you mean here please?
- nyssos 3y ago
  > By this, do you mean the typical EM formulation of Maxwell's laws produces Gauss's law and Faraday's law (which are instructive) where as the Geometric Algebra formula produces ∇F = J (less instructive?). > Can you explain a bit more what you mean here please? It's all downstream of geometric algebra leaving the metric implicit in its operations, basically - The metric is an extremely important physical quantity: it doesn't necessarily look that way when everything is classical and flat, but you have to start caring about it in curved spacetimes. - Even when the metric can be safely neglected, doing so makes it very easy to confuse degree (n-k) elements of your vector space V with degree k elements of its dual V. V and V are isomorphic but not canonically isomorphic - you have to choose a basis. And keeping track of where you introduce a choice of basis matters, because all physical quantities are basis-independent. Nature has no preferred coordinate system. - Geometric algebra doesn't make sense on a general manifold: you have to embed it in a sufficiently large geometric algebra and inherit structure from the embedding. This is better than working in a particular coordinate chart but worse than doing differential geometry in a purely geometric way. - Geometric algebra is not invariant under diffeomorphism, which means GR is dead in the water. You can build mostly equivalent theories by enforcing the equivalence principle in your dynamics instead, but it's more complicated and ironically far less geometric.
jiggawatts 3y ago
> baking in a lot of assumptions about the setting that fail to generalize That’s the point! That’s the entire point! Mathematicians want the most general, most abstract approach. They want to generalise to a wide range of problems and not be painted into any one specific example. Physics theories have an opposite goal to this: the ideal theory ought to take no parameters, and produce “reality” as the one and only possible outcome. The ideal theory ought not generalise to un-physical models. For example, the mathematics of general relativity have excess degrees of freedom that must be constrained through additional restrictions. Similar issues turn up almost anywhere matrices are used: they have too many degrees of freedom. Geometric Algebra is typically a better fit for what actually goes on in physics. For example, rotation matrices have precision issues, gimbal lock, and can’t be robustly interpolated. Rotations implemented using GA have none of these issues.
- nyssos 3y ago
  > The ideal theory ought not generalise to un-physical models. Sure - but a theory that fails to generalize to physical models is a bad one. A good classical theory should be a straightforward deformation of the corresponding quantum and/or relativistic theory. In this respect the best versions of classical mechanics are the standard Lagrangian and Hamiltonian approaches. > For example, rotation matrices have precision issues, gimbal lock, and can’t be robustly interpolated. No physicist was ever under the impression that euler angles were any realer than any other way of parameterizing SO(3), that matrices were the linear transformations they represent, that manifolds are their charts, or any other trivial map-territory confusion. Paying careful attention to the distinction between real physical objects on the one hand and their representations on the other is the central theme of the last century of physics: that's basically all a gauge theory is! This is exactly what I'm talking about with geometric algebra advocates only ever comparing it to the worst sort of high school coordinate-bashing imaginable. The argument always goes - Look at this horrible vector algebra with explicit charts and numbers all over the place - Now look at this nice coordinate-free geometric algebra construction - Therefore geometric algebra is the right setting for physics but it's a complete non sequitur, because the coordinate-freeness is doing all the heavy lifting. But everyone already works without coordinates wherever it's practical to do so! The question isn't whether you should work in terms of abstract objects or explicit coordinates, it's which abstract objects you should work with.

by nyssos 3y ago

guardingit 3y ago

Thanks for your comment. > They do not do a good job of tracking the physical structure of electromagnetism What do you mean by tracking the physical structure? By this, do you mean the typical EM formulation of Maxwell's laws produces Gauss's law and Faraday's law (which are instructive) where as the Geometric Algebra formula produces ∇F = J (less instructive?). > and in fact end up baking in a lot of assumptions about the setting that fail to generalize. Can you explain a bit more what you mean here please?

nyssos 3y ago
> By this, do you mean the typical EM formulation of Maxwell's laws produces Gauss's law and Faraday's law (which are instructive) where as the Geometric Algebra formula produces ∇F = J (less instructive?). > Can you explain a bit more what you mean here please? It's all downstream of geometric algebra leaving the metric implicit in its operations, basically - The metric is an extremely important physical quantity: it doesn't necessarily look that way when everything is classical and flat, but you have to start caring about it in curved spacetimes. - Even when the metric can be safely neglected, doing so makes it very easy to confuse degree (n-k) elements of your vector space V with degree k elements of its dual V. V and V are isomorphic but not canonically isomorphic - you have to choose a basis. And keeping track of where you introduce a choice of basis matters, because all physical quantities are basis-independent. Nature has no preferred coordinate system. - Geometric algebra doesn't make sense on a general manifold: you have to embed it in a sufficiently large geometric algebra and inherit structure from the embedding. This is better than working in a particular coordinate chart but worse than doing differential geometry in a purely geometric way. - Geometric algebra is not invariant under diffeomorphism, which means GR is dead in the water. You can build mostly equivalent theories by enforcing the equivalence principle in your dynamics instead, but it's more complicated and ironically far less geometric.

jiggawatts 3y ago

> baking in a lot of assumptions about the setting that fail to generalize That’s the point! That’s the entire point! Mathematicians want the most general, most abstract approach. They want to generalise to a wide range of problems and not be painted into any one specific example. Physics theories have an opposite goal to this: the ideal theory ought to take no parameters, and produce “reality” as the one and only possible outcome. The ideal theory ought not generalise to un-physical models. For example, the mathematics of general relativity have excess degrees of freedom that must be constrained through additional restrictions. Similar issues turn up almost anywhere matrices are used: they have too many degrees of freedom. Geometric Algebra is typically a better fit for what actually goes on in physics. For example, rotation matrices have precision issues, gimbal lock, and can’t be robustly interpolated. Rotations implemented using GA have none of these issues.

nyssos 3y ago
> The ideal theory ought not generalise to un-physical models. Sure - but a theory that fails to generalize to physical models is a bad one. A good classical theory should be a straightforward deformation of the corresponding quantum and/or relativistic theory. In this respect the best versions of classical mechanics are the standard Lagrangian and Hamiltonian approaches. > For example, rotation matrices have precision issues, gimbal lock, and can’t be robustly interpolated. No physicist was ever under the impression that euler angles were any realer than any other way of parameterizing SO(3), that matrices were the linear transformations they represent, that manifolds are their charts, or any other trivial map-territory confusion. Paying careful attention to the distinction between real physical objects on the one hand and their representations on the other is the central theme of the last century of physics: that's basically all a gauge theory is! This is exactly what I'm talking about with geometric algebra advocates only ever comparing it to the worst sort of high school coordinate-bashing imaginable. The argument always goes - Look at this horrible vector algebra with explicit charts and numbers all over the place - Now look at this nice coordinate-free geometric algebra construction - Therefore geometric algebra is the right setting for physics but it's a complete non sequitur, because the coordinate-freeness is doing all the heavy lifting. But everyone already works without coordinates wherever it's practical to do so! The question isn't whether you should work in terms of abstract objects or explicit coordinates, it's which abstract objects you should work with.