SR: GA Multivector vs. Tensor notation, Maxwell's equations

[Sagittarius A-Star]

TL;DR

Geometric Algebra (Clifford algebra) applied to Minkowski spacetime makes equations more compact than tensor notation, for example Maxwell's equation(s)

Geometric algebra (a subset of Clifford algebra) can be consistently used for all branches of physics, that are based on Euclidean space+time or Minkowski spacetime. It uses the following concepts for Minkowski spacetime:

• Geometric product (Dot product + Wedge product)
• Multivectors ( = scalar + vector (with 4 components) + bivector (with 6 components) + trivector (with 4 components) + pseudo-scalar)

Geometric product definition (edit: not valid for all kinds of multivectors):
$$ ab = a \cdot b + a\wedge b$$Dot product (commutative): $a \cdot b = \sum_{r} \sum_{s}a_\hat r \cdot b_\hat s$
Wedge product (anticommutative, generalizes the cross product to arbitrary dimensions): $a \wedge b = \sum_{r} \sum_{s} a_\hat r \wedge b_\hat s$
Example: Wedge product in 3D-space, resulting in a bivector:
$a∧b = (a_ie_i)∧(b_j e_j )= (a_2b_3 − b_3a_2)e_2 \wedge e_3 + (a_3b_1 − a_1b_3)e_3 \wedge e_1+ (a_1b_2 − a_2b_1)e_1 \wedge e_2$

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Spacetime​

Orthogonal basis vectors for t, x, y, and z: $\{\gamma_0, \gamma_1, \gamma_2, \gamma_3 \}$, satisfying $\gamma_0^2=1, \ \ \ \ \gamma_0 \cdot \gamma_i=0, \ \ \ \ \gamma_i \cdot \gamma_j= -\delta_{ij}$, where $i$ and $j$ run from 1 to 3 (West coast convention). This can be also written with the Minkowski metric:
$$\gamma_\mu \cdot \gamma_\nu=\eta_{\mu\nu}$$
Spacetime split:
Multiplying a 4-vector with the 4-velocity leads to a spacetime split. Example 4-momentum:
$$pv = p(v_\parallel + v_\perp) = p \cdot v + p\wedge v = E + \mathbf p$$
Electromagnetic field (with pseudoscalar $I:= \gamma_0 \gamma_1 \gamma_2 \gamma_3$ and spacetime vector derivative $\nabla = \gamma^\mu \partial_\mu$):
The EM field bivector is
$\begin {align} F = \nabla \wedge A & = \mathbf E + I \mathbf B \nonumber \\ & = E_1\gamma_1 \gamma_0 + E_2\gamma_2 \gamma_0 + E_3\gamma_3 \gamma_0 + I(B_1\gamma_1 \gamma_0 + B_2\gamma_2 \gamma_0 + B_3\gamma_3 \gamma_0) \nonumber \\ & = E_1\gamma_1 \gamma_0 + E_2\gamma_2 \gamma_0 + E_3\gamma_3 \gamma_0 + B_1\gamma_3 \gamma_2 + B_2\gamma_1 \gamma_3 + B_3\gamma_2 \gamma_1 \nonumber \end {align}$

Maxwell's equation:
$$ \nabla F = \nabla \cdot F + \nabla \wedge F = J + 0 $$

Geometric Algebra $\nabla F = J$	corresponding Tensor notation
$\nabla \cdot F = J$ (vector)	$\partial_\nu F^{\mu \nu} = J^\mu$ (4-tensor of rank 1)
$\nabla \wedge F= 0$ (no magnetic monopoles, trivector)	$\partial_\lambda F_{\mu \nu} + \partial_\mu F_{\nu \lambda} + \partial_\nu F_{\lambda \mu} = H_{\lambda \nu \mu} = 0$ (4-tensor of rank 3)

Lorentz force law:
$$m \dot v = qF \cdot v$$
I wonder, why the compact geometric algebra notation is not more often preferred over tensor notation.

Sources:
https://en.wikipedia.org/wiki/Spacetime_algebra
https://www.faculty.luther.edu/~macdonal/GAGC/GAGC.html
Video: GA in 2D and 3D
Video: GA in 4D
https://www.amazon.com/-/de/dp/0521715954?tag=pfamazon01-20

 

[PeterDonis]

I wonder, why the compact geometric algebra notation is not more often preferred over tensor notation.

You will find dot products and wedge products used in plenty of places in the literature; for example, Misner, Thorne & Wheeler use them extensively. You will also find plenty of discussion of Clifford algebra and the breakdown of the 16 possible combinations (scalar, vector, bivector, pseudovector, pseudoscalar).

What you will not find very much is the claim that somehow combining the dot and wedge products into a single "geometric product" adds anything useful. That's because many workers in the field don't appear to believe that it does add anything useful.

 

[weirdoguy]

I wonder, why the compact geometric algebra notation is not more often preferred over tensor notation.

For me tensors are way clearer, especially if you work with bundles.

 

[robphy]

TL;DR Summary: Geometric Algebra (Clifford algebra) applied to Minkowski spacetime makes equations more compact than tensor notation, for example Maxwell's equation(s)

I wonder, why the compact geometric algebra notation is not more often preferred over tensor notation.

I think part of the issue is a balance between
how much one has to learn in terms of mathematical preparation
in order to solve certain kinds of problems.

In addition, some notations are too-compact.
I like and am comfortable with the abstact-index-notation
because I can see what tensor-types I am working with.
(Needless to say, some instead want "indices that one sums over" or maybe just a list of components.)

Finally, I have to communicate my results in a language and notation
that my reader can understand.

Geometric algebra is interesting... but I'm not there yet.
(One stumbling block for me are various conventions I've seen in different references.
I'll sort that out when I can... but that's not a high priority right now.)

 

[Sagittarius A-Star]

What you will not find very much is the claim that somehow combining the dot and wedge products into a single "geometric product" adds anything useful.

One claim that I found for the usefulness is that GA contains STA elements, that have an inverse.
Source:
https://en.wikipedia.org/wiki/Spacetime_algebra#Division

An example is the nabla operator $\nabla$. It's inverse $\nabla^{-1} = {(\nabla \cdot \nabla)^{-1}\nabla}$ is an integral operator and can be applied i.e. to Maxwell's equation:
$$F = \nabla^{-1} J$$
Source (see page 26, via the above Wikipedia link):
https://davidhestenes.net/geocalc/pdf/SpacetimePhysics.pdf

Importantly, neither the dot product nor the wedge product alone may be inverted; only their combination as the sum (3.2) retains enough information to define an inverse.

Source (see page 20):
https://arxiv.org/abs/1411.5002

 

[Sagittarius A-Star]

(One stumbling block for me are various conventions I've seen in different references.
I'll sort that out when I can... but that's not a high priority right now.)

I found a corrections-video to the videos, linked in the OP. I edited the OP accordingly.

 

[Sagittarius A-Star]

For me tensors are way clearer, especially if you work with bundles.

There exist Clifford bundles, but I didn't look in detail into it.

Source:
https://en.wikipedia.org/wiki/Clifford_bundle#Clifford_bundle_of_a_Riemannian_manifold

 

[weirdoguy]

There exist Clifford bundles, but I didn't look in detail into it.

Yes, these are important when you consider spinor fields. I did look into details in the context of my plans for my phd years ago, but my clinical depression killed those, hehe... So... Anyways, the use of Clifford algebras in the context of spinors is still kind of a different matter than reformulating physics in terms of geometric algebra. First one is a necessity, because fermions exist, second one is an option.