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- TL;DR Summary
- 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:
$$ ab = a \cdot b + a\wedge b$$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##
$$\gamma_\mu \cdot \gamma_\nu=\eta_{\mu\nu}$$
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:
$$ \nabla F = \nabla \cdot F + \nabla \wedge F = J + 0 $$
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
- Geometric product (Dot product + Wedge product)
- Multivectors ( = scalar + vector (with 4 components) + bivector (with 6 components) + trivector (with 4 components) + pseudo-scalar)
$$ ab = a \cdot b + a\wedge b$$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##
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
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