For a Lagrangian, ##\mathcal{L}##, in which the ##\phi_{i}## are vector fields the stress-energy-momentum tensor is give by the following coordinate free geometric algebra expression -
$$T(n) = \dot{\nabla}\left < \dot{\phi}_{i}\partial_{\nabla\phi_{i}}\mathcal{L}n\right >-n\mathcal{L}$$
where...
Without regard to a coordinate system (I only wish to consider special relativity) the stress-energy-momentum tensor defines a linear transformation from a 4-vector to a 4-vector. Let T be the linear transformation then b = T(a), a and b are 4-vectors. What is the physical meaning of a and b...
Double Cover in Doran and Lasenby
It is mentioned in D&L. In GA rotations (in any dimension) are preformed with rotors which are scalars plus bivectors, R, with the condition that R*rev(R) = 1 (rev(R) is the reverse of R and * the geometric product). Then to rotate a vector, x, in the plane...
GA Notes and Software
If you are interested I have extensive geometric algebra/calculus notes (based on Doran and Lasenby) and symbolic software (python) at
https://github.com/brombo/GA
Applications of Geometric Algebra
Outside of Cambridge geometric algebra has mainly caught on for computer vision and robotics (in adopting something new engineers are usually ahead of physicists).
In physicis the main application is the gauge theory of gravity (Cambridge). The problem in...
Basic Geometric Algebra
See attached file DET.pdf. The thing to note is that in the case of an orthogonal basis e1,...,en we have e1^...^en = e1...en (wedge product is the same as the geometric/Clifford product).
Detailed note based on "Geometric Algebra for Physicists" by Doran and Laseby...
Problems in GA
Hestenes book is very dense. You might consider (less dense more understandable)
Geometric Algebra for Physcists by Doran and Lasenby
https://www.amazon.com/dp/0521715954/?tag=pfamazon01-20
I have notes and software that I based on the book at...
Gauge Theory of Gravity
A major alternative to GA is the gauge theory of gravity as presented in
"Geometric Algebra for Physicts" by Doran and Lasenby
https://www.amazon.com/dp/0521715954/?tag=pfamazon01-20
This is a first order theory including spin (there are two gauge fields) and...
From Clifford Algebra to Geometric Calculus
Geometric Algebra is the Clifford Algebra of a finite dimensional vector space. The best reference for this is
"Geometric Algebra for Physicists" by Doran and Lasenby
https://www.amazon.com/dp/0521715954/?tag=pfamazon01-20
I have a set of...
GA BAC CAB Relations
The generalized BAC CAB relations are shown in the attached file. All the relations were generation using software for the symbolic manipulation of multivectors to be found at
https://github.com/brombo/GA
This repository also contains notes on geometric algebra...
Relevant References
For Clifford/Geometric Algebra the most relevant reference is -
"Geometric Algebra for Physicsts" by Doran and Lasenby
https://www.amazon.com/dp/0521715954/?tag=pfamazon01-20
Which covers (among other things) both the Dirac equation and the Gauge Theory of Gravity...
GA Note and Software
You may wish to check out
https://github.com/brombo/GA
The repository contains my notes on GA (following Doran and Lasenby with the blanks filled in) and python software (detailed documentation in LaTeX docs) for symbolic manipulation of multivectors.