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1.6: The unimodular group SL(n, R) and the invariance of volume
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/01%3A_Algebraic_Preliminaries/1.06%3A_The_unimodular_group_SL(n_R)_and_the_invariance_of_volume
This result can be justified also in a more elegant way: The geometrical operations in figures b and c consist of adding the multiple of the vector $\vec{y}$ to the vector $\vec{x}$ , or adding the...This result can be justified also in a more elegant way: The geometrical operations in figures b and c consist of adding the multiple of the vector $\vec{y}$ to the vector $\vec{x}$ , or adding the multiple of the second row of the determinant to the first row, and we know that such operations leave the value of the determinant unchanged.
4.2: Rigid Body Rotation
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/04%3A_Spinor_Calculus/4.02%3A_Rigid_Body_Rotation
\[\begin{array}{c} {V(t) = U(\hat{x}_{3}, \frac{\dot{\alpha} t}{2}) V(0) U(\hat{e}_{3}, \frac{\dot{\gamma} t}{2})}\\ {\begin{pmatrix} {e^{-i \dot{\alpha} t/2}}&{0}\\ {0}&{e^{i \dot{\alpha} t/2}} \end{...\[\begin{array}{c} {V(t) = U(\hat{x}_{3}, \frac{\dot{\alpha} t}{2}) V(0) U(\hat{e}_{3}, \frac{\dot{\gamma} t}{2})}\\ { $\begin{pmatrix} {e^{-i \dot{\alpha} t/2}}&{0}\\ {0}&{e^{i \dot{\alpha} t/2}} \end{pmatrix}$ $\begin{pmatrix} {e^{-i \alpha (0)/2} \cos (\beta/2) e^{-i \gamma (0)/2}}&{-e^{-i \alpha (0)/2} \sin (\beta/2) e^{i \gamma (0)/2}}\\ {e^{i \alpha (0)/2} \sin (\beta/2) e^{-i \gamma (0)/2}}&{e^{i \alpha (0)/2} \cos (\beta/2) e^{i \gamma (0)/2}} \end{pmatrix}$ \times \begin{pmatrix} {e^{-i \do…
2.1: Introduction to Lorentz Group and the Pauli Algebra
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/02%3A_The_Lorentz_Group_and_the_Pauli_Algebra/2.01%3A_Introduction_to_Lorentz_Group_and_the_Pauli_Algebra
In the quantitative development of this idea we have to make a choice, whether to start with the classical wave concept and build in the corpscular aspects, or else start with the classical concept of...In the quantitative development of this idea we have to make a choice, whether to start with the classical wave concept and build in the corpscular aspects, or else start with the classical concept of the point particle, endowed with a constant and invariant mass, and modify these properties by means of the wave concept. The course of the present developments is set by the decision of following up the Einsteinian departure.
4.5: Review of SU(2) and preview of quantization
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/04%3A_Spinor_Calculus/4.05%3A_Review_of_SU(2)_and_preview_of_quantization
The rotation in the Poincare ́ space is associated with a phase shift between conjugate states, which is translated into a rotation of the Poincare ́ sphere, interpreted in turn as a change in orienta...The rotation in the Poincare ́ space is associated with a phase shift between conjugate states, which is translated into a rotation of the Poincare ́ sphere, interpreted in turn as a change in orientation, or change of shape of the vibrational patterns in ordinary space.
2.4: The Pauli Algebra
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/02%3A_The_Lorentz_Group_and_the_Pauli_Algebra/2.04%3A_The_Pauli_Algebra
The expressions in Equation $\ref{EQ2.4.49}$ are called the polar forms of N, the name being chosen to suggest that the representation of N by H and U is analogous to the representation of a complex num...The expressions in Equation $\ref{EQ2.4.49}$ are called the polar forms of N, the name being chosen to suggest that the representation of N by H and U is analogous to the representation of a complex number z by a positive number r and a phase factor:
1.1: Groups
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/01%3A_Algebraic_Preliminaries/1.01%3A_Groups
For an arbitrary element A of a finite $\mathcal{G}$ form the sequence: $A, A^{2}, A^{3} \cdots$ , let the numbers of distinct elements in the sequence be p. Thus we got the important result that t...For an arbitrary element A of a finite $\mathcal{G}$ form the sequence: $A, A^{2}, A^{3} \cdots$ , let the numbers of distinct elements in the sequence be p. Thus we got the important result that the order of a subgroup is a divisor of the order of the group. the three mirror planes of the regular triangle are in the same class and so are the four rotations by $2\pi / 3$ in a tetrahedron, or the eight rotations by $\pm 2\pi / 3$ in a cube.
5.5: On Parameterization and Integration
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/05%3A_Supplementary_Material_on_the_Pauli_Algebra/5.05%3A_On_Parameterization_and_Integration
\[\begin{array}{c} { $\begin{pmatrix} {k'_{1}}\\{k'_{2}}\\ {k'_{3}}\\ \end{pmatrix}$ = \begin{pmatrix} {l_{0}^{2}+l_{1}^{2}-l_{2}^{2}-l_{3}^{2}}&{2(l_{1}l_{2}-l_{0}l_{3})}&{2(l_{1}l_{3}+l_{0}l_{2})}\\ {2... $\begin{array}{c} {\begin{pmatrix} {k'_{1}}\\{k'_{2}}\\ {k'_{3}}\\ \end{pmatrix} = \begin{pmatrix} {l_{0}^{2}+l_{1}^{2}-l_{2}^{2}-l_{3}^{2}}&{2(l_{1}l_{2}-l_{0}l_{3})}&{2(l_{1}l_{3}+l_{0}l_{2})}\\ {2(l_{1}l_{2}+l_{0}l_{3})}&{l_{0}^{2}-l_{1}^{2}+l_{2}^{2}-l_{3}^{2}}&{2(l_{1}l_{3}-l_{0}l_{2})}\\ {2(l_{1}l_{3}-l_{0}l_{3})}&{2(l_{2}l_{3}+l_{0}l_{1})}&{l_{0}^{2}-l_{1}^{2}-l_{2}^{2}+l_{3}^{2}} \end{pmatrix} = \begin{pmatrix} {k_{1}}\\{k_{2}}\\ {k_{3}}\\ \end{pmatrix}} \end{array}$
References
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/zz%3A_Back_Matter/01%3A_References
Front Matter
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/00%3A_Front_Matter
4: Spinor Calculus
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/04%3A_Spinor_Calculus
Thumbnail: A spinor visualized as a vector pointing along the Möbius band, exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°. (CC ...Thumbnail: A spinor visualized as a vector pointing along the Möbius band, exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°. (CC BY-SA 3.0; Slawekb via Wikipedia)
4.1: From triads and Euler angles to spinors. A heuristic introduction
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/04%3A_Spinor_Calculus/4.01%3A_From_triads_and_Euler_angles_to_spinors._A_heuristic_introduction
\[\begin{array}{c} {\hat{e}_{31} = \langle \xi | \sigma_{1} | \xi \rangle = \xi_{0} \xi_{1}^{*}+\xi_{0}^{*} \xi_{1} = \mathcal{R} (\xi_{0}^{*} \xi_{1})}\\ {\hat{e}_{32} = \langle \xi | \sigma_{2} | \x... $\begin{array}{c} {\hat{e}_{31} = \langle \xi | \sigma_{1} | \xi \rangle = \xi_{0} \xi_{1}^{*}+\xi_{0}^{*} \xi_{1} = \mathcal{R} (\xi_{0}^{*} \xi_{1})}\\ {\hat{e}_{32} = \langle \xi | \sigma_{2} | \xi \rangle = i(\xi_{0} \xi_{1}^{*}-\xi_{0}^{*} \xi_{1}) = \mathcal{S} (\xi_{0}^{*} \xi_{1})}\\ {\hat{e}_{33} = \langle \xi | \sigma_{3} | \xi \rangle = \xi_{0} \xi_{1}^{*}+\xi_{1}^{*} \xi_{1}} \end{array}. \label{EQ4.1.36}$

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