Search
- https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Book%3A_Applied_Geometric_Algebra_(Tisza)/03%3A_The_Lorentz_Group_and_the_Pauli_Algebra/3.04%3A_The_Pauli_AlgebraWhereas the two-valuedness of the \(\mathcal{S U}(2)\) representation does not affect the transformation of the A vector based on the bilateral expression \ref{75}, the situation will be seen to be di...Whereas the two-valuedness of the \(\mathcal{S U}(2)\) representation does not affect the transformation of the A vector based on the bilateral expression \ref{75}, the situation will be seen to be different in the spinorial theory based on Equation \ref{62}, since under certain conditions the sign of the spinor \(|\xi\rangle\) is physically meaningful.
- https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Book%3A_Applied_Geometric_Algebra_(Tisza)/02%3A_Algebraic_Preliminaries/2.02%3A_The_geometry_of_the_three-dimensional_rotation_group._The_Rodrigues-Hamilton_theoremThe standard notation for the proper rotation group is \(\mathcal{O}^{+}, \text {or } \mathcal{S O}(3)\) short for “simple orthogonal group in three dimensions”. “Simple” means that the determinant of...The standard notation for the proper rotation group is \(\mathcal{O}^{+}, \text {or } \mathcal{S O}(3)\) short for “simple orthogonal group in three dimensions”. “Simple” means that the determinant of the transformation is \(+1\), we have proper rotations with the exclusion of the inversion of the coordinates:
- https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Book%3A_Applied_Geometric_Algebra_(Tisza)/04%3A_Pauli_Algebra_and_Electrodynamics/4.01%3A_Lorentz_transformation_and_Lorentz_forceIn order to characterize specifically the Lorentz force, we have to add that the characterization of the field is independent of the fourmomentum of the test charge, moreover it is independent of the ...In order to characterize specifically the Lorentz force, we have to add that the characterization of the field is independent of the fourmomentum of the test charge, moreover it is independent of the frame of reference of the observer. The effect of the Lorentz force on a particle (test charge) is represented as the transformation of the four-momentum space of the particle unto itself, and the transformations are elements of the active Lorentz group.
- https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Book%3A_Applied_Geometric_Algebra_(Tisza)/03%3A_The_Lorentz_Group_and_the_Pauli_Algebra/3.02%3A_The_corpuscular_aspects_of_lightIn the early days of relativity the consequences of Lorentz invariance involved mostly effects of the order of \((v/c)^2\), a quantity that is small for the velocities attainable at that time. Eq (\re...In the early days of relativity the consequences of Lorentz invariance involved mostly effects of the order of \((v/c)^2\), a quantity that is small for the velocities attainable at that time. Eq (\ref{5}) provides us with a definition of the four-momentum, but only for the case of the photon, that is for a particle with zero rest mass and the velocity c. and define the mass m of a particle as the invariant “length” of the four-momentum according to the Minkowski metric (with \(c = 1\)).
- https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Book%3A_Applied_Geometric_Algebra_(Tisza)/02%3A_Algebraic_Preliminaries/2.07%3A_On_alias_and_alibi._The_Object_GroupIt is fitting to conclude this review of algebraic preliminaries by formulating a rule that is to guide us in connecting the group theoretical concepts with physical principles, One of the concerns of...It is fitting to conclude this review of algebraic preliminaries by formulating a rule that is to guide us in connecting the group theoretical concepts with physical principles, One of the concerns of physicists is to observe, identify and classify particles. First, the same object may be observed by different inertial observers whose findings are connected by the transformations of the inertial group, to be called also the passive kinematic group.
- https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Book%3A_Applied_Geometric_Algebra_(Tisza)/03%3A_The_Lorentz_Group_and_the_Pauli_Algebra/3.03%3A_On_circular_and_hyperbolic_rotationsWe propose to develop a unified formalism for dealing with the Lorentz group SO(3,1) and its subgroup SO(3) . This program can be divided into two stages. First, consider a Lorentz transformation a...We propose to develop a unified formalism for dealing with the Lorentz group SO(3,1) and its subgroup SO(3) . This program can be divided into two stages. First, consider a Lorentz transformation as a hyperbolic rotation, and exploit the analogies between circular and hyperbolic trigonometric functions, and also of the corresponding exponentials. This simple idea is developed in this section in terms of the subgroups SO(2) and SO(1,1).
- https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Book%3A_Applied_Geometric_Algebra_(Tisza)/07%3A_Homework_Assignments
- https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Book%3A_Applied_Geometric_Algebra_(Tisza)/04%3A_Pauli_Algebra_and_Electrodynamics
- https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Book%3A_Applied_Geometric_Algebra_(Tisza)The mathematical tools employed by physicists have expanded considerably, from differential calculus, vector algebra and geometry, to advanced linear algebra, tensors, Hilbert space, spinors, Group th...The mathematical tools employed by physicists have expanded considerably, from differential calculus, vector algebra and geometry, to advanced linear algebra, tensors, Hilbert space, spinors, Group theory and many others. These course notes attempt at bringing conceptual clarity and unity to the application and interpretation of these advanced mathematical tools. In particular, there is an emphasis on the unifying role that Group theory plays in classical, relativistic, and quantum physics.
- https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Book%3A_Applied_Geometric_Algebra_(Tisza)/06%3A_Supplementary_Material_on_the_Pauli_Algebra
- https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Book%3A_Applied_Geometric_Algebra_(Tisza)/02%3A_Algebraic_Preliminaries/2.05%3A_A_short_survey_of_linear_groupsWe start with the group of nonsingular linear transformations defined by Equations 2.3.4 and 2.3.5 of Section 2.3 and designated as \(\mathcal{G L}(n, R)\), for “general linear group over the field F....We start with the group of nonsingular linear transformations defined by Equations 2.3.4 and 2.3.5 of Section 2.3 and designated as \(\mathcal{G L}(n, R)\), for “general linear group over the field F.” If the matrices are required to have unit determinants, they are called unimodular, and the group is \(\mathcal{S L}(n, F)\) for simple linear group.
