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The two-dimensional "spin 1/2" representation of the Lie algebra so(3), for example, does not correspond to an ordinary (single-valued) representation of the group SO(3). (This fact is the origin of statements to the effect that "if you rotate the wave function of an electron by 360 degrees, you get the negative of the original wave function.")
Representation theory depends upon the type of algebraic object being represented. There are several different classes of groups, associative algebras and Lie algebras, and their representation theories all have an individual flavour. Representation theory depends upon the nature of the vector space on which the algebraic object is represented.
It is fundamental in theoretical physics. In a physical theory having Minkowski space as the underlying spacetime, the space of physical states is typically a representation of the Poincaré group. (More generally, it may be a projective representation, which amounts to a representation of the double cover of the group.)
Many of the representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics. Representations appear in the description of fields in classical field theory, most importantly the electromagnetic field, and of particles in relativistic quantum mechanics, as well as of both particles and quantum fields in quantum field theory and of various objects in string ...
Lie groups — Many important Lie groups are compact, so the results of compact representation theory apply to them. Other techniques specific to Lie groups are used as well. Most of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields.
AP Physics C: Mechanics and AP Physics 1 are both introductory college-level courses in mechanics, with the former recognized by more universities. [8] The AP Physics C: Mechanics exam includes a combination of conceptual questions, algebra-based questions, and calculus-based questions, while the AP Physics 1 exam includes only conceptual and algebra-based questions.
From the perspective of representation theory, the significance of the roots and root vectors is the following elementary but important result: If : is a representation of , v is a weight vector with weight and X is a root vector with root , then
This is a list of representation theory topics, by Wikipedia page. See also list of harmonic analysis topics , which is more directed towards the mathematical analysis aspects of representation theory.