Physics
Adjoint Representation
The adjoint representation is a mathematical tool used in physics to describe the symmetries of a system. It involves associating each element of a Lie group with a linear transformation on the Lie algebra of that group. The adjoint representation is important in the study of gauge theories and particle physics.
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3 Key excerpts on "Adjoint Representation"
eBook - PDF- R Mirman(Author)
- 1995(Publication Date)
- WSPC(Publisher)
Thus group rep-resentation theory is a systematic way of solving quantum-mechanical problems (provided there is a group available). This indicates the role, and usefulness, of group theory in quantum mechanics. We also want the wavefunctions explicitly, their labels and physical significance, and wish to know how to organize the states into sets, all with the same energy. This provides most of what can be imderstood about the system; it comes from the representations of the group, this and more. The problem then is given a group of transformations, con-struct the representations: label and find the basis functions and ma-trices representing the group elements. Before doing this we must give a (reasonably) rigorous definition of representation. Problem V.2.C-1: In explaining why the states of the hydrogen atom are thebasis states of the rotation-group representations how was spher-ical symmetry used? Problem V.2.C-2: Here the appearance of representations in quantum mechanics has been emphasized. But they are introduced much earlier, at the beginning of elementary physics and in mathematics. The reader should note such occurrences and discuss the reasons for them. V.3 WHAT ARE REPRESENTATIONS? What does it mean to say a set of matrices forms a representation of the rotation group? By taking some physical object, say this book, and rotating it in various ways we can find experimentally the product of two rotations — the rotation that gives the same final orientation as the two performed in succession (although products are not usually found this way). If we can assign a matrix to each rotation such that the ma-trices representing (assigned to) the rotations have the same product as the rotations they represent, the set of matrices forms a representa-tion of the rotation group.
- A Barut, R Raczka;;;(Authors)
- 1986(Publication Date)
- WSPC(Publisher)
Chapter 13 Group Theory and Group Representations in Quantum Theory § 1. Group Representations in Physics Discrete and continuous groups occur in classical physics as transformation groups and express generally the symmetry of dynamical equations of particles or fields: crystal symmetry, the Galilei, PoincarS or Einstein group of transformations of space-time, the group of canonical transformations acting on the phase-space, the group of gauge transformation, or conformal transformations of the electro-magnetic potentials, symplectic transformation of thermodynamic functions, etc. We gave even a Lie algebra structure in the Poisson-brackets of classical mechanics and classical field theory, for which the canonical transformations act as the group of automorphisms. However, in all these only a defining repre-sentation of the groups occurs. The theory of group representations in linear spaces which is the subject matter of this book, is really the domain of quantum physics. The particular adaptability of the theory of group representations to quantum physics stems from the following basic kinematical difference between classical and quantum theories: In both classical and quantum theory a physical system can be described by the notion of 'state', and in both cases the system has continuously infinitely many states. In quantum theory, due to the linearity of the equations of motions, the states can be represented as linear combinations of a selected orthogonal sets of states, often a set of countable many elements; i.e., the states form a linear vector space. In contrast, in the classical case the dynamical equations in terms of coordinates and momenta are non-linear, hence there is no such set of basis states. The group-structure of physical theories may be studied along the following two different approaches: A.
eBook - PDFDark Matter
An Introduction
- Debasish Majumdar(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
In physics and in gauge theories in particular, where one would need to make calculations to obtain measurable results, the group elements are represented as matrices for convenience. The group operation is then matrix multiplications as in linear algebra. In fact, here the rep-resentation of a group means the representation by a set of matrices. For example, a rotation in space is represented by the rotation matri-ces. Thus if a system has rotational symmetry then, that symmetry or symmetry transformation can be represented by the representations of the group to which such rotational matrices belong. The symme-tries upon which the Standard Model and the gauge theory are based can then be formulated in terms of groups and studying such groups Particle Physics Basics 41 (e.g., generators, parameters of the group concerned), one can relate the fundamental particles and their properties and interactions. One can readily recognize the advantages of matrix representation of the group. For instance, the identity element will be a unit matrix 1 and the inverse of a group element (a matrix in this case) is the inverse of that matrix. Also, since matrix multiplication respects associativ-ity, the associativity criterion of the group is satisfied. One can also demonstrate how the closure property of a group is respected by con-sidering a simple example of the rotation matrix representing a rotation by an angle φ of a two-dimensional system of coordinates around the origin such as R = parenleftbigg cos φ − sin φ sin φ cos φ parenrightbigg , (3.47) (this in fact is the rotation of a 2D vector v 1 to another 2D vector v 2 by an angle φ ). All such rotation matrices form a group. Now, after two successive rotations by angles φ and θ (giving a rotation of ( φ + θ )), the above matrix takes the form R = parenleftbigg cos ( φ + θ ) − sin ( φ + θ ) sin ( φ + θ ) cos ( φ + θ ) parenrightbigg .
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