Using the cross product as a Lie bracket, the algebra of 3-dimensional real vectors is a Lie algebra isomorphic to the Lie algebras of SU(2) and SO(3). The structure constants are , where is the antisymmetric Levi-Civita symbol.
In mathematics, the structure constants or structure coefficients of an algebra over a field are the coefficients of the basis expansion (into linear combination of basis vectors) of the products of basis vectors.
Because the product operation in the algebra is bilinear, by linearity knowing the product of basis vectors allows to compute the product of any elements (just like a matrix allows to compute the action of the linear operator on any vector by providing the action of the operator on basis vectors).
Therefore, the structure constants can be used to specify the product operation of the algebra (just like a matrix defines a linear operator). Given the structure constants, the resulting product is obtained by bilinearity and can be uniquely extended to all vectors in the vector space, thus uniquely determining the product for the algebra.
Structure constants are used whenever an explicit form for the algebra must be given. Thus, they are frequently used when discussing Lie algebras in physics, as the basis vectors indicate specific directions in physical space, or correspond to specific particles (recall that Lie algebras are algebras over a field, with the bilinear product being given by the Lie bracket, usually defined via the commutator).
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In mathematics, the structureconstants or structure coefficients of an algebra over a field are the coefficients of the basis expansion (into linear combination...
physical constants, such as the fine-structureconstant α, which characterizes the strength of the electromagnetic interaction. Physical constant, as discussed...
physical constant is sometimes used to refer to some universal dimensionless constants. Perhaps the best-known example is the fine-structureconstant, α, which...
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discovery of energy quanta". In metrology, the Planck constant is used, together with other constants, to define the kilogram, the SI unit of mass. The SI...
the dimensionful constants (the elementary charge, the Planck constant, and the speed of light) of which the fine-structureconstant is composed is responsible...
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"Recommended values of the fundamental physical constants" (PDF). Symbols, Units, Nomenclature and Fundamental Constants in Physics. p. 54. Lalanne, J.-R.; Carmona...
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Arnold Sommerfeld, introducing the fine-structureconstant. The gross structure of line spectra is the structure predicted by the quantum mechanics of non-relativistic...
conditions eq. 6m are the only ones that the structureconstants must satisfy. But among the constants admissible by these conditions, there are equivalent...
such that the physical constants completely define the system of units and the relevant physical laws contain no conversion constants. While natural unit...
electron spin g-factor were the most accurately measured physical constants. The constant is expressed for either hydrogen as R H {\displaystyle R_{\text{H}}}...
underlying vector space, one may define the algebra in terms of structureconstants for multiplication: e i ∇ e j = ∑ k μ i j k e k {\displaystyle e_{i}\nabla...
involve several independent tensor structures, the structureconstants and conformal blocks depend on these tensor structures, and the primary field O p {\displaystyle...
five constants, Planck units. However, not all physical constants can be normalized in this fashion. For example, the values of the following constants are...
system, the a and b constants are equal, and the angles are 60°, 90°, and 90°, so the geometry is determined by the a and c constants alone. The lattice...
r_{\mathrm {e} }} ). Any one of these constants can be written in terms of any of the others using the fine-structureconstant α {\displaystyle \alpha } : r e...
is found at: "–: Electric constant, ε0". NIST reference on constants, units, and uncertainty: Fundamental physical constants. NIST. Retrieved 22 January...
field is therefore an entire function worth of coupling constants. These coupling constants are not pre-determined, adjustable, or universal parameters;...
This is a list of common physical constants and variables, and their notations. Note that bold text indicates that the quantity is a vector. List of letters...
downwards or upwards, depending on the type of tensor that is used. The structureconstants ( γ a b c {\displaystyle {\gamma _{ab}}^{c}} ) of a Lie algebra are...
In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate electronic energy levels and the resulting splittings in those...
derive equilibrium constant values from experimental data use association constants for both acids and bases. Because stability constants for a metal-ligand...
fields of a rational two-dimensional conformal field theory, whose structureconstants Nν λμ describe fusion of primary fields. In terms of the modular...
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