Charge (physics)

In physics, a charge is any of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges correspond to the time-invariant generators of a symmetry group, and specifically, to the generators that commute with the Hamiltonian. Charges are often denoted by , and so the invariance of the charge corresponds to the vanishing commutator , where H is the Hamiltonian. Thus, charges are associated with conserved quantum numbers; these are the eigenvalues of the generator Q . A "charge" can also refer to a point-shaped object with an electric charge and a position, such as in the method of image charges.

In physics, a charge is any of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges correspond to the time-invariant generators of a symmetry group, and specifically, to the generators that commute with the Hamiltonian. Charges are often denoted by , and so the invariance of the charge corresponds to the vanishing commutator , where H is the Hamiltonian. Thus, charges are associated with conserved quantum numbers; these are the eigenvalues of the generator Q . A "charge" can also refer to a point-shaped object with an electric charge and a position, such as in the method of image charges. The word "charge" is often used as a synonym for both the generator of a symmetry, and the conserved quantum number (eigenvalue) of the generator. Thus, letting the upper-case letter Q refer to the generator, one has that the generator commutes with the Hamiltonian . Commutation implies that the eigenvalues (lower-case) q are time-invariant: {{tmath|1= \textstyle \frac{ \mathrm{d}q }{ \mathrm{d}t } = 0 }}. So, for example, when the symmetry group is a Lie group, then the charge operators correspond to the simple roots of the root system of the Lie algebra; the discreteness of the root system accounting for the quantization of the charge. The simple roots are used, as all the other roots can be obtained as linear combinations of these. The general roots are often called raising and lowering operators, or ladder operators. The charge quantum numbers then correspond to the weights of the highest-weight modules of a given representation of the Lie algebra. So, for example, when a particle in a quantum field theory belongs to a symmetry, then it transforms according to a particular representation of that symmetry; the charge quantum number is then the weight of the representation. 2\otimes\overline{2}=3\oplus 1 . That is, the product of two (Lorentz) spinors is a (Lorentz) vector and a (Lorentz) scalar. Note that the complex Lie algebra sl(2,C) has a compact real form su(2) (in fact, all Lie algebras have a unique compact real form). The same decomposition holds for the compact form as well: the product of two spinors in su(2) being a vector in the rotation group O(3) and a singlet. The decomposition is given by the Clebsch–Gordan coefficients. A similar phenomenon occurs in the compact group SU(3), where there are two charge-conjugate but inequivalent fundamental representations, dubbed 3 and , the number 3 denoting the dimension of the representation, and with the quarks transforming under 3 and the antiquarks transforming under . The Kronecker product of the two gives 3\otimes\overline{3}=8\oplus 1 . That is, an eight-dimensional representation, the octet of the eight-fold way, and a singlet. The decomposition of such products of representations into direct sums of irreducible representations can in general be written as \Lambda \otimes \Lambda' = \bigoplus_i \mathcal{L}_i \Lambda_i for representations \Lambda . The dimensions of the representations obey the "dimension sum rule": d_\Lambda \cdot d_{\Lambda'} = \sum_i \mathcal{L}_i d_{\Lambda_i}. Here, d_\Lambda is the dimension of the representation , and the integers \mathcal{L} being the Littlewood–Richardson coefficients. The decomposition of the representations is again given by the Clebsch–Gordan coefficients, this time in the general Lie-algebra setting.

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