Operators charge conjugation

For a discussion of the transformation of the field operators under improper Lorentz transformations and discrete symmetry operations such as charge conjugation, see ... 

Invariance of Quantum Electrodynamics under Discrete Transformations.—In the present section we consider the invariance of quantum electrodynamics under discrete symmetry operations, such as space-inversion, time-inversion, and charge conjugation. 

The above transformation properties of the current operator make quantum electrodynamics invariant under the operation Ue, usually called charge conjugation, provided... 

Charge conjugate operator, 545 Charge-current density renormalized, 597... 

Use of the charge-exchange mechanism, reaction (8.13), to produce antihydrogen was first proposed by Deutch et al. (1986), and subsequently it was shown that the cross section for this process could be obtained by applying the charge conjugation and time reversal operators to the process of positronium formation in positron-hydrogen collisions (Humberston et al., 1987, and see section 4.2). Under time reversal, the positronium formation process equation (4.5) becomes... 

Among the mostly fundamentally assumed symmetries in nature are the Lorentz invariance and the validity of the CPT theorem which demands an invariance of nature under simultaneous charge conjugation (C), parity operation... 

In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

H and have been formulated in the commutator form, which ensures the correct behaviour under charge conjugation (Kalian 1958), although we will not dwell on this point in the following. The Hamiltonian commutes with the charge operator... 

The operator C is called charge conjugation. As an antiunitary transformation it is a symmetry transformation, that is, all transition probabilities are left invariant. 

A charge conjugation does not change the sign of the scalar potential relative to the other terms in the Dirac operator. Hence a scalar potential acts in the same way on electrons and positrons. If it is repulsive for electrons, so it is for positrons. 

It follows from a charge conjugation that for 0 < 7 < 1 the operator has the eigenvalues — n)- The corresponding radial eigenfunctions can be expressed in terms of hypergeometric functions ... 

The unitary transformation is known in exact analytic form only in the free particle case, when the operation of charge conjugation gives back the same Dirac operator, and is then known as the free-particle Foldy-Wouthuysen transformation [111]. For a general potential, block diagonalization can only be... 

The operators (166) and (167) are finite at this point, but they do not yet show the correct behavior under charge conjugation. Each individual field operator (161) transforms as [125,124]... 

T =transposition) with an unobservable phase 7), so that charge conjugation reorders the field operators in the current density,... 

Lorentz transformations is also invariant under the combined operations of charge conjugation, C, space inversion, P, and time reversal, T, taken in any order. 

The Hiickel model as applied to polyenes possesses a symmetry known as alternancy symmetry, since the polyene system can be subdivided into two sublattices such that the Hiickel resonance integral involves sites on different sublattices. In such systems, the Hamiltonian remains invariant when the creation and annihilation operators at each site are interchanged with a phase of +1 for sites on one sublattice and a phase of -1 on sites of the other. Even in interacting models this symmetry exists when the system is half-filled. The alternancy symmetry is known variously as electron-hole symmetry or charge-conjugation symmetry [16]. 

This therefore corresponds to exchanging particles and antiparticles. Such a symmetry operation is called the charge conjugation and denoted as C symmetry. This symmetry will be not marked in the wave function symbol (because as a rule, we are dealing with matter, not antimatter), but we will want to remember it later. Sometimes it may turn out unexpectedly to be useful (see Chapter 13, p. 820). After Wu s experiment, physicists tried to save the hypothesis that what is conserved is the CP symmetry i.e., the product of charge conjugation and inversion. However, analysis of experiments with the meson K decay has shown that even this symmetry is approximate (although the deviation is extremely small). 

CP invariance The symmetry generated by the combined operation of changing charge conjugation (Q and parity (P). CP violation occurs in weak interactions in kaon decay and in B-mesons. See also CPT theorem time reversal. 

If a Hamiltonian has particle-hole (or charge-conjugation) S3unmetry then it is invariant under the transformation of a particle into a hole under the action of the particle-hole operator, J ... 

This is the combined operation of charge conjugation and a rotation of it about the 2 axis of isospace ... 

---