Nuclear Creation and Annihilation Operators and Electromagnetic Multipole Fields

Abolfazl Safaei Bezgabadi, M. A. Bolorizadeh

In this paper, we present a quantum theory for field propagation through a three dimensional dielectric when the third order dispersion and the attenuation coefficients are included. A unique Lagrangian is defined leading to the correct equation of motion and the classical Hamiltonian. It is assumed that the dielectric has a combination of inhomogeneity, dispersion and nonlinearity. By employing constraint quantization approach the final Hamiltonian is expanded in terms of properly defined annihilation and creation operators. We obtain the quantum fields (quantum photon-polaritons fields) for propagation through the dielectric in the presence of the third order dispersion and the attenuation coefficients by using these annihilation and creation operators. The number operator in the final Hamiltonian indicates the number of photon-polaritons in the medium. The nonlinear part of the Hamiltonian could be derived by defining displacement field operator in terms of annihilation and creation operators. As a simple example, the present quantum theory is applied to field propagation through a one dimensional slab.

Hugo Reinhardt

Steven Weinberg

Up to this point we have not had much to say about the detailed structure of the Hamiltonian operator H . This operator can be defined by giving all its matrix elements between states with arbitrary numbers of particles. Equivalently, as we shall show here, any such operator may be expressed as a function of certain operators that create and destroy single particles. We saw in Chapter 1 that such creation and annihilation operators were first encountered in the canonical quantization of the electromagnetic field and other fields in the early days of quantum mechanics. They provided a natural formalism for theories in which massive particles as well as photons can be produced and destroyed, beginning in the early 1930s with Fermi's theory of beta decay. However, there is a deeper reason for constructing the Hamiltonian out of creation and annihilation operators, which goes beyond the need to quantize any pre-existing field theory like electrodynamics, and has nothing to do with whether particles can actually be produced or destroyed. The great advantage of this formalism is that if we express the Hamiltonian as a sum of products of creation and annihilation operators, with suitable non-singular coefficients, then the S -matrix will automatically satisfy a crucial physical requirement, the cluster decomposition principle, which says in effect that distant experiments yield uncorrelated results. Indeed, it is for this reason that the formalism of creation and annihilation operators is widely used in non-relativistic quantum statistical mechanics, where the number of particles is typically fixed.

Péter R. Śurján

Christopher Witte

Morgan, Peter

A random field that is empirically equivalent to the quantized electromagnetic field is constructed. A mapping between the creation and annihilation operator algebras of a random field and of the quantized electromagnetic field provides a functor between the algebras and the Hilbert spaces generated by the vacuum states over those algebras. The functor inevitably does not extend to a functorial relationship between the local algebras generated by the random field and by the quantized electromagnetic field, but the empirical content provided by the vacuum state restores an empirical equivalence through the Hilbert spaces. The isomorphism from one creation and annihilation algebra to the other is not translation invariant because it depends on mapping positive frequency modes of one helicity to equivalent negative frequency modes, but the two theories taken independently are presented in equally well-defined and manifestly Lorentz and translation covariant ways.