First-quantized Theory Research Articles

Bulk and surface plasmons: Wave-mechanical and second-quantized theories

In this paper we formulate the quantum theory for longitudinal (rotational free) collective electron modes, the so-called plasmons. Starting from the jellium dispersion relation, a first-quantized wave-mechanical theory is established. We show that plasmon quantum (quasi)particles are charged bosons described by complex Klein-Gordon fields satisfying a ``relativistic'' scalar Klein-Gordon equation in which the speed of light $c$ is replaced by a velocity $a$ on the order of the Fermi velocity. Based on a formally ($c\ensuremath{\rightarrow}a$) covariant description, the first-quantized theory is extended to the second-quantized level via a Lagrangian formalism. We show how the second-quantized theory enables the study of the plasmon quantum particles interaction with an electromagnetic gauge field via a modification of the free plasmon Lagrangian density by the minimal coupling principle. Utilizing the Weyl expansion, first- and second-quantized theories for surface-plasmon quantum particles are established in close analogy to those of the bulk plasmon quantum particle. This work opens up for the study of, e.g., squeezed, entangled, and coherent plasmon states in line with what has been studied theoretically and experimentally for photons for many years.

Wave function of the photon in a curved spacetime

It is known that the classical variables are replaced by operators for the dynamics of the electromagnetic field, thus the second quantization is traditionally used. There is a significant question in modern theoretical physics area: “Why is there no fully developed first-quantized theory of photons?” In the present work, we argue that a photon wave function can be obtained if one uses the massless Duffin-Kemmer-Petiau (mDKP) equation. Thus, we focus on exact solutions of the mDKP equation the Robertson–Walker type spacetime via the Teleparallel Theory (TPT) of Gravity. The mDKP equation is generally known as the spinor formulation of the photons. We also focus on solutions of the Maxwell equations (MEs) to verify the equivalence between the mDKP and the ME formulations of the photons. It is shown that photons present an oscillatory behavior.

First-quantized theory of expanding universe from field quantization in mini-superspace

We propose an improved variant of the third-quantization scheme, for the spatially homogeneous and isotropic cosmological models in Einstein gravity coupled with a neutral massless scalar field. Our strategy is to specify a semi-Riemannian structure on the mini-superspace and to consider the quantum Klein-Gordon field on the mini-superspace. Then, the Hilbert space of this quantum system becomes inseparable, which causes the creation of infinite number of universes. To overcome this issue, we introduce a vector bundle structure on the Hilbert space and the connection of the vector bundle. Then, we can define a consistent unitary time evolution of the quantum universe in terms of the connection field on the vector bundle. By doing this, we are able to treat the quantum dynamics of a single-universe state. We also find an appropriate observable set constituting the CCR-algebra, and obtain the Schr\"odinger equation for the wave function of the single-universe state. We show that the present quantum theory correctly reproduces the classical solution to the Einstein equation.

A SIMPLER SOLUTION OF THE NON-UNIQUENESS PROBLEM OF THE COVARIANT DIRAC THEORY

Although the standard generally covariant Dirac equation is unique in a topologically simple spacetime, it has been shown that it leads to non-uniqueness problems for the Hamiltonian and energy operators, including the non-uniqueness of the energy spectrum. These problems should be solved by restricting the choice of the Dirac gamma field in a consistent way. Recently, we proposed to impose the value of the rotation rate of the tetrad field. This is not necessarily easy to implement and works only in a given reference frame. Here, we propose that the gamma field should change only by constant gauge transformations. To get that situation, we are naturally led to assume that the metric can be put in a space-isotropic diagonal form. When this is the case, it distinguishes a preferred reference frame. We show that by defining the gamma field from the "diagonal tetrad" in a chart in which the metric has that form, the uniqueness problems are solved at once for all reference frames. We discuss the physical relevance of the metric considered and our restriction to first-quantized theory.

Light-cone gauge superstring field theory and dimensional regularization

We propose a dimensional regularization scheme in the light-cone gauge NSR superstring field theory to regularize the divergences caused by the colliding supercurrents inserted at the interaction points. We study the tree amplitudes and show that our scheme actually regularize the divergences. We examine the four-point amplitudes for the NS-NS closed strings and find that the results in the first-quantized theory are reproduced without any counterterms.

The First-Quantized Theory of Photons

In near-field optics and optical tunnelling theory, photon wavemechanics, i.e. the first-quantized theory of photons, allowsus to address the spatial field localization problem in a flexiblemanner which links smoothly to classical electromagnetics. We developphoton wave mechanics in a rigorous and unified way, based on whichfield quantization is obtained in a new way.

Massive relativistic particle model with spin from free two-twistor dynamics and its quantization

We consider a relativistic particle model in an enlarged relativistic phase space, M{sup 18}=(X{sub {mu}},P{sub {mu}},{eta}{sub {alpha}},{eta}{sub {alpha}},{sigma}{sub {alpha}},{sigma}{sub {alpha}},e,{phi}), which is derived from the free two-twistor dynamics. The spin sector variables ({eta}{sub {alpha}},{eta}{sub {alpha}},{sigma}{sub {alpha}},{sigma}{sub {alpha}}) satisfy two second class constraints and account for the relativistic spin structure, and the pair (e,{phi}) describes the electric charge sector. After introducing the Liouville one-form on M{sup 18}, derived by a nonlinear transformation of the canonical Liouville one-form on the two-twistor space, we analyze the dynamics described by the first and second class constraints. We use a composite orthogonal basis in four-momentum space to obtain the scalars defining the invariant spin projections. The first-quantized theory provides a consistent set of wave equations, determining the mass, spin, invariant spin projection and electric charge of the relativistic particle. The wave function provides a generating functional for free, massive higher spin fields.

Review Article

In the quantum physics of near-field optics and optical tunneling light–matter interactions are studied on a length scale (much) smaller than the wavelength of light, and questions regarding the possibilities for strong spatial localization of electromagnetic fields are here in focus. Some of these questions relate to the spatial resolution problem in optics, a problem which has gained considerable attention in connection to optical investigations of mesoscopic systems. Optics beyond the classical diffraction limit has renewed our interest in the various theories for spatial localization of single photons. In the present work aspects of these theories of particular importance for light–matter interaction on the microscopic and mesoscopic length scales are reviewed. Photon wave mechanics, i.e. the (rather unknown) first quantized theory of the photon, allows us to address the spatial field localization problem in a flexible manner which links smoothly to classical electromagnetics. The wave mechanics of free photons is discussed both in the momentum–time domain (Part A) and in the space–time domain (Part B). The first-quantized theory of spatial localization of photons subjected to field–matter interaction is described in Part C, paying particular attendance to the so-called photon energy wave function concept. In Part D, the spatial localization of photons are studied on a field theoretic (second-quantized) basis. The coarse-grained photon localization theory and the spatial localization perceived in various representations (gauges) here are core issues. In the two last parts of the review I describe photon fields in near-field optics (Part E), and the optical tunneling phenomenon, here seen as a fingerprint of weak photon localizability (Part F).

OPTICAL NEAR-FIELD INTERACTION ON THE BASIS OF PHOTON WAVE MECHANICS

Near-field optical aspects of classical electrodynamics are brought into focus by dividing the electromagnetic field into its transverse and longitudinal vector-field parts. A transverse electromagnetic propagator formalism thereafter is used to study the field-matter interaction in the transverse current density domain, the birth domain of the photon. Subsequently, a brief summary of photon wave mechanics, the first-quantized theory of the photon, is given, paying particular attention to the dynamics in the near-field zone of matter (atom, molecule, mesoscopic particle). In the wake of a discussion of the relativistic transformation properties of the covariant photon field matrix the photon energy wave function is introduced. In a central section, photon wave mechanics and near-field optics are brought in contact, and the photon embryo state, the polychromatic photon concept, and the quantum mechanical theory for the transverse one-photon current density discussed.

Optical tunneling: a fingerprint of the lack of photon localizability

It is argued that the photon tunneling process originates in an inability to localize photons completely in space. Seen in this perspective, optical tunneling experiments might allow one to obtain rich information on the photon localizability problem, that until now has been studied mainly theoretically. A rigorous analysis of the electrodynamics in the near-field zone of matter enables us to identify the transverse vector field and subsequently to use it to construct a first-quantized space–time theory for the photon birth process and to determine the source region of the photon. The present theory shows that optical tunneling never appears outside the photon’s source domain, and it is shown that an apparently superluminal response occurs as a consequence of the lack of complete photon localizability. No fundamental velocity is attached to this effect, which stems solely from quantum nonlocality. Starting from the Riemann–Silberstein vectors, which permit the introduction of a space–time description of a free polychromatic photon’s so-called energy wave function, a theoretical investigation of the near-field scattering of a single photon from a mesoscopic or microscopic (molecular, atomic) particle is presented. Inasmuch as the photon tunneling phenomenon appears to be an indispensable part of the near-field scattering process, it might be possible to establish a rigorous first-quantized theory of one-photon tunneling between macroscopic molecular solids by adding the tunneling contributions from the individual molecules.

Near-field optics: The nightmare of the photon

A first-quantized theory describing the birth process of a single photon in the near-field zone of a pointlike particle (atom, molecule, etc.) is established. The space-time description of the photon energy wave function embryo is shown to be useful for the understanding of the role played by (unborn) photons in near-field interactions where the spatial confinement of light plays a crucial role.

A comment on reparametrization invariance in witten's open string field theory

It is shown that Witten's theory of interacting open strings, which seems to depend on an arbitrary choice of a string midpoint, does in fact possess reparametrization invariance, which is hidden by a midpoint-dependent choice of phase in the transition from first-quantized theory to field theory. An equivalent version of this theory is constructed, which is manifestly reparametrization invariant, but is not manifestly local in spacetime.

First-quantized para-particle theory

The general formalism of first-quantized para-particle theory is presented in a way that clearly exposes the more important group-theoretic foundations. This is then used to give a simple and straightforward solution to a problem concerning possible changes in a particle's statistics through collisions, first posed by Pauli in 1927, and to show that para-particle statistics are different from those obeyed by particles in a “para-gas.” These arguments accord with the overall aim of demonstrating that para-particle theory is not as theoretically impenetrable as is usually thought, but in fact possesses a kind of generalized elegance.

Strong-coupling theory of a Dirac particle and a charged scalar field

We formulate the strong-coupling theory for a system of a Dirac particle interacting with a charged scalar field. The degrees of freedom of the Dirac particle are taken into account in the first-quantized theory. It is shown that the strong-coupling expansion ($\frac{1}{g}$ expansion) becomes possible if the mass of the Dirac particle is of order ${g}^{2}$. Recoil effects appear in the meson-fermion scattering amplitudes to leading order in the $\frac{1}{g}$ expansion and in the isobar energy levels to the next-to-leading order.

Correspondence between the first- and second-quantized theories of paraparticles

It is shown that the parabosons and parafermions of the second-quantized parafield theory are equivalent to the finite-order paraparticles of the first-quantized theory. We follow the prescription of Hartle and Taylor which eliminates the redundancy of the generalized ray from the first-quantized formalism. For each statistical type of particle this allows us to establish an isomorphism between the state spaces of the two theories, in such a way that all corresponding observables have identical matrix elements for all states. This means that any system of finite-order particles can equally well be described by either the first- or the second-quantized formalism, in the same sense as applies to ordinary bosons and fermions. We analyze the different conclusions reached by other authors and explain the reasons for our disagreement.

Parastatistics and a unified theory of identical particles

The simplest parafermi model is shown to be equivalent to a theory with two types of ordinary fermions that are dynamically indistinguishable. Also, a unified theory of identical particles is proposed that encompasses theories arising both from the permutation-symmetry approach of the first-quantized theory and from the parastatistics commutation-relation approach of second-quantized theory. Using this theory cluster properties are shown to be compatible with parastatistics.