Second-quantized Theory Research Articles

Surface plasmaritons: Wave-mechanical and second-quantized theories

In the wake of recently established quantum theories for bulk and surface plasmons, and for bulk plasmaritons [Jung and Keller, Phys. Rev. A 103, 063501 (2021); Phys. Rev. A 104, 053508 (2021)] wave-mechanical and second-quantized (QED) theories for surface plasmaritons are developed. Our photon wave-mechanical theory is based on the covariant four-potential in the Lorenz gauge, where dynamical equations for surface plasmariton quantum particles consisting of scalar ($S$), longitudinal ($L$), and transverse ($T1$, $T2$) photons and driven by a surface current density sheet are obtained. The second quantized QED theory of surface plasmaritons is formulated on the basis of the Heisenberg equations of motion for the contravariant four-potential annihilation operators. The connection of the surface plasmariton quantum theory to the quasiparticle picture of bulk plasmaritons is elucidated by studying the $T$-photon exponential decay lengths appearing in the description. The shortest decay length (equal on both sides of the surface) characterizes the one-dimensional spatial confinement of the $T$-photon source. Starting from a reexamination (and correction) of the dynamic boundary conditions for a flat jellium-vacuum interface carrying a surface current density, the dispersion relation for surface plasmaritons is obtained. The explicit form of the dispersion relation is derived upon a study of the local field in the selvedge region of the boundary. Fundamental aspects of the dispersion relation are pointed out, modeling the electron dynamics by the microscopic electrodynamics of a quantum well. Remarks on (i) our covariant surface plasmariton quantum theory seen in a broader perspective, (ii) suggestions for applications of the theory, and (iii) a comparison of the key ingredients in our trio of papers on plasmon and plasmariton quantum physics are presented.

Comment on “Bulk and surface plasmons: Wave-mechanical and second-quantized theories”

The effective Lagrangian for plasmons in an electron liquid is given and is further generalized to a mixture of electron and hole liquids, which we take to model the carriers in a semiconductor. The coupling between plasmon and external electromagnetic field is also discussed. These results show the main conclusions in the paper by Jung and Keller [Phys. Rev. A 103, 063501 (2021)] are uncorrect.

Polaritonic quantization in nonlocal polar materials.

In the Reststrahlen region, between the transverse and longitudinal phonon frequencies, polar dielectric materials respond metallically to light, and the resulting strong light-matter interactions can lead to the formation of hybrid quasiparticles termed surface phonon polaritons. Recent works have demonstrated that when an optical system contains nanoscale polar elements, these excitations can acquire a longitudinal field component as a result of the material dispersion of the lattice, leading to the formation of secondary quasiparticles termed longitudinal-transverse polaritons. In this work, we build on previous macroscopic electromagnetic theories, developing a full second-quantized theory of longitudinal-transverse polaritons. Beginning from the Hamiltonian of the light-matter system, we treat distortion to the lattice, introducing an elastic free energy. We then diagonalize the Hamiltonian, demonstrating that the equations of motion for the polariton are equivalent to those of macroscopic electromagnetism and quantize the nonlocal operators. Finally, we demonstrate how to reconstruct the electromagnetic fields in terms of the polariton states and explore polariton induced enhancements of the Purcell factor. These results demonstrate how nonlocality can narrow, enhance, and spectrally tune near-field emission with applications in mid-infrared sensing.

Third quantization of the electromagnetic field

We consider an approach in which the usual wave function in the quadrature representation of mode j of the electromagnetic field is further quantized to produce a field operator. Since the electromagnetic field is already second quantized, this corresponds to an additional or third quantization. The third-quantization approach can be used to perform certain quantum optics calculations in the Heisenberg picture that could only be performed in the Schrodinger picture when using the conventional second-quantized theory. This approach also allows an interesting generalization of quantum optics and quantum electrodynamics that is analogous to symmetry breaking in elementary particle theory. The predictions of the generalized theory could be tested using a proposed photon scattering experiment.

Bulk plasmaritons: Wave-mechanical and second-quantized theories

Inspired by a recently established quantum theory for bulk and surface plasmons [Jung and Keller, Phys. Rev. A 103, 063501 (2021).], wave-mechanical and second-quantized quantum electrodynamic theories for bulk plasmaritons in a homogeneous jellium are presented. Starting from the ``inner'' structure of the transversely polarized classical plasmariton mode, it is argued that a plasmariton quasiparticle may be formed by an always attached pair consisting of a transverse (gauge) photon and a never observable transverse plasmon. A first-quantized plasmariton theory is established from a plasmariton Klein-Gordon equation. It is shown that the plasmariton can be perceived as a diamagnetically driven spin-1 boson quasiparticle. A Lagrangian-Hamiltonian formalism is used to extend the first-quantized plasmariton theory to the second-quantized level. The minimal coupling principle is used to change the free QED theory into a theory coupling the plasmariton to an electromagnetic gauge field. Using the microscopic transverse Lindhard (random phase approximation) dielectric function, it is shown that for small wave numbers, the classical Boltzmann equation part of the Lindhard dielectric function results in a hydrodynamic Brewster branch, the quantization of which follows that of a vectorial Klein-Gordon equation. For plasmariton wave numbers larger than the Fermi wave number, a pure electrostatic model (with no magnetic-field component) results in a harmonic-oscillator description of the plasmariton's polarization field. With inspiration and elements from the Power-Zinau-Wolley description, the Pauli-Fierz theory for particle-tied transverse photons, and the propagator theory of Keller for the spatial localization of transverse photons, a formalism in which the plasmariton quasiparticle is dressed by a cloud of transverse (long-wavelength) photons is developed. The quantized radiated part of the electromagnetic field is described by a free-field quantization of the transverse part of the displacement field (${\mathbf{D}}_{T}$). The total Hamiltonian is diagonalized, and the resulting dispersion relation obtained. The interaction between the transverse radiated and attached photons implies that a strongly localized and charged plasmariton quasiparticle with its tied photons vibrates as a simple harmonic oscillator in the ``external'' radiative transverse photon field.

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.

Generation of Electroweak Interaction by Analogs of Dirac Gamma Matrices Constructed from Operators of the Creation and Annihilation of Spinors

Analogs of Dirac matrices are constructed from the creation and annihilation operators of spinors. They generate a chiral-symmetric version of electroweak fields and other fields. It is shown that the operator considered in the theory of second quantization to be the one of electric charge is a part of the hypercharge operator of the Pati–Salam theory.

Peculiarities of the electron energy spectrum in the Coulomb field of a superheavy nucleus

We consider the peculiarities of the electron energy spectrum in the Coulomb field of a superheavy nucleus and discuss the long history of an incorrect interpretation of this problem in the case of a pointlike nucleus and its current correct solution. We consider the spectral problem in the case of a regularized Coulomb potential. For some special regularizations, we derive an exact equation for the point spectrum in the energy interval (-m,m) and find some of its solutions numerically. We also derive an exact equation for charges yielding bound states with the energy E = -m; some call them supercritical charges. We show the existence of an infinite number of such charges. Their existence does not mean that the oneparticle relativistic quantum mechanics based on the Dirac Hamiltonian with the Coulomb field of such charges is mathematically inconsistent, although it is physically unacceptable because the spectrum of the Hamiltonian is unbounded from below. The question of constructing a consistent nonperturbative second-quantized theory remains open, and the consequences of the existence of supercritical charges from the standpoint of the possibility of constructing such a theory also remain unclear.

Quantum techniques for studying equilibrium in reaction networks

Anderson, Craciun and Kurtz have proved that a stochastically modelled chemical reaction system with mass-action kinetics admits of a stationary distribution when the deterministic model of the same system with mass-action kinetics admits of an equilibrium solution obeying a certain ‘complex balance’ condition. Here we present a proof of their theorem using tools from the theory of second quantization: Fock space, annihilation and creation operators and coherent states. This is an example of ‘stochastic mechanics’, where we take techniques from quantum mechanics and replace amplitudes by probabilities. We explain how the systems studied here can be described using either of two equivalent formalisms: reaction networks, as in chemistry, and stochastic Petri nets, as in some other disciplines.

No drama quantum electrodynamics?

Schr\"{o}dinger (Nature, v.169, 538 (1952)) noted that the complex matter field in the Klein-Gordon equation can be made real by a gauge transform, although charged fields are believed to require complex functions. Surprisingly, the result can be extended to the Dirac equation: three complex components of the Dirac spinor function can be algebraically eliminated, and the remaining component can be made real by a gauge transform. Therefore, the Dirac equation is generally equivalent to one fourth-order partial differential equation for one real function (A. Akhmeteli, J. Math. Phys. v.52, 082303 (2011)). These results both belong in textbooks and can be used for development of new efficient methods of quantum chemistry. The matter field can be algebraically eliminated both in scalar electrodynamics and in spinor electrodynamics in a certain gauge. The resulting equations describe independent dynamics of the electromagnetic field, which permits mathematical simplification and can be useful for interpretation of quantum theory. For example, in the Bohm interpretation, the electromagnetic field can replace the wave function as the guiding field. It is also shown that for these equations, generalized Carleman embedding generates systems of linear equations in the Hilbert space, which look like second-quantized theories and are equivalent to the original nonlinear systems on the set of solutions of the latter. Thus, the relevant local realistic models can be embedded into quantum field theories. These models are equivalent to scalar electrodynamics and spinor electrodynamics, so they correctly describe a large body of experimental data. Although they may need some modifications for better agreement with experiments, they may be of great interest as "no drama quantum theories", as simple (in principle) as classical electrodynamics. Possible issues with the Bell theorem are discussed.

Quantum Catastrophes and Ergodicity in the Dynamics of Bosonic Josephson Junctions

We study rainbow (fold) and cusp catastrophes that form in Fock space following a quench in a Bose Josephson junction. In the Gross-Pitaevskii mean-field theory, the rainbows are singular caustics, but in the second-quantized theory a Poisson resummation of the wave function shows that they are described by well-behaved Airy functions. The structural stability of these Fock space caustics against variations in the initial conditions and Hamiltonian evolution is guaranteed by catastrophe theory. We also show that the long-time dynamics are ergodic. Our results are relevant to the question posed by Berry [M. V. Berry, Nonlinearity 21, T19 (2008)]: Are there circumstances when it is necessary to second quantize wave theory in order to avoid singularities?

Coherent states of accelerated relativistic quantum particles, vacuum radiation and the spontaneous breakdown of the conformal SU(2,2) symmetry

ABSTRACTWe give a quantum mechanical description of accelerated relativistic particles in the framework of coherent states (CSs) of the (3+1)-dimensional conformal group SU(2, 2), with the role of accelerations and ‘kinematical redshift’ played by special conformal transformations (SCTs) and with the role of (proper) time translations played by dilations. The accelerated ground state of first quantization is a CS of the conformal group. We compute the distribution function giving the occupation number of each energy level in and, with it, the partition function , mean energy and entropy , which resemble that of an ‘Einstein solid’. An effective temperature can be assigned to this ‘accelerated ensemble’ through the thermodynamic expression , which leads to a (nonlinear) relation between acceleration and temperature different from Unruh’s (linear) formula. Then we construct the corresponding conformal-SU(2, 2)-invariant second-quantized theory and its spontaneous breakdown when selecting Poincaré-invariant degenerated θ-vacua (namely, CSs of conformal zero modes). SCTs (accelerations) destabilize the Poincaré vacuum and make it radiate.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

IS NO DRAMA QUANTUM THEORY POSSIBLE?

The matter field can be naturally eliminated from the equations of the Klein-Gordon-Maxwell electrodynamics in the unitary gauge. The resulting equations describe independent dynamics of the electromagnetic field: if components of the 4-potential of the electromagnetic field and their first derivatives with respect to time are known in the entire space at some time point, the values of their second derivatives with respect to time can be calculated for the same time point, so the Cauchy problem can be posed, and integration yields the 4-potential in the entire space-time. This surprising result both permits mathematical simplification and can be useful for interpretation of quantum theory. For example, in the Bohm interpretation, the electromagnetic field can replace the wave function as the guiding field. Independent of the interpretation, quantum phenomena can be described in terms of electromagnetic field only. For the system of nonlinear partial differential equations of the Klein-Gordon-Maxwell electrodynamics, a generalized Carleman linearization procedure generates a system of linear equations in the Hilbert space, which looks like a second-quantized theory and is equivalent to the original nonlinear system on the set of solutions of the latter. Similar, but less general results are obtained for the Dirac-Maxwell electrodynamics.

Multiple classical limits in relativistic and nonrelativistic quantum mechanics

The existence of a classical limit describing the interacting particles in a second-quantized theory of identical particles with bosonic symmetry is proved. This limit exists in addition to the previously established classical limit with a classical field behavior, showing that the limit ℏ→0 of the theory is not unique. An analogous result is valid for a free massive scalar field: two distinct classical limits are proved to exist, describing a system of particles or a classical field. The introduction of local operators in order to represent kinematical properties of interest is shown to break the permutation symmetry under some localizability conditions, allowing the study of individual particle properties.

THERMAL VACUUM RADIATION IN SPONTANEOUSLY BROKEN SECOND-QUANTIZED THEORIES ON CURVED PHASE SPACES OF CONSTANT CURVATURE

We construct second-quantized (field) theories on coset spaces of pseudo-unitary groups U(N+,N-). The existence of degenerated quantum vacua (coherent states of zero modes) leads to a breakdown of the original pseudo-unitary symmetry. The action of some spontaneously broken symmetry transformations destabilize the vacuum and make it to radiate. We study the structure of this thermal radiation for curved phase spaces of constant curvature: complex projective spaces ℂPN - 1 = SU(N)/U(N - 1) and open complex balls ℂDN - 1 = SU(1,N - 1)/U(N - 1). Positive curvature is related to generalized Fermi–Dirac (FD) statistics, whereas negative curvature is connected with generalized Bose–Einstein (BE) statistics, the standard cases being recovered for N = 2. We also make some comments on the contribution of the vacuum (dark) energy to the cosmological constant and the phenomenon of inflation.

GAUGE BOSON FAMILIES IN GRAND UNIFIED THEORIES OF FERMION MASSES: $E_6^4 \rtimes S_4$

In third quantization the origin of fermion families is easy to understand: the electron field, the muon field and the tau field are identical fields in precisely the same sense as three electrons are identical and indistinguishable particles of a theory of second quantization. In both cases, the permutation of these fields or particles leaves the Lagrangian invariant. One can also extend the concept of family to gauge bosons. This can be obtained through the semidirect product of the gauge group with the group of permutations of n objects. In this paper we have studied the group [Formula: see text]. We explain why we have chosen E6 as fundamental gauge group factor and why we start with a model with four gauge boson/fermion families to accommodate and to fit the Standard Model with only three fermion families. We suggest a possible symmetry breaking pattern of [Formula: see text] that could explain quark, lepton and neutrino masses and mixings.

Fractionalization of angular momentum at finite temperature around a magnetic vortex

Ambiguities in the definition of angular momentum of a quantum-mechanical particle in the presence of a magnetic vortex are reviewed. We show that the long-standing problem of the adequate definition is resolved in the framework of the second-quantized theory at nonzero temperature. Planar relativistic Fermi gas in the background of a point-like magnetic vortex with arbitrary flux is considered, and we find thermal averages, quadratic fluctuations, and correlations of all observables, including angular momentum, in this system. The kinetic definition of angular momentum is picked out unambiguously by the requirement of plausible behaviour for the angular momentum fluctuation and its correlation with fermion number.

Stochastic theory of self-induced transparency: linearized approach

Making use of c-number stochastic theory and soliton perturbation theory, we study the quantum fluctuations of a self-induced transparency (SIT) soliton propagating through a lossless two-level medium. Considering the fluctuations as small corrections to the classical soliton, we are able to construct and solve four stochastic equations that govern the evolution of four soliton parameters: photon number (intensity), phase, timing, and momentum (frequency). We find excellent agreement between our stochastic theory of SIT solitons and the second-quantized theory of Lai and Haus [Phys. Rev. A 42, 2925 (1990)].

Dirac spinors and tachyon quantization

Dirac spinors for spin-1/2 free tachyons are derived from the Dirac and Klein-Gordon equations in terms of the standard helicity representation of the Poincaré group and superluminal transformations (SLTs). It is shown that the proposed theory of second quantization is Lorentz invariant without inverting the spin-statistics relationship and without introducing indefinite pseudotachyonic states. Second quantized theory of spin-1/2 tachyons in the time-energy representation shows that tachyons are localized in time and that the most natural space for the description of tachyons is T4-space. It has been emphasized that on passing from R4-space (bradyons) to T4-space (tachyons) via either, i.e., complex or real SLTs, tachyons behave in T4-space exactly as bradyons do in R4-space. T4-space can be visualized as an internal space for a subluminal observer while R4-space appears as an internal space for a superluminal observer. PACS Nos.: 14.80K, 03.30t

Quasinormal-mode expansion for waves in open systems

An open system is not conservative because energy can escape to the outside. As a result, the time-evolution operator is not Hermitian in the usual sense and the eigenfunctions (factorized solutions in space and time) are no longer normal modes but quasinormal modes (QNMs) whose frequencies $\ensuremath{\omega}$ are complex. Qausinormal-mode analysis has been a powerful tool for investigating open systems. Previous studies have been mostly system specific, and use a few QNMs to provide approximate descriptions. Here the authors review developments that lead to a unifying treatment. The formulation leads to a mathematical structure in close analogy to that in conservative, Hermitian systems. Hence many of the mathematical tools for the latter can be transcribed. Emphasis is placed on those cases in which the QNMs form a complete set and thus give an exact description of the dynamics. In situations where the QNMs are not complete, the ``remainder'' is characterized. Applications to optics in microspheres and to gravitational waves from black holes are given as examples. The second-quantized theory is sketched. Directions for further development are outlined.