Relativistic Quantum Mechanical Equation Research Articles

QCD on the Light-Front

Light-Front Hamiltonian theory provides a rigorous frame-independent framework for solving nonperturbative QCD. The valence Fock-state wavefunctions of the light-front QCD Hamiltonian satisfy a single-variable relativistic equation of motion, analogous to the nonrelativistic radial Schr\"odinger equation, with an effective confining potential U which systematically incorporates the effects of higher quark and gluon Fock states. Remarkably, the potential U has a unique form of a harmonic oscillator potential if one requires that the chiral QCD action remains conformally invariant. A mass gap arises when one extends the formalism of de Alfaro, Fubini and Furlan to light-front Hamiltonian theory. The valence LF meson wavefunctions for zero quark mass satisfy a single-variable relativistic equation of motion in the invariant variable $\zeta^2=b^2_\perp x(1-x)$, which is conjugate to the invariant mass squared. The result is a nonperturbative relativistic light-front quantum mechanical wave equation which incorporates color confinement and other essential spectroscopic and dynamical features of hadron physics, including a massless pion for zero quark mass and linear Regge trajectories with the same slope in the radial quantum number n and orbital angular momentum L. The corresponding light-front Dirac equation provides a model of nucleons. The same light-front equations arise from the holographic mapping of the soft-wall model modification of AdS_5 space with a unique dilaton profile to QCD (3+1) at fixed light-front time. Light-front holography thus provides a precise relation between amplitudes in the fifth dimension of AdS space and light-front wavefunctions. We also discuss the implications of the underlying conformal template of QCD for renormalization scale-setting, and the implications of light-front quantization for the value of the cosmological constant.

Classical limit of relativistic quantum mechanical equations in the Foldy-Wouthuysen representation

It is shown that, under the Wentzel-Kramers-Brillouin approximation conditions, using the Foldy-Wouthuysen (FW) representation allows the problem of finding a classical limit of relativistic quantum mechanical equations to be reduced to the replacement of operators in the Hamiltonian and quantum mechanical equations of motion by the respective classical quantities.

A Spectral Integral Equation Solution of the Gross-Pitaevskii Equation

We present sixteen-component values "sedeons", generating associative noncommutative space-time algebra. The generalized second-order and first-order equations of relativistic quantum mechanics based on sedeonic wave function and sedeonic space-time operators are proposed. We also discuss the description of fields with massive quantum on the basis of second-order and first-order equations for sedeonic potentials.

Octonic second-order equations of relativistic quantum mechanics

We demonstrate a generalization of relativistic quantum mechanics using eight-component value “octons” that generate an associative noncommutative spatial algebra. It is shown that the octonic second-order equation for the eight-component octonic wave function, obtained from the Einstein relation for energy and momentum, describes particles with spin 1/2. It is established that the octonic wave function of a particle in the state with defined spin projection has a specific spatial structure that takes the form of an octonic oscillator with two spatial polarizations: longitudinal linear and transverse circular.

Strong-field approximation to the relativistic channeling of electrons in the presence of electromagnetic waves

We present a study of the interaction of a relativistically planar channeled electron with an intense electromagnetic field. Using a S-Matrix approach in the Strong Field Approximation, it is shown that the crystal periodicity affects drastically the excitation process, suppressing the possibility of multiphoton absorption except for some particular cases. This selective excitation opens the possibility to control the dynamics of the channeling process by means of an external field. Explicit expressions for the S-matrix N-photon excitation rates together with the corresponding conservation laws are obtained from the relativistic quantum mechanical Dirac equation.

Historical roots of gauge invariance

Gauge invariance is the basis of the modern theory of electroweak and strong interactions (the so called Standard Model). The roots of gauge invariance go back to the year 1820 when electromagnetism was discovered and the first electrodynamic theory was proposed. Subsequent developments led to the discovery that different forms of the vector potential result in the same observable forces. The partial arbitrariness of the vector potential A brought forth various restrictions on it. div A = 0 was proposed by J. C. Maxwell; 4-div A = 0 was proposed L. V. Lorenz in the middle of 1860's . In most of the modern texts the latter condition is attributed to H. A. Lorentz, who half a century later was one of the key figures in the final formulation of classical electrodynamics. In 1926 a relativistic quantum-mechanical equation for charged spinless particles was formulated by E. Schrodinger, O. Klein, and V. Fock. The latter discovered that this equation is invariant with respect to multiplication of the wave function by a phase factor exp(ieX/hc) with the accompanying additions to the scalar potential of -dX/cdt and to the vector potential of grad X. In 1929 H. Weyl proclaimed this invariance as a general principle and called it Eichinvarianz in German and gauge invariance in English. The present era of non-abelian gauge theories started in 1954 with the paper by C. N. Yang and R. L. Mills.

Time-like flows of energy momentum and particle trajectories for the Klein-Gordon equation

The Klein-Gordon equation is interpreted in the de Broglie-Bohm manner as a single-particle relativistic quantum mechanical equation that defines unique time-like particle trajectories. The particle trajectories are determined by the conserved flow of the intrinsic energy density which can be derived from the specification of the Klein-Gordon energy-momentum tensor in an Einstein-Riemann space. The approach is illustrated by application to the simple single-particle phenomena associated with square potentials.

Photon wave-particle duality and virtual electromagnetic waves

The question of the relation between the amplitude of the photon vector potential and its angular frequency is analyzed. The analogy between the relativistic quantum mechanical equations for a massles particle and those governing the photon vector potential appears clearly. Finally, the virtual electromagnetic waves associated with the photon and predicted by de Broglie, Bohr, and other appear naturally as a result of the photon vector potential quantification.

Relation between relativistic quantum mechanics and classical electromagnetic field theory.

The objective of this report is twofold. In the first place, it aims to demonstrate that a four-dimensional local U(1) gauge invariant relativistic quantum mechanical Dirac-type equation is derivable from the equations for the classical electromagnetic field. In the second place, the transformational consequences of this local U(1) invariance are used to obtain solutions of different Maxwell equations.

The two-body plus potential problem between quantum field theory and relativistic quantum mechanics (spinless case)

Starting with a pair of integrodifferential Bethe-Salpeter equations for two spinless particles interacting mutually and with an external static potential, we obtain a pair of compatible and separable coupled Klein-Gordon equations, between which the unwanted relative time variable can be easily eliminated. The method we use is a generalization of that proposed by Sazdijan for the two-particle problem, and the resulting equations are a generalization of the well-known Droz-Vincent-Todorov-Komar equations of relativistic quantum mechanics. We examine the instantaneous approximation and we test our methods in a simple case (Bethe-Salpeter kernel given by a single scalar particle exchange graph).

Bethe-Salpeter equation: Spin-0-spin-1/2 and spin-0-spin-0 bound states.

The Bethe-Salpeter equation is cast into a form in which the relationship between the relativistic quantum-mechanical two-body equation and the Bethe-Salpeter equation is established for a bound system whose constituents are either (i) a spinless particle and a particle of spin \textonehalf{} or (ii) both particles spinless. The connection between these equations and their nonrelativistic counterpart is immediately apparent from these forms. To lay the groundwork for physical applications for such systems in which the Coulomb interaction is the main source of the binding, scalar quantum electrodynamics in the radiation gauge is developed. Using the foregoing, the perturbation theory for the Bethe-Salpeter equation is formulated for electromagnetic bound systems consisting of spinless particles and for systems for which the constituents are a spin-\textonehalf{} and a spinless particle, and its application to the exotic bound state is discussed. Included as well is a calculation of the ground-state energy to ${\ensuremath{\alpha}}^{4}\mathrm{th}$ order of an electromagnetically bound system which is comprised of a particle of spin 0 and one of spin \textonehalf{}.

Evolution time Klein-Gordon equation and derivation of its nonlinear counterpart

Recently, relativistic quantum mechanical wave equations involving an extra scalar evolution parameter (analogous to proper time) have been increasingly studied in the literature. Since the physical meaning of this parameter is still very much a matter of discussion the authors aim to give it a clear physical meaning and then to proceed to a stochastic derivation of a nonlinear evolution time Klein-Gordon equation with a view to representing possible vacuum dissipative effects.

Low-frequency spontaneous radiation in a free-electron laser in the trapped regime.

The trapped regime of a free-electron laser is considered in terms of the relativistic quantum-mechanical Klein-Gordon equation in the presence of the undulator and the laser field. The electron which is trapped in the wells of the ponderomotive potential occupies discrete equidistant levels. Transitions between these quantum-mechanical levels are shown to lead to the well-known classical sidebands. As a new feature in spontaneous emission we obtain a low frequency \ensuremath{\delta}\ensuremath{\omega} emitted slightly off axis, where \ensuremath{\delta}\ensuremath{\omega} denotes the shift between the first sideband and the small-signal laser frequency. The low frequency is again classical and constitutes a sideband of the zeroth harmonic \ensuremath{\omega}=0. Under current experimental conditions this low-frequency spontaneous emission is at the borderline of detectability.

The connection of two-particle relativistic quantum mechanics with the Bethe–Salpeter equation

The formal equivalence between the wave equations of two-particle relativistic quantum mechanics, based on the manifestly covariant Hamiltonian formalism with constraints, and the Bethe–Salpeter equation are shown. This is achieved by algebraically transforming the latter so as to separate it into two independent equations that match the equations of Hamiltonian relativistic quantum mechanics. The first equation determines the relative time evolution of the system, while the second one yields a three-dimensional eigenvalue equation. A connection is thus established between the Bethe–Salpeter wave function and its kernel on the one hand and the quantum mechanical wave function and interaction potential on the other. For the sector of solutions of the Bethe–Salpeter equation having nonrelativistic limits, this relationship can be evaluated in perturbation theory. A generalized form of the instantaneous approximation that simplifies the various expressions involved in the above relations is also devised. It also permits the evaluation of the normalization condition of the quantum mechanical wave function as a three-dimensional integral.

Particles vs. events: The concatenated structure of world lines in relativistic quantum mechanics

The dynamical equations of relativistic quantum mechanics prescribe the motion of wave packets for sets of events which trace out the world lines of the interacting particles. Electromagnetic theory suggests thatparticle world line densities be constructed from concatenation of event wave packets. These sequences are realized in terms of conserved probability currents. We show that these conserved currents provide a consistent particle and antiparticle interpretation for the asymptotic states in scattering processes. The relation between current conservation and unitarity is used to establish relations between pair production and annihilation amplitudes and scattering. The discrete symmetriesC, T, P are studied and it is shown that no Dirac sea (for fermions where such a construction is possible, or bosons where it is not) is required for consistency of the theory. These currents, furthermore, represent the discrete symmetries in a way consistent with their interpretation as particle currents.

Generalized absorber theory and the Einstein-Podolsky-Rosen paradox

A generalized form of Wheeler-Feynman absorber theory is used to explain the quantum-mechanical paradox proposed by Einstein, Podolsky, and Rosen (EPR). The advanced solutions of the electromagnetic wave equation and of relativistic quantum-mechanical wave equations are shown to play the role of verifier in quantum-mechanical transactions, providing microscopic communication paths between detectors across spacelike intervals in violation of the EPR locality postulate. The principle of causality is discussed in the context of this approach, and possibilities for experimental tests of the theory are examined.

Invariance groups of certain equations of relativistic quantum mechanics

Invariance groups of certain equations of relativistic quantum mechanics

Factorized wave equations on homogeneous spaces of the Poincaré group

Relativistic quantum-mechanical wave equations are presented for spinor functions defined on the parameter manifold of homogeneous spaces of the Poincare group. These equations are linear in the momentum operators and may be iterated to define certain linear combinations of the Casimir invariants of the Poincare group. The additional variables, over and above those of the Minkowski manifold, are expected to be related to some internal structure which is effectively hidden in the free theory, but which may become observable when interactions are introduced. The minimal coupling to electromagnetism and to other external fields is presented.

Heuristic hypothesis on the relation between cosmology and elementary particles

Heuristic hypothesis on the relation between cosmology and elementary particles

A Manifestly Covariant Description of Arbitrary Dynamical Variables in Relativistic Quantum Mechanics

The concepts of instantaneous observables and dynamical variables are analyzed and generalized to arbitrary spacelike hyperplanes. A formalism is developed which gives the basic equations of relativistic quantum mechanics for dynamical variables on arbitrary hyperplanes a manifestly covariant form. A covariant linear transformation on the Poincaré generators introduces the hyperplane generators which yield commutation relations displaying a clear separation of the kinematical and dynamical properties of dynamical variables. An axiomatic study of the center-of-mass position operator yields the uniqueness of the operator and completes the physical interpretation of the hyperplane generators. The Poincaré invariance and hyperplane independence of the scattering operator is related to asymptotic conservation laws in the hyperplane formalism, and finally, a nonlocal, hyperplane-dependent, field theory of free spinless particles is considered.