Tachyonic Equation Research Articles

Pseudo-Hermitian quantum dynamics of tachyonic spin-1/2 particles

We investigate the spinor solutions, the spectrum and the symmetry properties of a matrix-valued wave equation whose plane-wave solutions satisfy the superluminal (tachyonic) dispersion relation , where E is the energy, is the spatial momentum and m is the mass of the particle. The equation reads (iγμ ∂μ − γ5 m)ψ = 0, where γ5 is the fifth current. The tachyonic equation is shown to be invariant and invariant. The tachyonic Hamiltonian breaks parity and is non-Hermitian but fulfils the pseudo-Hermitian property , where P is the parity matrix and is the full parity transformation. The energy eigenvalues and eigenvectors describe a continuous spectrum of plane-wave solutions (which correspond to real eigenvalues for ) and evanescent waves, which constitute resonances and anti-resonances with complex-conjugate pairs of resonance eigenvalues (for ). In view of additional algebraic properties of the Hamiltonian which supplement the pseudo-Hermiticity, the existence of a resonance energy eigenvalue E implies that E*, −E and −E* also constitute resonance energies of H5.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

ON SUPERLUMINAL FERMIONS WITHIN THE SECOND DERIVATIVE EQUATION

We postulate the second-order derivative equation with four parameters for spin-1/2 fermions possessing two mass states. For some choice of parameters fermions propagate with the superluminal speed. Thus, the novel tachyonic equation is suggested. The relativistic 20-component first-order wave equation is formulated and projection operators extracting states with definite energy and spin projections are obtained. The Lagrangian formulation of the first-order equation is presented and the electric current and energy–momentum tensor are found. The minimal and nonminimal electromagnetic interactions of fermions are considered and Schrödinger's form of the equation and the quantum-mechanical Hamiltonian are obtained. The canonical quantization of the field in the first-order formalism is performed and we find the vacuum expectation of chronological pairing of operators.

A note on tachyon dynamics

A suitable splitting of tachyon field equation is able to disclose non-trivial properties of Born–Infeld (some known, some unexpected) and Polyakov actions; the tachyon equation also can be analyzed in some detail. The analysis displays an intriguing connection between sine-Gordon theory and some crucial issues such as the emergence of perturbative string states when a D-brane and an anti-D-brane annihilate and the confinement of charged D-branes.

Gauge fields and scalars in rolling tachyon backgrounds

We investigate the dynamics of gauge and scalar fields on unstable D-branes with rolling tachyons. Assuming an FRW metric on the brane, we find a solution of the tachyon equation of motion which is valid for arbitrary tachyon potentials and scale factors. The equations of motion for a U(1) gauge field and a scalar field in this background are derived. These fields see an effective metric which differs from the original FRW metric. The field equations receive large corrections due to the curvature of the effective metric as well as the time variation of the gauge coupling. The equations of state for these fields resemble those of nonrelativistic matter rather than those of massless particles.

Non-critical Poincaré invariant bosonic string backgrounds and closed string tachyons

A new family of non critical bosonic string backgrounds in arbitrary space time dimension $D$ and with $ISO(1,D-2)$ Poincar\'e invariance are presented. The metric warping factor and dilaton agree asymptotically with the linear dilaton background. The closed string tachyon equation of motion enjoys, in the linear approximation, an exact solution of ``kink'' type interpolating between different expectation values. A renormalization group flow interpretation ,based on a closed string tachyon potential of type $-T^{2}e^{-T}$, is suggested.

String-motivated model.

The two-dimensional model which emerges from low-energy considerations of string theory is written down. Solutions of this classical model are noted, including some examples which have nontrivial tachyon field. One such represents the classical backreaction of the tachyon field on the black hole for a two parameter set of tachyon potentials. Assuming the classical black hole background in the `Eddington-Finkelstein' gauge, the tachyon equation is separable and the radial part is solved by a hypergeometric function, which is in general of complex argument. A semi-classical prescription for including the quantum effects of the tachyon field is described, and the resulting equations of motion are found. Special solutions of these equations are written down.

Antisymmetric tensor coupling and conformal invariance in sigma models corresponding to gauged WZNW theories.

String backgrounds associated with gauged $G/H$ WZNW models generically depend on $\alpha'$ or $1/k$. The exact expressions for the corresponding metric $G_{\m\n}$, antisymmetric tensor $B_{\m\n}$, and dilaton $\phi$ can be obtained by eliminating the $2d$ gauge field from the local part of the effective action of the gauged WZNW model. We show that there exists a manifestly gauge-invariant prescription for the derivation of the antisymmetric tensor coupling. When the subgroup $H$ is one-dimensional and $G$ is simple the antisymmetric tensor is given by the semiclassical ($\alpha'$-independent) expression. We consider in detail the simplest non-trivial example with non-trivial $B_{\m\n}$ -- the D=3 sigma model corresponding to the $[SL(2,R) x R]/R$ gauged WZNW theory (`charged black string') and show that the exact expressions for $G_{\m\n}$, $B_{\m\n}$ and $\phi$ solve the Weyl invariance conditions in the two-loop approximation. Similar conclusion is reached for the closely related $SL(2,R)/R$ chiral gauged WZNW model. We find that there exists a scheme in which the semiclassical background is also a solution of the two-loop conformal invariance equations (but the tachyon equation takes a non-canonical form). We discuss in detail the role of field redefinitions (scheme dependence) in establishing a correspondence between the sigma model and conformal field theory results.

The off-shell vertex in conformal field theory and string field theory

It is shown that while the usual β-functions approach gives interacting equations of motion that are only valid as small perturbations around the free equations, there is a simple modification that allows one to go off the free mass shell. This involves computing correlation functions of vertex operators that are separated by distances on the order of the lattice spacing — which is equivalent to studying the renormalization group equations with a finite cutoff. Applying this technique to the tachyon equation of motion, in the light cone gauge, we reproduce the off-shell three-tachyon aned vector-tachyon-tachyon vertices of (light cone gauge) open string field theory.