A relativistic superalgebra in a generalized Schrödinger picture

We consider a relativistic superalgebra in the picture in which the time and spatial derivative cannot be presented in the operators of the particle. The supersymmetry generators as well as the Hamilton operators for the massive relativistic particles with spin 0 and spin 1/2 are expressed in terms of the principal series of the unitary representations of the Lorentz group. We also consider the massless case. New Hamilton operators are constructed for the massless particles with spin 0 and spin 1/2. Received: 20 November 2002 / Published online: 14 April 2003 RID="a" ID="a" e-mail: rf@thp.uni-koeln.de     

Critical assessment of the Schrödinger picture of quantum mechanics

We provide an example in which the Heisenberg and the Schrödinger pictures of quantum mechanics give different results, thus confirming the statement of P.A.M. Dirac that the two pictures may lead to inequivalent results. We consider a one-dimensional nonrelativistic charged harmonic oscillator (frequency ω₀ and mass m), and take into account the action of the radiation reaction and the vacuum electromagnetic forces on the charged oscillator. We show that the Heisenberg picture gives the correct value, ℏω₀/2, for the ground state energy of the harmonic oscillator in both cases of classical and quantized vacuum fields. In the case of the Schrödinger picture, considering classical vacuum fields, and using a simple calculation for the classical radiation reaction force that is valid in the limit of large mass (mc²⪢ℏω₀), we obtain the value ℏω₀ for the ground state energy of the harmonic oscillator. We show that the vacuum electromagnetic forces play a very important role in the understanding of this discrepancy.     

Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

Collapse in the nonlinear Schrödinger equation

A new class of exact solutions with a singularity at finite time (collapse) is obtained for the nonlinear Schrödinger equation.     

On Gravitational Effects in the Schrödinger Equation

The Schrödinger  equation for a particle of rest mass $m$ and electrical charge $ne$ interacting with a four-vector potential $A_i$ can be derived as the non-relativistic limit of the Klein–Gordon  equation $\left( \Box '+m^2\right) \varPsi =0$ for the wave function $\varPsi $ , where $\Box '=\eta ^{jk}\partial '_j\partial '_k$ and $\partial '_j=\partial _j -\mathrm {i}n e A_j$ , or equivalently from the one-dimensional  action $S_1=-\int m ds +\int neA_i dx^i$ for the corresponding point particle in the semi-classical approximation $\varPsi \sim \exp {(\mathrm {i}S_1)}$ , both methods yielding the equation $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2m}\eta ^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + m + n e\phi \right) \varPsi $ in Minkowski  space–time  , where $\alpha ,\beta =1,2,3$ and $\phi =-A_0$ . We show that these two methods generally yield equations  that differ in a curved background  space–time   $g_{ij}$ , although they coincide when $g_{0\alpha }=0$ if $m$ is replaced by the effective mass $\mathcal{M}\equiv \sqrt{m^2-\xi R}$ in both the Klein–Gordon  action $S$ and $S_1$ , allowing for non-minimal coupling to the gravitational  field, where $R$ is the Ricci scalar and $\xi $ is a constant. In this case $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2\mathcal{M}'} g^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + \mathcal{M}\phi ^{(\mathrm g)} + n e\phi \right) \varPsi $ , where $\phi ^{(\mathrm g)} =\sqrt{g_{00}}$ and $\mathcal{M}'=\mathcal{M}/\phi ^{(\mathrm g)} $ , the correctness of the gravitational  contribution to the potential having been verified to linear order $m\phi ^{(\mathrm g)} $ in the thermal-neutron beam interferometry experiment due to Colella et al. Setting $n=2$ and regarding $\varPsi $ as the quasi-particle wave function, or order parameter, we obtain the generalization of the fundamental macroscopic Ginzburg-Landau equation of superconductivity to curved space–time. Conservation of probability and electrical current requires both electromagnetic gauge and space–time  coordinate conditions to be imposed, which exemplifies the gravito-electromagnetic analogy, particularly in the stationary case, when div ${{\varvec{A}}}=\hbox {div}{{\varvec{A}}}^{(\mathrm g)}=0$ , where ${{\varvec{A}}}^{\alpha }=-A^{\alpha }$ and ${{\varvec{A}}}^{(\mathrm g)\alpha }=-\phi ^{(\mathrm g)}g^{0\alpha }$ . The quantum-cosmological Schrödinger  (Wheeler–DeWitt) equation is also discussed in the $\mathcal{D}$ -dimensional  mini-superspace idealization, with particular regard to the vacuum potential $\mathcal V$ and the characteristics of the ground state, assuming a gravitational  Lagrangian   $L_\mathcal{D}$ which contains higher-derivative  terms up to order $\mathcal{R}^4$ . For the heterotic superstring theory  , $L_\mathcal{D}$ consists of an infinite series in $\alpha '\mathcal{R}$ , where $\alpha '$ is the Regge slope parameter, and in the perturbative approximation $\alpha '|\mathcal{R}| \ll 1$ , $\mathcal V$ is positive semi-definite for $\mathcal{D} \ge 4$ . The maximally symmetric ground state satisfying the field equations is Minkowski  space for $3\le {\mathcal {D}}\le 7$ and anti-de Sitter  space for $8 \le \mathcal {D} \le 10$ .     

Quantum Yang-Mills-Weyl Dynamics in the Schrödinger paradigm

Inspired by F. Wilczek’s QCD Lite, quantum Yang-Mills-Weyl Dynamics (YMWD) describes quantum interaction between gauge bosons (associated with a simple gauge group \(\mathbb{G}\) ) and larks (massless chiral fields charged by an irreducible unitary representation of \(\mathbb{G}\) ). Schrödinger representation of this quantum Yang-Mills-Weyl theory is based on a sesqui-holomorphic operator calculus of infinite-dimensional operators with variational derivatives. The spectrum of quantum YMWD in a compact bag is a sequence of eigenvalues convergent to +∞. The eigenvalues have finite multiplicities with respect to a von Neumann algebra with a regular trace. The spectrum is inversely proportional to the square of the running coupling constant. The rigorous mathematical theory is nonperturbative with a running coupling constant as the only ad hoc parameter. The application of the first mathematical principles is based on the properties of the compact simple Lie group \(\mathbb{G}\) .     

Long-time relaxation processes in the nonlinear Schrödinger equation

The nonlinear Schrödinger equation, known in low-temperature physics as the Gross-Pitaevskii equation, has a large family of excitations of different kinds. They include sound excitations, vortices, and solitons. The dynamics of vortices strictly depends on the separation between them. For large separations, some kind of adiabatic approximation can be used. We consider the case where an adiabatic approximation can be used (large separation between vortices) and the opposite case of a decay of the initial state, which is close to the double vortex solution. In the last problem, no adiabatic parameter exists (the interaction is strong). Nevertheless, a small numerical parameter arises in the problem of the decay rate, connected with an existence of a large centrifugal potential, which leads to a small value of the increment. The properties of the nonlinear wave equation are briefly considered in the Appendix A.