Vortex dynamics in a thin superfluid 4 He film as well as in a type II superconductor is describe... more Vortex dynamics in a thin superfluid 4 He film as well as in a type II superconductor is described by the classical counterpart of the model advocated by Peierls, and used for deriving the ground states of the Fractional Quantum Hall Effect. The model has non-commuting coordinates, and is obtained by reduction from a particle associated with the "exotic" extension of the planar Galilei group.
The quantum mechanically admissible definitions of the factor exp [i/ℏ S(y)] in Feynman's in... more The quantum mechanically admissible definitions of the factor exp [i/ℏ S(y)] in Feynman's integral—are put in bijection with the prequantisations of Kostant and Souriau. The different allowed expressions of this factor— the inequivalent prequantisations—are classified in terms of algebraic topology.
The anomalous velocity term in the semiclassical model of a Bloch electron deviates the trajector... more The anomalous velocity term in the semiclassical model of a Bloch electron deviates the trajectory from the conventional one. When the Berry curvature (alias noncommutative parameter) is a monopole in momentum space, as found recently in some ferromagnetic crystals while observing the anomalous Hall effect, we get a transverse shift, similar to that in the optical Hall effect.
A rotation by 2m around an Aharonov-Bohm solenoid is shown to change the wave function of a charg... more A rotation by 2m around an Aharonov-Bohm solenoid is shown to change the wave function of a charged spinless particle by a phase factor X=exp(ted/t) For fer.mions the effect manifests itself as a phase shift in addition to the well-known sign change. A Gedankenexperiment is proposed for observation.
For local variational systems (like a charged particle in the field of a Dirac monopole) a quantu... more For local variational systems (like a charged particle in the field of a Dirac monopole) a quantum mechanically well-defined action (Q.M.W.D.A.) can be introduced iff the system is prequantizable in the Kostant-Souriau sense. If the configuration space is multiply connected (as in the Bohm-Aharonov experiment), different expressions for the classical action may emerge; they are quantum mechanically equivalent (Q.M.E.) iff the corresponding prequantizations are equivalent. In both cases the situation depends on the behaviour of the non integrable phase factor of Wu and Yang.
The Biedenharn approach to the Dirac-Coulomb problem is applied to a system considered by D'Hoker... more The Biedenharn approach to the Dirac-Coulomb problem is applied to a system considered by D'Hoker and Vinet, which consists of a spin 1 2 particle in the combined field of a Dirac monopole plus a λ 2 /r 2 potential. The explicit solution is obtained by diagonalizing the 'Biedenharn-Temple operator', Γ.
The Maslov correction to the wave function is the jump of $$ \left( { - \frac{\pi } {2}} \right) ... more The Maslov correction to the wave function is the jump of $$ \left( { - \frac{\pi } {2}} \right) $$ in the phase when the system passes through a caustic. This can be explained by studying the second variation and the geometry of paths, as conveniently seen in Feynman’s path integral framework. The results can be extended to any system using the semiclassical approximation. The 1-dimensional harmonic oscillator is used to illustrate the different derivations reviewed here.
We relate here the geometric formulation of Hamilton's P r incinb presented in our previous paper... more We relate here the geometric formulation of Hamilton's P r incinb presented in our previous paper [ij to the usual one in terms of a Lagranqian .nation. The exact conditions for their equivalence are obtained and?a method evtpnd^H f°r const^Cti°n of a Lagrangian function. The formalism is extended to spin particles and a local Lagrangian is constructed in this case also. However, this function cannot be extended to a global S n ! ? Связываются различные способы определения принципа Гамилтона: гео метрический, введенной в предыдущей статье [l} и обыкновенный, в виде функ ции Лагранжа. Выводятся точные условия их эквивалентности и показывается ме тод построения функции Лагранжа. Формальность распространяется и на частицы со спином. Однако, построенную в этих условиях, локальную функцию Лагранжа нельзя глобально определить. bevezetett geometriai és szokásos Lagrange-függvényes megfogalmazása között. Pontos feltételeket kapunk azok ekvivalenciájára és módszert mutatunk a Lagrange-függvény konstrukciójára. A formalizmus kiterjeszthető a spines ré szecskékre is. Az ekkor konstruált lokális Lagrange-függvény azonban nem globalizálható.
A high-school exercise is used to get an insight into planetary motion. 1 Introduction: some hist... more A high-school exercise is used to get an insight into planetary motion. 1 Introduction: some history Science contests for high-school students have a hundred-years-old tradition in Hungary [1, 2]. The first one, open to students who just completed their high-school cursus, was organized in 1894 to mark the appointment of L. Eötvös as Minister of Education. This was also the year when the High-School Journal for Mathematics and Physics was first published. This Journal sets selected problems each month, and the best solutions are published under the name of the student who solved it. The early prizewinners of the Eötvös contest include such outstanding future scientists as L. Fejér (who became famous in mathematical analysis), Th. von Kármán (who made important contributions to hydro and aerodynamics), A. Haar (remembered for his invariant measure on group manifolds). .. followed by many others. In 1916, in the middle of World War I, a physics contest was held for the first time-and L. Szilárd (who later, with E. Fermi, patented the nuclear reactor) came second. E. Wigner (who got the Nobel prize for using group theory in atomic physics), and J. von Neumann (who built the first computer and contributed to many fields, ranging from game theory to quantum mechanics) only missed the contest because of the turmoil following the disaster of World War I. In 1925 Edward Teller not only got the first prize in physics, but also shared the mathematics prize with Laszlo Tisza. They became friends and their friendship lasted until Teller's recent death. Soon Tisza (then a mathematics student in Göttingen) changed to physics under the influence of Born, and published his very first paper jointly with Teller (who was preparing his Ph. D. in Leipzig with Heisenberg) on molecular spectrum. Later Tisza introduced the two-fluid model of superfluidity, further developped by Landau. At 96, he is still active at MIT. In the late twenties, von Neumann and Szilárd even suggested that the university examinations could be replaced by such a contest [2]. The tradition still continues, and contributes to forming future generations of scientists [1]. The contests have been internationalized with the International Olympiads for High-School Students: first in Mathematics, and, since 1967, also in Physics.
Vortex dynamics in a thin superfluid 4 He film as well as in a type II superconductor is describe... more Vortex dynamics in a thin superfluid 4 He film as well as in a type II superconductor is described by the classical counterpart of the model advocated by Peierls, and used for deriving the ground states of the Fractional Quantum Hall Effect. The model has non-commuting coordinates, and is obtained by reduction from a particle associated with the "exotic" extension of the planar Galilei group.
The quantum mechanically admissible definitions of the factor exp [i/ℏ S(y)] in Feynman's in... more The quantum mechanically admissible definitions of the factor exp [i/ℏ S(y)] in Feynman's integral—are put in bijection with the prequantisations of Kostant and Souriau. The different allowed expressions of this factor— the inequivalent prequantisations—are classified in terms of algebraic topology.
The anomalous velocity term in the semiclassical model of a Bloch electron deviates the trajector... more The anomalous velocity term in the semiclassical model of a Bloch electron deviates the trajectory from the conventional one. When the Berry curvature (alias noncommutative parameter) is a monopole in momentum space, as found recently in some ferromagnetic crystals while observing the anomalous Hall effect, we get a transverse shift, similar to that in the optical Hall effect.
A rotation by 2m around an Aharonov-Bohm solenoid is shown to change the wave function of a charg... more A rotation by 2m around an Aharonov-Bohm solenoid is shown to change the wave function of a charged spinless particle by a phase factor X=exp(ted/t) For fer.mions the effect manifests itself as a phase shift in addition to the well-known sign change. A Gedankenexperiment is proposed for observation.
For local variational systems (like a charged particle in the field of a Dirac monopole) a quantu... more For local variational systems (like a charged particle in the field of a Dirac monopole) a quantum mechanically well-defined action (Q.M.W.D.A.) can be introduced iff the system is prequantizable in the Kostant-Souriau sense. If the configuration space is multiply connected (as in the Bohm-Aharonov experiment), different expressions for the classical action may emerge; they are quantum mechanically equivalent (Q.M.E.) iff the corresponding prequantizations are equivalent. In both cases the situation depends on the behaviour of the non integrable phase factor of Wu and Yang.
The Biedenharn approach to the Dirac-Coulomb problem is applied to a system considered by D'Hoker... more The Biedenharn approach to the Dirac-Coulomb problem is applied to a system considered by D'Hoker and Vinet, which consists of a spin 1 2 particle in the combined field of a Dirac monopole plus a λ 2 /r 2 potential. The explicit solution is obtained by diagonalizing the 'Biedenharn-Temple operator', Γ.
The Maslov correction to the wave function is the jump of $$ \left( { - \frac{\pi } {2}} \right) ... more The Maslov correction to the wave function is the jump of $$ \left( { - \frac{\pi } {2}} \right) $$ in the phase when the system passes through a caustic. This can be explained by studying the second variation and the geometry of paths, as conveniently seen in Feynman’s path integral framework. The results can be extended to any system using the semiclassical approximation. The 1-dimensional harmonic oscillator is used to illustrate the different derivations reviewed here.
We relate here the geometric formulation of Hamilton's P r incinb presented in our previous paper... more We relate here the geometric formulation of Hamilton's P r incinb presented in our previous paper [ij to the usual one in terms of a Lagranqian .nation. The exact conditions for their equivalence are obtained and?a method evtpnd^H f°r const^Cti°n of a Lagrangian function. The formalism is extended to spin particles and a local Lagrangian is constructed in this case also. However, this function cannot be extended to a global S n ! ? Связываются различные способы определения принципа Гамилтона: гео метрический, введенной в предыдущей статье [l} и обыкновенный, в виде функ ции Лагранжа. Выводятся точные условия их эквивалентности и показывается ме тод построения функции Лагранжа. Формальность распространяется и на частицы со спином. Однако, построенную в этих условиях, локальную функцию Лагранжа нельзя глобально определить. bevezetett geometriai és szokásos Lagrange-függvényes megfogalmazása között. Pontos feltételeket kapunk azok ekvivalenciájára és módszert mutatunk a Lagrange-függvény konstrukciójára. A formalizmus kiterjeszthető a spines ré szecskékre is. Az ekkor konstruált lokális Lagrange-függvény azonban nem globalizálható.
A high-school exercise is used to get an insight into planetary motion. 1 Introduction: some hist... more A high-school exercise is used to get an insight into planetary motion. 1 Introduction: some history Science contests for high-school students have a hundred-years-old tradition in Hungary [1, 2]. The first one, open to students who just completed their high-school cursus, was organized in 1894 to mark the appointment of L. Eötvös as Minister of Education. This was also the year when the High-School Journal for Mathematics and Physics was first published. This Journal sets selected problems each month, and the best solutions are published under the name of the student who solved it. The early prizewinners of the Eötvös contest include such outstanding future scientists as L. Fejér (who became famous in mathematical analysis), Th. von Kármán (who made important contributions to hydro and aerodynamics), A. Haar (remembered for his invariant measure on group manifolds). .. followed by many others. In 1916, in the middle of World War I, a physics contest was held for the first time-and L. Szilárd (who later, with E. Fermi, patented the nuclear reactor) came second. E. Wigner (who got the Nobel prize for using group theory in atomic physics), and J. von Neumann (who built the first computer and contributed to many fields, ranging from game theory to quantum mechanics) only missed the contest because of the turmoil following the disaster of World War I. In 1925 Edward Teller not only got the first prize in physics, but also shared the mathematics prize with Laszlo Tisza. They became friends and their friendship lasted until Teller's recent death. Soon Tisza (then a mathematics student in Göttingen) changed to physics under the influence of Born, and published his very first paper jointly with Teller (who was preparing his Ph. D. in Leipzig with Heisenberg) on molecular spectrum. Later Tisza introduced the two-fluid model of superfluidity, further developped by Landau. At 96, he is still active at MIT. In the late twenties, von Neumann and Szilárd even suggested that the university examinations could be replaced by such a contest [2]. The tradition still continues, and contributes to forming future generations of scientists [1]. The contests have been internationalized with the International Olympiads for High-School Students: first in Mathematics, and, since 1967, also in Physics.
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