Properties of legendrian subvarieties of projective space

I prove that every smooth legendrian variety generated by quadrics is a homogeneous variety and further I give a list of all such legendrian varieties. A review of the subject is included, illustrated by examples. Another result is that no complete intersection is a legendrian variety.Comment: 44 pages, 11 figures. This is an extended translation of my MSc Degree disertatio

On single-photon wave function

We present in this paper how the single-photon wave function for transversal photons (with the direct sum of ordinary unitary representations of helicity 1 and -1 acting on it) is subsumed within the formalism of Gupta-Bleuler for the quantized free electromagnetic field in the Krein space (i.e. in the ordinary Hilbert space endowed with the Gupta-Bleuler operator $\eta$). Rigorous Gupta-Bleuler quantization of the free electromagnetic field is based on a generalization of ours (published formerly) of the Mackey theory of induced representations which includes representations preserving the indefinite Krein inner-product given by the Gupta-Bleuler operator and acting in the Krein space. The free electromagnetic field is constructed by application of the Segal's second quantization functor to the specific Krein-isometric representation. A closed subspace $\mathcal{H}_{\textrm{tr}}$ of the single-photon Krein space on which the indefinite Krein-inner-product is strictly positive is constructed such that the Krein-isometric single-photon representation generates modulo unphysical states precisely the action of a representation which preserves the positive inner product on $\mathcal{H}_{\textrm{tr}}$ induced by the Krein inner product, and is equal to the direct sum of ordinary unitary representations of helicity 1 and -1 respectively. Two states of single photon Krein space are physically equivalent whenever differ by a state of Krein norm zero and whose projection on $\mathcal{H}_{\textrm{tr}}$, in the sense of the Krein-inner-product, vanishes. In particulart it follows that the results of Bia{\l}ynicki-Birula on the single-photon wave function may be reconciled with the micro-local perturbative approach to QED initiated by St\"uckelberg and Bogoliubov.Comment: 34 pages, research paper, we have added some commets and removed several misprint

Surface states in zigzag and armchair graphene nanoribbons

This paper presents electronic spectra of zigzag and armchair graphene nanoribbons calculated within the tight-binding model for pi-electrons. Zigzag and armchair nanoribbons of different edge geometries are considered, with surface perturbation taken into account. The properties of surface states are discussed on the basis of their classification into Tamm states and Shockley states. In armchair nanoribbons surface states are shown to close the energy gap at the Dirac point for certain edge geometries.Comment: 10 pages, 10 figure

Causal perturbative QED and infra-red states

At the classical level the electromagnetic field can be well identified at the spatial infinity. Staruszkiewicz pointed out that the quantization of the electromagnetic field at spatial infinity is essentially unique and follows from the two fundamental principles: 1) gauge invariance and 2) canonical commutation relations for canonically conjugated generalized coordinates, and constructed a simple and mathematically transparent quantum theory of the Coulomb field, predicting (among other things) a relation between the theory of unitary representations of the $SL(2, \mathbb{C})$ group and the fine structure constant. Until now this theory has stayed outside the main stream of the perturbative development in QED, mainly due to the unsolved infra-red-type (IR) problems in the perturbative approach. Recently however there has been performed a more careful analysis of free fields including the mass less free gauge fields, such as the electromagnetic potential field, their Wick and chronological products, which revealed the need for a more careful and white noise construction of these fields, and which opened a way to resolve IR problems (at least those which shows up at each order separately). Comparison of the perturbatively constructed field at spatial infinity with the quantum phase field of the Staruszkiewicz theory leads to the proof of universality of the unit of charge.Comment: 23 page

Birational cobordisms and factorization of birational maps

In this paper we develop a Morse-like theory in order to decompose birational maps and morphisms of smooth projective varieties defined over a field of characteristic zero into more elementary steps which are locally \'etale isomorphic to equivariant flips, blow-ups and blow-downs of toric varieties. A crucial role in the considerations is played by K^*-actions where K is the base field. This paper serves as a basis for proving the weak factorization conjecture on factorization of birational maps in characteristic zero into blow-ups and blow-downs. This is carried out in two subsequent papers, one by the author (Combinatorial structures on toroidal varieties: a proof of the weak Factorization Theorem) and one joint with Abramovich, Karu and Matsuki (Torification and factorization of birational maps). In the last paper, the ideas of the present paper are discussed using geometric invariant theory.Comment: 23 page

The operator form of the effective potential governing the time evolution in n-dimensional subspace of states

This paper presents the operator form of the effective potential V governing the time evolution in 2 and 3 and n dimensional subspace of states. The general formula for the n dimensional case is considered the starting point for the calculation of the explicit formulae for 2 and 3 dimensional degenerate and non-degenerate cases. We relate the 2 and 3 dimensional cases to some physical systems which are currently investigated.Comment: 17 pages. To appear in Acta Physica Polonica

The effect of electron-electron interactions on the conditions of surface state existence

Electronic surface states in one-dimensional two-band TBA model are studied by use of the Green function method. The local density of states (LDOS) at successive atoms in a semi-infinite chain, even in the case of atoms distant from the surface, is found to be clearly different from that observed in an unperturbed (infinite) chain. The surface atom occupancy is calculated self-consistently, with the effect of electron-electron interactions taken into account. The electron-electron interactions are shown to have a significant impact on the conditions of surface state existence.Comment: submitted to Materials Science, (4 pages, 3 figures

Hypothetical first order ∣ΔS∣=2|\Delta S|=2 transitions in the K0−K0ˉK^{0}-\bar{K^{0}} complex

The influence of a hypothetical CP violating $|\Delta S|=2$ interaction on the time evolution of the $K^{0} - \bar{K^{0}}$ system is investigated. It is shown, that if we were to assume the existence of the superweak-like interaction then this would lead to the conclusion, that there might be observable effects in the masses of the neutral kaons. We address the possibility of experimental observation of these effects and perform a computer simulation of one of the parameters which describe such effects. Instead of the widely used Lee, Oehme and Yang approximation, which is not suitable to considering this kind of interaction we use a formalism based on the Krolikowski-Rzewuski equation.Comment: 16 pages, 4 figure

Toric Legendrian subvarieties

We give the full classification of smooth toric Legendrian subvarieties in projective space. We also prove that under some minor assumptions the group of linear automorphisms preserving given Legendrian subvariety preserves the contact structure of the ambient projective space.Comment: 19 pages, 5 figure

Hyperplane sections of Legendrian subvarieties

We prove that a general hyperplane section of a smooth Legendrian subvariety in a projective space admits Legendrian embedding into another projective space. This gives numerous new examples of smooth Legendrian subvarieties, some of which have positive Kodaira dimension.Comment: 9 page