The four-fermion interaction in D=2,3,4: a nonperturbative treatment

A new nonperturbative approach is used to investigate the Gross-Neveu model of four fermion interaction in the space-time dimensions 2, 3 and 4, the number $N$ of inner degrees of freedom being a fixed integer. The spontaneous symmetry breaking is shown to exist in $D=2,3$ and the running coupling constant is calculated. The four dimensional theory seems to be trivial.Comment: a minor correction: one more acknowledgement is added. Latex 2.09 file, 15 pages, no figures, accepted for publication to Int.J.Mod.Phys.

Remote atomic clock synchronization via satellites and optical fibers

In the global network of institutions engaged with the realization of International Atomic Time (TAI), atomic clocks and time scales are compared by means of the Global Positioning System (GPS) and by employing telecommunication satellites for two-way satellite time and frequency transfer (TWSTFT). The frequencies of the state-of-the-art primary caesium fountain clocks can be compared at the level of 10e-15 (relative, 1 day averaging) and time scales can be synchronized with an uncertainty of one nanosecond. Future improvements of worldwide clock comparisons will require also an improvement of the local signal distribution systems. For example, the future ACES (atomic clock ensemble in space) mission shall demonstrate remote time scale comparisons at the uncertainty level of 100 ps. To ensure that the ACES ground instrument will be synchronized to the local time scale at PTB without a significant uncertainty contribution, we have developed a means for calibrated clock comparisons through optical fibers. An uncertainty below 50 ps over a distance of 2 km has been demonstrated on the campus of PTB. This technology is thus in general a promising candidate for synchronization of enhanced time transfer equipment with the local realizations of UTC . Based on these experiments we estimate the uncertainty level for calibrated time transfer through optical fibers over longer distances. These findings are compared with the current status and developments of satellite based time transfer systems, with a focus on the calibration techniques for operational systems

From Quantum Universal Enveloping Algebras to Quantum Algebras

The ``local'' structure of a quantum group G_q is currently considered to be an infinite-dimensional object: the corresponding quantum universal enveloping algebra U_q(g), which is a Hopf algebra deformation of the universal enveloping algebra of a n-dimensional Lie algebra g=Lie(G). However, we show how, by starting from the generators of the underlying Lie bialgebra (g,\delta), the analyticity in the deformation parameter(s) allows us to determine in a unique way a set of n ``almost primitive'' basic objects in U_q(g), that could be properly called the ``quantum algebra generators''. So, the analytical prolongation (g_q,\Delta) of the Lie bialgebra (g,\delta) is proposed as the appropriate local structure of G_q. Besides, as in this way (g,\delta) and U_q(g) are shown to be in one-to-one correspondence, the classification of quantum groups is reduced to the classification of Lie bialgebras. The su_q(2) and su_q(3) cases are explicitly elaborated.Comment: 16 pages, 0 figures, LaTeX fil

Weyl approach to representation theory of reflection equation algebra

The present paper deals with the representation theory of the reflection equation algebra, connected with a Hecke type R-matrix. Up to some reasonable additional conditions the R-matrix is arbitrary (not necessary originated from quantum groups). We suggest a universal method of constructing finite dimensional irreducible non-commutative representations in the framework of the Weyl approach well known in the representation theory of classical Lie groups and algebras. With this method a series of irreducible modules is constructed which are parametrized by Young diagrams. The spectrum of central elements s(k)=Tr_q(L^k) is calculated in the single-row and single-column representations. A rule for the decomposition of the tensor product of modules into the direct sum of irreducible components is also suggested.Comment: LaTeX2e file, 27 pages, no figure

Manin matrices and Talalaev's formula

We study special class of matrices with noncommutative entries and demonstrate their various applications in integrable systems theory. They appeared in Yu. Manin's works in 87-92 as linear homomorphisms between polynomial rings; more explicitly they read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: $[M_{ij}, M_{kl}]=[M_{kj}, M_{il}]$ (e.g. $[M_{11}, M_{22}]=[M_{21}, M_{12}]$). We claim that such matrices behave almost as well as matrices with commutative elements. Namely theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, the Newton identities and so on and so forth) holds true for them. On the other hand, we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability. For instance, Manin matrices (and their q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and the so--called Cartier-Foata matrices. Also, they enter Talalaev's hep-th/0404153 remarkable formulas: $det(\partial_z-L_{Gaudin}(z))$, det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show that theorems of linear algebra, after being established for such matrices, have various applications to quantum integrable systems and Lie algebras, e.g in the construction of new generators in $Z(U(\hat{gl_n}))$ (and, in general, in the construction of quantum conservation laws), in the Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also discuss applications to the separation of variables problem, new Capelli identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints e.g. in Newton id-s fixed, normal ordering convention turned to standard one, refs. adde

Characteristic Relations for Quantum Matrices

General algebraic properties of the algebras of vector fields over quantum linear groups $GL_q(N)$ and $SL_q(N)$ are studied. These quantum algebras appears to be quite similar to the classical matrix algebra. In particular, quantum analogues of the characteristic polynomial and characteristic identity are obtained for them. The $q$-analogues of the Newton relations connecting two different generating sets of central elements of these algebras (the determinant-like and the trace-like ones) are derived. This allows one to express the $q$-determinant of quantized vector fields in terms of their $q$-traces.Comment: 11 pages, latex, an important reference [16] added