arXiv subjects

G. K. Savvidy

Publications and source records attributed to G. K. Savvidy.

29 records · Page 2Linked to original sources

arXiv1996

Alternative actions for quantum gravity and the intrinsic rigidity of the spacetime

Using the Steiner-Weyl expansion formula for parallel manifolds and the so called gonihedric principle we find a large class of discrete integral invariants which are defined on simplicial manifolds of various dimensions. These integral invariants include the discrete version of the Hilbert-Einstein action found by Regge and alternative actions which are linear with respect to the size of the manifold. In addition the concept of generalized deficit angles appear in a natural way and is related to higher order curvature terms. These angles may be used to introduce various aspects of rigidity in simplicial quantum gravity.

hep-th↗

arXiv1996

Period of K System Generator of Pseudorandom Numbers

We analyze the structure of the periodic trajectories of the matrix generator of pseudorandom numbers which has been proposed earlier. The structure of the periodic trajectories becomes more transparent when the rational sublattice coincides with the Galois field $GF[p]$. We are able to compute the period of the trajectories as a function of $p$ and the dimension of the matrix $d$.

hep-lat↗

arXiv1995

Interaction Hierarchy. Gonihedric String and Quantum Gravity

We have found that the Regge gravity \cite{regge,sorkin}, can be represented as a $superposition$ of less complicated theory of random surfaces with $Euler~character$ as an action. This extends to Regge gravity our previous result \cite{savvidy}, which allows to represent the gonihedric string \cite{savvidy1} as a superposition of less complicated theory of random paths with $curvature$ action. We propose also an alternative linear action $A(M_{4})$ for the four and high dimensional quantum gravity. From these representations it follows that the corresponding partition functions are equal to the product of Feynman path integrals evaluated on time slices with curvature and length action for the gonihedric string and with Euler character and gonihedric action for the Regge gravity. In both cases the interaction is proportional to the overlapping sizes of the paths or surfaces on the neighboring time slices. On the lattice we constructed spin system with local interaction, which have the same partition function as the quantum gravity. The scaling limit is discussed.

hep-th↗

arXiv1995

Two and Three-Dimensional Spin Systems with Gonihedric Action

We perform numerical simulations of the two and three-dimensional spin systems with competing interaction. They describe the model of random surfaces with linear-gonihedric action.The degeneracy of the vacuum state of this spin system is equal to $~~d \cdot 2^{N}~~$ for the lattice of the size $~N^{d}~$. We observe the second order phase transition of the three-dimensional system, at temperature $β_{c} \simeq 0.43932$ which almost coincides with $β_{c}$ of the 2D Ising model. This confirms the earlier analytical result for the case when self-interaction coupling constant $k$ is equal to zero. We suggest the full set of order parameters which characterize the structure of the vacuum states and of the phase transition.

hep-th↗

arXiv1995

GEOMETRICAL STRING and DUAL SPIN SYSTEMS

We are able to perform the duality transformation of the spin system which was found before as a lattice realization of the string with linear action. In four and higher dimensions this spin system can be described in terms of a two-plaquette gauge Hamiltonian. The duality transformation is constructed in geometrical and algebraic language. The dual Hamiltonian represents a new type of spin system with local gauge invariance. At each vertex $ξ$ there are $d(d-1)/2$ Ising spins $Λ_{μ,ν}= Λ_{ν,μ}$, $μ\neq ν= 1,..,d$ and one Ising spin $Γ$ on every link $(ξ,ξ+e_μ)$. For the frozen spin $Γ\equiv 1$ the dual Hamiltonian factorizes into $d(d-1)/2$ two-dimensional Ising ferromagnets and into antiferromagnets in the case $Γ\equiv -1$. For fluctuating $Γ$ it is a sort of spin glass system with local gauge invariance. The generalization to $p$-branes is given.

hep-th↗

arXiv1994

Interaction Hierarchy

We analyse a new class of statistical systems, which simulate different systems of random surfaces on a lattice. Geometrical hierarchy of the energy functionals on which these theories are based produces corresponding hierarchy of the surface dynamics and of the phase transitions. We specially consider 3D gonihedric system and have found that it is equivalent to the propagation of almost free 2D Ising fermions. We construct dual statistical system with new matchbox spin variable $G_ξ$, high temperature expansion of which equally well describe these surfaces.

hep-th↗

arXiv1994

Dual Statistical Systems and Geometrical String

We analyse statistical system with interface energy proportional to the length of the edges of interface. We have found the dual system high temperature expansion of which equally well generates surfaces with linear amplitude. These dual systems are in the same relation as 3D Ising ferromagnet to the 3D Gauge spin system.

hep-th↗

arXiv1994

A Lower Estimate for the Modified Steiner Functional

We prove inequality (1) for the modified Steiner functional A(M), which extends the notion of the integral of mean curvature for convex surfaces.We also establish an exression for A(M) in terms of an integral over all hyperplanes intersecting the polyhedralral surface M.

hep-th↗

arXiv1993

Self-Avoiding Gonihedric Srting and Spin Systems

We classify different theories of self-intersecting random surfaces assigning special weights to intersections. When self-intersection coupling constant $κ$ tends to zero, then the surface can freely inetrsect and it is completely self-avoiding when $κ$ tends to infinity. Equivalent spin systems for this general case were constructed. In two-dimension the system with $κ= 0$ is in complete disorder as it is in the case of 2D gauge Ising system.

hep-lat↗

arXiv1993

Geometrical String and Spin Systems

We formulate a new geometrical string on the euclidean lattice. It is possible to find such spin systems with local interaction which reproduce the same surface dynamics.In the three-dimensional case this spin system is a usual Ising ferromagnet with additional diagonal antiferromagnetic interaction and with specially adjusted coupling constants. In the four-dimensional case the spin system coincides with the gauge Ising system with an additional double-plaquette interaction and also with specially tuned coupling constants. We extend this construction to random walks and random hypersurfaces (membrane and p-branes) of high dimensionality. We compare these spin systems with the eight-vertex model and BNNNI models.

hep-th↗

arXiv1993

Gonihedric String and Asymptotic Freedom

Few natural basic principles allow to extend Feynman integral over the paths to an integral over the surfaces so, that they coincide at long time scale, that is when the surface degenerates into a single particle world line. In the first approximation the loop Green functions have perimeter behavior. That corresponds to the case when free quarks interact through one gluon exchange. Quantum fluctuations of the surface generate the area low. Thus in this string theory asymptotic freedom and confinement can coexist.

hep-th↗