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.Comment: 11 pages,Late

String fine tuning

We develop further a new geometrical model of a discretized string, proposed in [1] and establish its basic physical properties. The model can be considered as the natural extention of the usual Feynman amplitude of the random walks to random surfaces. Both amplitudes coinside in the case, when the surface degenarates into a single particle world line. We extend the model to open surfaces as well. The boundary contribution is proportional to the full length of the boundary and the coefficient of proportionality can be treated as a hopping parameter of the quarks. In the limit, when this parameter tends to infinity, the theory is essentialy simlplified. We prove that the contribution of a given triangulation to the partition function is finite and have found the explicit form for the upper bound. The question of the convergence of the full partition function remains open. In this model the string tension may vanish at the critical point, if the last one exists, and possess a nontrivial scaling limit. The model contains hidden fermionic variables and can be considered as an independent model of hadrons.Comment: 14 pages, pTeX fil

Gonihedric String Equation

We discuss the basic properties of the gonihedric string and the problem of its formulation in continuum. We propose a generalization of the Dirac equation and of the corresponding gamma matrices in order to describe the gonihedric string. The wave function and the Dirac matrices are infinite-dimensional. The spectrum of the theory consists of particles and antiparticles of increasing half-integer spin lying on quasilinear trajectories of different slope. Explicit formulas for the mass spectrum allow to compute the string tension and thus demonstrate the string character of the theory.Comment: 40 pages, Latex, 9 figure

Phase structure of four-dimensional gonihedric spin system

We perform Monte Carlo simulations of a gauge invariant spin system which describes random surfaces with gonihedric action in four dimensions. The Hamiltonian is a mixture of one-plaquette and additional two- and three-plaquette interaction terms with specially adjusted coupling constants. For the system with the large self-intersection coupling constant $k$ we observe the second-order phase transition at temperature $\beta_{c}\simeq 1.75$. The string tension is generated by quantum fluctuations as it was expected theoretically. This result suggests the existence of a noncritical string in four dimensions. For smaller values of $k$ the system undergoes the first order phase transition and for $k$ close to zero exhibits a smooth crossover.Comment: 14 pages, Latex, 10 figure

Stability of the Rotating Ellipsoidal D0-brane System

In this note we prove the complete stability of the classical fluctuation modes of the rotating ellipsoidal membrane. The analysis is carried out in the full SU(N) setting, with the conclusion that the fluctuation matrix has only positive eigenvalues. This proves that the solution will remain close to the original one for all time, under arbitrary infinitesimal perturbations of the gauge fields.Comment: 10 pages, LaTe

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 $\beta_{c} \simeq 0.43932$ which almost coincides with $\beta_{c}$ 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.Comment: 10 pages,Latex,The figures are availabl

Vacuum structure of gauge theory on lattice with two parallel plaquette action

We perform Monte Carlo simulations of a lattice gauge system with an action which contains two parallel plaquettes. The action is defined as a product of gauge group variables over two parallel plaquettes belonging to a given three-dimensional cube. The peculiar property of this system is that it has strong degeneracy of the vacuum state inherited from corresponding gonihedric $Z_2$ gauge spin system. These vacuua are well separated and can not be connected by a gauge transformation. We measure different observables in these vacuua and compare their properties.Comment: 9 pages, 6 figures, Late

Loop Transfer Matrix and Loop Quantum Mechanics

We extend the previous construction of loop transfer matrix to the case of nonzero self-intersection coupling constant $\kappa$. The loop generalization of Fourier transformation allows to diagonalize transfer matrices depending on symmetric difference of loops and express all eigenvalues of $3d$ loop transfer matrix through the correlation functions of the corresponding 2d statistical system. The loop Fourier transformation allows to carry out analogy with quantum mechanics of point particles, to introduce conjugate loop momentum P and to define loop quantum mechanics. We also consider transfer matrix on $4d$ lattice which describes propagation of memebranes. This transfer matrix can also be diagonalized by using generalized Fourier transformation, and all its eigenvalues are equal to the correlation functions of the corresponding $3d$ statistical system.Comment: 22 pages, Latex, psfig,eps