Euclidean time

In this equation, Dx(x) and S [x(x)] are, respectively, the position space path measure and the Euclidean time action. The centroid density also formally defines a classical-like effective potential, i.e., ... 

To find the values of A we shall use the instanton method turning to operate with the Euclidean time t —> —it. Taking initial and final times as Tj = -oo, Tf = 0 we present the amplitude A as... 

B. Beard and U. Wiese (1996) Simulations of discrete quantum systems in continuous Euclidean time. Phys. Rev. Lett. 77, p. 5130... 

The correlation function in Eq. (2.1) differs from the usual Euclidean time position correlation function C(t) because only paths with centroids at q contribute to the centroid-constrained propagator C (t, q ). However, one can obtain C(t) by averaging the centroid-constrained propagator over the normalized centroid density pXqc) that is. 

In general, it can be specified that the Euclidean time action functional for a reference system has a quadratic form in the path fluctuations variable q(r) such that... 

The BBM gas consists of an arbitrary number of hard spheres (or balls) of finite diameter that collide elastically both among themselves and with any solid walls (or mirrors) that they may encounter during their motion. Starting out on some site of a two-dimensional Euclidean lattice, each ball is allowed to move only in one of four directions (see figure 6.10). The lattice spacing, d = l/ /2 (in arbitrary units), is chosen so that balls collide while occupying adjacent sites. Unit time is... 

The one dimensional rules given in equations 8.105 and 8.106 can be readily generalized to a d dimensional Euclidean lattice. Let T] r, t) be the d dimensional analogue of the one dimensional local slope at lattice point r at time Addition of sand is then generated by the rule... 

To set up the problem and in order to appreciate more fully the difficulty in quantifying complexity, consider figure 12.1. The figure shows three patterns (a) an area of a regular two-dimensional Euclidean lattice, (b) a space-time view of the evolution of the nearest-neighbor one-dimensional cellular automata rule RllO, starting from a random initial state,f and (c) a completely random collection of dots. 

Another simple example is the traiditional two-dimensional random-walk on a four-neighbor Euclidean lattice [toff89]. Despite the fact that the underlying lattice is symmetric only with respect to rotations that are multiples of 90 deg, the probability distribution p(s, y) for a particle that begins its random walk at the origin becomes circularly symmetric in the limit as time t —> oo p x,y,t) —> (see figure 12.12). 

Fig. 12.12. A circularly symmetric Gaussian probability distribution p x,y) describing a two-dimensional random walk emerges for large times on the macroscopic level, despite the fact that the underlying Euclidean lattice is anisotropic.

Jourjine [jour85] generalizes Euclidean lattice field theory on a d-dimensional lattice to a cell complex. He uses homology theory to replace points by cells of various dimensions and fields by functions on cells, the cochains, in hopes of developing a formalism that treats space-time as a dynamical variable and describes the change in the dimension of space-time as a phase transition (see figure 12.19). 

The technology of proximity indices has been available and in use for some time. There are two general types of proximity indices (Jain and Dubes, 1988) that can be distinguished based on how changes in similarity are reflected. The more closely two patterns resemble each other, the larger their similarity index (e.g., correlation coefficient) and the smaller their dissimilarity index (e.g., Euclidean distance). A proximity index between the ith and th patterns is denoted by D(i, j) and obeys the following three relations ... 

Giddings (1990) presented a derivation applicable to both the planar format such as TLC that is distance-based and the comprehensive multidimensional separations that are time-based. The resolution was shown to be equal to the Euclidean norm of zone resolution components. This can be summarized as... 

The final objective is an equation that relates a geometrical object representing the curvature of space-time to a geometrical object representing the source of the gravitational field. The condition that all affine connections must vanish at a euclidean point, defines a tensor [41]... 

In the real world the stress tensor never vanishes and so requires a nonvanishing curvature tensor under all circumstances. Alternatively, the concept of mass is strictly undefined in flat Minkowski space-time. Any mass point in Minkowski space disperses spontaneously, which means that it has a space-like rather than a time-like world line. In perfect analogy a mass point can be viewed as a local distortion of space-time. In euclidean space it can be smoothed away without leaving any trace, but not on a curved manifold. Mass generation therefore resembles distortion of a euclidean cover when spread across a non-euclidean surface. A given degree of curvature then corresponds to creation of a constant quantity of matter, or a constant measure of misfit between cover and surface, that cannot be smoothed away. Associated with the misfit (mass) a strain field appears in the curved surface. 

Interpreted, as it is, within the standard model, Higgs theory has little meaning in the real world, failing, as it does to relate the broken symmetry of the field to the chirality of space, time and matter. Only vindication of the conjecture is expected to be the heralded observation of the field bosons at stupendous temperatures in monstrous particle accelerators of the future. However, the mathematical model, without cosmological baggage, identifies important structural characteristics of any material universe. The most obvious stipulation is to confirm that inertial matter cannot survive in high-symmetry euclidean space. 

There is no evidence that Minkowski space is flat on the large scale. The assumption of euclidean Minkowski space could therefore be, and probably is an illusion, like the flat earth. In fact, there is compelling evidence from observed spectroscopic red shifts that space is curved over galactic distances. These red shifts are proportional to distances from the source, precisely as required by a curved space-time[52j. An alternative explanation, in terms of an expanding-universe model that ascribes the red shifts to a Doppler... 

There also exists convincing internal evidence that real Minkowski space must be curved. Euclidean 4-space is commonly represented diagrammati-cally to distinguish between time and space axes as in figure 4. 

Figures 8a and 8b present the simulated current transients obtained from the self-affine fractal interfaces of r/ = 0.1 0.3 0.5 and r] = 1.0 2.0 4.0, respectively, embedded by the Euclidean two-dimensional space. It is well known that the current-time relation during the current transient experiment is expressed as the generalized Cottrell equation of Eqs. (16) and (24).154 So, the power exponent -a should have the value of - 0.75 for all the above self-affine fractal interfaces.

In figure 3 the dependence pA(t) in log-log coordinates, corresponding to the relationship (4), for the reesterification reaction in TBT presence is adduced. As can be seen, this dependence breaks down into two linear parts with different slopes. For the first part (/<90 min.) the slope is equal to -0,75, i.e., corresponded to the equation (6) for reaction proceeding in three-dimensional Euclidean space (d= 3). For the second part (/>90 min.) the slope is equal to 3, i.e., not corresponded to possible value of this exponent for recombination reaction or other analogous reactions, for which the value a is limited from above by the value 1,5 [2-4, 9], This means, that for the considered reesterification reaction times smaller of 90 min. it s necessary to identify as short times, i.e., on this temporal interval reactive particles concentration decay controls by local fluctuations of TBT distribution, and times equal or... 

The angular diameter of the observed patches is about 1°. Their (real) linear diameter can be estimated as ct, where c is the speed of light and t = 300 000 years is the time when the Universe became transparent. An angular diameter of 1° measured 14 billion years later means that the light rays remained parallel over the whole path and hence that the Universe is globally Euclidean. 

Pesic, P. D. (1993) Euclidean hyperspace and its physical significance. Nuovo Cimento B. 108B, ser. 2(10) 1145—53. (Contemporary approaches to quantum field theory and gravitation often use a 4-D space-time manifold of Euclidean signature called hyperspace as a continuation of the Lorentzian metric. To investigate what physical sense this might have, the authors review the history of Euclidean techniques in classical mechanics and quantum theory.)... 

Other careful computer simulations [33] focused more attention on the two diffusion-controlled reactions A + B —> 0 and A + A — 0, on both fractal and Id lattices. In the case of the Euclidean space it was well demonstrated that achievement of the theoretical limit of a = 0.25 is a quite long-time... 

Let us assume the existence of a four-dimensional (4D) flat Euclidean space E = (u,x,y,z), where the time dimension u = vut behaves exactly the same as the three spatial dimensions [102, 104]. Further, let S be filled with a fluid of preons (=tiny particles of mass m and Planck length dimensions). These particles are in continual motion with speed "V = (vH, vx, vy, vz) = (v , V). No a priori limits are set on the speed vu of preons along the u -axis.8... 

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