World line

As a particle moves in ordinary space its corresponding point in 4-space describes a path known as the world line. The vector dxM represents the change in the position 4-vector for differential motion along a world line. The absolute magnitude of the 4-vector is described by the dot product with itself and defines the invariant world scalar... 

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. 

Any stationary point is said to trace out a world line parallel to the time axis. A point in motion follows a worldline at an angle 9 with respect to t, with a maximum of 0 = r/4 rad for a velocity of v = c. The space-like region outside the so-called light cone is usually ignored as physically meaningless since v > c. There is no explanation of why a stationary point has a world line and why it s always time-like and never space-like. 

Molecular structures do not redraw the Darwinian tree of evolution but rather the clonal distribution of the Genomic Potential Hypothesis, and that forces us to realize that the model of descent with variation cannot be correct. Instead one observes clusters within clusters, both, in taxon development and molecular sequence similarity. One sees parallel world lines of proteins and species evolving as if from nothing, at different levels in antiquity, ending in extinction or breaking the surface to the present. 

The world-line of each macro-organism begins after branching and segregation. 

Relative motion according to Lorentz transformation refers specifically to unaccelerated uniform motion and is therefore known as special relativity (SR). The theory which developed to also take acceleration into account is known as general relativity (TGR). Based on the demonstration, by Eotvos and others, that there is no difference between the inertial and the gravitational mass of an object, TGR also became the theory of the gravitational field. The world line of an accelerated object appears curved in a Minkowski... 

Einstein s pioneering explanation of the photoelectric effect led to the perception of a photon as a particle that moves through the vacuum with constant speed c. Both of these conclusions are probably wrong6. The photon is not a particle but a standing i.e. stationary) wave, as explained before. The interacting charges remain on the same relativistic world line and therefore effectively in contact. 

Since world lines in the tangent spaces at neighbouring points on the manifold are no longer parallel, as shown in figure 13, a static distribution of mass points must be inherently unstable. In time, interactions occur and produce a chaotic situation such as the motion in an ideal gas. This... 

Figure 5.13 Arrows to indicate non-parallel world lines of objects in a curved manifold.

The standard representation of special relativity in terms of Minkowski space, as in figure 3.2, acquires significant new meaning in the Thierrin scheme. As in figure 6 the light cone now defines allowed directions of the u-axis of material objects and u itself represents the world-line of special relativity. The objection against non-local interaction disappears. Instead, the... 

To understand why there is no mutual annihilation of matter and antimatter that come into contact at the interface, it is noted that these phases are separated in the direction of the fourth coordinate u. By construction, points separated by the interface have their world lines (u) in opposite directions and therefore with perceived time flow inverted. This time difference prevents contact and hence annihilation across the interface. 

Quantum Monte Carlo World Line Algorithms.613... 

Equation (47) is now just a classical sum of integrals and can be evaluated by Monte Carlo sampling in the intitial states i), the order of the perturbation n and the times Ti i = 1,..., n). This is best done by considering a graphical world line representation of the partition function (47) shown in Fig. 9. The zero-th order terms in the sum exp(—/ 77o) ) are given by straight world... 

The spin stiffness ps can be obtained from fluctuations of the winding numbers of the world lines [45], a measurement which obviously requires nonlocal moves that can change these winding numbers. 

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