Euclidean symmetry

The unbounded domain causes difficulties, on the one hand, via continuous spectrum which may - and does - interfere with the pure point spectrum required for our bifurcation analysis. The unbounded domain is necessary, on the other hand, to correctly incorporate the Euclidean symmetry of the problem under translations and rotations. In section 3.2 below we will specifically address competing and coexisting pinning and drifting phenomena. Such phenomena are predicted for the light-sensitive BZ system, when full Euclidean symmetry is broken towards a mere translational lattice symmetry, by choosing a spatially periodic lighting. 

An important feature of dynamics and bifurcations of spiral waves is the Euclidean symmetry SE 2) of the plane. The special Euclidean symmetry group SE 2) consists of all planar translations and rotations. Barkley [5] was the first to notice the relevance of this group for meandering spiral wave dynamics. Indeed, let x K, be any solution of a spatially homo-... 

As was justified in section 3.2.4, the angle a denotes the phase and. 2 the position of the spiral tip. The Palais section coordinate n U is absent here, because the critical spectrum is now three-dimensional, only, and is accounted for by the three-dimensional group SE 2) itself. Therefore the center manifold M. is a graph over the group coordinates e ° ,z) G SE 2). A rigorous derivation of the reduced equation (3.22) has indeed been achieved in [25, 33], under the assumption that the unperturbed spiral wave n (-) is spectrally stabie with the exception of a triple critical eigenvalue due to symmetry see theorem 1. Note that the nonlinearities -y a,z,s) and h a,z,s) obey the lattice symmetry relic of full Euclidean symmetry, namely... 

D. Barkley. Euclidean symmetry and the dynamics of rotating spiral waves. Phys. Rev. Lett, 72 164-167, 1994. 

The important result is the obvious symmetry between TM1/ and R u as shown in (42) and (43). Both of these tensors vanish in empty euclidean space and a reciprocal relationship between them is inferred The presence of matter causes space to curl up and curvature of space generates matter. 

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. 

Cosmic structure based on a vacuum interface has been proposed before [49, 7] as a device to rationalize quantum events. To avoid partitioning the universe into regions of opposite chirality the two sides of the interface are joined together with an involution. The one-dimensional analogue is a Mobius strip. Matter on opposite sides of the interface has mutually inverted chirality - matter and anti-matter - but transplantation along the double cover gradually interconverts the two chiral forms. The amounts of matter and anti-matter in such a universe are equal, as required by symmetry, but only one form is observed to predominate in any local environment. Because of the curvature, which is required to close the universe, space itself is chiral, as observed in the structure of the electromagnetic field. This property does not appear in a euclidean Robertson-Walker sub-space. 

All the laws of physics are easier to accept (and even to understand) when the underlying symmetry of the problem is appreciated. For example, classical Euclidean space is isotropic and a physical system is invariant to any rotation in this space. By this we mean that all the measurable properties of the system are unaffected by the rotation. An investigation of the behaviour of a quantum state under such rotations allows the properties of the state to be defined. These properties are most succinctly expressed as quantum numbers. Although quantum numbers are frequently used to label the eigenstates or eigenvalues of a molecule, they really carry information about the symmetry properties of the associated eigenfunc-tions. 

The relationship between the exponent v, (v = lnp/lnfc), and the fractal dimension Dp of the excitation transfer paths may be derived from the proportionality and scaling relations by assuming that the fractal is isotropic and has spherical symmetry. The number of pores that are located along a segment of length Lj on the jth step of the self-similarity is / , — pi. The total number of pores in the cluster is S nj (pJf, where d is the Euclidean dimension... 

Time reversal transformation, t - — t This is like space inversion and most likely space-time inversion is a single symmetry that reflects the local euclidean topology of space, observed as the conservation of matter. 

Note that interpretations of the time-reversal experiments are only valid in strictly euclidean space-time. This condition is rarely emphasized by authors who state that all laws of physics are time-reversible, except for the law of entropy. Fact is that entropy is the only macroscopic state function which is routinely observed to be irreversible. One common explanation is to hint that entropy is an emergent property of macro systems and hence undefined for microsystems. Even so, the mystery of the microscopic origin of entropy remains. A plausible explanation may be provided if the assumed euclidean geometry of space-time is recognized as an approximate symmetry as demanded by general relativity. 

Not only the laws of Nature but also all major scientific theories are statements of observed symmetries. The theories of special and general relativity, commonly presented as deep philosophical constructs can, for instance, be formulated as representations of assumed symmetries of space-time. Special relativity is the recognition that three-dimensional invariances are inadequate to describe the electromagnetic field, that only becomes consistent with the laws of mechanics in terms of four-dimensional space-time. The minimum requirement is euclidean space-time as represented by the symmetry group known as Lorentz transformation. 

The universally observed flow of time is another example of a broken symmetry. A theoretical formulation of this proposition is not known, but in principle it should parallel the theory of superconductivity. A high-symmetry state could be associated with Euclidean Minkowski space that spontaneously transforms into a curved manifold of lower symmetry. In this case the hidden symmetry emerges from a Lagrangian which is invariant under the temporal evolution group... 

In essence, real world-space is not Euclidean and space is generally curved into the time dimension, consistent with the theory of general relativity. The curvature may not be sufficient to become obvious in a local context. However, it is sufficient to break the time-reversal symmetry that seems to characterize the laws of physics. Not only does it cause perpetual time flow with respect to all mass, but actually identifies a fixed direction for this flow. It creates an arrow of time and thereby eliminates an inconsistency in the logic of physics how reversible microscopic laws can underpin an irreversible macroscopic world. General curvature of space breaks the time-reversal symmetry and produces chiral space, manifest in the right-hand... 

Altmann considered two types of operations that belong to the Schrodinger subgroup the Euclidean and the discrete symmetry operations. Euclidean operations are those that change the laboratory axes, leaving the Hamiltonian operator invariant. They are translations and rotations of the whole molecule, in free space, in which the x,y,z molecular axes are kept constant. A discrete symmetry operation is a change of the molecular axes in such away as to induce permutations of the coordinates of identical particles [10]. 

Altmann remarked that, in free space, the Euclidean operations are not of physical interest. Therefore the Euclidean operations will be 2issimilated with the identity. Besides, he stated that the discrete symmetry operations are purely changes of labelling, especifically they are not motions of atoms. The Schrodinger subgroup may be then assimilated with the symmetry point group of the molecule in a fixed configuration. 

Now, any type of motion or symmetry operation which leaves a lattice invariant may be written in matrix notation. For example, if a lattice point is moved from point Q in Euclidean space with coordinates (xj, yi, Zi) to point P with coordinates (x2, ya, Z2), this can be written as ... 

Alternative symmetry deficiency measures of fuzzy sets are defined following the treatment of symmetry deficiency of ordinary subsets of finite n-dimensional Euclidean spaces, introduced earlier. To this end, we shall use certain concepts derived as generalizations of concepts in crisp set theory. 

Consider a crisp or fuzzy subset A of the Euclidean space X, a (possibly approximate) symmetry element R, and the associated symmetry operator R. A fixed point of R is chosen as a reference point c e A, and a local Cartesian coordinate system of origin c is specified, with coordinate axes oriented according to the usual conventions with respect to the symmetry operator R, as described for crisp sets in Section XIII. 

Choose m as the smallest positive integer that satisfies the condition R = E of Eq. (171) and take the powers R° = E,R,R, ...,R " of the symmetry operator R. Following the technique described in Section XIII, partition the Euclidean space X into m segments A q, Aj,..., j, where... 

The regular orbit displayed in Figure 2.7, is the geometry on the unit sphere such that the bond length , the Euclidean distance between adjacent vertices, is constant. This restriction is not necessary from a symmetry viewpoint it may be relaxed subject only to the requirement that the local four, three and two-fold symmetries are maintained. One important example of such a relaxation occurs for the regular orbit of the Oh Crystallographic point group. In the simplest model crystal of Oh point symmetry, the primitive cubic array, for example, as in cubium, lattice points are distributed as dictated by the lattice vector Rmnp such that... 

Conformal symmetry is very common in nature e.g. we can find it in the nautilus shell and the sunflower. These structures are clearly ordered, even if they do not give sharp diffraction patterns. Here the repetition is non-Euclidean, on a logarithmic spiral (nautilus), or on a torus (sunflower). We are inclined to say that any kind of repetition, conformal or isometric, even in non-Euclidean space, is ordered. However, classification of these more chaotic structures, as for liquids, is less certain. It may be that a liquid can be described as a structure with some of the characteristics of conformal symmetry or perhaps by a representation even more exotic, like a manifold of constant negative curvature. 

---