General Space-Time Symmetries

Based on the discussion so far we already understand in the context of classical Newtonian mechanics that the investigation of fundamental s)nnmetries of space, time and fundamental physical equations of motion provides deep insight into physical truth. We will come back to the question of the correct relativistic covariance principle valid for all physics in chapter 3. It may be anticipated here that the study of symmetry and covariance requirements will finally provide the means to construct the basic equations of motion for a freely moving electron in chapter 5 and for an electron in external scalar and vector fields in chapter 6. 

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The broken symmetries of chapter 1, assumed responsible for shaping the physical world, refer to the symmetry of the vacuum and thus finally to the geometry of space-time. It is not immediately obvious that chemical theories could also be reduced to the same cause. While physics produced quantum theory and general relativity, the fundamental contribution from chemistry was the periodic table of the elements. Although the structure of individual atomic nuclei may be considered shaped by local space-time symmetry, the functional relationship between different nuclides needs further study. 

Unlike other branches of physics, thermodynamics in its standard postulation approach [272] does not provide direct numerical predictions. For example, it does not evaluate the specific heat or compressibility of a system, instead, it predicts that apparently unrelated quantities are equal, such as (1 A"XdQ/dP)T = - (dV/dT)P or that two coupled irreversible processes satisfy the Onsager reciprocity theorem (L 2 L2O under a linear optimization [153]. Recent development in both the many-body and field theories towards the interpretation of phase transitions and the general theory of symmetry can provide another plausible attitude applicable to a new conceptual basis of thermodynamics, in the middle of Seventies Cullen suggested that thermodynamics is the study of those properties of macroscopic matter that follows from the symmetry properties of physical laws, mediated through the statistics of large systems [273], It is an expedient happenstance that a conventional simple systems , often exemplified in elementary thermodynamics, have one prototype of each of the three characteristic classes of thermodynamic coordinates, i.e., (i) coordinates conserved by the continuous space-time symmetries (internal energy, U), (ii) coordinates conserved by other symmetry principles (mole number, N) and (iii) non-conserved (so called broken ) symmetry coordinates (volume, V). 

It turns out that, in the CML, the local temporal period-doubling yields spatial domain structures consisting of phase coherent sites. By domains, we mean physical regions of the lattice in which the sites are correlated both spatially and temporally. This correlation may consist either of an exact translation symmetry in which the values of all sites are equal or possibly some combined period-2 space and time symmetry. These coherent domains are separated by domain walls, or kinks, that are produced at sites whose initial amplitudes are close to unstable fixed points of = a, for some period-rr. Generally speaking, as the period of the local map... 

Clearly, the above procedure can be continued (in principle) as many times as required. Thus, if the wave function includes n = —4 3 paths, we have simply to dehne the function I 4((t)) = —+ 8ti), and then map onto the (j) = 0 16ti cover space, which will unwind the function completely. In general, if there are h homotopy classes of Feynman paths that contribute to the Kernel, then one can unwind ihG by computing the unsymmetrised wave function ih in the 0 2hn cover space. The symmetry group of the latter will be a direct product of the symmetry group in the single space and the group... 

The absence of an E(3) field does not affect Lorentz symmetry, because in free space, the field equations of both 0(3) electrodynamics are Lorentz-invariant, so their solutions are also Lorentz-invariant. This conclusion follows from the Jacobi identity (30), which is an identity for all group symmetries. The right-hand side is zero, and so the left-hand side is zero and invariant under the general Lorentz transformation [6], consisting of boosts, rotations, and space-time translations. It follows that the B<3) field in free space Lorentz-invariant, and also that the definition (38) is invariant. The E(3) field is zero and is also invariant thus, B(3) is the same for all observers and E(3) is zero for all observers. 

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. 

It has already been shown that for constant a this invariance (symmetry) implies conservation of the charge of a free particle. In general relativity, which is based on a curved manifold rather than flat space with a globally fixed coordinate system, each point has its own coordinate system and hence its own gauge factor. This means that the gauge factor a is no longer a constant, but a function of space-time, i.e. 

The idea of Pollicott-Ruelle resonances relies on this mechanism of spontaneous breaking of the time-reversal symmetry [20, 21]. The Polhcott-Ruelle resonances are generalized eigenvalues sj of LiouviUian operator associated with decaying eigenstates which are singular in the stable phase-space directions but smooth in the unstable ones ... 

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