Lorentz invariance

For concreteness, let us suppose that the universe has a temporal depth of two to accommodate a Fi edkin-type reversibility i.e. the present and immediate past are used to determine the future, and from which the past can be recovered uniquely. The RUGA itself is deterministic, is applied synchronously at each site in the lattice, and is characterized by three basic dimensional units (1) digit transition, D, which represents the minimal informational change at a given site (2) the length, L, which is the shortest distance between neighboring sites and (3) an integer time, T, which, while locally similar to the time in physics, is not Lorentz invariant and is not to be confused with a macroscopic (or observed) time t. While there are no a priori constraints on any of these units - for example, they may be real or complex - because of the basic assumption of finite nature, they must all have finite representations. All other units of physics in DM are derived from D, L and T. 

Lorentz-Invariance on a Lattice One of the most obvious shortcomings of a CA-based microphysics has to do with the lack of conventional symmetries. A lattice, by definition, has preferred directions and so is structurally anisotropic. How can we hope to generate symmetries where none fundamentally exist A strong hint comes from our discussion of lattice gases in chapter 9, where we saw that symmetries that do not exist on the microscopic lattice level often emerge on the macroscopic dyneimical level. For example, an appropriate set of microscopic LG rules can spawn circular wavefronts on anisotropic lattices. 

It is easy to invent rules that conserve particle number, energy, momentum and so on, and to smooth out the apparent lack of structural symmetry (although we have cheated a little in our example of a random walk because the circular symmetry in this case is really a statistical phenomenon and not a reflection of the individual particle motion). The more interesting question is whether relativistically correct (i.e. Lorentz invariant) behavior can also be made to emerge on a Cartesian lattice. Toffoli ([toff89], [toffSOb]) showed that this is possible. 

We now sketch a simple deterministic lattice gas model of diffusion that becomes exactly Lorentz invariant in the continuum limit. We follow Toffoli ([toff89], [tofiSOb]) and Smith [smithm90]. 

A Lorentz invariant scalar product can be defined in the linear vector space formed by the positive energy solutions which makes this vector space into a Hilbert space. For two positive energy Klein-... 

Lorentz invariant scalar product, 499 of two vectors, 489 Lorentz transformation homogeneous, 489,532 improper, 490 inhomogeneous, 491 transformation of matrix elements, 671... 

The second derivative function11 is Lorentz invariant. The wave equation in this terminology is... 

It is to be expected that the equations relating electromagnetic fields and potentials to the charge current, should bear some resemblance to the Lorentz transformation. Stating that the equations for A and (j> are Lorentz invariant, means that they should have the same form for any observer, irrespective of relative velocity, as long as it s constant. This will be the case if the quantity (Ax, Ay, Az, i/c) = V is a Minkowski four-vector. Easiest would be to show that the dot product of V with another four-vector, e.g. the four-gradient, is Lorentz invariant, i.e. to show that... 

In order to give a physical interpretation of special relativity it is necessary to understand the implications of the Lorentz rotation. Within Galilean relativity the three-dimensional line element of euclidean space (r2 = r r) is an invariant and the transformation corresponds to a rotation in three-dimensional space. The fact that this line element is not Lorentz invariant shows that world space has more dimensions than three. When rotated in four-dimensional space the physical invariance of the line element is either masked by the appearance of a fourth coordinate in its definition, or else destroyed if the four-space is not euclidean. An illustration of the second possibility is the geographical surface of the earth, which appears to be euclidean at short range, although on a larger scale it is known to curve out of the euclidean plane. 

The KG equation is Lorentz invariant, as required, but presents some other problems. Unlike Schrodinger s equation the KG equation is a second order differential equation with respect to time. This means that its solutions are specified only after an initial condition on bothand d /dt has been given. However, in contrast to k, d /dt has no direct physical interpretation [61]. Should the KG equation be used to define an equation of continuity, as was done with Schrodinger s equation (4), it is found to be satisfied by... 

It is important to notice that in Equation 10.11, only the sum of the three terms is Lorentz invariant. The first term corresponds to the interaction of the charge density with the external Coulomb potential and the last term can be written in the form... 

Under specific assumptions which are met when Lorentz invariance is broken via the chemical potential. 

Afuc.qC is the usual Lorentz invariant matrix element and can be written in the lowest order as... 

A consequent 5-dimensional treatment would require Unified Theory of Quantum Mechanics and General Relativity. This unified theory is not available now, and we know evidences that present QM is incompatible with present GR. The well-known demonstrative examples are generally between QFT and GR (e.g. the notion of Quantum Field Theory vacua is only Lorentz-invariant and hence come ambiguities about the existence of cosmological Hawking radiations [19]). But also, it is a fundamental problem that the lhs of Einstein equation is c-number, while the rhs should be a quantum object. 

The present chapter is devoted mainly to one of these new theories, in particular to its possible applications to photon physics and optics. This theory is based on the hypothesis of a nonzero divergence of the electric field in vacuo, in combination with the condition of Lorentz invariance. The nonzero electric field divergence, with an associated space-charge current density, introduces an extra degree of freedom that leads to new possible states of the electromagnetic field. This concept originated from some ideas by the author in the late 1960s, the first of which was published in a series of separate papers [10,12], and later in more complete forms and in reviews [13-20]. 

This extended form of the field equations should remain Lorentz-invariant. Physical experience supports such a statement, as long as there are no results that conflict with it. 

Returning to the form (3) of the space-charge current density, and observing that (j, ) is a 4-vector, the Lorentz invariance thus leads to... 

The introduction of the current density (3) in 3-space is, in fact, less intuitive than what could appear at first glance. As soon as the charge density (4) is permitted to exist as the result of a nonzero electric field divergence, the Lorentz invariance of a 4-current (7) with the time part namely requires the associated space part to adopt the form (3), that is, by necessity. 

These central concepts of tachyon theory also come out of the present approach. An alternative way to satisfy the condition (8) of Lorentz invariance is thus to replace the form (70) of the velocity vector C by... 

The present theory has been developed in terms of an extended Lorentz invariant form of the electromagnetic field equations, in combination with an addendum of necessary basic quantum conditions. From the results of such a simplified approach, theoretical models have been obtained for a number of physical systems. These models could thus provide some hints and first... 

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. 

It follows that the transverse held 1 = 12 1 is Lorentz-invariant in free space, and so is the cyclic theorem ... 

In conclusion, the homogeneous field equation of 0(3) electrodynamics is Lorentz-invariant, and all its classical solutions must be also Lorentz-invariant. The same result is obtained therefore in QED. 

The B cyclic theorem is a Lorentz invariant construct in the vacuum and is a relation between angular momentum generators [42], As such, it can be used as the starting point for a new type of quantization of electromagnetic radiation, based on quantization of angular momentum operators. This method shares none of the drawbacks of canonical quantization [46], and gives photon creation and annihilation operators self-consistently. It is seen from the B cyclic theorem ... 

This is also the relation obtained in the hypothetical rest frame. Therefore, the B cyclic theorem is Lorentz-invariant in the sense that it is the same in the rest frame and in the light-like condition. This result can be checked by applying the Lorentz transformation rules for magnetic fields term by term [44], The equivalent of the B cyclic theorem in the particle interpretation is a Lorentz-invariant construct for spin angular momentum ... 

It is concluded that the B(3) component in the field interpretation is nonzero in the light-like condition and in the rest frame. The B cyclic theorem is a Lorentz-invariant, and the product B x B<2> is an experimental observable [44], In this representation, B(3> is a phaseless and fundamental field spin, an intrinsic property of the field in the same way that J(3) is an intrinsic property of the photon. It is incorrect to infer from the Lie algebra (796) that Ii(3) must be zero for plane waves. For the latter, we have the particular choice (803) and the algebra (796) reduces to... 

M. C. Combourieu and J. P. Vigier, Absolute space-time and realism in Lorentz invariant interpretation of quantum mechanics, Phys. Lett. A 175(5), 269-272 (1993). 

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