Lorentz proper

To make these notions precise, the transformation properties of the wavefunction x under spatial and time translations as well as under spatial rotations and pure Lorentz transformations must be specified and it must be shown that the generators of these transformations form a unitary representation of the group of translations and proper Lorentz transformations. This can in fact be shown5 but will not be here. 

We shall only mention the fact, that a unitary representation of the inhomogeneous proper Lorentz group is exhibited in this Hilbert space through the following identification of the generators of the... 

Now one readily verifies that for proper Lorentz transformations the matrices... 

Hence, under a proper homogeneous Lorentz transformation -without time inversion the quantity transforms like a scalar ... 

The previous results become somewhat more transparent when consideration is given to the manner in which matrix elements transform under Lorentz transformations. The matrix elements are c numbers and express the results of measurements. Since relativistic invariance is a statement concerning the observable consequences of the theory, it is perhaps more natural to state the requirements of invariance as a requirement that matrix elements transform properly. If Au(x) is a vector field, call... 

Lorentz group, inhomogeneous proper, unitary representation in Hilbert space, 497... 

This expression demonstrates use of the Einstein summation convention 6. The significance of r is made clear by examining a particle momentarily at rest in a Lorentz system. The components of the vector, transformed dx = (0, 0, 0, icdt ) and dr2 = —(1 /c dx dx = [dt )2. Thus dr is the time interval on a clock travelling with the particle and is therefore referred to as the interval of the particle s proper time or world time. The relationship between dr and an interval of time as measured in a given Lorentz system can be derived directly by expanding the equation... 

The variational formalism makes it possible to postulate a relativistic Lagrangian that is Lorentz invariant and reduces to Newtonian mechanics in the classical limit. Introducing a parameter m, the proper mass of a particle, or mass as measured in its own instantaneous rest frame, the Lagrangian for a free particle can be postulated to have the invariant form A = mulxiilx = — mc2. The canonical momentum is pf, = iiiuj, and the Lagrangian equation of motion is... 

The leading idea is that the custoinarj combination of space and time — kinematics— mnst be abandoned. There is no absolute time, but just as every moving system has its proper co-ordinates x, y, z, so it has also a proper time t, which has to be transformed as well as the co-ordinates when we pass to a new system. The equations defining this so-called Lorentz transformation for two systems moving in the sc-direction with the relative velocity v are... 

We can arrive at this result from the consideration that the velocity defined by the components dxjdt, dyjdt, dzjdt (or the momentum obtained from this velocity on multiplication by the mass) cannot, in view of Lorentz s transformation, be regarded as a vector, since the differential dt in the denominator is also transformed. We obtain a serviceable covariant definition if we replace dt by dt, where dt is the element of the proper time of the particle, i.e. the time measured in the system of reference in which the particle is at rest. The relation between dt and dt is found by taking the derivative of t y... 

The group of Poincare transformations consists of coordinate transformations (rotations, translations, proper Lorentz transformations...) linking the different inertial frames that are supposed to be equivalent for the description of nature. The free Dirac equation is invariant under these Poincare transformations. More precisely, the free Dirac equation is invariant under (the covering group of) the proper orthochronous Poincare group, which excludes the time reversal and the space-time inversion, but does include the parity transformation (space reflection). 

We finally note that more general Lorentz transformations are now obtained easily because any proper Lorentz transformation can be written in a unique way as the product of a boost and a rotation. 

Lorentz transformations with detA = 1 are said to be proper Lorentz transformations, and those with A q > 0 are said to be orthochronous. The class of proper orthochronous transformations is a subgroup of -Sf. 

Infinitesimal transformations The proper inhomogeneous Lorentz transformations in close to the identity are of particular importance they have the form... 

In Section VI we study in detail two fast short-lived vibration mechanisms b and c, which concern item 2. The dielectric response to the elastic rotational vibrations of hydrogen-bonded (HB) polar molecules and to translational vibrations of charges, formed on these molecules, is revealed in terms of two interrelated Lorentz lines. A proper force constant corresponds to each line. The effect of these constants on the spectra of the complex susceptibility is considered. The dielectric response of the H-bonded molecules to elastic vibrations is shown to arise in the far IR region. Namely, the translational band (T-band) at the frequency v about 200 cm-1 is caused by vibration of charges, while the neighboring V-band at v about 150 cm-1 arises due to elastic rigid-dipole reorientations. In the case of water these bands overlap, and in the case of ice they are resolved due to longer vibration lifetime. 

Even that is not absolutely clear. There are at least two preferred frames moving with respect to each other one is related to the local DM cluster, while the other is related to the isotropic CMB. They suggest a different distance scale and both can in principle lead to periodic variations. Any periodic effect induced by the dark- matter-determined frame has no relation to a violation of the Lorentz invariance. With the CMB that is not clear. CMB proper is a kind of environment. Meanwhile, if there is any fundamental violation of the Lorentz invariance, we would expect that violation determined the frame where the Big Bang happened and thus where the CMB is isotropic. So this frame is specific because of a possible violation and because of environmental effects, related to violations of the Lorentz invariance in the remote past... 

By relating the rest mass to the internal motion, quantum theory brings an insight into the bearing of such relativistic concepts as Lorentz-invariant, Minkowski s proper interval Xq. As the property moC is the residual momentum when the linear partp is subtracted from the total entity m c (Eq. 2.7b), the property xo is the residual interval when the space coordinate is subtracted from the time coordinate c f (Eq. 2.5b). 

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