Covariant Form

The last feature requires a new definition and formulation of SSP or FM in relativistic systems since spin is no more a good quantum number in relativistic theories spin couples with momentum and its direction changes during the motion. It is well known that the Pauli-Lubanski vector W1 is the four vector to represent the spin degree of freedom in a covariant form,... 

Here p, v = 0,1,2,3 x = (xo, x) and , , , dyl are arbitrary parameters that satisfy relations (50). Thus, we have represented Poincare-invariant ansatzes (52) in the explicitly covariant form. 

Ansatzes (53)-(55) are given in explicitly covariant form. This fact enables us to perform symmetry reduction of Eqs. (46) in a unified way. First, we give without derivation three important identities for the tensor [35] ... 

The last term vanishes since the A(3) photon is found to be very massive in an examination of this approach to electromagnetism embedded in an extended standard model. These issues will be discussed later. This photon decays away and so the A(3) potential is very short ranged 10 17 cm and is of no consequence to quantum optics. Let V x A( l = By11. Now compute Maxwell s equation, where 3> = V + ie/h) A + A(2>) is a covariant form of V... 

Using plane-wave solutions we get the following dispersion relation in a covariant form... 

Expressions (3.32) and (3.33) are solutions of equations which are written below in the simplest covariant form (see Section 8.4 and Appendix D)... 

The external resistance force of a particle in equation (7.1) is split into two terms, the first of which is equal to ((v,J — v3i r ) - the resistance in a corresponding monomer liquid, and the second one, T) , is connected with the neighbouring macromolecules and satisfies the equation, which can be written in the simplest covariant form (see Section 8.4 and Appendix D). 

Similarly, the internal resistance force Gf satisfies the covariant-form equation / c C1a ... 

This PWE renormalization method was also called noncovariant contrary to the covariant approach described above where the covariant procedure in 4-dimensional momentum space was used to separate out and cancel the divergent terms. In principle, the noncovariant procedure should not lead to any differences provided that both bound -term and counterterm are described in the same way. Such a difference may arise only if the counterterm, unlike the bound -term is written in covariant form [12]. 

Here the are the hydrodynamical forces and — are mtemal torques which may include interactions between non-neighboring segments that prevent overlapping configurations. The last term represents the diffusion due to thermal motions. Eq. (6.3) is written in a covariant form to show the invariance of the form under linear transformations. 

NONMEM was used to estimate the parameters for each bootstrap data set. Individual Bayesian parameters were generated. These estimates along with covariates formed a new data set. 

The key feature of the theory of QED—whether it is cast in nonrelativis-tic or fully covariant forms is that the electromagnetic field obeys quantum mechanical laws. A frequent first step in the construction of either version of the theory is the writing of the classical Lagrangian function for the interaction of a charged particle with a radiation field. For a particle of mass m, electronic charge —e, located at position vector q, and moving with velocity d /df c in a position-dependent potential V( ) subject to electromagnetic radiation described by scalar and vector potentials cp0) and a(r), at field point... 

Eh is the covariant form of the Hartree energy, which can be split into the Coulomb contribution e j and a transverse part Ej,... 

As required by special relativity, space and time variables should appear in a symmetric way and this requirement is most obvious in the covariant form of the Dirac equation ... 

From 7q = I4 we see immediately that V = cjoVcov- The equation (83) is called the Dirac equation in covariant form. It is best suited for investigations concerning relativistic invariance, because it me is a scalar (which by definition of a scalar is invariant under Lorentz transformations) and the term (7,5) is written in the form of a Minkowski scalar product (if 7 and d were ordinary vectors in Minkowski space, the invariance of this term would be already guaranteed by (81). 

The action of a relativistic free particle can be written on manifestly covariant form[41]... 

The Breit interaction may be regarded as the 0(a ) correction to the Coulomb interaction, and the covariant form vfj represents the complete interaction... 

Next, let s explore the consequences of these new relations of space and time relativistic transformations. If we rewrite (as above) the Lorentz transformation in a covariant form... 

So far we have just defined another four-component quantity Af, but by now it is not clear whether it properly transforms under Lorentz transformations in order to justify the phrase 4-vector. In order to prove the transformation property of the gauge field, we re-express the inhomogeneous Maxwell equations in Lorenz gauge as given by Eq. (2.138) in explicitly covariant form by employment of the charge-current density and the gauge field A, ... 

Eq. (3.198) just represents the inhomogeneous Maxwell equations in covariant form, cf. Eq. (3.172). We have thus derived the inhomogeneous Maxwell equations as the natural equations of motion for the gauge potential A. The sources, as described by the charge-current density are considered as external variables which do not represent dynamical degrees of freedom, i.e., only the action (or effect) of the sources on the gauge fields is taken into account. 

The Klein-Gordon equation for a freely moving particle can also be written in a compact, explicitly covariant form. 

Here, it is important to understand that is a four-component object consisting of (4x4)-matrices rather than a Lorentz scalar, since is not a Lorentz 4-vector. Moreover, although the equation already seems to be in covariant form, this still needs to be shown because we do not yet know how Y transforms under Lorentz transformations. In the following, we must determine the transformation properties of Y so that Eq. (5.54) is covariant under Lorentz transformations, which is a mandatory constraint for any true law of Nature. 

The covariant form of the Dirac equation of a freely moving particle, Eq. (5.54), allows us to incorporate arbitrary external electromagnetic fields. These fields can then be used to describe the interaction of electrons with light. But it must be noted that the treatment of light is then purely classical. A fully quantized description of light and matter on an equal footing is introduced only in quantum electrodynamics and discussed in chapter 7. 

For the Dirac equation with external electromagnetic fields in covariant form it is appropriate to define suitable 4-quantities, In this way, time and spa-... 

Now, we have a unified substitution pattern at hand, which also comprises the time-like coordinates. Substitution of Eq. (5.116) in the field-free Dirac equation as written in Eq. (5.54) yields the covariant form of the Dirac equation with external electromagnetic fields. 

Eq. (7.8) is the most general covariant form of the inhomogeneous Maxwell equations, which immediately imply the continuity equation dy.j = + div = 0 of section 5.2.3, and Eq. (7.9) is the covariant time-dependent Dirac equation in the presence of external electric and magnetic fields. The homogeneous Maxwell equations are automatically satisfied by the sole existence of... 

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