Four-vectors

Now the Lagrangean associated with the nuclear motion is not invariant under a local gauge transformation. Eor this to be the case, the Lagrangean needs to include also an interaction field. This field can be represented either as a vector field (actually a four-vector, familiar from electromagnetism), or as a tensorial, YM type field. Whatever the form of the field, there are always two parts to it. First, the field induced by the nuclear motion itself and second, an externally induced field, actually produced by some other particles E, R, which are not part of the original formalism. (At our convenience, we could include these and then these would be part of the extended coordinates r, R. The procedure would then result in the appearance of a potential interaction, but not having the field. ) At a first glance, the field (whether induced internally... 

Throughout, the space coordinates and other vectorial quantities are written either in vector fomi x, or with Latin indices k— 1,2,3) the time it) coordinate is Ap = ct. A four vector will have Greek lettered indices, such as Xv (v = 0,1,2,3) or the partial derivatives 0v- m is the electronic mass, and e the charge. 

Contravariant and covairant four vectors are connected through the metric = diag (1,-1,-1,-1) by... 

As noted above, jC in Eq. (154) arises from terras in which p 7 v. The corresponding contribution to the four current was evaluated in [104,323] and was shown to yield the polarization cuirent. Our result is written in teims of the magnetic field H and the electric field E, as well as the spinor four-vector v / and the vectorial 2x2 sigma raatiices given in Eq. (151). 

In going from the Schrodinger equation to the Klein-Gordon equation, we obtain the neeessary symmetry between spaee and time by having seeond-order derivatives throughout. It is usually written in a form that brings out its relativistic invarianee by using what is ealled/our-vector notation. We define a four-vector X to have components... 

Vierervektor, m. four-component vector, four-vector. 

Each side of this rule can be completed to form a Lorentz four-vector by including the following equation ... 

We shall denote the space time coordinates by a (which as a four-vector is denoted by a light face x) with x° — t, x1 = x, af = y, xz = z x — ai0,x. We shall use a metric tensor grMV = gliV with components... 

The defining equations (9-133) and (9-134) for H and P reflect the fact that the particles are free and do not interact with one another. The total energy and total momentum of the system is, therefore, the sum of energies and momenta of the individual particles as indicated by Eqs. (9-135) to (9-137). We shall see that we may consider the operators H, P as the time and space components of the four-vector P ... 

One verifies that upon representing the gBV as the four-dimensional curl of the four vector StB(a )... 

We shall adopt Eqs. (9-510) and (9-511) as the covariant wave equation for the covariant four-vector amplitude 9ttf(a ) describing a photon. The physically realizable amplitudes correspond to positive frequency solutions of Eq. (9-510), which in addition satisfy the subsidiary condition (9-511). In other words the admissible wave functions satisfy... 

We sliall call a four-vector ( ) transversal if it satisfies a 0, in... 

We have noted that if is the energy-momentum four vector of a photon (i.e., P = 0, k0 > 0) there exist only two other linearly independent vectors orthogonal to ku. We shall denote these as tft k) and ejf fc). They satisfy... 

The (four-vector). au(x) represents the result of some local measurement at the point x performed by a (Lorentz) observer 0. An observer O (related to 0 by a Lorentz transformation x = Ax) describes this measurement by... 

This is verified by applying U(a,A) to the left and U(a,A) l to the right of both sides of Eqs. (11-142) and (11-97), mid making use of the transformation properties of (11-184) and (11-185) of the fields. Equations (11-241) and (11-242) were to be expected. They guarantee that transformation properties of the one-particle states. Consider the one-negaton state, p,s,—e>. Upon taking the adjoint of Eq. (11-241) and multiplying by ya we obtain... 

Consider next the current operator ju(x). The correspondence principle suggests that its form is j ( ) = — (e/2)( ( )yB, (a )]. Such a form foijn(x) does not satisfy Eq. (11-477). In fact, due to covariance, the spectral representation of the vacuum expectation value of 8u(x)Av(x), where ( ) is an arbitrary four-vector, is given by... 

Chaise-current four vector, 545 Chebyshev approximation, 96 Chebychev inequality, 124 Chemoft, H., 102,151 Cholesky method, 67 Circuit, 256 matrix, 262... 

Introduction of the half-integral spin of the electrons (values h/2 and —fe/2) alters the above discussion only in that a spin coordinate must now be added to the wavefunctions which would then have both space and spin components. This creates four vectors (three space and one spin component). Application of the Pauli exclusion principle, which states that all wavefunctions must be antisymmetric in space and spin coordinates for all pairs of electrons, again results in the T-state being of lower energy [equations (9) and (10)]. 

The value of the dot product is a measure of the coalignment of two vectors and is independent of the coordinate system. The dot product therefore is a true scalar, the simplest invariant which can be formed from the two vectors. It provides a useful form for expressing many physical properties the work done in moving a body equals the dot product of the force and the displacement the electrical energy density in space is proportional to the dot product of electrical intensity and electrical displacement quantum mechanical observations are dot products of an operator and a state vector the invariants of special relativity are the dot products of four-vectors. The invariants involving a set of quantities may be used to establish if these quantities are the components of a vector. For instance, if AiBi forms an invariant and Bi are the components of a vector, then Az must be the components of another vector. 

Four-vectors for which the square of the magnitude is greater than or equal to zero are called space-like when the squares of the magnitudes are negative they are known as time-like vectors. Since these characteristics arise from the dot products of the vectors with reference to themselves, which are world scalars, the designations are invariant under Lorentz transformation[17], A space-like 4-vector can always be transformed so that its fourth component vanishes. On the other hand, a time-like four-vector must always have a fourth component, but it can be transformed so that the first three vanish. The difference between two world points can be either space-like or time-like. Let be the difference vector... 

The invariance of the general wave equation (22) is a special consequence of a more general four-vector invariance involving the four-gradient defined as... 

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... 

By defining a four-vector with the components of the current-density vector J and charge density, i.e. 

The generation of invariants in the Lorentz transformation of four-vectors has been interpreted to mean that the transformation is equivalent to a rotation. The most general rotation of a four-vector, defined as the quaternion q = w + ix + jy + kz is given by [39]... 

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,... 

From the values x(t) the Jacobian matrix J[x t) can be calculated. For example, in a system of order n = 4, four vectors of dimension four must be determined. To carry out this, a matrix of 16 x 16 elements is defined as follows ... 

---