Null vector

If we operate on the vacuum state with any annihilation operator the result is the null vector ... 

It therefore follows from the transversality condition [Eq. (9-516)] that a physically admissible wave function u(k) can only be spacelike or a multiple of k , since k is a null vector. In the coordinate system in which... 

Every n vector can be represented as a point in an -dimensional coordinate space. The n elements of the vector are the coordinates along n basis vectors, such as defined in the previous section. The null vector 0 defines the origin of the coordinate space. Note that the origin together with an endpoint define a directed line segment or axis, which also represents a vector. Hence, there is an equivalence between points and axes, which can both be thought as geometrical representations of vectors in coordinate space. (The concepts discussed here are extensions of those covered previously in Sections 9.2.4 to 9.2.5.)... 

Figure 5.6. The molar concentration vector c of length K can be partitioned by a linear transformation into three parts ipT, a reacting vector of length Ny ycv, a conserved-variable vector of length /V ( >,o, and 0, a null vector of length K — Ny — The linear transformation matrix...

The only nonzero components of the PL vectors and are the longitudinal and time-like components. It follows that since is null, its magnitude is zero, and so 1 and Evl are null. This result is, in turn, consistent with the fact that the PL vector is a pseudovector, whereas is a null vector whose dual is null. 

Each such null vector may be considered an invariant or symmetry of the thermodynamic system, because it corresponds to an operation (change of extensive variables Xt) that produces no response in any intensive state variable and thus leaves the thermodynamic state unaltered (Sidebar 7.2). As described in Sidebar 10.3, these invariants also play a role somewhat analogous to overall rotations and translations ( null eigenmodes of the Hessian matrix) in the theory of molecular vibrations. 

Each such GD vector is a null vector satisfying the linear dependence condition [cf. 12.54b]... 

With T(GS> and T, one has the opportunity of removing from consideration any vector subspace containing near-null vectors, i.e., such vectors v that Sv w o. (There may be some difficulty in practice, if a library subroutine was used instead of handcoding, in the case of t(gs1). 

By substituting the previous definition for these spinors into (56) and using (79), it follows that A is a null vector, (e.g., AjAJ = 0), which, according to (78), implies l = l2= 0. 

Turning now to the case of polarization/rotation modulation, or continuous rotation of iT i 2rl2 corresponding to a continuous rotation of Cj through 20, there is a rotation of the resultant through 0. This correspondence is a consequence of the A 1A = 7 relation, namely, that if the unitary transformation of A or A 1 is applied separately the identity matrix will not be obtained. However, if the unitary transformation is applied twice, then the identity matrix is obtained and from this follows the remarkable properties of spinors that corresponding to two unitary transformations of, for example, 27i, namely, 471, one null vector rotation of 271 is obtained. This is a bisphere correspondence and is shown in Fig 3b. This figure also represents the case of polarization/rotation modulation—as opposed to static polarization/rotation. 

In eq. (1), v is not necessarily a lattice translation t, since w may be either the null vector 0 or the particular non-lattice translation associated with some screw axis or glide plane. If v C a VR, then there are no screw axes or glide planes among the symmetry elements... 

Repeat to obtain a trapezoidal array of m orthogonal null vectors xfU), each with leading coefficient unity. 

This agrees with the form of an MST null-vector expanded in the energy-dependent basis at k = < o + Ae. Because of numerical approximations inherent in practical calculations, it is generally inconvenient to diagonalize matrix co directly. The... 

The reactance matrix K is ai Q. Exact solutions require the matrix to be of rank n0, implying n linearly independent null-vectors as solutions of the homogeneous equations ma = 0. Because this algebraic condition is not satisfied in general by approximate wave functions, a variational method is needed in order to specify in some sense an optimal approximate solution matrix a. 

---