Spinor invariant

Brinkman, H.C., 1956, Applications of Spinor Invariants in Atomic Physics (North-Holland, Amsterdam). 

The requirement of Postulate 3 that the equations of motion be form-invariant (i.e., that fix ) and A u(x ) satisfy the same equation of motion with respect to a as did fx) and Au x) with respect to x demands that the field variables transform under such transformations according to a finite dimensional representation of the Lorentz group. In other words it demands that transform like a spinor... 

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. 

It is easy to see that Eq. (1) is invariant with respect to a similarity transformation in the four-dimensional spinor space. If k and E are solutions of Eq. (1) then... 

Thus, to completely solve the problem of symmetry reduction within the framework of the formulated algorithm above, we need to be able to perform steps 3-5 listed above. However, solving these problems for a system of partial differential equations requires enormous amount of computations moreover, these computations cannot be fully automatized with the aid of symbolic computation routines. On the other hand, it is possible to simplify drastically the computations, if one notes that for the majority of physically important realizations of the Euclid, Galileo, and Poincare groups and their extensions, the corresponding invariant solutions admit linear representation. It was this very idea that enabled us to construct broad classes of invariant solutions of a number of nonlinear spinor equations [31-33]. 

Noether s theorem will be proved here for a classical relativistic theory defined by a generic field , which may have spinor or tensor indices. The Lagrangian density (, 9/x) is assumed to be Lorentz invariant and to depend only on scalar forms defined by spinor or tensor fields. It is assumed that coordinate displacements are described by Jacobi s theorem S(d4x) = d4x 9/xgeneral variation of the action integral, evaluated over a closed space-time region 2, is... 

Consequently the spinors < that make up Q must transform according to the irreducible representations or corepresentations (if the invariance group contains antiunitary operators) of the invariance group. 

A significant point here is that it is not the squared invariant ds2 that is to underlie the covariance of the laws of nature. It is rather the linear invariant ds that plays this role. How, then, do we proceed from the squared metric to the linear metric That is to say, how does one take the square root of ds2l The answer can be seen in Dirac s procedure, when he factorized the Klein-Gordon equation to yield the spinor form of the electron equation in wave mechanics -the Dirac equation. Indeed, Dirac s result indicated that by properly taking the square root of ds2 in relativity theory, extra spin degrees of freedom are revealed that were previously masked. 

Thus the identification (17) <[)a (/<), ) is not to be understood as form-invariant regarding the dependence (17) of the spinor variables < )a on the tensor variables FpV in any other Lorentz frame. In other words, the Lorentz transformation of... 

The terms of the respective (reducible) tensor and the (irreducible) spinor expressions of the electromagnetic laws that must correspond in all Lorentz frames are those that identify with physical observations. These are the conservation laws of electromagnetism. They derive, in mm, from the invariants of the theory. 

According to the spinor calculus [19], further invariants, in addition to (22), that correspond with the standard invariants of electromagnetic field theory, are... 

The starting point then to achieve the factorization of the Einstein equations is the factorized differential line element in the quaternion form, ds = q,1(x)dxll, where qyi are a set of four quaternion-valued components of a 4-vector. Thus ds is, geometrically, a scalar invariant, but it is algebraically a quaternion. As such, it behaves like a second-rank spinor of the type v / v /, where / is a two-component spinor variable [17]. 

In the presence of an external field the Dirac equation will not be invariant, because an external field is not invariant under all Poincare transformations (unless it is a constant). But at least we can expect that the Poincare transformed spinor < (x) — M t/j(A (x — a)) is a solution of the Dirac equation with an appropriately transformed potential matrix Here it has to be assumed that... 

To one who is familiar with nonrelativistic quantum mechanic it may appear quite clear what is meant by a spherically symmetric potential— any potential that actually only depends on x, so that it is invariant under any rotation applied to the system. It is indeed true that a scalar function

R ) for all rotation matrices R, if and only if it is a function of r = [x]. But a general potential in quantum mechanics is given by a Hermitian matrix, and the unitary operators (86) representing the rotations in the Hilbert space of the Dirac equation can also affect the spinor-components. Hence it is not quite straightforward to tell, which potentials are spherically symmetric. 

Before deriving the explicit form of the matrix U in terms of the operator X it should be mentioned that the spectrum of the Dirac operator Hd is invariant under arbitrary similarity transformations, i.e., non-singular (invertible) transformations U, whether they are unitary or not. But only unitary transformations conserve the normalisation of the Dirac spinor and leave scalar products and matrix elements invariant. Therefore a restriction to unitary transformations is inevitable as soon as one is interested in properties of the wavefunction. Furthermore, the problem experiences a great technical simplification by the choice of a unitary transformation, since the inverse transformation U can in general hardly be accomplished if U was not unitary. 

Levy-Leblond [16] has realized that not only the Lorentz group (or rather the homomorphic group SL(2) [32, 7]), but also the Galilei group has spinor-field representations. While the simplest possible spinor field with s = I and m 0 in the Lorentz framework is described by the Dirac equation, the corresponding field in a Galilei-invariant theory satisfies the Levy-Leblond equation (LLE)... 

This factor 2 arises in the Pauli equation, only if one derives the latter either from the DE or the LLE. However, if one formulates the Pauli equation as the SE with an additional spin-dependent term, without any reference to the DE or the LLE, the gyromagnetic ratio 2 must be postulated in an ad hoc way. A direct derivation of a Galilei invariant theory for spin- particles in terms of two-component spinors does not appear to be possible [16]. This does require four-component spinors. Slight deviations from g = 2 are caused by QED effects (radiative corrections), that are outside the scope of this chapter. 

This transformation is in accordance with the Condon-Shortley phase conventions for the spherical basis functions [7]. In fact, our initial Hamiltonian matrix in Eq. (7.21) was constructed in this way. The resulting vector corresponds to the triplet spin functions, which we used in Sect. 6.4. The total spinor product space has dimension 4. The remainder after extraction of the three triplet functions corresponds to the spin singlet, which is invariant and transforms as a scalar. Spinors are thus the fundamental building blocks of 3D space. Their transformation properties were known to Rodrigues as early as 1840. It was some ninety years before Pauli realized that elementary particles, such as electrons, had properties that could be described... 

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