Spinors

Kaufman B 1949 Orystal statistics II. Partition function evaluated by Spinor analysis Phys. Rev. 65 1232... 

Similarides Between Potential Ruid Dynamics and Quantum Mechanics Electrons in the Dirac Theory The Nearly Nonrelativistic Limit The Lagrangean-Density Correction Term Topological Phase for Dirac Electrons What Have We Learned About Spinor Phases ... 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

In a non-Abelian theory (where the Hamiltonian contains noncommuting matrices and the solutions are vector or spinor functions, with N in Eq. (90) >1) we also start with a vector potential Af, [In the manner of Eq. (94), this can be decomposed into components A, in which the superscript labels the matrices in the theory). Next, we define the field intensity tensor through a covaiiant curl by... 

Here, v / is a four-component spinor, is a four potential, and the 4x4 matrices 7 are given by... 

Introducing the moduli ai and phases <]) for the four spinor components t]ij (i = 1,2,3,4), we note the following relations (in which no summations over i are implied) ... 

The result of interest in the expressions shown in Eqs. (160) and (162) is that, although one has obtained expressions that include corrections to the nonrelativistic case, given in Eqs. (141) and (142), still both the continuity equations and the Hamilton-Jacobi equations involve each spinor component separately. To the present approximation, there is no mixing between the components. 

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 Eq. (168), the first, magnetic-field term admixes different components of the spinors both in the continuity equation and in the Hamilton-Jacobi equation. However, with the z axis chosen as the direction of H, the magnetic-field temi does not contain phases and does not mix component amplitudes. Therefore, there is no contribution from this term in the continuity equations and no amplitude mixing in the Hamilton-Jacobi equations. The second, electric-field term is nondiagonal between the large and small spinor components, which fact reduces its magnitude by a further small factor of 0 particle velocityjc). This term is therefore of the same small order 0(l/c ), as those terms in the second line in Eqs. (164) and (166) that refer to the upper components. 

H. What Have We Learned About Spinor Phases ... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

In this paper, for functions (pi r) we shall use the four-component spinors r) being solutions of the Dirac equation... 

The proof takes different forms in different representations. Here we assume that quantum states are column vectors (or spinors ) iji, with n elements, and that the scalar product has the form ft ip. If ip were a Schrodinger function, J ftipdr would take the place of this matrix product, and in Dirac s theory of the electron, it would be replaced by J fttpdr, iji being a four-component spinor. But the work goes through as below with only formal changes. Use of the bra-ket notation (Chapter 8) would cover all these cases, but it obscures some of the detail we wish to exhibit here. 

Corson, E. M., Tensors, Spinors and Relativistic Wave Equations, Hafher Publish ing Co., New York, 1953. 

Similarly, for wave functions, hereafter called spinors, we define operations isomorphic to, r and complex conjugation. Thus if u is a column spinor we define... 

Note that p and / + are then column spinors.) One again verifies that = , = etc. 

It should be noted that these two solutions are orthogonal to each other uftPWP) = 0. By noting that the projection operator P+(p) operating on the remaining two basis spinors... 

They, in fact, form a complete set for the spinors of momentum p, in the sense that an arbitrary spinor w of momentum p can be expanded in terms of the above solutions as follows ... 

These relations were derived on the assumption that the normalization of the spinors was u u =1. It is oftentimes useful (for covariance reasons) to normalize the spinors so that uu = constant. The relation between these two normalizations is readily obtained. Upon multiplying Eq. (9-307) by (p E(p),s) on the left, we obtain... 

If wx and w2 are spinors corresponding to definite energy, momentum, and helicity, the matrices ww are explicitly given by Eqs. (9-344) or (9-345). Finally the resulting traces involving y-matrices can always be evaluated using the commutation relations [y ,yv]+ — 2gr"v. Thus, for example... 

Equation (9-369) allows us to infer that the transformation of the adjoint spinor under Lorentz transformation is given by... 

Transformation properties of Dirac spinors in particular under inversions Marshak, R. E., and Sudarshan, E. C. G., Introduction to Elementary Particle Physics, Interscience Publishers, Inc., New York, 1961. 

Recall that the scalar product (10-222) is independent of a (and hence of time) for spinors satisfying Eq. (10-221). If we now define the operators... 

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