Foldy Representation

Foldy representation for finite mass So far we have ignored the discrete operators P and T which generate the other 3 classes of It proves necessary to double the dimension of each irrep, so that for a representation (m,y) for mass m and spin y the dimension is 2(2y-l-1). Foldy [57] gives a canonical form in terms of a representation space of vector-valued functions P(ct,x) with inner product... 

Equation (37) implies that H has energy eigenvalues (p) and that the position X in the Foldy representation leads to a velocity... 

Equation (9-392) together with (9-394) and (9-395) are the proofs of the assertions that x is the position operator in the Foldy-Wouthuysen representation.16 (Note also that x commutes with /J the sign of the energy.) We further note that in the FTP-representation the operators x x p and Z commute with SFW separately and, hence, are constants of the motion. In the F W-representation the orbital and spin angular momentum operators are thus separately constants of the motion. The fact that... 

Foldy, L, L., 497,498,536,539 Foldy-Wouthuysen representation, 537 "polarization operator in, 538 position operator in, 537 Ford, L. R., 259 Four-color problem, 256 Fourier transforms of Schrodinger operators, 564... 

Ponderomotive force, 382 Position operator, 492 in Dirac representation, 537 in Foldy-Wouthuysen representation, 537 spectrum of, 492 Power, average, 100 Power density spectrum, 183 Prather, J. L., 768 Predictability, 100 Pressure tensor, 21 Probabilities addition of, 267 conditional, 267 Probability, 106... 

The Foldy-Wouthuysen and Dirac representations for a free particle [1 85... 

In performing the similarity transformation above, we have changed our representation of the particle from a Dirac representation to the so-called Foldy-Wouthuysen representation. This new representation provides a very simple link with the non-relativistic Schrodinger-Pauli representation. The latter, which is a two-component representation, just corresponds to the two upper components of the Foldy-Wouthuysen representation (3.125). It must be noted that under a similarity transformation such as (3.105), the operators which represent physical observables are also transformed. For example, the position observable whose operator is R in the Dirac representation is transformed to R " in the Foldy-Wouthuysen representation where... 

The explicit form of the position operator R " in the Foldy-Wouthuysen representation can be obtained by substituting S from equations (3.104) and (3.120) into (3.126). After some manipulation, the final result is... 

We must therefore ask what is the significance of the operator R in the Foldy-Wouthuysen representation. This new observable, which is called the mean position, has an operator representative in the Dirac representation R where... 

Four-component methods are computationally expensive since one has to deal with small-component integrals. Therefore, various two-component methods in which small-component degrees of freedom are removed have flourished in the literature. We focus the present discussion on the X2c theory at one-electron level (X2c-le). The X2c-le scheme consists of a one-step block diagonalization of the Dirac Hamiltonian in its matrix representation via a Foldy-Wouthuysen-type matrix unitary transformation ... 

Since Uq is a unitary matrix, it preserves the orthonormality condition. Ho is then diagonalized by a standard hermitean eigenvalue solver. The eigenvalue equation has the same structure as Eq. (14.35) with the primed labels (L) and (S) replaced by doubly primed ones, ( )" and (S)", to indicate the change of basis by the free-particle Foldy-Wouthuysen transformation. The X matrix in this basis representation is obtained by Eq. (14.36) with the same label replacement. Analogously, the renormalization matrix reads R" = I + The final decoupling transformation... 

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