# Foldy–Wouthuysen transformation

The Foldy–Wouthuysen transform is widely used in high energy physics. It was historically formulated by Leslie Lawrance Foldy and Siegfried Adolf Wouthuysen in 1949 to understand the nonrelativistic limit of the Dirac equation, the equation for spin-1/2 particles.[1][2][3][4] A detailed general discussion of the Foldy–Wouthuysen-type transformations in particle interpretation of relativistic wave equations is in Acharya and Sudarshan (1960).[5]

## A canonical transform

The FW transformation is a unitary transformation of the orthonormal basis in which both the Hamiltonian and the state are represented. The eigenvalues do not change under such a unitary transformation, that is, the physics does not change under such a unitary basis transformation. Therefore, such a unitary transformation can always be applied: in particular a unitary basis transformation may be picked which will put the Hamiltonian in a more pleasant form, at the expense of a change in the state function, which then represents something else. See for example the Bogoliubov transformation, which is an orthogonal basis transform for the same purpose. The suggestion that the FW transform is applicable to the state or the Hamiltonian is thus not correct.

Foldy and Wouthuysen made use of a canonical transform that has now come to be known as the Foldy–Wouthuysen transformation. A brief account of the history of the transformation is to be found in the obituaries of Foldy and Wouthuysen[6][7] and the biographical memoir of Foldy.[8] Before their work, there was some difficulty in understanding and gathering all the interaction terms of a given order, such as those for a Dirac particle immersed in an external field. With their procedure the physical interpretation of the terms was clear, and it became possible to apply their work in a systematic way to a number of problems that had previously defied solution.[9][10] The Foldy–Wouthuysen transform was extended to the physically important cases of spin-0 and spin-1 particles,[11] and even generalized to the case of arbitrary spins.[12]

## Description

The Foldy–Wouthuysen (FW) transformation is a unitary transformation on a fermion wave function of the form:

${\displaystyle \psi \to \psi '=U\psi }$

(1)

where the unitary operator is the 4 × 4 matrix:

${\displaystyle U=e^{\beta {\boldsymbol {\alpha }}\cdot {\hat {p}}\theta }=\cos \theta +\beta {\boldsymbol {\alpha }}\cdot {\hat {p}}\sin \theta =e^{{\boldsymbol {\gamma }}\cdot {\hat {p}}\theta }=\cos \theta +{\boldsymbol {\gamma }}\cdot {\hat {p}}\sin \theta }$

(2)

Above,

${\displaystyle {\hat {p}}^{i}\equiv {\frac {p^{i}}{|p|}}}$

is the unit vector oriented in the direction of the fermion momentum. The above are related to the Dirac matrices by β = γ0 and αi = γ0γi, with i = 1, 2, 3. A straightforward series expansion applying the commutativity properties of the Dirac matrices demonstrates that 2 above is true. The inverse

${\displaystyle U^{-1}=e^{-\beta {\boldsymbol {\alpha }}\cdot {\hat {p}}\theta }=\cos \theta -\beta {\boldsymbol {\alpha }}\cdot {\hat {p}}\sin \theta }$

so it is clear that U−1U = I, where I is a 4 × 4 identity matrix.

## Foldy–Wouthuysen transformation of the Dirac Hamiltonian for a free fermion

This transformation is of particular interest when applied to the free-fermion Dirac Hamiltonian operator

${\displaystyle {\hat {H}}_{0}\equiv \alpha \cdot p+\beta m}$

in biunitary fashion, in the form:

{\displaystyle {\begin{aligned}{\hat {H}}_{0}\to {\hat {H}}'_{0}&\equiv U{\hat {H}}_{0}U^{-1}\\&=U(\alpha \cdot p+\beta m)U^{-1}\\&=(\cos \theta +\beta {\boldsymbol {\alpha }}\cdot {\hat {p}}\sin \theta )(\alpha \cdot p+\beta m)(\cos \theta -\beta {\boldsymbol {\alpha }}\cdot {\hat {p}}\sin \theta )\end{aligned}}}

(3)

Using the commutativity properties of the Dirac matrices, this can be massaged over into the double-angle expression:

{\displaystyle {\begin{aligned}{\hat {H}}'_{0}&=(\alpha \cdot p+\beta m)(\cos \theta -\beta {\boldsymbol {\alpha }}\cdot {\hat {p}}\sin \theta )^{2}\\&=(\alpha \cdot p+\beta m)e^{-2\beta {\boldsymbol {\alpha }}\cdot {\hat {p}}\theta }\\&=(\alpha \cdot p+\beta m)(\cos 2\theta -\beta {\boldsymbol {\alpha }}\cdot {\hat {p}}\sin 2\theta )\end{aligned}}}

(4)

This factors out into:

${\displaystyle {\hat {H}}'_{0}=\alpha \cdot p\left(\cos 2\theta -{\frac {m}{|p|}}\sin 2\theta \right)+\beta (m\cos 2\theta +|p|\sin 2\theta )}$

(5)

## Choosing a particular representation: Newton–Wigner

Clearly, the FW transformation is a continuous transformation, that is, one may employ any value for θ which one chooses. Now comes the distinct question of choosing a particular value for θ, which amounts to choosing a particular transformed representation.

One particularly important representation, is that in which the transformed Hamiltonian operator Ĥ0 is diagonalized. Clearly, a completely diagonalized representation can be obtained by choosing θ such that the α · p term in 5 is made to vanish. Such a representation is specified by defining:

${\displaystyle \tan 2\theta \equiv {\frac {|p|}{m}}}$

(6)

so that 5 is reduced to the diagonalized (this presupposes that β is taken in the Dirac–Pauli representation (after Paul Dirac and Wolfgang Pauli) in which it is a diagonal matrix):

${\displaystyle {\hat {H}}'_{0}=\beta (m\cos 2\theta +|p|\sin 2\theta )}$

(7)

By elementary trigonometry, 6 also implies that:

${\displaystyle \sin 2\theta ={\frac {|p|}{\sqrt {m^{2}+|p|^{2}}}}\quad {\text{and}}\quad \cos 2\theta ={\frac {m}{\sqrt {m^{2}+|p|^{2}}}}}$

(8)

so that using 8 in 7 now leads following reduction to:

${\displaystyle {\hat {H}}'_{0}=\beta {\sqrt {m^{2}+|p|^{2}}}}$

(9)

This calculation can be examined in further detail in the following link.

Prior to Foldy and Wouthuysen publishing their transformation, it was already known that 9 is the Hamiltonian in the Newton–Wigner (NW) representation (named after Theodore Duddell Newton and Eugene Wigner) of the Dirac equation. What 9 therefore tells us, is that by applying a FW transformation to the Dirac–Pauli representation of Dirac's equation, and then selecting the continuous transformation parameter θ so as to diagonalize the Hamiltonian, one arrives at the NW representation of Dirac's equation, because NW itself already contains the Hamiltonian specified in (9). See this link.

If one considers an on-shell mass—fermion or otherwise—given by m2 = pσpσ, and employs a Minkowski metric tensor for which diag(η) = (+1, −1, −1, −1), it should be apparent that the expression

${\displaystyle p^{0}={\sqrt {m^{2}+|p|^{2}}}}$

is equivalent to the Ep0 component of the energy-momentum vector pμ, so that 9 is alternatively specified rather simply by Ĥ0 = βE.

## Correspondence between the Dirac–Pauli and Newton–Wigner representations, for a fermion at rest

Now let us consider a fermion at rest, which we may define in this context as a fermion for which | p | = 0. From 6 or 8, this means that cos 2θ = 1, so that θ = 0, ±π, ±2π and, from 2, that the unitary operator U = ±I. Therefore, any operator O in the Dirac-Pauli representation upon which we perform a biunitary transformation, will be given, for an at-rest fermion, by:

${\displaystyle O\to O'\equiv UOU^{-1}=(\pm I)(O)(\pm I)=O}$

(10)

Contrasting the original Dirac–Pauli Hamiltonian operator

${\displaystyle {\hat {H}}_{0}\equiv \alpha \cdot p+\beta m}$

with the NW Hamiltonian 9, we do indeed find the | p | = 0 "at rest" correspondence:

${\displaystyle {\hat {H}}_{0}={\hat {H}}'_{0}=\beta m}$

(11)

## The velocity operator in the Dirac–Pauli representation

Now, let us consider the velocity operator. To obtain this operator, we must commute the Hamiltonian operator Ĥ0 with the canonical position operators xi, i.e., we must calculate

${\displaystyle {\hat {v_{i}}}\equiv i\left[{\hat {H}}_{0},x_{i}\right]}$

One good way to approach this calculation, is to start by writing the scalar rest mass m as

${\displaystyle m=\gamma ^{0}{\hat {H}}_{0}+\gamma ^{j}p_{j}}$

and then to mandate that the scalar rest mass commute with the xi. Thus, we may write:

${\displaystyle 0=[m,x_{i}]=\left[\left(\gamma ^{0}{\hat {H}}_{0}+\gamma ^{j}p_{j}\right),x_{i}\right]=\left[\gamma ^{0}{\hat {H}}_{0},x_{i}\right]+i\gamma _{i}}$

(12)

where we have made use of the Heisenberg canonical commutation relationship [xi,pj] = −ij to reduce terms. Then, multiplying from the left by γ0 and rearranging terms, we arrive at:

${\displaystyle {\frac {d{\hat {x}}_{i}}{dt}}={\hat {v_{i}}}\equiv i\left[{\hat {H}}_{0},x_{i}\right]=\alpha _{i}}$

(13)

Because the canonical relationship

${\displaystyle i\left[{\hat {H}}_{0},{\hat {v}}_{i}\right]\neq 0}$

the above provides the basis for computing an inherent, non-zero acceleration operator, which specifies the oscillatory motion known as zitterbewegung.

## The velocity operator in the Newton–Wigner representation

In the Newton–Wigner representation, we now wish to calculate

${\displaystyle {\hat {v}}_{i}'\equiv i\left[{\hat {H}}'_{0},x_{i}\right]}$

If we use the result at the very end of section 2 above, Ĥ0 = βp0, then this can be written instead as:

${\displaystyle {\hat {v}}_{i}'\equiv i\left[{\hat {H}}'_{0},x_{i}\right]=i\beta \left[p_{0},x_{i}\right]}$

(14)

Using the above, we need simply to calculate [p0,xi], then multiply by .

The canonical calculation proceeds similarly to the calculation in section 4 above, but because of the square root expression in p0 = m2 + | p |2, one additional step is required.

First, to accommodate the square root, we will wish to require that the scalar square mass m2 commute with the canonical coordinates xi, which we write as:

${\displaystyle 0\equiv \left[m^{2},x_{i}\right]=\left[\left(p^{0}p_{0}+p^{j}p_{j}\right),x_{i}\right]=\left[p^{0}p_{0},x_{i}\right]+2ip_{i}}$

(15)

where we again use the Heisenberg canonical relationship [xi,pj] = −ij. Then, we need an expression for [p0,xi] which will satisfy 15. It is straightforward to verify that:

${\displaystyle i\left[p_{0},x_{i}\right]={\frac {p_{i}}{p^{0}}}=v_{i}}$

(16)

will satisfy 15 when again employing [xi,pj] = −ij. Now, we simply return the factor via 14, to arrive at:

${\displaystyle {\frac {d{\hat {x}}_{i}'}{dt}}={\hat {v}}_{i}'\equiv i\left[{\hat {H}}'_{0},x_{i}\right]=\beta {\frac {p_{i}}{p^{0}}}=\beta v_{i}}$

(17)

This is understood to be the velocity operator in the Newton–Wigner representation. Because:

${\displaystyle i\left[{\hat {H}}'_{0},{\hat {v}}_{i}'\right]=i\left[\beta p_{0},\beta v_{i}\right]=0}$

(18)

it is commonly thought that the zitterbewegung motion arising out of 12 vanishes when a fermion is transformed into the Newton–Wigner representation.

## The velocity operators for a fermion at rest

Now, let us compare equations 13 and 17 for a fermion at rest, defined earlier in section 3 as a fermion for which | p | = 0. Here, (13) remains:

${\displaystyle {\hat {v}}_{i}\equiv i\left[{\hat {H}}_{0},x_{i}\right]=\alpha _{i}}$

(19)

while 17 becomes:

${\displaystyle {\hat {v}}_{i}'\equiv i\left[{\hat {H}}'_{0},x_{i}\right]=\beta {\frac {p_{i}}{p^{0}}}=0}$

(20)

In equation 10 we found that for a fermion at rest, O′ = O for any operator. One would expect this to include:

${\displaystyle {\hat {v}}_{i}'={\hat {v}}_{i}}$

(21)

however, equations 19 and 20 for a | p | = 0 fermion appear to contradict 21.

## Similar alternatives – perturbative schemes

Starting with the one-particle Dirac equation written earlier with Ĥ0 and rewritten here as:

${\displaystyle \left(c{\vec {\alpha }}\cdot {\vec {p}}+\beta m_{o}c^{2}+IV\right)\Phi =E\Phi }$

where I = I4 is the 4 × 4 unit matrix. This Hamiltonian is rewritten, namely divided into two parts:

${\displaystyle {\hat {H}}\equiv {\hat {H}}_{0}+\lambda {\hat {H}}_{1}}$

where

${\displaystyle {\hat {H}}_{0}=c{\vec {\alpha }}\cdot {\vec {p}}+\beta m_{o}c^{2}+{\tfrac {1}{2}}(I+\beta )V}$

and

${\displaystyle {\hat {H}}_{1}={\frac {1}{\alpha ^{2}}}{\hat {H}}'_{1}={\frac {1}{2\alpha ^{2}}}(I-\beta )V}$

where α1/137 is the fine-structure constant (not to be confused with the Dirac alpha matrices). Letting

${\displaystyle \phi ^{0}=A\left({\begin{array}{c}\Psi _{1}^{0}\\\Psi _{2}^{0}\end{array}}\right)}$

into the zero order equation for Ĥ0 and using a particular but known representation of the Dirac operators, yields:

${\displaystyle \Psi _{2}^{0}=c{\frac {{\vec {\sigma }}\cdot {\vec {p}}}{\left(E_{0}+m_{0}c^{2}\right)}}\Psi _{1}^{0}}$

where σi are the 2 × 2 Pauli matrices. Note that the potential V does not appear in the equation above. The equation for the other spinor is:

${\displaystyle c\sigma \cdot {\vec {p}}\Psi _{2}^{0}+V\Psi _{1}^{0}=\varepsilon _{0}\Psi _{1}^{0}}$

where ε0 = E0m0c2. Eliminating Ψ0
2
gives:

${\displaystyle \left\{{\frac {p^{2}}{2m_{0}}}+\left(1+{\frac {\varepsilon _{0}}{2m_{0}c^{2}}}\right)(V-\varepsilon _{0})\right\}\Psi _{1}^{0}\equiv 0.}$

This is simply the nonrelativistic equation for a system with a renormalized potential and energy eigenvalue:

${\displaystyle E'=\left(1+{\frac {\varepsilon _{0}}{2m_{0}c^{2}}}\right)\varepsilon _{0}\quad {\mbox{and}}\quad V'=\left(1+{\frac {\varepsilon _{0}}{2m_{0}c^{2}}}\right)V}$

The higher-order corrections can be obtained by conventional perturbation theory. This is known as Moore's decoupling technique. Though it resembles the FW transformation, it is computationally and conceptually much simpler. Though misunderstood at first, in part because the fine structure constant appears in both the equations and the order parameter λ requiring care in the bookkeeping of the perturbative scheme, Moore's decoupling technique was vindicated for the (relativistic) hydrogen atom using conventional Rayleigh Schrödinger perturbation theory and computer algebra and proven to converge to the correct solution.[13]

It has been applied successfully to relativistic calculations on alkali metals and represents one of many relativistic perturbative schemes investigated by Werner Kutzelnigg.[14][15]

## Other applications

The powerful machinery of the Foldy–Wouthuysen transform originally developed for the Dirac equation has found applications in many situations such as acoustics, and optics.

It has found applications in very diverse areas such as atomic systems[16][17] synchrotron radiation[18] and derivation of the Bloch equation for polarized beams.[19]

The application of the Foldy–Wouthuysen transformation in acoustics is very natural; comprehensive and mathematically rigorous accounts.[20][21][22]

In the traditional scheme the purpose of expanding the optical Hamiltonian

${\displaystyle {\hat {H}}=-\left(n^{2}(r)-{\hat {p}}_{\perp }^{2}\right)^{\frac {1}{2}}}$

in a series using

${\displaystyle {\frac {{\hat {p}}_{\perp }^{2}}{n_{0}^{2}}}}$

as the expansion parameter is to understand the propagation of the quasi-paraxial beam in terms of a series of approximations (paraxial plus nonparaxial). Similar is the situation in the case of charged-particle optics. Let us recall that in relativistic quantum mechanics too one has a similar problem of understanding the relativistic wave equations as the nonrelativistic approximation plus the relativistic correction terms in the quasi-relativistic regime. For the Dirac equation (which is first-order in time) this is done most conveniently using the Foldy–Wouthuysen transformation leading to an iterative diagonalization technique. The main framework of the newly developed formalisms of optics (both light optics and charged-particle optics) is based on the transformation technique of Foldy–Wouthuysen theory which casts the Dirac equation in a form displaying the different interaction terms between the Dirac particle and an applied electromagnetic field in a nonrelativistic and easily interpretable form.

In the Foldy–Wouthuysen theory the Dirac equation is decoupled through a canonical transformation into two two-component equations: one reduces to the Pauli equation[23] in the nonrelativistic limit and the other describes the negative-energy states. It is possible to write a Dirac-like matrix representation of Maxwell's equations. In such a matrix form the Foldy–Wouthuysen can be applied.[24][25][26][27][28]

There is a close algebraic analogy between the Helmholtz equation (governing scalar optics) and the Klein–Gordon equation; and between the matrix form of the Maxwell's equations (governing vector optics) and the Dirac equation. So it is natural to use the powerful machinery of standard quantum mechanics (particularly, the Foldy–Wouthuysen transform) in analyzing these systems.

The suggestion to employ the Foldy–Wouthuysen Transformation technique in the case of the Helmholtz equation was mentioned in the literature as a remark.[29]

It was only in the recent works, that this idea was exploited to analyze the quasiparaxial approximations for specific beam optical system.[30] The Foldy–Wouthuysen technique is ideally suited for the Lie algebraic approach to optics. With all these plus points, the powerful and ambiguity-free expansion, the Foldy–Wouthuysen Transformation is still little used in optics. The technique of the Foldy–Wouthuysen Transformation results in what is known as nontraditional prescriptions of Helmholtz optics[31] and Maxwell optics[32] respectively. The nontraditional approaches give rise to very interesting wavelength-dependent modifications of the paraxial and aberration behaviour. The nontraditional formalism of Maxwell optics provides a unified framework of light beam optics and polarization. The nontraditional prescriptions of light optics are closely analogous with the quantum theory of charged-particle beam optics.[33][34][35][36] In optics, it has enabled the deeper connections in the wavelength-dependent regime between light optics and charged-particle optics to be seen (see Electron optics).[37][38]

## Notes

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2. ^ Foldy, L. L. (1952). "The Electromagnetic Properties of the Dirac Particles". Physical Review. 87 (5): 682–693.
3. ^ Pryce, M. H. L. (1948). "The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles". Proceedings of the Royal Society of London A. 195: 62–81.
4. ^ Tani, S. (1951). "Connection between particle models and field theories. I. The case spin 12". Progress of Theoretical Physics. 6: 267–285.
5. ^ Acharya, R.; Sudarshan, E. C. G. (1960). "Front Description in Relativistic Quantum Mechanics". Journal of Mathematical Physics. 1: 532–536.
6. ^ Brown, R. W.; Krauss, L. M.; Taylor, P. L. (2001). "Obituary of Leslie Lawrence Foldy". Physics Today. 54 (12): 75.
7. ^ Leopold, H. (1997). "Obituary of Siegfried A Wouthuysen". Physics Today. 50 (11): 89.
8. ^ Foldy, L. L. (2006). "Origins of the FW Transformation: A Memoir". In Fickinger, William. Physics at a Research University: Case Western Reserve University 1830–1990. pp. 347–351.
9. ^ Bjorken, J. D.; Drell, S. D. (1964). Relativistic Quantum Mechanics. New York, San Francisco: McGraw-Hill.
10. ^ Costella, J. P.; McKellar, B. H. J. (1995). "The Foldy–Wouthuysen transformation". American Journal of Physics. 63: 1119–1124. arXiv:. doi:10.1119/1.18017.
11. ^ Case, K. M. (1954). "Some generalizations of the Foldy–Wouthuysen transformation". Physical Review. 95: 1323–1328.
12. ^ Jayaraman, J. (1975). "A note on the recent Foldy–Wouthuysen transformations for particles of arbitrary spin". Journal of Physics A. 8: L1–L4.
13. ^ Scott, T. C.; Moore, R. A.; Fee, G. J.; Monagan, M. B.; Vrscay, E. R. (1990). "Perturbative Solutions of Quantum Mechanical Problems by Symbolic Computation". Journal of Computational Physics. 87: 366–395. doi:10.1016/0021-9991(90)90258-3.
14. ^ Kutzelnigg, W. (1990). "Perturbation theory of relativistic corrections. II. Analysis and classification of known and other possible methods". Zeitschrift für Physik D. 15: 27.
15. ^ Kutzelnigg, W. (2002). "Perturbation theory of relativistic effects". In Schwerdtfeger, P. Relativistic Electronic Structure Theory, Part I. Elsevier.
16. ^ Asaga, T.; Fujita, T.; Hiramoto, M. (2000). "EDM operator free from Schiff's theorem". Progress in Theoretical Physics. 106 (6): 1223–1238. arXiv:. doi:10.1143/PTP.106.1223.
17. ^ Pachucki, K. (2004). "Higher-order effective Hamiltonian for light atomic systems". Physical Review A. 71: 012503. arXiv:. doi:10.1103/PhysRevA.71.012503.
18. ^ Lippert, M.; Bruckel, Th.; Kohler, Th.; Schneider, J. R. (1994). "High-Resolution Bulk Magnetic Scattering of High-Energy Synchrotron Radiation". Europhysics Letters. 27 (7): 537–541.
19. ^ Heinemann, K.; Barber, D. P. (1999). "The semiclassical Foldy–Wouthuysen transformation and the derivation of the Bloch equation for spin-12 polarized beams using Wigner functions". In Chen, P. Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics, 4–9 January 1998, Monterey, California, USA. Singapore: World Scientific. arXiv:.
20. ^ Fishman, L. (1992). "Exact and operator rational approximate solutions of the Helmholtz, Weyl composition equation in underwater acoustics—the quadratic profile". Journal of Mathematical Physics. 33 (5): 1887–1914.
21. ^ Fishman, L. (2004). "One-way wave equation modeling in two-way wave propagation problems". In Nilsson, B.; Fishman, L. Mathematical Modelling of Wave Phenomena 2002, Mathematical Modelling in Physics, Engineering and Cognitive Sciences. 7. Växjö, Sweden: Växjö University Press. pp. 91–111.
22. ^ Wurmser, D. (2004). "A parabolic equation for penetrable rough surfaces: using the Foldy–Wouthuysen transformation to buffer density jumps". Annals of Physics. 311: 53–80.
23. ^ Osche, G. R. (1977). "Dirac and Dirac–Pauli equation in the Foldy–Wouthuysen representation". Physical Review D. 15 (8): 2181–2185.
24. ^ Białynicki-Birula, I. (1996). "Photon wave function". Progress in Optics. 36: 245–294. arXiv:.
25. ^ Khan, Sameen Ahmed (2002). "Maxwell Optics: I. An exact matrix representation of the Maxwell equations in a medium". Physica Scripta. 71: 440–442. arXiv:. doi:10.1238/Physica.Regular.071a00440.
26. ^ Laporte, O.; Uhlenbeck, G. E. (1931). "Applications of spinor analysis to the Maxwell and Dirac Equations". Physical Review. 37: 1380–1397.
27. ^ Majorana, E. (1974). Unpublished notes, quoted in Mignani, R.; Recami, E.; Baldo, M. "About a Dirac-like Equation for the Photon, According to Ettore Majorana". Lettere al Nuovo Cimento. 11: 568–572.
28. ^ Moses, E. (1959). "Solutions of Maxwell's equations in terms of a spinor notation: the direct and inverse problems". Physical Review. 113 (6): 1670–1679.
29. ^ Fishman, L.; McCoy, J. J. (1984). "Derivation and Application of Extended Parabolic Wave Theories. Part I. The Factored Helmholtz Equation". Journal of Mathematical Physics. 25: 285–296.
30. ^ Khan, Sameen Ahmed; Jagannathan, Ramaswamy; Simon, Rajiah (2002). "Foldy–Wouthuysen transformation and a quasiparaxial approximation scheme for the scalar wave theory of light beams". arXiv:.
31. ^ Khan, Sameen Ahmed (2005). "Wavelength-dependent modifications in Helmholtz Optics". International Journal of Theoretical Physics. 44 (1): 95–125. doi:10.1007/s10773-005-1488-0.
32. ^ Khan, Sameen Ahmed (2006). "Wavelength-Dependent Effects in Light Optics". In Krasnoholovets, Volodymyr; Columbus, Frank. New Topics in Quantum Physics Research. New York: Nova Science Publishers. pp. 163–204.
33. ^ Jagannathan, R.; Simon, R.; Sudarshan, E. C. G.; Mukunda, N. (1989). "Quantum theory of magnetic electron lenses based on the Dirac equation". Physics Letters A. 134: 457–464. doi:10.1016/0375-9601(89)90685-3.
34. ^ Jagannathan, R. (1990). "Quantum theory of electron lenses based on the Dirac equation". Physical Review A. 42: 6674–6689. doi:10.1103/PhysRevA.42.6674.
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