4794:
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.
1462:
138:. A brief account of the history of the transformation is to be found in the obituaries of Foldy and Wouthuysen and the biographical memoir of Foldy. 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. The Foldy–Wouthuysen transform was extended to the physically important cases of
1125:
4835:
optics 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. In optics, it has enabled the deeper connections in the wavelength-dependent regime between light optics and charged-particle optics to be seen (see
1151:
441:
1457:{\displaystyle {\begin{aligned}{\hat {H}}'_{0}&=({\boldsymbol {\alpha }}\cdot \mathbf {p} +\beta m)(\cos \theta -\beta {\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}\sin \theta )^{2}\\&=({\boldsymbol {\alpha }}\cdot \mathbf {p} +\beta m)e^{-2\beta {\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}\theta }\\&=({\boldsymbol {\alpha }}\cdot \mathbf {p} +\beta m)(\cos 2\theta -\beta {\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}\sin 2\theta )\end{aligned}}}
823:
4865:
234:
1120:{\displaystyle {\begin{aligned}{\hat {H}}_{0}\to {\hat {H}}'_{0}&\equiv U{\hat {H}}_{0}U^{-1}\\&=U({\boldsymbol {\alpha }}\cdot \mathbf {p} +\beta m)U^{-1}\\&=(\cos \theta +\beta {\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}\sin \theta )({\boldsymbol {\alpha }}\cdot \mathbf {p} +\beta m)(\cos \theta -\beta {\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}\sin \theta )\end{aligned}}}
4537:
25:
1656:
436:{\displaystyle U=e^{\beta {\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}\theta }=\mathbb {I} _{4}\cos \theta +\beta {\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}\sin \theta =e^{{\boldsymbol {\gamma }}\cdot {\hat {\mathbf {p} }}\theta }=\mathbb {I} _{4}\cos \theta +{\boldsymbol {\gamma }}\cdot {\hat {\mathbf {p} }}\sin \theta }
712:
2095:
4834:
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 and Maxwell
4793:
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
118:
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
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Heinemann, K.; Barber, D. P. (1999). "The semiclassical Foldy–Wouthuysen transformation and the derivation of the Bloch equation for spin-1⁄2 polarized beams using Wigner functions". In Chen, P (ed.).
2366:
4830:
It was only in the recent works, that this idea was exploited to analyze the quasiparaxial approximations for specific beam optical system. The Foldy–Wouthuysen technique is ideally suited for the
94:
particles. A detailed general discussion of the Foldy–Wouthuysen-type transformations in particle interpretation of relativistic wave equations is in
Acharya and Sudarshan (1960). Its utility in
4121:
1156:
828:
3991:
3566:
1651:{\displaystyle {\hat {H}}'_{0}={\boldsymbol {\alpha }}\cdot \mathbf {p} \left(\cos 2\theta -{\frac {m}{|\mathbf {p} |}}\sin 2\theta \right)+\beta (m\cos 2\theta +|\mathbf {p} |\sin 2\theta )}
707:{\displaystyle U^{-1}=e^{-\beta {\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}\theta }=\mathbb {I} _{4}\cos \theta -\beta {\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}\sin \theta }
4173:
3189:
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208:
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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
43:
3458:
2090:{\displaystyle \sin 2\theta ={\frac {|\mathbf {p} |}{\sqrt {m^{2}+|\mathbf {p} |^{2}}}}\quad {\text{and}}\quad \cos 2\theta ={\frac {m}{\sqrt {m^{2}+|\mathbf {p} |^{2}}}}}
4817:
5532:
Fishman, L. (1992). "Exact and operator rational approximate solutions of the
Helmholtz, Weyl composition equation in underwater acoustics—the quadratic profile".
2258:
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 (
4624:
2594:
754:
3231:
4827:
The suggestion to employ the Foldy–Wouthuysen
Transformation technique in the case of the Helmholtz equation was mentioned in the literature as a remark.
4824:. So it is natural to use the powerful machinery of standard quantum mechanics (particularly, the Foldy–Wouthuysen transform) in analyzing these systems.
4802:
1821:
4797:
In the Foldy–Wouthuysen theory the Dirac equation is decoupled through a canonical transformation into two two-component equations: one reduces to the
5456:
Lippert, M.; Bruckel, Th.; Kohler, Th.; Schneider, J. R. (1994). "High-Resolution Bulk
Magnetic Scattering of High-Energy Synchrotron Radiation".
2133:
5880:
Fishman, L.; McCoy, J. J. (1984). "Derivation and
Application of Extended Parabolic Wave Theories. Part I. The Factored Helmholtz Equation".
5582:
Wurmser, D. (2004). "A parabolic equation for penetrable rough surfaces: using the Foldy–Wouthuysen transformation to buffer density jumps".
98:
is now limited due to the primary applications being in the ultra-relativistic domain where the Dirac field is treated as a quantised field.
5501:
Proceedings of the 15th
Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics, 4–9 January 1998, Monterey, California, USA
465:
4615:
The application of the Foldy–Wouthuysen transformation in acoustics is very natural; comprehensive and mathematically rigorous accounts.
3463:
the above provides the basis for computing an inherent, non-zero acceleration operator, which specifies the oscillatory motion known as
4875:
1734:
6238:
6151:
5687:
4572:
61:
6167:
Conte, M.; Jagannathan, R.; Khan, S. A.; Pusterla, M. (1996). "Beam optics of the Dirac particle with anomalous magnetic moment".
4733:
2900:
6026:
Khan, Sameen Ahmed (2006). "Wavelength-Dependent Effects in Light Optics". In Krasnoholovets, Volodymyr; Columbus, Frank (eds.).
3750:
The canonical calculation proceeds similarly to the calculation in section 4 above, but because of the square root expression in
2480:
5567:
Fishman, L. (2004). "One-way wave equation modeling in two-way wave propagation problems". In Nilsson, B.; Fishman, L. (eds.).
3598:
2472:
in the Dirac–Pauli representation upon which we perform a biunitary transformation, will be given, for an at-rest fermion, by:
123:, which is an orthogonal basis transform for the same purpose. The suggestion that the FW transform is applicable to the state
111:
2302:
4848:
2265:
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pleasant form, at the expense of a change in the state function, which then represents something else. See for example the
4994:"The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles"
4033:
5810:
Mignani, R.; Recami, E.; Baldo, M. (2008). "About a Dirac-like Equation for the Photon, According to Ettore Majorana".
3796:
4801:
in the nonrelativistic limit and the other describes the negative-energy states. It is possible to write a Dirac-like
1143:
Using the commutativity properties of the Dirac matrices, this can be massaged over into the double-angle expression:
5845:
Moses, E. (1959). "Solutions of Maxwell's equations in terms of a spinor notation: the direct and inverse problems".
5711:
Khan, Sameen Ahmed (2005). "Maxwell Optics: I. An exact matrix representation of the Maxwell equations in a medium".
5921:"Foldy–Wouthuysen transformation and a quasiparaxial approximation scheme for the scalar wave theory of light beams"
5569:
Mathematical Modelling of Wave Phenomena 2002, Mathematical Modelling in Physics, Engineering and Cognitive Sciences
4341:{\displaystyle {\frac {d{\hat {x}}_{i}'}{dt}}={\hat {v}}_{i}'\equiv i\left=\beta {\frac {p_{i}}{p^{0}}}=\beta v_{i}}
83:
4813:
3481:
6050:
2990:
2803:
120:
5315:
Jayaraman, J. (1975). "A note on the recent Foldy–Wouthuysen transformations for particles of arbitrary spin".
131:
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6046:
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173:
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This transformation is of particular interest when applied to the free-fermion Dirac Hamiltonian operator
4916:
5946:
2778:
Now, consider the velocity operator. To obtain this operator, we must commute the Hamiltonian operator
79:
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is the unit vector oriented in the direction of the fermion momentum. The above are related to the
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5262:
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4809:
5191:
Foldy, L. L. (2006). "Origins of the FW Transformation: A Memoir". In Fickinger, William (ed.).
5074:
Acharya, R.; Sudarshan, E. C. G. (1960). "Front Description in Relativistic Quantum Mechanics".
1694:
One particularly important representation is that in which the transformed Hamiltonian operator
2411:
Correspondence between the Dirac–Pauli and Newton–Wigner representations, for a fermion at rest
6259:
6234:
6147:
6116:
5683:
5617:
Osche, G. R. (1977). "Dirac and Dirac–Pauli equation in the Foldy–Wouthuysen representation".
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This is understood to be the velocity operator in the Newton–Wigner representation. Because:
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First, to accommodate the square root, we will wish to require that the scalar square mass
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Now consider a fermion at rest, which we may define in this context as a fermion for which
5959:
5805:
4836:
737:
143:
139:
6091:
Jagannathan, R. (1990). "Quantum theory of electron lenses based on the Dirac equation".
2230:
Prior to Foldy and Wouthuysen publishing their transformation, it was already known that
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4717:{\displaystyle {\hat {H}}=-\left(n^{2}(r)-{\hat {p}}_{\perp }^{2}\right)^{\frac {1}{2}}}
5350:
Asaga, T.; Fujita, T.; Hiramoto, M. (2000). "EDM operator free from Schiff's theorem".
5210:
4821:
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4605:
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4514:
3464:
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147:
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Khan, Sameen Ahmed (2006). "The Foldy–Wouthuysen Transformation Technique in Optics".
6143:
6138:. Advances in Imaging and Electron Physics. Vol. 97. Elsevier. pp. 257–358.
5742:
5679:
4585:
The powerful machinery of the Foldy–Wouthuysen transform originally developed for the
2645:{\displaystyle {\hat {H}}_{0}\equiv {\boldsymbol {\alpha }}\cdot \mathbf {p} +\beta m}
805:{\displaystyle {\hat {H}}_{0}\equiv {\boldsymbol {\alpha }}\cdot \mathbf {p} +\beta m}
6253:
6225:. Advances in Imaging and Electron Physics. Vol. 152. Elsevier. pp. 49–78.
6077:
5973:
Khan, Sameen Ahmed (2005). "Wavelength-dependent modifications in Helmholtz Optics".
5831:
5750:
5485:
5477:
3357:{\displaystyle {\frac {d{\hat {x}}_{i}}{dt}}={\hat {v_{i}}}\equiv i\left=\alpha _{i}}
2241:
570:
162:
6012:
5697:
5403:
Pachucki, K. (2004). "Higher-order effective Hamiltonian for light atomic systems".
5389:
5336:
5266:
5920:
5916:
5764:
5442:
5280:
Case, K. M. (1954). "Some generalizations of the Foldy–Wouthuysen transformation".
5227:
Costella, J. P.; McKellar, B. H. J. (1995). "The Foldy–Wouthuysen transformation".
1706:
is diagonalized. A completely diagonal representation can be obtained by choosing
4864:
1907:{\displaystyle {\hat {H}}'_{0}=\beta (m\cos 2\theta +|\mathbf {p} |\sin 2\theta )}
6207:
4831:
4601:
107:
5434:
6004:
5771:(1931). "Applications of spinor analysis to the Maxwell and Dirac Equations".
5603:
4957:
Foldy, L. L. (1952). "The Electromagnetic Properties of the Dirac Particles".
4523:
vanishes when a fermion is transformed into the Newton–Wigner representation.
115:
6112:
5866:
5638:
5035:"Connection between particle models and field theories. I. The case spin 1⁄2"
2887:
One good way to approach this calculation, is to start by writing the scalar
5792:
5301:
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3207:
where we have made use of the Heisenberg canonical commutation relationship
2888:
5194:
Physics at a Research University: Case Western Reserve University 1830–1990
5018:
4993:
4978:
4618:
In the traditional scheme the purpose of expanding the optical Hamiltonian
6120:
5059:
5034:
4943:
4917:"On the Dirac Theory of Spin 1⁄2 Particles and its Non-Relativistic Limit"
158:
The Foldy–Wouthuysen (FW) transformation is a unitary transformation on a
5662:
5381:
2236:
is the Hamiltonian in the Newton–Wigner (NW) representation (named after
91:
6051:"Quantum theory of magnetic electron lenses based on the Dirac equation"
5987:
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5725:
5509:
5417:
2207:{\displaystyle {\hat {H}}'_{0}=\beta {\sqrt {m^{2}+|\mathbf {p} |^{2}}}}
5823:
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It has found applications in very diverse areas such as atomic systems
159:
5571:. Vol. 7. Växjö, Sweden: Växjö University Press. pp. 91–111.
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525:{\displaystyle {\hat {p}}^{i}\equiv {\frac {p^{i}}{|\mathbf {p} |}}}
5192:
5656:. Progress in Optics. Vol. 36. Elsevier. pp. 245–294.
3743:
Using the above, we need simply to calculate , then multiply by
2271:
If one considers an on-shell mass—fermion or otherwise—given by
2975:
and then to mandate that the scalar rest mass commute with the
1780:{\displaystyle \tan 2\theta \equiv {\frac {|\mathbf {p} |}{m}}}
4858:
4530:
3475:
In the Newton–Wigner representation, we now wish to calculate
18:
4805:. In such a matrix form the Foldy–Wouthuysen can be applied.
1691:
amounts to choosing a particular transformed representation.
4783:{\displaystyle {\frac {{\hat {p}}_{\perp }^{2}}{n_{0}^{2}}}}
2965:{\displaystyle m=\gamma ^{0}{\hat {H}}_{0}+\gamma ^{j}p_{j}}
2588:
Contrasting the original Dirac–Pauli Hamiltonian operator
2565:{\displaystyle O\to O'\equiv UOU^{-1}=(\pm I)(O)(\pm I)=O}
226:
where the unitary operator is the 4 × 4 matrix:
5503:. Singapore: World Scientific. pp. physics/9901044.
4009:
where we again use the Heisenberg canonical relationship
3720:{\displaystyle {\hat {v}}_{i}'\equiv i\left=i\beta \left}
3571:
If we use the result at the very end of section 2 above,
146:
particles, and even generalized to the case of arbitrary
106:
The FW transformation is a unitary transformation of the
5652:
Białynicki-Birula, I. (1996). "V Photon Wave Function".
2361:{\displaystyle p^{0}={\sqrt {m^{2}+|\mathbf {p} |^{2}}}}
6223:
The Foldy–Wouthuysen Transformation Technique in Optics
4883:
4019:. Then, we need an expression for which will satisfy
86:
in 1949 to understand the nonrelativistic limit of the
39:
6030:. New York: Nova Science Publishers. pp. 163–204.
2739:{\displaystyle {\hat {H}}_{0}={\hat {H}}'_{0}=\beta m}
1683:
transformation, that is, one may employ any value for
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Transforming the Dirac Hamiltonian for a free fermion
588:
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to reduce terms. Then, multiplying from the left by
1687:
which one chooses. Choosing a particular value for
5109:Brown, R. W.; Krauss, L. M.; Taylor, P. L. (2001).
4116:{\displaystyle i\left={\frac {p_{i}}{p^{0}}}=v_{i}}
1675:
Choosing a particular representation: Newton–Wigner
573:properties of the Dirac matrices demonstrates that
78:was historically significant and was formulated by
34:
may be too technical for most readers to understand
5209:
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4589:has found applications in many situations such as
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569:. A straightforward series expansion applying the
524:
435:
202:
6136:Quantum theory of the optics of charged particles
3986:{\displaystyle 0\equiv \left=\left=\left+2ip_{i}}
4808:There is a close algebraic analogy between the
8:
5975:International Journal of Theoretical Physics
5915:Khan, Sameen Ahmed; Jagannathan, Ramaswamy;
4998:Proceedings of the Royal Society of London A
4803:matrix representation of Maxwell's equations
3561:{\displaystyle {\hat {v}}_{i}'\equiv i\left}
2391:is alternatively specified rather simply by
2296:, it should be apparent that the expression
3184:{\displaystyle 0==\left=\left+i\gamma _{i}}
2877:{\displaystyle {\hat {v_{i}}}\equiv i\left}
4555:remove low-quality or irrelevant citations
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4573:Learn how and when to remove this message
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62:Learn how and when to remove this message
46:, without removing the technical details.
4915:Foldy, L. L.; Wouthuysen, S. A. (1950).
2381:component of the energy-momentum vector
1803:In the Dirac-Pauli representation where
1726:vanishes. This is arranged by choosing:
5216:. New York, San Francisco: McGraw-Hill.
4907:
3781:commute with the canonical coordinates
3590:, then this can be written instead as:
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1080:
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934:
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6028:New Topics in Quantum Physics Research
5955:
5944:
4818:matrix form of the Maxwell's equations
2790:with the canonical position operators
1813:is then reduced to a diagonal matrix:
203:{\displaystyle \psi \to \psi '=U\psi }
5808:(1974). Unpublished notes, quoted in
5208:Bjorken, J. D.; Drell, S. D. (1964).
3223:and rearranging terms, we arrive at:
134:that has now come to be known as the
127:the Hamiltonian is thus not correct.
44:make it understandable to non-experts
16:Used to understand the Dirac equation
7:
5152:"Obituary of Siegfried A Wouthuysen"
4366:
4167:
4027:
3790:
3592:
3225:
2984:
2673:
2474:
2127:
1938:
1815:
1728:
1679:Clearly, the FW transformation is a
1482:
1145:
817:
228:
167:
5111:"Obituary of Leslie Lawrence Foldy"
3772:, one additional step is required.
3471:In the Newton–Wigner representation
3380:Because the canonical relationship
2125:and then simplifying now leads to:
815:in biunitary fashion, in the form:
130:Foldy and Wouthuysen made use of a
114:and the state are represented. The
4820:(governing vector optics) and the
4812:(governing scalar optics) and the
2763:Transforming the velocity operator
14:
5743:10.1238/Physica.Regular.071a00440
2768:In the Dirac–Pauli representation
4863:
4604:radiation and derivation of the
4535:
4513:it is commonly thought that the
2629:
2340:
2186:
2068:
2002:
1972:
1880:
1762:
1624:
1562:
1526:
1425:
1376:
1339:
1297:
1246:
1200:
1091:
1045:
1011:
942:
789:
685:
628:
510:
414:
360:
321:
264:
23:
5882:Journal of Mathematical Physics
5534:Journal of Mathematical Physics
5352:Progress of Theoretical Physics
5076:Journal of Mathematical Physics
5039:Progress of Theoretical Physics
4490:{\displaystyle i\left=i\left=0}
2027:
2021:
136:Foldy–Wouthuysen transformation
76:Foldy–Wouthuysen transformation
5212:Relativistic Quantum Mechanics
4849:Relativistic quantum mechanics
4746:
4681:
4668:
4662:
4634:
4416:
4391:
4259:
4226:
4190:
3642:
3609:
3525:
3492:
3427:
3405:
3311:
3287:
3248:
3135:
3052:
3019:
3000:
2927:
2844:
2820:
2712:
2690:
2605:
2553:
2544:
2541:
2535:
2532:
2523:
2487:
2346:
2335:
2192:
2181:
2144:
2074:
2063:
2008:
1997:
1977:
1967:
1901:
1885:
1875:
1853:
1832:
1767:
1757:
1645:
1629:
1619:
1597:
1567:
1557:
1499:
1447:
1429:
1392:
1389:
1364:
1343:
1310:
1285:
1266:
1250:
1216:
1213:
1188:
1166:
1110:
1095:
1061:
1058:
1033:
1030:
1015:
981:
955:
930:
892:
860:
850:
838:
765:
689:
632:
515:
505:
476:
418:
364:
325:
268:
180:
1:
6231:10.1016/S1076-5670(08)00602-2
6144:10.1016/S1076-5670(08)70096-X
5680:10.1016/S0079-6638(08)70316-0
4548:excessive number of citations
6078:10.1016/0375-9601(89)90685-3
6041:Jagannathan, R.; Simon, R.;
4519:
4161:
4155:. Now, we simply return the
4141:
4021:
3453:{\displaystyle i\left\neq 0}
2458:, that the unitary operator
1930:By elementary trigonometry,
6221:Khan, Sameen Ahmed (2008).
6208:10.1016/j.ijleo.2005.11.010
5229:American Journal of Physics
2657:
2454:
2433:
2427:
2387:
2260:
2250:
2232:
2121:
2115:
1932:
1809:
1722:
579:above is true. The inverse
575:
6281:
5478:10.1209/0295-5075/27/7/008
5435:10.1103/PhysRevA.71.012503
2797:, i.e., we must calculate
2771:
2671:"at rest" correspondence:
2468:. Therefore, any operator
84:Siegfried Adolf Wouthuysen
6005:10.1007/s10773-005-1488-0
5604:10.1016/j.aop.2003.11.006
5337:10.1088/0305-4470/8/1/001
121:Bogoliubov transformation
6113:10.1103/PhysRevA.42.6674
5867:10.1103/PhysRev.113.1670
5812:Lettere al Nuovo Cimento
5639:10.1103/PhysRevD.15.2181
4992:Pryce, M. H. L. (1948).
2661:, we do indeed find the
2655:with the NW Hamiltonian
110:basis in which both the
5793:10.1103/PhysRev.37.1380
5302:10.1103/PhysRev.95.1323
2982:. Thus, we may write:
2238:Theodore Duddell Newton
1480:This factors out into:
5954:Cite journal requires
5019:10.1098/rspa.1948.0103
4979:10.1103/PhysRev.87.688
4784:
4718:
4517:motion arising out of
4491:
4342:
4117:
3987:
3721:
3562:
3454:
3358:
3185:
2966:
2878:
2740:
2646:
2566:
2362:
2208:
2091:
1908:
1807:is a diagonal matrix,
1781:
1652:
1458:
1121:
806:
708:
526:
437:
204:
6169:Particle Accelerators
4944:10.1103/PhysRev.78.29
4814:Klein–Gordon equation
4785:
4719:
4492:
4343:
4145:when again employing
4118:
3988:
3788:, which we write as:
3722:
3563:
3455:
3359:
3186:
2967:
2879:
2772:Further information:
2741:
2647:
2567:
2371:is equivalent to the
2363:
2209:
2092:
1909:
1782:
1653:
1459:
1122:
807:
736:is a 4 × 4
709:
527:
438:
205:
102:A canonical transform
80:Leslie Lawrance Foldy
6134:Khan, S. A. (1996).
5654:Photon wave function
5382:10.1143/PTP.106.1223
5317:Journal of Physics A
5150:Leopold, H. (1997).
4878:for the books listed
4734:
4625:
4373:
4174:
4034:
3797:
3599:
3482:
3387:
3232:
2991:
2901:
2804:
2680:
2595:
2481:
2303:
2294:) = (+1, −1, −1, −1)
2134:
1945:
1822:
1735:
1489:
1152:
824:
755:
717:so it is clear that
586:
466:
235:
174:
6200:2006Optik.117..481K
6105:1990PhRvA..42.6674J
6070:1989PhLA..134..457J
6043:Sudarshan, E. C. G.
5997:2005IJTP...44...95K
5939:2002physics...9082K
5923:: physics/0209082.
5894:1984JMP....25..285F
5859:1959PhRv..113.1670M
5785:1931PhRv...37.1380L
5735:2005PhyS...71..440K
5672:2005quant.ph..8202B
5631:1977PhRvD..15.2181O
5596:2004AnPhy.311...53W
5546:1992JMP....33.1887F
5519:1999physics...1044H
5470:1994EL.....27..537L
5458:Europhysics Letters
5427:2005PhRvA..71a2503P
5374:2001PThPh.106.1223A
5329:1975JPhA....8L...1J
5294:1954PhRv...95.1323C
5251:1995AmJPh..63.1119C
5197:. pp. 347–351.
5168:1997PhT....50k..89H
5127:2001PhT....54l..75B
5088:1960JMP.....1..532A
5060:10.1143/ptp/6.3.267
5051:1951PThPh...6..267T
5010:1948RSPSA.195...62P
4971:1952PhRv...87..688F
4936:1950PhRv...78...29F
4777:
4762:
4697:
4430:
4405:
4273:
4240:
4204:
3656:
3623:
3539:
3506:
2726:
2158:
1936:also implies that:
1846:
1513:
1180:
874:
132:canonical transform
96:high energy physics
90:, the equation for
5824:10.1007/bf02812391
4816:; and between the
4810:Helmholtz equation
4780:
4763:
4739:
4727:in a series using
4714:
4674:
4527:Other applications
4487:
4409:
4384:
4338:
4252:
4219:
4183:
4113:
3983:
3717:
3635:
3602:
3558:
3518:
3485:
3450:
3354:
3181:
2962:
2874:
2736:
2705:
2642:
2562:
2437:, this means that
2358:
2204:
2137:
2087:
1904:
1825:
1777:
1648:
1492:
1454:
1452:
1159:
1117:
1115:
853:
802:
704:
522:
433:
200:
6099:(11): 6674–6689.
6093:Physical Review A
6058:Physics Letters A
5779:(11): 1380–1397.
5619:Physical Review D
5584:Annals of Physics
5405:Physical Review A
5235:(12): 1119–1124.
5136:10.1063/1.1445566
5096:10.1063/1.1703689
5033:Tani, S. (1951).
4905:
4904:
4778:
4749:
4711:
4684:
4637:
4583:
4582:
4575:
4511:
4510:
4419:
4394:
4362:
4361:
4320:
4262:
4229:
4214:
4193:
4137:
4136:
4098:
4007:
4006:
3741:
3740:
3645:
3612:
3528:
3495:
3430:
3408:
3378:
3377:
3314:
3290:
3269:
3251:
3205:
3204:
3138:
3055:
2930:
2847:
2823:
2760:
2759:
2715:
2693:
2608:
2586:
2585:
2356:
2288:tensor for which
2228:
2227:
2202:
2147:
2111:
2110:
2085:
2084:
2025:
2019:
2018:
1928:
1927:
1835:
1801:
1800:
1775:
1672:
1671:
1572:
1502:
1478:
1477:
1432:
1346:
1253:
1169:
1141:
1140:
1098:
1018:
895:
863:
841:
768:
692:
635:
520:
479:
457:
456:
421:
367:
328:
271:
224:
223:
72:
71:
64:
6272:
6245:
6244:
6218:
6212:
6211:
6183:
6177:
6176:
6164:
6158:
6157:
6131:
6125:
6124:
6088:
6082:
6081:
6064:(8–9): 457–464.
6055:
6038:
6032:
6031:
6023:
6017:
6016:
5990:
5970:
5964:
5963:
5957:
5952:
5950:
5942:
5932:
5912:
5906:
5905:
5902:10.1063/1.526149
5877:
5871:
5870:
5853:(6): 1670–1679.
5842:
5836:
5835:
5803:
5797:
5796:
5769:Uhlenbeck, G. E.
5761:
5755:
5754:
5728:
5708:
5702:
5701:
5665:
5663:quant-ph/0508202
5649:
5643:
5642:
5625:(8): 2181–2185.
5614:
5608:
5607:
5579:
5573:
5572:
5564:
5558:
5557:
5554:10.1063/1.529666
5540:(5): 1887–1914.
5529:
5523:
5522:
5512:
5496:
5490:
5489:
5453:
5447:
5446:
5420:
5400:
5394:
5393:
5367:
5358:(6): 1223–1238.
5347:
5341:
5340:
5312:
5306:
5305:
5288:(5): 1323–1328.
5277:
5271:
5270:
5244:
5224:
5218:
5217:
5215:
5205:
5199:
5198:
5188:
5182:
5181:
5179:
5177:10.1063/1.882018
5147:
5141:
5140:
5138:
5106:
5100:
5099:
5071:
5065:
5064:
5062:
5030:
5024:
5023:
5021:
4989:
4983:
4982:
4954:
4948:
4947:
4921:
4912:
4900:
4897:
4891:
4867:
4859:
4789:
4787:
4786:
4781:
4779:
4776:
4771:
4761:
4756:
4751:
4750:
4742:
4738:
4723:
4721:
4720:
4715:
4713:
4712:
4704:
4702:
4698:
4696:
4691:
4686:
4685:
4677:
4661:
4660:
4639:
4638:
4630:
4578:
4571:
4567:
4564:
4558:
4539:
4538:
4531:
4505:
4496:
4494:
4493:
4488:
4480:
4476:
4475:
4474:
4459:
4458:
4435:
4431:
4426:
4421:
4420:
4412:
4401:
4396:
4395:
4387:
4367:
4356:
4347:
4345:
4344:
4339:
4337:
4336:
4321:
4319:
4318:
4309:
4308:
4299:
4291:
4287:
4286:
4285:
4269:
4264:
4263:
4255:
4236:
4231:
4230:
4222:
4215:
4213:
4205:
4200:
4195:
4194:
4186:
4178:
4168:
4165:, to arrive at:
4158:
4154:
4131:
4122:
4120:
4119:
4114:
4112:
4111:
4099:
4097:
4096:
4087:
4086:
4077:
4072:
4068:
4067:
4066:
4054:
4053:
4028:
4018:
4001:
3992:
3990:
3989:
3984:
3982:
3981:
3963:
3959:
3958:
3957:
3945:
3944:
3935:
3934:
3917:
3913:
3912:
3911:
3899:
3895:
3894:
3893:
3884:
3883:
3871:
3870:
3861:
3860:
3838:
3834:
3833:
3832:
3820:
3819:
3791:
3787:
3780:
3771:
3770:
3769:
3767:
3746:
3735:
3726:
3724:
3723:
3718:
3716:
3712:
3711:
3710:
3698:
3697:
3674:
3670:
3669:
3668:
3652:
3647:
3646:
3638:
3619:
3614:
3613:
3605:
3593:
3589:
3578:
3567:
3565:
3564:
3559:
3557:
3553:
3552:
3551:
3535:
3530:
3529:
3521:
3502:
3497:
3496:
3488:
3459:
3457:
3456:
3451:
3443:
3439:
3438:
3437:
3432:
3431:
3423:
3416:
3415:
3410:
3409:
3401:
3372:
3363:
3361:
3360:
3355:
3353:
3352:
3340:
3336:
3335:
3334:
3322:
3321:
3316:
3315:
3307:
3292:
3291:
3286:
3285:
3276:
3270:
3268:
3260:
3259:
3258:
3253:
3252:
3244:
3236:
3226:
3222:
3216:
3199:
3190:
3188:
3187:
3182:
3180:
3179:
3164:
3160:
3159:
3158:
3146:
3145:
3140:
3139:
3131:
3127:
3126:
3109:
3105:
3104:
3103:
3091:
3087:
3086:
3085:
3076:
3075:
3063:
3062:
3057:
3056:
3048:
3044:
3043:
3018:
3017:
2985:
2981:
2971:
2969:
2968:
2963:
2961:
2960:
2951:
2950:
2938:
2937:
2932:
2931:
2923:
2919:
2918:
2893:
2883:
2881:
2880:
2875:
2873:
2869:
2868:
2867:
2855:
2854:
2849:
2848:
2840:
2825:
2824:
2819:
2818:
2809:
2796:
2789:
2785:
2754:
2745:
2743:
2742:
2737:
2722:
2717:
2716:
2708:
2701:
2700:
2695:
2694:
2686:
2674:
2670:
2668:
2651:
2649:
2648:
2643:
2632:
2624:
2616:
2615:
2610:
2609:
2601:
2580:
2571:
2569:
2568:
2563:
2519:
2518:
2497:
2475:
2471:
2467:
2451:
2444:
2424:
2422:
2406:
2398:
2384:
2380:
2367:
2365:
2364:
2359:
2357:
2355:
2354:
2349:
2343:
2338:
2330:
2329:
2320:
2315:
2314:
2295:
2286:Minkowski metric
2284:, and employs a
2283:
2257:
2222:
2213:
2211:
2210:
2205:
2203:
2201:
2200:
2195:
2189:
2184:
2176:
2175:
2166:
2154:
2149:
2148:
2140:
2128:
2105:
2096:
2094:
2093:
2088:
2086:
2083:
2082:
2077:
2071:
2066:
2058:
2057:
2048:
2044:
2026:
2023:
2020:
2017:
2016:
2011:
2005:
2000:
1992:
1991:
1982:
1981:
1980:
1975:
1970:
1964:
1939:
1922:
1913:
1911:
1910:
1905:
1888:
1883:
1878:
1842:
1837:
1836:
1828:
1816:
1806:
1795:
1786:
1784:
1783:
1778:
1776:
1771:
1770:
1765:
1760:
1754:
1729:
1719:
1709:
1705:
1701:
1690:
1686:
1666:
1657:
1655:
1654:
1649:
1632:
1627:
1622:
1590:
1586:
1573:
1571:
1570:
1565:
1560:
1551:
1529:
1521:
1509:
1504:
1503:
1495:
1483:
1472:
1463:
1461:
1460:
1455:
1453:
1434:
1433:
1428:
1423:
1417:
1379:
1371:
1357:
1353:
1352:
1348:
1347:
1342:
1337:
1331:
1300:
1292:
1278:
1274:
1273:
1255:
1254:
1249:
1244:
1238:
1203:
1195:
1176:
1171:
1170:
1162:
1146:
1135:
1126:
1124:
1123:
1118:
1116:
1100:
1099:
1094:
1089:
1083:
1048:
1040:
1020:
1019:
1014:
1009:
1003:
974:
970:
969:
945:
937:
920:
916:
915:
903:
902:
897:
896:
888:
870:
865:
864:
856:
849:
848:
843:
842:
834:
818:
811:
809:
808:
803:
792:
784:
776:
775:
770:
769:
761:
735:
729:
713:
711:
710:
705:
694:
693:
688:
683:
677:
657:
656:
651:
642:
641:
637:
636:
631:
626:
620:
601:
600:
568:
561:
548:
531:
529:
528:
523:
521:
519:
518:
513:
508:
502:
501:
492:
487:
486:
481:
480:
472:
451:
442:
440:
439:
434:
423:
422:
417:
412:
406:
389:
388:
383:
374:
373:
369:
368:
363:
358:
352:
330:
329:
324:
319:
313:
293:
292:
287:
278:
277:
273:
272:
267:
262:
256:
229:
218:
209:
207:
206:
201:
190:
168:
67:
60:
56:
53:
47:
27:
26:
19:
6280:
6279:
6275:
6274:
6273:
6271:
6270:
6269:
6250:
6249:
6248:
6241:
6220:
6219:
6215:
6194:(10): 481–488.
6185:
6184:
6180:
6166:
6165:
6161:
6154:
6133:
6132:
6128:
6090:
6089:
6085:
6053:
6040:
6039:
6035:
6025:
6024:
6020:
5988:physics/0210001
5972:
5971:
5967:
5953:
5943:
5930:physics/0209082
5914:
5913:
5909:
5879:
5878:
5874:
5847:Physical Review
5844:
5843:
5839:
5818:(12): 568–572.
5809:
5804:
5800:
5773:Physical Review
5763:
5762:
5758:
5726:physics/0205083
5713:Physica Scripta
5710:
5709:
5705:
5690:
5651:
5650:
5646:
5616:
5615:
5611:
5581:
5580:
5576:
5566:
5565:
5561:
5531:
5530:
5526:
5510:physics/9901044
5498:
5497:
5493:
5455:
5454:
5450:
5418:physics/0411168
5402:
5401:
5397:
5349:
5348:
5344:
5314:
5313:
5309:
5282:Physical Review
5279:
5278:
5274:
5259:10.1119/1.18017
5226:
5225:
5221:
5207:
5206:
5202:
5190:
5189:
5185:
5149:
5148:
5144:
5108:
5107:
5103:
5073:
5072:
5068:
5032:
5031:
5027:
5004:(1040): 62–81.
4991:
4990:
4986:
4959:Physical Review
4956:
4955:
4951:
4924:Physical Review
4919:
4914:
4913:
4909:
4901:
4895:
4892:
4881:
4868:
4857:
4845:
4837:Electron optics
4732:
4731:
4652:
4651:
4647:
4646:
4623:
4622:
4579:
4568:
4562:
4559:
4552:
4546:may contain an
4540:
4536:
4529:
4503:
4466:
4450:
4446:
4442:
4383:
4379:
4371:
4370:
4354:
4328:
4310:
4300:
4277:
4251:
4247:
4206:
4179:
4172:
4171:
4156:
4152:
4146:
4129:
4103:
4088:
4078:
4058:
4045:
4044:
4040:
4032:
4031:
4016:
4010:
3999:
3973:
3949:
3936:
3926:
3925:
3921:
3903:
3885:
3875:
3862:
3852:
3851:
3847:
3846:
3842:
3824:
3811:
3810:
3806:
3795:
3794:
3786:
3782:
3776:
3763:
3758:
3756:
3751:
3744:
3733:
3702:
3689:
3688:
3684:
3660:
3634:
3630:
3597:
3596:
3588:
3581:
3576:
3572:
3543:
3517:
3513:
3480:
3479:
3473:
3420:
3398:
3397:
3393:
3385:
3384:
3370:
3344:
3326:
3304:
3303:
3299:
3277:
3261:
3241:
3237:
3230:
3229:
3218:
3214:
3208:
3197:
3171:
3150:
3128:
3118:
3117:
3113:
3095:
3077:
3067:
3045:
3035:
3034:
3030:
3029:
3025:
3009:
2989:
2988:
2980:
2976:
2952:
2942:
2920:
2910:
2899:
2898:
2891:
2859:
2837:
2836:
2832:
2810:
2802:
2801:
2795:
2791:
2788:
2783:
2779:
2776:
2770:
2765:
2752:
2683:
2678:
2677:
2664:
2662:
2598:
2593:
2592:
2578:
2507:
2490:
2479:
2478:
2469:
2459:
2446:
2438:
2418:
2416:
2413:
2401:
2396:
2392:
2382:
2372:
2344:
2321:
2306:
2301:
2300:
2289:
2281:
2272:
2255:
2220:
2190:
2167:
2132:
2131:
2103:
2072:
2049:
2006:
1983:
1965:
1943:
1942:
1920:
1820:
1819:
1804:
1793:
1755:
1733:
1732:
1711:
1707:
1704:
1699:
1695:
1688:
1684:
1677:
1664:
1555:
1534:
1530:
1487:
1486:
1470:
1451:
1450:
1355:
1354:
1313:
1276:
1275:
1265:
1181:
1150:
1149:
1133:
1114:
1113:
972:
971:
958:
918:
917:
904:
885:
875:
831:
822:
821:
758:
753:
752:
746:
738:identity matrix
731:
718:
646:
605:
589:
584:
583:
563:
550:
540:
503:
493:
469:
464:
463:
449:
378:
343:
282:
244:
233:
232:
216:
183:
172:
171:
156:
104:
68:
57:
51:
48:
40:help improve it
37:
28:
24:
17:
12:
11:
5:
6278:
6276:
6268:
6267:
6265:Dirac equation
6262:
6252:
6251:
6247:
6246:
6239:
6213:
6178:
6159:
6152:
6126:
6083:
6033:
6018:
5965:
5956:|journal=
5907:
5888:(2): 285–296.
5872:
5837:
5798:
5756:
5719:(5): 440–442.
5703:
5688:
5644:
5609:
5574:
5559:
5524:
5491:
5464:(7): 537–541.
5448:
5395:
5365:hep-ph/0005314
5342:
5307:
5272:
5242:hep-ph/9503416
5219:
5200:
5183:
5142:
5101:
5082:(6): 532–536.
5066:
5045:(3): 267–285.
5025:
4984:
4965:(5): 688–693.
4949:
4906:
4903:
4902:
4871:
4869:
4862:
4856:
4853:
4852:
4851:
4844:
4841:
4822:Dirac equation
4799:Pauli equation
4791:
4790:
4775:
4770:
4766:
4760:
4755:
4748:
4745:
4725:
4724:
4710:
4707:
4701:
4695:
4690:
4683:
4680:
4673:
4670:
4667:
4664:
4659:
4655:
4650:
4645:
4642:
4636:
4633:
4606:Bloch equation
4587:Dirac equation
4581:
4580:
4543:
4541:
4534:
4528:
4525:
4515:zitterbewegung
4509:
4508:
4499:
4497:
4486:
4483:
4479:
4473:
4469:
4465:
4462:
4457:
4453:
4449:
4445:
4441:
4438:
4434:
4429:
4425:
4418:
4415:
4408:
4404:
4400:
4393:
4390:
4382:
4378:
4360:
4359:
4350:
4348:
4335:
4331:
4327:
4324:
4317:
4313:
4307:
4303:
4297:
4294:
4290:
4284:
4280:
4276:
4272:
4268:
4261:
4258:
4250:
4246:
4243:
4239:
4235:
4228:
4225:
4218:
4212:
4209:
4203:
4199:
4192:
4189:
4182:
4150:
4135:
4134:
4125:
4123:
4110:
4106:
4102:
4095:
4091:
4085:
4081:
4075:
4071:
4065:
4061:
4057:
4052:
4048:
4043:
4039:
4014:
4005:
4004:
3995:
3993:
3980:
3976:
3972:
3969:
3966:
3962:
3956:
3952:
3948:
3943:
3939:
3933:
3929:
3924:
3920:
3916:
3910:
3906:
3902:
3898:
3892:
3888:
3882:
3878:
3874:
3869:
3865:
3859:
3855:
3850:
3845:
3841:
3837:
3831:
3827:
3823:
3818:
3814:
3809:
3805:
3802:
3784:
3739:
3738:
3729:
3727:
3715:
3709:
3705:
3701:
3696:
3692:
3687:
3683:
3680:
3677:
3673:
3667:
3663:
3659:
3655:
3651:
3644:
3641:
3633:
3629:
3626:
3622:
3618:
3611:
3608:
3586:
3579:
3569:
3568:
3556:
3550:
3546:
3542:
3538:
3534:
3527:
3524:
3516:
3512:
3509:
3505:
3501:
3494:
3491:
3472:
3469:
3465:zitterbewegung
3461:
3460:
3449:
3446:
3442:
3436:
3429:
3426:
3419:
3414:
3407:
3404:
3396:
3392:
3376:
3375:
3366:
3364:
3351:
3347:
3343:
3339:
3333:
3329:
3325:
3320:
3313:
3310:
3302:
3298:
3295:
3289:
3284:
3280:
3273:
3267:
3264:
3257:
3250:
3247:
3240:
3212:
3203:
3202:
3193:
3191:
3178:
3174:
3170:
3167:
3163:
3157:
3153:
3149:
3144:
3137:
3134:
3125:
3121:
3116:
3112:
3108:
3102:
3098:
3094:
3090:
3084:
3080:
3074:
3070:
3066:
3061:
3054:
3051:
3042:
3038:
3033:
3028:
3024:
3021:
3016:
3012:
3008:
3005:
3002:
2999:
2996:
2978:
2973:
2972:
2959:
2955:
2949:
2945:
2941:
2936:
2929:
2926:
2917:
2913:
2909:
2906:
2885:
2884:
2872:
2866:
2862:
2858:
2853:
2846:
2843:
2835:
2831:
2828:
2822:
2817:
2813:
2793:
2786:
2774:Zitterbewegung
2769:
2766:
2764:
2761:
2758:
2757:
2748:
2746:
2735:
2732:
2729:
2725:
2721:
2714:
2711:
2704:
2699:
2692:
2689:
2653:
2652:
2641:
2638:
2635:
2631:
2627:
2623:
2619:
2614:
2607:
2604:
2584:
2583:
2574:
2572:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2517:
2514:
2510:
2506:
2503:
2500:
2496:
2493:
2489:
2486:
2412:
2409:
2399:
2369:
2368:
2353:
2348:
2342:
2337:
2333:
2328:
2324:
2318:
2313:
2309:
2279:
2246:Dirac equation
2226:
2225:
2216:
2214:
2199:
2194:
2188:
2183:
2179:
2174:
2170:
2164:
2161:
2157:
2153:
2146:
2143:
2113:so that using
2109:
2108:
2099:
2097:
2081:
2076:
2070:
2065:
2061:
2056:
2052:
2047:
2042:
2039:
2036:
2033:
2030:
2015:
2010:
2004:
1999:
1995:
1990:
1986:
1979:
1974:
1969:
1962:
1959:
1956:
1953:
1950:
1926:
1925:
1916:
1914:
1903:
1900:
1897:
1894:
1891:
1887:
1882:
1877:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1845:
1841:
1834:
1831:
1799:
1798:
1789:
1787:
1774:
1769:
1764:
1759:
1752:
1749:
1746:
1743:
1740:
1710:such that the
1702:
1676:
1673:
1670:
1669:
1660:
1658:
1647:
1644:
1641:
1638:
1635:
1631:
1626:
1621:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1589:
1585:
1582:
1579:
1576:
1569:
1564:
1559:
1554:
1549:
1546:
1543:
1540:
1537:
1533:
1528:
1524:
1520:
1516:
1512:
1508:
1501:
1498:
1476:
1475:
1466:
1464:
1449:
1446:
1443:
1440:
1437:
1431:
1427:
1420:
1416:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1378:
1374:
1370:
1366:
1363:
1360:
1358:
1356:
1351:
1345:
1341:
1334:
1330:
1326:
1323:
1320:
1316:
1312:
1309:
1306:
1303:
1299:
1295:
1291:
1287:
1284:
1281:
1279:
1277:
1272:
1268:
1264:
1261:
1258:
1252:
1248:
1241:
1237:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1202:
1198:
1194:
1190:
1187:
1184:
1182:
1179:
1175:
1168:
1165:
1158:
1157:
1139:
1138:
1129:
1127:
1112:
1109:
1106:
1103:
1097:
1093:
1086:
1082:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1047:
1043:
1039:
1035:
1032:
1029:
1026:
1023:
1017:
1013:
1006:
1002:
998:
995:
992:
989:
986:
983:
980:
977:
975:
973:
968:
965:
961:
957:
954:
951:
948:
944:
940:
936:
932:
929:
926:
923:
921:
919:
914:
911:
907:
901:
894:
891:
884:
881:
878:
876:
873:
869:
862:
859:
852:
847:
840:
837:
830:
829:
813:
812:
801:
798:
795:
791:
787:
783:
779:
774:
767:
764:
745:
742:
715:
714:
703:
700:
697:
691:
687:
680:
676:
672:
669:
666:
663:
660:
655:
650:
645:
640:
634:
630:
623:
619:
615:
612:
608:
604:
599:
596:
592:
537:Dirac matrices
533:
532:
517:
512:
507:
500:
496:
490:
485:
478:
475:
455:
454:
445:
443:
432:
429:
426:
420:
416:
409:
405:
401:
398:
395:
392:
387:
382:
377:
372:
366:
362:
355:
351:
346:
342:
339:
336:
333:
327:
323:
316:
312:
308:
305:
302:
299:
296:
291:
286:
281:
276:
270:
266:
259:
255:
251:
247:
243:
240:
222:
221:
212:
210:
199:
196:
193:
189:
186:
182:
179:
155:
152:
103:
100:
88:Dirac equation
70:
69:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
6277:
6266:
6263:
6261:
6258:
6257:
6255:
6242:
6240:9780123742193
6236:
6232:
6228:
6224:
6217:
6214:
6209:
6205:
6201:
6197:
6193:
6189:
6182:
6179:
6174:
6170:
6163:
6160:
6155:
6153:9780120147397
6149:
6145:
6141:
6137:
6130:
6127:
6122:
6118:
6114:
6110:
6106:
6102:
6098:
6094:
6087:
6084:
6079:
6075:
6071:
6067:
6063:
6059:
6052:
6048:
6044:
6037:
6034:
6029:
6022:
6019:
6014:
6010:
6006:
6002:
5998:
5994:
5989:
5984:
5981:(1): 95–125.
5980:
5976:
5969:
5966:
5961:
5948:
5940:
5936:
5931:
5926:
5922:
5918:
5917:Simon, Rajiah
5911:
5908:
5903:
5899:
5895:
5891:
5887:
5883:
5876:
5873:
5868:
5864:
5860:
5856:
5852:
5848:
5841:
5838:
5833:
5829:
5825:
5821:
5817:
5813:
5807:
5802:
5799:
5794:
5790:
5786:
5782:
5778:
5774:
5770:
5766:
5760:
5757:
5752:
5748:
5744:
5740:
5736:
5732:
5727:
5722:
5718:
5714:
5707:
5704:
5699:
5695:
5691:
5689:9780444825308
5685:
5681:
5677:
5673:
5669:
5664:
5659:
5655:
5648:
5645:
5640:
5636:
5632:
5628:
5624:
5620:
5613:
5610:
5605:
5601:
5597:
5593:
5589:
5585:
5578:
5575:
5570:
5563:
5560:
5555:
5551:
5547:
5543:
5539:
5535:
5528:
5525:
5520:
5516:
5511:
5506:
5502:
5495:
5492:
5487:
5483:
5479:
5475:
5471:
5467:
5463:
5459:
5452:
5449:
5444:
5440:
5436:
5432:
5428:
5424:
5419:
5414:
5411:(1): 012503.
5410:
5406:
5399:
5396:
5391:
5387:
5383:
5379:
5375:
5371:
5366:
5361:
5357:
5353:
5346:
5343:
5338:
5334:
5330:
5326:
5322:
5318:
5311:
5308:
5303:
5299:
5295:
5291:
5287:
5283:
5276:
5273:
5268:
5264:
5260:
5256:
5252:
5248:
5243:
5238:
5234:
5230:
5223:
5220:
5214:
5213:
5204:
5201:
5196:
5195:
5187:
5184:
5178:
5173:
5169:
5165:
5161:
5157:
5156:Physics Today
5153:
5146:
5143:
5137:
5132:
5128:
5124:
5120:
5116:
5115:Physics Today
5112:
5105:
5102:
5097:
5093:
5089:
5085:
5081:
5077:
5070:
5067:
5061:
5056:
5052:
5048:
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4896:February 2017
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4832:Lie algebraic
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4139:will satisfy
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2264:). See this
2263:
2262:
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2252:
2247:
2243:
2242:Eugene Wigner
2239:
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571:commutativity
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177:
170:
169:
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165:of the form:
164:
163:wave function
161:
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55:
52:February 2017
45:
41:
35:
32:This article
30:
21:
20:
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5968:
5947:cite journal
5910:
5885:
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5840:
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5811:
5806:Majorana, E.
5801:
5776:
5772:
5759:
5716:
5712:
5706:
5653:
5647:
5622:
5618:
5612:
5590:(1): 53–80.
5587:
5583:
5577:
5568:
5562:
5537:
5533:
5527:
5500:
5494:
5461:
5457:
5451:
5408:
5404:
5398:
5355:
5351:
5345:
5323:(1): L1–L4.
5320:
5316:
5310:
5285:
5281:
5275:
5232:
5228:
5222:
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5203:
5193:
5186:
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5075:
5069:
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4987:
4962:
4958:
4952:
4930:(1): 29–36.
4927:
4923:
4910:
4893:
4888:citation bot
4882:Please help
4873:
4829:
4826:
4807:
4796:
4792:
4726:
4617:
4614:
4599:
4584:
4569:
4560:
4553:Please help
4545:
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2654:
2587:
2576:
2464:
2460:
2453:
2450:= 0, ±π, ±2π
2447:
2440:
2432:
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2419:
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58:
49:
33:
6047:Mukunda, N.
5765:Laporte, O.
4886:or run the
4602:synchrotron
4159:factor via
154:Description
116:eigenvalues
112:Hamiltonian
108:orthonormal
6254:Categories
5162:(11): 89.
5121:(12): 75.
2669:| = 0
2452:and, from
2445:, so that
2423:| = 0
2385:, so that
1681:continuous
6175:: 99–126.
5832:122510061
5751:250793483
5486:250889471
4754:⊥
4747:^
4689:⊥
4682:^
4672:−
4644:−
4635:^
4610:polarized
4591:acoustics
4464:β
4448:β
4417:^
4392:^
4326:β
4296:β
4260:^
4242:≡
4227:^
4191:^
3804:≡
3682:β
3643:^
3625:≡
3610:^
3526:^
3508:≡
3493:^
3445:≠
3428:^
3406:^
3346:α
3312:^
3294:≡
3288:^
3249:^
3173:γ
3136:^
3120:γ
3069:γ
3053:^
3037:γ
2944:γ
2928:^
2912:γ
2889:rest mass
2845:^
2827:≡
2821:^
2731:β
2713:^
2691:^
2637:β
2626:⋅
2622:α
2618:≡
2606:^
2548:±
2527:±
2513:−
2499:≡
2488:→
2244:) of the
2163:β
2145:^
2038:θ
2032:
1958:θ
1952:
1899:θ
1893:
1869:θ
1863:
1851:β
1833:^
1751:≡
1748:θ
1742:
1643:θ
1637:
1613:θ
1607:
1595:β
1584:θ
1578:
1548:−
1545:θ
1539:
1523:⋅
1519:α
1500:^
1445:θ
1439:
1430:^
1419:⋅
1415:α
1411:β
1408:−
1405:θ
1399:
1384:β
1373:⋅
1369:α
1350:θ
1344:^
1333:⋅
1329:α
1325:β
1319:−
1305:β
1294:⋅
1290:α
1263:θ
1260:
1251:^
1240:⋅
1236:α
1232:β
1229:−
1226:θ
1223:
1208:β
1197:⋅
1193:α
1167:^
1108:θ
1105:
1096:^
1085:⋅
1081:α
1077:β
1074:−
1071:θ
1068:
1053:β
1042:⋅
1038:α
1028:θ
1025:
1016:^
1005:⋅
1001:α
997:β
991:θ
988:
964:−
950:β
939:⋅
935:α
910:−
893:^
880:≡
861:^
851:→
839:^
797:β
786:⋅
782:α
778:≡
766:^
702:θ
699:
690:^
679:⋅
675:α
671:β
668:−
665:θ
662:
639:θ
633:^
622:⋅
618:α
614:β
611:−
595:−
567:= 1, 2, 3
489:≡
477:^
431:θ
428:
419:^
408:⋅
404:γ
397:θ
394:
371:θ
365:^
354:⋅
350:γ
338:θ
335:
326:^
315:⋅
311:α
307:β
301:θ
298:
275:θ
269:^
258:⋅
254:α
250:β
198:ψ
185:ψ
181:→
178:ψ
6260:Fermions
6049:(1989).
6013:55537377
5919:(2002).
5698:17695022
5390:17118044
5267:16766114
4843:See also
4428:′
4403:′
4271:′
4238:′
4202:′
3762:+ |
3654:′
3621:′
3537:′
3504:′
2724:′
2495:′
2425:. From
2156:′
1844:′
1720:term in
1511:′
1178:′
872:′
730:, where
188:′
92:spin-1/2
6196:Bibcode
6121:9903968
6101:Bibcode
6066:Bibcode
5993:Bibcode
5935:Bibcode
5890:Bibcode
5855:Bibcode
5781:Bibcode
5731:Bibcode
5668:Bibcode
5627:Bibcode
5592:Bibcode
5542:Bibcode
5515:Bibcode
5466:Bibcode
5443:5376899
5423:Bibcode
5370:Bibcode
5325:Bibcode
5290:Bibcode
5247:Bibcode
5164:Bibcode
5123:Bibcode
5084:Bibcode
5047:Bibcode
5006:Bibcode
4967:Bibcode
4932:Bibcode
4612:beams.
3757:√
2248:. What
562:, with
459:Above,
160:fermion
38:Please
6237:
6150:
6119:
6011:
5830:
5749:
5696:
5686:
5484:
5441:
5388:
5265:
4874:lacks
4595:optics
4593:, and
3768:|
2663:|
2417:|
144:spin-1
140:spin-0
6188:Optik
6054:(PDF)
6009:S2CID
5983:arXiv
5925:arXiv
5828:S2CID
5747:S2CID
5721:arXiv
5694:S2CID
5658:arXiv
5505:arXiv
5482:S2CID
5439:S2CID
5413:arXiv
5386:S2CID
5360:arXiv
5263:S2CID
5237:arXiv
4920:(PDF)
4876:ISBNs
4855:Notes
2439:cos 2
2290:diag(
148:spins
6235:ISBN
6148:ISBN
6117:PMID
5960:help
5684:ISBN
4608:for
2266:link
2240:and
549:and
142:and
82:and
74:The
6227:doi
6204:doi
6192:117
6140:doi
6109:doi
6074:doi
6062:134
6001:doi
5898:doi
5863:doi
5851:113
5820:doi
5789:doi
5739:doi
5676:doi
5635:doi
5600:doi
5588:311
5550:doi
5474:doi
5431:doi
5378:doi
5356:106
5333:doi
5298:doi
5255:doi
5172:doi
5131:doi
5092:doi
5055:doi
5014:doi
5002:195
4975:doi
4940:doi
4839:).
4147:= −
4011:= −
3209:= −
2894:as
2463:= ±
2443:= 1
2431:or
2119:in
2029:cos
2024:and
1949:sin
1890:sin
1860:cos
1739:tan
1634:sin
1604:cos
1575:sin
1536:cos
1436:sin
1396:cos
1257:sin
1220:cos
1102:sin
1065:cos
1022:sin
985:cos
696:sin
659:cos
539:by
425:sin
391:cos
332:sin
295:cos
42:to
6256::
6233:.
6202:.
6190:.
6173:56
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