View source for Foldy–Wouthuysen transformation

{{Short description|Used to understand the Dirac equation}}
{{Technical|date=February 2017}}
The '''Foldy–Wouthuysen transformation''' was historically significant and was 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.<ref>{{cite journal|last1=Foldy|first1=L. L.|last2=Wouthuysen|first2=S. A.|date=1950|title=On the Dirac Theory of Spin 1⁄2 Particles and its Non-Relativistic Limit|url=http://www.physics.drexel.edu/~bob/Quantum_Papers/Foldy-Wouthuysen.pdf|journal=Physical Review|volume=78|issue=1|pages=29–36|bibcode=1950PhRv...78...29F|doi=10.1103/PhysRev.78.29}}</ref><ref>{{cite journal|last1=Foldy|first1=L. L.|date=1952|title=The Electromagnetic Properties of the Dirac Particles|journal=Physical Review|volume=87|issue=5|pages=688–693|bibcode=1952PhRv...87..688F|doi=10.1103/PhysRev.87.688}}</ref><ref>{{cite journal|last1=Pryce|first1=M. H. L.|date=1948|title=The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles|journal=Proceedings of the Royal Society of London A|volume=195|issue=1040|pages=62–81|bibcode=1948RSPSA.195...62P|doi=10.1098/rspa.1948.0103|doi-access=free}}</ref><ref>{{cite journal|last1=Tani|first1=S.|date=1951|title=Connection between particle models and field theories. I. The case spin 1⁄2|journal=Progress of Theoretical Physics|volume=6|issue=3|pages=267–285|bibcode=1951PThPh...6..267T|doi=10.1143/ptp/6.3.267|doi-access=free}}</ref> A detailed general discussion of the Foldy–Wouthuysen-type transformations in particle interpretation of relativistic wave equations is in Acharya and Sudarshan (1960).<ref>{{cite journal|last1=Acharya|first1=R.|last2=Sudarshan|first2=E. C. G.|date=1960|title=Front Description in Relativistic Quantum Mechanics|journal=Journal of Mathematical Physics|volume=1|issue=6|pages=532–536|bibcode=1960JMP.....1..532A|doi=10.1063/1.1703689}}</ref> Its utility in [[high energy physics]]  is now limited due to the primary applications being in the ultra-relativistic domain where the Dirac field is treated as a quantised field.

==A canonical transform==
The FW transformation is a unitary transformation of the [[orthonormal]] basis in which both the [[Hamiltonian (quantum mechanics)|Hamiltonian]] and the state are represented. The [[eigenvalue]]s 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 transformation|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<ref>{{cite journal|last1=Brown|first1=R. W.|last2=Krauss|first2=L. M.|last3=Taylor|first3=P. L.|date=2001|title=Obituary of Leslie Lawrence Foldy|journal=Physics Today|volume=54|issue=12|page=75|doi=10.1063/1.1445566|bibcode=2001PhT....54l..75B|doi-access=free}}</ref><ref>{{cite journal|last1=Leopold|first1=H.|date=1997|title=Obituary of Siegfried A Wouthuysen|journal=Physics Today|volume=50|issue=11|page=89|doi=10.1063/1.882018|bibcode=1997PhT....50k..89H|doi-access=free}}</ref> and the biographical memoir of Foldy.<ref>{{cite book|last1=Foldy|first1=L. L.|date=2006|contribution=Origins of the FW Transformation: A Memoir|pages=347–351|title=Physics at a Research University: Case Western Reserve University 1830–1990|url=http://www.phys.cwru.edu/history|editor-first=William|editor-last=Fickinger}}</ref> 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.<ref>{{cite book|last1=Bjorken|first1=J. D.|last2=Drell|first2=S. D.|date=1964|title=Relativistic Quantum Mechanics|url=https://archive.org/details/relativisticquan0000bjor|url-access=registration|publisher=McGraw-Hill|location=New York, San Francisco}}</ref><ref>{{cite journal|last1=Costella|first1=J. P.|last2=McKellar|first2=B. H. J.|date=1995|title=The Foldy–Wouthuysen transformation|arxiv=hep-ph/9503416|journal=American Journal of Physics|volume=63|issue=12|pages=1119–1124|doi=10.1119/1.18017|bibcode=1995AmJPh..63.1119C|s2cid=16766114}}</ref> The Foldy–Wouthuysen transform was extended to the physically important cases of [[scalar particle|spin-0]] and [[vector particle|spin-1]] particles,<ref>{{cite journal|last1=Case|first1=K. M.|date=1954|title=Some generalizations of the Foldy–Wouthuysen transformation|journal=Physical Review|volume=95|issue=5|pages=1323–1328|bibcode=1954PhRv...95.1323C|doi=10.1103/PhysRev.95.1323}}</ref> and even generalized to the case of arbitrary [[Spin (physics)|spins]].<ref>{{cite journal|last1=Jayaraman|first1=J.|date=1975|title=A note on the recent Foldy–Wouthuysen transformations for particles of arbitrary spin|journal=Journal of Physics A|volume=8|issue=1|page=L1–L4|bibcode=1975JPhA....8L...1J|doi=10.1088/0305-4470/8/1/001}}</ref>

==Description==

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

{{NumBlk|:|<math>\psi \to \psi '=U\psi </math>|{{EquationRef|1}}}}

where the unitary operator is the 4&nbsp;×&nbsp;4 matrix:

{{NumBlk|:|<math>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 </math>|{{EquationRef|2}}}}

Above,

:<math>\hat{p}^i \equiv \frac{p^i}{|\mathbf{p}|}</math>

is the unit vector oriented in the direction of the fermion momentum. The above are related to the [[Dirac matrices]] by {{math|''β'' {{=}} ''γ''<sup>0</sup>}} and {{math|''α<sup>i</sup>'' {{=}} ''γ''<sup>0</sup>''γ''<sup>''i''</sup>}}, with {{math|''i'' {{=}} 1, 2, 3}}. A straightforward series expansion applying the [[commutativity]] properties of the Dirac matrices demonstrates that {{EquationNote|2}} above is true. The inverse

:<math>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</math>

so it is clear that {{math|''U''<sup>−1</sup>''U'' {{=}} ''I''}}, where {{math|''I''}} is a 4&nbsp;×&nbsp;4 [[identity matrix]].

==Transforming the Dirac Hamiltonian for a free fermion==

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

:<math>\hat{H}_0 \equiv \boldsymbol{\alpha} \cdot \mathbf{p} + \beta m</math>

in biunitary fashion, in the form:

{{NumBlk|:|<math>\begin{align}\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{align}</math>|{{EquationRef|3}}}}

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

{{NumBlk|:|<math>\begin{align}\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{align}</math>|{{EquationRef|4}}}}

This factors out into:

{{NumBlk|:|<math>\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)</math>|{{EquationRef|5}}}}

===Choosing a particular representation: Newton–Wigner===

Clearly, the FW transformation is a ''continuous'' transformation, that is, one may employ any value for {{mvar|θ}} which one chooses.  Choosing a particular value for {{mvar|θ}} amounts to choosing a particular transformed representation.

One particularly important representation is that in which the transformed Hamiltonian operator {{math|''Ĥ''{{prime}}<sub>0</sub>}} is diagonalized.  A completely diagonal representation can be obtained by choosing {{mvar|θ}} such that the {{math|'''α''' · '''p'''}} term in {{EquationNote|5}} vanishes. This is arranged by choosing:

{{NumBlk|:|<math>\tan 2\theta \equiv \frac{|\mathbf{p}|}{m}</math>|{{EquationRef|6}}}}

In the Dirac-Pauli representation where {{mvar|β}} is a diagonal matrix, {{EquationNote|5}} is then reduced to a diagonal matrix:

{{NumBlk|:|<math>\hat{H}'_0= \beta (m \cos 2\theta + |\mathbf{p}| \sin 2\theta)</math>|{{EquationRef|7}}}}

By elementary trigonometry, {{EquationNote|6}} also implies that:

{{NumBlk|:|<math>\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} </math>|{{EquationRef|8}}}}

so that using {{EquationNote|8}} in {{EquationNote|7}} and then simplifying now leads to:

{{NumBlk|:|<math>\hat{H}'_0= \beta \sqrt{m^2+|\mathbf{p}|^2}</math>|{{EquationRef|9}}}}

Prior to Foldy and Wouthuysen publishing their transformation, it was already known that {{EquationNote|9}} is the Hamiltonian in the Newton–Wigner (NW) representation (named after [[Theodore Duddell Newton]] and [[Eugene Wigner]]) of the [[Dirac equation]]. What {{EquationNote|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 {{mvar|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 ({{EquationNote|9}}).  See this [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.27.3209&rep=rep1&type=pdf link].

If one considers an on-shell mass—fermion or otherwise—given by {{math|''m''<sup>2</sup> {{=}} ''p<sup>σ</sup>p<sub>σ</sub>''}}, and employs a [[Minkowski metric]] tensor for which {{math|diag(''η'') {{=}} (+1, −1, −1, −1)}}, it should be apparent that the expression
:<math> p^0 = \sqrt{m^2 + |\mathbf{p}|^2} </math>
is equivalent to the {{math|''E'' ≡ ''p''<sup>0</sup>}} component of the energy-momentum vector {{mvar|p<sup>μ</sup>}}, so that {{EquationNote|9}} is alternatively specified rather simply by {{math|''Ĥ''{{prime}}<sub>0</sub> {{=}} ''βE''}}.

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

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

{{NumBlk|:|<math>O \to O' \equiv U O U^{-1} = (\pm I) (O) (\pm I) = O</math>|{{EquationRef|10}}}}

Contrasting the original Dirac–Pauli Hamiltonian operator

:<math>\hat{H}_0 \equiv \boldsymbol{\alpha} \cdot \mathbf{p} + \beta m</math>

with the NW Hamiltonian {{EquationNote|9}}, we do indeed find the {{math|{{abs|'''p'''}} {{=}} 0}} "at rest" correspondence:

{{NumBlk|:|<math>\hat{H}_0 = \hat{H}'_0= \beta m</math>|{{EquationRef|11}}}}

== Transforming the velocity operator ==

===In the Dirac–Pauli representation===
{{Further|Zitterbewegung}}
Now, consider the velocity operator.  To obtain this operator, we must commute the Hamiltonian operator {{math|''Ĥ''{{prime}}<sub>0</sub>}} with the canonical position operators {{mvar|x<sub>i</sub>}}, i.e., we must calculate

:<math>\hat{v_i}\equiv i\left[\hat{H}_0,x_i\right]</math>

One good way to approach this calculation, is to start by writing the scalar [[rest mass]] {{mvar|m}} as

:<math>m=\gamma^0\hat{H}_0+\gamma^jp_j</math>

and then to mandate that the scalar rest mass commute with the {{mvar|x<sub>i</sub>}}.  Thus, we may write:

{{NumBlk|:|<math>0=[m,x_i]=\left[\left(\gamma^0\hat{H}_0+\gamma^jp_j\right),x_i\right]=\left[\gamma^0\hat{H}_0,x_i\right]+i\gamma_i</math>|{{EquationRef|12}}}}

where we have made use of the Heisenberg canonical commutation relationship {{math|[''x<sub>i</sub>'',''p<sub>j</sub>''] {{=}} −''iη<sub>ij</sub>''}} to reduce terms. Then, multiplying from the left by {{math|''γ''<sup>0</sup>}} and rearranging terms, we arrive at:

{{NumBlk|:|<math>\frac{d\hat{x}_i}{dt}=\hat{v_i}\equiv i\left[\hat{H}_0,x_i\right]=\alpha_i</math>|{{EquationRef|13}}}}

Because the canonical relationship

:<math>i\left[\hat{H}_0,\hat{v}_i\right] \ne 0</math>

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

===In the Newton–Wigner representation===

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

:<math>\hat{v}_i'\equiv i\left[\hat{H}'_0,x_i\right]</math>

If we use the result at the very end of section 2 above, {{math|''Ĥ''{{prime}}<sub>0</sub> {{=}} ''βp''<sub>0</sub>}}, then this can be written instead as:

{{NumBlk|:|<math>\hat{v}_i'\equiv i\left[\hat{H}'_0,x_i\right]=i \beta \left[p_0,x_i\right] </math>|{{EquationRef|14}}}}

Using the above, we need simply to calculate {{math|[''p''<sub>0</sub>,''x<sub>i</sub>'']}}, then multiply by {{mvar|iβ}}.

The canonical calculation proceeds similarly to the calculation in section 4 above, but because of the square root expression in {{math|''p''<sup>0</sup> {{=}} {{sqrt|''m''<sup>2</sup> + {{abs|''p''}}<sup>2</sup>}}}}, one additional step is required.

First, to accommodate the square root, we will wish to require that the scalar square mass {{math|''m''<sup>2</sup>}} commute with the canonical coordinates {{mvar|x<sub>i</sub>}}, which we write as:

{{NumBlk|:|<math>0 \equiv \left[m^2,x_i\right] = \left[\left(p^0p_0+p^jp_j\right),x_i\right] = \left[p^0p_0,x_i\right]+2ip_i</math>|{{EquationRef|15}}}}

where we again use the Heisenberg canonical relationship {{math|[''x<sub>i</sub>'',''p<sub>j</sub>''] {{=}} −''iη<sub>ij</sub>''}}.  Then, we need an expression for {{math|[''p''<sub>0</sub>,''x<sub>i</sub>'']}} which will satisfy {{EquationNote|15}}.  It is straightforward to verify that:

{{NumBlk|:|<math>i\left[p_0,x_i\right]=\frac{p_i}{p^0}=v_i</math>|{{EquationRef|16}}}}

will satisfy {{EquationNote|15}} when again employing {{math|[''x<sub>i</sub>'',''p<sub>j</sub>''] {{=}} −''iη<sub>ij</sub>''}}. Now, we simply return the {{mvar|iβ}} factor via {{EquationNote|14}}, to arrive at:

{{NumBlk|:|<math>\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 </math>|{{EquationRef|17}}}}

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

{{NumBlk|:|<math>i\left[\hat{H}'_0,\hat{v}_i'\right]=i\left[\beta p_0,\beta v_i\right]=0</math>|{{EquationRef|18}}}}

it is commonly thought that the [[zitterbewegung]] motion arising out of {{EquationNote|12}} vanishes when a fermion is transformed into the Newton–Wigner representation.

==Other applications==
{{Excessive citations|section|date=February 2017}}

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<ref>{{cite journal|last1=Asaga|first1=T.|last2=Fujita|first2=T.|last3=Hiramoto|first3=M.|date=2000|title=EDM operator free from Schiff's theorem|arxiv=hep-ph/0005314|journal=Progress of Theoretical Physics|volume=106|issue=6|pages=1223–1238|doi=10.1143/PTP.106.1223|bibcode=2001PThPh.106.1223A|s2cid=17118044}}</ref><ref>{{cite journal|last1=Pachucki|first1=K.|date=2004|title=Higher-order effective Hamiltonian for light atomic systems|arxiv=physics/0411168|journal=Physical Review A|volume=71|issue=1|page=012503|doi=10.1103/PhysRevA.71.012503|bibcode=2005PhRvA..71a2503P|s2cid=5376899}}</ref> [[synchrotron]] radiation<ref>{{cite journal|last1=Lippert|first1=M.|last2=Bruckel|first2=Th.|last3=Kohler|first3=Th.|last4=Schneider|first4=J. R.|date=1994|title=High-Resolution Bulk Magnetic Scattering of High-Energy Synchrotron Radiation|journal=Europhysics Letters|volume=27|issue=7|pages=537–541|bibcode=1994EL.....27..537L|doi=10.1209/0295-5075/27/7/008|s2cid=250889471 }}</ref> and derivation of the [[Bloch equation]] for [[spin polarization|polarized]] beams.<ref>{{cite book|last1=Heinemann|first1=K.|title=Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics, 4–9 January 1998, Monterey, California, USA|last2=Barber|first2=D. P.|date=1999|publisher=World Scientific|editor-last=Chen|editor-first=P|location=Singapore|pages=physics/9901044|contribution=The semiclassical Foldy–Wouthuysen transformation and the derivation of the Bloch equation for spin-1⁄2 polarized beams using Wigner functions|bibcode=1999physics...1044H|arxiv=physics/9901044}}</ref>

The application of the Foldy–Wouthuysen transformation in acoustics is very natural; comprehensive and mathematically rigorous accounts.<ref>{{cite journal|last1=Fishman|first1=L.|date=1992|title=Exact and operator rational approximate solutions of the Helmholtz, Weyl composition equation in underwater acoustics—the quadratic profile|journal=Journal of Mathematical Physics|volume=33|issue=5|pages=1887–1914|bibcode=1992JMP....33.1887F|doi=10.1063/1.529666}}</ref><ref>{{cite book|last1=Fishman|first1=L.|date=2004|contribution=One-way wave equation modeling in two-way wave propagation problems|editor1-first=B.|editor1-last=Nilsson|editor2-first=L.|editor2-last=Fishman|title=Mathematical Modelling of Wave Phenomena 2002, Mathematical Modelling in Physics, Engineering and Cognitive Sciences|volume=7|publisher=Växjö University Press|location=Växjö, Sweden|pages=91–111}}</ref><ref>{{cite journal|last1=Wurmser|first1=D.|date=2004|title=A parabolic equation for penetrable rough surfaces: using the Foldy–Wouthuysen transformation to buffer density jumps|journal=Annals of Physics|volume=311|issue=1|pages=53–80|bibcode=2004AnPhy.311...53W|doi=10.1016/j.aop.2003.11.006}}</ref>

In the traditional scheme the purpose of expanding the optical Hamiltonian

:<math> \hat{H} = - \left( n^2 (r) - \hat{p}_\perp^2 \right)^\frac12</math>

in a series using

:<math>\frac{\hat{p}_\perp^2}{n_0^2}</math>

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]]<ref>{{cite journal|last=Osche|first=G. R.|date=1977|title=Dirac and Dirac–Pauli equation in the Foldy–Wouthuysen representation|journal=Physical Review D|volume=15|issue=8|pages=2181–2185|bibcode=1977PhRvD..15.2181O|doi=10.1103/PhysRevD.15.2181}}</ref> in the nonrelativistic limit and the other describes the negative-energy states. It is possible to write a Dirac-like [[Matrix Representation of Maxwell Equations|matrix representation of Maxwell's equations]]. In such a matrix form the Foldy–Wouthuysen can be applied.<ref>{{cite book|last=Białynicki-Birula|first=I.|date=1996|arxiv=quant-ph/0508202|title=Photon wave function|chapter=V Photon Wave Function |series=Progress in Optics|volume=36|issue=1996|pages=245–294|publisher=Elsevier |bibcode=2005quant.ph..8202B|doi=10.1016/S0079-6638(08)70316-0|isbn=9780444825308|s2cid=17695022 }}</ref><ref>{{cite journal|first=Sameen Ahmed|last=Khan|year=2005|arxiv=physics/0205083|title=Maxwell Optics: I. An exact matrix representation of the Maxwell equations in a medium|journal=Physica Scripta|volume=71|issue=5|pages=440–442|doi=10.1238/Physica.Regular.071a00440|bibcode=2005PhyS...71..440K|s2cid=250793483 }}</ref><ref>{{cite journal|author1-link=Otto Laporte|last1=Laporte|first1=O.|author2-link=George Uhlenbeck|last2=Uhlenbeck|first2=G. E.|date=1931|title=Applications of spinor analysis to the Maxwell and Dirac Equations|journal=Physical Review|volume=37|issue=11|pages=1380–1397|bibcode=1931PhRv...37.1380L|doi=10.1103/PhysRev.37.1380}}</ref><ref>[[Ettore Majorana|Majorana, E.]] (1974). Unpublished notes, quoted in {{cite journal|last1=Mignani|first1=R.|last2=Recami|first2=E.|last3=Baldo|first3=M.|title=About a Dirac-like Equation for the Photon, According to Ettore Majorana|journal=Lettere al Nuovo Cimento|volume=11|issue=12|pages=568–572|doi=10.1007/bf02812391|year=2008|s2cid=122510061}}</ref><ref>{{cite journal|last=Moses|first=E.|date=1959|title=Solutions of Maxwell's equations in terms of a spinor notation: the direct and inverse problems|journal=Physical Review|volume=113|issue=6|pages=1670–1679|bibcode=1959PhRv..113.1670M|doi=10.1103/PhysRev.113.1670}}</ref>

There is a close algebraic analogy between the [[Helmholtz equation]] (governing scalar optics) and the [[Klein–Gordon equation]]; and between the [[Matrix representation of Maxwell's equations|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.<ref>{{cite journal|last1=Fishman|first1=L.|last2=McCoy|first2=J. J.|date=1984|title=Derivation and Application of Extended Parabolic Wave Theories. Part I. The Factored Helmholtz Equation|journal=Journal of Mathematical Physics|volume=25|issue=2|pages=285–296|bibcode=1984JMP....25..285F|doi=10.1063/1.526149}}</ref>

It was only in the recent works, that this idea was exploited to analyze the quasiparaxial approximations for specific beam optical system.<ref>{{cite journal|first1=Sameen Ahmed|last1=Khan|first2=Ramaswamy|last2=Jagannathan|author3-link=Rajiah Simon|first3=Rajiah|last3=Simon|date=2002|title=Foldy–Wouthuysen transformation and a quasiparaxial approximation scheme for the scalar wave theory of light beams|url=https://archive.org/details/arxiv-physics0209082|pages=physics/0209082|arxiv=physics/0209082|bibcode=2002physics...9082K}}</ref> The Foldy–Wouthuysen technique is ideally suited for the [[Lie algebra]]ic 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<ref>{{cite journal|first1=Sameen Ahmed|last1=Khan|year=2005|title=Wavelength-dependent modifications in Helmholtz Optics|journal=International Journal of Theoretical Physics|volume=44|issue=1|pages=95–125|doi=10.1007/s10773-005-1488-0|arxiv=physics/0210001|bibcode=2005IJTP...44...95K|s2cid=55537377}}</ref> and Maxwell optics<ref>{{cite book|first1=Sameen Ahmed|last1=Khan|date=2006|contribution=Wavelength-Dependent Effects in Light Optics|title=New Topics in Quantum Physics Research|editor1-first=Volodymyr|editor1-last=Krasnoholovets|editor2-first=Frank|editor2-last=Columbus|publisher=Nova Science Publishers|location=New York|pages=163–204}}</ref> 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.<ref>{{cite journal|first1=R.|last1=Jagannathan|first2=R.|last2=Simon|author3-link=George Sudarshan|first3=E. C. G.|last3=Sudarshan|author4-link=Narasimhaiengar Mukunda|first4=N.|last4=Mukunda|doi=10.1016/0375-9601(89)90685-3|title=Quantum theory of magnetic electron lenses based on the Dirac equation|journal=Physics Letters A|volume=134|issue=8–9|pages=457–464|date=1989|bibcode=1989PhLA..134..457J|url=http://eprints.iisc.ac.in/964/1/Quantum_theory_of_magnetic_electron_lenses.pdf}}</ref><ref>{{cite journal|first1=R.|last1=Jagannathan|doi=10.1103/PhysRevA.42.6674|pmid=9903968|title=Quantum theory of electron lenses based on the Dirac equation|journal=Physical Review A|volume=42|issue=11|pages=6674–6689|date=1990|bibcode=1990PhRvA..42.6674J}}</ref><ref>{{Cite book|first1=S. A.|last1=Khan|doi=10.1016/S1076-5670(08)70096-X|title=Quantum theory of the optics of charged particles|volume=97|pages=257–358|date=1996|series=Advances in Imaging and Electron Physics|publisher=Elsevier |isbn=9780120147397}}</ref><ref>{{cite journal|last1=Conte|first1=M.|last2=Jagannathan|first2=R.|last3=Khan|first3=S. A.|last4=Pusterla|first4=M.|date=1996|title=Beam optics of the Dirac particle with anomalous magnetic moment|journal=Particle Accelerators|volume=56|pages=99–126}}</ref> 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]]).<ref>{{cite journal|first1=Sameen Ahmed|last1=Khan|date=2006|doi=10.1016/j.ijleo.2005.11.010|title=The Foldy–Wouthuysen Transformation Technique in Optics|journal=Optik|volume=117|issue=10|pages=481–488|bibcode=2006Optik.117..481K}}</ref><ref>{{Cite book|first1=Sameen Ahmed|last1=Khan|doi=10.1016/S1076-5670(08)00602-2|title=The Foldy–Wouthuysen Transformation Technique in Optics|volume=152|pages=49–78|date=2008|series=Advances in Imaging and Electron Physics|publisher=Elsevier |isbn=9780123742193}}</ref>

==See also==

*[[Relativistic quantum mechanics]]

==Notes==
{{lacking ISBN|date=February 2017}}
{{reflist|30em}}

{{DEFAULTSORT:Foldy-Wouthuysen transformation}}
[[Category:Fermions]]
[[Category:Dirac equation]]