# Foldy-Wouthuysen（FW）変換

Relativistic quantum mechanics of a Proca particle in

Riemannian spacetimes

I.はじめに リーマン時空におけるProca（スピン1）粒子の一般的な量子力学的記述を提示します。その電磁相互作用が分析されます。異常磁気モーメント（AMM）と電気双極子モーメント（EDM）が考慮されます。 Foldy-WouthuysenFW）変換[1]は、非相対論的ディラック粒子に対して実行されます。プランク定数のゼロ乗に比例する項の正確な式を取得します。この目的のために、Refsで開発および実証された相対論的FW変換法を適用します。 [2–4]。相対論的FW変換を使用すると、相対論的量子力学QM）をSchr̈odinger形式で表現できます。 FW変換のさまざまなプロパティとアプリケーションがRefsで検討されています。 [5–7]FW変換は、電気力学289]、場の量子論[10]、光学[11–13]、物性物理学[14]原子核物理学[1516]、重力[17–19]で広く使用されています。 、弱い相互作用の理論[20]および量子化学[21–25]ディラックフェルミオンだけでなく、スピンのある粒子にも適用できます[1126–35]。 最近、FW変換がうまく利用され[36]、いくつかの純粋なボソン量子系における隠れた超対称性と超共形対称性[37]の起源を明らかにしました。 Proca量子力学の先行研究では、スピン1粒子の電磁相互作用の詳細な分析は、参考文献で開発されたアプローチに基づいています。 [31]AMMを使用したスピン1粒子の相対論的FWハミルトニアンは参考文献[33]で導出されています。しかし、FW変換[3133353839]を使用したすべての先行調査は、特殊相対性理論の枠組みの中で実行されました。参考文献で作成されたミンコフスキー空間におけるProca粒子のQMの分析についても言及することができます。 [40]。湾曲した時空におけるProca粒子のQMのいくつかの研究は、Refsで実施されました。 [41–46]。これらの作品では、カルタン時空のねじれも考慮されています。アインシュタイン-プロカ解の非計量性は、参考文献で研究されています。 [44]。参照で。 [41–43]、リーマン・カルタン時空におけるプロカ粒子のラグランジアンが得られました。電磁相互作用を除く対応するプロカ方程式は、参考文献に示されています。 [42]。電磁相互作用を含むプロカ方程式は、参考文献で得られています。 [43]。ただし、FW変換はRefsでは使用されていません。 [41–46]Wentzel-Kramers-Brillouin近似と準古典的軌道コヒーレント近似がRefsに適用されています。 [4142]および参考文献。 [43]、それぞれ。本研究では、強い電磁気場と重力場における相対論的スピン1粒子のProcaQMの古典極限を取得する可能性を示します。この目的のために、後続の坂田畠谷[47]およびFW変換を実行します。 FW表現では、古典極限への移行は通常、量子力学ハミルトニアン運動方程式演算子を対応する古典量に置き換えることになります[48]。以前は、リーマン時空におけるスカラー粒子の詳細な量子力学的記述は、参考文献で満たされていました。 [34]ディラック粒子の場合、対応する問題はRefsで解決されています。 [17

I. INTRODUCTION

We present a general quantum-mechanical description of a Proca (spin-1) particle in Riemannian spacetimes. Its electromagnetic interactions are analyzed. The anomalous magnetic moment (AMM) and the electric dipole moment (EDM) are taken into account.

The Foldy-Wouthuysen (FW) transformation [1] is performed for a nonrelativistic Dirac particle. We obtain exact expressions for terms proportional to the zero power of the Planck constant. For this purpose, we apply the relativistic FW transformation method developed and substantiated in Refs. [2–4]. The use of the relativistic FW transformation allows one to express the relativistic quantum mechanics (QM) in the Schr ̈odinger form.

Various properties and applications of the FW transformation have been considered in Refs. [5–7]. The FW transformation is widely used in electrodynamics 2, 8, 9], quantum field theory [10], optics [11–13], condensed matter physics [14], nuclear physics [15, 16],gravity [17–19], the theory of the weak interaction [20] and also quantum chemistry [21–25].

It is applicable not only for Dirac fermions but also for particles with any spins [11, 26–35].

Recently, the FW transformation has been successfully employed [36] to clarify the origin of the hidden supersymmetry and superconformal symmetry [37] in some purely bosonic quantum systems.

In precedent studies of Proca quantum mechanics, a detailed analysis of electromagnetic interactions of a spin-1 particle has been based on the approach developed in Ref. [31]. The relativistic FW Hamiltonian of a spin-1 particle with the AMM has been derived in Ref.[33]. However, all precedent investigations using the FW transformation [31, 33, 35, 38, 39] have been fulfilled in the framework of special relativity. We can also mention an analysis of QM of a Proca particle in the Minkowski space made in Ref. [40]. Some studies of QM of a Proca particle in curved spacetimes have been carried out in Refs. [41–46]. In these works,the Cartan spacetime torsion has also been considered. The nonmetricity in Einstein-Proca solutions has been studied in Ref. [44]. In Refs. [41–43], Lagrangians of a Proca particle in Riemann-Cartan spacetimes have been obtained. The corresponding Proca equations ex-cluding electromagnetic interactions have been presented in Ref. [42]. The Proca equations with an inclusion of electromagnetic interactions have been obtained in Ref. [43]. However,the FW transformation has not been used in Refs. [41–46]. The Wentzel-Kramers-Brillouin approximation and the quasiclassical trajectory-coherent approximation have been applied in Refs. [41, 42] and in Ref. [43], respectively. In the present work, we demonstrate the possibility to obtain the classical limit of Proca QM for a relativistic spin-1 particle in strong electromagnetic and gravitational fields. For this purpose, we perform the subsequent Sakata-Taketani [47] and FW transformations. In the FW representation, the passage to the classical limit usually reduces to a replacement of the operators in quantum-mechanical Hamiltonians and equations of motion with the corresponding classical quantities [48]. Pre-viously, a detailed quantum-mechanical description of a scalar particle in Riemannian space-times has been fulfilled in Ref. [34]. For a Dirac particle, the corresponding problem hasbeen solved in Refs. [17,

VII. UNIFICATION AND CLASSICAL LIMIT OF RELATIVISTIC QUANTUM

MECHANICS IN THE FOLDY-WOUTHUYSEN REPRESENTATION

The Hamiltonian (53) also fully agrees with the corresponding FW Hamiltonians for a scalar particle [34] and for a Dirac one [18]. In the latter Hamiltonians,we can disregard terms of the first and higher orders in the Planck constant. The Hamiltonians differ only in the dimensions of contained matrices defined by the dimensions of t

he corresponding wave functions. For states with a positive total energy, lower spinors (or lower parts of spinorlike wave functions) are equal to zero for any particles. Their nullification unifies a

normalization of the wave functions. For any spin, the FW wave func

tions are normalized to unit, and their probabilistic interpretation is restored. It should be underlined that the quantum-mechanical Hamiltonians become rather similar for bosons and fermions. A

difference between the Hermiticity of the initial Hamiltonians for fermions [18] and the β-pseudo-Hermiticity of the corresponding Hamiltonians for bosons [34] disappears after the FW transformation. These properties indicate the unification of relativistic QM for particles with different spins in the FW representation.

In this paper, we do not analyze terms of the first order in the Planck constant. Such terms define spin interactions. However, it has been shown in Ref. [33] that the spin-dependent terms in the FW Hamiltonians for spin-1/2 and spin-1 particles with the AMMs and EDMs interacting with arbitrary electromagnetic fields in Minkowski spacetimes perfectly agree.

These terms define equations of spin motion, which coincide with each

other in the classical limit. These equations also coincide with the corresponding classical equations (see Ref. [72] and references therein). Of course, the FW Hamiltonian for spin-1 particles additionally contains bilinear in spin terms which also influence spin dynamics [39, 7375].

It can be concluded that the use of the FW representation allows on

e to unify the main equations of relativistic QM for particles with different spins and to demonstrate that their classical limit agrees with the corresponding classical equations. This conclusion fully agrees with the results obtained in Ref. [76] in which the specific quantum-mechanical approach has been used.

VIII. SUMMARY

A comparison of fundamentals of Dirac and Proca QM shows that the

problem of quanti-zation with an introduction of interactions can be solved more easy for a Dirac particle than for a Proca (spin-1) one. However, the solution of this problem for the Proca particle is possible [57–63, 65–67] while it meets some difficulties. A consideration of the results obtained for the Proca particle in electromagnetic fields [31, 33, 39] demonstrates an importance of the ST and FW transformations which result in the Schr ̈odinger form of the PCS equations.

After this, the classical limit of Proca QM in electromagnetic fields can be easily determined.

Therefore, a development of Proca QM needs not only a formulation

of general covariant Proca equations in electromagnetic and gravitational fields but also a determination of the Hamiltonian form and of the classical limit of these equations with the use of the ST and

FW transformations. These results in turn allow one to establish a connection of QM of the Proca particle with QM of particles with other spins.

The present work proposes the extension of relativistic QM of a Proca particle on Rie-mannian spacetimes. The formulated covariant Proca equations take into account the AMM and the EDM of a spin-1 particle and are based on the PCS equations in special relativity and precedent studies of the Proca particle in curved spacetimes.

It is important to mention that the covariant derivatives in the Dirac and Proca equations substantially differ. As an example, the relativistic FW transformation with allowance for terms proportional to the zero power of the Planck constant has been performed. The Hamiltonian obtained agrees with the corresponding Hamiltonians derived for scalar and Dirac particles and with their classical counterpart. This conclusion is in agreement with the results obtained in Ref.

[76]. The consideration presented demonstrate the unification of r

elativistic QM in the FW representation.

VII。相対論的量の統一と古典極限 FOLDY-WOUTHUYSEN表現のメカニズム ハミルトニアン53）は、スカラー粒子[34]およびディラック粒子[18]の対応するFWハミルトニアンとも完全に一致します。後者のハミルトニアンでは、プランク定数1次以上の項は無視できます。ハミルトニアンは、対応する波動関数の次元によって定義される含まれる行列の次元のみが異なります。正の総エネルギーを持つ状態の場合、下部スピノール（またはスピノールのような波動関数の下部）は、どの粒子でもゼロに等しくなります。それらの無効化は、 波動関数の正規化。どのスピンでも、FW波動関数は単位に正規化され、確率論的解釈が復元されます。量子力学ハミルトニアンはボソンとフェルミ粒子でかなり似ていることを強調する必要があります。フェルミ粒子の最初のハミルトニアンのエルミティシティ[18]と、ボソンの対応するハミルトニアンβ疑似エルミティシティ[34]の違いは、FW変換後に消えます。これらの特性は、FW表現でスピンが異なる粒子の相対論的QMが統一されていることを示しています。 この論文では、プランク定数1次の項を分析しません。このような用語は、スピン相互作用を定義します。しかし、それは参考文献に示されています。 [33]ミンコフスキー時空の任意の電磁場と相互作用するAMMおよびEDMを持つスピン1/2およびスピン1粒子のFWハミルトニアンのスピン依存項は完全に一致します。 これらの用語は、古典極限で互いに一致するスピン運動の方程式を定義します。これらの方程式は、対応する古典的な方程式とも一致します（参考文献[72]およびその中の参考文献を参照）。もちろん、スピン1粒子のFWハミルトニアンには、スピンダイナミクスにも影響を与えるスピン項の双線形が追加で含まれています[3973–75]FW表現を使用すると、スピンが異なる粒子の相対論的QMの主方程式を統一し、それらの古典極限が対応する古典方程式と一致することを実証できると結論付けることができます。この結論は、参考文献で得られた結果と完全に一致しています。 [76]特定の量子力学的アプローチが使用されています。 VIII。概要 ディラックQMとプロカQMの基本を比較すると、相互作用の導入による量子化の問題は、プロカ（スピン1）粒子よりもディラック粒子の方が簡単に解決できることがわかります。ただし、Proca粒子のこの​​問題の解決は可能ですが[57–6365–67]、いくつかの問題があります。電磁界内のProca粒子について得られた結果の考察[313339]は、PCS方程式のSchr̈odinger形式をもたらすSTおよびFW変換の重要性を示しています。 この後、電磁界におけるProcaQMの古典的な限界を簡単に決定できます。 したがって、Proca QMの開発には、電磁気場および重力場における一般共変Proca方程式の定式化だけでなく、STおよびFW変換を使用したこれらの方程式のハミルトニアン形式および古典極限の決定も必要です。これらの結果により、Proca粒子のQMと他のスピンを持つ粒子のQMとの接続を確立することができます。 本研究は、リーマン時空におけるプロカ粒子の相対論的量子力学の拡張を提案している。公式化された共変Proca方程式は、スピン1粒子のAMMEDMを考慮に入れており、特殊相対性理論PCS方程式と、湾曲した時空におけるProca粒子の先行研究に基づいています。 ディラック方程式とプロカ方程式の共変微分は実質的に異なることに言及することが重要です。例として、プランク定数のゼロパワーに比例する項を考慮した相対論的FW変換が実行されました。得られたハミルトニアンは、スカラー粒子とディラック粒子に対して導出された対応するハミルトニアン、およびそれらの古典的な対応物と一致します。この結論は、参考文献で得られた結果と一致しています。 [76]。提示された考察は、FW表現における相対論的QMの統一を示しています。

もともとディラック方程式のために開発されたFoldy–Wouthuysen変換の強力な機械は、音響や光学などの多くの状況でアプリケーションを見つけました。 原子系[13] [14]放射光[15]や偏光ビームのブロッホ方程式の導出など、非常に多様な分野での応用が見出されています。[16] 音響学におけるFoldy–Wouthuysen変換の適用は非常に自然です。包括的で数学的に厳密な説明。[17] [18] [19] 従来のスキームでは、光学ハミルトニアンを拡張する目的 拡張パラメータは、一連の近似（近軸と非近軸）の観点から準近軸ビームの伝搬を理解することです。荷電粒子光学系の場合も同様です。相対論的量子力学においても、非相対論的近似と準相対論的領域における相対論的補正項と同様に、相対論的波動方程式を理解するという問題があることを思い出してください。ディラック方程式（時間的に1次）の場合、これは、反復対角化手法につながるFoldy-Wouthuysen変換を使用して最も便利に実行されます。新しく開発された光学の形式（光光学と荷電粒子光学の両方）の主なフレームワークは、ディラック方程式ディラック粒子と非相対論的で簡単に解釈できる形で電磁界を印加します。 Foldy-Wouthuysen理論では、ディラック方程式2つの2成分方程式への正準変換によって分離されます。1つは非相対論的極限でパウリ方程式[20]に還元され、もう1つは負のエネルギー状態を記述します。マクスウェルの方程式のディラックのような行列表現を書くことができます。このようなマトリックス形式では、Foldy–Wouthuysenを適用できます。[21] [22] [23] [24] [25] ヘルムホルツ方程式（スカラー光学系を支配する）とクライン-ゴルドン方程式の間には、密接な代数的類似性があります。マクスウェルの方程式（ベクトル光学系を支配する）の行列形式とディラック方程式の間。したがって、これらのシステムの分析には、標準的な量子力学の強力な機構（特に、Foldy-Wouthuysen変換）を使用するのが自然です。 ヘルムホルツ方程式の場合にFoldy-Wouthuysen変換手法を採用するという提案は、文献に注釈として記載されていました。[26] このアイデアが特定のビーム光学システムの準近軸近似を分析するために利用されたのは、最近の研究でのみでした。[27] Foldy–Wouthuysen手法は、光学に対するリー代数アプローチに最適です。これらすべてのプラスポイント、強力で曖昧さのない拡張により、Foldy-Wouthuysen変換はまだ光学でほとんど使用されていません。 Foldy-Wouthuysen変換の手法により、それぞれヘルムホルツ光学系[28]とマクスウェル光学系[29]の非伝統的な処方として知られるものが生まれます。非伝統的なアプローチは、近軸および収差の振る舞いの非常に興味深い波長依存の変更を引き起こします。マクスウェル光学系の非伝統的な形式は、光ビーム光学系と偏光の統一されたフレームワークを提供します。光光学の非伝統的な処方は、荷電粒子ビーム光学の量子論と非常に類似しています。[30] [31] [32] [33]光学では、光光学と荷電粒子光学の間の波長依存領域でのより深い接続を見ることができます（電子光学を参照）。[34] [35]

# Foldy–Wouthuysen transformation

https://en.wikipedia.org/wiki/Foldy%E2%80%93Wouthuysen_transformation

Foldy-Wouthuysen変換は歴史的に重要であり、1949年にLeslie LawranceFoldySiegfriedAdolf Wouthuysenによって定式化され、スピン1/2粒子の方程式であるディラック方程式の非相対論的限界を理解しました。[1] [2] [3] [4]相対論的波動方程式の粒子解釈におけるFoldy-Wouthuysenタイプの変換の詳細な一般的な議論は、Acharya and Sudarshan1960）にあります。[5]ディラック場が量子化場として扱われる超相対論的領域に主な用途があるため、高エネルギー物理学におけるその有用性は現在制限されています。

## 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]

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[13][14] synchrotron radiation[15] and derivation of the Bloch equation for polarized beams.[16]

The application of the Foldy–Wouthuysen transformation in acoustics is very natural; comprehensive and mathematically rigorous accounts.[17][18][19]

In the traditional scheme the purpose of expanding the optical Hamiltonian

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[20] 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.[21][22][23][24][25]

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.[26]

It was only in the recent works, that this idea was exploited to analyze the quasiparaxial approximations for specific beam optical system.[27] 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[28] and Maxwell optics[29] 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.[30][31][32][33] 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).[34][35]

Maxwell Optics: II. An Exact Formalism

Abstract

We present a formalism for light optics starting with the Maxwell equa

tions and casting them into an exact matrix form taking into account the spatial and temporal variations of the permittivity and permeability. This 8×8 matrix representation is used to construct the optical Hamiltonian. This has a close analogy with the algebraic structure of the Dirac equation, enabling the use of the rich machinery of the Dirac electron theory. We get interesting wavelength-dependent contributions which can not be obtained in any of the traditional approaches.

1 Introduction

The traditional scalar wave theory of optics (including aberrations

to all orders) is based on the beam-optical Hamiltonian derived using the Fermat’s principle. This approach is purely geometrical and works adequately in the scalar regime. The other approach is based on the Helmholtz equation which is derived from the Maxwell equations. Then one makes the square-root of the Helmholtz operator followed by an expansion of the radical [1, 2].

This approach works to all orders and the resulting expansion is no different from the one obtained using the geometrical approach of the Fermat’s principle.

Another way of obtaining the aberration expansion is based on the al-gebraic similarities between the Helmholtz equation and the Klein-Gordon equation. Exploiting this algebraic similarity the Helmholtz equation is lin-earized in a procedure very similar to the one due to Feschbach-Villars, for linearizing the Klein-Gordon equation. This brings the Helmholtz equation to a Dirac-like form and then follows the procedure of the Foldy-Wouthuysen expansion used in the Dirac electron theory. This approach, which uses the algebraic machinery of quantum mechanics, was developed recently[3], pro-viding an alternative to the traditional square-root procedure. This scalar formalism gives rise to wavelength-dependent contributions modifying the aberration coefficients [4]. The algebraic machinery of this formalism is very similar to the one used in the quantum theory of charged-particle beam optics,based on the Dirac [5] and the Klein-Gordon [6] equations respectively. The detailed account for both of these is available in [7]. A treatment of beam optics taking into account the anomalous magnetic moment is available in [8].

As for the polarization: A systematic procedure for the passage from scalar to vector wave optics to handle paraxial beam propagation p

roblems,completely taking into account the way in which the Maxwell equations cou-ple the spatial variation and polarization of light waves, has been formu-lated by analysing the basic Poincar ́e invariance of the system, and this procedure has been successfully used to clarify several issues in Maxwell op-tics [9, 10, 11].

In all the above approaches, the beam-optics and the polarization

are studied separately, using very different machineries. The derivation of the Helmholtz equation from the Maxwell equations is an approximation as one neglects the spatial and temporal derivatives of the permittivity and perme-ability of the medium. Any prescription based on the Helmholtz equation is bound to be an approximation, irrespective of how good it may be in cer-tain situations. It is very natural to look for a prescription based fully on the Maxwell equations. Such a prescription is sure to provide a deeper un-derstanding of beam-optics and polarization in a unified manner. With this as the chief motivation we construct a formalism starting with the Maxwell equations in a matrix form: a single entity containing all the four Maxwell equations.

In our approach we require an exact matrix representation of the

Maxwell equations in a medium taking into account the spatial and temporal varia-tions of the permittivity and permeability. It is necessary and sufficient to use 8×8 matrices for such an exact representation. The derivation of the required matrix representation, and how it differs from the numerous other ones is presented in Part-I [12].

In the present Part (Part-II) we proceed with the exact matrix r

epresen-tation of the Maxwell equations derived in Part-I, and construct a general formalism. The derived representation has a very close algebraic correspon-dence with the Dirac equation. This enables us to apply the machinery of the Foldy-Wouthuysen expansion used in the Dirac electron theory. The Foldy-Wouthuysen transformation technique is outlined in Appendix-A. General expressions for the Hamiltonians are derived without assuming any specific form for the refractive index. These Hamiltonians are shown to contain the extra wavelength-dependent contributions which arise very naturally in our approach. In Part-III [13] we apply the general formalism to the specific examples: A.

Medium with Constant Refractive Index. This example is es-sentially for illustrating some of the details of the machinery used.

The other application, B.

Axially Symmetric Graded Index Medium is used to demonstrate the power of the formalism. Two points are worth mentioning,Image Rotation

: Our formalism gives rise to the image rotation (proportional to the wavelength) and we have derived an explicit relationship for the angle of the image rotation. The other pertains to the aberrations: In our formalism we get all the nine aberrations permitted by the axial symme-try. The traditional approaches give six aberrations. Our formalism modifies these six aberration coefficients by wavelength-dependent contributions and also gives rise to the remaining three permitted by the axial symmetry. The existence of the nine aberrations and image rotation are well-known in axi-ally symmetric magnetic lenses, even when treated classically. The quantum treatment of the same system leads to the wavelength-dependent modifica-tions [7]. The alternate procedure for the Helmholtz optics in [3, 4] gives the usual six aberrations (though modified by the wavelength-dependent contri-butions) and does not give any image rotation. These extra aberrations and the image rotation are the exclusive outcome of the fact that the formalism is based on the Maxwell equations, and done exactly.

The traditional beam-optics is completely obtained from our approach in the limit wavelength,̄λ−→0, which we call as the traditional limit of our formalism. This is analogous to the classical limit obtained by taking ̄ h−→0 in the quantum prescriptions. The scheme of using the Foldy-Wouthuysen machinery in this formalism is very similar to the one used in the quantum theory of charged-particle beam optics

[5, 6, 7]. There too one recovers the classical prescriptions in the limit λ0−→0 where λ0= ̄h/p0 is the de Broglie wavelength and p0 is the design momentum of the system under study.

The studies on the polarization are in progress. Some of the results

in [11] have been obtained as the lowest order approximation of the more general framework presented here. These will be presented in Part-IV soon [14]

5 Concluding Remarks

We start with the Maxwell equations and express them in a matrix for

m in a medium with varying permittivity and permeability in presence of sources using 8×8 matrices. From this exact matrix representation we construct the exact optical Hamiltonian for a monochromatic quasiparaxial light beam.

The optical Hamiltonian has a very close algebraic similarity with the Dirac equation. We exploit this similarity to adopt the standard machinery,namely the Foldy-Wouthuysen transformation technique of the Dirac theory. This enabled us to obtain the beam-optical Hamiltonian to any desired degree of accuracy. We further get the wavelength-dependent contributions to at each order, starting with the lowest-order paraxial paraxial Hamiltonian.

The beam-optical Hamiltonians also have the wavelength-dependent ma-trix terms which are associated with the polarization. In this approach we have been able to derive a Hamiltonian which contains both the beam-optics and the polarization. In Part-III [13] we shall apply the formalism to the specific examples and see how the beam-optics (paraxial behaviour and the aberrations) gets modified by the wavelength-dependent contributions. In Part-IV [14] we shall examine the polarization component of the formalism presented here.

5おわりに マクスウェル方程式から始めて、8×8行列を使用して、ソースの存在下で誘電率透磁率を変化させる媒体で行列形式で表現します。この正確な行列表現から、単色の準近軸光ビームの正確な光学ハミルトニアンを構築します。 光学ハミルトニアンは、ディラック方程式と非常に近い代数的類似性を持っています。この類似性を利用して、標準的な機構、つまりディラック理論のFoldy-Wouthuysen変換手法を採用します。これにより、ビーム光学ハミルトニアンを任意の精度で取得することができました。さらに、最低次の近軸近軸ハミルトニアンから始めて、各次数での波長依存の寄与を取得します。 ビーム光学ハミルトニアンには、偏光に関連する波長依存のマトリックス項もあります。このアプローチでは、ビーム光学と偏光の両方を含むハミルトニアンを導出することができました。パートIII [13]では、特定の例に形式を適用し、波長に依存する寄与によってビーム光学系（近軸挙動と収差）がどのように変更されるかを確認します。 Part-IV [14]では、ここに提示されている形式の分極成分を調べます。