High-order Foldy-Wouthuysen transformations of the Dirac and Dirac-Pauli Hamiltonians in the weak-field limit
aa r X i v : . [ qu a n t - ph ] J un High-order Foldy-Wouthuysen transformations of the Dirac and Dirac-PauliHamiltonians in the weak-field limit
Tsung-Wei Chen ∗ Department of Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
Dah-Wei Chiou † Department of Physics and Center for Condensed Matter Sciences,National Taiwan University, Taipei 10617, Taiwan
The low-energy and weak-field limit of Dirac equation can be obtained by an order-by-order blockdiagonalization approach to any desired order in the parameter π /mc ( π is the kinetic momentumand m is the mass of the particle). In the previous work, it has been shown that, up to the or-der of ( π /mc ) , the Dirac-Pauli Hamiltonian in the Foldy-Wouthuysen (FW) representation maybe expressed as a closed form and consistent with the classical Hamiltonian, which is the sum ofthe classical relativistic Hamiltonian for orbital motion and the Thomas-Bargmann-Michel-Telegdi(T-BMT) Hamiltonian for spin precession. In order to investigate the exact validity of the corre-spondence between classical and Dirac-Pauli spinors, it is necessary to proceed to higher orders.In this paper, we investigate the FW representation of the Dirac and Dirac-Pauli Hamiltonians byusing Kutzelnigg’s diagonalization method. We show that the Kutzelnigg diagonalization methodcan be further simplified if nonlinear effects of static and homogeneous electromagnetic fields areneglected (in the weak-field limit). Up to the order of ( π /mc ) , we find that the FW transforma-tion for both Dirac and Dirac-Pauli Hamiltonians is in agreement with the classical Hamiltonianwith the gyromagnetic ratio given by g = 2 and g = 2 respectively. Furthermore, with higher-orderterms at hand, it is demonstrated that the unitary FW transformation admits a closed form in thelow-energy and weak-field limit. PACS numbers: 03.65.Pm, 11.10.Ef, 71.70.Ej
I. INTRODUCTION
The relativistic quantum theory for a spin-1/2 parti-cle is described by a spinor satisfying the Dirac equa-tion [1, 2]. The four-component spinor of the Dirac par-ticle is composed of two two-component Weyl spinorswhich correspond to the particle and antiparticle parts.Rigourously, because of the non-negligible probability ofcreation/annihilation of particle-antiparticle pairs, theDirac equation is self-consistent only in the context ofquantum field theory. For the purpose of obtaining thelow-energy limit of Dirac equation without accountingfor the field-theory particle-antiparticle interaction, theDirac equation is converted to a two-component equa-tion.The Pauli substraction method eliminates the twosmall components from the four-component spinor ofDirac equation and leads to the block-diagonal butenergy-dependent effective Hamiltonian in which somenon-hermitian terms may appear. Apart from the diffi-culties, in the seminal paper [3], Foldy and Wouthuysen(FW) established a series of successive unitary transfor-mations via decomposing the Hamiltonian into even andodd matrices; a block-diagonalized effective Hamiltoniancan be constructed up to a certain order of π /mc . The ∗ Electronic address: [email protected] † Electronic address: [email protected] series of successive unitary transformations in the FWmethod can be replaced by a single transformation viathe L¨owding partitioning method [4, 5]. Furthermore, Eriksen developed a systematic deriva-tion of the unitary transformation and gave an exact FWtransformation for a charged spin-1/2 particle in interac-tion with non-explicitly time-dependent field [6]. The va-lidity of the Eriksen method is investigated in Ref. [7]. InRef. [8], Kutzelnigg developed a single unitary transfor-mation that allows one to obtain the block-diagonalizedDirac Hamiltonian without evoking the decompositionof even and odd matrices used in FW method. Alterna-tively, the Dirac Hamiltonian can also be diagonalized viaexpansion in powers of Planck constant ~ [9–11], whichenables us to investigate the influences of quantum cor-rections on the classical dynamics in strong fields [12].On the other hand, the classical relativistic dynamicsfor a charged particle with intrinsic spin in static andhomogeneous electromagnetic fields is well understood. It should be emphasized that The FW transformation is notmeant to be used for the second quantization, and furthermoreit exists only in the weak-field limit or in some special cases (asstudied in Sec. II A). If we ruthlessly try to second quantize thetheory in the FW representation, whether we can succeed ornot, we should perform the second quantization in a way verydifferent from the conventional approach. This is because, inthe FW representation, we encounter the non-locality due to zitterbewegung (see also P. Strange in Ref. [5]).
The orbital motion is governed by the classical relativisticHamiltonian H c orbit = p c π + m c + V ( x ) , (1.1)where π = p − q A /c is the kinetic momentum opera-tor with A being the magnetic vector potential [13] and V ( x ) the electric potential energy. The spin motion isgoverned by the Thomas-Bargmann-Michel-Teledgi (T-BMT) equation which describes the precession of spin asmeasured by the laboratory observer [14], d s dt = qmc s × F ( x ) (1.2)with F = (cid:18) g − γ (cid:19) B − (cid:16) g − (cid:17) γγ + 1 ( β · B ) β − (cid:18) g − γγ + 1 (cid:19) β × E , (1.3)where g is the gyromagnetic ratio, β the boost veloc-ity, γ = 1 / p − β the Lorentz factor and E and B are electric and magnetic fields measured in the labo-ratory frame. The intrinsic spin s in Eq. (1.2) is be-ing observed in the rest frame of the particle. Because { s i , s j } = ǫ ijk s k , Eq. (1.2) can be recast as the Hamil-ton’s equation: d s dt = { s , H cspin } (1.4)with H cspin = − qmc s · F (1.5)called the T-BMT Hamiltonian. The combination of Eqs(1.1) and (1.5) is hereafter called the classical Hamilto-nian H c , H c = H corbit + H cspin = p c π + m c + V − µ · F , (1.6)where µ = q s /mc is the intrinsic magnetic moment of anelectron.The connection between the Dirac equation and classi-cal Hamiltonian has been investigated by several authors[3, 15–18]. For a free Dirac particle, it has been shownthat the exactly diagonalized Dirac Hamiltonian corre-sponds to the classical relativistic Hamiltonian [3]. InRefs. [15, 16], it was shown that the T-BMT equationmay be derived from the WKB wavefunction solutionsto the Dirac equation. In the presence of external elec-tromagnetic fields, the Dirac Hamiltonian in the FW rep-resentation has been block-diagonalized up to the orderof ( π /mc ) , but the connection is not explicit [17]. Re-cently, in Ref. [19], it has been shown that up to ( π /mc ) ,the resulting FW transformed Dirac, or more generic,Dirac-Pauli [20] Hamiltonian in the presence of static and homogeneous electromagnetic fields may agree with theclassical Hamiltonian [Eq. (1.6)] in the weak-field limit.The order-by-order block-diagonalization methods tohigher orders of π /mc can be used to investigate the va-lidity of the connection. Furthermore, if the connectionis indeed establishable (in a closed form), corrections tothe classical T-BMT equation due to field inhomogene-ity, if any, could also be included. Motivated by theseregards, we adopt a systematic method that can sub-stantially simplify the calculation of FW transformationto any higher orders in the FW representation of DiracHamiltonian. It must be stressed that block diagonalia-tion of a four-component Hamiltonian into two uncoupledtwo-component Hamiltonian is not unique, as any com-position with additional unitary transformations that actseparately on the positive and negative energy blocks willalso do the job. Different block-diagonalization transfor-mations are however unitarily equivalent to one another,and thus yield the same physics. The truly vexed ques-tion is: whether does the unitary bock-diagonalizationtransformation exist at all? In the absence of electricfields, we will show that the answer is affirmative. Onthe other hand, in the presence of electric fields, theanswer seems to be negative, as the energy interact-ing with electromagnetic fields renders the probability ofcreation/annihilation of particle-antiparticle pairs non-negligible and thus the particle-antiparticle separationinconsistent. Nevertheless, in the weak-field limit, theinteracting energy is well below the Dirac energy gap(2 mc ) and we will demonstrate that the unitary trans-formation exists and indeed admits a closed form in thelow-energy and weak-field limit.In this article, we derive the FW transformed DiracHamiltonian up to the order of ( π /mc ) by usingKutzelnigg’s diagonalization method [8]. The key featureof the Kutzelnigg approach is that it provides an exactblock-diagonalized form of Dirac Hamiltonian involvinga self-consistent equation [see Eq. (2.9)]. The explicitform of the FW transformed Dirac Hamiltonian can beobtained by solving the self-consistent equation. We willshow that the Kutzelnigg method can be further simpli-fied in the weak-field limit, and this simplification enablesus to obtain the higher-order terms systematically. Wewill show that the block diagonalization of Dirac andDirac-Pauli Hamiltonians up to the order of ( π /mc ) in the Foldy-Wouthuysen representation is in agreementwith classical Hamiltonian, and the closed form of theunitary transformation can be found.This article is organized as follows. In Sec. II, weconstruct a unitary operator based on the Kutzelniggmethod to obtain the exact FW transformed DiracHamiltonian and the self-consistent equation. The ex- Once the Hamiltonian is block-diagonalized, further unitarytransformations that do not mix the positive and negative en-ergy blocks merely rotate the 2 × act solution of the self-consistent equation is discussed.The FW transformed Dirac Hamiltonian in the presenceof inhomogeneous electromagnetic fields are derived inSec. III. The effective Hamiltonian up to ( π /mc ) forthe inhomogeneous electromagnetic field is in agreementwith the previous result shown in Refs. [3, 17]. The staticand homogeneous electromagnetic fields are considered inSec. IV, where the simplification of the effective Hamilto-nian is discussed and the FW transformed Dirac Hamil-tonian is obtained up to ( π /mc ) . In Sec. V, the com-parison with the classical relativistic Hamiltonian andT-BMT equation with g = 2 is discussed. The FW trans-formed Dirac-Pauli Hamiltonian is shown in Sec. VI. InSec. VII, we demonstrate that the exact unitary transfor-mation matrix in the low-energy and weak-field limit canbe formally obtained. The conclusions are summarizedin Sec. VIII. Some calculational details are supplementedin Appendices. II. KUTZELNIGG DIAGONALIZATIONMETHOD FOR DIRAC HAMILTONIAN
In this section, we use the unitary operator basedon the Kutzelnigg diagonalization method [8] and applythe unitary operator to the Dirac Hamiltonian. We ob-tain the formally exact Foldy-Wouthuysen transformedHamiltonian by requiring that the unitary transforma-tion yields a block-diagonal form.The Dirac Hamiltonian in the presence of electromag-netic fields can be written as H D = (cid:18) V + mc c σ · π c σ · π V − mc (cid:19) ≡ (cid:18) h + h h h − (cid:19) (2.1)where π = p − q A /c is the kinetic momentum operatorand V = qφ . The electric field and magnetic field are E = −∇ φ and B = ∇ × A , respectively. We note that inthe static case, ∇ × E = 0, and thus, π × E = − E × π .The wave function of the Dirac equation H D ψ = i ~ ∂∂t ψ is a two two-spinors ψ = (cid:18) ψ + ψ − (cid:19) . (2.2)A unitary operator U which formally decouples positiveand negative energy states can be written as the followingform [8] U = (cid:18) Y Y X † − ZX Z (cid:19) , (2.3)where operators Y and Z are defined as Y = 1 √ X † X , Z = 1 √ XX † . (2.4) Applying the unitary transformation Eq. (2.3) toEq. (2.1), U H D U † is of the form U H D U † = (cid:18) H FW H X † H X H ′ (cid:19) (2.5)The unitary transformation transforms the wave function ψ to a two-spinor, U (cid:18) ψ + ψ − (cid:19) = (cid:18) ψ FW (cid:19) , (2.6)where the wave function for the negative energy statemust be zero and the FW transformed wave function isgiven by ψ FW = p X † X ψ + . (2.7)We require that the transformed Hamiltonian takes theblock-diagonal form: U H D U † = (cid:18) H FW H ′ (cid:19) . (2.8)We find that the requirement of the vanishing off-diagonal term H X = 0 yields the constraint on the X operator: X = 12 mc {− Xh X + h + [ V, X ] } . (2.9)Equation (2.9) is a self-consistent formula for operator X . The resulting FW transformed Hamiltonian H FW isgiven by H FW = Y (cid:0) h + + X † h + h X + X † h − X (cid:1) Y. (2.10)Because the operator X plays an important role in gen-erating the FW transformed Hamiltonian and the corre-sponding unitary operator, the operator X for the DiracHamiltonian is hereafter called the Dirac generating op-erator . To our knowledge, the exact solution of Eq. (2.9)for a general potential is still unknown except for thetwo cases: a free particle and a particle subject only tomagnetic fields. For the case with a nontrivial electricpotential, we assume that the solution of Eq. (2.9) canbe obtained by using series expansion. The series solution of the Dirac generating operator X is notunique because any unitary transformation would lead to a sat-isfactory X as long as it does not mix positive and negative en-ergy states. This implies that the form of the block-diagonalizedHamiltonian is not unique. In this regard, we focus only on theseries solution of X that can correctly generate the FW trans-formed Dirac Hamiltonian linear in EM fields and up to order of( π /mc ) , as shown in Sec.III. A. Exact solution of Dirac generating operator
For a free particle ( A = 0 and V = 0), it can be shownthat Eq. (2.9) has an exact solution X = c ( σ · p ) mc + E p , (2.11)where E p = p m c + p c . Using Eqs. (2.3) and (2.4),the unitary transformation matrix can be written as U = 1 p E p ( E p + mc ) (cid:18) E p + mc c σ · p − c σ · p E p + mc (cid:19) . (2.12)Equation (2.12) is the same with the result obtainedfrom the standard FW transformation [3]. The result-ing FW transformed free-particle Dirac Hamiltonian isblock-diagonalized, H FW = (cid:18) E p − E p (cid:19) . (2.13)It is interesting to note that in the absence of electricfield (i.e., V = const), Eq. (2.9) also admits an exactsolution X = 1 mc + E π ( c σ · π ) , (2.14)where E π = p m c + c ( σ · π ) . This can be provedby directly substituting Eq. (2.14) into Eq. (2.9) with V = const. The exact unitary transformation matrixcan be formally constructed and the resulting FW trans-formed Hamiltonian can be obtained. In the presenceof a nontrivial electric potential, it is difficult to ob-tain an exact solution because the term [ V, X ] does notvanish. Therefore, the diagonalization procedure for theDirac Hamiltonian must be performed order-by-order. Itis necessary to choose a dimensionless quantity as theorder-expanding parameter. We note that the form ofEq. (2.14) can be rewritten as { / [1 + ( σ · ξ ) ] } σ · ξ withthe dimensionless quantity ξ = π /mc . The order-by-order block-diagonalized Hamiltonian can be expressedin terms of the order parameter ξ . In this paper, we fur-ther focus on the weak-field limit in order to compare theblock-diagonalizaed Hamiltonian with the classical coun-terpart. B. Series expansion of Dirac generating operator
The upper-left diagonal term of
U H D U † is the FWtransformed Hamiltonian H FW under the constraintEq. (2.9) and it can be written as (see Appendix A) H FW = mc + e G/ Ae − G/ , (2.15)where operators A (hereafter called the Dirac energy op-erator ) and G (hereafter called the Dirac exponent oper-ator ) are defined as A ≡ V + h X, G ≡ ln (cid:0) X † X (cid:1) . (2.16) The other requirement H X † = 0 gives the constraint onthe hermitian of the Dirac generating operator X † , whichis simply the hermitian conjugate of Eq. (2.9), namely, H † X = H X † . It can be shown that H X = 0 and H X † = 0imply that (see Appendix A) H † FW = mc + e − G/ A † e G/ (2.17)and then Eq. (2.15) is a hermitian operator since theDirac exponent operator is a hermitian operator. Equa-tion (2.15) can be simplified if we rewrite the Dirac en-ergy operator A as the sum of its hermitian ( A H ) andanti-hermitian ( A N ) parts, A = A H + A N , where A H = A + A † , A N = A − A † . (2.18)Combining Eqs. (2.15) together with (2.17), the FWtransformed Dirac Hamiltonian H FW is made up of H FW = (cid:16) H FW + H † FW (cid:17) /
2, and it can be written as H FW = mc + A H + S, (2.19)where the Dirac string operator S is given by S = 12 [ G, A N ] + 12!2 [ G, [ G, A H ]] + 13!2 [ G, [ G, [ G, A N ]]]+ 14!2 [ G, [ G, [ G, [ G, A H ]]]] + · · · , (2.20)where we have used the Baker-Campbell-Hausdorff for-mula [21]: e B De − B = D + [ B, D ] + [ B, [ B, D ]] /
2! +[ B, [ B, [ B, D ]]] / · · · . We note that the anti-hermitianpart of the Dirac energy operator always appears in thoseterms with odd numbers of Dirac exponent operators,and the hermitian part of the Dirac energy operator al-ways appears in those terms with even numbers of Diracexponent operators. Since the commutator of two hermi-tian operators must be an anti-hermitian operator, it canbe shown that the Dirac string operator is a hermitianoperator.In order to compare FW transformed Dirac Hamil-tonian to the classical Hamiltonian, we solve the self-consistent equation for X [Eq. (2.9)] by power series ex-pansion in terms of orders of 1 /c , X = X c + X c + X c + · · · . (2.21) X is the first order of the Dirac generating operator, X the second, and so on. Substituting Eq. (2.21) intoEq. (2.9), we can obtain each order of the Dirac generat-ing operator X ℓ . For order of 1 /c and 1 /c , we have2 mX = σ · π , mX = 0 . (2.22)The expanding terms of the Dirac generating operatorwith third and higher orders can be determined by thefollowing equations,2 mX j = − X k + k =2 j − X k σ · π X k + [ V, X j − ] , mX j +1 = − X k + k =2 j X k σ · π X k + [ V, X j − ] , (2.23)where j = 1 , , · · · . Consider terms of even order of 1 /c ,namely, X j . The fourth order of the Dirac generatingoperator X is determined by 2 mX = − ( X σ · π X + X σ · π X ) + [ V, X ]. Since the second order of X is zero( X = 0), the fourth order of X also vanishes, i.e., X =0. The sixth order of X is obtained by 2 mX = − ( X σ · π X + X σ · π X + X σ · π X + X σ · π X ) + [ V, X ].Because both the second and fourth order of the Diracgenerating operators vanish, we find that the sixth orderof Dirac generating operator X is also zero, as well as X , X , and so on. Therefore, we have X = X = X = · · · = 0 , (2.24)and the non-zero terms are those expanding terms of theDirac generating operators with odd subscripts, namely, X = X /c + X /c + X /c + · · · . Furthermore, sincethe operator h = c σ · π is of the order of c , the seriesexpansion of A = V + h X has only even powers of c : A = A + A c + A c + · · · , (2.25)where the ℓ th order of the Dirac energy operator A ℓ isrelated to the ℓ th order of Dirac generating operator X ℓ by A = V + h c X ,A ℓ = h c X ℓ +1 , (2.26)where ℓ = 2 , , , · · · . On the other hand, the series ex-pansion of ln(1+ y ) is ln(1+ y ) = y − y / y / − y / · · · . Because y = X † X , the power series of y containsonly even powers of c : y = y /c + y /c + y /c + · · · ,where y ℓ are given by y ℓ = X k + k = ℓ X † k X k . (2.27)For example, y = X † X + X † X + X † X . Consequently,the Dirac exponent operator can only have terms witheven powers of c (we note that G = 0) G = G c + G c + G c + · · · , (2.28)where the ℓ th order of the Dirac exponent operator G ℓ can be expressed in terms of y ℓ : G ℓ = y ℓ − X k + k = ℓ y k y k + 13 X k + k + k = ℓ y k y k y k − X k + ··· + k = ℓ y k y k y k y k + · · · . (2.29) For example, G = y − ( y y + y y ) / y /
3. Therefore,the FW transformed Dirac Hamiltonian can be expandedin terms of A ℓ and G ℓ and has only even powers of c .That is, H FW = mc + X ℓ H ( ℓ )FW , (2.30)where the ℓ th order of the FW transformed Dirac Hamil-tonian denoted as H ( ℓ )FW ( ℓ = 0 , , , , · · · ) are given by(up to c ) H (0)FW = A H ,c ℓ H ( ℓ )FW = A Hℓ + S ℓ , (2.31)where ℓ = 2 , , , · · · ,
12. The ℓ th order of Dirac stringoperator S ℓ is given by S ℓ = 12 X ℓ + ℓ = ℓ [ G ℓ , A Nℓ ]+ 12!2 X ℓ + ℓ + ℓ = ℓ [ G ℓ , [ G ℓ , A Hℓ ]]+ 13!2 X ℓ + ··· + ℓ = ℓ [ G ℓ , [ G ℓ , [ G ℓ , A Nℓ ]]]+ 14!2 X ℓ + ··· + ℓ = ℓ [ G ℓ , [ G ℓ , [ G ℓ , [ G ℓ , A Hℓ ]]]]+ · · · . (2.32)As mentioned above, any unitary transformationwould lead to a satisfactory generating operator as longas it does not mix positive and negative energy states.The non-uniqueness property of generating operator canbe easily seen as follows. If we perform the Kutzel-nigg diagonalization method upon Eq. (2.8) again, thenwe obtain another block-diaogonalized Hamiltonian withnew operator equation for the generating operator. Thenew diagonalized Hamiltonian H ′ FW is determined by Eq.(2.10) with the replacements: h − → H ′ , h = 0, and h + → H FW . The form of new diagonalized Hamiltoniandepends on the solution of the new generating operator.We use series expansion to construct the generatingoperator and require that the resulting generating oper-ator can go back to the exact solution in the free-particlecase where the Hamiltonian is block-diagonalized to Eq.(2.13). In this representation, the positive and negativeenergies are decoupled and have classical relativistic en-ergy representation [ c.f. Eqs. (2.13) and (1.1)], which is the
FW representation obtained in this article. Impor-tantly, we will show that the series expansion of gener-ating operator [Eq. (2.21)] can indeed generate the FWrepresentation. In this sense, interestingly, we can obtainan exact solution of generating operator and find that thespin part of the resulting block-diagonalized Hamiltonianis equivalent to the T-BMT Hamiltonian.In the next section, we will show that by using Eqs.(2.31) and (2.32) the effective Hamiltonian resulting fromFoldy-Wouthuysen diagonalization method is equivalentto that from the Kutzelnigg diagonalization method up toterms with order of ( π /mc ) , from which the fine struc-ture, Darwin term and spin-orbit interaction can be de-duced. III. INHOMOGENEOUS FIELDS
Up to this step, only two assumptions are made: (1)the electromagnetic fields are static, and (2) the Diracgenerating operator X can be solved by series expansion.We calculate the first two terms H (0)FW + H (2)FW and showthat the resulting Hamiltonian H FW = mc + H (0) F W + H (2)FW is in agreement with the previous result. The zerothorder of the FW transformed Dirac Hamiltonian is H (0)FW = A H , (3.1)where A = V + ( h /c ) X . The first order of theDirac generating operator X is given in Eq. (2.22),which is valid for inhomogeneous fields. Using [ π i , π j ] = iq ~ c ǫ ijk B k , we have ( σ · π ) = π − q ~ c σ · B , and H (0) F W [Eq. (3.1)] becomes H (0)FW = V + π m − q ~ mc σ · B . (3.2)We note that A is already a hermitian operator, andthus A N = 0. The second and third terms of Eq. (3.2)are the kinetic energy and Zeeman energy. The secondorder H (2)FW is given by c H (2)FW = A H + S , (3.3)where A = ( h /c ) X , S = [ G , A N ] /
2. For X , fromEq. (2.23) we have X = − X σ · π X + [ V, X ]. Using[ V, σ · π ] = − iq ~ σ · E , we obtain X = − Tm σ · π − i ~ m σ · E , (3.4)where T ≡ ( σ · π ) / m (3.5)is the kinetic energy operator. The operators X is validfor inhomogeneous fields. From Eqs. (2.27) and (2.29),the operator G is X † X = T / m . Since A N = 0, wehave S = [ G , A N ] / A , we have A = − ( σ · π ) m − iq ~ m ( σ · π )( σ · E ) . (3.6)The hermitian part of A is given by A H = ( A + A † ) / A H = − ( σ · π ) m − iq ~ m [ σ · π , σ · E ] . (3.7) Using σ i σ j = δ ij + iǫ ijk σ k , we have [ σ · π , σ · E ] = − i ~ ∇ · E + i σ · π × E − i σ · E × π . For static case, we have π × E = − E × π . Therefore, up to the second order ofmagnetic field, Eq. (3.7) becomes H (2)FW = − π m c + q ~ m c (cid:2) π ( σ · B ) + ( σ · B ) π (cid:3) − q ~ m c σ · ( E × π ) − q ~ m c ∇ · E . (3.8)The first term of Eq. (3.8) is the relativistic correction tothe kinetic energy. The second term of Eq. (3.8) is therelativistic correction to the Zeeman energy. The fourthand fifth terms of Eq. (3.8) are the spin-orbit interactionand the Darwin term which provides heuristic evidenceof the Zitterbewegung phenomenon [22]. Combining Eqs.(3.2) and (3.8), we obtain the Foldy-Wouthuysen trans-formed Dirac Hamiltonian up to terms with ( π /mc ) : H FW = mc + H (0)FW + H (2)FW = mc + V + π m − q ~ mc σ · B − π m c + q ~ m c (cid:2) π ( σ · B ) + ( σ · B ) π (cid:3) − q ~ m c ∇ · E − q ~ m c σ · ( E × π ) . (3.9)Equation (3.9) is in agreement with the earlier results[3, 17] which are obtained by the standard FW method.If we take into account the terms of the second or-der in electromagnetic fields, our result gives − e ~ m c B .However, the FW diagonalization method shows that theterms of the second order in electromagnetic field shouldbe e ~ m c ( E − B ). In comparison with the standard FWtransformation method, this discrepancy suggests thatthe assumption that the series expansion of X [Eq. (2.21)]exists is valid only in the low-energy and weak-field limitor in the absence of an electric field.In the following sections, we will obtain the FW trans-formed Dirac and Dirac-Pauli Hamiltonians in the low-energy and weak-field limit. IV. FW TRANSFORMED DIRACHAMILTONIAN
The previous section shows that the Kutzelnigg diag-onalization method is valid when we consider only termswith linear electromagnetic fields. We focus only on lin-ear terms of electromagnetic fields in comparison with theT-BMT equation. In this section, we consider the staticand homogeneous electromagnetic field and neglect theproduct of fields in the FW transformed Dirac Hamilto-nian. The FW transformed Dirac Hamiltonian containsthe Dirac energy operator and Dirac string operator [seeEqs. (2.31) and (2.32)]. We will calculate H ( ℓ )FW from ℓ = 0 to ℓ = 12. Nevertheless, we have to emphasize thatEq. (2.26) implies that the ℓ th order of Dirac energy op-erator is obtained from the next order of the Dirac gen-erating operator. Therefore, we have to obtain the termof the generating operator up to the order of 1 /c , i.e. X . The explicit forms of the expanding terms of thegenerating operators can be derived by using Eqs. (2.22)and (2.23). Up to the order of 1 /c , we have X = σ · π m , X = − Tm σ · π − iq ~ m σ · E , X = 14 T m σ · π + 316 iq ~ m π ( σ · E ) + 18 iq ~ m ( E · π )( σ · π ) ,X = − T m σ · π − iq ~ m π ( σ · E ) − iq ~ m π ( E · π )( σ · π ) ,X = 716 T m σ · π + 35256 iq ~ m π ( σ · E ) + 29128 iq ~ m π ( E · π )( σ · π ) ,X = − T m σ · π − iq ~ m π ( σ · E ) − iq ~ m π ( E · π )( σ · π ) ,X = 3332 T m σ · π + 2312048 iq ~ m π ( σ · E ) + 2811024 iq ~ m π ( E · π )( σ · π ) , (4.1)where T = ( σ · π ) / m . The forms of X and X inEq. (4.1) are also valid for inhomogeneous fields. Theexpanding terms of the Dirac generating operator from X to X in Eq. (4.1) are valid only for homogeneousfields. Inserting Eqs. (4.1) into Eq. (2.26), we can obtaineach order of the Dirac energy operator. Furthermore,we rewrite each order of the Dirac energy operator A ℓ as the combination of the hermitian part ( A Hℓ ) and anti-hermitian part ( A Nℓ ), A ℓ = A Hℓ + A Nℓ , (4.2)where A Hℓ and A Nℓ satisfy A H † ℓ = A Hℓ and A N † ℓ = − A Nℓ .The hermitian parts of the Dirac energy operator from A H to A H are given by A H = T + V, A H = − T m − q ~ m σ · ( E × π ) ,A H = T m + 316 q ~ m π σ · ( E × π ) ,A H = − T m − q ~ m π σ · ( E × π ) ,A H = 78 T m + 35256 q ~ m π σ · ( E × π ) ,A H = − T m − q ~ m π σ · ( E × π ) ,A H = 3316 T m + 2312048 q ~ m π σ · ( E × π ) , (4.3) The anti-hermitian parts of the Dirac energy operatorfrom A N to A N are given by A N = 0 , A N = − iq ~ m E · π , A N = + 516 iq ~ m π E · π ,A N = − iq ~ m π E · π , A N = + 93256 iq ~ m π E · π ,A N = − iq ~ m π E · π , A N = + 7932048 iq ~ m π E · π . (4.4)We emphasize that the second and higher order of elec-tromagnetic field will be neglected in Eqs. (4.3) and (4.4).In order to simplify the present expression, the formof A ℓ still contains terms with non-linear electromag-netic fields because the operator T can be written as T = ( σ · π ) / m = m ( π − q ~ c σ · B ). We will neglectthese higher order terms when constructing Hamiltonian.On the other hand, to evaluate the Dirac string opera-tor, we have to obtain the Dirac exponent operator by ex-panding ln(1+ X † X ). After straightforward calculations,the expanding terms of the Dirac exponent operator G ℓ (up to 1 /c ) are given by G = T m , G = − T m − q ~ m σ · ( E × π ) ,G = 1112 T m + 516 q ~ m π σ · ( E × π ) ,G = − T m − q ~ m π σ · ( E × π ) ,G = 19380 T m + 93256 q ~ m π σ · ( E × π ) ,G = − T m − q ~ m π σ · ( E × π ) . (4.5)Since we always neglect terms with E , B , EB and mul-tiple products of them, the kinetic energy operator T commutes with π k σ · ( E × π ) and π k E · π , and we have[ T, π k σ · ( E × π )] = 0 + o ( f ) , [ T, π k E · π ] = 0 + o ( f ) , [ π k σ · ( E × π ) , π n σ · ( E × π )] = 0 + o ( f ) , (4.6)where o ( f ) represents the second order and higher or-ders of homogeneous electromagnetic fields.Applying Eqs. (4.3), (4.4), (4.5) and (4.6) to the Diracstring operators [Eq. (2.32)], we find that all the non-vanishing Dirac string operators S ℓ (from ℓ = 2 to ℓ = 12)are proportional to second and higher orders of electricand magnetic fields which are being neglected. This canalso be proved as follows.Firstly, consider the Dirac string operator with onlyone Dirac exponent operator, S = [ G, A N ] / o ( G ).The Dirac exponent operator is G = P ℓ G ℓ /c ℓ = G T + G so , where G T is the term with collections ofthe kinetic energy operator T , i.e., G T = T / mc − (5 / T /m c + (11 / T /m c + · · · , and G so =( − / m c + 5 π / m c − π / m c + · · · ) ~ σ · ( E × π ) = F ( π ) σ · ( E × π ), where F ( π ) represents thepower series of π . The anti-hermitian part of the Diracenergy operator is A N = P ℓ A Nℓ /c ℓ = ( − / m c +5 π / m c − π / m c + · · · ) i ~ E · π = g ( π ) E · π ,where g ( π ) represents the power series of π . There-fore, [ G, A N ] can be written as [ G, A N ] = [ G T , A N ] +[ G so , A N ]. Since we have [ T, π k ] = [ T, E · π ] = 0 + o ( f ), thus [ G T , A N ] = 0 + o ( f ). The commutator[ G so , A N ] = [ F ( π ) σ · ( E × π ) , g ( π ) E · π ] also vanishes upto second-order terms of homogeneous electromagneticfield, because we have [ σ · ( E × π ) , g ( π )] = 0 + o ( f ),[ σ · ( E × π ) , E · π ] = 0 + o ( f ), [ F ( π ) , E · π ] = 0 + o ( f )and [ F ( π ) , g ( π )] = 0. We obtain [ G, A N ] = 0 + o ( f ),and thus, the terms containing odd numbers of G in theDirac string operator [see Eq. (2.20)] always vanishes upto second-order terms of homogeneous electromagneticfields.Secondly, consider the term containing two Dirac expo-nent operators in the Dirac string operator, [ G, [ G, A H ]].The hermitian part of the Dirac energy operator can bewritten as A H = P ℓ A ℓ /c ℓ = V + A HT + A H so , where A HT = T − T / mc + T / m c − T / m c + · · · and A H so = K ( π ) σ · ( E × π ), where K ( π ) is thepower series of π . The commutator [ G, A H ] becomes[ G, A H ] = [ G T , V ] + [ G T , A HT ] + [ G T , A H so ] + [ G so , V ] +[ G so , A HT ]+[ G so , A H so ]. Since we have [ T, π k σ · ( E × π )] =0 + o ( f ) and [ π k σ · ( E × π ) , π n σ · ( E × π )] = 0 + o ( f ),the commutators [ G T , A H so ], [ G so , A HT ] and [ G so , A H so ] van- ish up to second-order terms of homogeneous electro-magnetic fields as well as [ G so , V ]. For the commuta-tor [ G T , V ], using [ T, V ] = − i ~ E · π /m , we find that[ G T , V ] = R ( π ) E · π + o ( f ), where R ( π ) is the powerseries of π . That is, [ G, A H ] = R ( π ) E · π + o ( f ). Sim-ilar to the commutator [ G, A N ], where A N = g ( π ) E · π ,we find that this implies that [ G, [ G, A H ]] = 0 + o ( f ).Therefore, the terms containing even numbers of theDirac exponent operators in the Dirac string operator[see Eq. (2.20)] always vanishes up to second-order termsof homogeneous electromagnetic fields. In short, it canbe shown that from ℓ = 0 to ℓ = 12, the expanding termsof the Dirac string operator satisfies S ℓ = 0 + o ( f ) , (4.7)where o ( f ) represents the second and higher orders ofelectromagnetic fields.Consider Eq. (2.31) together with (4.7), we find that H ( ℓ ) F W is exactly equal to A Hℓ , i.e. c ℓ H ( ℓ ) F W = A Hℓ + o ( f ) , (4.8)where ℓ = 0 , , , · · · ,
12. Equation (4.8) is the main re-sult of this paper. This implies that the FW transformedDirac Hamiltonian is only determined by the hermitianpart of the Dirac energy operator regardless of the Diracexponent operator G . We have shown that Eq. (4.8)is valid at least up to 1 /c . We believe that this re-sult is valid to all higher orders of 1 /c . Equation (4.8)enables us to solely focus on the hermitian part of theDirac energy operator since the anti-hermitian part canbe exactly cancelled by the remaining string operators.As a consequence, this result provides us a method toobtain higher order terms faster than traditional Foldy-Wouthuysen transformation.Comparing the form of the resulting FW transformedDirac Hamiltonian with the classical Hamiltonian, we de-fine the magnetic moment µ and the scaled kinetic mo-mentum ξ as µ = q ~ mc σ , ξ = π mc . (4.9)On the other hand, the kinetic energy operator T inEq. (4.3) can be replaced by T = π m − q ~ mc σ · B , andafter neglecting second and higher orders of electromag-netic fields, the FW transformed Hamiltonian [Eq. (4.8)]becomes H (0)FW = V + 12 mc ξ − µ · B ,H (2)FW = − mc ξ + 12 ξ µ · B − µ · ( E × ξ ) ,H (4)FW = + 116 mc ξ − ξ µ · B + 38 ξ µ · ( E × ξ ) ,H (6)FW = − mc ξ + 516 ξ µ · B − ξ µ · ( E × ξ ) ,H (8)FW = + 7256 mc ξ − ξ µ · B + 35128 ξ µ · ( E × ξ ) ,H (10)FW = − mc ξ + 63256 ξ µ · B − ξ µ · ( E × ξ ) ,H (12)FW = + 332048 mc ξ + 2311024 ξ µ · B − ξ µ · ( E × ξ ) . (4.10)After substituting Eq. (4.10) into Eq. (2.30), the FWtransformed Dirac Hamiltonian becomes a sum of twoterms: H FW = X ℓ H ( ℓ )FW = H orbit + H spin , (4.11)where the orbital Hamiltonian H orbit is the kinetic en-ergy (including the rest mass energy) plus the potentialenergy, H orbit = mc (1 + 12 ξ − ξ + 116 ξ − ξ + 7256 ξ − ξ + 332048 ξ ) + V, (4.12)and the spin Hamiltonian H spin is the Hamiltonian ofintrinsic magnetic moment in electromagnetic fields, H spin = − (1 − ξ + 38 ξ − ξ + 35128 ξ − ξ + 2311024 ξ ) µ · B + ( −
12 + 38 ξ − ξ + 35128 ξ − ξ + 2311024 ξ ) µ · ( E × ξ ) . (4.13)In the following section, we will show that the FW trans-formed Dirac Hamiltonian is equivalent to the Hamilto-nian obtained from T-BMT equation with g = 2. V. FW TRANSFORMED DIRACHAMILTONIAN AND CLASSICALHAMILTONIAN
The orbital Hamiltonian H orbit [Eq. (4.12)] is expectedto be equivalent to the classical relativistic energy γmc + V . However, the boost velocity in T-BMT equation is not ξ [19]. Take the series expansion of (1 + ξ ) / intoaccount,(1 + ξ ) / = 1 + 12 ξ − ξ + 116 ξ − ξ + 7256 ξ − ξ + 332048 ξ − ξ + · · · , (5.1)we find that the series of ξ in Eq. (4.12) is exactly equalto Eq. (5.1) up to ξ . This enable us to define the boostoperator b β via the Lorentz operator b γ ,(1 + ξ ) / = b γ = 1 q − b β . (5.2)In this sense, the orbital Hamiltonian can now be writ-ten as H orbit = b γmc + V . In classical relativistic the-ory, the Lorentz factor γ is related to boost velocity by γ = 1 / p − β . However, in the relativistic quantummechanics since different components of the kinetic mo-mentum operator π do not commute with one another,the boost operator b β should not simply satisfy the form b γ = 1 / q − b β . We will go back to this point whendiscussing the spin Hamiltonian. The boost operator b β plays an important role on showing the agreement be-tween the spin Hamiltonian H spin and the T-BMT equa-tion.The spin Hamiltonian can be written as a sum of Zee-man Hamiltonian H ze and spin-orbit interaction H so , H spin = H ze + H so . (5.3)The Zeeman Hamiltonian H ze is the relativistic correc-tion to the Zeeman energy: H ze = − (1 − ξ + 38 ξ − ξ + 35128 ξ − ξ + 2311024 ξ ) µ · B . (5.4)0The spin-orbit interaction H so is the interaction of elec-tric field and the electric dipole moment arising from theboost on the intrinsic spin magnetic moment: H so = − ( 12 − ξ + 516 ξ − ξ + 63256 ξ − ξ ) µ · ( E × ξ ) . (5.5)We first focus on the series in the Zeeman Hamiltonian.Consider the series expansion of (1 + ξ ) − / ,(1 + ξ ) − / = 1 − ξ + 38 ξ − ξ + 35128 ξ − ξ + 2311024 ξ − ξ + · · · , (5.6)we find that the series in H ze is exactly equal to (1 + ξ ) − / up to ξ . Therefore, the Zeeman HamiltonianEq. (5.4) can be written as H ze = − b γ µ · B . (5.7)On the other hand, the the spin-orbit term in the T-BMTHamiltonian transforms like [ g/ − γ/ (1 + γ )] and g = 2for the Dirac Hamiltonian. Therefore, consider the seriesexpansion of (1 − b γ/ (1 + b γ ))(1 / b γ ), we have (cid:18) − b γ b γ (cid:19) b γ = 1 p ξ −
11 + p ξ = 12 − ξ + 516 ξ − ξ + 63256 ξ − ξ + 4292048 ξ + · · · , (5.8)where b γ (1 + b γ ) − = (1 + b γ ) − b γ was used. The seriesin Eq. (5.5) is in agreement with Eq. (5.8) up to ξ .Therefore, we have H so = − (cid:18) − b γ b γ (cid:19) b γ µ · ( E × ξ ) . (5.9)We note that if Eq. (5.9) is in complete agreement withthe T-BMT equation, the boost velocity operator b β mustbe defined by b β = b γ ξ . In general, the commutator[ ξ , / b γ ] is not equal to zero, and Eq. (5.2) cannot besatisfied. However, since we require that the FW trans-formed Dirac Hamiltonian H FW is linear in electromag-netic fields, the magnetic field obtained from the operator[ ξ , / b γ ] should be neglected. In that sense, the commuta-tor [ ξ , / b γ ] should be identified as zero in this case, andthe boost operator can be written as b β = 1 b γ ξ = ξ b γ . (5.10) The identity can be shown as follows. b γ (1 + b γ ) − = [(1 + b γ ) b γ − ] − = ( b γ − + 1) − = [ b γ − (1 + b γ )] − = (1 + b γ ) − b γ . It can be shown that Eq. (5.10) satisfies Eq. (5.2). There-fore, the spin Hamiltonian Eq. (4.13) with substitutionsof Eqs. (5.7) and (5.9) becomes H spin = H ze + H so = − µ · (cid:20) b γ B − (cid:18) − b γ b γ (cid:19) b β × E (cid:21) . (5.11)Up to the twentieth order there is a complete agree-ment between the spin part of the FW transformed DiracHamiltonian and the T-BMT equation with g = 2. TheFW transformed Hamiltonian is given by H FW = H orbit + H spin = V + b γmc − µ · (cid:20) b γ B − (cid:18) − b γ b γ (cid:19) b β × E (cid:21) , (5.12)which is in agreement with the classical Hamiltonian with g = 2. In the next section, we take into account thePauli anomalous magnetic moment and show that theclassical correspondence of the Dirac-Pauli Hamiltonianis the classical Hamiltonian with g = 2. VI. FW TRANSFORMATION FORDIRAC-PAULI HAMILTONIAN
In the previous section, the agreement to the classi-cal Hamiltonian is shown to be complete up to terms ofthe order ( π /mc ) in the absence of anomalous electronmagnetic moment, i.e., g = 2. The Dirac electron in-cluding the Pauli anomalous magnetic moment can bedescribed by the Dirac-Pauli Hamiltonian denoted by H which contains the Dirac Hamiltonian as well as anoma-lous magnetic interaction V B and anomalous electric in-teraction V E , H = H D + (cid:18) V B iV E − iV E − V B (cid:19) = (cid:18) H + H H † H − (cid:19) (6.1)where H + = V + V B + mc , H − = V − V B − mc and H = h + iV E . The Dirac Hamiltonian H D is given inEq. (2.1), and V B = − µ ′ σ · B , V E = µ ′ σ · E . (6.2)The coefficient µ ′ is defined as µ ′ = (cid:16) g − (cid:17) q ~ mc . (6.3)For an electron with g = 2, we have V B = 0 and V E = 0.Applying the unitary transformation Eq. (2.3) U = (cid:18) Y YX † −ZX Z (cid:19) (6.4)1to the Dirac-Pauli Hamiltonian, the self-consistent equa-tion for the Dirac-Pauli generating operator X is given bythe requirement of vanishing off-diagonal term of U H U † ,i.e. 2 mc X = [ V, X ] + h − X h X− iV E − i X V E X − {X , V B } , (6.5)where h = c σ · π . The FW transformed Dirac-PauliHamiltonian can be obtained from the upper-left blockdiagonal term of U H U † , and it is given by H FW = Y (cid:16) H + + X † H † + H X + X † H − X (cid:17) Y . (6.6)Similar to the derivation of Eq. (2.15), we find that theFW transformed Dirac-Pauli Hamiltonian [Eq. (6.6)] canalso be simplified as (see Appendix B) H FW = mc + e G / A e −G / , (6.7)where the Dirac-Pauli energy operator A and the Pauli-Dirac exponent operator G are given by A = V + h X + V B + iV E X , G = ln (cid:0) X † X (cid:1) . (6.8)Similar to Eq. (2.19) obtained from the requirement ofhermiticity of H FW , we find that the FW transformedDirac-Pauli Hamiltonian also satisfies H FW = H † FW = mc + e −G / A e G / and the FW transformed Dirac-PauliHamiltonian can be rewritten as H FW = mc + A H + S , (6.9)where A H is the hermitian part of the Dirac-Pauli energyoperator and the Dirac-Pauli string operator S is thesame as Eq. (2.20) by the replacements A → A and G →G , i.e. S = 12 [ G , A N ] + 12!2 [ G , [ G , A H ]] + 13!2 [ G , [ G , [ G , A N ]]]+ 14!2 [ G , [ G , [ G , [ G , A H ]]]] + · · · . (6.10)Similar to the Dirac string operator, the anti-hermitianpart of the Dirac-Pauli energy operator always appears in those terms with odd numbers of Dirac-Pauli exponentoperators, and the hermitian part of the Dirac-Pauli en-ergy operator always appears in those terms with evennumbers of Dirac exponent operators. The power se-ries solutions to the Dirac-Pauli generating operator canbe obtained by means of Eq. (6.5) via substitution ofthe series expansion X = P i X k /c k , k = 1 , , , · · · ,and each order of Dirac-Pauli energy operator A k canbe obtained from A = P k A k /c k by using Eq. (6.8).Each order of Dirac-Pauli energy operators can be de-composed into hermitian ( A Hk ) and anti-hermitian ( A Nk )parts, A k = A Hk + A Nk . As a result, the FW transformedDirac-Pauli Hamiltonian can be written as H F W = mc + X k =0 , , , ··· H ( k ) F W (6.11)with c k H ( k ) F W = A Hk + S k . (6.12)To obtain the FW transformed Dirac-Pauli Hamiltonianup to k = 12, the largest order of the Dirac-Pauli gener-ating operator must have the order of k = 13, i.e., X .This is because the operator h = c σ · π is of the order of c , the order of the Dirac-Pauli energy operator is lowerthan that of the Dirac-Pauli generating operator. Fur-thermore, since each order of the Dirac-Pauli generatingoperator must equal that of the Dirac generating operatorwhen g = 2, we can rewrite the Dirac-Pauli generatingoperator ( X k ) as the sum of the Dirac generating oper-ator ( X k ) and the anomalous generating operator ( X ′ k ),namely, X k = X k + X ′ k + o ( f ) , (6.13)where the anomalous generating operator X ′ k vanisheswhen g = 2. Similar to the derivation of power seriessolution to the Dirac generating operator shown in theprevious section, the explicit forms of different orders ofthe anomalous generating operator X ′ k are given by ( k =1 , , · · · , X ′ = 0 , X ′ = 0 , X ′ = − iµ ′′ m σ · E , X ′ = µ ′′ m B · π , X ′ = 38 iµ ′′ m π ( σ · E ) − iµ ′′ m ( σ · π )( E · π ) X ′ = − µ ′′ m π ( B · π ) , X ′ = − iµ ′′ m π ( σ · E ) + 14 iµ ′′ m π ( σ · π )( E · π ) ,X ′ = 516 µ ′′ m π ( B · π ) , X ′ = 35128 iµ ′′ m π ( σ · E ) − iµ ′′ m π ( σ · π )( E · π ) ,X ′ = − µ ′′ m π ( B · π ) , X ′ = − iµ ′′ m π ( σ · E ) + 732 iµ ′′ m π ( σ · π )( E · π ) ,X ′ = 63256 µ ′′ m π ( B · π ) , X ′ = 2311024 iµ ′′ m π ( σ · E ) − iµ ′′ m π ( σ · π )( E · π ) , (6.14)where µ ′′ = ( g/ − q ~ / m = cµ ′ . We note that sincethe gyromagnetic ratio always accompanies linear-orderterms of electric or magnetic fields, the operator X ′ k isproportional to electromagnetic fields and contains thekinetic momentum operator.Substituting Eq. (6.13) into Eq. (6.8), the Dirac-Paulienergy operator ( A k ) can be written as the sum of the en-ergy operator for the Dirac Hamiltonian ( A k ) and anoma-lous energy operator ( A ′ k ), A k = A k + A ′ k + o ( f ) , (6.15)where the expanding terms of the Dirac energy operator A k from k = 0 to k = 12 are given in Eqs. (4.3) and(4.4). The k th order of the anomalous energy operatoris related to k th orders of Dirac generating operator andanomalous generating operator by A ′ = 0 , A ′ = cV B ,A ′ k = ( h /c ) X ′ k +1 + icV E X k − , k = 2 , , · · · ,A ′ k = ( h /c ) X k +1 , k = 3 , , · · · , . (6.16)Using Eqs. (4.1) and (6.14), the hermitian parts of theexpanding terms of the anomalous energy operator from zeroth order to twentieth orders are given by A ′ H = 0 , A ′ H = − µ ′′ σ · B ,A ′ H = − µ ′′ m σ · ( E × π ) ,A ′ H = 12 µ ′′ m ( σ · π )( B · π ) ,A ′ H = µ ′′ m π σ · ( E × π ) ,A ′ H = − µ ′′ m π ( σ · π )( B · π ) ,A ′ H = − µ ′′ m π σ · ( E × π ) ,A ′ H = 516 µ ′′ m π ( σ · π )( B · π ) ,A ′ H = 516 µ ′′ m π σ · ( E × π ) A ′ H = − µ ′′ m π ( σ · π )( B · π ) ,A ′ H = − µ ′′ m π σ · ( E × π ) ,A ′ H = 63256 µ ′′ m π ( σ · π )( B · π ) ,A ′ H = 63256 µ ′′ m π σ · ( E × π ) . (6.17)For example, consider the twentieth order of the anoma-lous operator A ′ , which is given by A ′ = ( h /c ) X ′ + icV E X . Substituting V E , X ′ and X into A ′ andneglecting the second order of homogeneous electromag-3netic fields, we find that A ′ = 2311024 iµ ′′ m π ( σ · π )( σ · E ) − iµ ′′ m T ( σ · E )( σ · π ) − iµ ′′ m π σ · E = (cid:18) × (cid:19) µ ′′ m π σ · ( E × π )+ (cid:18) − − × (cid:19) iµ ′′ m π σ · E = 63256 µ ′′ m π σ · ( E × π ) , (6.18)where in the second equality we have used ( σ · E )( σ · π ) = E · π + i σ · ( E × π ) and E × π = − π × E forhomogeneous fields, and the kinetic energy operator isreplaced by T → π / m . The anti-hermitian part of A ′ is iµ ′′ π σ · E /m and its numerical coefficient is zero.Interestingly, we find that all the anti-hermitian part of A ′ k from k = 0 to k = 12 vanish up to second-order termsof homogeneous electromagnetic fields, i.e. A ′ Nk = 0 + o ( f ) . (6.19)On the other hand, the series expansion of the Dirac-Pauli exponent operator G = ln(1 + X † X ) can also bewritten as G = P k G k /c k and G k = G k + G ′ k , where G k (the k th order of the Dirac exponent operator) is givenin Eq. (4.5) and G ′ k is the k th order of the anomalousexponent operator. The expanding terms of the anoma-lous exponent operators G ′ k from k = 1 to k = 12 are asfollows: G ′ = 0 , G ′ = 0 , G ′ = 0 , G ′ = − µ ′′ m σ · ( E × π ) ,G ′ = µ ′′ m ( σ · π )( B · π ) , G ′ = 58 µ ′′ m π σ · ( E × π ) ,G ′ = − µ ′′ m π ( σ · π )( B · π ) ,G ′ = − µ ′′ m π σ · ( E × π ) ,G ′ = 1116 µ ′′ m π ( σ · π )( B · π ) ,G ′ = 93128 µ ′′ m π σ · ( E × π ) ,G ′ = − µ ′′ m π ( σ · π )( B · π ) ,G ′ = − ~ m π σ · ( E × π ) , (6.20)which all vanish when g = 2 and each order of the anoma-lous exponent operator must be proportional to electro-magnetic fields. Substitute Eqs. (6.17), (6.19) and (6.20)into Eq. (6.10), it can be shown that similar to the re-sult of the Dirac string operator, the Dirac-Pauli string operator also vanishes up to second-order terms of homo-geneous electromagnetic fields; i.e., we have S k = 0 + o ( f ) . (6.21)This can be proved as follows.Firstly, consider the term containing only one Dirac-Pauli exponent operator in the Dirac-Pauli string oper-ator [see Eq. (6.10)]. It is given by [ G , A N ] /
2. Since G = G + G ′ and A N = A N + A ′ N , we have [ G , A H ] / G, A N ] / G, A ′ N ] / G ′ , A N ] / G ′ , A ′ N ] /
2, where[
G, A N ] / G, A N ] / o ( f ). The anomalous exponent operatorcan be written as G ′ = P k G ′ k /c k = F ( π ) σ · ( E × π ) + F ( π )( σ · π )( B · π ), where F ( π ) represents a powerseries of π as well as F ( π ). We note that A ′ N = P k A ′ N /c k = 0 + o ( f ) [see Eq. (6.19)]. It is obviousthat the second term [ G, A ′ N ] and fourth term [ G ′ , A ′ N ]vanish up to second-order terms of homogeneous electro-magnetic fields. The third term [ G ′ , A N ] = [ f ( π ) σ · ( E × π ) + f ( π )( σ · π )( B · π ) , g ( π ) E · π ] also vanishesbecause we have [ f ( π ) , g ( π )] = [ f ( π ) , g ( π )] = 0and [ σ · ( E × π ) , E · π ] = [( σ · π )( B · π ) , E · π ] = [ σ · ( E × π ) , g ( π )] = [( σ · π )( B · π ) , g ( π )] = [ f ( π ) , E · π ] =[ f ( π ) , E · π ] = 0 + o ( f ).Therefore, we have [ G , A N ] / o ( f ). Since thecommutator [ G , A N ] / G always vanish up to second-order terms ofhomogeneous electromagnetic fields.Secondly, consider the terms with two Dirac-Pauliexponent operators in the Dirac-Pauli string operator. Itis given by [ G , [ G , A H ]] / = [ G, [ G, A H ]] / +[ G, [ G, A ′ H ]] / + [ G, [ G ′ , A H ]] / +[ G ′ , [ G, A H ]] / + o ( f ), where we have neglectedthe second-order terms of electromagnetic fields,such as [ G, [ G ′ , A ′ H ]], [ G ′ , [ G, A ′ H ]], [ G ′ , [ G ′ , A H ]] and[ G ′ , [ G ′ , A ′ H ]]. The first term [ G, [ G, A H ]] is the Diracstring operator containing only two Dirac exponent oper-ators and it has been show that [ G, [ G, A H ]] = 0 + o ( f ).The anomalous energy operator can be written as A ′ H = P k A ′ Hk /c k = K ( π ) σ · ( E × π ) + K ( π )( σ · π )( B · π ),where K ( π ) represents the power series of π as well as K ( π ). The commutator [ G, A ′ H ]can be written as [ G, A ′ H ] = [ G T + G so , A ′ H ] =[ G T , A ′ H ]+[ G so , A ′ H ], where G so = F ( π ) σ · ( E × π ) and G T = T / m − (5 / T /m + (11 / T /m + · · · . UsingEq. (4.6), it can be shown that [ G T , A ′ H ] = 0 + o ( f )and [ G so , A ′ H ] = 0 + o ( f ), and thus, the sec-ond term [ G, [ G, A ′ H ]] vanishes up to second-order terms of homogeneous electromagnetic fields.Consider the third term [ G, [ G ′ , A H ]], wherethe commutator [ G ′ , A H ] becomes [ G ′ , A H ] =[ F ( π ) σ · ( E × π )+ F ( π )( σ · π )( B · π ) , V + A HT + A H so ] =[ F ( π ) σ · ( E × π ) , V ] + [ F ( π )( σ · π )( B · π ) , V ] + o ( f ).However, the two commutators [ F ( π ) σ · ( E × π ) , V ]and [ F ( π )( σ · π )( B · π ) , V ] are proportional to the4second order of homogeneous electromagnetic fields since[ π , V ] = iq ~ E , and thus we have [ G, [ G ′ , A H ]] = 0+ o ( f ).The fourth term [ G ′ , [ G, A H ]] also vanishes up to o ( f )since it has been shown that [ G, A H ] = R ( π ) E · π + o ( f )and G ′ contains first-order terms of homogeneous elec-tromagnetic fields.Therefore, we have shown that [ G , [ G , A H ]] / =0 + o ( f ). Since the commutator [ G , [ G , A H ]] always ap-pears in those terms with even number of the Dirac-Pauliexponent operator [see Eq. (6.10)], this implies that theterms with even numbers of G always vanishes up to sec-ond order of homogeneous electromagnetic fields.As a consequence, the FW transformed Dirac-PauliHamiltonian is determined only by the hermitian partof Dirac-Pauli energy operator, i.e. c k H ( k )FW = A Hk + o ( f ) . (6.22)Since the Dirac-Pauli energy operator is composed ofthe Dirac energy operator and anomalous energy oper-ator, A Hk = A Hk + A ′ Hk , and the Dirac energy operatoris related to the FW transformed Dirac Hamiltonian by c k H ( k )FW = A Hk , the FW transformed Dirac-Pauli Hamil-tonian can be written as the sum of the FW transformedDirac Hamiltonian and the anomalous Hamiltonian: H FW = H FW + H ′ FW . (6.23)The k th order of the FW transformed Pauli-Dirac Hamil-tonian can be written as H ( k )FW = H ( k )FW + H ′ ( k )FW , (6.24)where the the k th order of the anomalous Hamiltonian H ′ FW denoted as H ′ ( k )FW is determined by the k th order ofthe anomalous energy operator: c k H ′ ( k )FW = A ′ Hk . (6.25) Using Eqs. (6.17), (6.19) and (4.9), the terms H ( k ) F W from k = 0 to k = 12 are given by H ′ (0)FW = 0 , H ′ (1)FW = − (cid:16) g − (cid:17) µ · B ,H ′ (2)FW = − (cid:16) g − (cid:17) µ · ( E × ξ ) ,H ′ (3)FW = 12 (cid:16) g − (cid:17) ( µ · ξ )( B · ξ ) ,H ′ (4)FW = (cid:16) g − (cid:17) ξ µ · ( E × ξ ) ,H ′ (5)FW = − (cid:16) g − (cid:17) ξ ( µ · ξ )( B · ξ ) ,H ′ (6)FW = − (cid:16) g − (cid:17) ξ µ · ( E × ξ ) ,H ′ (7)FW = 516 (cid:16) g − (cid:17) ξ ( µ · ξ )( B · ξ ) ,H ′ (8)FW = 516 (cid:16) g − (cid:17) ξ µ · ( E × ξ ) ,H ′ (9)FW = − (cid:16) g − (cid:17) ξ ( µ · ξ )( B · ξ ) ,H ′ (10)FW = − (cid:16) g − (cid:17) ξ µ · ( E × ξ ) ,H ′ (11)FW = 63256 (cid:16) g − (cid:17) ξ ( µ · ξ )( B · ξ ) ,H ′ (12)FW = 63256 (cid:16) g − (cid:17) ξ µ · ( E × ξ ) . (6.26)By using Eqs. (5.6) and (5.8), the anomalous Hamilto-nian can be written as H ′ FW = X k =0 H ′ ( k )FW = − (cid:16) g − (cid:17) µ · B − (cid:16) g − (cid:17) b γ µ · ( E × ξ )+ (cid:16) g − (cid:17) (cid:18) − b γ b γ (cid:19) b γ ( µ · ξ )( B · ξ ) . (6.27)Combining the FW transformed Dirac Hamiltonian[Eq. (5.12)] and the anomalous Hamiltonain [Eq. (6.27)],we have (up to terms of ( π /mc ) ) H FW = H FW + H ′ FW = V + b γmc − (cid:18) g − b γ (cid:19) µ · B + (cid:18) g − b γ b γ (cid:19) µ · ( b β × E )+ (cid:16) g − (cid:17) b γ b γ ( µ · b β )( B · b β ) . (6.28)Equation (6.28) is in agreement with the classical Hamil-tonian with g = 2.The FW transformed Dirac-Pauli Hamiltonian[Eq. (6.27)] can also be obtained by directly evaluatingEq. (6.6). Since Eq. (6.6) is explicitly hermitian,the calculation can be done without accounting for5the Dirac-Pauli string operator and the separation ofhermitian and anti-hermitian parts of the Dirac-Paulienergy operator [23]. Up to ( π /mc ) , we find thatthe result shown in Ref. [23] is in agreement with thepresent result. To find the classical correspondence ofthe quantum theory of charged spin-1/2 particle, wehave to perform the FW transformation on the quantumHamiltonian. The procedure presented in this paperprovides us a more systematic and efficient method toobtain higher order expansion in the FW representation. VII. EXACT UNITARY TRANSFORMATION
We now turn to the discussion of the exact series ex-pansions of the Dirac and Dirac-Pauli generating oper-ators. The exact unitary transformation of a free par-ticle Dirac Hamiltonian has been given in Eq. 2.12. Inthe presence of electromagnetic fields, the series of suc-cessive FW transformations becomes much more com-plicated. However, it is still possible to obtain the ex-act unitary transformation by deducing the close formfrom the finite-order series expansion, if the order weobtained is high enough. For example, the exact unitarytransformation of the free particle Dirac Hamiltonian canbe obtained from the successive FW transformations, ifterms in the series expansion is many enough to deter-mine the closed form. Therefore, in order to find the closed form for generic cases, we must proceed to higherorders. On the other hand, it has been proposed thatthe low-energy and weak-field limit of the Dirac (resp.Dirac-Pauli) Hamiltonian is consistent with the classicalHamiltonian, which is the sum of the classical relativisticHamiltonian and T-BMT Hamiltonian with g = 2 (resp. g = 2). This suggests that there exists an exact unitarytransformation for the low-energy and weak-field limit.In this section we will find the closed form of the unitarytransformation from the high-order series expansions ofthe generating operators.The unitary transformation matrix is related to thegenerating operator by Eqs. (2.3) and (2.4) in Kutzel-nigg’s diagonalization method. If the closed form of gen-erating operator is found, the exact unitary transforma-tion matrix can be obtained.For the low-energy and weak-field limit of the DiracHamiltonian, the Dirac generating operator can be writ-ten as X = X c + X c + X c + · · · . (7.1)In Sec. IV, we have obtained the terms X ℓ up to order of ℓ = 13, which are given in Eq. (4.1). We find that Eq.(7.1) with Eq. (4.1) can be incorporated into the closedform X = 11 + p σ · ξ ) σ · ξ + p ξ −
11 + p ξ ! − imc µ · E + p ξ
11 + p ξ ! ( µ · ξ )( E · ξ ) , (7.2)The magnetic field generated from the operator ( σ · π ) is included in the first term of Eq. (7.2). In the absenceof electromagnetic fields, the kinetic momentum π is re-placed by the canonical momentum p . In this case, Eq.(7.2) becomes c σ · p / [ mc + p m c + c p ], which is thesame as Eq. (2.11), and the resulting unitary transfor-mation is exactly Eq. (2.12). We also note that in theabsence of an electric field, Eq. (7.2) becomes Eq. (2.14). Taking the anomalous magnetic moment into account,the Dirac-Pauli generating operator can be written as X = X + X ′ , (7.3)where X is given in Eq. (7.2). We find that the anoma-lous generating operator X ′ with Eq. (6.14) can be in-corporated into the closed form X ′ = X ′ c + X ′ c + X ′ c + · · · = p ξ −
11 + p ξ ! (cid:16) g − (cid:17) mc (cid:18) − i µ · E + q ~ mc B · ξ (cid:19) + 1 p ξ
11 + p ξ ! (cid:20) − imc (cid:16) g − (cid:17) ( µ · ξ )( E · ξ ) (cid:21) . (7.4)6The closed forms of X and X ′ have been deduced fromthe high-order series expansions Eq. (4.1) and Eq. (6.14)respectively, but the rigorous proofs are stilling missing.The merit of obtaining the closed forms is neverthelessenormous: it allows us to guess the generic forms of X ℓ and X ′ ℓ in the series expansions, which in turn enableus to conduct rigorous proofs by mathematical induction[24].With Eqs. (7.2) and (7.4) at hand, we can formallyconstruct the exact unitary transformation. However,the main problem to be addressed is that the resultingexact unitary matrix is valid only in the low-energy andweak-field limit. In this regard, when we apply the ex-act unitary transformation to the Dirac or Dirac-PauliHamiltonian, we have to neglect nonlinear electromag-netic effects. In strong fields, the particle’s energy in-teracting with electromagnetic fields could exceed theDirac energy gap (2 mc ) and it is no longer adequateto describe the relativistic quantum dynamics withouttaking into account the field-theory interaction to theantiparticle. In fact, some doubts have been thrown onthe mathematical rigour of the FW transformation [25].The study of this paper nevertheless suggests that theexact FW transformation indeed exists and is valid inthe low-energy and weak-field limit and furthermore theFW transformed Hamiltonian agrees with the classicalcounterpart (see [24] for closer investigations). VIII. CONCLUSIONS AND DISCUSSION
The motion of a particle endowed with charge and in-trinsic spin is governed by the classical Lorentz equationand the T-BMT equation. Assuming that the canoni-cal relation of classical spins (via Poisson brackets) is thesame as that of quantum spins (via commutators), the T-BMT equation can be recast as the Hamilton’s equationand the T-BMT Hamiltonian is obtained. By treatingpositions, momenta and spins as independent variablesin pase space, the classical Hamiltonian describing themotion of spin-1/2 charged particle is the sum of theclassical relativistic Hamiltonian and T-BMT Hamilto-nian.On the other hand, the correspondence between theclassical Hamiltonian and the low-energy and weak-fieldlimit of Dirac equation has been investigated by severalauthors. For a free particle, the Foldy-Wouthuysen trans-formation of Dirac equation was shown to exactly leadto the classical relativistic Hamiltonian of a free particle.Intriguingly, when spin precession and interaction withelectromagnetic fields are also taken into account, it wasfound that the connection between Dirac equation andclassical Hamiltonian becomes explicit if the order-by-order block diagonalization of the the Dirac Hamiltoniancan be proceed to higher-order terms.The low-energy and weak-field limit of the relativis-tic quantum theory of spin-1/2 charged particle is in-vestigated by performing the Kutzelnigg diagonalisation method on the Dirac Hamiltonian. We show that inthe presence of inhomogeneous electromagnetic fields theFoldy-Wouthuysen transformed Dirac Hamiltonian up toterms with ( π /mc ) can be reproduced by the Kutzelniggdiagonalisation method.When the electromagnetic fields are homogeneous andnonlinear effects are neglected, the Foldy-Wouthuysentransformation of the Dirac Hamiltonian is obtained upto terms of ( π /mc ) . The series expansion of the or-bital part of the transformed Dirac Hamiltonian in termsof the kinetic momentum enables us to define the boostvelocity operator. According to the correspondence be-tween the kinetic momentum and the boost velocity op-erator, we found that up to terms of ( π /mc ) the Foldy-Wouthuysen transformed Dirac Hamiltonian is consistentwith the classical Hamiltonian with the gyromagnetic ra-tio given by g = 2. Furthermore, when the anomalousmagnetic moment is considered as well, we found that upto terms of ( π /mc ) the Foldy-Wouthuysen transformedDirac-Pauli Hamiltonian is in agreement with the classi-cal Hamiltonian with g = 2.The investigation in this paper reveals the fact thatthe classical Hamiltonian (classical relativistic Hamilto-nian plus the T-BMT Hamiltonian) must be the low-energy and weak-field limit of the Dirac-Pauli equation.As shown in the above sections, we can establish the con-nection order-by-order in the FW representation. More-over, this implies that, in the low-energy and weak-fieldlimit, there must exist an exact FW transformation thatcan block-diagonalize the Dirac-Pauli Hamiltonian to theform corresponding to the classical Hamiltonian. For afree particle, the exact unitary transformation has beenobtained by Foldy and Wouthuysen, which alternativelycan also be obtained by the order-by-order method. Wefound that the generating operators can be written asclosed forms, and consequently we can formally constructthe exact unitary transformation that block-diagonalizesthe Dirac and Dirac-Pauli Hamiltonians. However, itshould be emphasized that the exact unitary transfor-mation is valid only in the low-energy and weak-fieldlimit and existence of the exact unitary transformationdemands a rigours proof [24].On the other hand, it is true that even if the uni-tary FW transformation exists, it is far from unique, asone can easily perform further unitary transformationswhich preserve the block decomposition upon the block-diagonalized Hamiltonian (see also Sec. II). While dif-ferent block-diagonalization transformations are unitar-ily equivalent to one another and thus yield the samephysics, however, the pertinent operators σ , x , and p may represent very different physical quantities in differ-ent representations. To figure out the operators’ physicalinterpretations, it is crucial to compare the resulting FWtransformed Hamiltonian to the classical counterpart ina certain classical limit via the correspondence principle .In Kutzelnigg’s method, σ , x , and p simply representthe spin, position, and conjugate momentum of the par-ticle (as decoupled from the antiparticle) in the resulting7FW representation. In other words, Kutzelnigg’s methoddoes not give rise to further transformations that obscurethe operators’ interpretations other than block diagonal-ization.The correspondence we observed may be extended tothe case of inhomogeneous electromagnetic fields (exceptthat the Darwin term has no classical correspondence)[26], but inhomogeneity gives rise to complications whichmake it cumbersome to obtain the FW transformationin an order-by-order scenario, including the Kutzelniggmethod. We wish to tackle this problem in further re-search. Acknowledgments
The authors are grateful to C.-L. Chang for sharinghis calculations. T.W.C. would like to thank G. Y. Guo,R. Winkler and M.-C. Chang for valuable discussions.T.W.C. is supported by the National Science Council ofTaiwan under Contract No. NSC 101-2112-M-110-013-MY3; D.W.C. is supported by the Center for AdvancedStudy in Theoretical Sciences at National Taiwan Uni-versity.
Appendix A: Hermiticity of FW transformed DiracHamiltonian
Under the unitary transformation [Eq. (2.5)], theFoldy-Wouthuysen transformed Dirac Hamiltonian isgiven by the upper-left term of
U H D U † , which is H FW = (cid:0) Y h + + Y X † h (cid:1) Y + (cid:0) Y h + Y X † h − (cid:1) XY = Y (cid:0) h + + X † h + h X + X † h − X (cid:1) Y. (A1)Since the operators Y , h + and h are hermitian, it is easyto show that H F W also satisfies H F W = H † F W . The twooff-diagonal terms are given by H X = Z ( − Xh + + h − Xh X + h − X ) Y,H X † = Y (cid:0) − h + − X † h X † + h + X † h − (cid:1) Z. (A2)Equation (A1) can be further simplified by using H X = 0and H X † = 0.The brackets in the second equality of Eq. (A1) can berewritten as (cid:0) h + + X † h + h X + X † h − X (cid:1) = V + mc + X † h + h X + X † (cid:0) V − mc (cid:1) X = V + h X + mc (cid:0) − X † X (cid:1) + (cid:0) X † V X + X † h (cid:1) . (A3) On the other hand, we have (cid:0) X † V X + X † h (cid:1) = X † ( V X + h )= X † ([ V, X ] + XV + h )= X † (cid:0) mc X + Xh X + XV (cid:1) = 2 mc X † X + X † Xh X + X † XV, (A4)where Eq. (2.9) was used in the third equality of Eq. (A4).Substituting Eq. (A4) into Eq. (A3), we have (cid:0) h + + X † h + h X + X † h − X (cid:1) = V + h X + mc (cid:0) − X † X (cid:1) + 2 mc X † X + X † Xh X + X † XV = (cid:0) X † X (cid:1) V + (cid:0) X † X (cid:1) h X + mc (cid:0) X † X (cid:1) = Y − (cid:0) V + h X + mc (cid:1) . (A5)Inserting Eq. (A5) into Eq. (A1), we obtain H FW = Y Y − (cid:0) V + h X + mc (cid:1) Y = mc + Y − ( V + h X ) Y. (A6)The condition H X † = 0 implies X † = 12 mc (cid:0) h − X † h X † + [ X † , V ] (cid:1) . (A7)Applying Eq. (A7) to Eq. (A1), we have H FW = Y (cid:0) h + + X † h + h X + X † h − X (cid:1) Y = Y (cid:2) V + X † h + mc (1 − X † X ) + (cid:0) X † V + h (cid:1) X (cid:3) Y = Y [ V + X † h + mc (1 − X † X ) + 2 mc X † X + V X † X + X † h X † X ] Y = Y (cid:0) V Y − + X † h Y − + mc Y − (cid:1) Y, (A8)where Eq. (A7) was used in the second equality. Weobtain H FW = mc + Y (cid:0) V + X † h (cid:1) Y − . (A9)Because Y , h and V are hermitian operators, this im-plies that the hermitian of Eq. (A6) is H † FW = mc + Y (cid:0) V + X † h (cid:1) Y − , and this is the same as Eq. (A9). Asa consequence, we have H † FW = H FW . Appendix B: Hermiticity of FW transformedDirac-Pauli Hamiltonian
In this appendix, we will show that the FW trans-formed Dirac-Pauli Hamiltonian can be written asEq. (6.7) and show that Eq. (6.7) is a hermitian oper-ator. Under the unitary transformation [Eq. (6.4)], the8Foldy-Wouthuysen transformed Dirac-Pauli Hamiltonianis given by the upper-left term of U H U † : H FW = Y (cid:16) H + + X † H † + H X + X † H − X (cid:17) Y , (B1)where H + = V + V B + mc , H = h + iV E and H = V − V B − mc . The operator h is h = c σ · π . Since theoperator Y is hermitian, it is easy to show that Eq. (B1)also satisfies H FW = H † FW . The two off-diagonal termsare required to vanish and they are given by − X H + + H † − X H X + H − X = 0 , (B2)and − H + X † − X † H † X † + H + X † H − = 0 . (B3)By multiplying X † on the left-hand side of Eq. (B2), wehave (cid:16) X † H † + X † H − X (cid:17) = X † X H + + X † X H X . (B4)Substituting Eq. (B4) into Eq. (B1) by eliminating (cid:16) X † H † + X † H − X (cid:17) , we obtain H FW = Y − ( H + + H X ) Y , (B5) where the definition of the operator Y = 1 / √ X † X was used. On the other hand, multiplying X on the right-hand side of Eq. (B3), we have (cid:0) H X + X † H − X (cid:1) = H + X † X + X † H † X † X . (B6)Substituting Eq. (B6) into Eq. (B1) and eliminating theterm (cid:0) H X + X † H − X (cid:1) , we obtain H FW = Y (cid:16) H + + X † H † (cid:17) Y − . (B7)Because Y is a hermitian operator and so is H + , thisimplies that the hermitian of Eq. (B5) is H † FW = Y (cid:16) H + X † H † (cid:17) Y − , which is the same as Eq. (A9). Asa consequence, we have H † FW = H FW . On the otherhand, H + + H X can be written as ( H + + H X ) = mc + V + V B + ( h + iV E ) X . Equation (B5) can besimplified as H FW = mc + e G / A e −G / , (B8)where the operators A and G are defined as A = V + h X + V B + iV E X and G = ln (cid:0) X † X (cid:1) , respectively. [1] P. A. M. Dirac, Proc. R. Soc. London , 610 (1928).[2] P. A. M. Dirac, Principles of Quantum Mechanics (Clarendon Oxford, 1982, Fourth Edition);[3] L. L. Foldy and S. A. Wouthuysen, Phys. Rev. , 29(1950).[4] P. O. L¨owding, J. Chem. Phys. , 1396 (1951); J. M.Luttinger and W. Kohn, Phys. Rev. , 896 (1955).[5] R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electrons and Hole Systems (Springer, 2003,First Edition); P. Strange,
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