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Exact form of the exponential Foldy-Wouthuysen transformation operator for an arbitrary-spin particle

The exact exponential Foldy-Wouthuysen transformation operator applicable for a particle with an arbitrary spin is derived. It can be successfully utilized for verifying any Foldy-Wouthuysen transformation method based on the exponential operator. When a verified method is relativistic, the relativistic exponential operator should be expanded in the semirelativistic power series. The use of the obtained exponential operator for a derivation of the Foldy-Wouthuysen Hamiltonian makes it possible to check the validity of any other method of the Foldy-Wouthuysen transformation, while this possibility may need a greater computational effort.

The Foldy-Wouthuysen (FW) transformation [1] restoring the Schrödinger form of relativistic wave equations is one of the basic methods of contemporary quantum mechanics (QM). An importance of the FW transformation for physics has significantly increased nowadays due to the great progress of the art of analytic computer calculations. A great advantage of the FW representation is the simple form of operators corresponding to classical observables. In this representation, the Hamiltonian and all operators are even, i.e., block-diagonal (diagonal in two spinors). 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. The possibility of such a replacement, explicitly or implicitly used in practically all works devoted to the FW transformation, has been rigorously proved for the stationary case in Ref. [2]. Thanks to these properties, the FW representation provides the best possibility of obtaining a meaningful classical limit of relativistic QM not only for the stationary case [1][2][3][4] but also for the nonstationary one [5,6].
The general form of an initial Hamiltonian for arbitrary-spin particles is given by [39] The even operators M and E and the odd operator O are diagonal and off-diagonal in two spinors, respectively. Equation (1) is applicable for a particle with any spin if the number of components of a corresponding wave function is equal to 2(2s + 1), where s is the spin quantum number. For a Dirac particle, the M operator usually reduces to the particle rest energy mc 2 : The Hamiltonian H is Hermitian for fermions and pseudo-Hermitian (more exactly, βpseudo-Hermitian, H = H ‡ ≡ βH † β) for bosons. We assume that the operators βM, E, and O also possess this property. The transformation operator for bosons is therefore β- We mention the existence of bosonic symmetries of the Dirac equation [40].
The FW transformation operator can be presented in the two forms, nonexponential (U F W ) and exponential (S F W ): This operator transforms the initial Hamiltonian H to the FW representation: The FW Hamiltonian obtained with this operator is even. The FW transformation vanishes either lower or upper spinor for positive and negative energy states, respectively.
There is an infinite set of representations different from the FW representation whose distinctive feature is a block-diagonal form of the Hamiltonian. The FW transformation is uniquely defined by the condition that the exponential operator S F W is odd, and Hermitian [41,42] (β-pseudo-Hermitian for bosons). This condition is equivalent to [41,42] Eriksen [41] has found the exact expression for the nonexponential FW transformation operator. It is convenient to present this expression in the form [4] The initial Hamiltonian operator H is arbitrary. It is easy to see that [41] where [. . . , . . . ] means a commutator.
The equivalent form of the operator U E [4] shows that it is properly unitary (βpseudounitary for bosons): The Eriksen formula is an important achievement of the theory of the FW transformation.
However, the FW transformation method proposed by Eriksen [41] is semirelativistic. We use the term "semirelativistic" for methods applying an expansion of a derived block-diagonal Hamiltonian in a series of even terms of ascending order in 1/c. For the semirelativistic and the relativistic methods, the zeroth order Hamiltonian is the Schrödinger one and the relativistic FW Hamiltonian of a free particle, respectively. The Eriksen method is not practically used in specific calculations. Since the exact equation (7) contains the square roots of Dirac matrices, it excludes a possibility to obtain a series of relativistic terms with the relativistic FW Hamiltonian of a free particle [1] as the zero-order approximation (see Ref. [43] for more details). Contemporary methods of the FW transformation are relativistic. The use of all semirelativistic methods is restricted due to their divergence at large momenta, when p/(mc) > 1. In this case (which takes place for an electron near a nucleus), the semirelativistic methods become inapplicable [44] (see also Ref. [43]).
Nevertheless, the calculation of the FW Hamiltonian by the perfectly substantiated Eriksen method [45][46][47] is very important for checking results obtained by other semirelativistic and relativistic methods (see the examples given in Ref. [43]).
We should mention that the condition (5) is used much more often than the condition (6) to check the correspondence of the final Hamiltonian to the FW representation. For example, this is one of thebasic conditions for the generalized Douglas-Kroll-Hess method which is rather widely applied in quantum chemistry [31,44,48,49]. This fact obviously demonstrates the importance of the exact derivation of the exponential FW transformation operator. However, the problem of the exact form of S F W was never investigated. In the present study, we solve this problem.
Evidently, the exact form of S F W should be based on the operator λ. It is convenient to use the following relations (see, e.g., Ref. [47]): To obtain an explicit expression for S F W , it is instructive to consider the special case the odd terms are eliminated and the Hamiltonian H is transformed to the FW representation [47]. In the considered special case, and the operator λ is determined exactly [50]: Equation (11) has two solutions, Θ 1 and Θ 2 , differing in π/2 [10]. Since they are defined by the relations As cos 2Θ = 1 − tan 2 Θ 1 + tan 2 Θ , sin 2Θ = tan 2Θ cos 2Θ, the needed trigonometrical operators are given by This equation shows that cos 2Θ 1 > 0, cos 2Θ 2 < 0.
Thus, there are two unitary transformations of the operator H to an even form. They are characterized by the angles Θ 1 and Θ 2 , where the angle Θ 1 corresponds to the FW transformation. As a result of the both transformations, one of the spinors (lower for Θ 1 and upper for Θ 2 ) becomes zero as for free particles [10]. Since we consider the FW transformation, Θ = Θ 1 .
A comparison with Eq. (11) shows that The corresponding exponential FW transformation operator is given by Let us check that Eqs. (16) and (17) obtained in the above-mentioned special case remain valid in the general case of a relativistic particle in arbitrary stationary fields. We can utilize the operator relations (cf. Ref. [47]) sin Θ = sin 2Θ 2 (1 + cos 2Θ) = sin 2Θ cos Θ = 1 + cos 2Θ 2 (1 + cos 2Θ) These relations repeat the corresponding trigonometrical ones. We can mention that the FW transformation operator U F W is equal to unit when the initial Hamiltonian does not contain odd terms (O = 0).
Equation (17)  Indeed, a calculation of a semirelativistic series for the exponential operator is straightforward.
The numerator and denominator in the formula for λ commute. Therefore, it is convenient to present this quantity in the form where {. . . , . . . } means an anticommutator. Then, we apply the expansion of the square root in the power series. It is important that this procedure can be performed for arbitrary-spin particles because the initial Hamiltonians can be presented in the form (2). For example, the operator M for a spin-1 particle in a magnetic field reads [38,46] where S is the spin matrix for the spin-1 particle, π = p − (e/c)A is the kinetic momentum operator, A is the vector potential, and e is the charge of a particle. For the electron, it is negative (e = −|e|). The two last terms in Eq. (23) can be added to the operator E and therefore the operator M can be reduced to mc 2 .
For analytic calculations with computer, one can use the formula and the well-known expansions The result of the expansion in the power series can be written as follows: where q E and q O denote the sums of even and odd terms, respectively. The resulting expansion of the operator (λ − βλβ)/2 in the semirelativistic power series is given by The final equation for the exponential FW transformation operator takes the form This operator can be calculated with any necessary precision while the computational effort depends on this precision.
We should also add that the FW Hamiltonian can be calculated with the exponential operator as follows (cf. Ref. [1]): where Unlike the original FW approach [1], this approach needs not subsequent iterations.
This result allows one to verify any other FW transformation method which is not based on the exponential operator. For this purpose, one has to expand all terms of the relativistic series for the FW Hamiltonian calculated by the checked method in powers of E/(mc 2 ), O/(mc 2 ) and then has to compare the obtained expression with the FW Hamiltonian given by Eqs. (29) and (30). The computational effort, of course, grows when the maximum powers increase. However, any error usually manifests itself in noncoincidence of the first few terms (see, for example, Refs. [43,51]).
Let us verify the original method by Foldy and Wouthuysen [1] as an example of the application of Eqs. (24) - (28). A calculation of the exponential FW transformation operator in the stationary case with allowance for all terms up to the order of m −4 results in The original method [1] belongs to iterative methods. The result of successive iterations expressed by the equation To verify the original method [1], it is sufficient to take into account three first iterations and to hold terms up to the order of m −3 . In the considered stationary case, they are given by [7,9,43] The result of these iterations can be approximately presented as follows (see Ref. [43]): The operator S differs from S F W due to the even last term. Therefore, the original method of the FW transformation [1] does not lead to the FW representation. This paradoxical fact has been first mentioned by Eriksen and Korlsrud [42] (see also Refs. [9,43,45,46]).
The original method [1] can be corrected due to an additional transformation [43]. This correction allows one to reach the FW representation.
Any representation which block-diagonalizes the Hamiltonian but differs from the FW representation can be successfully used for a calculation of energy spectrum of particles in stationary states. However, an application of such a representation for other purposes is restricted. In particular, the representation which differs from the FW one is, at least, inconvenient for description of spin processes (see, for example, Refs. [4,18,52]).
Thus, we have derived the exact exponential FW transformation operator which is applicable for a particle with an arbitrary spin. This result provides an opportunity to verify any FW transformation method based on the exponential operator. Moreover, the validity of any other FW transformation method can also be checked. For this purpose, one has to use Eq. (29) in order to calculate the FW Hamiltonian and to compare it with the corresponding Hamiltonian found by the checked method. The latter possibility may need a greater computational effort.
Ministry for Science and Technology (Bundesministerium für Bildung und Forschung).