What is the relativistic spin operator?

Although the spin is regarded as a fundamental property of the electron, there is no universally accepted spin operator within the framework of relativistic quantum mechanics. We investigate the properties of different proposals for a relativistic spin perator. It is shown that most candidates are lacking essential features of proper angular momentum operators, leading to spurious Zitterbewegung (quivering motion) or violating the angular momentum algebra. Only the Foldy-Wouthuysen operator and the Pryce operator qualify as proper relativistic spin operators. We demonstrate that ground states of highly charged hydrogen-like ions can be utilized to identify a legitimate relativistic spin operator experimentally.


Introduction
Quantum mechanics forms the universally accepted theory for the description of physical processes on the atomic scale. It has been validated by countless experiments and is used in many technical applications. However, even today quantum mechanics presents physicists with some conceptual difficulties. In particular, the concept of spin is related to such difficulties and myths [1,2]. Although there is consensus that elementary particles have a quantum mechanical property called spin, the understanding of the physical nature of the spin is still incomplete [3].
Historically, the concept of spin was introduced in order to explain some experimental findings such as the emission spectra of alkali metals and the Stern-Gerlach experiment. A direct measure of the spin (or more precisely the electronʼs magnetic moment) was, however, missing until the pioneering work by Dehmelt [4]. Nevertheless, spin measurement experiments [5][6][7][8][9][10] still require sophisticated methods. Pauli and Bohr even claimed that the spin of free electrons was impossible to measure for fundamental reasons [11]. Recent renewed interest in the fundamental aspects of the spin arose, for example, from the growing field of (relativistic) quantum information [12][13][14][15][16][17], quantum spintronics [18], spin effects in graphene [19][20][21] and in light-matter interaction at relativistic intensities [22][23][24].
According to the formalism of quantum mechanics, each measurable quantity is represented by a Hermitian operator. Taking the experiments that aim to measure bare electron spins seriously, we have to ask the question: what is the correct (relativistic) spin operator? Although the spin is regarded as a fundamental property of the electron, a universally accepted spin operator for the Dirac theory is still missing. The pivotal question we try to tackle is: which mathematical operator corresponds to an experimental spin measurement? This question may be answered by comparing the experimental results with the theoretical predictions originating from different spin operators, and testing which operator is compatible with the experimental data.
A relativistic spin operator may be introduced by splitting the undisputed total angular momentum operatorĴ into an external partL and an internal partŜ, commonly referred to as the orbital angular momentum and the spin, viz.ˆ=ˆ+Ĵ L S. The question for the right splitting of the total angular momentum into an orbital part and a spin part is closely related to the quest for the right relativistic position operator [25][26][27]. This becomes evident by writingˆ=ˆ×L r p with the position operatorr and the kinematic momentum operatorp, which is in the atomic units as used in this paper,   = − p i . Thus, different definitions of the spin operatorŜ induce different relativistic position operators,r.
Introducing the position vector r and the operator Σ with i j k ( , , ) being a cyclic permutation of (1, 2, 3) and the matrices α T obeying the algebra the operator of the relativistic total angular momentum is given by Σ = ×ˆ+Ĵ r p 2. Thus, the most obvious way of splittingĴ is to define the orbital angular momentum operatorˆ= ×L r p P and the spin operator Σ =Ŝ 2 P , which is a direct generalization of the orbital angular momentum operator and the spin operator of the nonrelativistic Pauli theory. This naive splitting, however, suffers from several problems, e.g.L P andŜ P do not commute with the free Dirac Hamiltonian nor with the Dirac Hamiltonian for central potentials. Thus, in contrast to classical and nonrelativistic quantum theory, the angular momentaL P andŜ P are not conserved. This has consequences, e.g. for the labeling of the eigenstates of the hydrogen atom. In nonrelativistic theory, bound hydrogen states may be constructed as simultaneous eigenstates of the Pauli-Coulomb Hamiltonian, the squared orbital angular momentum, the z-components of the orbital angular momentum and the spin. In the Dirac theory, however, the squared total angular momentumĴ 2 , the total angular momentum in the z-directionĴ 3 , and the so-called spinorbit operatorK (or the parity) are utilized [28,29]. In particular, it is not possible to construct simultaneous eigenstates of the Dirac-Coulomb Hamiltonian and some component ofŜ P .

Relativistic spin operators
To overcome conceptual problems with the naive splitting ofĴ intoL P andŜ P , several alternatives for a relativistic spin operator have been proposed. However, there is no single commonly accepted relativistic spin operator, leading to the unsatisfactory situation that the relativistic spin operator is not unambiguously defined. We will investigate the properties of different popular definitions of the spin operator that result from different splittings ofĴ with the aim of finding means that allow us to identify the legitimate relativistic spin operator by experimental methods. Table 1 summarizes various proposals for a relativistic spin operatorŜ. These operators are often motivated by abstract group theoretical considerations rather than by experimental evidence. For example, Wigner showed in his seminal work [54][55][56] that the spin degree of freedom can be associated with irreducible representations of the sub-group of the inhomogeneous Lorentz group that leaves the four-momentum invariant. We will denote individual components ofŜ byŜ i with index ∈ } { i 1, 2, 3 . The spin operators are defined in terms of the particleʼs rest mass m 0 , the speed of light c, the matrix β such that i i 2 the free particle Dirac Hamiltonian In the nonrelativistic limit, i.e. when the plane wave expansion of a wave packet has only components with momenta that are small compared to m c 0 , the expectation values for all operators in table 1 converge to the same value. Note that the nomenclature in table 1 is not universally adopted in the literature and other authors may utilize different operator names. Furthermore, the spin operators can be formulated by various different but algebraically equivalent expressions. For example, the so-called Gürsey-Ryder operator in [46,47] is equivalent to the Chakrabarti operator of table 1.
One may conclude that an operator can not be considered as a relativistic spin operator if it does not inherit the key properties of the nonrelativistic Pauli spin operator. In particular, we demand the following features from a proper relativistic spin operator.
(i) It is required to commute with the free Dirac Hamiltonian.
with ε i j k , , denoting the Levi-Civita symbol. The first property is required to ensure that the relativistic spin operator is a constant of motion if forces are absent, such that spurious Zitterbewegung of the spin is prevented. The second requirement is commonly regarded as the fundamental property of angular momentum operators of spin-half particles [57]. The physical quantity that is represented by the operatorŜ should not depend on the orientation of the chosen coordinate system. This can be ensured by fulfilling [57] εˆ=⎡ The angular momentum algebra (6) and the relation (7) determine the properties of the spin and the orbital angular momentum as well as the relationship between them. As a consequence of (7), the orbital angular momentumˆ=ˆ−L J S that is induced by a particular choice of the spin . Thus,L is a physical vector operator, too. AsL represents an angular momentum operator, it must obey the angular momentum algebra. Furthermore, we may say that the total angular momentumĴ is split into an internal partŜ and an external partL only if internal and external angular momenta can be measured independently, i.e.Ŝ andL commute.
follow from (7). All spin operators in table 1 fulfill (7). The Czachor spin operatorŜ Cz , the Frenkel spin operatorŜ F , and the Fradkin-Good operatorŜ FG , are however, disqualified as relativistic spin operators by violating the angular momentum algebra (6). Furthermore, the Pauli spin operatorŜ P and the Chakrabarti spin operatorŜ Ch do not commute with the free Dirac Hamiltonian, ruling them out as meaningful relativistic spin operators. According to our criteria, only the Foldy-Wouthuysen spin operatorŜ FW and the Pryce spin operatorŜ Pr remain as possible relativistic spin operators.

Electron spin of hydrogen-like ions
The question of which of the proposed relativistic spin operators (if any) in table 1 provides the correct mathematical description of spin can be answered definitely only by comparing theoretical predictions with experimental results. For this purpose, one needs a physical setup that shows strong relativistic effects and is as simple as possible. Such a setup is provided by the bound eigenstates of highly charged hydrogen-like ions, i.e. atomic systems with an atomic core of Z protons and a single electronic charge. These ions can be produced at storage rings [58] or by utilizing electron beam ion traps [59,60] C n j m n j m , , , , , , n j m n j m 2 , , , , n j m n j m 3 , , , , , , n j m n j m , , , , The eigenenergies are given with α el denoting the fine structure constant by In order to establish a close correspondence between the nonrelativistic Schrödinger-Pauli theory and the relativistic Dirac theory, one may desire to find a splitting ofĴ into a sum =ˆ+Ĵ L S of commuting operators such that bothL andŜ (i) fulfill the angular momentum algebra, and (ii) form a complete set of commuting operators that containsĤ C as well asŜ 3 and/ orL 3 .
The latter property would ensure that all hydrogenic energy eigenstates are spin eigenstates and/or orbital angular momentum eigenstates, too. Such hypothetical eigenstates would be superpositions of ψ κ n j m , , , of the same energy. Consequently, these superpositions are eigenstates ofĴ 2 , too, because the energy (15) depends on the principal quantum number n as well as the quantum number j. Thus, any complete set of commuting operators for specifying hydrogenic quantum states necessarily includesĴ

An experimental test for relativistic spin operators
Theoretical considerations have led to several proposals for a relativistic spin operator, as illustrated in table 1. The identification of the correct relativistic spin operator, however, demands an experimental test. The inequality (17) may serve as a basis for such an experimental test. More precisely, the inequality (17) allows falsification of the hypothesis that the spin measurement procedure is an experimental realization of some operatorŜ, whereŜ is one of the operators in table 1. In this test, the electron of a highly charged hydrogen-like ion is prepared in its ground state ψ ↑ first, e.g. by exposing the ion to a strong magnetic field in the z-direction and turning it off adiabatically. (Preparing a superposition of ψ ↑ and ψ ↓ will reduce the sensitivity of the experimental test.) Afterwards, the spin will be measured along the z-direction, e.g. by a Stern-Gerlach-like experiment, yielding the experimental expection value s. . excluded as a relativistic spin operator by experimental evidence. In particular, realizing full spin-polarization, i.e. = ± s 1 2, eliminates all operators in table 1 except the Pryce operator.

Conclusions
We investigated the properties of various proposals for a relativistic spin operator. Only the Fouldy-Wouthuysen operator and the Pryce operator fulfill the angular momentum algebra, and are constants of motion in the absence of forces. While different theoretical considerations lead to different spin operators, the definite relativistic spin operator has to be justified by experimental evidence. The energy eigenstates of highly charged hydrogen-like ions, in particular the ground states, can be utilized to exclude candidates for a relativistic spin operator experimentally. The proposed spin operators predict different maximal degrees of spin polarization. Only the Pryce spin operator allows for a complete polarization of spin in the hydrogenic ground state.