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What is the relativistic ?

1, 1 1 1, 2 Heiko Bauke, ∗ Sven Ahrens, Christoph H. Keitel, and Rainer Grobe 1Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany 2Intense Theory Unit and Department of Physics, Illinois State University, Normal, IL 61790-4560 USA (Dated: April 14, 2014)

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

Keywords: spin, relativistic , hydrogen-like ions

PACS numbers: 03.65.Pm, 31.15.aj, 31.30.J-

Introduction Quantum mechanics forms the universally A relativistic spin operator may be introduced by splitting accepted theory for the description of physical processes on the undisputed total angular Jˆ into an the atomic scale. It has been validated by countless experi- external part Lˆ and an internal part Sˆ, commonly referred to as ments and it is used in many technical applications. However, the orbital angular momentum and the spin, viz. Jˆ = Lˆ +Sˆ. The quantum mechanics presents with some conceptual question for the right splitting of the total angular momentum difficulties even today. In particular, the concept of spin is into an orbital part and a spin part is closely related to the related to such difficulties and myths [1, 2]. Although there is quest for the right relativistic [25–27]. This consensus that elementary particles have some quantum me- becomes evident by writing Lˆ = rˆ pˆ with the position operator × chanical property that is called spin, the understanding of the rˆ and the kinematic momentum operator pˆ, which is in atomic physical of the spin is still incomplete [3]. units as used in this paper pˆ = i∇. Thus, different definitions Sˆ − ff Historically, the concept of spin was introduced in order of the spin operator induce di erent relativistic position r to explain some experimental findings such as the emission operators ˆ. r Σˆ = spectra of alkali metals and the Stern-Gerlach experiment. A Introducing the position vector and the operator Σˆ , Σˆ , Σˆ T direct measuring of the spin (or more precisely the electron’s ( 1 2 3) via ), however, was missing until the pioneering Σˆi = iα jαk (1) work by Dehmelt [4]. Nevertheless, spin measurement exper- − iments [5–10] still require sophisticated methods. Pauli and with (i, j, k) being a cyclic permutation of (1, 2, 3) and the T Bohr even claimed that the spin of free was impossi- matrices (α1, α2, α3) = α obeying the algebra

ble to measure for fundamental reasons [11]. Recent renewed 2 interest in fundamental aspects of the spin arose, for example, αi = 1 , αiαk + αkαi = 2δi,k , (2) from the growing field of (relativistic) the operator of the relativistic total angular momentum is given ff [12–17], quantum [18], spin e ects in graphene by Jˆ = r pˆ + Σˆ /2. Thus, the most obvious way of splitting Jˆ [19–21] and in light- interaction at relativistic intensities × is to define the orbital angular momentum operator Lˆ P = r pˆ arXiv:1303.3862v2 [quant-ph] 11 Apr 2014 [22–24]. × and the spin operator SˆP = Σˆ /2, which is a direct generaliza- According to the formalism of quantum mechanics, each tion of the orbital angular momentum operator and the spin measurable quantity is represented by a Hermitian operator. operator of the nonrelativistic Pauli theory. This naive splitting, Taking the experiments that aim to measure bare electron spins however, suffers from several problems, e. g., Lˆ P and SˆP do not seriously, we have to ask the question: What is the correct commute with the free Dirac Hamiltonian nor with the Dirac (relativistic) spin operator. Although the spin is regarded as a Hamiltonian for central potentials. Thus, in contrast to classi- fundamental property of the electron, a universally accepted cal and nonrelativistic quantum theory the angular momenta spin operator for the Dirac theory is still missing. The pivotal Lˆ P and SˆP are not conserved. This has consequences, e. g., for question we try to tackle is: Which mathematical operator cor- the labeling of the eigenstates of the hydrogen . In nonrel- responds to an experimental spin measurement? This question ativistic theory, bound hydrogen states may be constructed as may be answered by comparing experimental results with theo- simultaneous eigenstates of the Pauli-Coulomb Hamiltonian, retical predictions originating from different spin operators and the squared orbital angular momentum, the z-components of testing which operator is compatible with experimental data. the orbital angular momentum and the spin. In the Dirac theory, 2

2 however, the squared total angular momentum Jˆ , the total an- with εi, j,k denoting the Levi-Civita symbol. gular momentum in z-direction Jˆ3, and the so-called spin- operator Kˆ (or the ) are utilized [28, 29]. In particular, The first property is required to ensure that the relativistic spin it is not possible to construct simultaneous eigenstates of the operator is a constant of motion if are absent, such that spurious Zitterbewegung of the spin is prevented. The Dirac-Coulomb Hamiltonian and some component of SˆP. Relativistic spin operators To overcome conceptual requirement is commonly regarded as the fundamental property problems with the naive splitting of Jˆ into Lˆ and Sˆ , several of angular momentum operators of spin-half particles [57]. The P P ˆ alternatives for a relativistic spin operator have been proposed. physical quantity that is represented by the operator S should However, there is no single commonly accepted relativistic not depend on the orientation of the chosen coordinate system. spin operator, leading to the unsatisfactory situation that the This can be ensured by fulfilling [57] relativistic spin operator is not unambiguously defined. We [Jˆ, Sˆ ] = iε Sˆ . (7) will investigate the properties of different popular definitions i j i, j,k k of the spin operator which result from different splittings of Jˆ The angular momentum algebra (6) and the relation (7) de- with the aim to find means that allow to identify the legitimate termine the properties of the spin and the orbital angular mo- relativistic spin operator by experimental methods. mentum as well as the relationship between them. As a con- Table 1 summarizes various proposals for a relativistic spin sequence of (7), the orbital angular momentum Lˆ = Jˆ Sˆ operator Sˆ. These operators are often motivated by abstract − that is induced by a particular choice of the spin obeys theoretical considerations rather than by experimental [Jˆ, Lˆ ] = iε Lˆ . Thus, Lˆ is a physical vector operator, too. evidence. For example, Wigner showed in his seminal work i j i, j,k k As Lˆ represents an angular momentum operator, it must obey [54–56] that the spin degree of freedom can be associated with the angular momentum algebra. Furthermore, we may say that irreducible representations of the sub-group of the inhomoge- the total angular momentum Jˆ is split into an internal part Sˆ neous that leaves the four-momentum invariant. and an external part Lˆ only if internal and external angular We will denote individual components of Sˆ by Sˆ with index i momenta can be measured independently, i. e., Sˆ and Lˆ com- i 1, 2, 3 . The spin operators are defined in terms of the ∈ { } mute. Both conditions are fulfilled if, and only if, the spin particle’s rest mass m , the c, the matrix β such 0 operator Sˆ satisfies the angular momentum algebra (6) because that the relations 2 β = 1 , αiβ + βαi = 0 , (3) [Lˆ , Lˆ ] = iε Lˆ + [Sˆ , Sˆ ] iε Sˆ , (8) i j i, j,k k i j − i, j,k k the free particle Dirac Hamiltonian [Lˆ , Sˆ ] = iε Sˆ [Sˆ , Sˆ ] (9) i j i, j,k k − i j Hˆ = cα pˆ + m c2β , (4) 0 · 0 follow from (7). All spin operators in Tab. 1 fulfill (7). The ˆ ˆ and the operator Czachor spin operator SCz, the Frenkel spin operator SF, and the Fradkin-Good operator SˆFG, however, are disqualified as 2 2 2 1/2 relativistic spin operators by violating the angular momentum pˆ0 = (m0c + pˆ ) . (5) algebra (6). Furthermore, the Pauli spin operator SˆP and the In the nonrelativistic limit, i. e., when the plane wave expansion Chakrabarti spin operator SˆCh do not commute with the free of a wave packet has only components with momenta which are Dirac Hamiltonian, ruling them out as meaningful relativis- small compared to m0c, expectation values for all operators in tic spin operators. According to our criteria, only the Foldy- Tab. 1 converge to the same value. Note that the nomenclature Wouthuysen spin operator SˆFW and the Pryce spin operator SˆPr in Tab. 1 is not universally adopted in the literature and other remain as possible relativistic spin operators. authors may utilize different operator names. Furthermore, Electron spin of hydrogen-like ions The question of the spin operators can be formulated by various different but which of the proposed relativistic spin operators (if any) in algebraically equivalent expressions. For example, the so- Tab. 1 provides the correct mathematical description of spin called Gürsey-Ryder operator in [46, 47] is equivalent to the can be answered definitely only by comparing theoretical pre- Chakrabarti operator of Tab. 1. dictions with experimental results. For this purpose one needs One may conclude that an operator can not be considered as a physical setup that shows strong relativistic effects and is a relativistic spin operator if it does not inherit the key proper- as simple as possible. Such a setup is provided by the bound ties of the nonrelativistic Pauli spin operator. In particular, we eigenstates of highly charged hydrogen-like ions, i. e., atomic demand from a proper relativistic spin operator the following systems with an atomic core of Z and a single elec- features: tronic charge. These ions can be produced at storage rings [58] = 1. It is required to commute with the free Dirac Hamiltonian. or by utilizing electron beam ion traps [59, 60] up to Z 92 (hydrogen-like uranium). The degenerate bound eigenstates of 2. A spin operator must feature the two eigenvalues 1/2 and the corresponding Coulomb-Dirac Hamiltonian ± it has to obey the angular momentum algebra Z HˆC = Hˆ0 (10) [Sˆ i, Sˆ j] = iεi, j,kSˆ k (6) − r | | 3

TABLE 1: Definitions and commutation properties of various relativistic spin operators.

[Hˆ 0, Sˆ][Sˆ i, Sˆ j] Eigenvalues Operator name Definition = 0? = iεi, j,kSˆ k? = 1/2? 1 ± Pauli [30–32] Sˆ = Σˆ no yes yes P 2 1 iβ pˆ (Σˆ pˆ) Foldy-Wouthuysen [33–37] SˆFW = Σˆ + pˆ α × × yes yes yes 2 2p ˆ0 × − 2p ˆ0(p ˆ0 + m0c) m2c2 im cβ pˆ Σˆ Sˆ = 0 Σˆ + 0 p α + p Czachor [38] Cz 2 2 ˆ · 2 ˆ yes no no 2p ˆ0 2p ˆ0 × 2p ˆ0 1 iβ Frenkel [39–41] SˆF = Σˆ + pˆ α yes no no 2 2m0c × 1 i pˆ (Σˆ pˆ) Chakrabarti [42–48] SˆCh = Σˆ + α pˆ + × × no yes yes 2 2m0c × 2m0c(m0c + pˆ0) 1 1 pˆ Pryce [47, 49–52] Sˆ = βΣˆ + Σˆ pˆ(1 β) yes yes yes Pr 2 2 · − pˆ2 ! 1 1 Hˆ pˆ Sˆ = βΣˆ + Σˆ p 0 β Fradkin-Good [46, 53] FG ˆ 2 yes no yes 2 2 · cpˆ0 − pˆ

are commonly expressed as simultaneous eigenstates ψn, j,m,κ [61] excluding Sˆ 3 and Lˆ 3 from any complete set of commut- 2 of HˆC, Jˆ , Jˆ3, and the so-called spin-orbit operator Kˆ = ing operators for specifying relativistic hydrogenic eigenstates. β Σˆ [r ( i∇) + 1)] fulfilling the eigenequations [28, 29] In conclusion, hydrogenic eigenstates are generally { · × − } not eigenstates of any spin operator that fulfills the angular Hˆ ψ = (n, j)ψ n = 1, 2,..., (11) C n, j,m,κ E n, j,m,κ momentum algebra. ˆ2 1 3 1 In momentum space, the relativistic spin operators intro- J ψn, j,m,κ = j( j + 1)ψn, j,m,κ j = 2 , 2 ,..., n 2 , (12) − duced in Tab. 1 are simple matrices. Thus, by employ- Jˆ3ψn, j,m,κ = mψn, j,m,κ m = j, ( j 1),..., j , (13) − − ing the momentum space representation of ψn, j,m,κ, spin ex- Kˆ ψ = κψ κ = j 1 , j + 1 . (14) n, j,m,κ n, j,m,κ − − 2 2 pectation values of the degenerate hydrogenic ground states ψ = ψ1,1/2,1/2,1 and ψ = ψ1,1/2, 1/2, 1 can be evaluated [62]. The eigenenergies are given with α denoting the fine structure ↑ ↓ − − el For simplicity, we measure spin along the z-direction for the re- constant by minder of this section. The spin expectation value of a general    1/2 iζ − superposition ψ = cos(η/2)ψ + sin(η/2)e ψ of the hydro-   α2 Z2  ↑ ↓ 2   el  genic ground states ψ and ψ is given by (n, j) = m0c 1 +  q  . ↑ ↓ E   2 2 2   n j 1/2 + ( j 1/2) α Z  D E el 2 η 2 η − − − − ψ Sˆ 3 ψ = cos ψ Sˆ 3 ψ + sin ψ Sˆ 3 ψ (15) 2 h ↑| | ↑i 2 h ↓| | ↓i In order to establish a close correspondence between the η η + 2 cos sin cos ζ Re ψ Sˆ 3 ψ . (16) nonrelativistic Schrödinger-Pauli theory and the relativistic 2 2 h ↑| | ↓i ˆ Dirac theory, one may desire to find a splitting of J into a sum For all spin operators introduced in Tab. 1, the mixing Jˆ = Lˆ + Sˆ of commuting operators such that both Lˆ and Sˆ term Re ψ Sˆ 3 ψ vanishes and, furthermore, ψ Sˆ 3 ψ = h ↑| | ↓i h ↑| | ↑i ψ Sˆ 3 ψ > 0. Thus, the expectation value (16) is maximal 1. fulfill the angular momentum algebra, and − h ↓| | ↓i for η = 0 and minimal for η = π and the inequality 2. form a complete set of commuting operators that contains Hˆ as well as Sˆ and/or Lˆ . ψ Sˆ 3 ψ ψ Sˆ 3 ψ ψ Sˆ 3 ψ (17) C 3 3 h ↓| | ↓i ≤ h | | i ≤ h ↑| | ↑i The latter property would ensure that all hydrogenic energy holds for all hydrogenic ground states ψ. eigenstates are spin eigenstates and/or orbital angular momen- For every proposed spin operator in Tab. 1 we get different tum eigenstates, too. Such hypothetical eigenstates would be values for the upper and lower bounds in (17). The spin ex- superpositions of ψn, j,m,κ of the same energy. Consequently, pectation values ψ Sˆ 3 ψ and implicitly ψ Sˆ 3 ψ for the h ↑| | ↑i h ↓| | ↓i these superpositions are eigenstates of Jˆ2, too, because the operators of Tab. 1 are displayed as a of the atomic energy (15) depends on the principal n as number Z in Fig. 1. None of the spin operators in Tab. 1 com- well as the quantum number j. Thus, any complete set of com- mutes with HˆC. Thus, the expectation values ψ Sˆ 3 ψ and h ↑| | ↑i muting operators for specifying hydrogenic quantum states ψ Sˆ 3 ψ generally do not equal one of the eigenvalues of Sˆ 3. h ↓| | ↓i necessarily includes Jˆ2. As a consequence of the postulated For small atomic numbers (Z < 20), all spin operators yield angular momentum algebra for Lˆ and Sˆ, the operator Jˆ2 com- about 1/2. For larger Z, however, expectation values differ 2 2 ± mutes with Lˆ as well as with Sˆ , but with neither Sˆ 3 nor Lˆ 3 significantly from each other. In particular, spin expectation 4

imental evidence. In particular, realizing full spin-, 0.55 i. e., s = 1/2, eliminates all operators in Tab. 1 except the ± Pryce operator. 0.50 Conclusions We investigated the properties of various 0.45 proposals for a relativistic spin operator. Only the Fouldy- i ↑

ψ Wouthuysen operator and the Pryce operator fulfill the angular | 3 ˆ S 0.40 | ˆ ˆ momentum algebra and are constants of motion in the absence ↑ SP,3 SCh,3 ψ

h of forces. While different theoretical considerations led to dif- ˆ ˆ 0.35 SFW,3 SPr,3 ferent spin operators, the definite relativistic spin operator has SˆCz,3 SˆFG,3 0.30 to be justified by experimental evidence. Energy eigenstates SˆF,3 of highly charged hydrogen-like ions, in particular the ground 0.25 states, can be utilized to exclude candidates for a relativistic 0 20 40 60 80 100 120 Z spin operator experimentally. The proposed spin operators predict different maximal degrees of . Only the Pryce spin operator allows for a complete polarization of FIG. 1: Spin expectation values ψ Sˆ 3 ψ of various relativistic spin h ↑| | ↑i spin in the hydrogenic . operators for the hydrogenic ground state ψ as a function of the ↑ atomic number Z measured in z-direction. Spin expectation values We have enjoyed helpful discussions with Prof. C. Müller, S. Meuren, Prof. Q. Su and E. Yakaboylu. R. G. acknowledges for ψ follow by via ψ Sˆ 3 ψ = ψ Sˆ 3 ψ . ↓ h ↑| | ↑i − h ↓| | ↓i the warm hospitality during his sabbatical leave in Heidelberg. This work was supported by the NSF. values differ from 1/2 even for spin operators with eigenval- ± ues 1/2. This means, it is possible to discriminate between ± different relativistic spin operator candidates. The spin expec- tation value’s magnitude decreases with growing Z when the ∗ [email protected] Pauli, the Fouldy-Wouthuysen, the Czachor, the Chakrabarti, [1] H. Nikolic,´ Foundations of Physics 37, 1563 (2007). or the Fradkin-Good spin operator is applied. The Frenkel spin [2] D. Giulini, Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics 39, 557 operator yields spin expectation values with modulus exceed- (2008). ing 1/2 which is due to the violation of the angular momentum [3] M. Morrison, Studies In History and Philosophy of Science Part algebra. 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