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What Is the Relativistic Spin Operator?

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What Is the Relativistic Spin Operator? What is the relativistic spin operator? 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 Laser Physics 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 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 operator. 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. Keywords: spin, relativistic quantum mechanics, 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 momentum operator 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 physicists 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 position operator [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 nature of the spin is still incomplete [3]. units as used in this paper pˆ = ir. 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 magnetic moment), 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 electrons 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) quantum information the operator of the relativistic total angular momentum is given ff [12–17], quantum spintronics [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-matter 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 atom. 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 "i; j;k denoting the Levi-Civita symbol. gular momentum in z-direction Jˆ3, and the so-called spin-orbit operator Kˆ (or the parity) 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 forces are absent, such that spurious Zitterbewegung of the spin is prevented. The second 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 ] = i" 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 group theoretical considerations rather than by experimental [Jˆ; Lˆ ] = i" 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 Lorentz group 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 2 f g mute. Both conditions are fulfilled if, and only if, the spin particle’s rest mass m , the speed of light c, the matrix β such 0 operator Sˆ satisfies the angular momentum algebra (6) because that the commutator relations 2 β = 1 ; αiβ + βαi = 0 ; (3) [Lˆ ; Lˆ ] = i" Lˆ + [Sˆ ; Sˆ ] i" Sˆ ; (8) i j i; j;k k i j − i; j;k k the free particle Dirac Hamiltonian [Lˆ ; Sˆ ] = i" 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.
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