Singularity of a relativistic vortex beam and proper relativistic observables

We have studied the phase singularity of relativistic vortex beams for two sets of relativistic operators using circulation. One set includes new spin and orbital angular momentum (OAM) operators, which are derived from the parity-extended Poincaré group, and the other set consists of the (usual) Dirac spin and OAM operators. The first set predicts the same singularity in the circulation as in the case of nonrelativistic vortex beams. On the other hand, the second set anticipates that the singularity of the circulation is spin-orientation-dependent and can disappear, especially for a relativistic paraxial electron beam with spin parallel to the propagating direction. These contradistinctive predictions suggest that a relativistic electron beam experiment with spin-polarized electrons could for the first time answer a long-standing fundamental question, i.e., what are the proper relativistic observables, raised from the beginning of relativistic quantum mechanics following the discovery of the Dirac equation.


Singularity of a relativistic vortex beam and proper relativistic observables
Yeong Deok Han 1 , taeseung choi 2,3 ✉ & Sam Young cho 4 We have studied the phase singularity of relativistic vortex beams for two sets of relativistic operators using circulation. one set includes new spin and orbital angular momentum (oAM) operators, which are derived from the parity-extended poincaré group, and the other set consists of the (usual) Dirac spin and OAM operators. The first set predicts the same singularity in the circulation as in the case of nonrelativistic vortex beams. on the other hand, the second set anticipates that the singularity of the circulation is spin-orientation-dependent and can disappear, especially for a relativistic paraxial electron beam with spin parallel to the propagating direction. these contradistinctive predictions suggest that a relativistic electron beam experiment with spin-polarized electrons could for the first time answer a long-standing fundamental question, i.e., what are the proper relativistic observables, raised from the beginning of relativistic quantum mechanics following the discovery of the Dirac equation.
Nonrelativistic electron vortex beams carrying orbital angular momentum (OAM) have recently been studied and are well-understood using the paraxial approximation of the Schrödinger equation [1][2][3][4][5][6] . The wavefunction of a nonrelativistic electron vortex includes a phase singularity factor, φ e il , where φ is the azimuthal angle around the axis of the vortex, and the electron vortex beam can carry orbital angular momentum of l in which l is an integer known as the topological charge 1 . As the energy of electron vortex beams reaches the relativistic regime of 200~300 keV 1,2,7,8 , the validity of the interpretation of the rotational motion of the high-energy electrons in the experiment as a relativistic electron vortex is questioned [9][10][11][12][13][14][15] .
To understand such relativistic electrons, one should use the Dirac equation 16 , which successfully describes relativistic electrons, instead of the Schrödinger equation. However, in the usual Dirac theory, the spin angular momentum and orbital angular momentum of an electron are not separately conserved unlike in the Schrödinger theory. As a result, Bialynicki-Birula et al. proved the assertion that any acceptable solutions of the Dirac equation cannot be eigenstates of the (usual) Dirac OAM, and showed that the vortex lines continuously smeared out into all space for their exponential solutions, which become the standard vortex wavefunction in the nonrelativistic limit 12 . This raised the question of whether a relativistic vortex can be generated from high-energy electron beams. In contrast, Barnett 13 argued that relativistic electron vortices with a well-defined OAM and phase singularity truly exist using the so-called Foldy-Woutheysen (FW) representation 17 in which the vortex charge is related to the eigenvalues of OAM, as is the case for nonrelativistic vortices.
The controversal results of Bialynicki-Birula et al. and Barnett 12,13 originated from the use of different spin and OAM operators as relativistic operators, as indicated by Bliokh et al. 18 . Explicitly, Bialynicki-Birula et al. used the usual Dirac spin and OAM operators, but Barnett used the FW mean spin and the FW mean OAM operators. Essentially, such other choices are due to a lack of understanding of relativistic operators. As an unsolved fundamental issue, obtaining a proper relativistic spin operator for massive spin-1/2 particles has been a long-standing problem from the beginning of relativistic quantum mechanics 17,[19][20][21][22][23][24][25][26][27] . Many discussions have attempted to suggest possible proper description of spin for massive elementary particles.
Recently, two of the present authors, i.e., Choi and Cho 28 rigorously derived a spin operator for the Dirac field that transforms covariantly under the Lorentz transformation. We call this spin operator the new spin operator to distinguish it from the other spin operators such as the Dirac and the FW mean spin operators. The new spin operator was shown to be the generator of the SU(2) little group of the Poincaré group and admits a

Results
Dirac spin and orbital angular momentum. In this section, for a clear comparison with the new spin and the corresponding OAM in the study of a relativistic vortex, we will briefly review the original Dirac theory 16 . As introduced by Dirac 16 , the total angular momentum is the sum of the Dirac spin and the corresponding Dirac OAM, i.e., × r p D , where r D is the Dirac position operator, which is the canonical operator represented by ∇ i p , and σ k ( = k x y z { , , }) are the Pauli matrices. This total angular momentum is a constant of motion under the following Dirac Hamiltonian We use the Einstein summation convention in which we sum over repeated indices. For the Dirac Hamiltonian, the Dirac matrices are in the standard representation 16 . Here we use the natural unit = = c 1  . However, as noticed, the Dirac spin and the Dirac OAM are not separately conserved with the Dirac Hamiltonian, which is the reason why Dirac introduced spin angular momentum. That is, Table 1. Table of the properties, operator-representatives, and relations of the new spin S N , the particle spin S P , and the antiparticle spin S AP . Detailed explanations are given, and other notations are adapted from the Results. S N gives the 2 nd Casimir of the Poincaré group, neither S P nor S AP . All three spins satisfy the su(2) algebra, i.e.,

Particle Antiparticle
www.nature.com/scientificreports www.nature.com/scientificreports/ The commutator of the Dirac OAM and the Dirac Hamiltonian in Eq. (4b) is not zero because the Dirac velocity operator is not proportional to the momentum p. This suggests that the existence of Zitterbewegung 29,30 , which is fast trembling motion first observed by Schrödinger, is closely related to the non-conservation of the Dirac OAM.
The Dirac Hamiltonian (2) gives the following well-known four solutions: the new spin and corresponding orbital angular momentum. Recently we derived the covariant spin operator of the parity-extended Poincaré group whose eigenstates provide the representation corresponding to a free massive elementary field with spin s 28 . The representation space of the parity-extended Poincaré group for free massive spin-1/2 fields is the four-spinor space in which the usual Dirac particle and antiparticle spinors reside 28,31 . In this section, we introduce the covariant spin operator as the new spin operator and the corresponding OAM in association with the new position operator.
The new spin operator was originally constructed by the generators of the Poincaré group; however to compare the differences between the new spin and the Dirac spin explicitly, it is convenient to represent the new spin operator in the Dirac four-spinor space by using the Dirac spin operator as 28 and I is the 2-dimensional identity matrix. We use upper case P for a momentum operator and lower case p for the eigenvalue of a momentum operator.
The following two fundamental dynamical equations were derived for a free massive spin-1/2 particle and antiparticle from the property of the parity operation 28 AP where ψ µ p ( ) P and ψ µ p ( ) AP are the particle and antiparticle spinors, respectively. These two Eqs. (8a) and (8b) are the same as the two covariant Dirac equations for particle and antiparticle spinors, where the Dirac gamma matrices are γ β = 0 and γ βα = k k in the standard representation 31 . There are two positive energy and two negative energy solutions for each of Eqs. (8a) and (8b). Among these 8 solutions, the two particle eigenspinors are the same as  satisfy the same orthogonality relation as The new spin S N k in Eq. (7) can also be expressed by using the following relation 28 Then, it is easily seen that the four spinors are also eigenstates of the new spin S N k with the same eigenvalues of the rest spin Σ k /2 for the rest spinors However, S N k is not a good observable because it is not Hermitian, as seen from the last term in Eq. (7). This does not mean that S N k is not a proper spin operator. In fact, S N k becomes equivalent to the Hermitian particle spin operator S P k and antiparticle spin operator S AP k as they act on the particle states with the explicit form of the particle and antiparticle spin operators in the momentum representation. Note that the particle and the antiparticle spin operators can be expressed by using the FW transformation matrix U p ( ) FW /2 0 5 Thus, the particle spin S P k is straightforwardly shown to be the same as the FW mean spin operator discussed in refs. 13,17,23 .
It has been shown that S N k gives Noether's conserved spin angular momentum 28 . The conservation of spin can be also confirmed by using the commutators between spins and the corresponding Hamiltonians. Since the antiparticle spinors , the corresponding Hamiltonian for is also different from the original Dirac Hamiltonian, which is obtained as 31 N D /2 /2 5 5 R N satisfies the same commutation relations as those of the Dirac position operator, i.e, The new position operator acting on the particle (2020) 10:7417 | https://doi.org/10.1038/s41598-020-64168-0 www.nature.com/scientificreports www.nature.com/scientificreports/ and the antiparticle states becomes the following Hermitian particle and antiparticle position operators, respectively, similar to the new particle and antiparticle spin operators S P k and S AP k . Subsequently, the velocity operators for the particle and the antiparticle are determined as are conserved by themselves. The OAM conservation of the free massive particles and antiparticles is also verified by the commutation relations: existence of singular relativistic vortices and its implication. In the nonrelativistic case, free electron vortex states (with a phase singularity) carry a well-defined OAM, which requires the conservation of the OAM 1,2,33 . It is natural to expect that the conserved OAM is also essential for the existence of singular relativistic electron (Dirac particle) vortices. As studied in the previous sections, because of the Zitterbewegung of the Dirac position operator, the Dirac OAM is not conserved as shown in Eq. (4b). In constrast, the particle position operator shows no Zitterbewegung and, as a result, gives the conserved particle OAM in Eq. (20a). Therefore, the eigenstates of the particle OAM operator would compose the eigenstates of the particle Hamiltonian with a well-defined particle OAM such as those of the nonrelativistic case, but this is not the case for the eigenstates of the Dirac OAM. This raises the question: "could the existence of a singular relativistic vortex in an experiment be a probe for proper spin and position operators?" We call this question the'which operator question' . To answer the'which operator question' , a specific solution for relativistic beams is needed. Here, we focus on a Dirac particle (electron), because the reasoning for an antiparticle (positron) is parallel to that of a particle and straightforward. Let us first consider the particle spin and the particle OAM, which admit the vortex solutions with well-defined OAM. We assume the relativistic beam to be paraxial, which propagates mainly along z-direction, i.e., The vortex solutions expressed in terms of the eigenstates of the particle OAM can be most easily studied in the FW representation for electrons, because the particle position and the particle OAM operators are represented in the usual canonical form in the FW representation as , and φ is the azimuthal angle of the cylindrical coordinate ρ φ z ( , , ) in the FW representation. The FW transformations of the state ψ x ( ) and the Dirac Hamiltonian H D in the original representation are performed as . We consider the solution ψ x ( ) FW to be monoenergetic with energy E for simplicity. Then, it is sufficient to analyse only the spatial dependence of ψ x ( ) FW , i.e., . The expectation value of the operator O at x in the original representation given by is the operator representative in the FW representation. The right expression in the FW representation is (Methods 0.1.2) to the left. This relation provides a nontrivial spin-orbit interaction effect in terms of the Dirac spin and the Dirac OAM rather than the new (particle) spin and the new (particle) OAM. The normalized expectation value of O is divided by the probability amplitude ψ ψ † x x ( ) ( ) , which is also not the same as

FW FW
in general (Methods 0.1.2). The singularity of the vortex is encoded in the rotational property of the velocity around the propagating direction. The expectation value of the particle velocity operator in Eq. (19a) at x, which we call the particle velocity at x, is written as  www.nature.com/scientificreports www.nature.com/scientificreports/ are zero, which shows the SOI effects. This implies that the LG solution could be a suitable expression of an experimental relativistic paraxial electron beam.
In the paraxial approximation, the state ψ x ( ) FW in Eq. (28) varies only gradually along the z-axis such that That is, the z dependence of the solution can be considered solely by e ip z 0 and w z ( ) can be replaced by ≡ w w (0) . We are interested in the singular behaviour of the relativistic wave solutions for ρ → 0, hence the region of ρ < w/ 2 will be considered. On the other hand, ρ should be greater than 1/m, the Compton wavelength, because one particle theory is not valid in the region less than the Compton wavelength in which pair production is inevitable. Thus, in our study, we refer to the region of the vortex solution determined by ρ < < m w 1 2 (30) as the physical vortex region for simplicity. In the physical vortex region, the wavefunction ψ x ( ) FW in Eq. (28) can be written as x y z l l 0 1 As a result, the particle velocity at x is given as This particle velocity at x describes that electrons move along the z-direction with spiral circular motion, which represents the singular vortex motion along the z-axis. This result shows that the relativistic vortex solution interpreted by the particle velocity supports the singular vortex like the nonrelativistic vortex with the following circulation 14,15 where C is an arbitrary closed path around the z-axis. Next, we study the singularity of the vortex solutions in Eq. (28) by using the Dirac position and the Dirac velocity. The Dirac velocity at x is obtained as (Methods 0.2) .
y , which can also be expected from the eigenspinors in Eq. (5). Similar to the particle velocity, the Dirac velocity at x indicates that electrons move along the z-direction with a spiral motion. However, in contrast to the particle velocity, the spiral motion described by the xand y-components of the Dirac velocity depends on the expectation value of the z-component of the Dirac spin, i.e., the Dirac spin orientation. The circulation for the Dirac velocity shows that the Dirac spin orientation determines whether a singular vortex exists.
For a comparison with the circulation of the particle velocity Γ P , we plot the circulation of the Dirac velocity Γ D as a function of twice the Dirac spin orientation S 2 D z for positive l in Fig. 1(b). Figure 1(b) clearly shows that Γ P does not depend on the Dirac spin orientation, but Γ D depends on the Dirac spin orientation. If the Dirac spin is unpolarized or the Dirac spin orientation is in the xy-plane perpendicular to the propagating direction of the electron beam, Γ D is the same as Γ P . However, compared with Γ P , Γ D can be stronger if the angle between the Dirac spin orientation and the propagating direction of the electron beam is obtuse or weaker if acute. In particular, if the Dirac spin orientation is parallel to the propagating direction of the electron beam, the spiral circular motion of the electron beam disappears, i.e., Γ D = 0 ( Fig. 1(b)).
The Dirac velocity may be distinguished from the particle velocity experimentally by the characteristic spin orientation-dependent property. In particular, in order to answer the'which operator question' , for instance, two different setups can be considered in using spin-polarized electron beams moving along the z-direction in relativistic vortex experiments. In one setup, the spin is antiparallel to the propagating direction, and in the other setup, the spin is parallel to the propagating direction. As discussed in Fig. 1(b), for the particle velocity, the two setups will give the same vortex structure independent of the spin. However, for the Dirac velocity, the two setups will give very different observations of electron beams; i.e., the antiparallel spin gives spiral circular currents leading a vortex structure, but the parallel spin gives non-spiral circular currents resulting in no vortex structure. The probability current given by the particle velocity in Eq. (35) forms a whirlpool around the singularity, but the current given by the Dirac velocity in Eq. (37) with spin parallel to the propagating direction does not form a whirlpool. The probability density in Eq. (33) shows typical behaviour near the phase singularity 1 . The magnetic field induced by the whirlpool motion can be detected in an experiment. Thus, whether the non-vortex structure exists can play the role of a smoking-gun in distinguishing which ones can be proper relativistic operators. Consequently, distinguishable experimental observation results of a relativistic vortex in such two different setups could provide a clear answer to the question regarding proper relativistic observables, i.e., position, spin, and OAM. In addition, such an experimental answer to the long-standing question regarding proper relativistic observables could also provide reliable evidence to clarify whether Zitterbewegung is a real physical effect. Similar results are expected in relativistic proton and positron vortices for the new operators using parallel logic.

conclusion
We have studied the singularity of relativistic electron vortex beams using two different sets of relativistic operators. The first set includes the particle position, spin, and OAM operators that admit well-defined OAM l for an LG vortex solution in the FW representation. The particle operators predict a singularity in the circulation of the inverse FW transformed relativistic LG vortex solution to the original representation, which shows the same feature of the Schrödinger nonrelativistic vortex. The second set consists of the usual Dirac position, spin, and OAM operators by which the spin orientation-dependent singularity of the same vortex solution is anticipated.
It was predicted that spin seems to have little effect in the study of a relativistic electron vortex beam for typical parameters in state-of-the art transmission electron microscopy experiments based on the estimation of the particle density ψ ψ † x x ( ) ( ) in the paraxial regime 35 . Actually, our study shows a similar result where ψ ψ † x x ( ) ( ) has a considerable spin effect in Eq. (47) (Methods 0.1.2) if ρ is smaller than the Compton wavelength, i.e., ρ < m 1/ , while the spin effect can be negligible in ψ ψ † x x ( ) ( ) in Eq. (33) for the physical region, i.e., ρ < < m w 1/ / 2 , of interest to us. However, in sharp contrast to the behaviour of the particle density ψ ψ † x x ( ) ( ) in the paraxial regime, as discussed in the Results, the behaviours of the particle velocity and the Dirac velocity exhibit crucial differences due to the spin. Then, we discussed a possible experimental setup to probe a proper set of relativistic observables based on the very different predictions from the two sets of relativistic operators for the singularity of the LG vortex solution. Especially for a paraxial electron beam with spin parallel to the propagating direction, it could be experimentally distinguished that for the Dirac operators, singularity-and vortex-like motion do not exist, but for the particle operators, a singular vortex exists. Therefore, such spin-polarized relativistic electron vortex beam experiments could provide an answer to the question: which relativistic observables are proper? Methods equivalent expressions between the original and the fW representation. New spin, particle spin and antipar ticle spin.
Then the operators (1 )/2 0 γ ± project the rest spinors onto particle and antiparticle subspace, respectively, i.e., By using the relation The first line of the above equation corresponds to the same relation in ref. 23   for Σ φ = −sin φ Σ x + cos φ Σ y with the paraxial condition in the physical region. The second term in the last line becomes greater than the first when the ρ satisfies ρ < + < l E m m /( ) 1/ , i.e., less than the Compton wavelength. Therefore, the final expression in the physical vortex region becomes