## Abstract

This paper concerns the spacetime locations of massive spinning particles in relativistic quantum mechanics. Using techniques from twistor theory and the unitary representations of the Lorentz group, it is shown that, for a particle of mass *m* and spin *j*, the radial distance from the particle to the observer is bounded below by . If the particle is the source of a Kerr–Minkowski field, the analogue of a Kerr field in general relativity, it is further shown that quantization removes the ring singularity if *j* is Fermionic or zero.

## 1. Introduction

In general relativity, there is a standard solution (Kerr 1963; Hawking & Ellis 1973; Misner *et al.* 1973) to Einstein's field equations that represents the gravitational field of a stationary particle with mass and intrinsic spin. However, such a solution possesses a ring singularity. As the singularity is approached, the tidal forces, epitomized by the scalar invariants of the Riemann tensor, grow without bound. The question has therefore been posed by several authors (e.g. Misner *et al.* 1973, §34.6) as to whether quantization would remove the singularity. Of course, this would require a theory consistent, at least in appropriate domains of application, with both quantum mechanics and general relativity.

An attempt to attack the problem involved a study of *Kerr–Newton particles* (Bramson 1992), particles with mass and spin whose Newtonian gravitational fields are the non-relativistic analogues of Kerr's solution in general relativity. The classical Kerr–Newton potential satisfies Laplace's equation and exhibits a ring singularity. Remarkably, however, quantization removes the singularity for Fermionic sources. Furthermore, the gravitational binding energy associated with the field of such a source is finite (Bramson 2007).

With the extension to general relativity in mind, the question arises as to what can be said in special relativity. Accordingly, a study (Bramson 1993) established two results for a spin particle in Minkowski spacetime. First, the distance (a quantum operator) from the observer or sensor to the particle's centre of mass has eigenvalues that are bounded below by of the particle's Compton wavelength. (In fact, this holds true for *any* spin system.) The second result concerned Kerr–Minkowski fields, fields on Minkowski spacetime whose classical versions exhibit ring singularities and satisfy Maxwell's equations or the linearized Einstein equations, the linearized analogue of Kerr's solution in general relativity. Then, quantization removes the singularity.

The purpose of this paper is to generalize the results just described. It therefore concerns itself with systems in relativistic quantum mechanics, in particular with their spins and centres of mass. Accordingly, it concerns the Poincaré and Lorentz groups. The commutation relations for the relevant observables are well known (Schweber 1962, §2*c*) and are presented in §2.

However, a subtlety exists and it concerns whether a quantum observable is regarded as a geometric object (a spinor, vector or tensor) or as a set of components of that object in some frame of reference. If the frame is owned wholly by the observer, so that it commutes with itself, and lies entirely outside the system under scrutiny, so that it commutes with the system's observables, then the algebra of observables and the algebra of their sets of components are identical. However, if the frame is intrinsic to the system, so that it fails to commute with the system's observables, then the algebra of components differs from the algebra of observables.

In the case of non-relativistic angular momentum, the difference is merely one of sign (see appendix A). However, in relativistic theory, the difference is more pronounced. Indeed, the requirement for a comoving viewpoint leads to a frame that inevitably fails to commute with the system's observables. Such a frame may be constructed within twistor theory (Penrose & MacCallum 1973; Penrose & Rindler 1988; Huggett & Tod 1994) by decomposing a massive particle into a pair of massless constituents and this is described in §3.

In §4, the particle's observables are investigated in the comoving frame. It turns out that the particle's Pauli–Lubanski spin vector and the displacement of its mass centre from the observer define a pair of vectors whose components in the comoving frame obey the algebra of the (homogeneous) Lorentz group. The irreducible unitary representations of that group have been studied by several authors (Harish-Chandra 1947; Naimark 1957, 1964; Gel'fand *et al.* 1963) and the main results are outlined in §5. A surprising analogy emerges between the Casimir operators of the Lorentz group and Boyer–Lindquist coordinates (Hawking & Ellis 1973, §5.6).

Sections 6 and 7 use the results of §5. In §6, the particle's range from the observer is shown to be bounded below by an expression involving its mass and spin. In §7, Kerr–Minkowski fields are revisited for sources of arbitrary spin. It is shown that quantization removes the singularity for all Fermions and also for spin 0.

A word should be added concerning the use of Penrose's abstract index notation (Penrose 1968*a*, §3; Penrose & Rindler 1987, §2). In this notation, a vector ** v** is written synonymously as

*v*

^{a}, the index

*a*depicting the

*type*of geometric object involved.

*v*

^{a}is

*not*a set of components, but the abstract index notation enables the easy performance of various tensor manipulations such as contraction, symmetrization and skew symmetrization; and, in quantum theory, it becomes straightforward to keep track of the order in which operators appear.

The components of *v*^{a} with respect to some basis will be written *v*^{a}. Thus, bold indices like **a** will take values 0, 1, 2 and 3. Note that if is a dual basis in classical theory, then(1.1)However, in quantum theory, the quantities and may not necessarily be the same. This issue will require the greatest care later.

Throughout, natural units will be adopted. Both Planck's constant and the speed of light *c* will be chosen to be unity.

## 2. Poincaré algebra

In Minkowski spacetime , consider a stationary system with mass *m*, (4-)momentum *P*^{a} and angular-momentum/mass-dipole tensor ^{ab}(*X*), the latter being dependent on the choice of origin *X* in . The velocity of the system's centre of mass defines a unit, future-pointing, timelike vector *t*^{a} according to(2.1)

In relativistic quantum mechanics, ^{ab} and *P*^{a} form the generators of the Poincaré group, while the mass and the magnitude of the spin are the two Casimir operators (Schweber 1962, §2*c*). The momentum commutes with itself while, if *g*_{ab} is the Minkowski metric,(2.2)(2.3)

The spinor formalism (Penrose & Rindler 1987, §3) provides the decompositions(2.4)where *μ*^{AB} and are symmetric, and these, in turn, yield the dual of ^{ab},(2.5)

The system's mass-dipole vector *D*^{a} and Pauli–Lubanski spin vector *J*^{a} (some authors use *W*) are given by(2.6)

Note that, although *D*_{a}*D*^{a} is Hermitian, *D*^{a} is not (Bramson 1993, §4),(2.7)use having been made of equation (2.3). Further, although *D*^{a}*t*_{a} vanishes, *t*_{a}*D*^{a} does not. Next, it proves convenient to decompose ^{ab} into its ‘electric’ and ‘magnetic’ parts with respect to *t*^{a}. To this end, define another pair of bivectors,(2.8)

Then, ^{ab} and ^{*ab} may be expressed in terms of *D*^{ab} and *J*^{ab},(2.9)and the algebra for *D*^{a} and *J*^{a} (Bramson 1993, §4) becomes(2.10)

Importantly, the Hermitian operators *D*_{a}*D*^{a}, *J*_{a}*J*^{a} and *J*_{a}*D*^{a} commute with each other. In classical theory, the spacelike vector *D*^{a}/*m* lies orthogonal to *t*^{a} and, for a given *X*, defines the transverse displacement of the system's centre of mass worldline from *X* (e.g. Bramson 1993, §2), while the three scalars have the following meanings: (i) −*D*_{a}*D*^{a}/*m*^{2} is the square of the radial distance from *X* to the system's centre of mass worldline, (ii) −*J*_{a}*J*^{a} is the square of the system's intrinsic spin, and (iii) *J*_{a}*D*^{a}, along with the two scalars just described, serves to define the cosine of the polar angle between the spin vector and the radial vector. In relativistic quantum theory, the last of these holds true only when *X* is far removed from the system. In the near zone, things are more subtle and this will be discussed in §5. Also, in classical theory, the three scalars define the ring singularities of the fields produced by Kerr–Minkowski particles. To decide whether these singularities persist at the quantum level, it will be necessary to understand the consequences of the algebra depicted in equation (2.10). Surprisingly, this algebra, when perceived in a sensible frame, is that of the *homogeneous* Lorentz group.

## 3. Twistorially defined comoving frame

The purpose of this section is to show, via a brief foray into twistor theory (Penrose & MacCallum 1973; Penrose & Rindler 1988; Huggett & Tod 1994), that an orthonormal tetrad may be chosen as reference frame with the following properties: (i) one of its members is the velocity *t*^{a} of the given system's centre of mass and (ii) its members commute with themselves and satisfy a commutation relation with ^{ab} analogous to equation (2.3) so that(3.1)

(3.2)

To this end, consider the decomposition of the massive system into a pair of massless constituents represented by twistors and their complex conjugates , where *α* is an abstract index and *Γ* takes values 1 and 2. This follows the approach of Penrose (1975, 1977) and Perjés (1975, 1977) with the exception that upper case Greek indices have replaced lower case Roman ones. With respect to an origin *X*, the twistors may be decomposed into their momentum (*π*) and angular-momentum (*ω*) spinor parts,(3.3)

(3.4)

The system's angular-momentum/mass-dipole tensor ^{ab} takes the form depicted in equation (2.4) with(3.5)a summation over *Γ* understood, while the system's momentum *P*^{a} is given by(3.6)

The twistor commutation relations(3.7)guarantee that ^{ab} and *P*^{a} satisfy the Poincaré algebra presented in equations (2.2) and (2.3). In addition, it may be seen that replacing *P*^{a} by any of the four null vectors,(3.8)replicates the algebra of equations (3.1) and (3.2). The quantity defines a standard null tetrad with the property that(3.9)

Of course, this defines a quantum-mechanical frame because the *π*s and s do not commute with the *ω*s and s. However, the *π*s and s commute among themselves; so, there is no ambiguity in the definition of the four vectors . From the null tetrad, a proper, orthochronous, orthonormal tetrad may be defined by taking suitable linear combinations (Penrose & Rindler 1987, §3.1) and this trivially satisfies the algebra of equation (3.1). To be explicit,(3.10)

## 4. Quantum algebra using the comoving frame

To proceed, employ the comoving frame defined at the end of §3. This amounts to an orthonormal tetrad and its dual such that(4.1)

Furthermore, the commutation relations presented in equations (3.1) and (3.2) are satisfied. Note that *t*^{a}, being one of the frame members, commutes with all frame members. In particular, and may equivalently be used to define *t*_{a}. It follows from the definitions (2.6) that(4.2)

(4.3)

Noting that the metric *g*_{ab} commutes with *D*^{c}, *J*^{c} and , and that the same is true of its frame components *g*_{ab}, indices may be raised, lowered and contracted without worrying about factor ordering. Hence,(4.4)

Next, define two sets of components,(4.5)the order in the first case being critical. It is straightforward to show that *J*^{a} is Hermitian and orthogonal to *t*^{a}. This is true also of *D*^{a},(4.6)use having been made of equations (2.7) and (4.4). Further,(4.7)

In fact, both *D*^{a} and *J*^{a} are left and right orthogonal to . In a similar way, it follows that(4.8)

The commutation relations for *D*^{a} and *J*^{a} employ a generic result that, if *A*, *B*, *C* and *D* are operators such that *C* and *D* commute with each other,(4.9)

Using this, together with the commutation relations (2.10) and (4.4), yields(4.10)

Apart from a change in sign for each of the operators *D*^{a} and *J*^{a} (compare the quantum mechanics of non-relativistic spin discussed in appendix A), this is the algebra of the homogeneous Lorentz group and will be discussed in §5.

## 5. Unitary irreducible representations of the homogeneous Lorentz group

At the end of §2, a set of Hermitian scalar operators, *D*_{a}*D*^{a}, *J*_{a}*J*^{a} and *D*_{a}*J*^{a}, was presented. In equation (4.8), these were rewritten in terms of their comoving frame components. The underlying operators, *D*^{a} and *J*^{a}, satisfy the algebra of the homogeneous Lorentz group presented in equation (4.10). It would seem natural to treat the three scalar operators as system observables and to describe the system in terms of the simultaneous eigenstates of these observables. Owing to the probabilistic nature of quantum mechanics, it follows that *unitary* representations must be employed.

The unitary representations of the homogeneous Lorentz group have been studied by several authors (Harish-Chandra 1947; Naimark 1957, 1964; Gel'fand *et al.* 1963). Reverting to three-dimensional notation and incorporating the change in sign by writing(5.1)the commutation relations presented in equation (4.10) amount to(5.2)

The scalar Hermitian operators are(5.3)

The Casimir operators are (*d*^{2}−*j*^{2}) and ** j**.

**. To these,**

*d*

*j*^{2}may be added to form a commuting set. As usual, the eigenvalues of

*j*^{2}are

*j*(

*j*+1), where

*j*=0, , 1, , …. Following Barrett & Crane (2000), only the principal series of the unitary irreducible representations will be employed here. These are labelled by a pair (

*ρ*,

*s*), where(5.4)For each (

*ρ*,

*s*) pair,(5.5)

To unravel the physical meanings of *ρ* and *s*, define(5.6)where *b* is the Compton wavelength of a particle of mass *m*, but note that *r*^{2} is not the same as *x*^{2}. The relations (5.5) now become(5.7)

For *r* large compared with *a*, *x*^{2} and *r*^{2} are roughly equal so that, in the asymptotic region, *r* is a reasonable radial coordinate while *s* measures the component of ** j** in the radial direction. Further, when

*j*≠0, define(5.8)(5.9)with the constraint that the eigenvalues of

*θ*are to lie in the range (0,

*π*). Eliminating

*s*between equations (5.7) and (5.8) yields a relation surprisingly familiar in general relativity,(5.10)

Think of *r* and *θ* as Boyer–Lindquist coordinates (Hawking & Ellis 1973, §5.6) while the triple ** x** defines Kerr–Schild coordinates. The −

*b*

^{2}term in equation (5.10) distinguishes this quantum expression from the classical relation of Hawking & Ellis (1973, below eqn (5.30)).

## 6. Bounded range

Note that *x*^{2}, the square of the radial distance from the observer to the system's mass centre, can never vanish, a result already known for spin (Bramson 1993, §5). Indeed, equation (5.7) implies that it is bounded below,(6.1)the bound being attained when *s*=±*j* and *r*=0. For spin 0,(6.2)

The measured range of a spin 0 particle is bounded below by its Compton wavelength *b*. We may therefore imagine the particle to be contained within a spherical exclusion zone of radius *b*. It is almost as if the space defined by the particle has a sphere removed. For non-zero spins, it proves useful to employ the commuting coordinates *z* and *w* with *z* defined in equation (5.8) and satisfying(6.3)and *w* defined, up to sign, by(6.4)

Equations (6.3) and (6.4) define the legal locations of the particle in the (*w*, *z*) plane. With *r* running from 0 to ∞, the discrete values of *s*, given in equation (5.4), define the ‘radials’ on which the particle may be found. In non-relativistic quantum mechanics, these are straight lines radiating from the origin. Here, however, they are hyperbolae. There are two cases. First, when *s*≠0, for Fermions and some Boson states, eliminating *r* between equations (6.3) and (6.4) yields(6.5)

Second, for those Boson states with *s*=0,(6.6)

The legal locations of particles are illustrated in figures 1 and 2.

## 7. Singularity removal for Kerr–Minkowski particles

In Bramson (1993, §5), a discussion concerning the fields generated by stationary spinning sources revealed that the classical ring singularity disappeared for spin . The question arises as to whether this holds true for other spins. The fields in question comprise a hypothetical massless spin-0 (scalar) field, a spin-1 (Maxwell) field and a spin-2 (linearized Einstein) field. Each of these involves the quantity with *μ*^{AB} given in equation (2.4) and *n* taking the respective values , and . The fields are singular when *μ*_{AB}*μ*^{AB} vanishes or, equivalently, when the complex vector (*D*^{a}+i*J*^{a}) is null and this is true when(7.1)Classically, this defines the Kerr ring, the intersection of a sphere of radius |** J**|/

*m*with the equatorial plane orthogonal to

**. Using the results of §5, an equivalent condition for the singularity involves the Casimir operators of the Lorentz group,(7.2)**

*J*In terms of the parameters that label the irreducible unitary representations of that group, this equates to(7.3)

The singularity therefore occurs when(7.4)

For Fermions, *s* is half-odd integral in which case there is no singularity and this replicates the result found for Kerr–Newton particles (Bramson 1992). In addition, there is no singularity in the spinless case for, in that case, both *j* and *s* vanish.

## 8. Conclusion

Having discovered that quantum mechanics removed the ring singularity from the gravitational fields produced by Fermionic, Kerr–Newton particles, it seemed natural to ask whether this remarkable feature survived a relativistic extension. Specifically, if Kerr–Minkowski fields (e.g. the linearized Kerr solution) are examined within the framework of *relativistic* quantum mechanics, what can be said? This paper shows that the singularity again disappears for Fermionic sources. Surprisingly, it also disappears for sources with no spin.

A second result concerns the displacement of the mass centre of a relativistic system from an observer. In non-relativistic quantum mechanics, the eigenvalues of the radial range operator include the whole of the non-negative real line; but not so for relativistic quantum mechanics. For a system with mass *m* and spin *j*, the distance from the observer to the mass centre is bounded below by and this holds true for both Fermions and Bosons.

Suppose, in addition, that the observer measures the orientation of the system's spin with regard to his line of sight. In both non-relativistic and relativistic quantum theory, this orientation is specified by an angle, the polar angle, between the system's spin vector and the radial vector from the observer to the system's mass centre; in both cases, the eigenvalues form a discrete set.

What is the observer to make of the spacetime perceived in this way? If his concept of spacetime is conditioned solely by measuring the location and orientation of some material object or, equivalently, by his set of possible locations with respect to that object, his perception will be that: (i) he lies on one of a discrete set of radials from that object and (ii) he is excluded from approaching that object too closely. With regard to (i), the number of radials increases with the magnitude of *j* so that, for large *j*, the observer will have the illusion that the polar angle, his bearing from the reference object, has a continuous spectrum. With regard to (ii), he will deduce, after making repeated measurements, that the spacetime possesses an exclusion zone from which he is forbidden. To put it another way, the spacetime appears to have had a region removed.

Finally, what might be expected of a theory that reconciles quantum mechanics (and eventually quantum field theory) with general relativity? Two points are worth making. First, focusing on the observables ‘location’ and ‘spin’ led inevitably to the homogeneous Lorentz group and, owing to quantum considerations, to the unitary irreducible representations of that group. The Casimir operators of that group and the magnitude of the spin then yielded a pair of parameters that bore a striking resemblance to Boyer–Lindquist coordinates in general relativity. Perhaps these are natural structures with a common source in both theories; but the possibility arises of a deeper connection between general relativity and relativistic quantum mechanics (see also Penrose 1968*b*). Within the latter framework, it would be necessary to include at least linearized gravitation if only to accommodate the existence of Newton's gravitational constant. Second, when it comes to stationary, asymptotically flat spacetimes (isolated systems) in general relativity, the spacetime perceived by remote observers is quite different from the one usually postulated. The latter is Einstein's curved manifold; the former is a Minkowski observation space; and the usual classical observables such as spin, momentum, dipole moment and quadrupole moment reside in the observation space and not in the original space. This fact alone reveals something of the nature of a successor theory.

## Acknowledgments

I am grateful for a conversation with Dick Harris at the end of which I realized that the bound on range stood independently of the issue of singularity removal. I thank QinetiQ for library support and two referees for their constructive comments that have helped add clarity to my text.

## Footnotes

- Received November 16, 2006.
- Accepted April 5, 2007.

- © 2007 The Royal Society