## Abstract

Highly localized positive-energy states of the free Dirac electron are constructed and shown to evolve in a simple way under the action of Dirac's equation. When the initial uncertainty in position is small on the scale of the Compton wavelength, there is an associated uncertainty in the mean energy that is large compared with the rest mass of the electron. However, this does not lead to any breakdown of the one-particle description, associated with the possibility of pair-production, but rather leads to a rapid expansion of the probability density outwards from the point of localization, at speeds close to the speed of light.

## 1. Introduction

Various difficulties arise in the description of the localization of a particle in relativistic quantum mechanics, as described in the large literature on the subject stretching back to the birth of relativistic quantum theory (Schrödinger 1930, 1931, 1995; Landau & Peierls 1931, 1965; Pryce 1935, 1948; Papapetrou 1940; Newton & Wigner 1949; Foldy & Wouthuysen 1950; Tani 1951; Wigner 1952; Wightman & Schweber 1955; Dirac 1958; Wightman 1962; Currie *et al*. 1963; Jordan & Mukunda 1963; Licht 1963; Weidlich & Mitra 1963; Bacry 1964, 1988; Knight 1964; Philips 1964; Berg 1965; Fleming 1965, 1966; Galindo 1965; Barut & Malin 1968; Kálnay 1971; Kálnay & Torres 1973; Hegerfeldt 1974, 1985, 1989, 1998; Haba 1976; Skagerstam 1976; Perez & Wilde 1977; Wigner & O'Connell 1977, 1978; Hegerfeldt & Ruijsenaars 1980; Haag 1992; Thaller 1992; Dodonov & Mizrahi 1993; Jaekel & Reynaud 1996; Omnes 1997; Ali 1998). It is sometimes said that these difficulties are inevitable because any attempt to localize a particle of rest-mass *m*, on a scale small compared with its Compton wavelength *λ*_{C}=*ℏ*/*mc*, is associated with an uncertainty in energy that is large compared with its rest-energy *mc*^{2}, so that the creation of particle–antiparticle pairs in the measurement process becomes possible, leading to the breakdown of a one-particle description.

However, it has recently been argued (Bracken & Melloy 1999) that, in the case of the free Dirac electron (Dirac 1958), a sensible notion of arbitrarily precise localization at an instant is possible at the one-particle level, in terms of the familiar Dirac position operator which multiplies the covariant Dirac wavefunction *ψ*(* x*,

*t*) by

*. This had long been thought impossible, because*

**x***has no positive-energy (generalized) eigenvectors (Newton & Wigner 1949). However positive-energy, normalized states can be constructed (Bracken & Melloy 1999) with any prescribed value*

**x***for the expectation value 〈*

**a***〉=(*

**x***ψ*,

*) at a prescribed instant, and with arbitrarily small positive*

**x**ψ*Δ*

_{x}, defined in the usual way as(1.1)Furthermore, these states can be chosen such that as

*Δ*

_{x}approaches zero, the mean value of the velocity approaches any prescribed value with magnitude less than

*c*, while the eigenvalue of a suitable spin operator can be held fixed. The existence of such asymptotically localizing positive-energy states appears to define a notion of instantaneous localizability for the electron that is free of the serious difficulties that have dogged earlier attempts to resolve the ‘localization problem’. In particular, localization in this sense has natural transformation properties with respect to spatial translations, rotations and Lorentz boosts, unlike localization in the Newton–Wigner sense (Newton & Wigner 1949).

It must be emphasized that no positive-energy state of the electron has compact support in the coordinate representation (Hegerfeldt & Ruijsenaars 1980; Hegerfeldt 1985, 1989, 1998); every positive-energy state has ‘tails’ extending to infinity in all directions, that typically die off like exp(−|* x*|/

*λ*

_{C}). This is therefore true of asymptotically localizing states, but—and this is the crucial point that seems to have been overlooked for so long—that does not preclude arbitrarily small values of

*Δ*

_{x}being achieved, and in that sense, the electron can be localized as sharply as desired, using only positive-energy normalized states and staying within a one-particle picture.

Before the localization problem for the electron can be considered solved, it is important to show that states which asymptotically localize the electron at an initial time evolve subsequently in an appropriate way under the action of Dirac's equation,(1.2)where * α*,

*β*are the usual Dirac matrices. (In what follows, we choose the standard form with

*β*diagonal (Dirac 1958).)

When positive-energy states are highly localized at an instant, more precisely when they correspond to *Δ*_{x}≪*λ*_{C}, they do indeed have an uncertainty in energy that is large compared with *mc*^{2}. But the effect of this is not for the one-particle picture to break down because of any possibility to create particle–antiparticle pairs. Rather, as we shall see, what happens is that such a one-particle positive-energy state has a very large mean energy, which causes the associated wavefunction and probability density to expand rapidly away from the centre of localization, at speeds that approach *c* more and more closely as the initial value of *Δ*_{x} is made smaller and smaller.

To illustrate this point, we have examined the time-evolution of the probability density *ρ*(* x*,

*t*)=

*ψ*(

*,*

**x***t*)

^{†}

*ψ*(

*,*

**x***t*) partly analytically, partly numerically, for suitably chosen positive-energy states, as described in the next section.

## 2. Theory

From this point onwards we take *ℏ*=1, *c*=1, *m*=1. (Equivalently, * x*,

*and*

**p***t*now represent

*/*

**x***λ*

_{C},

*/*

**p***mc*and

*ct*/

*λ*

_{C}, respectively.) We consider the sequence of positive-energy normalized solutions of Dirac's equation (1.2) defined for

*n*=1, 2, 3,…, by(2.1)where the integral is over all momentum space. Here

*is an arbitrary constant coordinate, and(2.2)where*

**a***u*(

*) is the positive-energy bispinor (Schweber 1961)(2.3)and*

**p***f*(

*) is an arbitrary normalized square integrable function,(2.4)Because*

**p***u*

^{†}(

*)*

**p***u*(

*)=1, we then have(2.5)at all times*

**p***t*as required.

The choice of *f*(* p*) subject to equation (2.4) is largely arbitrary. For simplicity in what follows, we make the special choice(2.6)where

**p**_{0}is an arbitrary constant momentum. Note that as

*n*increases,

*f*(

*/*

**p***n*), and hence

*Φ*

_{n}(

*), becomes more and more spread out in momentum space. As this happens, becomes more and more localized near*

**p***=*

**x***, in the sense that the sequence of corresponding uncertainties in position*

**a***Δ*

_{x,n}(0) approaches zero as

*n*increases. Then is a sequence of positive-energy solutions of Dirac's equation (1.2) that localize the electron more and more closely about

*=*

**x***at time*

**a***t*=0. For more details, see Bracken & Melloy (1999) and Melloy (2002).

Unfortunately it is not possible to obtain an expression for *Δ*_{x,n}(*t*) or *Δ*_{x,n}(0) in closed form, even with the simple choice (2.6) for *f*(* p*), but it is easily seen that this choice does lead to(2.7)In the first of these equations, is the (time-independent) expectation value of the Dirac velocity operator in the state

*Ψ*

_{n}(

*,*

**x***t*). The lack of dependence on

*t*in the right-hand side of the second and third equations reflects the fact that is a constant of the motion. The third equation shows that the uncertainty in the momentum of the particle, expressed here as a multiple of

*mc*, increases without bound as

*n*→∞, as the localization at

*t*=0 becomes sharper and sharper. It follows that the mean energy and the uncertainty in the energy also increase without bound.

A short calculation shows that the choice (2.6) leads also to(2.8)as *n*→∞, where ‘erf’ is the error function (Abramowitz & Stegun 1965). Note that the magnitude of **v**_{0} is less than *c* (=1), and approaches 1 as |**p**_{0}|→∞. Rather than prescribe we could alternatively, as indicated in Bracken & Melloy (1999), have chosen *f* so that approaches a prescribed value with magnitude less than 1 as *n*→∞, but the simple choice (2.6) leading to (2.7) and (2.8) is more convenient for our purposes here.

There is also some arbitrariness in the choice of *u*(* p*). The present choice corresponds to fixing the eigenvalue of the component

*Σ*

_{3}of Pryce's spin operator (Pryce 1935, 1948; Bracken & Melloy 1999), which is a constant of the motion, to have the value +1/2 at every value of

*n*, at all values of

*t*.

To summarize: at *t*=0 and as *n*→∞, the positive-energy state (2.1) localizes the electron more and more closely about 〈* x*〉=

*, with approaching*

**a**

**v**_{0}, and with

*Σ*

_{3}having eigenvalue 1/2. The first picture in figure 5 shows a cross-section of the density

*ρ*

_{n}(

*, 0) in the*

**x***x*

_{1}

*x*

_{2}-plane, in the case

**p**_{0}=

**0**and

*n*=3. Localization well within the Compton wavelength |

*|=1 is evident in this picture, even for such a small value of*

**x***n*. (Note the choice

**p**_{0}=

**0**leads to a spherically symmetric

*ρ*

_{n}(

*,*

**x***t*) at all times, as seen in the subsequent plots in figure 5.)

We now apply the method of stationary phase (Bender & Orszag 1978) to equation (2.1), and consider *t*→∞ in that formula with (* x*−

*)/*

**a***t*fixed. Accordingly, we define(2.9)and consider

*φ*to be independent of

*t*in the limiting process, so that equation (2.1) takes the form(2.10)Because(2.11)the function

*φ*is stationary at just the one point(2.12)Now let (

*) denote the negative of the 3×3 Jacobian matrix of second derivatives of*

**p***φ*(

*),(2.13)Then we have(2.14)as*

**p***→*

**p***, and equation (2.10) gives(2.15)as*

**k***t*→∞. The matrix (

*) is symmetric and is diagonalized by a suitable orthogonal transformation(2.16)In equation (2.15) we make the change of variable(2.17)and noting that is positive, we get(2.18)as*

**k***t*→∞. Because (Abramowitz & Stegun 1965)(2.19)we get(2.20)as

*t*→∞, using (2.9) and (2.12). Note that this asymptotic approximation to

*ρ*

_{n}is correctly normalized(2.21)Bear in mind that there are two asymptotic limits involved here: one is the large

*t*limit in which the method of stationary phase assumes validity, while the other is the large

*n*limit in which the density is increasingly localized at

*=*

**x***at*

**a***t*=0.

## 3. Illustrative results

For the purpose of obtaining representative plots, with no significant loss of generality we take * a*=

**0**and

**p**_{0}=(0,

*p*

_{0}, 0), with

*f*(

*) as in equation (2.6). Consider firstly the spherically symmetric case when*

**p***p*

_{0}=0. Then the asymptotic (long-time) form of the density is given from equation (2.20) by(3.1)where

*=*

**ξ***/*

**x***t*as before. Figure 1 shows a plot of

*F*

_{2}(

*) along any one direction (here the*

**ξ***ξ*

_{2}-direction) through the origin, and figure 2 shows a contour plot of a cross-section of

*F*

_{2}(

*) in the*

**ξ***ξ*

_{1}

*ξ*

_{2}-plane.

These figures, obtained using Maple 9.5, show clearly how at large times the density radiates outwards at almost the speed of light, which defines the surface **ξ**^{2}=1. Note that in this stationary phase approximation, there is no density at all outside this surface, whereas in an exact representation there would be exponentially small such contributions, decreasing in size with increasing time, and arising from propagation of the tails of the initial density. Note also that these figures are for only a small value of *n*. For larger values of *n*, the density is so highly localized near **ξ**^{2}=1 that not only is there no density outside this surface in this approximation, but it is also hard to distinguish anything inside that surface in the corresponding pictures.

In the asymmetric case with *p*_{0}≠0, we have instead of equation (3.1)(3.2)With the choice *p*_{0}=0.5, figures 3 and 4 now replace figures 1 and 2.

In this case, the density again spreads out at close to the speed of light, but now does so asymmetrically in order to give , the appropriate non-zero (constant) value.

In order to see how the density evolves at small times, from *t*=0 onwards, we used Matlab 7 to evaluate the three-dimensional Fourier transform in equation (2.1) and to calculate *ρ*_{n}(* x*,

*t*). Density plots of cross-sections of

*ρ*

_{3}(

*,*

**x***t*) in the

*x*

_{1}

*x*

_{2}-plane at times increasing from

*t*=0 are shown in figures 5 and 6 for the symmetric and asymmetric cases, respectively. The pictures are consistent with those obtained using the stationary phase approximation, which is accurate from surprisingly small times

*t*. The spherical surface is shown as a dashed circle in our cross-sectional plots; at

*t*=0 its radius is the Compton wavelength, and at large

*t*it is indistinguishable from the surface |

*|/*

**x***t*=1 mentioned in connection with figures 1–4.

If the initial localization, as characterized by *Δ*_{x}, is large compared with *λ*_{C}, then the rate of spreading is slow compared with the speed of light, and the behaviour described by Schrödinger's non-relativistic equation is recovered. This can be achieved by giving *n* in equation (2.6), a positive non-integral value close to 0. A plot of the density along one axis at successive times in the case **p**_{0}=**0**, *n*=0.1 is shown in figure 7.

## 4. Concluding remarks

The results obtained here using stationary phase and numerical methods are consistent with the conclusions of Bracken & Melloy (1999), that arbitrarily precise localization of the relativistic electron can be described consistently using Dirac's position operator * x*. Positive-energy states that are initially highly localized on the scale of the Compton wavelength do not lead to pair-production, but rather give rise to densities that expand radially outwards at speeds close to that of light because of their large mean energy.

In our opinion, the pictures shown above, and the associated analysis, provide convincing evidence of the existence of positive-energy states of the relativistic electron that are initially as highly localized as one wishes to make them, and that evolve subsequently in a natural way, consistent with a one-particle picture. It is true that the infinite tails associated with all these densities are obscured in these figures because they die off so rapidly as |* x*|→∞. And it is true that in the non-relativistic limit, one can find localizing states without such tails. But even in the non-relativistic case, that can only be achieved instantaneously, at the moment of measurement, and immediately afterwards such non-relativistic states also develop tails extending to infinity—an instantaneous spreading that is obviously incompatible with the requirements of special relativity. The situation for the relativistic electron differs qualitatively only in the non-existence of this instantaneous, causality-violating effect.

This note will have served its purpose if it lays to rest once and for all, at least for the Dirac electron, the ‘folk theorem’ that arbitrarily precise instantaneous localization of a free particle in relativistic quantum mechanics is impossible. Failure of a one-particle picture is *not* an inevitable consequence of localization on the scale of the Compton wavelength.

## Footnotes

- Received December 15, 2004.
- Accepted June 20, 2005.

- © 2005 The Royal Society