Quantum mechanics of Klein–Gordon fields I: Hilbert Space, localized states, and chiral symmetry

Annals of Physics 321 (2006) 2183–2209 www.elsevier.com/locate/aop

Quantum mechanics of Klein–Gordon fields I: Hilbert Space, localized states, and chiral symmetry A. Mostafazadeh a b

a,*

, F. Zamani

b

Department of Mathematics, Koc¸ University, Rumelifeneri Yolu, 34450 Sariyer, Istanbul, Turkey Department of Physics, Institute for Advanced Studies in Basic Sciences, 45195-1159 Zanjan, Iran Received 16 November 2005; accepted 9 February 2006 Available online 23 March 2006

Abstract We derive an explicit manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein–Gordon equation. This expression involves a one-parameter family of conserved current densities J la , with a 2 (1, 1), that are analogous to the chiral current density for spin half fields. The conservation of J la is related to a global gauge symmetry of the Klein–Gordon fields whose gauge group is U (1) for rational a and the multiplicative group of positive real numbers for irrational a. We show that the associated gauge symmetry is responsible for the conservation of the total probability of the localization of the field in space. This provides a simple resolution of the paradoxical situation resulting from the fact that the probability current density for free scalar fields is neither covariant nor conserved. Furthermore, we discuss the implications of our approach for free real scalar fields offering a direct proof of the uniqueness of the relativistically invariant positive-definite inner product on the space of real Klein–Gordon fields. We also explore an extension of our results to scalar fields minimally coupled to an electromagnetic field.  2006 Elsevier Inc. All rights reserved.

*

Corresponding author. Fax: +90212 3381559. E-mail addresses: [email protected] (A. Mostafazadeh), [email protected] (F. Zamani).

0003-4916/$ - see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2006.02.007

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1. Introduction Every textbook on non-relativistic quantum mechanics has a discussion of how one should compute the probability of the presence of a free spin-zero particle in a region V of space. The same cannot be said for textbooks on relativistic quantum mechanics, because the (inner product on the) Hilbert space for such a particle (i.e., a free scalar/ Klein–Gordon field) and the nature of its position operator are not universally fixed by the basic axioms of quantum theory. It is well-known that such a particle can be safely described by a first-quantized scalar field. Therefore, the above problem of finding the probability of its localization in V is a physical problem demanding a consistent formulation within first-quantized relativistic quantum mechanics. The aim of this paper is to provide an explicit formulation of quantum mechanics of both real and complex Klein– Gordon fields that would enable one to address the relativistic analogs of typical quantum mechanical problems. Specifically, we • determine the Hilbert space by providing an explicit manifestly covariant expression for the most general relativistically invariant positive-definite inner product on the space of all solutions of the Klein–Gordon equation (including positive- and negative-energy solutions and their superpositions), • construct the observables of the theory, • formulate it in terms of the position wave functions, • identify the probability current density for the localization of the field in space, • offer a resolution of the conceptual difficulty caused by the local non-conservation and non-covariance of the probability current density and the global conservation and covariance of the total probability. A by-product of our analysis is a simple proof of the uniqueness of the relativistically invariant positive-definite inner product on the space of real Klein–Gordon fields. Our general results have applications in the study of relativistic coherent states for real and complex Klein–Gordon fields. This is the subject of a companion paper [1]. The demand for devising a probabilistic interpretation for Klein–Gordon fields is among the oldest problems of modern physics. Though this problem was never fully resolved, it provided the motivation for some of the most important developments of the 20th-century theoretical physics. The most notable of these are the discovery of the Dirac equation and the advent of the method of second-quantization which eventually led to the formulation of the quantum field theories. The latter, in turn, provided the grounds for completely disregarding the original problem of finding a probabilistic interpretation for Klein–Gordon fields as there were sufficient evidence that the interacting first-quantized (scalar) field theories involved certain inconsistencies such as the Klein paradox [2].1 There were also some general arguments suggesting that the ‘‘localization’’ of bosonic fields in space was not possible [4].2 These led to the consensus that the correct physical picture was provided by second-quantized field theories.

1

See however [3]. Here by impossibility of the localization of a field, one means the non-existence of a covariant current density whose time-like (zeroth) component takes non-negative values. 2

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This point of view, which is almost universally accepted, is not in conflict with the demand for a consistent formulation of first-quantized relativistic quantum mechanics of a free Klein–Gordon field. This is actually required in the study of the relationship between non-relativistic and relativistic quantum mechanics. The conventional interpretation of Klein–Gordon fields is provided in terms of the Klein–Gordon current density $

J lKG ¼ igwðx0 ;~ xÞ ol wðx0 ;~ xÞ.

ð1Þ

Here g 2 Rþ is a constant, w is a solution of the Klein–Gordon equation ½ol ol  M2 wðx0 ;~ xÞ ¼ 0;

ð2Þ

M :¼ mc= h is the inverse of the Compton’s wave length, m is the mass of w, olol = glmolom, glm are components of the inverse of the Minkowski metric (glm) with signature (1, 1, 1, 1), and for any pair of Klein–Gordon fields w1 and w2, $ w1 ol w2 :¼ w1 ol w2  ðol w1 Þw2 . As it is discussed in most textbooks on relativistic quantum mechanics, J lKG is a conserved four-vector current density, i.e., it is a four-vector field satisfying the continuity equation ol J lKG ¼ 0. Therefore,

J 0KG

Z

Q :¼ R3

ð3Þ

may be used to define a conserved quantity, namely

d3~ xÞ. xJ 0KG ðx0 ;~

ð4Þ

The fact that Q (respectively, J 0KG ) takes positive as well as negative values does not allow one to identify it with a probability (respectively, probability density). Instead, one identifies Q with the electric charge of the field and views J lKG as the corresponding four-vector charge density [5]. In this way the continuity Eq. (3) provides a differential manifestation of the electric charge conservation. It can be shown that Q takes positive values for positive-energy Klein–Gordon fields and that one can define a Lorentz-invariant positive-definite inner product on the set of positive-energy fields. This is the basis of the point of view according to which one identifies the physical Hilbert space with the subspace of positive-energy fields [6]. This approach however fails for the cases that the Klein–Gordon field is subject to a time-dependent background field, for in this case the notion of a positive-energy Klein–Gordon field is ill-defined. Even in the absence of time-dependent background fields, the above restriction to the positive-energy fields limits the choice of possible observables to those that do not mix positive- and negative-energy fields. This leads to certain difficulties in modelling relativistic measurements that involves postulating local interactions between the field and the measuring device [7]. There are also well-known (and related) problems regarding the violation of causality [8].3 The safest way out of all these difficulties seems to be a total abandonment of the first-quantized field theories as viable physical theories [11]. Nevertheless, one should also accept that there is no rigorous argument showing that these difficulties cannot be resolved for non-interacting first-quantized fields. Indeed, as we shall 3

See however [9,10,7] and references therein.

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demonstrate below one can develop a consistent quantum theory for the first-quantized scalar fields that are either free or minimally coupled to a time-independent magnetic field. The issue of finding a probabilistic interpretation for first-quantized Klein–Gordon fields has also been extensively studied in the context of canonical quantum gravity. There it emerges as a fundamental obstacle in developing a quantum theory of gravity [12]. This time neither Dirac’s trick of considering an associated first order field equation nor the application of the method of second-quantization can be applied satisfactorily [13]. It is also not possible to define a subset of positive-energy solutions which would serve as the underlying vector space for the ‘physical Hilbert space’ of the theory. This necessitates dealing with the above-described problems with first-quantized fields directly. The lack of a probabilistic interpretation for canonical quantum gravity and its simplified version known as quantum cosmology is widely referred to as the Hilbert-space problem [13]. This terminology reflects the view that the problem of devising a probabilistic interpretation for the wave functions appearing in these theories (namely the Wheeler– DeWitt fields) is equivalent to constructing a Hilbert space to which these fields belong. In this way one can identify the observables of the theory with Hermitian operators acting in this Hilbert space and utilize Born’s probabilistic interpretation of quantum mechanics. The Klein–Gordon charge density may be used to define an inner product on the space of solutions of the Klein–Gordon equation. This is known as the Klein–Gordon inner product Z   ðw1 ; w2 ÞKG ¼ ig d3~ xÞ w_ 2 ðx0 ;~ xÞ  w_ 1 ðx0 ;~ xÞ w2 ðx0 ;~ xÞ; ð5Þ x½w1 ðx0 ;~ R3

where w1 and w2 are Klein–Gordon fields, g is a non-zero positive and an R real 0constant, overdot stands for a x0-derivative, i.e., o0. Clearly, ðw; wÞKG ¼ d3~ xÞ ¼ Q. As xJ KG ðx0 ;~ Q may be negative, the Klein–Gordon inner product is indefinite [14]. Therefore, endowing the (vector) space V of solutions of the Klein–Gordon equation with the Klein–Gordon inner product does not produce a genuine inner product space. One may pursue the approach of the indefinite-metric quantum theories [15] and identify the subspace V þ of positive-energy solutions as the physical space of state vectors. Restricting the Klein–Gordon inner product to this subspace one obtains a (definite) inner product space that can be extended to a separable Hilbert space Hþ through Cauchy completion [16]. This is the basis of developing quantum field theories in curved background spacetimes [17,18]. To ensure that the right-hand side of (5) is convergent, it is sufficient to assume that for _ 0 Þ : R3 ! C defined by all x0 2 R, the functions wðx0 Þ; wðx _ 0 Þð~ wðx xÞ :¼ o0 wðx0 ;~ xÞ; 3 2 0 0 _ Þ 2 L ðR Þ. This is an assumption that we shall make are square-integrable, i.e., wðx Þ; wðx throughout this paper. It is supported by the fact that w tends to a solution of the free Schro¨dinger equation in the non-relativistic limit (c fi 1). We can respectively express the Klein–Gordon equation (2), the space V of its solutions, _ 0 Þ accordand the Klein–Gordon inner product (5) in terms of the functions w (x0) and wðx ing to: wðx0 Þð~ xÞ :¼ wðx0 ;~ xÞ;

€ 0 Þ þ Dwðx0 Þ ¼ 0; wðx n o  wðx0 Þ þ Dwðx0 Þ ¼ 0 ; V ¼ w : R ! L2 ðR3 Þ€

ð6Þ ð7Þ

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h i ðw1 ; w2 ÞKG ¼ ig hw1 ðx0 Þjw_ 2 ðx0 Þi  hw_ 1 ðx0 Þjw2 ðx0 Þi ;

2187

ð8Þ

where D : L2 ðR3 Þ ! L2 ðR3 Þ is the operator defined by ðD/Þð~ xÞ :¼ ðr2 þ M2 Þ/ð~ xÞ 8/ 2 L2 ðR3 Þ;

ð9Þ

w1 ; w2 2 V are arbitrary, and ÆÆ|Ææ stands for the inner product of L ðR Þ. Recently, it has been noticed that one can devise a systematic method of endowing V with a positive-definite inner product [19–21]. In particular, one may define a positive-definite and relativistically invariant inner product on V according to i 1 h hw1 ðx0 ÞjD1=2 w2 ðx0 Þi þ hw_ 1 ðx0 ÞjD1=2 w_ 2 ðx0 Þi . ðw1 ; w2 Þ :¼ ð10Þ 2M 2

3

Note that D and consequently D1 are positive-definite operators (Hermitian operators with strictly positive spectra) and D±1/2 is the unique positive square root of D±1. In addition to being positive-definite and relativistically invariant, the inner product (10) is a conserved quantity in the sense that the x0-derivative of the right-hand side of (10) vanishes. As explained in [20], this is required to make the inner product (10) well defined. The expression (10) was originally obtained in [19] using the results of the theory of pseudo-Hermitian operators [22]. The construction of a positive-definite and relativistically invariant inner product for Klein–Gordon fields has been previously considered by Wald [18], Halliwell and Ortiz [23] and Woodard [24].4 The construction proposed by Wald [18] (see also [27]) uses the symplectic structure of the solution space of the Klein–Gordon equation [28]. We shall discuss this construction in some detail in Section 5. Here we suffice to mention that it yields an implicit expression for a positive-definite and relativistically invariant inner product whose explicit form coincides with (10). In contrast to Wald, Halliwell and Ortiz [23] study various Green’s functions for the Klein–Gordon equation and use the Hadamard Green function to introduce a positive-definite inner product. The approach of Woodard [24] is also completely different. He employs the idea of gauge-fixing the inner product of the auxiliary Hilbert space obtained in the quantization of the classical relativistic particle within the framework of constraint quantization. A detailed application of this idea for Klein–Gordon fields is given in [29,30,9].5 The approaches of Halliwell–Ortiz and Woodard yield the same expression for the inner product, namely  w1 ; w2 :¼ ðw1þ ; w2þ ÞKG  ðw1 ; w2 ÞKG ;

ð11Þ

where, for i = 1, 2, wi+ and wi are, respectively, the positive-energy and negative-energy parts of the Klein–Gordon field wi. As we shall see below, (11) is yet another implicit form of the inner product (10). The advantage of the approach of [19] over those of [18,23,24,29,30,9] is that not only it yields an explicit expression for a positive-definite and relativistically invariant inner product, but it also allows for a construction of all such inner products: The most general positive-definite, relativistically invariant, and conserved inner product on the space V of solutions of the Klein–Gordon equation (6) is given by [19,20] 4

See also [25,26]. The method of refined algebraic quantization also known as the group-averaging [31,32] is an extension of this idea. 5

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ðw1 ; w2 Þa ¼

j n hw1 ðx0 ÞjD1=2 w2 ðx0 Þi þ hw_ 1 ðx0 ÞjD1=2 w_ 2 ðx0 Þi 2Mh io þia hw ðx0 Þjw_ 2 ðx0 Þi  hw_ 1 ðx0 Þjw ðx0 Þi ; 1

2

ð12Þ

where j 2 Rþ and a 2 (1, 1) are arbitrary constants. If we set g ¼ 1=ð2MÞ in (8), we can express (12) in the form ðw1 ; w2 Þa ¼ j½ðw1 ; w2 Þ þ aðw1 ; w2 ÞKG .

ð13Þ

Note that j is an unimportant multiplicative constant that can be absorbed in the definition of the Klein–Gordon fields w1 and w2. The existence of the positive-definite inner products (13) and their relativistic invariance and conservation raise the natural question whether there is a conserved four-vector current density associated with these inner products. A central result of the present article is that such a current density exists. In fact, we will construct a one-parameter family J la (with a 2 (1, 1)) of current densities and show by direct computation that not only they transform as vector fields but they also satisfy the continuity equation and yield the wellknown Schro¨dinger probability current density in non-relativistic limit. As we will show below, J la is analogous to the chiral current density for spin half fields. It has two remarkable applications. First, it may be used to yield a manifestly covariant expression for the inner products (12). Second, its local conservation law may be linked with the conservation of the total probability of the localization of the field in space. This turns out to be a consequence of a previously unnoticed Abelian global gauge symmetry of the Klein– Gordon equation. The organization of the article is as follows. In Section 2, we outline a derivation of the current densities J la and explore their properties. In Section 3, we derive the probability current density for the localization of a Klein–Gordon field in space and show how it relates to the current densities J la . In Section 4, we study the underlying gauge symmetry associated with the conservation of J la . In Section 5, we describe the implications of our findings for real scalar fields. In Section 6, we discuss a generalization of our results to Klein–Gordon fields interacting with a background electromagnetic field. In Section 7, we present our concluding remarks. The appendices include some useful calculations that are, however, not of primary interest. 2. Derivation and properties of J la Let w 2 V be a Klein–Gordon field. Then in view of (12), we have Z n j _ 0 ;~ _ 0 ;~ ^ 1=2 wðx0 ;~ ^ 1=2 wðx ðw; wÞa ¼ d3~ xÞ D xÞ þ wðx xÞ D xÞ x wðx0 ;~ 2M R3 h io _  0 _ 0 ;~ þia wðx0 ;~ xÞ wðx ;~ xÞ  wðx xÞ wðx0 ;~ xÞ

ð14Þ

^ :¼ r2 þ M2 . Now, using the analogy with non-relativistic quantum mechanics, with D we define the current density J 0a associated with w as the integrand in (14). That is h io j n  ^ 1=2 _ _ _  wðxÞ ; ð15Þ _ D ^ 1=2 wðxÞ wðxÞ D þ ia wðxÞ wðxÞ  wðxÞ J 0a ðxÞ :¼ wðxÞ þ wðxÞ 2M where we have set x :¼ ðx0 ;~ xÞ.

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To obtain the spatial components J ia (with i 2 {1, 2, 3}) of J la , we follow the procedure outlined in [33]. Namely, we perform an infinitesimal Lorentz boost transformation that changes the reference frame to the one moving with a velocity ~ v. That is we consider 0 b ~ x; ~ x !~ x0 ¼ ~ x ~ bx0 ; ð16Þ x0 ! x0 ¼ x0  ~ ~ where b :¼ ~ v=c, and we ignore second and higher order terms in powers of the components of ~ b. Assuming that J la is indeed a four-vector field, we obtain the following transformation rule for J 0a 0 0 ~ ~ J 0a ðxÞ ! J 00 a ðx Þ ¼ J a ðxÞ  b  J a ðxÞ.

ð17Þ 0 J 00 a ðx Þ,

Next, we recall that we can use (15) to read off the expression for namely n h io j w0 ðx0 Þ D^01=2 w0 ðx0 Þ þ w_ 0 ðx0 Þ D^01=2 w_ 0 ðx0 Þ þ ia w0 ðx0 Þ w_ 0 ðx0 Þ  w_ 0 ðx0 Þ w0 ðx0 Þ ; J 00a ðx0 Þ :¼ 2M ð18Þ 2 0 0 2 0 0 00 0 0 0 0 0 0 _ ^ ¼ r þ M and w ðx Þ :¼ ow ðx Þ=ox . This reduces the determiwhere x :¼ ðx ;~ x Þ, D nation of ~ J a to expressing the right-hand side of (18) in terms of the quantities associated with the original (unprimed) frame and comparing the resulting expression with (17). It is obvious that w is a scalar field w0 ðx0 Þ ¼ wðxÞ.

ð19Þ

A less obvious fact is that D1=2 w_ is also a scalar field. This can be directly checked by performing an infinitesimal Lorentz transformation as we demonstrate in Appendix A. Alternatively, we may appeal to the observation that the generator h of x0-translations, that is defined [20] as the operator hw :¼ i hw_ acting in the space V of Klein–Gordon fields, squar2 es to  h D. Hence, as noted in [21],  h1D1/2h is nothing but the charge-grading operator C. This in turn means that iD1=2 w_ ¼ Cw ¼: wc .

ð20Þ

Clearly wc is also a scalar field, and consequently D1=2 w_ is Lorentz invariant _ ^ 01=2 w_ 0 ðx0 Þ ¼ D ^ 1=2 wðxÞ. D

ð21Þ

Next, we use (16) and (19) to deduce: _ ~ w_ 0 ðx0 Þ ¼ wðxÞ þ~ b  rwðxÞ; 0a

~D ^ ¼D ^ a  2a~ ^ a1 o0 D br

ð22Þ 8a 2 R.

ð23Þ

In view of Eqs. (21)–(23), we then have:  ^ 01=2 0 0  ^ 1=2  _ ~D ^ 1=2 wðxÞ; br w ðx Þ ¼ wðxÞ D wðxÞ  wðxÞ ~ w0 ðx0 Þ D ^ w_ 0 ðx0 Þ D

01=2

 ^ 1=2 _ _ _ D ~ ^ 1=2 wðxÞ w_ 0 ðx0 Þ ¼ wðxÞ þ ½~ b  rwðxÞ wðxÞ. D

ð24Þ ð25Þ

Now, we substitute (19), (22), (24), and (25) in (18) and make use of (17) to obtain h io j n  ^ 1=2 _  _ ~ ~D ~ ~ ~ ^ 1=2 wðxÞ wðxÞ r  ½rwðxÞ wðxÞ  ia wðxÞ rwðxÞ J a ðxÞ ¼ D wðxÞ .  ½rwðxÞ 2M ð26Þ This relation suggests

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J la ðxÞ

  $ $ j  l ^ 1=2 _  l wðxÞ o D wðxÞ  iawðxÞ o wðxÞ . ¼ 2M

ð27Þ

It is not difficult to check (using the Klein–Gordon equation) that the expression for J 0a obtained using this equation agrees with the one given in (15). We can use (20) to further simplify (27). This yields6   $ ij ~ a ðxÞ ; wðxÞ ol w J la ðxÞ ¼  ð28Þ 2M where ~ a :¼ w þ aw. w c The current density

ð29Þ J la

has the following remarkable properties.

~ a are scalar fields, J l is 1. The expression (28) for J la is manifestly covariant; since w and w a indeed a four-vector field. ~ a satisfy the Klein–Gordon equation (2), one can show 2. Using the fact that both w and w (by a direct calculation) that the following continuity equation holds: ol J la ¼ 0.

ð30Þ

J la

Hence is a conserved current density. 3. As we show in Appendix B, in the non-relativistic limit as c fi 1, J la tends to the Schro¨dinger’s probability current density for a free particle. Specifically, setting j = 1/(1 + a), we find: xÞ ¼ .ðx0 ;~ xÞ; lim J 0a ðx0 ;~

c!1

ð31Þ

1 0 lim ~ J a ðx0 ;~ xÞ ¼ ~ xÞ; ð32Þ jðx ;~ c!1 c where . and ~ j are, respectively, the non-relativistic scalar and current probability densities [11]: .ðx0 ;~ xÞ :¼ jwðx0 ;~ xÞj2 ; i i hh  ~  0 ~ 0 ;~ ~ wðx0 ;~ xÞ :¼  xÞ rwðx ;~ xÞ  wðx0 ;~ xÞrwðx xÞ . jðx0 ;~ 2m

ð33Þ ð34Þ

4. Although J 0a ðxÞ has been constructed out of a positive-definite inner product, namely (12), it is in general not even real. This can be easily checked by computing J 0a ðxÞ for a linear combination of two plane wave solutions of the Klein–Gordon equation with different and oppositely signed energies. Similarly J la is complex-valued. The real and imaginary parts of J la are by construction real-valued conserved four-vector current densities. They are further studied in Appendix C.

6

Note that the results of [34] pertaining the uniqueness of the Klein–Gordon current density do not rule out the existence of the current density (27), because these results are obtained under the assumption that the current ~ a in (28)) is a clear ^ 1=2 in (27) (alternatively w involves only the field and its first derivatives. The appearance of D indication that this assumption is violated.

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We can use the relation (27) for the current density J la and Eq. (12) to yield a manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein–Gordon equation (2), namely   Z $ $ ij  l  l ðw1 ; w2 Þa ¼  drðxÞnl ðxÞ w1 ðxÞ o Cw2 ðxÞ þ aw1 ðxÞ o w2 ðxÞ ; ð35Þ 2M r where r is an arbitrary spacelike (Cauchy) hypersurface of the Minkowski space with volume element dr and unit (future) timelike normal four-vector nl. Note that in deriving (35) we have also made an implicit use of the polarization principle [35], namely that any inner product is uniquely determined by the corresponding norm. Next, we recall that in view of (2) and (20), C 2 w ¼ w. Hence, C viewed as a linear operator acting on V is an involution, C 2 ¼ 1. We can use it to split V into the subspaces V  of ±-energy Klein–Gordon fields according to V  :¼ fw 2 VjCw ¼ w g.

ð36Þ

Clearly, for all w 2 V, we can introduce the corresponding ±-energy components: w :¼ 12ðw  CwÞ 2 V 

ð37Þ

with the property w = w+ + w. If we decompose the fields w1 and w2 appearing in (35) into their ±-energy components w1± and w2± and use (5) with g ¼ 1=ð2MÞ, we find   ð38Þ ðw1 ; w2 Þa ¼ j ð1 þ aÞðw1þ ; w2þ ÞKG  ð1  aÞðw1 ; w2 ÞKG . If we set j = 1 and a = 0, we have (w1, w2)a = (w1, w2), and (38) shows that the inner product (11) coincides with the inner product (10). Another consequence of (38) is the identity ðw1 ; Cw2 Þa ¼ ðCw1 ; w2 Þa ; which holds for all a 2 (1, 1). Therefore, if we identify (35) with the inner product of the physical Hilbert space for Klein–Gordon fields, the charge-grading operator C is a Hermitian involution acting in the physical Hilbert space. In this sense, it is the analog of the chirality operator c5 of the Dirac theory of spin 1/2 fields. As we shall see in Section 4, the conserved current J la is associated with certain global gauge transformations of the Klein–Gordon fields that are analogous to the chiral transformations of the Dirac spinors [11] for a = 0 and a twisted version of the latter for a „ 0. This suggests the terminology: ‘twisted chiral current’ for J la . 3. Probability current density for localization of Klein–Gordon fields in space 2

In non-relativistic quantum mechanics, the interpretation of jwð~ x; tÞj as the probability density for the localization of a particle in (configuration) space relies on the following basic premises. 1. The state of the particle is described by an element |w (t)æ of the Hilbert space L2 ðR3 Þ. 2. There is a Hermitian operator ~ x representing the position observable whose eigenvectors j~ xi form a basis.

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3. The position wave function wð~ x; tÞ uniquely determines the state vector |w(t)æ and consequently the corresponding state, because wð~ x; tÞ are the coefficients of the expansion of |w (t)æ in the position basis: Z jwðtÞi ¼ d3~ xwð~ x; tÞj~ xi. ð39Þ R3

4. The probability of localization of the particle in a region V R3 at time t 2 R is given by Z 2 P V ðtÞ ¼ d3~ ð40Þ xkK~x jwðtÞik ; V

where Kx :¼ j~ xih~ xj is the projection operator onto j~ xi, iÆi2 := ÆÆ|Ææ, and the state vector |w(t)æ is supposed to be normalized, i|w(t)æi = 1. It is because of the orthonormality of the position eigenvectors, i.e., h~ xj~ x0 i ¼ d3 ð~ x ~ x0 Þ, that we can write (40) in the form Z 2 P V ðtÞ ¼ d3~ ð41Þ xjwð~ x; tÞj . V

We maintain that the same ingredients are necessary for defining the probability density for the localization of the Klein–Gordon fields in space, i.e., one must first define a genuine ~ for the Klein–Gordon fields and then use the posiHilbert space and a position operator X tion eigenvectors to define a position wave function associated with each Klein–Gordon field. Refs. [20,21] give a thorough discussion of how one can construct the Hilbert space, a position operator, and the corresponding position wave functions for the Klein–Gordon and similar fields. For completeness, here we include a brief summary of this construction, elaborate on its consequences, and present its application in our attempt to determine the probability density for the localization of a Klein–Gordon field in space. 3.1. The Hilbert space Endowing the vector space V of Eq. (7) with the inner product (12) and performing the Cauchy completion of the resulting inner product space yield a separable Hilbert space Ha for each choice of the parameter a 2 (1, 1). However, as discussed in great detail in [20], the choice of a is physically irrelevant, because different choices yield unitarily equivalent Hilbert spaces Ha . In particular, for all a 2 (1, 1), there is a unitary transformation7 U a : Ha ! L2 ðR3 Þ L2 ðR3 Þ. In fact, we can obtain the explicit form of Ua rather easily. In view of the general results of [20], for all w 2 Ha U a w :¼

1 2

! ! pffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 1=4 rffiffiffiffiffiffi _ 0 Þ 1 þ a½wðx00 Þ þ wc ðx00 Þ 1 þ a½D wðx00 Þ þ iD1=4 wðx j 1 j 1=4 0 D ¼ ; p ffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffi 1=4 _ 0 Þ M 2 M 1  a½wðx00 Þ  wc ðx00 Þ 1  a½D wðx00 Þ  iD1=4 wðx 0

ð42Þ

where

x00

0

2 R is a fixed initial value for x .

The unitarity of Ua means that for all w1 ; w2 2 Ha , ÆUaw1, Uaw2æ = (w1, w2)a where ÆÆ, Ææ stands for the inner product of L2 ðR3 Þ L2 ðR3 Þ [16]. 7

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As far as the physical properties of the system are concerned we can confine our attention to the simplest choice for a, namely a = 0. In this way we obtain the Hilbert space H :¼ H0 that is mapped to L2 ðR3 Þ L2 ðR3 Þ via the unitary operator U := U0. Setting a = 0 in (42) we find [21] ! ! rffiffiffiffiffiffi rffiffiffiffiffiffi _ 0Þ 1 j D1=4 wðx00 Þ þ iD1=4 wðx 1 j 1=4 wðx00 Þ þ wc ðx00 Þ 0 U w :¼ ¼ D . ð43Þ _ 0Þ 2 M D1=4 wðx00 Þ  iD1=4 wðx 2 M wðx00 Þ  wc ðx00 Þ 0 We can also calculate the inverse of U. The result is [21] rffiffiffiffiffiffi i  1  0 M 1=4 h iðx0 x0 ÞD1=2 0 0 1=2 0 U n ðx Þ ¼ e n1 þ eiðx x0 ÞD n2 ; D ð44Þ j

 n1 where n ¼ 2 L2 ðR3 Þ L2 ðR3 Þ and x0 2 R are arbitrary. n2 It is important to note that Ua (and in particular U) depend on the choice of x00 . Therefore they fail to be unique. 3.2. Position and momentum operators Let ~ x and ~ p be the usual position and momentum operators acting in L2 ðR3 Þ, respectively, j~ xi be the position eigenvectors satisfying Z ~ x ~ x0 Þ; d3~ xj~ xih~ xj ¼ 1; ð45Þ xj~ xi ¼ ~ xj~ xi; h~ xj~ x0 i ¼ d3 ð~ R3

ri with i 2 {1, 2, 3} be the Pauli matrices





0 1 0 i 1 r1 ¼ ; r2 ¼ ; r3 ¼ 1 0 i 0 0

0 1

 ;

ð46Þ

3 3 2 2 and r0 be the P32 · 2 identity matrix. Then any observable acting in L ðR 2Þ 3L ðR Þ is of the form O ¼ l¼0 Ol  rl where Ol are Hermitian operators acting in L ðR Þ. This in turn implies that the general form of the observables (Hermitian operators) acting in the Hilbert space H is given by U1OU, for U : H ! L2 ðR3 Þ L2 ðR3 Þ is a unitary operator. In ~; ~ particular, as proposed in [21], we may identify the operators X P : H ! H, defined by

~ :¼ U 1 ð~ X x  r0 ÞU ;

~ P :¼ U 1 ð~ p  r0 ÞU

ð47Þ

with position and momentum operators for the Klein–Gordon fields, respectively. It turns ~ out that ð~ P wÞðx0 Þ ¼ ~ pwðx0 Þ which one may abbreviate as ~ P ¼~ p. The position operator X has the following more complicated expression whose derivation we give in Appendix D. ~ ¼~ X xþ

i h~ p i hðx0  x00 Þ~ p  o0 . 2 ~ p2 þ m2 þm Þ

2ð~ p2

ð48Þ

~ to the subspace of positive-energy Klein–GorAs pointed out in [21] the restriction of X don fields coincides with the Newton–Wigner position operator [36]. 3.3. Localized states and position wave functions Clearly the operators ~ x  r0 and 1  r3 from a maximal commuting set of observables acting in L2 ðR3 Þ L2 ðR3 Þ. Hence their common eigenvectors

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nð;~xÞ :¼ j~ xi  e ð49Þ



 1 0 with eþ :¼ and e :¼ , form a complete orthonormal basis of L2 ðR3 Þ L2 ðR3 Þ. 0 1 This together with the fact that U is a unitary transformation imply that the fields wð;~xÞ :¼ U 1 nð;~xÞ

ð50Þ

form a complete orthonormal basis of H, i.e. X Z ð;~ xÞ ð0 ;~ x0 Þ 3 0 Þ0 ¼ d;0 d ð~ x ~ x Þ; d3~ xjwð;~xÞ Þðwð;~xÞ j ¼ I; ðw ; w ¼1

ð51Þ

R3

where for all w 2 H, |w)(w| is the projection operator defined by |w)(w|/ := (w,/)0w and I is the identity map acting in H. Furthermore, we have for both  = ±1 ~wð;~xÞ ¼ ~ X xwð;~xÞ .

ð52Þ

It is also not difficult to see [21] that the charge-grading or chirality operator is given by C ¼ U 1 ð1  r3 ÞU . Hence Cwð;~xÞ ¼ wð;~xÞ .

ð53Þ ð;~ xÞ

represent spatially localized Klein–Gordon In view of Eqs. (51)–(53), the state vectors w fields with definite (charge-) grading .8 They can be employed to associate each Klein– Gordon field w 2 H with a unique position wave function, namely f ð;~ xÞ :¼ ðwð;~xÞ ; wÞ0 .

ð54Þ

As shown in [21], one can use these wave functions to represent all the physical quantities associated with the Klein–Gordon fields. In particular, the transition amplitudes between two states (inner product of two state vectors) take the simple form X Z  ðw1 ; w2 Þ0 ¼ d3~ xÞ f2 ð;~ xÞ; ð55Þ xf1 ð;~ ¼1

R3

where w1 ; w2 2 H and f1, f2 are the corresponding wave functions. As suggested by (55), the wave functions f ð;~ xÞ belong to L2 ðR3 Þ. Moreover due to the ð;~ xÞ implicit dependence of w on x00 appearing in the expression for U, f ð;~ xÞ depend on x00 . This dependence becomes explicit once we express f ð;~ xÞ in terms of w directly. To see this, we first substitute (44) and (49) in (50) to obtain rffiffiffiffiffiffi M 1=4 iðx0 x0 ÞD1=2 ð;~ xÞ 0 0 w ðx Þ ¼ e j~ xi. ð56Þ D j We then use this equation and (12) to compute the right-hand side of (54). This yields rffiffiffiffiffiffi j ^ 1=4 iðx0 x0 ÞD^1=2 0 f ð;~ xÞ ¼ w ðx0 ;~ xÞ; ð57Þ D e M where w :¼ 12ð1 þ CÞw ¼ 12ðw þ wc Þ 8

The important observation that there is a classical observable associated with  is made in [25].

ð58Þ

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2195

is the definite-charge (definite-energy) component of w with charge-grading . Note however that w satisfies the Foldy equation [37] ^ 1=2 w ðx0 ;~ xÞ ¼ D xÞ. io0 w ðx0 ;~

ð59Þ

This in turn implies 0 x0 ÞD ^ 1=2 0

eiðx

w ðx0 ;~ xÞ ¼ w ðx00 ;~ xÞ.

Hence (57) takes the simple form rffiffiffiffiffiffi j ^ 1=4 f ð;~ xÞ ¼ xÞ. D w ðx00 ;~ M

ð60Þ

As seen from this equation the wave functions f ð;~ xÞ depend on x00 . It is also interesting to note that one can use (56) to compute wð;~yÞ ðx0 ;~ xÞ :¼ h~ xjwð;~yÞ ðx0 Þi h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii9 8 rffiffiffiffiffiffi 0 0 Z 1
1

5=4 M 3=4 3=2 1 M wð;~yÞ ðx00 ;~ xÞ ¼ K 5=4 ðMj~ x ~ yjÞ; ð61Þ 2 p C j 4 j~ x ~ yj where C stands for the Gamma function. Eq. (61) provides an explicit demonstration of the curious fact that wðþ;~xÞ are indeed identical with the Newton–Wigner localized states [36] and that wð;~xÞ are the negative-energy analogs of the latter. It is remarkable that we have obtained these localized states without pursuing the axiomatic approach of [36]. A perhaps more important observation is that actually one does not need to use the rather complicated expression (61) in calculating physical quantities [20,21]. One can instead employ the corresponding wave functions which are simply delta functions: The wave function fð;~xÞ ð0 ;~ x0 Þ for wð;~xÞ ðx00 Þ has the form d;0 dð~ x ~ x0 Þ. 3.4. Probability density for spatial localization of a field ~ and position wave funcHaving obtained the expression for the position operator X tions f ð;~ xÞ, we may proceed as in non-relativistic quantum mechanics and identify the probability of the localization of a Klein–Gordon field w in a region V R3 , at time t0 ¼ x00 =c, with Z 2 PV ¼ d3~ ð62Þ xkP~x wk0 ; V

~ with eigenvalue ~ where P~x is the projection operator onto the eigenspace of X x, i.e. X ð;~xÞ ð;~xÞ P~x ¼ jw Þðw j; ¼1

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k  k0 :¼ ð; Þ0 is the square of the norm of H, and we assume iwi0 = 1. Substituting this relation in (62) and making use of (51) and (54), we have X Z 3 2 PV ¼ d~ xjf ð;~ xÞj . ¼1

V

Therefore, the probability density is given by o X j n ^ 1=4 2 2 2 _ 0 ;~ ^ 1=4 wðx jD wðx00 ;~ xÞ ¼ jf ð;~ xÞj ¼ xÞj þ jD qðx00 ;~ . 0 xÞj 2M ¼1

ð63Þ

To establish the second equality in (63), we have made use of (60), (58), and (20). For a position measurement to be made at time t = x0/c, we have the probability density o j n ^ 1=4 2 2 _ 0 ;~ ^ 1=4 wðx xÞ ¼ xÞj þ jD xÞj . jD wðx0 ;~ ð64Þ qðx0 ;~ 2M We can use (20) to express q in the following slightly more symmetrical form: o j n ^ 1=4 2 ^ 1=4 wc ðxÞj2 . jD wðxÞj þ jD qðxÞ ¼ 2M

ð65Þ

Although the above discussion is based on a particular choice for the parameter a, namely a = 0, it is generally valid. To see this, suppose we choose to work with the inner product (12) and hence the Hilbert space Ha for some a „ 0. Then we have a different posi~ 1 ~a :¼ U 1 ð~ tion operator: X a x  r0 ÞU a ¼ U a X U a where U a :¼ U 1 a U

ð66Þ

~a are clearly related is a unitary operator mapping H onto Ha . The eigenvectors of X xÞ ð;~ xÞ 1 to wð;~xÞ by wð;~ ¼ U w . Now, given w 2 H , we define w :¼ U w a a a a and check that the a a position wave function for wa is given by xÞ wð;~ a

xÞ ð;~ xÞ xÞ :¼ ðwð;~ ; U a wÞa ¼ ðwð;~xÞ ; wÞ ¼ f ð;~ xÞ. fa ð;~ a ; wa Þa ¼ ðU a w

ð67Þ

Here we made use of the fact that U a : H ! Ha is a unitary operator. As seen from (67), the position wave functions for w and wa coincide. As a result so do the corresponding probability densities. If we are to compute the probability density qa of the spatial localization of a Klein– ~a for a „ 0, we have, Gordon field w with the position operator being identified with X 0 for a measurement made at t0 ¼ x0 =c o j n ^ 1=4 0 0 2 ^ 1=4 w_ 0 ðx0 ;~ jD wa ðx0 ;~ qa ðx00 ;~ xÞ ¼ xÞj2 þ jD xÞj ; ð68Þ a 0 2M where w0a :¼ U 1 a w. We can compute the latter using (42), (43), and (66). This leads to _ 0 Þ; ^ 1=2 wðx w0a ðx00 Þ ¼ aþ wðx00 Þ þ ia D 0 where a :¼ 12

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 1þa 1a .

_ 0 Þ; ^ 1=2 wðx0 Þ þ aþ wðx w_ 0a ðx00 Þ ¼ ia D 0 0

ð69Þ

ð70Þ

Now, substituting (69) and (70) in (68) and doing the necessary algebra, we find for a measurement that is made at t = x0/c the following remarkably simple result

A. Mostafazadeh, F. Zamani / Annals of Physics 321 (2006) 2183–2209

2197

h io j n ^ 1=4 2  _ _ _ ^ 1=4 wðxÞÞðD ^ 1=4 wðxÞj ^ 1=4 wðxÞÞ D ^ 1=4 wðxÞÞ ^ 1=4 wðxÞ  ðD þ ia ðD jD wðxÞj2 þ jD 2M  1=4 o j n ^ 1=4 ^ 1=4 wc ðxÞj2 þ a ðD ^ wðxÞÞ D ^ 1=4 wðxÞÞðD ^ 1=4 wc ðxÞÞ . ^ 1=4 wc ðxÞ þ ðD ¼ jD wðxÞj2 þ jD 2M

qa ðxÞ ¼

ð71Þ For a positive-energy Klein–Gordon field (65) reduces to a probability density that was originally obtained by Rosenstein and Horwitz [33] by restricting the second-quantized scalar field theory to its one-particle sector. This coincidence may be viewed as a verification of the validity of our approach: The first-quantized theory formulated by an explicit construction of the Hilbert space and a position observable reproduces a result obtained from the second-quantized theory. We can use the method discussed in Section 2 to also define a current density J la such that J 0a ¼ qa . As we show in Appendix E, this yields j  ^ 1=4  ^ 1=4 wc ðxÞ  ðD ^ 1=4 wc ðxÞÞol ðD ^ 1=4 wðxÞÞ I ðD wðxÞÞ ol D J la ðxÞ ¼ 2M  1=4  ^ 1=4 wðxÞ  ðD ^ 1=4 wc ðxÞÞol ðD ^ 1=4 wc ðxÞÞ ; ^ wðxÞÞ ol D þa ðD ð72Þ where I stands for the imaginary part of its argument. Appendix E also includes a proof that the probability current density J la has the correct non-relativistic limit: setting j = 1/(1 + a) yields lim J 0a ðx0 ;~ xÞ ¼ .ðx0 ;~ xÞ;

c!1

1 0 ~ a ðx0 ;~ lim J xÞ ¼ ~ xÞ; jðx ;~ c!1 c

ð73Þ

where . and j are the non-relativistic scalar and current probability densities given by (33) and (34), respectively. If we restrict to the positive-energy Klein–Gordon fields and set a = 0, Eq. (72) reduces to the probability current density obtained by Rosenstein and Horwitz in [33]. As also indicated by these authors, the resulting current density, namely J l0 , does not satisfy the continuity equation. Hence it is not a conserved current. Furthermore, as we show in Appendix E, J l0 is indeed not even a four-vector field. The same lack of covariance and conservation applies to J la for a „ 0. The only advantage of J la over J la is that, unlike the latter which is generally complex-valued, the former is manifestly real-valued and positive-definite. The non-conservation (respectively, non-covariance) of the probability current density J la raises the paradoxical possibility of the non-conservation (respectively, frame-dependence) of the total probability: Z P a :¼ d3~ xÞ. ð74Þ xqa ðx0 ;~ R3

It turns out that indeed the latter is a frame-independent conserved quantity, thanks to the covariance and conservation of the current density J la and the identity Z Z 3 0 d~ xÞ ¼ d3~ xÞ; ð75Þ xqa ðx ;~ xJ 0a ðx0 ;~ R3

R3

which follows from (15), (71) and the fact that D±1/4 is a self-adjoint operator acting in L2 ðRÞ. In a sense, qa(x) and J 0a ðxÞ differ only by a ‘‘boundary term.’’ Combining (74) and (75), we have

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Pa ¼

Z 3

d3~ xÞ. xJ 0a ðx0 ;~

ð76Þ

R

This relation implies that although the probability density qa is not the zero-component of a conserved four-vector current density, its integral over the whole space that yields the total probability (74) is nevertheless conserved. Furthermore, this global conservation law stems from a local conservation law, i.e., a continuity equation for a four-vector current density namely J la . 4. Gauge symmetry associated with the conservation of the total probability The fact that the conservation of the total probability P a has its root in the local conservation of the covariant current J la suggests, by virtue of the No¨ther’s theorem, the presence of an underlying gauge symmetry. To determine the nature of this symmetry, we make use of the well-known fact that the conserved charge associated with any conserved current is the generator of the infinitesimal gauge transformations [38]. The specific form of the latter is most conveniently obtained in the Hamiltonian formulation. The Lagrangian L for a free Klein–Gordon field w and the corresponding canonical ð~ momenta pð~ xÞ, p xÞ associated with wð~ xÞ :¼ wðx0 ;~ xÞ and w ð~ xÞ :¼ w ðx0 ;~ xÞ are, respectively, given by [39]: Z   k L :¼  d3~ xÞ ol wð~ xÞ þ M2 wð~ xÞ wð~ xÞ ; ð77Þ x ol wð~ 2 R3 pð~ xÞ :¼

dL k xÞ; ¼ w_  ð~ _ dwð~ xÞ 2

ð~ p xÞ :¼

dL k_ xÞ ¼ p ð~ xÞ; ¼ wð~  _ xÞ 2 dw ð~

ð78Þ

where k :¼  hc=M ¼  h2 =m and we have suppressed the x0-dependence of the fields for simplicity. In terms of the canonical phase space variables (w, p) and (w*, p*), the conserved charge for J la , namely the total probability (76), takes the form Z   j ^ 1=2 wð~ ^ 1=2 p ð~ Pa ¼ d3~ xÞ þ 4k2 pð~ xÞD xÞ þ 2ik1 a½wð~ xÞ pð~ xÞ  wð~ xÞpð~ xÞ ; x wð~ xÞ D 2M R3 ð79Þ where we have made use of (15) and (78). We can obtain the infinitesimal symmetry transformation w ! w þ dw;

ð80Þ

generated by P a using dwð~ xÞ ¼ fwð~ xÞ; P a gdn;

ð81Þ

where {Æ,Æ} is the Poisson bracket   Z dA dB dB dA dA dB dB dA  þ   d3~ x ; fA; Bg :¼ dwð~ xÞ dpð~ xÞ dwð~ xÞ dpð~ xÞ dw ð~ xÞ dp ð~ xÞ dw ð~ xÞ dp ð~ xÞ R3 ð82Þ A; B are observables, and dn is an infinitesimal real parameter. In view of (78)–(82), we have

A. Mostafazadeh, F. Zamani / Annals of Physics 321 (2006) 2183–2209

dwð~ xÞ ¼

2199

i dP a j h ^ 1=2 _ D dn ¼ wð~ xÞ  iawð~ xÞ dn. Mk dpð~ xÞ

We may employ (20) to further simplify this expression. The result is dwð~ xÞ ¼ idhðC þ aÞwð~ xÞ;

ð83Þ

where dh :¼ jdn=ðMkÞ ¼ jdn=ðhcÞ. According to (83), the symmetry transformations (80) are generated by the operator C þ a. One can easily exponentiate the latter to obtain the following expression for the corresponding non-infinitesimal symmetry transformations: w ! eihðCþaÞ w ¼ eiah eihC w ¼ eiah ½cos h  i sin hCw;

ð84Þ

where h 2 R is arbitrary and we have made use of C 2 ¼ 1. In terms of the ±-energy components w± of w, Eq. (84) takes the form X w ¼ wþ þ w ! eiðaþ1Þh wþ þ eiða1Þh w ¼ eiðaþÞh w . ð85Þ ¼

The analogy between the charge-grading operator C and the chirality operator c5 suggests that for a = 0 the gauge transformation (84) is a spin-zero counterpart of the chiral transformation of the Dirac spinors [11]. It is not difficult to see from (84) and (85) that the gauge group9 Ga associated with these transformations is a one-dimensional connected Abelian Lie group. Therefore, it is isomorphic to either of U (1) or Rþ , the latter being the non-compact multiplicative group of positive real numbers [40]. We can construct a simple model

for (faithful representation of) the group Ga using the  wþ . Then C is represented by the Pauli matrix r3 two-component representation w ¼ w and a typical element of Ga takes the form ! 0 eiðaþ1Þh ga ðhÞ :¼ . ð86Þ 0 eiða1Þh This expression suggests that the gauge group Ga is a subgroup of U (1) · U (1). It is not difficult to show that Ga is a compact subgroup of this group and consequently isomorphic to U (1) if and only if the parameter a is a rational number. This in turn implies that for irrational a the group Ga is isomorphic to Rþ .10 We can easily construct a concrete example of these isomorphisms. For a rational a, we have a = m/n where m and n are relatively prime integers with n positive. In this case we let ua:Ga fi U (1) be defined by ua ðeihðaþCÞ Þ :¼ eih=n . For an irrational a, we define va : Ga ! Rþ according to va ðeihðaþCÞ Þ :¼ eh . Then it is an easy exercise to show that both ua and va are (Lie) group isomorphisms. Clearly, the Ga gauge symmetry associated with the conservation of the total probability, alternatively the current density J la , is a global gauge symmetry. Similarly to the U (1) 9

Here we identify the gauge group with its connected component that includes the identity and is obtained by exponentiating the generator C þ a. 10 In this case, although Ga is (isomorphic to) an abstract subgroup of U (1) · U (1) it fails to be a Lie subgroup of this group [40].

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gauge symmetry associated with the Klein–Gordon current, namely the one responsible for the electric charge conservation, one may consider allowing for ðx0 ;~ xÞ-dependent gauge parameters: h = h(x), i.e., consider local Ga gauge transformations. One then expects that the imposition of this local gauge symmetry should lead to a gauged Klein–Gordon equation involving a gauge field that couples to the current J la . The naive minimal coupling prescription, however, fails because it makes the generator C þ a of Ga generally ðx0 ;~ xÞ-dependent. In this respect the local Ga gauge symmetry is different from the usual local Yang–Mills-type gauge symmetries. The group U (1) · U (1) that enters the above discussion of the gauge group Ga as an embedding group is also a group of gauge transformations of the Klein–Gordon fields. It corresponds to the global Ga gauge transformations supplemented with the global U (1) gauge transformations associated with the conservation of the electric charge (4). 5. Real scalar fields Let R V be the space of real Klein–Gordon fields and w 2 R has the ±-energy components w±. Clearly wþ ðxÞ þ w ðxÞ ¼ wðxÞ ¼ wðxÞ ¼ wþ ðxÞ þ w ðxÞ . Because w+ is a positive-energy field, it’s complex conjugate, This together with (87) imply 

xÞ ¼ w ðx0 ;~ xÞ . w ðx0 ;~

ð87Þ wþ ,

is a negative-energy field. ð88Þ

Combining this equation and (57) (or equivalently (60)) shows that the position wave functions f ð;~ xÞ for w satisfy f ð;~ xÞ ¼ f ð;~ xÞ .

ð89Þ

This is a characteristic feature of real Klein–Gordon fields in their position representation. Another feature of real Klein–Gordon fields is that, up to a trivial multiplicative constant, there is a unique positive-definite and relativistically invariant inner product on the space R of real Klein–Gordon fields and that the latter is indeed a real inner product. This follows from the observation that the expression (12) for the most general positive-definite and relativistically invariant inner product (Æ,Æ)a reduces to that of (Æ,Æ)0 which is, up to a trivial scaling, the same as the inner product (10). It is also easy to see that because D is both real and Hermitian the right-hand side of (10) for real fields w1 and w2 is real. This shows that the restriction of (10) onto R yields a real inner product. In [18] Wald outlines a method of constructing a positive-definite inner product on the space R of real Klein–Gordon fields. In the following we shall offer a brief review of Wald’s approach and show that in accordance with the above uniqueness property of (10) the resulting inner product coincides with (10). Let RCþ denote the subspace of V spanned by the positive-energy part, w+, of real fields w 2 R. The restriction of the Klein–Gordon inner product (5) onto RCþ is positive-definite. Hence endowing RCþ with this inner product yields a genuine inner product space which may be Cauchy completed to a complex Hilbert space K. The association w fi w+ yields a (real) linear one-to-one map K : R ! K which takes R into a dense subspace of K. One can use K to define a real inner product ð; Þ : R R ! K on R according to

A. Mostafazadeh, F. Zamani / Annals of Physics 321 (2006) 2183–2209

  ðw1 ; w2 Þ :¼ R ðKw1 ; Kw2 ÞKG .

2201

ð90Þ

To obtain the explicit form of this inner product, which to the best of our knowledge has not been previously provided, we use the charge-grading operator C to express K in the form Kw ¼ wþ ¼ 12ð1 þ CÞw.

ð91Þ

This together with (5), (20), and (6) allow us to compute h i ðKw1 ; Kw2 ÞKG ¼ ig hw1þ jw_ 2þ i  hw_ 1þ jw2þ i h io gn hw1 jD1=2 w2 i þ hw_ 1 jD1=2 w_ 2 Þi þ i hw1 jw_ 2 i  hw_ 1 jw2 i . ¼ 2

ð92Þ

Substituting the latter relation in (90) yields i gh ðw1 ; w2 Þ ¼ hw1 jD1=2 w2 i þ hw_ 1 jD1=2 w_ 2 i ; 2 which coincides with (10) if we set g ¼ 1=M. Hence, as expected, Wald’s construction leads to the same inner product as the one found using the theory of pseudo-Hermitian operators in [19,21]. In particular, Eq. (35) with a = 0 provides an explicit and manifestly covariant expression for Wald’s inner product (90). It is instructive to note that the inner product (90) is directly related to the real part of the standard two-point (Wightman-) function, Æ0|w (x)w (y)|0æ, of the quantized Klein– Gordon fields [18,27].11 This provides another argument for its relativistic invariance: since the (standard) vacuum state is Poincare´ invariant, the inner product (Æ,Æ) is also relativistically invariant. 6. Klein–Gordon fields in a background electromagnetic field A scalar field w minimally coupled to a background electromagnetic field Al satisfies   ð93Þ ½iol  qAl ðxÞ½iol  qAl ðxÞ þ M2 wðxÞ ¼ 0; where q := e/(hc), e is the electric charge, and Al is assumed to be real-valued. _ 0 ;~ Supposing that for all x0, wðx0 ;~ xÞ and wðx xÞ are square-integrable functions, we can easily write Eq. (93) as an ordinary differential equation in L2 ðR3 Þ. The latter takes the form € 0 Þ þ 2iquðx0 ;~ _ 0 Þ þ Dwðx0 Þ ¼ 0; wðx xÞwðx

ð94Þ

where u := A is the scalar potential, D : L ðR Þ ! L ðR Þ is the operator: n o 2 2 ~  iq~ _ 0 ;~ ðD/Þð~ xÞ :¼ ½r Aðx0 ;~ xÞ þ iquðx xÞ  q2 uðx0 ;~ xÞ þ M2 /ð~ xÞ 8/ 2 L2 ðR3 Þ 0

2

3

2

3

ð95Þ and ~ A ¼ ðA ; A ; A Þ is the vector potential. 1

11

2

3

This defines another Green’s function for the Klein–Gordon operator that is called the Hadamard or Schwinger Green function [23].

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Eq. (94) takes the form of the Klein–Gordon equation (6) for a free scalar field provided that we make the gauge transformation " Z 0 # x wðx0 Þ ! vðx0 Þ :¼ uðx0 ;~ xÞwðx0 Þ; uðx0 ;~ xÞ :¼ exp iq dsuðs;~ xÞ ; ð96Þ x00

where x00 2 R is an arbitrary but fixed initial value of x0. Substituting (95) and (96) in (94) and doing the necessary algebra, we find € vðx0 Þ þ Dq vðx0 Þ ¼ 0; 2

3

2

ð97Þ 3

where Dq : L ðR Þ ! L ðR Þ is the operator ^ q /ð~ xÞ :¼ D xÞ ðDq /Þð~ and

8/ 2 L2 ðR3 Þ

n o 2 1 ~  iqAðx ~ 0 ;~ ^ q :¼ uðx0 ;~ D xÞ ½r xÞ þ M2 uðx0 ;~ xÞ .

ð98Þ

ð99Þ

It is not difficult to observe that indeed Dq is a positive-definite operator acting in L2 ðR3 Þ. Therefore, according to the terminology of Refs. [19,20], Eq. (97) is an example of a Klein–Gordon-type field equation. The main difference between the free Klein–Gordon equation (6) and Eq. (97) is that the latter is a non-stationary Klein–Gordon-type _ equation unless u is constant and ~ A ¼ 0. These conditions are fulfilled only if the electric _ ~ vanishes and the magnetic field ~ ~ ~ field ~ E ¼ ð~ A þ ruÞ B¼r A is time-independent. For a scalar field interacting with an arbitrary stationary magnetic field we can apply the results of Sections 2 and 3 by simply replacing the operator D by D (and noting that ~ ¼ Að~ ~ xÞ) or equivalently by enforcing the minimal coupling prescription: u = 0 and A ~ ~ ~ r ! r  iqAð~ xÞ. If either a non-zero electric field or a non-stationary magnetic field is present, then the transformed field v is a non-stationary Klein–Gordon-type field and one must employ the quantum mechanics of such fields as outlined in [20]. 7. Concluding remarks The first-quantized relativistic quantum mechanics for free scalar fields may be formulated by constructing a genuine Hilbert space of the solutions of the Klein–Gordon equation. This involves endowing the solution space of this equation with a positive-definite inner product. The requirements that this inner product be well-defined, positive-definite, and relativistically invariant fix it up to an arbitrary real parameter a 2 (1, 1) and an overall trivial coefficient j 2 Rþ . The resulting family of inner products, (Æ,Æ)a, define unitarily equivalent Hilbert spaces Ha and therefore are physically identical [20]. They admit a manifestly covariant expression that involves a conserved four-vector current density J la . In view of the unitary-equivalence of Ha and L2 ðR3 Þ L2 ðR3 Þ, one can use the ordinary position operator for a non-relativistic spin-1/2 particle, that acts in L2 ðR3 Þ L2 ðR3 Þ, to define a relativistic position operator for Klein–Gordon fields w. This in turn yields a position basis in which w is uniquely determined by a set of wave functions f ð;~ xÞ. In terms of these wave functions the probability density for the localization of w in space takes the same form as in non-relativistic quantum mechanics. By expressing f ð;~ xÞ directly in terms of w, one obtains a manifestly positive-definite probability density qa. This turns out to

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coincide with the Rosenstein–Horwitz probability density [33], if one sets a = 0 and restricts to the positive-energy Klein–Gordon fields. One can define a current density J la whose zero-component equals qa. But J la is neither covariant nor conserved. The probability density qa ¼ J 0a may be linked with the zero-component J 0a of the conserved current density J la in the sense that their integrals over the whole space are identical. This in particular ensures the conservation and frame-independence of the total probability. It also allows for the interpretation of the continuity equation satisfied by J la as a local manifestation of the (global) conservation law for the total probability. The latter stems from an underlying Abelian global gauge-symmetry of the Klein–Gordon equation which resembles the chiral symmetry for Dirac spinor fields for a = 0 and a ‘twisted’ analog of it for a „ 0. The nature of the corresponding gauge group Ga depends on the parameter a. For rational values of a, Ga = U (1); for irrational values of a, Ga ¼ Rþ . For real Klein–Gordon fields the parameter a drops out of the expression for the inner product. In view of the fact (established in [20]) that (Æ,Æ)a is the most general positive-definite, relativistically invariant, and conserved inner product for Klein–Gordon fields, the preceding observation shows that for a real scalar field such an inner product is (up to an irrelevant scale factor) unique. We independently checked, by direct calculation, that indeed the inner products considered in earlier publications, in particular by Wald [18], coincide with the unique inner product (Æ,Æ)0 for which we have given an explicit and manifestly covariant expression for the first time. The expression for the current densities J la obtained for free Klein–Gordon fields may be easily generalized to scalar fields interacting with a stationary background magnetic field. For a more general background electromagnetic field, the scalar field may be gauge-transformed to a non-stationary Klein–Gordon-type field. Therefore, in order to understand first-quantized scalar fields interacting with such an electromagnetic field one should employ the quantum mechanics of non-stationary Klein–Gordon-type fields [20]. This requires a separate investigation of its own and will be dealt with elsewhere. Perhaps a more interesting subject of future study is the local analog of the above-described global Ga gauge symmetry. A simple application of the general results presented in this paper is in the construction and investigation of the relativistic coherent states. This is the subject of the second paper of this series [1]. Acknowledgment A.M. wishes to acknowledge the support of the Turkish Academy of Sciences in the _ ¨ BA-GEBIP/2001-1-1). framework of the Young Researcher Award Program (EA-TU Appendix A. Lorentz-Invariance of D1=2 w_ Consider performing an infinitesimal Lorentz transformation xl ! x0l ¼ Klm xm ; xlm

Klm ¼ dlm þ xlm ;

ð100Þ

are the antisymmetric generators of the Lorentz transformation [39] and where jxlm j  1. The latter condition means that we can safely neglect the second and higher order terms in powers of xlm . It is not difficult to show using (100) and its inverse transformation, namely

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A. Mostafazadeh, F. Zamani / Annals of Physics 321 (2006) 2183–2209 l

x0l ! xl ¼ ðK1 Þm x0m ¼ x0l  xlm x0m ;

ð101Þ

that r02 :¼ o0i o0i ¼ r2  2xli ol oi ¼ r2 þ 2xl0 ol o0 . Here to establish the last equality we have added and subtracted the antisymmetry of xlm . Now, we substitute (102) in D0 ¼ M2  r02 to obtain ^ 0a ¼ D ^ a1 ol o0 ^ a  2axl0 D D

ð102Þ 2xl0 ol o0

and employed

8a 2 R.

ð103Þ

Moreover, using (101) we have _ w_ 0 ðx0 Þ ¼ o00 w0 ðx0 Þ ¼ wðxÞ  xl0 ol wðxÞ.

ð104Þ

Finally, in view of (103), (104), and the Klein–Gordon equation (2), we find

 1=2  _ ^ 01=2 w_ 0 ðx0 Þ ¼ D ^ 3=2 ol o0 wðxÞ ^ D  xl0 ol wðxÞ þ xl0 D

_ € _ ^ 1=2 ol wðxÞ ¼ D ^ 3=2 ol wðxÞ ^ 1=2 wðxÞ ^ 1=2 wðxÞ.  xl0 D þD ¼D _ ^ 1=2 wðxÞ Therefore D is Lorentz-invariant. Appendix B. Nonrelativistic limit of J la Let w be a Klein–Gordon field and define v : R4 ! C according to 0

vðx0 ;~ xÞ :¼ eiMx wðx0 ;~ xÞ.

ð105Þ

Then as it is well-known [5], in the non-relativistic limit as c fi 1, v(x) satisfies the noni relativistic free Schro¨dinger equation: v_ ¼ 2M r2 v, and   i 0 _ r2 vðxÞ : ¼ eiMx iMvðxÞ þ lim wðxÞ ð106Þ c!1 2M Furthermore ^ 1=2 ¼ M1 þ 1M3 r2 . lim D 2

c!1

ð107Þ

Eqs. (20, 29, 106), and (107) imply lim wc ¼ w;

c!1

~ a ¼ ð1 þ aÞw. lim w

c!1

Substituting these relations in (28), we find   $ ijð1 þ aÞ  lim wðxÞ ol wðxÞ ; lim J la ðxÞ ¼  c!1 2M c!1

ð108Þ

ð109Þ

If we set j = 1/(1 + a) and l = 1, 2, 3 in this expression, we obtain (32). Similarly using (106) and (109), we arrive at (31). This ends our demonstration that J la has the correct non-relativistic limit.

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Appendix C. Real and imaginary parts of J la Given a free Klein–Gordon field w, we can use (58) to write w ¼ wþ þ w ;

wc ¼ wþ  w

Substituting these relations and (29) in (28) and carrying out the necessary calculations, we find the following expressions for the real and imaginary parts of J la :  j  I ð1 þ aÞwþ ol wþ  ð1  aÞw ol w þ aðwþ ol w þ w ol wþ Þ RðJ la Þ ¼ M   ð110Þ $ $ $ ij  l  l  l ð1 þ aÞwþ o wþ  ð1  aÞw o w þ 2iaIðwþ o w Þ ; ¼ 2M $ j j Rðwþ ol w  w ol wþ Þ ¼ Rðwþ ol w Þ. ð111Þ IðJ la Þ ¼ M M Here R and I stand for the real and imaginary part of their arguments, respectively. Note that for a Klein–Gordon field with a definite charge-grading , i.e., for a positiveor negative-energy field, J la is real and up to a real coefficient, namely ±(1 ± a), coincides with the Klein–Gordon current density. However, due to the particular sign of this coefficient and the fact that |a| < 1, J 0a is positive-definite for both the positive- and negativeenergy plane-wave solutions of the Klein–Gordon equation. Another interesting case is that of the real Klein–Gordon fields for which w ¼ wþ . $ Then in view of (110) and (111), J la ¼ ðij=MÞwþ ol wþ which is again real, but unlike the Klein–Gordon current density it does not vanish. Note that in this case J la is independent of a. This was to be expected, because a enters in the expression (27) for J la as the coefficient of a term which is essentially the Klein–Gordon current density. ~ Appendix D. Derivation of the position operator X ~ be the position operator defined in (47) and w be a Klein–Gordon field. Then as Let X ~wÞðx0 Þ is determined by the initial conditions shown in [21], ðX ~ wðx0 Þ; ~wÞðx0 Þ ¼ X ðX 0 0

_ 0 Þ; ~ y wðx ~wÞðx0 Þ ¼ X o0 ð X 0 0

ð112Þ

where ~ :¼ ~ X xþ

i h~ p . 2ð~ p2 þ m2 Þ

ð113Þ

~, we first express w(x0) in terms of the initial data To derive an explicit expression for X 0 0 _ ðwðx0 Þ; wðx0 ÞÞ. This yields [20,21] _ 0 Þ. wðx0 Þ ¼ cos½ðx0  x00 ÞD1=2 wðx00 Þ þ sin½ðx0  x00 ÞD1=2 D1=2 wðx 0

ð114Þ

~w is also a Klein–Gordon field, we can use this equation to express ðX ~wÞðx0 Þ Noting that X in terms of the initial data (112). Doing the necessary calculation, we then find [21] i hs~ p h _ 0Þ ; ~ wðx0 Þ þ i ~wÞðx0 Þ ¼ X ðX sin½sD1=2 D1=2 wðx00 Þ  cos½sD1=2 wðx ð115Þ 0 2 2 ð~ p þm Þ where s :¼ ðx0  x00 Þ. Next, we recall that wc as given by (20) is also a Klein–Gordon field. Thus, in view of (114), the terms in brackets in (115) is equal to iD1/2wc(x0). This implies

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hs~ p _ 0 ~ wðx0 Þ  i ~wÞðx0 Þ ¼ X wðx Þ ðX ð~ p2 þ m2 Þ

 i h~ p i hs~ p  2 o0 wðx0 Þ; ¼ ~ xþ 2ð~ p2 þ m2 Þ ð~ p þ m2 Þ

ð116Þ

which establishes (48). Appendix E. Derivation and properties of J la The derivation of J la mimics that of J la . First, we identify J 0a with qa of Eq. (71). To obtain the spatial components of J ia of J la , we then perform an infinitesimal Lorentz boost transformation (16) which yields l 0 ~ ~ J 0a ðxÞ ! J 00 a ðx Þ ¼ J 0 ðxÞ  b  J a ðxÞ.

ð117Þ

0 Next, we use (71) to read off the expression for J 00 a ðx Þ, namely  01=4 0 0  01=4 0 0 j n ^ 01=4 0 0 2 0 ^ w ðx Þ ^ 01=4 w0 ðx0 Þj2 þ a ðD ^ w ðx ÞÞ D jD w ðx Þj þ jD J 00 ðx Þ ¼ c c a 2M o   ^ 01=4 w0 ðx0 ÞÞðD ^ 01=4 w0 ðx0 ÞÞ . þðD c

ð118Þ

In view of Eq. (23) and the fact that w(x) and wc(x) are scalars, we further have: n o  _ _ ~ D ~D ^ 1=4 wðxÞÞ r ^ 1=4 wðxÞrð ^ 01=4 w0 ðx0 Þj2 ¼ jD ^ 1=4 wðxÞj2  1~ ^ 3=4 wðxÞÞ ^ 3=4 wðxÞ b  ðD þD jD ; 2 ð119Þ o ~ D ~D ^ 1=4 wc ðxÞÞ r ^ 01=4 w0 ðx0 Þj2 ¼ jD ^ 1=4 wc ðxÞj2  1~ ^ 1=4 wc ðxÞrð ^ 3=4 w_ c ðxÞÞ ; ^ 3=4 w_ c ðxÞ þ D b  ð D jD c 2 n

n

ð120Þ

~D ^ 1=4 wðxÞÞ r ^ 01=4 w0 ðx0 Þ ¼ ðD ^ 1=4 wc ðxÞ  1~ ^ 01=4 w0 ðx0 ÞÞ D ^ 1=4 wðxÞÞ D ^ 3=4 w_ c ðxÞ b  ðD ðD c 2 o  _ ~ D ^ 1=4 wc ðxÞrð ^ 3=4 wðxÞÞ þD ; ð121Þ n _ ~D ^ 1=4 wc ðxÞÞ r ^ 01=4 w0 ðx0 ÞÞðD ^ 01=4 w0 ðx0 ÞÞ ¼ ðD ^ 1=4 wðxÞÞðD ^ 1=4 wc ðxÞÞ  1~ ^ 3=4 wðxÞ b  ðD ðD c 2 o ~ D ^ 1=4 wðxÞrð ^ 3=4 w_ c ðxÞÞ . þD ð122Þ Now, substituting (119)–(122) in (118) and making use of (20) and (117), we obtain n ~ D ~D ~ a ðxÞ ¼  ij ðD ^ 1=4 wðxÞÞ r ^ 1=4 wðxÞrð ^ 1=4 wc ðxÞÞ ^ 1=4 wc ðxÞ  D J 4M ~ D ~D ^ 1=4 wc ðxÞrð ^ 1=4 wðxÞÞ ^ 1=4 wðxÞ  D ^ 1=4 wc ðxÞÞ r þðD h ~ D ~D ^ 1=4 wðxÞÞ r ^ 1=4 wðxÞrð ^ 1=4 wðxÞÞ ^ 1=4 wðxÞ  D þa ðD io ~ D ~D ^ 1=4 wc ðxÞÞ r ^ 1=4 wc ðxÞrð ^ 1=4 wc ðxÞÞ . ^ 1=4 wc ðxÞ  D þðD ð123Þ This relation suggests

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ij  ^ 1=4  ^ 1=4 wc ðxÞ  D ^ 1=4 wðxÞol ðD ^ 1=4 wc ðxÞÞ ðD wðxÞÞ ol D 4M ^ 1=4 wðxÞ  D ^ 1=4 wc ðxÞol ðD ^ 1=4 wðxÞÞ ^ 1=4 wc ðxÞÞ ol D þðD  1=4 ^ 1=4 wðxÞ  D ^ wðxÞÞ ol D ^ 1=4 wðxÞol ðD ^ 1=4 wðxÞÞ þa ðD  ^ 1=4 wc ðxÞ  D ^ 1=4 wc ðxÞol ðD ^ 1=4 wc ðxÞÞ ; ^ 1=4 wc ðxÞÞ ol D þðD

ð124Þ

J la ðxÞ ¼ 

which we can also write as j  ^ 1=4 ^ 1=4 wc ðxÞ  ðD ^ 1=4 wc ðxÞÞol ðD ^ 1=4 wðxÞÞ J la ðxÞ ¼ I ðD wðxÞÞ ol D 2M  1=4  ^ 1=4 wðxÞ  ðD ^ 1=4 wc ðxÞÞol ðD ^ 1=4 wc ðxÞÞ . ^ wðxÞÞ ol D þa ðD

ð125Þ J 0a ðxÞ

Using Klein–Gordon equation, we can easily check that the expression for ¼ qa ðxÞ obtained by setting l = 0 in (125) agrees with the one given in (71). Next, we explore the non-relativistic limit of the probability current density (125). Following the treatment of Appendix B, we first derive ^ 1=4 ¼ M1=2  1 M3=2 r2 ; lim D 4

^ 1=4 ¼ M1=2 þ 1M5=2 r2 . lim D 4

c!1

c!1

Substituting these relations in (125), we find   $ ijð1 þ aÞ  l l lim wðxÞ o wðxÞ . lim J a ¼  c!1 2M c!1 Now, setting j = 1/(1 + a), considering l = 0 and l „ 0 separately, and making use of (105) and (106), we obtain (73): lim J 0a ðx0 ;~ xÞ ¼ .ðx0 ;~ xÞ;

c!1

1 0 ~ a ðx0 ;~ lim J xÞ ¼ ~ xÞ; jðx ;~ c

c!1

where . and ~ j are the non-relativistic scalar and current probability densities given by (33) and (34), respectively. If we confine our attention to the positive-energy Klein–Gordon fields (for which wc = w) and set a = 0, the expression (124) coincides with the Rosenstein–Horwitz’s current [33]:  ij  ^ 1=4  ^ 1=4 wðxÞÞ  ðD ^ 1=4 wðxÞÞol ðD ^ 1=4 wðxÞÞ . ðD wðxÞÞ ol ðD ð126Þ J lRH ðxÞ ¼  2M As we show below, in general ol J la 6¼ 0. Hence J la is not a conserved current density. This was noticed by Rosenstein and Horwitz [33] for the probability current (126). A more dramatic result that seems to be missed by these authors is that J lRH is not even a fourvector field. The same holds for J la . This can be most conveniently shown by computing J la for a superposition of a pair of positive-energy plane-wave Klein–Gordon fields 0

~

0

~

wðxÞ ¼ c1 eik1 x þ c2 eik2 x ¼ c1 eix1 x eik1 ~x þ c2 eix2 x eik2 ~x ;

ð127Þ l

where for ‘ 2 {1,2}, c‘ 2 C  f0g, k‘ is a constant four-vector, k‘ Æ x := (k‘)lx , and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x‘ :¼ ~ k 2‘ þ M2 . Note that for this choice of the field we have wc = w and J l0 ¼ J lRH . Hence, in view of (127) and (126), o j n 2 l 2 jc1 j k 1 þ jc2 j k l2 þ Rðc1 c2 eiðk1 k2 Þ.x ÞK l ; J l0 ðxÞ ¼ J lRH ðxÞ ¼ ð128Þ M

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A. Mostafazadeh, F. Zamani / Annals of Physics 321 (2006) 2183–2209

where rffiffiffiffiffiffi rffiffiffiffiffiffi x2 l x1 l K ¼ k1 þ k . x1 x2 2 l

ð129Þ

Moreover, setting wc = w in (125), we find J la ðxÞ ¼ ð1 þ aÞJ l0 ðxÞ.

ð130Þ

A direct implication of Eqs. (128) and (130) is that J la is a vector field if and only if Kl is a four-vector. But as we show next the latter fails to be the case. Suppose (by contradiction) that Kl is a four-vector, then KlKl must be a scalar. It is not difficult to show that

 x1 2 x2 l K l K ¼ 2k 1  k 2  M þ ; ð131Þ x1 x2 where we have made use of k ‘  k ‘ ¼ M2 . For x1 „ x2, the term multiplying M2 on the right-hand side of (131) fails to be a scalar. This shows that KlKl is not a scalar; Kl is not a four-vector; and in general J la is not a vector field. Next, we wish to point out that computing the current density J la for the field (127) we find the following manifestly covariant expression: J la ðxÞ ¼

o   jð1 þ aÞ n 2 l jc1 j k 1 þ jc2 j2 k l2 þ R c1 c2 eiðk1 k2 Þ.x ðk l1 þ k l2 Þ . M

ð132Þ

Having obtained the explicit form of both J la and J la for the field (127), we can easily check their conservation property. A simple calculation shows that:

rffiffiffiffiffiffi rffiffiffiffiffiffi x1 x2 2 l ol J a ðxÞ ¼ ðM þ k 1  k 2 Þ  F ðxÞ; ð133Þ x2 x1 ol J la ðxÞ ¼ ½ðk 1  k 2 Þ  ðk 1 þ k 2 ÞF ðxÞ ¼ 0;

ð134Þ

where F ðxÞ :¼ 

jð1 þ aÞ   iðk1 k2 Þ.x  I c1 c2 e M

and we have made use of (130), (128), (129), and (132) and the fact that the term in the square bracket on the right-hand side of (134) vanishes identically by virtue of k ‘  k ‘ ¼ M2 . According to Eq. (133), for x1 „ x2, ol J la ðxÞ 6¼ 0. Therefore, unlike J la , the probability current density J la fails to be conserved. References [1] [2] [3] [4] [5] [6] [7]

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