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Causality and spatial correlations of the relativistic scalar field in the presence of a static source
2 October 1995 PHYSICS
ELSEYIER
LETTERS
A
Physics Letters A 206 (1995) l-6
Causality and spatial correlations of the relativistic scalar field in the presence of a static source A. la Barbera a, R. Passante b a Istitato di Fisica dell’llniuersitir, Via Archirafi 36, I-90123 Palenno, Italy b Istituto per le Applicazioni Interdisciplinari della Fisica, Consiglio Nazionale delle Ricerche, Via Archirafi 36, I-90123 Palermo, Italy Received 22 May 1995; accepted for publication 2 August 1995 Communicated by P.R. Holland
Abstract The spatial correlations of the energy density of the virtual cloud around a static source of the relativistic bosonic scalar field are considered. An exact expression for the correlation function in the dressed ground state of the source is obtained which indicates antibunching of the virtual bosons. Subsequently the time-dependent undressing process is considered and superluminal changes of the spatial correlations are found. The problem of causality is discussed in detail in this framework. PACS: 03.70. + k
1. Introduction The dynamics of the cloud of virtual quanta surrounding a field source after instantaneous switching on or off of the source-field coupling has been recently investigated from the point of view of the relativistic causality [l]. The problem of causality has also been discussed in the dynamics of a pair of atoms interacting with the electromagnetic radiation field [2]. In this case it has been shown that, during the time-dependent dressing of the two atoms separated by a distance R, the dynamics of each atom is strictly causal because the presence of one atom affects the dynamics of the other one only after the “causality” time t = R/c. Instantaneous interatomic correlations appear well before the time R/c, but it can be shown that they do not violate relativistic causality. Work along this line has also led to a better insight into the connection between causality and correlations and seems conceptually and practically relevant, particularly in view of recent suggestions that QED might exhibit a non-casual behaviour [3]. In contrast with interatomic correlations, field correlations in the presence of a source have not yet been fully investigated from the point of view of causality, and a satisfactory casual interpretation of recent experiments on photon propagation in the presence of atoms [4] has not been provided yet. In order to show that also field correlations in the presence of sources evolve in time in a rigorously causal way, in this paper we consider the spatial correlations of the cloud of virtual quanta (mesons) around a structureless source interacting with the relativistic scalar field, described by the Klein-Gordon equation. This model can be solved exactly and it is the simplest model of a nucleon interacting with the meson field 151. As such, it can provide guidance in a future investigation of more complex QED phenomena. We focus our 0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9601(95)00581-l
2
A. la Barbera, R. Passante/Physics
Letters A 206 (1995) l-6
attention on the spatial correlations of the energy density of the virtual field around the source. We preliminarily consider the time-independent case of the dressed ground state of the system and obtain exact expressions for the correlation function. Then we discuss the time-dependent case in which the source-field interaction is suddenly switched off (undressing) and we consider the consequent dynamics of the initial spatial correlations of the cloud. We show that non-local changes appear (they are the field counterpart of the instantaneous atomic correlations discussed in Ref. [2]), which, however, do not violate relativistic causality. The discussion of these results gives new insights into the subtle connection between field correlations and causality.
2. The correlation
function in the dressed ground state
Our model consists of a static field source described by a density p(r) field of mass m. In units in which c = 1 = h, the Hamiltonian is [5] H=t/
d3r [b”(r)
=$
[~+(r)]*+m*+*(r)
+
WkatRak -&
and interacting
with a Klein-Gordon
-2gp(r)c$(r)]
( pia, + pkaL),
k (2oJ)“*l
(2.1)
where V is the quantization volume, ok = (k* + m*)‘/*, a, and ai satisfy the and pk is the Fourier transform of the source density p(r). This Hamiltonian the best of our knowledge has never been used for the purposes of the present Hamiltonian (2.1) can be diagonalized by a unitary transformation, and operators,
usual bosonic commutation rules is, of course, well known, but to paper. expressed in terms of dressed
H= &A:A~+E~,
(2.2)
k
where A, = ak - gp,/(20,3V)‘/* A, 16) = 0.
and
E, = - &‘C, 1pk 1“/w,“. The dressed ground state is defined by
We have evaluated the correlation properties of the virtual meson cloud in the dressed ground particular we have considered the spatial correlations of the field energy density A?‘(r),
~(r,r’)~(~l~~(r)~*,(r’)l8)-(~l~~(r)l~)(~l~~(r’)I~)=~~(r,r’)+~~(r,J).
state. In
(2.3)
x,&r, r’) is independent of g by definition. Thus it represents the spatial correlation function of the bare vacuum in the absence of the source. On the other hand x,(r, t’) is the contribution due to the presence of the source. We have evaluated explicitly these terms assuming a point-like source located at the origin p(r) = 6(r)
( pk = 1).
For the g-independent
term in (2.3) we find exactly 1
x0(
r,
r’>
=
X0(R)
=
2
mg
-
-
2(27r)4
( mR)2 [
G(4
+
Kd4
+
-&2w
+$K,2(mR) + i
&Z&(mR) -K3(m) 2+2K:(d?) i
-&(~I
1
(
I
1 3
(2.4)
where R = r- r’ and K,(x) are modified Bessel functions 161. Asymptotic results are easily obtained from the exact result (2.4). For distances larger than the Compton wavelength of the field quanta (mR Z+ 1) we obtain e-2mR
mg Xo(J9
=
-
2(27Q3
-
(mR)3 .
(2.5)
A. la Barbera, R. Passante / Physics Letters A 206 (I 995) l-6
3
An exponential decrease with a decay factor N m-l for R zs=m-’ is to be expected for a massive field, in contrast with the inverse power laws of a massless field [1,7]. For R <
(2.6)
which is independent of m. This Rm8 dependence is reminiscent of the QED case, where an R-4 law is found for the spatial correlation of the fields [7] and consequentely an R -8 dependence is expected for the e.m. energy density which is related to the square of the fields. For the g-dependent part of (2.3) we find 8
,-m(r+r’)
x&a r’) =g24(;*)4 (&)(mr)(mr’)
+ 1+ G
i
[
K,(d)
KlW)
cos[L(R,
r)] - (I+
--$)K,(mR)
cos[L(R,r')]
1
+(1+;)(I+ -$)(X2($)cos[i(r, +>I -K,(mR)
(2.7)
r)] cos[L( R, r')]
cos[~(R,
II and, for I r I = I r’ 1 = r,
xg(c r’> =x@,a>=g2--
e-2mr
m8
4(2~)~
2( mr)3 sin(ia)
+2 1+ i K,(2mr i 1 1
2 K,(2mr
( Ii
+ 1+-
mr
K,(2mr sin(icY))
sin(icz))
sin(+a)
sin(+,))
2mr sin( +a)
cos a+K,(2mr
sin(i,))
sin2(ia)
II,
(2.8)
where (Y is the angle between r and r’. (2.7) and (2.8) are proportional to g2; terms proportional to g4 are present in (6 I ~?“(r)Z~(r’) 16) and in (6 I Z’(r) 16X6 I Z”(r’> 16) but they cancel each other when the correlation function is evaluated. Limiting cases of (2.8) may give physical insights into the behaviour of the correlation function. For (Yz+ 1 and mR = 2mr sin(ia> < 1 with mr arbitrary (i.e. separation of the two points in which the correlation is evaluated smaller than m-l, but arbitrary distance from the source) we have m8 XJ’,
e-2mr
a) =g2-2(27r)4 (mr)6
2
1+’ i
mr
I
1 a4 .
P-9)
This expression exhibits an exponential decrease with the distance from the source and a l/a4 dependence on the angle. The former feature is typical of massive fields and the latter is due to the bosonic character of the quanta of the field. The rapid growth of the correlation function as (Y+ 0 indicates that, analogously to the real bosons, also the virtual bosons of the cloud have a tendency to bunching. This is the first time that an effect of this hind is suggested in connection with virtual mesons, and it may have important observable consequences
4
A. la Barbera, R. Passante / Physics Letters A 206 (1995) l-6
when, following a traumatic event, the quanta of the cloud are stripped off the source and are emitted as real [5,8]. In this case it is natural to expect that after emission the quanta maintain the correlations they had in the cloud. If this expectation is correct, these considerations may provide a qualitative model to explain the angular correlations of the mesons emitted in nucleon-antinucleon collisions that have been observed 191.
3. The correlation function during the undressing process In order to discuss the problem of causality at the level of correlation functions, we now consider their time dependence during the undressing process of the source. We consider a fully dressed source of the Klein-Gordon field, as in Section 2, and we assume that at t = 0 the source is suddenly decoupled from the field. This is a simplified model of the time-dependence of the virtual cloud following annihilation of the source and its transformation into a pulse of free quanta [5,8]. The problem of causality have been discussed in previous papers at the level of energy density [8]. In this paper we will discuss the problem of causality during undressing at the level of correlation functions, calculating the time evolution of the spatial correlations of the energy density. We will show that this gives new insights into the causality problem in quantum field theory. We assume that for t < 0 our system is described by the interacting Hamiltonian (2.1) and that g = 0 for t > 0. In other words we assume that at t = 0, when the interaction is switched off, the system is in its dressed ground state 16). For t > 0 the state 16) is not a stationary state, and we can easily find its time evolution
(3.1) Evaluating xI(r,
the energy density correlation r’) = M(t)
function
I $(t)> -
I&(r)&(J)
= x0( r, r’) + xi(r,
in the state I I/J(~)) we find that it can be partitioned
I&“,(4 I ~(Wtw
(W)
I&W
as
W)>
r’),
(3.2)
where x&r, f ), which is time-independent and does not depend on the coupling constant g, has been obtained in Section 2. xi(r, r’) is the time-dependent part of the correlation arising from the “disappearance” of the source at t = 0; after some lengthy algebra it can be cast in the form (for a point-like source pk = 1)
x,‘!r,
r’) =g2[4
r, r’, t) +B(r,
r’, t) +C(r,
r’, t) +D(r,
r’, t) +E(r,
r’, r)],
(3.3)
where
m4 A(r, f, t) = ___
mK,(mR)B(t,
4(2~)~
m2 B( r, r’, t) = - ___
m2
m2 C(r, r’, t) = ___-
=yt,
g’(
R
aqt, K*(d)g7(tY
r)
D(r, r’, t) = ___
cos[L(r,
t, r) aqt,
r’)
X ar
t,
r’)
cos[L(
R,
r)],
R
m*
4(2rr)4
r>
ar
--K,(M)
4(2~)~
aq
r’),
r)g(t,
R
ar’
’
r’) ar,
f)]
cos[L(R,
- gZ&(mR)
r’)],
cos[L(R,
r)] cos[L(R,
f)]
A. la Barbera, R. Passante / Physics Letters A 206 (I 995) I-6
E( r, r’, t) = ____
(3.4) In these expressions
J,,(x) are Bessel functions,
0(x)
is the step function
and
(3.5) The main results of this paper stem from the form of (3.3). For r, r’ > t, F(t, r) = F(t, r’) = 0, and Eq. (3.3) reduces to Eq. (2.7), that is, to the correlation existing before the switching off of the source, X,‘
r’) =/&(r,
r’).
(3.6)
This shows that for t < r, r’ the correlation function between the points r and J does not change. This indicates a causal behaviour in the correlation function of the energy density. A more interesting result is obtained if the distance of one or both points from the source is less than t. In this case inspection of Eqs. (3.31, (3.4) easily shows that in general the correlation at time t is not the same as the correlation at time t G 0. An essential point is that this happens even if the distance R = I r - r’ I between the two points is larger than t. This means that a physical event occurring at I = 0, r = 0 (i.e. the switching off of the source) can modify at time t the field correlations at two points r and t’ even if their distance R is larger than the “causality time” t, provided r < 1 and/or r’ < t. This result is new and indicates nonlocal features in the behaviour of the field correlation functions we have considered; similar nonlocalities were recently predicted in the dynamics of interatomic correlations for a pair of two-level atoms in QED [2]. These nonlocal correlations, however, do not imply any violation of relativistic causality as we will now argue. In fact, in order to measure the correlations, one needs to set up two observers performing local energy density measurements at points r and r’; for simplicity, we can assume r = r’. The two observers can find a change in the energy density only after the causality time t = r because the dynamics of the energy density is causal [8], but in order to obtain the correlation function they must transmit the results of the measurements to each other or to a common it is easy to convince oneself that this step requires a time, which depends on the separation of “supervisor”; the two points, which ensures overall relativistic causality. This confirms that, in a physical system, the development of “nonlocal” correlations is not a signature of a violation of causality, because it does not involve transmission of information; we see that also in field dynamics, as well as in atomic dynamics, causality and spatial correlations, although interrelated, must be kept conceptually well separated. We wish to suggest that experiments apparently indicating superluminal photon propagation [4] should be investigated from the point of view of field correlations. We hope to discuss this point in the future.
4. Summary and conclusions We have considered a structureless pointlike source of the relativistic scalar field. This model can be solved exactly and it may be regarded as a simplified model of the source-field interaction. In order to discuss the problem of causality at the level of correlation functions we have considered the spatial correlations of the energy density of the virtual field around the source, both in a time-independent and in a time-dependent situation. The main results of our investigation are the following: (i) Antibunching of the virtual mesons in the ground state of the system. We have suggested that this may give rise to observable effects in the case of sudden annihilation of the source.
6
A. la Barbera, R. Passante / Physics Letters A 206 (1995) l-6
(ii) Superluminal changes in the field-field correlation function. We have argued that this behaviour does not contradict relativistic causality, but that it may help in explaining apparent experimental superluminal propagation of photons in the presence of field sources.
Acknowledgement The authors wish to thank F. Persico for fruitful discussions and suggestions on the subject of this paper and for reading the manuscript. Partial financial support by the Comitato Regionale Ricerche Nucleari e Struttura della Materia and by the Istituto Nazionale di Fisica della Materia is also acknowledged.
References [l] G. Compagno, G.M. Palma, R. Passante and F. Persico, in: New frontiers in quantum electrodynamics and quantum optics, ed. A.O. Barut (Plenum, New York, 1990) p. 129. [2] A.K. Biswas, G. Compagno, G.M. Palma, R. Passante and F. Persico, Phys. Rev. A 42 (1990) 4291. [3] M.H. Rubin, Phys. Rev. D 35 (1987) 3836; G.C. Hegerfeldt, Phys. Rev. Lett. 72 (1994) 596; D. Buchholz and 3. Yngvason, Phys. Rev. Lett. 73 (1994) 612. (41 A.M. Steinberg, P.G. Kwiat and R.Y. Chiao, Phys. Rev. Lett. 71 (1993) 708; W. Heitmann and G. Nimtx, Phys. Lett. A 196 (1994) 154. [5] E.M. Henley and W. Thirring, Elementary quantum field theory (McGraw-Hill, New York, 1962); K. Gottfried and V.F. Weisskopf, Concepts of particle physics (Oxford Univ. Press, Oxford, 1986). [6] M. Abramowitz and I. Stegun, Handbook of mathematical functions (Dover, New York, 1965). [7] C. Cohen-Tannoudji, J. Dupont-Rot and G. Grynberg, Photons and atoms (Wiley, New York, 1989). [8] F. Persico and E.A. Power, Phys. Rev. A 36 (1987) 475; G. Compagno, R. Passante and F. Per&o, Phys. Rev. A 38 (1988) 600; G. Compagno, G.M. Palma, R. Passante and F. Persico, Europhys. Lett. 9 (1989) 215. [9] M. Gyulassy, SK Kauffmann and L.W. Wilson, Phys. Rev. C 20 (1979) 2267; C. Albajer et al. (UAl Collaboration), Phys. Lett. B 226 (1989) 410; R.D. Amado, F. Cannata, J.P. Dedonder, M.P. Lecher and Y. Lu, Phys. Lett. B 339 (1994) 201.
ELSEYIER
LETTERS
A
Physics Letters A 206 (1995) l-6
Causality and spatial correlations of the relativistic scalar field in the presence of a static source A. la Barbera a, R. Passante b a Istitato di Fisica dell’llniuersitir, Via Archirafi 36, I-90123 Palenno, Italy b Istituto per le Applicazioni Interdisciplinari della Fisica, Consiglio Nazionale delle Ricerche, Via Archirafi 36, I-90123 Palermo, Italy Received 22 May 1995; accepted for publication 2 August 1995 Communicated by P.R. Holland
Abstract The spatial correlations of the energy density of the virtual cloud around a static source of the relativistic bosonic scalar field are considered. An exact expression for the correlation function in the dressed ground state of the source is obtained which indicates antibunching of the virtual bosons. Subsequently the time-dependent undressing process is considered and superluminal changes of the spatial correlations are found. The problem of causality is discussed in detail in this framework. PACS: 03.70. + k
1. Introduction The dynamics of the cloud of virtual quanta surrounding a field source after instantaneous switching on or off of the source-field coupling has been recently investigated from the point of view of the relativistic causality [l]. The problem of causality has also been discussed in the dynamics of a pair of atoms interacting with the electromagnetic radiation field [2]. In this case it has been shown that, during the time-dependent dressing of the two atoms separated by a distance R, the dynamics of each atom is strictly causal because the presence of one atom affects the dynamics of the other one only after the “causality” time t = R/c. Instantaneous interatomic correlations appear well before the time R/c, but it can be shown that they do not violate relativistic causality. Work along this line has also led to a better insight into the connection between causality and correlations and seems conceptually and practically relevant, particularly in view of recent suggestions that QED might exhibit a non-casual behaviour [3]. In contrast with interatomic correlations, field correlations in the presence of a source have not yet been fully investigated from the point of view of causality, and a satisfactory casual interpretation of recent experiments on photon propagation in the presence of atoms [4] has not been provided yet. In order to show that also field correlations in the presence of sources evolve in time in a rigorously causal way, in this paper we consider the spatial correlations of the cloud of virtual quanta (mesons) around a structureless source interacting with the relativistic scalar field, described by the Klein-Gordon equation. This model can be solved exactly and it is the simplest model of a nucleon interacting with the meson field 151. As such, it can provide guidance in a future investigation of more complex QED phenomena. We focus our 0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9601(95)00581-l
2
A. la Barbera, R. Passante/Physics
Letters A 206 (1995) l-6
attention on the spatial correlations of the energy density of the virtual field around the source. We preliminarily consider the time-independent case of the dressed ground state of the system and obtain exact expressions for the correlation function. Then we discuss the time-dependent case in which the source-field interaction is suddenly switched off (undressing) and we consider the consequent dynamics of the initial spatial correlations of the cloud. We show that non-local changes appear (they are the field counterpart of the instantaneous atomic correlations discussed in Ref. [2]), which, however, do not violate relativistic causality. The discussion of these results gives new insights into the subtle connection between field correlations and causality.
2. The correlation
function in the dressed ground state
Our model consists of a static field source described by a density p(r) field of mass m. In units in which c = 1 = h, the Hamiltonian is [5] H=t/
d3r [b”(r)
=$
[~+(r)]*+m*+*(r)
+
WkatRak -&
and interacting
with a Klein-Gordon
-2gp(r)c$(r)]
( pia, + pkaL),
k (2oJ)“*l
(2.1)
where V is the quantization volume, ok = (k* + m*)‘/*, a, and ai satisfy the and pk is the Fourier transform of the source density p(r). This Hamiltonian the best of our knowledge has never been used for the purposes of the present Hamiltonian (2.1) can be diagonalized by a unitary transformation, and operators,
usual bosonic commutation rules is, of course, well known, but to paper. expressed in terms of dressed
H= &A:A~+E~,
(2.2)
k
where A, = ak - gp,/(20,3V)‘/* A, 16) = 0.
and
E, = - &‘C, 1pk 1“/w,“. The dressed ground state is defined by
We have evaluated the correlation properties of the virtual meson cloud in the dressed ground particular we have considered the spatial correlations of the field energy density A?‘(r),
~(r,r’)~(~l~~(r)~*,(r’)l8)-(~l~~(r)l~)(~l~~(r’)I~)=~~(r,r’)+~~(r,J).
state. In
(2.3)
x,&r, r’) is independent of g by definition. Thus it represents the spatial correlation function of the bare vacuum in the absence of the source. On the other hand x,(r, t’) is the contribution due to the presence of the source. We have evaluated explicitly these terms assuming a point-like source located at the origin p(r) = 6(r)
( pk = 1).
For the g-independent
term in (2.3) we find exactly 1
x0(
r,
r’>
=
X0(R)
=
2
mg
-
-
2(27r)4
( mR)2 [
G(4
+
Kd4
+
-&2w
+$K,2(mR) + i
&Z&(mR) -K3(m) 2+2K:(d?) i
-&(~I
1
(
I
1 3
(2.4)
where R = r- r’ and K,(x) are modified Bessel functions 161. Asymptotic results are easily obtained from the exact result (2.4). For distances larger than the Compton wavelength of the field quanta (mR Z+ 1) we obtain e-2mR
mg Xo(J9
=
-
2(27Q3
-
(mR)3 .
(2.5)
A. la Barbera, R. Passante / Physics Letters A 206 (I 995) l-6
3
An exponential decrease with a decay factor N m-l for R zs=m-’ is to be expected for a massive field, in contrast with the inverse power laws of a massless field [1,7]. For R <
(2.6)
which is independent of m. This Rm8 dependence is reminiscent of the QED case, where an R-4 law is found for the spatial correlation of the fields [7] and consequentely an R -8 dependence is expected for the e.m. energy density which is related to the square of the fields. For the g-dependent part of (2.3) we find 8
,-m(r+r’)
x&a r’) =g24(;*)4 (&)(mr)(mr’)
+ 1+ G
i
[
K,(d)
KlW)
cos[L(R,
r)] - (I+
--$)K,(mR)
cos[L(R,r')]
1
+(1+;)(I+ -$)(X2($)cos[i(r, +>I -K,(mR)
(2.7)
r)] cos[L( R, r')]
cos[~(R,
II and, for I r I = I r’ 1 = r,
xg(c r’> =x@,a>=g2--
e-2mr
m8
4(2~)~
2( mr)3 sin(ia)
+2 1+ i K,(2mr i 1 1
2 K,(2mr
( Ii
+ 1+-
mr
K,(2mr sin(icY))
sin(icz))
sin(+a)
sin(+,))
2mr sin( +a)
cos a+K,(2mr
sin(i,))
sin2(ia)
II,
(2.8)
where (Y is the angle between r and r’. (2.7) and (2.8) are proportional to g2; terms proportional to g4 are present in (6 I ~?“(r)Z~(r’) 16) and in (6 I Z’(r) 16X6 I Z”(r’> 16) but they cancel each other when the correlation function is evaluated. Limiting cases of (2.8) may give physical insights into the behaviour of the correlation function. For (Yz+ 1 and mR = 2mr sin(ia> < 1 with mr arbitrary (i.e. separation of the two points in which the correlation is evaluated smaller than m-l, but arbitrary distance from the source) we have m8 XJ’,
e-2mr
a) =g2-2(27r)4 (mr)6
2
1+’ i
mr
I
1 a4 .
P-9)
This expression exhibits an exponential decrease with the distance from the source and a l/a4 dependence on the angle. The former feature is typical of massive fields and the latter is due to the bosonic character of the quanta of the field. The rapid growth of the correlation function as (Y+ 0 indicates that, analogously to the real bosons, also the virtual bosons of the cloud have a tendency to bunching. This is the first time that an effect of this hind is suggested in connection with virtual mesons, and it may have important observable consequences
4
A. la Barbera, R. Passante / Physics Letters A 206 (1995) l-6
when, following a traumatic event, the quanta of the cloud are stripped off the source and are emitted as real [5,8]. In this case it is natural to expect that after emission the quanta maintain the correlations they had in the cloud. If this expectation is correct, these considerations may provide a qualitative model to explain the angular correlations of the mesons emitted in nucleon-antinucleon collisions that have been observed 191.
3. The correlation function during the undressing process In order to discuss the problem of causality at the level of correlation functions, we now consider their time dependence during the undressing process of the source. We consider a fully dressed source of the Klein-Gordon field, as in Section 2, and we assume that at t = 0 the source is suddenly decoupled from the field. This is a simplified model of the time-dependence of the virtual cloud following annihilation of the source and its transformation into a pulse of free quanta [5,8]. The problem of causality have been discussed in previous papers at the level of energy density [8]. In this paper we will discuss the problem of causality during undressing at the level of correlation functions, calculating the time evolution of the spatial correlations of the energy density. We will show that this gives new insights into the causality problem in quantum field theory. We assume that for t < 0 our system is described by the interacting Hamiltonian (2.1) and that g = 0 for t > 0. In other words we assume that at t = 0, when the interaction is switched off, the system is in its dressed ground state 16). For t > 0 the state 16) is not a stationary state, and we can easily find its time evolution
(3.1) Evaluating xI(r,
the energy density correlation r’) = M(t)
function
I $(t)> -
I&(r)&(J)
= x0( r, r’) + xi(r,
in the state I I/J(~)) we find that it can be partitioned
I&“,(4 I ~(Wtw
(W)
I&W
as
W)>
r’),
(3.2)
where x&r, f ), which is time-independent and does not depend on the coupling constant g, has been obtained in Section 2. xi(r, r’) is the time-dependent part of the correlation arising from the “disappearance” of the source at t = 0; after some lengthy algebra it can be cast in the form (for a point-like source pk = 1)
x,‘!r,
r’) =g2[4
r, r’, t) +B(r,
r’, t) +C(r,
r’, t) +D(r,
r’, t) +E(r,
r’, r)],
(3.3)
where
m4 A(r, f, t) = ___
mK,(mR)B(t,
4(2~)~
m2 B( r, r’, t) = - ___
m2
m2 C(r, r’, t) = ___-
=yt,
g’(
R
aqt, K*(d)g7(tY
r)
D(r, r’, t) = ___
cos[L(r,
t, r) aqt,
r’)
X ar
t,
r’)
cos[L(
R,
r)],
R
m*
4(2rr)4
r>
ar
--K,(M)
4(2~)~
aq
r’),
r)g(t,
R
ar’
’
r’) ar,
f)]
cos[L(R,
- gZ&(mR)
r’)],
cos[L(R,
r)] cos[L(R,
f)]
A. la Barbera, R. Passante / Physics Letters A 206 (I 995) I-6
E( r, r’, t) = ____
(3.4) In these expressions
J,,(x) are Bessel functions,
0(x)
is the step function
and
(3.5) The main results of this paper stem from the form of (3.3). For r, r’ > t, F(t, r) = F(t, r’) = 0, and Eq. (3.3) reduces to Eq. (2.7), that is, to the correlation existing before the switching off of the source, X,‘
r’) =/&(r,
r’).
(3.6)
This shows that for t < r, r’ the correlation function between the points r and J does not change. This indicates a causal behaviour in the correlation function of the energy density. A more interesting result is obtained if the distance of one or both points from the source is less than t. In this case inspection of Eqs. (3.31, (3.4) easily shows that in general the correlation at time t is not the same as the correlation at time t G 0. An essential point is that this happens even if the distance R = I r - r’ I between the two points is larger than t. This means that a physical event occurring at I = 0, r = 0 (i.e. the switching off of the source) can modify at time t the field correlations at two points r and t’ even if their distance R is larger than the “causality time” t, provided r < 1 and/or r’ < t. This result is new and indicates nonlocal features in the behaviour of the field correlation functions we have considered; similar nonlocalities were recently predicted in the dynamics of interatomic correlations for a pair of two-level atoms in QED [2]. These nonlocal correlations, however, do not imply any violation of relativistic causality as we will now argue. In fact, in order to measure the correlations, one needs to set up two observers performing local energy density measurements at points r and r’; for simplicity, we can assume r = r’. The two observers can find a change in the energy density only after the causality time t = r because the dynamics of the energy density is causal [8], but in order to obtain the correlation function they must transmit the results of the measurements to each other or to a common it is easy to convince oneself that this step requires a time, which depends on the separation of “supervisor”; the two points, which ensures overall relativistic causality. This confirms that, in a physical system, the development of “nonlocal” correlations is not a signature of a violation of causality, because it does not involve transmission of information; we see that also in field dynamics, as well as in atomic dynamics, causality and spatial correlations, although interrelated, must be kept conceptually well separated. We wish to suggest that experiments apparently indicating superluminal photon propagation [4] should be investigated from the point of view of field correlations. We hope to discuss this point in the future.
4. Summary and conclusions We have considered a structureless pointlike source of the relativistic scalar field. This model can be solved exactly and it may be regarded as a simplified model of the source-field interaction. In order to discuss the problem of causality at the level of correlation functions we have considered the spatial correlations of the energy density of the virtual field around the source, both in a time-independent and in a time-dependent situation. The main results of our investigation are the following: (i) Antibunching of the virtual mesons in the ground state of the system. We have suggested that this may give rise to observable effects in the case of sudden annihilation of the source.
6
A. la Barbera, R. Passante / Physics Letters A 206 (1995) l-6
(ii) Superluminal changes in the field-field correlation function. We have argued that this behaviour does not contradict relativistic causality, but that it may help in explaining apparent experimental superluminal propagation of photons in the presence of field sources.
Acknowledgement The authors wish to thank F. Persico for fruitful discussions and suggestions on the subject of this paper and for reading the manuscript. Partial financial support by the Comitato Regionale Ricerche Nucleari e Struttura della Materia and by the Istituto Nazionale di Fisica della Materia is also acknowledged.
References [l] G. Compagno, G.M. Palma, R. Passante and F. Persico, in: New frontiers in quantum electrodynamics and quantum optics, ed. A.O. Barut (Plenum, New York, 1990) p. 129. [2] A.K. Biswas, G. Compagno, G.M. Palma, R. Passante and F. Persico, Phys. Rev. A 42 (1990) 4291. [3] M.H. Rubin, Phys. Rev. D 35 (1987) 3836; G.C. Hegerfeldt, Phys. Rev. Lett. 72 (1994) 596; D. Buchholz and 3. Yngvason, Phys. Rev. Lett. 73 (1994) 612. (41 A.M. Steinberg, P.G. Kwiat and R.Y. Chiao, Phys. Rev. Lett. 71 (1993) 708; W. Heitmann and G. Nimtx, Phys. Lett. A 196 (1994) 154. [5] E.M. Henley and W. Thirring, Elementary quantum field theory (McGraw-Hill, New York, 1962); K. Gottfried and V.F. Weisskopf, Concepts of particle physics (Oxford Univ. Press, Oxford, 1986). [6] M. Abramowitz and I. Stegun, Handbook of mathematical functions (Dover, New York, 1965). [7] C. Cohen-Tannoudji, J. Dupont-Rot and G. Grynberg, Photons and atoms (Wiley, New York, 1989). [8] F. Persico and E.A. Power, Phys. Rev. A 36 (1987) 475; G. Compagno, R. Passante and F. Per&o, Phys. Rev. A 38 (1988) 600; G. Compagno, G.M. Palma, R. Passante and F. Persico, Europhys. Lett. 9 (1989) 215. [9] M. Gyulassy, SK Kauffmann and L.W. Wilson, Phys. Rev. C 20 (1979) 2267; C. Albajer et al. (UAl Collaboration), Phys. Lett. B 226 (1989) 410; R.D. Amado, F. Cannata, J.P. Dedonder, M.P. Lecher and Y. Lu, Phys. Lett. B 339 (1994) 201.