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A naive remark concerning the bohm-Aharonov experiment
Volume 79A, number 2,3
PHYSICS LETTERS
29 September 1980
A NAIVE REMARK CONCERNING THE BOHM-AHARONOV EXPERIMENT TrevorW. MARSHALL Department of Mathematics, Manchester University, Manchester M13 9PL, UK Received 24 May 1980
It is shown that correlation measurements on annihilation photons cannot be used to establish the correctness, or otherwise, of local realistic theories.
The Bohm—Aharonov experiment [1,2] was proposed to decide whether the pair of photons emitted in positronium annihilation is correctly described by quantum electrodynamics or by the Furry [3] model. According to the former, the pair is emitted in the singlet state I,LI =
2—’/2(IXY)
—
I YX)),
(1)
while, according to the latter, the state of the pair is
where F(0) =
2
+ “~—
~
,
G(0) =
sin20
(6) (2 cos 0)~ (2 cos 0)2 We are here using the standard notation [5] for the Klein—Nishina formula. Note that the Furry expres/
___________
—
.
—
sion is obtained by a simple integration of this formula: C(0 1, Oi, 02, 02)
a density matrix, for which the probability of the wave function being ili(a) = IXcos a + Y sin a,
—
X sin a + Y cos a), (2)
is
=—f [F(01)
(0 ~ a <
(3)
~•
G(01) cos 2(~~ a)] —
(7)
0
X [F(02)
p(a) = ~
—
—
G(02) cos 2(02
—
a)] da
(FM)
The expressions (4) and (5) should be compared with
The method proposed by Bohm and Aharonov for deciding between these alternatives was to observe the Compton scattering of the two photons at two widely separated locations. If the polar and azimuthal angles at these two locations are denoted (Oi, Oi) and (02, 02 + ~j ir), the coincidence rate has a different angular factor according to which model is taken, namely
the coincidence rates obtained when a singlet pair of optical photons, emitted in an atomic cascade, are analysed with conventional polarimeters. For the ideal case, these give 2(0 C(01, 02) = cos 1 02) (QED) (8) = cos~(01 02) + ~ (FM), (9)
C(01, Oi, 02, 02) = F(0 1)F(02)
and, again, the second expression may be obtained by simple integration of the transmission probabilities for
+
G(01)G(02) cos 2(Oi
C(01,
—
02)
(QED),
(4)
(FM),
(5)
0i’ °2’02) = F(01)F(02)
~
—
~
,
—
the separate polarizers: C(Ø1, 02) 2(0
+
~G(01)G(02) cos 2(Oi —02)
i-f cos
2(0 1
—
a) cos
2
—
a) da
(FM). (10)
147
Volume 79A, number 2,3
PHYSICS LETTERS
29 September 1980
Any local realistic theory gives a coincidence rate which is a generalization of eq. (10), namely:
One such representation has been given by Bell [6], namely
C(0i,02)=fP(0i,X)P(0,X)P(X)dX~
C(01,01,02,02)
(11)
r [F(o1) It
where P and p are non-negative functions of their arguments. The theorem of Bell [6] established that such a representation is not possible if C(01, 02) is given by eq. (8). We therefore have the possibility of a crucial experiment against all local realistic theories, though it should be stressed [7,8] that no experiment so far performed falls into this category. The situation is quite different with annthilation 2(0 photons, because the term containing cos 1 02) is largely masked by F(01)F(02). For example, 01 I = 02 ~ir gives 2(0 ir, Ø~,~ ~ 0~)= + 2 cos 1 02) (QED) 2(~i—~) (FM),(12) = ~ +c (13) so the Compton polarimeter is much less efficient in analysing polarization than the devices used for optical —
— —
photons. Attempts have been made [4,9—11]to use eq. (4) as the basis for an experimental test for local realistic theories, and the whole eq.of (4)the is Compton indeed corroborated. But the lowondiscrimination polarimeter is such that, as a result, only a subclass of local realistic theories can be excluded. This subclass has been characterized [121by two assumptions: “(i) in principle, ideal linear polarisers can be constructed for high-energy photons; (ii) the results, which would be obtained in an experiment using ideal analysers, and those obtained in a Compton scattering experiment, are conectly related by quantum theory”. But these two assumptions together restrict the angular factor to rather trivial generalizations of eq. (7), in which the uniform density ~ is replaced by a non-uniform p(a). We now show, by explicit counterexample, that a representation of form (11) is possible for eq. (4). This is enough to show that even an ideal experiment based on Compton polarimeters cannot discriminate between quantum electrodynamics and local realistic theories in general. Indeed the Bohm—Aharonov experiment can do little more than it was originally designed for, namely, discriminate between quantum electrodynamics and the Furry model, 148
=
i ir
v2G(o ~)~
—
a)I (14)
J
0
X [F(07) \/~G(02)cos 2(02 a)] dcs It involves a modification in the Klein—Nishina formula for polarized photons, but note that this formula cannot, at present, be subjected to experimental test in the energy range in question, because of the impossibility of constructing suitable polarizers. The scattering for unpolarized photons is F(0), in agreement with both quantum electrodynamics and cx—
—
periment. Another representation is
+
iT
(15)
iT
l—r
I h(0
2(0 1)h(07)cos
J
2(0 1
—
a) cos
2 —a)da,
=
F(0)+\/~G(0)
0
where =
f(0)
F(o)_ \‘~G(O) 12
I
(I
—
g(0)
r)1
1
+ (I
—
(16)
h(0)
(I r) and r is an arbitrary parameter in the interval (0, 1). This expression represents a model in which (i) either a pair of circularly polarized photons is emitted, with probability r, or a pair of linearly polarized photons is emitted, with probability I r, (ii) there are two types of circularly polarized photons, with scattering cross sectionsf(0) andg(0), (iii) the linearly polarized photons have scattering cross section h(0) cos2(Ø a). If only one photon is observed, its scattering cross section is ‘[f(O) +g(0)] + ~(l r)h(0) = F(0), (17) —
—
—
which, again, is the Klein—Nishina formula for unpolarized photons. There seems little doubt that a wider family of representations exists for C(0 1, 01, 02, 02), combining features from eqs. (14) and (15), and having at least two adjustable parameters. It may be, therefore, that
Volume 79A, number 2,3
PHYSICS LETTERS
values may be chosen for these parameters which effect a smooth transition from the cross sections proposed here to the Klein—Nishina formulas which have been verified experimentally in the low-energy X me2) range [5] Kasday et al. [4], commenting on their experimental findings, said: “It would be pleasing to be able to say that the results of this experiment rule out hidden variables theories. We cannot say that.” Setting aside the value judgement contalned in the first sentence, the above analysis confirms their statement. These authors also stated that “any theory that satisfies Bell’s theorem and reproduces our results looks quite artificial”. The last two words conceal another value judgement, and a full reply to them would involve a discussion of scientific methodology which is not ap propnate here. The above discussion shows that what is at stake is essentially the belief of these authors that the Klein—Nishina formula for polarized photons still holds good outside the range of frequencies for which it has been experimentally verified. There have recently been some extravagant claims [13—15]made in the popular scientific literature about the interpretation of results obtained in Bohm— Aharonov type experiments. It is unfortunate that the above quoted words of Kasday et al. have been so widely disregarded, and I hope that this note will contribute towards a correct appraisal of these experiments. ~
29 September 1980
References [1]
D. Bohm, Quantum theory (Prentice Hall, New York,
Ch. XXII. [21 1951) D. Bohm andY. Aharonov, Phys. Rev. 108 (1957) 1070. [3] W.H. Furry, Phys. Rev. 49 (1936) 393,476.
[4] L.R. Kasday, J.D. Ullman and CS. Wu, Nuovo Cimento 25B (1975) 633. [51W. Heitler, The quantum theory of radiation (Oxford U. P., 1954) pp. 211 ff.
[6] J.S. Bell, Physics 1 (1964) 195.
[71 T.W. [8] T.W.
Marshall, Phys. Lett. 75A (1980) 265. Marshall, Phys. Lett. 78A (1980) 15. [9] G. Farad, D. Gutkowski, S. Notarrigo and A.R. Pennisi, Lett. Nuovo Cimento 9 (1974) 607. [10] A.R. Wilson, J. Lowe and D.K. Butt, J. Phys. G2 (1976) 613. [11] M. Bruno, M. d’Agostino and C. Maroni, Nuovo Cimento 40B (1977) 142. [12] J.F. aauser and A. Shimony, Rep. Prog. Phys. 41 (1978) 1881. [13] B. d’Espagnat, Quantum theory and reality, Sri. Am. (Nov. 1979). [14] B.J. Hiley, Ghostly interactions in physics, New Sri. (March 6th, 1980). [15] D. Zohar, Why things really can go bump in the night, Sunday Times (April 20th, 1980). [16] L.R. Kasday, Foundations of quantum mechanics, ed. B. d’Espagnat (Academic Press, New York, 1971) p. 208.
149
PHYSICS LETTERS
29 September 1980
A NAIVE REMARK CONCERNING THE BOHM-AHARONOV EXPERIMENT TrevorW. MARSHALL Department of Mathematics, Manchester University, Manchester M13 9PL, UK Received 24 May 1980
It is shown that correlation measurements on annihilation photons cannot be used to establish the correctness, or otherwise, of local realistic theories.
The Bohm—Aharonov experiment [1,2] was proposed to decide whether the pair of photons emitted in positronium annihilation is correctly described by quantum electrodynamics or by the Furry [3] model. According to the former, the pair is emitted in the singlet state I,LI =
2—’/2(IXY)
—
I YX)),
(1)
while, according to the latter, the state of the pair is
where F(0) =
2
+ “~—
~
,
G(0) =
sin20
(6) (2 cos 0)~ (2 cos 0)2 We are here using the standard notation [5] for the Klein—Nishina formula. Note that the Furry expres/
___________
—
.
—
sion is obtained by a simple integration of this formula: C(0 1, Oi, 02, 02)
a density matrix, for which the probability of the wave function being ili(a) = IXcos a + Y sin a,
—
X sin a + Y cos a), (2)
is
=—f [F(01)
(0 ~ a <
(3)
~•
G(01) cos 2(~~ a)] —
(7)
0
X [F(02)
p(a) = ~
—
—
G(02) cos 2(02
—
a)] da
(FM)
The expressions (4) and (5) should be compared with
The method proposed by Bohm and Aharonov for deciding between these alternatives was to observe the Compton scattering of the two photons at two widely separated locations. If the polar and azimuthal angles at these two locations are denoted (Oi, Oi) and (02, 02 + ~j ir), the coincidence rate has a different angular factor according to which model is taken, namely
the coincidence rates obtained when a singlet pair of optical photons, emitted in an atomic cascade, are analysed with conventional polarimeters. For the ideal case, these give 2(0 C(01, 02) = cos 1 02) (QED) (8) = cos~(01 02) + ~ (FM), (9)
C(01, Oi, 02, 02) = F(0 1)F(02)
and, again, the second expression may be obtained by simple integration of the transmission probabilities for
+
G(01)G(02) cos 2(Oi
C(01,
—
02)
(QED),
(4)
(FM),
(5)
0i’ °2’02) = F(01)F(02)
~
—
~
,
—
the separate polarizers: C(Ø1, 02) 2(0
+
~G(01)G(02) cos 2(Oi —02)
i-f cos
2(0 1
—
a) cos
2
—
a) da
(FM). (10)
147
Volume 79A, number 2,3
PHYSICS LETTERS
29 September 1980
Any local realistic theory gives a coincidence rate which is a generalization of eq. (10), namely:
One such representation has been given by Bell [6], namely
C(0i,02)=fP(0i,X)P(0,X)P(X)dX~
C(01,01,02,02)
(11)
r [F(o1) It
where P and p are non-negative functions of their arguments. The theorem of Bell [6] established that such a representation is not possible if C(01, 02) is given by eq. (8). We therefore have the possibility of a crucial experiment against all local realistic theories, though it should be stressed [7,8] that no experiment so far performed falls into this category. The situation is quite different with annthilation 2(0 photons, because the term containing cos 1 02) is largely masked by F(01)F(02). For example, 01 I = 02 ~ir gives 2(0 ir, Ø~,~ ~ 0~)= + 2 cos 1 02) (QED) 2(~i—~) (FM),(12) = ~ +c (13) so the Compton polarimeter is much less efficient in analysing polarization than the devices used for optical —
— —
photons. Attempts have been made [4,9—11]to use eq. (4) as the basis for an experimental test for local realistic theories, and the whole eq.of (4)the is Compton indeed corroborated. But the lowondiscrimination polarimeter is such that, as a result, only a subclass of local realistic theories can be excluded. This subclass has been characterized [121by two assumptions: “(i) in principle, ideal linear polarisers can be constructed for high-energy photons; (ii) the results, which would be obtained in an experiment using ideal analysers, and those obtained in a Compton scattering experiment, are conectly related by quantum theory”. But these two assumptions together restrict the angular factor to rather trivial generalizations of eq. (7), in which the uniform density ~ is replaced by a non-uniform p(a). We now show, by explicit counterexample, that a representation of form (11) is possible for eq. (4). This is enough to show that even an ideal experiment based on Compton polarimeters cannot discriminate between quantum electrodynamics and local realistic theories in general. Indeed the Bohm—Aharonov experiment can do little more than it was originally designed for, namely, discriminate between quantum electrodynamics and the Furry model, 148
=
i ir
v2G(o ~)~
—
a)I (14)
J
0
X [F(07) \/~G(02)cos 2(02 a)] dcs It involves a modification in the Klein—Nishina formula for polarized photons, but note that this formula cannot, at present, be subjected to experimental test in the energy range in question, because of the impossibility of constructing suitable polarizers. The scattering for unpolarized photons is F(0), in agreement with both quantum electrodynamics and cx—
—
periment. Another representation is
+
iT
(15)
iT
l—r
I h(0
2(0 1)h(07)cos
J
2(0 1
—
a) cos
2 —a)da,
=
F(0)+\/~G(0)
0
where =
f(0)
F(o)_ \‘~G(O) 12
I
(I
—
g(0)
r)1
1
+ (I
—
(16)
h(0)
(I r) and r is an arbitrary parameter in the interval (0, 1). This expression represents a model in which (i) either a pair of circularly polarized photons is emitted, with probability r, or a pair of linearly polarized photons is emitted, with probability I r, (ii) there are two types of circularly polarized photons, with scattering cross sectionsf(0) andg(0), (iii) the linearly polarized photons have scattering cross section h(0) cos2(Ø a). If only one photon is observed, its scattering cross section is ‘[f(O) +g(0)] + ~(l r)h(0) = F(0), (17) —
—
—
which, again, is the Klein—Nishina formula for unpolarized photons. There seems little doubt that a wider family of representations exists for C(0 1, 01, 02, 02), combining features from eqs. (14) and (15), and having at least two adjustable parameters. It may be, therefore, that
Volume 79A, number 2,3
PHYSICS LETTERS
values may be chosen for these parameters which effect a smooth transition from the cross sections proposed here to the Klein—Nishina formulas which have been verified experimentally in the low-energy X me2) range [5] Kasday et al. [4], commenting on their experimental findings, said: “It would be pleasing to be able to say that the results of this experiment rule out hidden variables theories. We cannot say that.” Setting aside the value judgement contalned in the first sentence, the above analysis confirms their statement. These authors also stated that “any theory that satisfies Bell’s theorem and reproduces our results looks quite artificial”. The last two words conceal another value judgement, and a full reply to them would involve a discussion of scientific methodology which is not ap propnate here. The above discussion shows that what is at stake is essentially the belief of these authors that the Klein—Nishina formula for polarized photons still holds good outside the range of frequencies for which it has been experimentally verified. There have recently been some extravagant claims [13—15]made in the popular scientific literature about the interpretation of results obtained in Bohm— Aharonov type experiments. It is unfortunate that the above quoted words of Kasday et al. have been so widely disregarded, and I hope that this note will contribute towards a correct appraisal of these experiments. ~
29 September 1980
References [1]
D. Bohm, Quantum theory (Prentice Hall, New York,
Ch. XXII. [21 1951) D. Bohm andY. Aharonov, Phys. Rev. 108 (1957) 1070. [3] W.H. Furry, Phys. Rev. 49 (1936) 393,476.
[4] L.R. Kasday, J.D. Ullman and CS. Wu, Nuovo Cimento 25B (1975) 633. [51W. Heitler, The quantum theory of radiation (Oxford U. P., 1954) pp. 211 ff.
[6] J.S. Bell, Physics 1 (1964) 195.
[71 T.W. [8] T.W.
Marshall, Phys. Lett. 75A (1980) 265. Marshall, Phys. Lett. 78A (1980) 15. [9] G. Farad, D. Gutkowski, S. Notarrigo and A.R. Pennisi, Lett. Nuovo Cimento 9 (1974) 607. [10] A.R. Wilson, J. Lowe and D.K. Butt, J. Phys. G2 (1976) 613. [11] M. Bruno, M. d’Agostino and C. Maroni, Nuovo Cimento 40B (1977) 142. [12] J.F. aauser and A. Shimony, Rep. Prog. Phys. 41 (1978) 1881. [13] B. d’Espagnat, Quantum theory and reality, Sri. Am. (Nov. 1979). [14] B.J. Hiley, Ghostly interactions in physics, New Sri. (March 6th, 1980). [15] D. Zohar, Why things really can go bump in the night, Sunday Times (April 20th, 1980). [16] L.R. Kasday, Foundations of quantum mechanics, ed. B. d’Espagnat (Academic Press, New York, 1971) p. 208.
149