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Are 2π rotations observable?
Volume 96A, number 9
PHYSICS LETTERS
1 August 1983
ARE 2ic ROTATIONS OBSERVABLE? Thomas F. JORDAN Physics Department, University of Minnesota, Duluth, MN 55812, USA Received 28 April 1983 Revised manuscript received 1 June 1983
Two beams of indistinguishablefermions from independent sources cannot produce an interference pattern that changes when the wavefunction of each single-fermion state for one beam is multiplied by —1; they cannot produce interference the way photons from independent sources can. This means the changes in neutron interference caused by a magnetic field applied to one of the beams cannot be interpreted as equivalent to changes that could be made by rotating the source of one beam by 2ir radians. If it is assumed that rotation by 2ir radians cannot be observed, the argument made here becomes a simple proof that particles with half-integral spin cannot be bosons.
From the time they were first suggested, experiments with neutron interference [1] have been interpreted [2] as observations of the factor—i that represents rotation by 2ir radians for a particle with spin 1/2. It would be interesting if this factor were observable; it would indicate that rotations in space involve more than has been assumed. However, it has been shown that the fermion antisymmetry of neutron states complicates the interpretation of these experiments [3]. This will be shown here in a different way, simply by pointing out that two beams of indistinguishable fermions from independent sources cannot produce an interference pattern that changes when the wavefunction of each single-fermion state for one beam is multiplied by —1; they cannot produce interference the way photons from independent sources can. This means the changes in neutron interference observed in experiments cannot be interpreted as equivaient to changes that could be made by rotating the source of one beam by 2ir radians. In these experiments [1] a beam of neutrons is split into two separated beams and a magnetic field is applied to one of them. For a neutron spin ~tT with magnetic moment ~ moving through a uniform magnetic field B for a time t, the state is changed by exp[—it(---iøB)/h] which is the same as it would be changed by rotation ,
0 031-9163/83/0000—0000/$ 03.00 © 1983 North-Holland
through the angle = 2t ~ around an axis in the direction of—B. In the Heisenberg picture the transformation of the spin matrices is the same as this rotation. When 0 is 2x the wavefunction is multiplied by —1. The effect of this change in one beam relative to the other is observed as a change in the interference pattern produced by the recombined beam. About all this there is no question. The question is whether observing this is equivalent to observing a rotation [4]. Can the magnetic field be regarded as a handy experimental tool for producing a change that could also be produced~at least in principie, by actually rotating one of the beams? It appears not. Such an interpretation cannot be made in a clear way because quantum mechanics predicts this change cannot be observed in an interference pattern if the beams are independent. In this regard, neutron interference is fundamentally different from interference of light. A two-slit interference pattern produced by a split beam of photons from a single source can also be produced, at least in theory, by independent beams represented by coherent states [5,6], so an experiment done with a single source is also conceivable with independent sources. Thus one can imagine rotating one of the beams by rotatmg its source. This is not possible with fermions. .
.
.
.
457
Volume 96A, number 9
PHYSICS LETTERS
If the beams are not independent, it would not be remarkable to observe the effect of rotating one of them by 2ir radians. It is a common everyday experience to observe the effect of rotation by 2ir radians of a subsystem that is attached to another part of a larger system. For example, consider turning a crank to wind up a spring that drives a motor. In quantum mechanics, subsystems that are separated are not necessarily independent. That is demonstrated by Einstein—Podolsky--Rosen correlations, Consider an expansion of the positive-frequency part of the neutron field operator,
1 August 1983
Interference can also be explained as correlations of the numbers of quanta counted at different points [9]. When there is no interference described by the mean number, there can be interference due to correlations of fluctuations. Such interference might be observed with two independent neutron beams. The fluctuations would make it more difficult to observe the effect of a magnetic field applied to one of the beams. Rotations by 2ir radians could not be observed because if the beams are independent the correlations are not changed when the wavefunction of each single-neutron state for one beam is multiplied by -~-l.The correlations are described by expectation values
~(x t)=~a ~ (x t) n
~
(i~it(x
1,t1) ~1i(x1,t1)~1it(x2,t2) ~i(x2,t2))
in terms of orthonormal single-neutron wavefunctions ,Li~(x,t) and corresponding annihilation operators a~ the adjoints a~are the creation operators for the singleneutron states with wavefunctions ~0(x, t). Since neutrons are fermions, the annihilation and creation operators for different single-neutron states anticommute. The mean number of neutrons counted at position x at time t is proportional to the expectation value, (~jt~x,t) ~x,t)>=~(a~an>I~n~x,t)I2
i~
+2Re m(a m n
)(a) m
n
and the fermion antisymmetry implies (am> ~n>
=
(aman>
=
~n~2m> = —(an) (am)
so either (am> or (an> must be zero, and *
(am an)
(am> (an) = (am) ~n> = 0.
There are no interference terms in the formula for the mean number of neutrons counted where the beams combine. In contrast, for independent beams of photons represented by coherent states [5], the interference terms are the same as for a split beam from a single source [61; this can explain [6] interference observed [7] with two lasers [8]. 458
whose expansions contain expectation values (at a at a / k m n of four creation and annihilation operators. When the beams are independent, the fermion antisymmetry implies the expectation values are zero if three of the creation and annihilation operators are for one beam and one is for the other. For example, if j, k, m are for one beam and n is for the other, then t t t t t t (a1 akam> (an) = (a1 aka~an)= (—a~a1akam)
—(aisfl )(ata If there an ieven k at) mnumber of operators for each beam, the expectation values are not changed when the wavefunction of each single-neutron state for one beam is multiplied by —1. The connection between spin and statistics is evident here. When the particles are bosons, independent beams can produce interference that changes when the wavefunction of each single-particle state is multiplied by —1. This is not possible for fermions. On the other hand, rotation by 2ir radians multiplies a wavefunction by —l only for half-integral spin. A wavefunction for integral spin is not changed. If it is assumed that rotation by 2n radians cannot be observed, what we have here is a simple proof that particles with half-integral spin cannot be bosons. References
Ill
H. Rauch eta!., Phys. Lett. 54A (1975) 425 S.A. Werner, R. Colella, A.W. Overhauser and C.F. Eagen, Phys. Rev. Lett. 35 (1975) 1053.
Volume 96A, number 9
PHYSICS LETTERS
[2] H.J. Bernstein, Phys. Rev. Lett. 18(1967)1102; Y. Aharonov and L. Susskind, Phys. Rev. 158 (1967) 1237; R. Mirman,Phys. Rev. Dl (1970) 3349 [3] J. Byrne, Nature 275 (1978) 188. [4] G.C. Hegerfeldt and K. Kraus, Phys. Rev. 170 (1968) 1185. [5] Ri. G!auber, Phys. Rev. Lett. 10(1963) 84;Phys. Rev. 130 (1963) 2529; 131 (1963) 2766.
1 August 1983
[6] T.F. Jordan and F. Ghielmetti, Phys. Rev. Lett. 12 (1964) 607. [7] G. Magyar and L. Mandel, Nature 198 (1963) 255; M.5. Lipsett and L. Mande!, Nature 199 (1963) 553; R.L. Pflegor and L. Mandel, Phys. Rev. 159 (1967) 1084. [8] H. Haken, Phys. Rev. Lett. 13 (1964) 329a. [91 L.Mandel,Phys.Rev. 134 (1964) AlO.
459
PHYSICS LETTERS
1 August 1983
ARE 2ic ROTATIONS OBSERVABLE? Thomas F. JORDAN Physics Department, University of Minnesota, Duluth, MN 55812, USA Received 28 April 1983 Revised manuscript received 1 June 1983
Two beams of indistinguishablefermions from independent sources cannot produce an interference pattern that changes when the wavefunction of each single-fermion state for one beam is multiplied by —1; they cannot produce interference the way photons from independent sources can. This means the changes in neutron interference caused by a magnetic field applied to one of the beams cannot be interpreted as equivalent to changes that could be made by rotating the source of one beam by 2ir radians. If it is assumed that rotation by 2ir radians cannot be observed, the argument made here becomes a simple proof that particles with half-integral spin cannot be bosons.
From the time they were first suggested, experiments with neutron interference [1] have been interpreted [2] as observations of the factor—i that represents rotation by 2ir radians for a particle with spin 1/2. It would be interesting if this factor were observable; it would indicate that rotations in space involve more than has been assumed. However, it has been shown that the fermion antisymmetry of neutron states complicates the interpretation of these experiments [3]. This will be shown here in a different way, simply by pointing out that two beams of indistinguishable fermions from independent sources cannot produce an interference pattern that changes when the wavefunction of each single-fermion state for one beam is multiplied by —1; they cannot produce interference the way photons from independent sources can. This means the changes in neutron interference observed in experiments cannot be interpreted as equivaient to changes that could be made by rotating the source of one beam by 2ir radians. In these experiments [1] a beam of neutrons is split into two separated beams and a magnetic field is applied to one of them. For a neutron spin ~tT with magnetic moment ~ moving through a uniform magnetic field B for a time t, the state is changed by exp[—it(---iøB)/h] which is the same as it would be changed by rotation ,
0 031-9163/83/0000—0000/$ 03.00 © 1983 North-Holland
through the angle = 2t ~ around an axis in the direction of—B. In the Heisenberg picture the transformation of the spin matrices is the same as this rotation. When 0 is 2x the wavefunction is multiplied by —1. The effect of this change in one beam relative to the other is observed as a change in the interference pattern produced by the recombined beam. About all this there is no question. The question is whether observing this is equivalent to observing a rotation [4]. Can the magnetic field be regarded as a handy experimental tool for producing a change that could also be produced~at least in principie, by actually rotating one of the beams? It appears not. Such an interpretation cannot be made in a clear way because quantum mechanics predicts this change cannot be observed in an interference pattern if the beams are independent. In this regard, neutron interference is fundamentally different from interference of light. A two-slit interference pattern produced by a split beam of photons from a single source can also be produced, at least in theory, by independent beams represented by coherent states [5,6], so an experiment done with a single source is also conceivable with independent sources. Thus one can imagine rotating one of the beams by rotatmg its source. This is not possible with fermions. .
.
.
.
457
Volume 96A, number 9
PHYSICS LETTERS
If the beams are not independent, it would not be remarkable to observe the effect of rotating one of them by 2ir radians. It is a common everyday experience to observe the effect of rotation by 2ir radians of a subsystem that is attached to another part of a larger system. For example, consider turning a crank to wind up a spring that drives a motor. In quantum mechanics, subsystems that are separated are not necessarily independent. That is demonstrated by Einstein—Podolsky--Rosen correlations, Consider an expansion of the positive-frequency part of the neutron field operator,
1 August 1983
Interference can also be explained as correlations of the numbers of quanta counted at different points [9]. When there is no interference described by the mean number, there can be interference due to correlations of fluctuations. Such interference might be observed with two independent neutron beams. The fluctuations would make it more difficult to observe the effect of a magnetic field applied to one of the beams. Rotations by 2ir radians could not be observed because if the beams are independent the correlations are not changed when the wavefunction of each single-neutron state for one beam is multiplied by -~-l.The correlations are described by expectation values
~(x t)=~a ~ (x t) n
~
(i~it(x
1,t1) ~1i(x1,t1)~1it(x2,t2) ~i(x2,t2))
in terms of orthonormal single-neutron wavefunctions ,Li~(x,t) and corresponding annihilation operators a~ the adjoints a~are the creation operators for the singleneutron states with wavefunctions ~0(x, t). Since neutrons are fermions, the annihilation and creation operators for different single-neutron states anticommute. The mean number of neutrons counted at position x at time t is proportional to the expectation value, (~jt~x,t) ~x,t)>=~(a~an>I~n~x,t)I2
i~
+2Re m
)(a) m
n
and the fermion antisymmetry implies (am> ~n>
=
(aman>
=
~n~2m> = —(an) (am)
so either (am> or (an> must be zero, and *
(am an)
(am> (an) = (am) ~n> = 0.
There are no interference terms in the formula for the mean number of neutrons counted where the beams combine. In contrast, for independent beams of photons represented by coherent states [5], the interference terms are the same as for a split beam from a single source [61; this can explain [6] interference observed [7] with two lasers [8]. 458
whose expansions contain expectation values (at a at a / k m n of four creation and annihilation operators. When the beams are independent, the fermion antisymmetry implies the expectation values are zero if three of the creation and annihilation operators are for one beam and one is for the other. For example, if j, k, m are for one beam and n is for the other, then t t t t t t (a1 akam> (an) = (a1 aka~an)= (—a~a1akam)
—(aisfl )(ata If there an ieven k at) mnumber of operators for each beam, the expectation values are not changed when the wavefunction of each single-neutron state for one beam is multiplied by —1. The connection between spin and statistics is evident here. When the particles are bosons, independent beams can produce interference that changes when the wavefunction of each single-particle state is multiplied by —1. This is not possible for fermions. On the other hand, rotation by 2ir radians multiplies a wavefunction by —l only for half-integral spin. A wavefunction for integral spin is not changed. If it is assumed that rotation by 2n radians cannot be observed, what we have here is a simple proof that particles with half-integral spin cannot be bosons. References
Ill
H. Rauch eta!., Phys. Lett. 54A (1975) 425 S.A. Werner, R. Colella, A.W. Overhauser and C.F. Eagen, Phys. Rev. Lett. 35 (1975) 1053.
Volume 96A, number 9
PHYSICS LETTERS
[2] H.J. Bernstein, Phys. Rev. Lett. 18(1967)1102; Y. Aharonov and L. Susskind, Phys. Rev. 158 (1967) 1237; R. Mirman,Phys. Rev. Dl (1970) 3349 [3] J. Byrne, Nature 275 (1978) 188. [4] G.C. Hegerfeldt and K. Kraus, Phys. Rev. 170 (1968) 1185. [5] Ri. G!auber, Phys. Rev. Lett. 10(1963) 84;Phys. Rev. 130 (1963) 2529; 131 (1963) 2766.
1 August 1983
[6] T.F. Jordan and F. Ghielmetti, Phys. Rev. Lett. 12 (1964) 607. [7] G. Magyar and L. Mandel, Nature 198 (1963) 255; M.5. Lipsett and L. Mande!, Nature 199 (1963) 553; R.L. Pflegor and L. Mandel, Phys. Rev. 159 (1967) 1084. [8] H. Haken, Phys. Rev. Lett. 13 (1964) 329a. [91 L.Mandel,Phys.Rev. 134 (1964) AlO.
459