- Email: [email protected]
Significance of Aharonov-Bohm effect measurements
Physica B 151 (1988) 384-385 North-Holland, Amsterdam
SIGNIFICANCE OF AHARONOV-BOHM EFFECT MEASUREMENTS
Murray PESHKIN Argonne National Laboratory, Argonne, IL 60439-4843, USA
Experiments have proved decisively that the Aharonov-Bohm effect (AB) is correctly described by standard quantum mechanics based on a single-valued solution of the Schr6dinger equation. What does this tell us about the theory that was not obvious before AB? Firstly, we have a direct and unambiguous demonstration that the physics is determined by the quantities exp{ie/hc) f A . d x ) as required by gauge theory. The importance of this test has been stressed for some years by C.N. Yang. Only in the AB geometry is the effect of local magnetic fields eliminated in principle. Independently of the gauge theory discussion, AB raises questions about the locality of the electromagnetic interaction and about quantum mechanics in a multiply connected region. Consider what happens when the current in an infinite solenoid is suddenly turned on at time t = 0. The vector potential A o(r, t) is given by the retarded solution of the Maxwell equations as a cylindrical wave that expands with the velocity of light. An electron at distance r suffers torque equal to (e/c)r(OAo/Ot) whose time integral is independent of r and given by eeb/2~c, where q~ is the flux through the solenoid after a long time. Thus every electron in the world has its kinetic angular momentum ( r x mv)z incremented by the amount eq)/2wc, which is not generally an integer multiple of h. This changes the height of the centrifugal barrier and influences bound state energies and scattering amplitudes. From this time-dependent point of view, there is no problem with causality or locality. In assembling the incident wave packet to perform the AB scattering experiment, one is compelled to use electrons having fractional kinetic angular
momentum due to their past encounter with a local electric field. From another point of view, the theory is nevertheless nonlocal. We can imagine creating a pair, long after the magnetic flux was established, and using the electron from the pair for the scattering experiment. The theory is then instructing us that the Hilbert space available to the newly-created electron contains only states whose kinetic angular momentum relates correctly to a distant magnetic flux. We cannot assemble a wave packet in the incoming region from the same kind of plane waves that we had without the distant flux. It is interesting to note that this question of available partial waves for a scattering experiment is not unique to AB. The scattering of an alpha particle from a second alpha particle exhibits the same feature; only even partial waves are allowed. Any nonlocality that we infer from this appears to be a feature of quantum mechanics even without the electromagnetic interaction. Multiple connectedness of the space available to the electron raises another issue of principle. In a multiply connected region, the commutation relations have inequivalent representations which lead to inequivalent physics. That point goes back at least to Weyl in the early days of quantum mechanics. Those inequivalent representations appear in the language of the Schr6dinger equation as alternative Hilbert spaces with specific multiple-valued wave functions. That we should always choose the single-valued functions is an independent assumption, not easily tested by experiment. It is easier to give theoretical reasons why the Hilbert space should be unchanged by changing the applied magnetic field
0378-4363 / 88 / $03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M. Peshkin / Significance of Aharonov-Bohm effect measurements
than to justify the assertion that we should use the single-valued wave functions in the absence of a magnetic field in all cases. T h e experiments of T o n o m u r a a p p e a r to justify the single-valued choice with high precision w h e n the electrons are excluded f r o m the interior of his metal micro-rings. O n e m a y still speculate a b o u t o t h e r realizations of a multiply c o n n e c t e d space. F o r example, Wilczek p r o p o s e s hypothetical " a t o m s " m a d e o f an electron b o u n d to a flux line, m a k e s assumptions equivalent to a multiple-valued representation o f the c o m m u t a t i o n relations, and finds that his atoms have unusual spins and
385
possibly unusual statistics. F r o m the present point of view, there is n o t h i n g w r o n g with that and in fact one does not n e e d the flux. It is e n o u g h to exclude electrons f r o m a tube of vanishing d i a m e t e r and i m p o s e the necessary b o u n d a r y condition.
Acknowledgement This w o r k was s u p p o r t e d by the U.S. D e p a r t m e n t of E n e r g y , N u c l e a r Physics Division, u n d e r contract W-31-109-ENG-38.
DISCUSSION (Q) D. Greenberger: It seems to me that a multiply-connected world is merely a mathematical abstraction. One can always drill a hole in it and then the wave function will be single-valued and the problem disappears. So the singlevaluedness of the wave function is the appropriate situation in all cases and the multi-valued cases are only approximations.
(A) M. Peshkin: You maybe right, but maybe not. It's an experimental question. For example, phase transitions do not exist for finite numbers of particles, but the idealization of an infinite system is the best zero-order approximation. Unknown fields may well be represented best by excluded volumes in the real world.
SIGNIFICANCE OF AHARONOV-BOHM EFFECT MEASUREMENTS
Murray PESHKIN Argonne National Laboratory, Argonne, IL 60439-4843, USA
Experiments have proved decisively that the Aharonov-Bohm effect (AB) is correctly described by standard quantum mechanics based on a single-valued solution of the Schr6dinger equation. What does this tell us about the theory that was not obvious before AB? Firstly, we have a direct and unambiguous demonstration that the physics is determined by the quantities exp{ie/hc) f A . d x ) as required by gauge theory. The importance of this test has been stressed for some years by C.N. Yang. Only in the AB geometry is the effect of local magnetic fields eliminated in principle. Independently of the gauge theory discussion, AB raises questions about the locality of the electromagnetic interaction and about quantum mechanics in a multiply connected region. Consider what happens when the current in an infinite solenoid is suddenly turned on at time t = 0. The vector potential A o(r, t) is given by the retarded solution of the Maxwell equations as a cylindrical wave that expands with the velocity of light. An electron at distance r suffers torque equal to (e/c)r(OAo/Ot) whose time integral is independent of r and given by eeb/2~c, where q~ is the flux through the solenoid after a long time. Thus every electron in the world has its kinetic angular momentum ( r x mv)z incremented by the amount eq)/2wc, which is not generally an integer multiple of h. This changes the height of the centrifugal barrier and influences bound state energies and scattering amplitudes. From this time-dependent point of view, there is no problem with causality or locality. In assembling the incident wave packet to perform the AB scattering experiment, one is compelled to use electrons having fractional kinetic angular
momentum due to their past encounter with a local electric field. From another point of view, the theory is nevertheless nonlocal. We can imagine creating a pair, long after the magnetic flux was established, and using the electron from the pair for the scattering experiment. The theory is then instructing us that the Hilbert space available to the newly-created electron contains only states whose kinetic angular momentum relates correctly to a distant magnetic flux. We cannot assemble a wave packet in the incoming region from the same kind of plane waves that we had without the distant flux. It is interesting to note that this question of available partial waves for a scattering experiment is not unique to AB. The scattering of an alpha particle from a second alpha particle exhibits the same feature; only even partial waves are allowed. Any nonlocality that we infer from this appears to be a feature of quantum mechanics even without the electromagnetic interaction. Multiple connectedness of the space available to the electron raises another issue of principle. In a multiply connected region, the commutation relations have inequivalent representations which lead to inequivalent physics. That point goes back at least to Weyl in the early days of quantum mechanics. Those inequivalent representations appear in the language of the Schr6dinger equation as alternative Hilbert spaces with specific multiple-valued wave functions. That we should always choose the single-valued functions is an independent assumption, not easily tested by experiment. It is easier to give theoretical reasons why the Hilbert space should be unchanged by changing the applied magnetic field
0378-4363 / 88 / $03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M. Peshkin / Significance of Aharonov-Bohm effect measurements
than to justify the assertion that we should use the single-valued wave functions in the absence of a magnetic field in all cases. T h e experiments of T o n o m u r a a p p e a r to justify the single-valued choice with high precision w h e n the electrons are excluded f r o m the interior of his metal micro-rings. O n e m a y still speculate a b o u t o t h e r realizations of a multiply c o n n e c t e d space. F o r example, Wilczek p r o p o s e s hypothetical " a t o m s " m a d e o f an electron b o u n d to a flux line, m a k e s assumptions equivalent to a multiple-valued representation o f the c o m m u t a t i o n relations, and finds that his atoms have unusual spins and
385
possibly unusual statistics. F r o m the present point of view, there is n o t h i n g w r o n g with that and in fact one does not n e e d the flux. It is e n o u g h to exclude electrons f r o m a tube of vanishing d i a m e t e r and i m p o s e the necessary b o u n d a r y condition.
Acknowledgement This w o r k was s u p p o r t e d by the U.S. D e p a r t m e n t of E n e r g y , N u c l e a r Physics Division, u n d e r contract W-31-109-ENG-38.
DISCUSSION (Q) D. Greenberger: It seems to me that a multiply-connected world is merely a mathematical abstraction. One can always drill a hole in it and then the wave function will be single-valued and the problem disappears. So the singlevaluedness of the wave function is the appropriate situation in all cases and the multi-valued cases are only approximations.
(A) M. Peshkin: You maybe right, but maybe not. It's an experimental question. For example, phase transitions do not exist for finite numbers of particles, but the idealization of an infinite system is the best zero-order approximation. Unknown fields may well be represented best by excluded volumes in the real world.