Circumstantial Evidence for a Realistic Interpretation of Quantum Mechanics

Mystery

Richard Feynman [1] was very straightforward about the conceptual problem of quantum mechanics. He said "we cannot make the mystery go away … we will just tell you how it works." He used the double-slit experiment as an example and discussed it in great detail because "it contains the only mystery." The mystery is that particles appear to behave like waves in this experiment. This wave-like behavior appears as interference fringes in the distribution of particles that pass through the screen vs. the angle between the incoming and outgoing momentum of the particles. In the history of physics, a mystery has usually turned out to be an opportunity for a deeper understanding of how nature works. Why is this mystery different?

Uncertainty

According to Tolman's account [2], Planck derived the correct formula for the spectral distribution of black-body radiation by assuming that matter emits and absorbs electromagnetic energy in quanta. The energy of these quanta is proportional to the frequency of the radiation, and the ratio of energy over frequency, which has the units of action, is called Planck's constant. Now the energy and momentum of radiation are known classically, so one can discover that quanta must carry momentum equal to the product of their wave number with the same constant. Knowing that, you can motivate Heisenberg's uncertainty principle by considering a measurement of a particle's position under a microscope. The resolution of a microscope is proportional to the wavelength of the light source, but the deflection of each light quantum that contributes to the contrast in the image must change the particle's momentum by an amount that can't be completely determined. Thus Planck's quantum hypothesis implies that the experimental uncertainty in the particle's momentum after the measurement will be proportional to the wave number of the light source. Therefore the product of the uncertainties in momentum and position cannot be reduced below a multiple of Planck's constant.

When this kind of reasoning about uncertainty was explained to me in school, I imagined that the instructor would go on to analyze other situations by considering the incalculable effects of the quantized exchange of momentum and energy between matter and light. It seemed plausible that the classical description of precise position and momentum of a particle was experimentally unsuitable simply because experiments depended on electromagnetic interactions that were now to be quantized. However, the teaching followed the historical development of quantum mechanics, which was substantially different.

A Path Not Taken

What if you had been around at the beginning and had approached the double-slit experiment the way I imagined? You have particles approaching the mask along some direction, and you want to calculate the probability that the ones getting through are deflected by a given angle. You would have to settle for a probability because experimenters can't pin down all the details of each interaction. However, you can argue from symmetry that there is no effect along the direction of the slits. Along the direction perpendicular to the slits, the particle momentum changes, and this change in momentum is evidently supplied by the mask..

The mask is made up of a large number of atoms, and we hope we can describe the average effect of all these interactions in terms of a classical transparency of the mask as a function of position. If you thought about the problem in this way, you might guess that the probability would have something to do with the Fourier transform of this transparency function, at the spatial frequency corresponding to the momentum transferred, using the known relation between spatial frequency and momentum for light. A Fourier transform is generally a complex number, and a probability has to be a positive number, so your first guess might be to use the square of the absolute magnitude of this transform. The fact that you are asking for a probability also means that you don't have to worry about finding an expression that has the right units because you are going to normalize it anyway. Of course, the procedure suggested here gives exactly the same answer as the wave theory of matter.

The Wave Theory of Matter

Historically, the problem of accounting for the general exchange of quanta between matter and light got put on the back burner for awhile. It turned out, perhaps fortuitously, to be possible to get the right answer for the double-slit experiment and in many other situations by ascribing wave properties to the particles. However, the wave theory makes us add together amplitudes representing processes that are mutually exclusive before squaring the absolute magnitude of the sum to get a probability. When these amplitudes are added, they can interfere, either constructively or destructively. Nobody really understands this, and the wave theory of matter may have turned out to be a Faustian bargain if intuitive understanding is the soul of physics.

The wave theory of matter is most elegantly expressed in Feynman's path-integral formulation, where the quantum expression of the principle of least action appears in the foreground. The path-integral formulation also provides a relatively convenient way to do calculations that take into account all possible interactions such as the exchange of virtual photons and the creation and annihilation of particle-antiparticle pairs. The fact that these calculations turn out to be extremely accurate suggests that quantum electrodynamics is a better representation of how nature works than the preceding wave theory of matter. Therefore one could ask, does QED give the same answer as the wave theory of matter in the double-slit experiment? In QED, a particle is scattered from an atom by exchanging one or more photons with it. There is no more classical transparency function, but just a distribution of the target atoms.

Virtual Photons

I want to suggest that it isn't really necessary to think of a particle as being in a superposition of the mutually exclusive states of going through both of the slits. Probably, this conventional (surrealistic) interpretation of quantum mechanics was needed because the concept of virtual photons was not available at the right time. The Fourier transform alluded to above may represent a supply of virtual photons that can be exchanged with the mask. Can it be a coincidence that this supply is greatest at the angles of constructive interference? The intuitive power of this realistic interpretation, is that, when one of the slits is closed, the supply of virtual photons at the spatial frequency corresponding to the distance between the slits disappears. There is no longer a need to say that the interference goes away because you could know which slit the particles are going through.

Conspiracy

The possibility of a realistic interpretation of quantum mechanics depends on a conspiracy in which the wave theory of matter gives the same answer as QED when scattering from all the atoms in a classical target are properly added up. Instead of the wave theory of matter, we should try to explain interference in terms of the distribution of virtual photons represented by the classical transparency function. For a long time, I wondered if the wave theory of matter is merely a convenient way of summarizing the effects of quantization in the exchange of energy and momentum with the electromagnetic field. Then I happened to re-read Freeman Dyson's article [3] in the special issue of Physics Today devoted to Feynman's memory. Dyson recalled how Feynman had astounded him by deriving Maxwell's equations from quantum mechanics! If the derivation is correct, as Dyson said, then there is a unsuspected relation between quantum mechanics and electromagnetism. Dyson added that Feynman's derivation could be generalized to interactions mediated by guage fields and that he regretted that he was never able to persuade Feynman to publish it.

Acknowledgement

I want to thank Professor Anton Zeilinger of the University of Vienna for taking time at a difficult moment to discuss interpretations of quantum mechanics. I enjoyed the meeting very much, and I hope Anton will appreciate my attempt to respond to a challenge I found in one of his articles [4]. The challenge is actually a quotation of Rabi, who said "The problem is that the theory is too strong, too compelling. I feel we are missing a basic point. The next generation, as soon as they will have found that point, will knock on their heads and say: How could they have missed that?"

References

1. R. P. Feynman, R. B. Leighton and M. Sands, "The Feynman Lectures on Physics," Addison Wesley, Reading, MA (1965) Volume III, Chapter 1.

2. R. C. Tolman, "The Principles of Statistical Mechanics," Part 7, Section 52, Oxford University Press, London (1938) Part 2, Chapter VII, Section 52.

3. Freeman Dyson, "Feynman at Cornell," Physics Today, February 1989, pp. 32-38.

4. A. Zeilinger, "On the Interpretation and Philosophical Foundation of Quantum Mechanics," "Vastakohtien todellisuus", Festschrift for K.V. Laurikainen U. Ketvel et al. (Eds.), Helsinki University Press, 1996.

(last updated May 21, 1999)

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Copyright 2003, Terence J. Nelson