- Email: [email protected]
Prospects
PROSPECTS Notwithstanding the elegance and simplicity of its present form, the conceptual framework of quantum field theory is likely to undergo dramatic changes in future years if quantum field theory is not superseded entirely by a new and different dynamical theory for elementary particles. In quantum electrodynamics the technical procedure called "renormalization" is required to derive finite predictions for experimentally observable quantities from the mathematical formulations. The status of quantum electrodynamics with renormalization has been described by HeitIer and by Jauch and Rohrlich: The ambiguities can always be settled by applying a certain amount of .' wishful mathematics," namely, by using additional conditions for the evaluation of ambiguous integrals. Unless such conditions are used, the results of the theory may contradict its very foundation .... Clearly such a mathematical situation is unacceptable. On the other hand, these difficulties do not prevent us from giving a theoretical answer to every legitimate question concerning observable effects. These answers are, wherever they can be tested, always in excellent agreement with the facts, and no serious discrepancy exceeding the limits of accuracy of the calculation has so far been discovered. 1 With respect to the applications, i.e., the actual description of physical quantities, we have here one of the best-established physical theories. Whenever the theory is subjected to an experimental test, we find the theoretical prediction in complete agreement with the experimental result. The accuracy 1 W. Hertler, "The Quantum Theory of Radiation," p. 354. Oxford Univ. Press (Clarendon), London and New York, 1954.
93
94
Prospects
in the agreement is limited only by the experimental error and the endurance and ingenuity of the computer. With respect to the fundamental concepts, on the contrary, we are not so fortunate. The theory is incomplete insofar as we are forced to introduce the charge of the electron and the masses of electron and photon as phenomenological quantities. The value of these quantities cannot be deduced from the theory but must be accepted as empirically given. This point of view enabled us to carry through the program of renormalization which was essential for the removal of the divergences in the iteration solution." Thus, the renormalization procedure in quantum electrodynamics can at best be accepted as tentative and heuristic. The alternative negative attitude, outright rejection of renormalized quantum electrodynamics, is in fact advocated by Dirac: It seems to be quite impossible to put this theory on a mathematically sound basis. At one time physical theory was all built on mathematics that was inherently sound. I do not say that physicists always use sound mathematics; they often use unsound steps in their calculations. But previously when they did so it was simply because of, one might say, laziness. They wanted to get results as quickly as possible without doing unnecessary work. It was always possible for the pure mathematician to come along and make the theory sound by bringing in further steps, and perhaps by introducing quite a lot of cumbersome notation and other things that are desirable from a mathematical point of view in order to get everything expressed rigorously but do not contribute to the physical ideas. The earlier mathematics could always be made sound in that way, but in the renormalization theory we have a theory that has defied all the attempts of the mathematician to make it sound. I am inclined to suspect that the renormalization theory is something that will not survive in the future, and that the remarkable agreement between its results and experiment should be looked on as a fluke."
If the renormalization procedure in quantum electrodynamics is in essence illegitimate, the entire conceptual framework of quantum field theory must be suspect. With regard to this possibility, we recall the opinion of extreme dissent by Einstein: 2 J. M. Jauch and F. Rohrlich, "The Theory of Photons and Electrons," pp. 415--416. Addison-Wesley, Cambridge, Massachusetts, 1955. Reprinted by permission. 3 P. A. M. Dirac, Sci. Amer. 208, 50 (May 1963).
Prospects
95
Is it conceivable that a classical field theory permits one to understand the atomistic and quantum structure of reality? Almost everybody will answer this question with" no." But I believe that at the present time nobody knows anything reliable about it. This is so because we cannot judge in what manner and how strongly the exclusion of singularities reduces the manifold of solutions. We do not possess any method at all to derive systematically solutions that are free of singularities. Approximation methods are of no avail since one never knows whether or not there exists to a particular appoximate solution an exact solution free of singularities. For this reason we cannot at present compare the content of a nonlinear classical field theory with experience. Only a significant progress in the mathematical methods can help here. At the present time the opinion prevails that a field theory must first, by "quantization," be transformed into a statistical theory of field probabilities according to more or less established rules. I see in this method only an attempt to describe relationships of an essentially nonlinear character by linear methods." 4 A. Einstein, "The Meaning of Relativity," p. 165. Princeton Univ. Press, Princeton, New Jersey, 1966. Reprinted by permission.
93
94
Prospects
in the agreement is limited only by the experimental error and the endurance and ingenuity of the computer. With respect to the fundamental concepts, on the contrary, we are not so fortunate. The theory is incomplete insofar as we are forced to introduce the charge of the electron and the masses of electron and photon as phenomenological quantities. The value of these quantities cannot be deduced from the theory but must be accepted as empirically given. This point of view enabled us to carry through the program of renormalization which was essential for the removal of the divergences in the iteration solution." Thus, the renormalization procedure in quantum electrodynamics can at best be accepted as tentative and heuristic. The alternative negative attitude, outright rejection of renormalized quantum electrodynamics, is in fact advocated by Dirac: It seems to be quite impossible to put this theory on a mathematically sound basis. At one time physical theory was all built on mathematics that was inherently sound. I do not say that physicists always use sound mathematics; they often use unsound steps in their calculations. But previously when they did so it was simply because of, one might say, laziness. They wanted to get results as quickly as possible without doing unnecessary work. It was always possible for the pure mathematician to come along and make the theory sound by bringing in further steps, and perhaps by introducing quite a lot of cumbersome notation and other things that are desirable from a mathematical point of view in order to get everything expressed rigorously but do not contribute to the physical ideas. The earlier mathematics could always be made sound in that way, but in the renormalization theory we have a theory that has defied all the attempts of the mathematician to make it sound. I am inclined to suspect that the renormalization theory is something that will not survive in the future, and that the remarkable agreement between its results and experiment should be looked on as a fluke."
If the renormalization procedure in quantum electrodynamics is in essence illegitimate, the entire conceptual framework of quantum field theory must be suspect. With regard to this possibility, we recall the opinion of extreme dissent by Einstein: 2 J. M. Jauch and F. Rohrlich, "The Theory of Photons and Electrons," pp. 415--416. Addison-Wesley, Cambridge, Massachusetts, 1955. Reprinted by permission. 3 P. A. M. Dirac, Sci. Amer. 208, 50 (May 1963).
Prospects
95
Is it conceivable that a classical field theory permits one to understand the atomistic and quantum structure of reality? Almost everybody will answer this question with" no." But I believe that at the present time nobody knows anything reliable about it. This is so because we cannot judge in what manner and how strongly the exclusion of singularities reduces the manifold of solutions. We do not possess any method at all to derive systematically solutions that are free of singularities. Approximation methods are of no avail since one never knows whether or not there exists to a particular appoximate solution an exact solution free of singularities. For this reason we cannot at present compare the content of a nonlinear classical field theory with experience. Only a significant progress in the mathematical methods can help here. At the present time the opinion prevails that a field theory must first, by "quantization," be transformed into a statistical theory of field probabilities according to more or less established rules. I see in this method only an attempt to describe relationships of an essentially nonlinear character by linear methods." 4 A. Einstein, "The Meaning of Relativity," p. 165. Princeton Univ. Press, Princeton, New Jersey, 1966. Reprinted by permission.