R. P. Feynman: Renormalization - a dippy process [1].

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Inherent Gauge Inconsistencies Within Electrodynamics

The subject discussed on this page is closely related to the above mentioned Feynman's statement. Indeed, renormalization and gauge transformations are indispensable elements of the present structure of QED.

The following short items describe the origin of the problem and justify the title of this page.
  1. The following theorem plays a crucial role in the analysis.

    Theorem A': If a Lagrangian function is invariant under a certain transformation then the entire theory that is derived from this Lagrangian is invariant under the transformation provided the Lagrangian and the transformation are free of mathematical contradictions.

  2. An important element of the analysis is the distinction between classical electrodynamics that is strictly based on Maxwell equations together with the Lorentz force (MLE) versus electrodynamic theories that are a part of the variational principle (VE). MLE and VE have a different structure. Indeed, the fundamental equations of MLE, namely, Maxwell equations and the Lorentz force, are independent of the 4-potential whereas the Lagrangian function of VE depends explicitly on the 4-potential.
  3. In their celebrated textbook: The Classical Theory of Fields, Landau and Lifshitz prove that MLE can be derived from VE. Hence, fig. 1 describes the relationship between these theories.
    Fig1. Relationship between two electromagnetic theories.
  4. The meaning of this relationship is that if MLE is true then VE may be either true or false. (If you do not trust me then please see the truth table here ). The significance of this result is that although it is well known that {MLE + gauge} are true, this fact does not rule out the possibility that {VE + gauge} are false.

  5. Gauge transformation adds the following phase factor to the expression of a quantum function

    exp(-ieΛ(x))  =  1 - ieΛ(x) + ... (1)

    Here e is the electronic charge, which is a dimensionless Lorentz scalar in the units where ℏ=c=1, Λ(x) is the gauge function and x denotes the four space-time coordinates. Now, a very well known law of physics states that all terms of a physically acceptable expression must have the same dimension, and, in a relativistic theory, they must also satisfy covariance. Since the number 1 of (1) is a dimensionless Lorentz scalar, one concludes that the gauge function Λ(x) must also be a dimensionless Lorentz scalar. It can be proved that under these constraints Λ(x) must be a numerical constant and its associated gauge 4-vector ∂Λ(x)/∂xμ vanishes identically. (For reading a paper that proves this conclusion, click here .)

  6. The present form of VE takes the gauge function Λ(x) as an arbitrary function of the four space-time coordinates. This matter is found in the quantum version of VE (See [2], p. 207 after (23b); [3], p.192; [4], p. 342 after (8.1.13); [5], p. 482, after (15.1); [6], p. 316, after (14.21).) It can also be found in the Landau and Lifshitz famous VE textbook on classical electrodynamics (see [7], p.52, before (18.1)). This is a clear error because it is inconsistent with the conclusion of the previous item. Hence, the presently accepted form of gauge theory violates theorem A'.
Using the above mentioned discrepancy of gauge transformation, one finds that it is just a matter of working out some details in order to prove that gauge transformation leads to uncorrectable errors in VE. The following few lines explain briefly the root of the problem. The concepts of Lagrangian and of the wave nature of a charged particle do not exist in MLE. Hence, also the associated concepts of Hamiltonian, interference and parity are irrelevant to the description of a charged particle in MLE. On this basis one can prove that although gauge transformations hold in MLE, this fact does not settle the inconsistencies of gauge transformation in VE. Indeed, it can be proved that in the case of a charged particle, each of the VE notions of Hamiltonian, interference and parity conservation are inconsistent with gauge transformations. It is also proved that gauge transformations are inconsistent with the Hamiltonian of classical electrodynamics. For reading an article that discusses these issues in detail - click here .

Gauge transformation is an indispensable element of the present structure of QED. It turns out that the foregoing analysis provides yet another reason for correcting this theory. The primary title of this page indicates that other problematic issues of QED have already been pointed out a long time ago.

Note also the following experimental aspect of QED. For many decades QED has acquired the reputation of an amazingly accurate theory whose predictions fit the data by 8 decimal digits (see [5], top of p. 197). However, recent measurements of muonic hydrogen show a discrepancy of several percents (for details click here ). This outcome means that the above mention 8 digits QED accuracy does not hold any more and that the present version of QED poorly fits experimental data. This is yet another good reason for the need to reexamine its theoretical structure.

The following lines are relevant to the subject discussed herein. Quite a few people who are known for their important contribution to QED have made unfavorable remarks on renormalization. For example, P. A. M. Dirac has described it as a procedure of an "illogical character" [8]. He continued and said: "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". A similar approach has been expressed by R. P. Feynman, who is another eminent QED figure. He has used a more colorful terminology and called renormalization "a dippy process" [1]. Feynman continued and stated: "I suspect that renormalization is not mathematically legitimate". See also the section Attitudes and interpretation here .


References:

[1] R. P. Feynman, QED, The Strange Theory of Light and Matter (Penguin, London, 1990). p. 128.

[2] W. Pauli, Rev. Mod. Phys., 13, 203, (1941).

[3] C.N. Yang and R. Mills, Phys. Rev., 96, 191 (1954).

[4] S. Weinberg The Quantum Theory of Fields, Vol. I (Cambridge Univ. Press, Cambridge, 1995).

[5] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, Mass., 1995).

[6] F. Halzen and A. D. Martin, Quarks and Leptons, An Introductory Course in Modern Particle Physics (John Wiley, New York,1984).

[7] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Elsevier, Amsterdam, 2005).

[8] P. A. M. Dirac, Scientific American, 208, 45, May 1963. ( see here ).