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Introduction
A well known QED problem is relates to the infinities which
are obtained from its calculations. The current resolution of
this problem is called renormalization. This procedure
looks unacceptable because
"it seemed illegitimate to do something tantamount to subtracting
infinities from infinities to get finite answers" (see Wikipedia).
Two eminent QED figures have objected renormalization. P. A. M. Dirac
has described it as a procedure of an "illogical character" [1].
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.
He has used a more colorful terminology and called renormalization
"a dippy process" [2]. Feynman continued and stated: "I suspect that
renormalization is not mathematically legitimate".
A similar remark can be found in Ryder's textbook [3]: "...the feeling
remains that there ought to be a more satisfactory way of doing things."
By contrast, no serious physicist has used such expressions with
respect to special relativity. It can be concluded that the final
word probably has not yet been said on QED.
The need for a mathematically legitimate procedure indicates that
other serious problems may undermine QED's self-consistency
and that it deserves a
further analysis of its structure. Evidently, any debate where
some people play the role of the devil's advocate can only enhance
our understanding of the debated issue. Unfortunately,
an initiative aiming to organize this
kind of debate is not included in the agenda of journals of
the present establishment. On the contrary, apparently too many people are
quite sure that rejecting "dissident" papers is one of their important
duties.
The purpose of this page is to show that besides its renormalization
problem, QED suffers from other contradictions.
The QED Lagrangian Density
The Lagrangian density is the cornerstone of any quantum field
theory [4,5]. Contrary to the common belief, the QED Lagrangian density
contains erroneous elements. A clear and short proof of this claim is shown
here
.
This recently published
paper concludes that "the well known infinities of QED show that
mathematics screams when stumbling
upon something which is inherently wrong."
Gauge Transformations
Gauge transformations are legitimate procedures in a classical
theory that takes Maxwell equations
and the Lorentz force as its cornerstone.
This statement is correct because the 4-potential is not
directly used in these equations. However,
it can be proved that this is not true for electrodynamics
that is derived from the variational principle. Here
the 4-potential is explicitly used in the Lagrangian density of
the system. (The following expressions are written in
units where ħ=c=1 and x denotes the four space-time
coordinates.) This issue is clearly seen in a quantum theory where
the primary gauge function Λ(x) appears as
an exponential factor of the wave function
ψ(x) (see [5], p. 78).
|
ψ(x) →
exp(ieΛ(x))ψ(x)
| |
(1)
|
Regarding the power series expansion
|
exp(ieΛ(x)) =
1 + ieΛ(x) + ... |
|
(2)
|
one finds that the first term is a pure number. Hence, all terms of the
expansion must be pure numbers. Now,
the imaginary unit i is a pure number and, in
the units used herein, the electric charge e is also a pure
number where e2 ≈ 1/137.
For this reason one
concludes that the gauge function Λ(x) must be a
dimensionless Lorentz scalar. This outcome denies the ordinary
definition of Λ(x) as an arbitrary function
of space-time coordinates. For a further discussion
click here
.
New Data
Muon dependent measurements of the proton's charge radius
show a discrepancy of several percents with respect to the
corresponding data that
have been obtained from electron dependent measurements (see
here
).
Thus, the QED amazing precision of seven or more decimal digits has
deteriorated into just one decimal digit.
This evidence indicates that Dirac was probably right when he said
that the remarkable agreement between the results of QED's
renormalization and experiment should be looked on as a fluke
(see above).
This experimental discrepancy
provides another good reason for a reexamination of the
present QED structure.
References:
[1] P. A. M. Dirac, Scientific American, 208, 45, May 1963.
(see
here
.)
[2] R. P. Feynman, QED, The Strange Theory of Light and Matter (Penguin,
London, 1990). (See p. 128.)
[3] L. H. Ryder, Quantum Field Theory (Cambridge University Press,
Cambridge, 1997). (See p. 390.)
[4] S. Weinberg, The Quantum Theory of Fields, Vol. I, (Cambridge
University Press, Cambridge, 1995).
[5] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum
Field Theory (Addison-Wesley, Reading, Mass., 1995).
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