[time 59] RE: [time 58] Re: your "Is there a better way than renormalization?" post

Hitoshi Kitada (hitoshi@kitada.com)
Fri, 26 Mar 1999 18:18:51 +0900

Dear Doug,

Stephen forwarded his mail to you to our time list. Let me make a comment on
the following point:

>> Since I am so operationally focused, I did not find Kitada's papers very
>> engaging. Some parts struck me as inaccurate. Schrodinger first wrote
>> down a relativistic wave equation, but it did not agree with the data
>> from the non-relativistic hydrogen atom. QM and special relativity are
>> completely compatible, honest.

As an explanation of incompatibility of QM and special relativity, I quote a
passage form my paper (http://www.kitada.com/time_IV.tex, Introduction):

> One of the features of the Schr\"odinger equation is that it yields
> the stability of matter (of the first kind, see \cite{[Li]}), which is
> violated in the classical framework of Maxwell's equations and
> Rutherford model of atoms, where atoms collapse by the continuous
> radiation of light from the electrons around the nucleus according
> to classical electromagnetism so that the electrons fall
> into the nucleus. However the Schr\"odinger operator $H$ defined in
> (SO) is bounded from below by some constant $- L > - \infty$ in the
> sense that $L^2$-inner product $(Hf,f)$ satisfies
> $$
> (Hf,f) \ge -L (f,f)
> $$
> for any $f$ belonging to the domain ${\cal D}(H)$ of $H$
> under a suitable assumption on the pair potentials $V_{ij}$.
> This means that the total energy of the quantum system does not
> decrease below $- L$, therefore the system does not collapse. In
> this respect, quantum mechanics remedies the difficulty of
> classical theory, while it is not Lorentz invariant.
> In 1928, Dirac \cite{[Di]} introduced a system of equations, which is
> invariant under Lorentz transformation, and could explain some of
> the relativistic quantum-mechanical phenomena. However, Dirac
> operator is not bounded from below, and Dirac equation does not
> imply the stability of matter unlike the Schr\"odinger equation.
> Dirac thus proposed an idea that the vacuum is filled with electrons
> with negative energy so that the electrons around the nucleus
> cannot fall into the negative energy anymore by the Pauli
> exclusion principle, which explains the stability of matter.
> However, if one has to consider plural kinds of elementary
> particles at a time, one has to introduce the vacuum which
> is filled with those plural kinds of particles, and the vacuum
> can depend on the number of the kinds of particles which one
> takes into account. The vacuum then may not be determined to be unique.
> Further if one has to include Bosons into consideration,
> the Pauli exclusion principle does not hold and the stability
> of matter does not follow. In this sense, the idea of
> ````filling the negative energy sea;" unfortunately," ``is
> ambiguous in the many-body case," as Lieb writes in
> \cite{[Li]}, p.33.
> Quantum field theory is introduced (see \cite{[St]} for a review)
> to overcome this difficulty as well as to explain the
> annihilation-creation phenomena of particles, which are
> familiar in elementary particle physics. Quantum field
> theory is a theory of infinite degrees of freedom. In
> the case of the Schr\"odinger equation (SE), the
> degree of freedom is $3N$, the number of coordinates
> $x_{11}, x_{12}, x_{13}, \cdots, x_{N1}, x_{N2}$, and $x_{N3}$ of
> $N$ particles. Contrary to this, quantum field theory deals with
> the infinite number of particles, which makes it possible to
> discuss creation-annihilation processes inside the theory.
> However, since it deals with infinite number of freedom, even at
> the first step of the definition of the Hamiltonian of the system
> obtained by second quantization, there is a difficulty, the
> difficulty of divergence. This sort of difficulty appears at almost
> every stage of the development of the theory, and physicists had
> to find clever ways to avoid the difficulties at each step after the
> theory was introduced. Mathematically, the difficulty of divergence
> has not been overcome yet at all. Physicists however noticed that
> if one could get finite quantities in a systematic way by
> extracting some infinite quantities from the divergent quantities,
> then those finite quantities might express the reality. Actually
> in their explanation of Lamb shift, they seemed to have succeeded
> going in this way and to have been able to give predictions
> outstandingly close to experiments. However, the calculation
> done is up to the 6th or 8th order of a series giving Lamb shift
> or anomalous magnetic moment of electrons (\cite{[K-L]}). Dyson noticed
> (\cite{[Dy]}) that the series has symptom to diverge to infinity.
> The procedure mentioned in the above to yield finite quantities from
> infinite ones is called process of ``renormalization," and still
> forms active areas of researches in theoretical
> physics. In the mathematical attempt, called ``axiomatic quantum
> field theory," which was planned to clarify the meaning of
> quantum field theory and construct the theory consistently, it is
> known that in some mathematical but important examples (see, e.g.,
> \cite{[Fr]}), renormalizability conditions and the axioms of quantum
> field theory yield that the theory must not involve interaction
> terms inside the theory. I.e., the theory is void as a physics.
> These are the situation currently understood as an incompatibility
> problem between quantum theory and special theory of relativity.

In addition to these, Dirac equation and other relativistic equations (either
special or general relativistic ones) are not able to describe the many-body
situation, at least in the sense that no one has been successful in finding
such a formalism.

Best regards,
Hitoshi Kitada

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