**Hitoshi Kitada** (*hitoshi@kitada.com*)

*Fri, 26 Mar 1999 18:18:51 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Stephen Paul King: "[time 60] Re: duality (sci.physics.research)"**Previous message:**Stephen P. King: "[time 58] Re: your "Is there a better way than renormalization?" post"

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
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*> the stability of matter (of the first kind, see \cite{[Li]}), which is
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*> violated in the classical framework of Maxwell's equations and
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*> Rutherford model of atoms, where atoms collapse by the continuous
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*> radiation of light from the electrons around the nucleus according
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*> to classical electromagnetism so that the electrons fall
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*> into the nucleus. However the Schr\"odinger operator $H$ defined in
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*> (SO) is bounded from below by some constant $- L > - \infty$ in the
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*> sense that $L^2$-inner product $(Hf,f)$ satisfies
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*> $$
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*> (Hf,f) \ge -L (f,f)
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*> $$
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*> for any $f$ belonging to the domain ${\cal D}(H)$ of $H$
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*> under a suitable assumption on the pair potentials $V_{ij}$.
*

*> This means that the total energy of the quantum system does not
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*> decrease below $- L$, therefore the system does not collapse. In
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*> this respect, quantum mechanics remedies the difficulty of
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*> classical theory, while it is not Lorentz invariant.
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*>
*

*> In 1928, Dirac \cite{[Di]} introduced a system of equations, which is
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*> invariant under Lorentz transformation, and could explain some of
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*> the relativistic quantum-mechanical phenomena. However, Dirac
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*> operator is not bounded from below, and Dirac equation does not
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*> imply the stability of matter unlike the Schr\"odinger equation.
*

*>
*

*> Dirac thus proposed an idea that the vacuum is filled with electrons
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*> with negative energy so that the electrons around the nucleus
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*> cannot fall into the negative energy anymore by the Pauli
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*> exclusion principle, which explains the stability of matter.
*

*> However, if one has to consider plural kinds of elementary
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*> particles at a time, one has to introduce the vacuum which
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*> is filled with those plural kinds of particles, and the vacuum
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*> can depend on the number of the kinds of particles which one
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*> takes into account. The vacuum then may not be determined to be unique.
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*> Further if one has to include Bosons into consideration,
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*> the Pauli exclusion principle does not hold and the stability
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*> of matter does not follow. In this sense, the idea of
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*> ````filling the negative energy sea;" unfortunately," ``is
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*> ambiguous in the many-body case," as Lieb writes in
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*> \cite{[Li]}, p.33.
*

*>
*

*> Quantum field theory is introduced (see \cite{[St]} for a review)
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*> to overcome this difficulty as well as to explain the
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*> annihilation-creation phenomena of particles, which are
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*> familiar in elementary particle physics. Quantum field
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*> theory is a theory of infinite degrees of freedom. In
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*> the case of the Schr\"odinger equation (SE), the
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*> degree of freedom is $3N$, the number of coordinates
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*> $x_{11}, x_{12}, x_{13}, \cdots, x_{N1}, x_{N2}$, and $x_{N3}$ of
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*> $N$ particles. Contrary to this, quantum field theory deals with
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*> the infinite number of particles, which makes it possible to
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*> discuss creation-annihilation processes inside the theory.
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*> However, since it deals with infinite number of freedom, even at
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*> the first step of the definition of the Hamiltonian of the system
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*> obtained by second quantization, there is a difficulty, the
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*> difficulty of divergence. This sort of difficulty appears at almost
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*> every stage of the development of the theory, and physicists had
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*> to find clever ways to avoid the difficulties at each step after the
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*> theory was introduced. Mathematically, the difficulty of divergence
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*> has not been overcome yet at all. Physicists however noticed that
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*> if one could get finite quantities in a systematic way by
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*> extracting some infinite quantities from the divergent quantities,
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*> then those finite quantities might express the reality. Actually
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*> in their explanation of Lamb shift, they seemed to have succeeded
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*> going in this way and to have been able to give predictions
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*> outstandingly close to experiments. However, the calculation
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*> done is up to the 6th or 8th order of a series giving Lamb shift
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*> or anomalous magnetic moment of electrons (\cite{[K-L]}). Dyson noticed
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*> (\cite{[Dy]}) that the series has symptom to diverge to infinity.
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*>
*

*> The procedure mentioned in the above to yield finite quantities from
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*> infinite ones is called process of ``renormalization," and still
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*> forms active areas of researches in theoretical
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*> physics. In the mathematical attempt, called ``axiomatic quantum
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*> field theory," which was planned to clarify the meaning of
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*> quantum field theory and construct the theory consistently, it is
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*> known that in some mathematical but important examples (see, e.g.,
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*> \cite{[Fr]}), renormalizability conditions and the axioms of quantum
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*> field theory yield that the theory must not involve interaction
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*> terms inside the theory. I.e., the theory is void as a physics.
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*>
*

*>
*

*> These are the situation currently understood as an incompatibility
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*> 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

**Next message:**Stephen Paul King: "[time 60] Re: duality (sci.physics.research)"**Previous message:**Stephen P. King: "[time 58] Re: your "Is there a better way than renormalization?" post"

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