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As an example of practicing the grammatical physics,
here I propose a new coordinate system
in place of the coordinate system of the quantum mechanics for a particle.
This is equivalent to changing a quantization prescription fundamentally.
In 2006, I propose this coordinate system
mainly to explain the methodology of grammatical physics
and I was not then the person
who claims that this new coordinate system
definitely fits real situations better,
but even then I believed that this new coordinate system is physically interesting
and that some theory constructed on it may succeed.
The coordinate system is Mnew which is described below.
Mnew is the mapping
which maps each functional Φ from {χ:R→R3} to C
to a quantum history Mnew(Φ)
and so it is a coordinate system in my broad meaning,
where Mnew must have the following relationship with Mq.
“If Ψ(x,y,z,t)=expψ(x,y,z,t) and Φ[χ]=exp[α∫dtψ(χ(t),t)], then Mnew(Φ)=Mq(Ψ).
Therefore the range of Mnew includes the range of Mq as a subset.”
The physical implication of Mnew is the entanglement of a quantum history.
The entanglement is a well-known concept for a quantum state.
For example, as for a physical system with n degrees of freedom,
a quantum state is generally represented by a mapping (the wave function)
from Rn to C.
Especially when this wave function Ψ can be factorized to the form
Ψ(x1, ・・・, xn) ＝ Πi ψi(xi),
the quantum state is called a disentangled quantum state.
Because a wave function can not be factorized to such a form generally,
a general quantum state is entangled.
When we replace the name i of a degree of freedom with the name t of a time in the above discussion,
Rn is replaced with {χ:R→R},
Ψ is replaced with Φ,
(x1, ･･･, xn) is replaced with χ,
ψi is replaced with φ(□,t)
and xi is replaced with χ(t).
Therefore, as for the system with one degree of freedom,
the quantum history in the meaning of the existing quantum mechanics
is the special disentangled quantum history in the meaning of the new grammar,
and the new grammar implies that a general quantum history is entangled one.
But, I replaced the infinite product Πi with ‘exp[α∫dt’
because I felt that this infinite product didn't seem real,
and α is a very large new physical constant.
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The content of this article was presented by me at APS+JPS 2006 Autumn Meeting.
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The new quantum grammar was first thought out in the following article written in the Japanese language.
メッセージ1 @ 1-12 @ 時間の量子化 @ 中心問題解決案 @ 文法レベルでの自然 @ Japanese @ Vintage(2005-2008)
It was written on 2005/05/27.
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