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Let Φ be a normalized functional representing a quantum history.
Let the twist remove normalization be used there.

Then the function, representing a transition amplitude of the new grammatical quantum mechanics, corresponding to the Green's function of the old quantum mechanics, is given by the following formula.

Γ(g, t2; f, t1) = ∫Dχ dar(g; t2, t2 + 1/α)Φ[χ]・{dar(f; t1, t1 + 1/α)Φ[χ]}*.
This function corresponds to <g, t2| f, t1> of the old quantum mechanics.

dar(f; a, b) is the stacked Daruma game operator defined as follows.

[dar(f; a, b)]pre Φ[χ] = ∫dx f(x)* Φ[ξ(□,x)]
where ξ is a function such that
ξ(t,x) = χ(t) (t < a or t ≧ b),
ξ(t,x) = x (a ≦ t < b),
ξ(t,x) must be smoothened at t = a and t =b.

[dar(f; a, b)] Φ[χ(a,b)] = [dar(f; a, b)]pre Φ[χ]
χ(a,b)(t) = χ(t) (t < a),
χ(a,b)(t) = χ(t + (b - a)) (t ≧ a).

The twist remove normalization for a quantum history written in the form:
Φ[χ] = Σj exp[α∫dt φj(χ(t),t)]
is defined by the condition:
(∂/∂ε)∫dx [exp φj(x,t)]* exp φj(x,t + ε) |ε = 0 = 0.
This condition is satisfied with the function:
φj(x,t) = φ'j(x,t) + (i/h)t∫dx' [exp φ'j(x',t)]* H exp φ'j(x',t)
where exp φ'j is a solution of the old Schrödinger equation and H is the Hamiltonian of it.
That is to say,
φj(x,t) = φ'j(x,t) + (i/h)t<j|H|j>.

However to define the twist remove normalization generally is a remaining problem.
Twist Remove Normalization


This article is a rewrite of the article 'Probability Formula in New Grammar' in the following page.
Later Edition@Theory of Quantum History Entangled in Time-like Direction@Products of Grammatical Physics@Grammatical Physics@Forum@Vintage(2008-2014)

The content of this article was presented by me at JPS 2013 Autumn Meeting.

Author Yuichi Uda, Write start at 2015/05/21/19:48JST, Last edit at 2015/05/23/17:50JST