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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, t_{2}; f, t_{1}) = ∫Dχ dar(g; t_{2}, t_{2} + 1/α)Φ[χ]・{dar(f; t_{1}, t_{1} + 1/α)Φ[χ]}^{*}. This function corresponds to ＜g, t_{2} f, t_{1}＞ 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] where 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] = Σ_{j} exp[α∫dt φ_{j}(χ(t),t)] is defined by the condition: (∂/∂ε)∫dx [exp φ_{j}(x,t)]^{*} exp φ_{j}(x,t + ε) _{ε = 0} = 0. This condition is satisfied with the function: φ_{j}(x,t) = φ'_{j}(x,t) + (i/ where exp φ'_{j} is a solution of the old Schrödinger equation and H is the Hamiltonian of it. That is to say, φ_{j}(x,t) = φ'_{j}(x,t) + (i/ 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 Timelike Direction@Products of Grammatical Physics@Grammatical Physics@Forum@Vintage(20082014) 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  






