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In this page, I write exact solutions of Energy Representation of Uda Equation (1). Any unentangled solution is functional Φ such that where ∀n ∈ Z; c_{n} ∈ C. The quantum history represented by this solution is the same history as represented by wave function ψ such that ψ(n, t) ＝ exp(iωtn + c_{n}) ＝ λ_{n} exp(iωnt) in the old grammar. Here, λ_{n} ≡ exp(c_{n}). If A†An＞ ＝ nn＞, Σ_{n=0}^{∞} ψ(n, t)n＞ ＝ Σ_{n=0}^{∞} λ_{n}exp(iωnt)n＞, Σ_{n=0}^{∞} ψ(n, 0)n＞ ＝ Σ_{n=0}^{∞} λ_{n}n＞. Of course, any superposition of unentangled solutions is also a solution. Moreover, any entangled solution perhaps is a superposition of unentangled solutions. It is functional Φ such that where ∀n ∈ Z; a_{n} ∈ R and b_{n} ∈ R and f(a_{0}, b_{0}, a_{1}, b_{1}, a_{2}, b_{2}, ・・・) ∈ C. c_{n} = a_{n} + ib_{n}, λ_{n} = exp(a_{n} + ib_{n}). Normalization condition for initial state is that . To let the integral converge, b and ν is restricted. However, I do not put such a condition because there is no ambiguity even if the integral diverges.  The content of this page was developed in Number Representation of Uda Equation @ Quantum History Theory @ Problems. 

Author Yuichi Uda, Write start at 2019/11/08/17:22JST, Last edit at 2019/11/22/18:45JST  






