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In this page, I write exact solutions of Uda Equation in case of V = 0. . where f: R → C. Φ[χ(□  ε)] = ∫Dπ exp{α∫_{∞}^{∞}dt[(i/ = ∫Dπ exp{α∫_{∞}^{∞}dt[(i/ = ∫Dπ' exp{α∫_{∞}^{∞}dt[(i/ = ∫Dπ' exp{α∫_{∞}^{∞}dt[(i/ where π'(t) = π(t + ε). i = ∫Dπ' {α∫_{∞}^{∞}dt[π'(t)^{2}/(2m)]} exp{α∫_{∞}^{∞}dt[(i/ {α∫_{∞}^{∞}dt[(i = ∫Dπ' {α∫_{∞}^{∞}dt[π(t)^{2}/(2m)]} exp{α∫_{∞}^{∞}dt[(i/ ∴ i ∫_{∞}^{∞}dp exp{α∫_{a}^{a +ε}dt[(i/ =∫_{∞}^{∞}dp exp{αε[(i/ = exp{αεf([i = exp{αεf([i = √[2m If f(p) = 0, ψ(x, t) = exp{[im/(2 If f(p) = kp, ∫_{∞}^{∞}dp exp{αε[(i/ = ∫_{∞}^{∞}dp exp[αε{(i/ = c exp{αε[(i/ = c exp{αε[im/(2 ∴ ∫Dπ exp{α∫_{a}^{a+1/α}dt[(i/ ≒ c'[∫_{∞}^{∞}dp exp{αε[(i/ = c''exp{[im/(2 If f(p) = K(p  b)^{2}, ∫_{∞}^{∞}dp exp{αε[(i/ = ∫_{∞}^{∞}dp exp{αε[K + (i/ = C exp{αε(1/4)[(i/ = C exp{αε(1/2)(m/ ∴ ∫Dπ exp{α∫_{a}^{a+1/α}dt[(i/ ≒ C'[∫_{∞}^{∞}dp exp{αε[(i/ = C''exp{(1/2)(m/ I can not calculate for general f but I assume that the following functional integral converges except for constant factor for wide variety of f. ∫Dπ exp{α∫_{a}^{a+1/α}dt[(i/ Any superposition of solutions of the form at the top of this page is also a solution. It is as follows. where F is an arbitrary complex valued functional. This perhaps covers all general solutions.  The original form of the solution shown in this page started being displayed at the top of this page at 2019/11/11/21:05JST. It was of the same meaning as the meaning of the present form though it was transformed into the present form later. Gaussian integration was done on 2019/11/15. 

Author Yuichi Uda, Write start at 2019/11/11/20:15JST, Last edit at 2019/12/06/17:13JST  






