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< Exact Solution of Uda Equation (6) >

Uda Equation in case of V = 0.

.
where f: RC.

Φ[χ(□ - ε)] = ∫Dπ exp{α∫-∞dt[-(i/h)tπ(t)2/(2m) + f(π(t))]} exp[(i/h)α∫-∞dt'π(t')χ(t' - ε)]
= ∫Dπ exp{α∫-∞dt[-(i/h)tπ(t)2/(2m) + f(π(t))]} exp[(i/h)α∫-∞dt'π(t' + ε)χ(t')]
= ∫Dπ' exp{α∫-∞dt[-(i/h)tπ'(t - ε)2/(2m) + f(π'(t - ε))]} exp[(i/h)α∫-∞dt'π'(t')χ(t')]
= ∫Dπ' exp{α∫-∞dt[-(i/h)(t + ε)π'(t)2/(2m) + f(π'(t))]} exp[(i/h)α∫-∞dt'π'(t')χ(t')]
where π'(t) = π(t + ε).

ih limε→0 (1/ε) {Φ[χ(□ - ε)] - Φ[χ]}
= ∫Dπ' {α∫-∞dt[π'(t)2/(2m)]} exp{α∫-∞dt[-(i/h)(t + ε)π'(t)2/(2m) + f(π'(t))]} exp[(i/h)α∫-∞dt'π'(t')χ(t')].

{α∫-∞dt[-(ih/α)δ/δχ(t)]2/(2m)]}Φ[χ]
= ∫Dπ' {α∫-∞dt[π(t)2/(2m)]} exp{α∫-∞dt[-(i/h)tπ(t)2/(2m) + f(π(t))]} exp[(i/h)α∫-∞dt'π(t')χ(t')].

∴ ih limε→0 (1/ε) {Φ[χ(□ - ε)] - Φ[χ]} = {α∫-∞dt[-(ih/α)δ/δχ(t)]2/(2m)]}Φ[χ].

-∞dp exp{α∫aa +εdt[-(i/h)tp2/(2m) + (i/h)xp + f(p)]} (ε→ +0)
=∫-∞dp exp{αε[-(i/h)ap2/(2m) + (i/h)xp + f(p)]}
= exp{αεf([-ih/(αε)]∂/∂x)} ∫-∞dp exp{αε[-(i/h)ap2/(2m) + (i/h)xp]}
= exp{αεf([-ih/(αε)]∂/∂x)} √{π/[αε(i/h)a/(2m)]}・exp{(-1/h22ε2x2/[4αε(i/h)a/(2m)]}
= √[2mhπ/(iαεa)]・exp{αεf([-ih/(αε)]∂/∂x)} exp{[imαε/(2ha)]x2}.

If f(p) = 0, ψ(x, t) = exp{[im/(2h)](x2/t)}.

If f(p) = kp,
-∞dp exp{αε[-(i/h)ap2/(2m) + (i/h)xp + f(p)]}
= ∫-∞dp exp[αε{-(i/h)ap2/(2m) + [(i/h)x + k]p}]
= c exp{αε[(i/h)x + k]2/[4(i/h)a/(2m)]}
= c exp{αε[im/(2h)](x - ihk)2/a}
∴ ∫Dπ exp{α∫aa+1/αdt[-(i/h)tπ(t)2/(2m) + (i/h)xπ(t) + f(π(t))]}
≒ c'[-∞dp exp{αε[-(i/h)ap2/(2m) + (i/h)xp + f(p)]}]1/(αε)
= c''exp{[im/(2h)](x - ihk)2/a}.

If f(p) = -K(p - b)2,
-∞dp exp{αε[-(i/h)ap2/(2m) + (i/h)xp + f(p)]}
= ∫-∞dp exp{-αε[K + (i/h)a/(2m)]p2 + αε[(i/h)x + 2bK]p - αεKb2}
= C exp{αε(1/4)[(i/h)x + 2bK]2/[K + (i/h)a/(2m)]}
= C exp{αε(-1/2)(m/h)(x - 2ihbK)2/(2hmK + ia)}
∴ ∫Dπ exp{α∫aa+1/αdt[-(i/h)tπ(t)2/(2m) + (i/h)xπ(t) + f(π(t))]}
≒ C'[-∞dp exp{αε[-(i/h)ap2/(2m) + (i/h)xp + f(p)]}]1/(αε)
= C''exp{-(1/2)(m/h)(x - 2ihbK)2/(2hmK + ia)}.

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∫aa+1/αdt[-(i/h)tπ(t)2/(2m) + (i/h)xπ(t) + f(π(t))]}.

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.

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