Theory of Quantum History Entangled in Time-like Direction

ξ(□, x) represents a function which is a point in the function space for each x.

So, ξ is a parametric representation of a line in the function space, and x is a parameter of it.

By thinking of t as an index of a coordinate in the function space, η(x) = √{α∫dt [∂ξ(t,x)/∂x]²}
is understood as ds/dx where ds is the length of a line element.

η(x) is chosen to be 1 in the definition of the probability formula.

[dar(f; a, a + 1/α)]_pre Φ[χ]
= ∫dx η(x) f(x)^* Φ[ξ(□,x)]
= ∫dx f(x)^* Φ[ξ(□,x)]

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Last edited at 2013/05/31/17:34JST

I suppose that a general definition of the twist remove normalization for a functional Φ is given by the condition:
∫Dχ Φ[χ]^* Φ[χ] = 1,
(d/dε) ∫Dχ Φ[χ]^* Φ[χ(□ - ε)] |_{ε = 0} = 0,
where χ(□ - ε) is a function such that
[χ(□ - ε)](t) = χ(t - ε) for all t.

Possibly, a general definition of the twist remove normalization for a functional Φ is given by the condition:
∫Dχ Φ[χ]^* Φ[χ] = 1,
lim_{ε → 0} ∫Dχ Φ[χ]^* Φ[χ(□ - ε)] = 1
because the phase factor exp[-⊿t(i/h)＜j|H]|j＞] in Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > Quadratic Formula corresponds not to (d/dε) ∫Dχ Φ[χ]^* Φ[χ(□ - ε)] |_{ε = 0} but to lim_{ε → 0} ∫Dχ Φ[χ]^* Φ[χ(□ - ε)].

Possibly, the twist remove normalization may not be necessary for Φ in the case that we use exp∫dt instead of Π_t.
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Last edited at 2013/05/28/16:05JST

Φ[χ] = Σ_j exp[α∫dt φ_j(χ(t), t)],

[dar(f; a, a + 1/α)]_pre Φ[χ]
= ∫dx f(x)^* Φ[ξ(□, x)]
= ∫dx f(x)^* Σ_j exp[α∫dt φ_j(ξ(t, x), t)]
= Σ_j ∫dx f(x)^* exp[α∫_a^a+1/α dt φ_j(x, t)] exp[α∫_{t'＜a or t'≧a+1/α} dt' φ_j(χ(t'), t')]

dar(f; a, a + 1/α) Φ[χ_(a,a+1/α)]
= [dar(f; a, a + 1/α)]_pre Φ[χ]
= Σ_j ∫dx f(x)^* exp[α∫_a^a+1/α dt φ_j(x, t)] exp[α∫_{t'＜a or t'≧a+1/α} dt' φ_j(χ(t'), t')]
= Σ_j ∫dx f(x)^* exp[α∫_a^a+1/α dt φ_j(x, t)] exp[α∫_-∞^a dt' φ_j(χ_(a,a+1/α)(t'), t') + α∫_a^∞ dt' φ_j(χ_(a,a+1/α)(t'), t'+1/α)]
≒ Σ_j ∫dx f(x)^* exp φ_j(x, a)
　　　exp[ Σ_k＜0 φ_j(χ_(a,a+1/α)(a + k/α), a + k/α) + Σ_k≧0 φ_j(χ_(a,a+1/α)(a + k/α), a + (k + 1)/α)] ∵ α ≫ 1

dar(f; a, a + 1/α) Φ[χ]
≒ Σ_j ∫dx f(x)^* exp φ_j(x, a) exp[ Σ_k＜0 φ_j(χ(a + k/α), a + k/α) + Σ_k≧0 φ_j(χ(a + k/α), a + (k + 1)/α)]

Let exp φ_j be a solution of the old Schrodinger equation for each j.
Let {exp φ₁(□, t), exp φ₂(□, t), ・・・} be an orthonormal basis of the state space of the old quantum mechanics for each t.

∫Dχ dar(g; b, b + 1/α) Φ[χ]・{dar(f; a, a + 1/α) Φ[χ]}^*
≒ Σ_j,j' [Π_k∫dχ(a+k/α)]
　　∫dx' g(x')^* exp φ_j'(x', b) exp[ Σ_k'＜0 φ_j'(χ(b + k'/α), b + k'/α) + Σ_k'≧0 φ_j'(χ(b + k'/α), b + (k' + 1)/α)]
　　{∫dx f(x)^* exp φ_j(x, a) exp[ Σ_k＜0 φ_j(χ(a + k/α), a + k/α) + Σ_k≧0 φ_j(χ(a + k/α), a + (k + 1)/α)]}^*
= Σ_j,j' ∫dx' g(x')^* exp φ_j'(x', b) ∫dx f(x) [exp φ_j(x, a)]^*
　　Π_k＜0 ∫dχ(a+k/α) [exp φ_j(χ(a + k/α), a + k/α)]^* [exp φ_j(χ(a + k/α), a + k/α)]
　　Π_{0≦k＜α(b-a)} ∫dχ(a+k/α) [exp φ_j(χ(a + k/α), a + (k + 1)/α)]^* [exp φ_j(χ(a + k/α), a + k/α)]
　　Π_k≧α(b-a) ∫dχ(a+k/α) [exp φ_j(χ(a + k/α), a + k/α)]^* [exp φ_j(χ(a + k/α), a + k/α)]
= Σ_j,j' ∫dx' g(x')^* exp φ_j'(x', b) ∫dx f(x) [exp φ_j(x, a)]^*
　　(δ_j,j')^∞ Π_{0≦k＜α(b-a)} ∫dχ(a+k/α) [exp φ_j(χ(a + k/α), a + (k + 1)/α)]^* [exp φ_j(χ(a + k/α), a + k/α)]
= Σ_j ＜g|j,b＞＜j,a|f＞ Π_{0≦k＜α(b-a)} ＜j,a+(k+1)/α|j,a+k/α＞
≒ Σ_j ＜g|j,b＞＜j,a|f＞ ∵ ※
= Σ_j ＜g|U(b,a)|j,a＞＜j,a|f＞
= ＜g|U(b,a)|f＞
if α(b - a) ∈ N.

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Π_{0≦k＜α(b-a)} ∫dχ(a+k/α) [exp φ_j(χ(a + k/α), a + (k + 1)/α)]^* [exp φ_j(χ(a + k/α), a + k/α)]
= Π_{0≦k＜α(b-a)} ＜j,a+(k+1)/α|j,a+k/α＞
≒ Π_{0≦k＜α(b-a)} ＜j,a+k/α|j,a+k/α＞ ∵the twist remove normalization
= ＜j,a|j,a＞^α(b-a)
= 1
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However, notice that the new grammar version of the Schrödinger equation has no solution such that Φ[χ] = Σ_j exp[α∫dt φ_j(χ(t), t)].

A few approximations caused by the finiteness of α shift the result from the one in the old quantum mechanics.
Such an approximation is not needed if α is infinity.
I don't think that it is a prediction of the new theory distinct from the old quantum mechanics.
I think that it should be understood as the fact that a quantum history and a measurement are less related to each other than in the old quantum mechanics.

A prediction of the new theory distinct from the old quantum mechanics will be caused by the entanglement of a quantum history in a time-like direction.

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Last edited at 2013/06/03/16:48JST