Theory of Quantum History Entangled in Time-like Direction

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 Schrödinger equation for each j.
Let {exp φ'₁(□, t), exp φ'₂(□, t), ・・・} be an orthonormal basis of the state space of the old quantum mechanics for each t.
Let exp φ_j be a wave function given from exp φ'_j by the twist remove normalization for each j.

∫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 2013/06/04/15:35JST

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.


When
Φ[χ] = Σ_j Π_k φ_j(χ(k/α), k/α),
change of the phase factor of φ_j
φ'_j(χ(k/α), k/α) = exp[iF_j(k/α)]φ_j(χ(k/α), k/α)
changes only the whole phase factor by exp[iΣ_kF_j(k/α)].
Different paterns of changing phase factors without changing the whole phase factor can not be distinguished.
This concerns the energy ambiguity problem.

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Last edited at 2013/05/28/17:08JST

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

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, t₂; f, t₁) = ∫Dχ dar(g; t₂, t₂ + 1/α)Φ[χ]・{dar(f; t₁, t₁ + 1/α)Φ[χ]}^*.
This function corresponds to ＜g, t₂| f, t₁＞ 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/h)t∫dx' [exp φ'_j(x',t)]^* H exp φ'_j(x',t)
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/h)t＜j|H|j＞.

However to define the twist remove normalization generally is a remaining problem.
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Last edited at 2013/06/04/15:58JST

I showed a problem of classical mechanics and its solutions as a cartoon of the contents of the reduced form of the new equation at JPS 2013 Spring Meeting.

It originated from the idea on 2011/08/02.

When time is descretized and V = 0, the reduced form of the new grammar version of the Schrodinger equation

is rewitten as follows.


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Last edited at 2013/04/09/16:32JST

The equation for the function fⁿ_{e, k} is rewritten as

This is the Schrodinger equation in the old quantum mechanics for a 2-dimensional system with Hamiltonian:

The classical Hamiltonian of this system is

So the classical equations of motion of this system are

To solve this equation is not difficult.
The solution is as follows.

and

Thus the first equation in this message seems not to be pathological.
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Last edited at 2011/11/05/14:41JST

If we choose functions f⁰_k0, f¹_e1,k1, f²_e2,k2,・・・ and coefficients c(k₀; e₁, k₁; e₂, k₂; ・・・) as

then

is a solution of the new grammar version of Schrodinger equation in the case V=0.
The equation for the function f⁰_k is easily solved as follows.

where k is an arbitrary real number.
The equation for the function fⁿ_{e, k} is solved as follows which was presented at JPS 2010 Autumn Meeting by Yuichi Uda.
By using Fourier expansion of the function fⁿ_{e, k}:

the equation for the function fⁿ_{e, k} is rewritten as

To solve this equation, I introduced polar coordinates r and θ in pq-plane as p = r cosθ and q = r sinθ.
Then the equation for the function fⁿ_{e, k} reduces to the following expression.

This equation has the following solution.

Because must be continuous, g must be periodic in θ.

where e is an arbitrary real number and k is an arbitrary integer which satisfies:

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Last edited at 2011/10/26/15:18JST

The circular time was introduced by Yuichi Uda at JPS 2009 Autumn Meeting 13pSH-3 to solve the new grammar version of Schrödinger equation.

When we adopt the circular time, a functional Φ representing a quantum history can be expressed by a function of Fourier coefficients as follows.
Φ[χ] = F(a₀[χ],a₁[χ],a₂[χ],・・・;b₁[χ],b₂[χ],・・・)
where

Then the functional derivative with respect to χ(t) is written in terms of partial derivatives with respect to the Fourier coefficients.

By using orthogonality of trigonometric functions:

it follows that

On the other hand,

Then the new grammar version of Schrödinger equation is rewritten as follows.

Reducing this equation, it follows that

That is,

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Last edited at 2011/10/11/14:41JST

Yuichi Uda found an unentangled fundamental solution on 2011/04/13.
This solution is written as

for the new grammar version of Schrödinger equation:

when V(x;t) = -[1/(2m)][p(t)]² - [dp(t)/dt]x.
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Last edited at 2011/08/23/15:07JST

The reduced form of the new grammar version of Schrödinger equation is analogous to the ordinary Schrödinger equation:

This equation is for a stationary quantum state with energy eigenvalue zero.
mdχ(t)/dt is analogous to qA^k(x) and -S[χ] is analogous to qφ(x).
This correspondence is fully understood by writing
　qA^t[χ] = mdχ(t)/dt, qφ[χ] = -S[χ].
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Last edited at 2011/08/22/10:26JST