Theory of Quantum History Entangled in Time-like Direction

Thinking of each value of t as a name of a degree of freedom, I made a grammar for describing quantum histories entangled in time-like direction.

This grammar first appeared as a post written in Japanese language in http://groups.msn.com/grammar/ on 2005/05/27.
(Links > Japanese > ビンテージ > 文法レベルでの自然 > 中心問題解決案 > 時間の量子化 > 第1件から第12件まで)

After that, it was proposed at JPS 2006 Spring Meeting 27pXA-6 by Yuuichi Uda.

And a new equation corresponding to this grammar instead of the Schrödinger equation first appeared as a post written in Japanese language in http://groups.msn.com/grammar/ on 2005/06/03.
(Links > Japanese > ビンテージ > 文法レベルでの自然 > 中心問題解決案 > 時間の量子化 > 第13件から第27件まで)

After that, it was proposed at APS+JPS 2006 Autumn Meeting by Yuuichi Uda.
He also proposed it at JPS 2007 Spring Meeting 28pSL-11.

Because MSN Group Service was closed on 2009/02, we can not see the above messages now in http://groups.msn.com/grammar/
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Last edited on 2011/08/16/14:42JST

A functional Φ representing an entangled quantum history obeys the new grammar version of Schrödinger equation:

where α is a new physical constant with the dimension of (time)^-1 introduced by Yuichi Uda and χ(□-ε) is a function defined as
∀t∈R; [χ(□-ε)](t) = χ(t-ε).

This equation describes the physical law for a particle with mass m in one-dimensional space.
V is a function representing a potential energy.
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Last edited at 2011/08/17/15:17JST

Yuichi Uda derived the following equation from the new grammar version of Schrödinger equation on 2011/07/01.

where S is the action functional such that

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Last edited at 2011/08/22/15:26JST

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

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 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

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 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

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

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(f, t₁; g, t₂) = ∫Dχ dar(f; t₁, t₁ + 1/α)Φ[χ]・{[dar(g; t₂, t₂ + 1/α)]Φ[χ]}^*.
This function corresponds to ＜f, t₁| g, t₂＞ 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) if t ≦ a or t ≧ b,
ξ(t,x) = x if a < t < b,
it 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) if t ≦ a,
χ_(a,b)(t) = χ(t + (b - a)) if 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}.
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.

However to define the twist remove normalization generally is a remaining problem.
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Last edited at 2013/05/31/17:35JST