# Theory of Quantum History Entangled in Time-like Direction

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SourceCodeOf_HumanGenome > Theory of Quantum History Entangled in Time-like Direction @ 2008/12/6 11:52 |
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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/ --- Last edited on 2011/08/16/14:42JST |

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SourceCodeOf_HumanGenome > New Grammar Version of Schrödinger Equation @ 2011/8/14 14:51 |
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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. --- Last edited at 2011/08/17/15:17JST |

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SourceCodeOf_HumanGenome > Reduced Form of the New Equation @ 2011/8/16 14:58 |
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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 --- Last edited at 2011/08/22/15:26JST |

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SourceCodeOf_HumanGenome > Meta Gauge Field @ 2011/8/21 10:32 |
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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]. --- Last edited at 2011/08/22/10:26JST |

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SourceCodeOf_HumanGenome > Unentangled Fundamental Solution @ 2011/8/22 15:32 |
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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.--- Last edited at 2011/08/23/15:07JST |

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SourceCodeOf_HumanGenome > Circular Time Regularization @ 2011/8/24 10:54 |
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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(a _{0}[χ],a_{1}[χ],a_{2}[χ],・・・;b_{1}[χ],b_{2}[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, --- Last edited at 2011/10/11/14:41JST |

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SourceCodeOf_HumanGenome > Solution by using circular time in the case V=0 @ 2011/10/12 10:38 |
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If we choose functions
f ^{0}_{k0},
f^{1}_{e1,k1},
f^{2}_{e2,k2},・・・ and
coefficients c(k_{0}; e_{1}, k_{1};
e_{2}, k_{2}; ・・・) as then is a solution of the new grammar version of Schrodinger equation in the case V=0. The equation for the function f ^{0}_{k} is easily solved as
follows.where k is an arbitrary real number. The equation for the function f ^{n}_{e, k} is solved as
follows which was presented at JPS 2010 Autumn Meeting by Yuichi Uda.By using Fourier expansion of the function f ^{n}_{e,
k}:the equation for the function f ^{n}_{e, k} is rewritten
asTo 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 f ^{n}_{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: --- Last edited at 2011/10/26/15:18JST |

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SourceCodeOf_HumanGenome > Similar problem in the old theories @ 2011/10/26 15:44 |
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The equation for the function
f ^{n}_{e, k} is rewritten asThis 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. --- Last edited at 2011/11/05/14:41JST |

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SourceCodeOf_HumanGenome > Cartoon of the Equation @ 2013/4/3 14:39 |
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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. --- Last edited at 2013/04/09/16:32JST |

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SourceCodeOf_HumanGenome > Probability Formula in New Grammar @ 2013/5/18 13:32 |
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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, t _{1}; g,
t_{2}) = ∫Dχ dar(f; t_{1}, t_{1} +
1/α)Φ[χ]・{[dar(g; t_{2}, t_{2} +
1/α)]Φ[χ]}^{*}.This function corresponds to ＜f, t _{1}| g, t_{2}＞ 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/_{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. --- Last edited at 2013/05/31/17:35JST |

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