# Theory of Quantum History Entangled in Time-like Direction

1 | 2 | Next Page>

SourceCodeOf_HumanGenome > Examine Probability Formula @ 2013/5/28 16:29 |
---|

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 ≫ 1dar(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.---※--- 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 --- 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. --- Last edited at 2013/06/04/15:35JST |

Edit Delete Reply |

SourceCodeOf_HumanGenome > Twist Remove Normalization @ 2013/5/27 11:23 |
---|

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)] = 1because 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 _{k}F_{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. --- Last edited at 2013/05/28/17:08JST |

Edit Delete Reply |

SourceCodeOf_HumanGenome > Line Integral in Function Space @ 2013/5/26 16:32 |
---|

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)]--- Last edited at 2013/05/31/17:34JST |

Edit Delete Reply |

SourceCodeOf_HumanGenome > Probability Formula in New Grammar @ 2013/5/18 13:32 |
---|

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, t _{2}; f,
t_{1}) = ∫Dχ dar(g; t_{2}, t_{2} +
1/α)Φ[χ]・{dar(f; t_{1}, t_{1} +
1/α)Φ[χ]}^{*}.This function corresponds to ＜g, t _{2}| f, t_{1}＞ 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/_{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/However to define the twist remove normalization generally is a remaining problem. --- Last edited at 2013/06/04/15:58JST |

Edit Delete Reply |

SourceCodeOf_HumanGenome > Cartoon of the Equation @ 2013/4/3 14:39 |
---|

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 |

Edit Delete Reply |

SourceCodeOf_HumanGenome > Similar problem in the old theories @ 2011/10/26 15:44 |
---|

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 |

Edit Delete Reply |

SourceCodeOf_HumanGenome > Solution by using circular time in the case V=0 @ 2011/10/12 10:38 |
---|

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 |

Edit Delete Reply |

SourceCodeOf_HumanGenome > Circular Time Regularization @ 2011/8/24 10:54 |
---|

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 |

Edit Delete Reply |

SourceCodeOf_HumanGenome > Unentangled Fundamental Solution @ 2011/8/22 15:32 |
---|

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 |

Edit Delete Reply |

SourceCodeOf_HumanGenome > Meta Gauge Field @ 2011/8/21 10:32 |
---|

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 |

Edit Delete Reply |

1 | 2 | Next Page>