1. |∫Dχ(t≠a and t≠b) Φ[χ]|^2

　I had thought since 2011 spring that this approach is hopeful.
　That is to say, I had thought that an adequate functional representing a certain quantum history plays the same role as the integrand of the Feynman path integral.
　The following tries belong to this approach or a means of this approach.

・日本物理学会2011年春季大会＠学会発表＠活動報告＠学問＠宇田英才教室
・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > Probability Interpretation
・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > A Sketch of an Entangled Solution
・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > Correspondence between Solution and Action
・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > Path Integral of Solution
・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > To Change Centipede of Centipedical Starfish
・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > Another Type of an Entangled Solution


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After 1, before 2.

　Being faced with the deadlock of the first approach, I considered how the problem is essentially.

　The following articles belong to this stage.

・News > 2013/05/15/14:47:50
・News > 2013/05/15/16:17:05
・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > The Case that Only Two Times Exist
・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > The Case that Only Three Times Exist
・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > The Case of General Descrete Time

　In this stage, I came to be sure that an entangled quantum history can contain the full time development of the old quantum mechanics and the failure of my past tries is only because of bad choices.


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2. |∫Dχ(t≠a and t≠b) ∫dχ(a)f(χ(a))<SUP>*</SUP> ∫dχ(b)g(χ(b))<SUP>*</SUP> Φ[χ]|<SUP>2</SUP>

　This approach is a generalization of the first approach.
　If f and g are the delta functions, this formula reduces to the formula of the first approach.
　By noticing that the directions of bra and ket in this approach are incorrect, I decided to abandon this approach at last.
　The following try belong to this approach.

・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > Probability Formula

　In this approach, a quantum history Φ receives an initial state f and a final state g from the same side as <f|<g|Φ>.
　The directions of bra and ket in this are different from the directions of bra and ket in the existing Green's function <f|U|g>.
　So, this approach can not reproduce the existing Green's function.




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3. ∫dy ∫dχ(a)f(χ(a))*Φ[χ]|<SUB>χ(b)=y</SUB> [∫dχ(b)g(χ(b))*Φ[χ]|<SUB>χ(a)=y</SUB>]*

　This approach follows the idea that Σ<SUB>j</SUB>＜g|j,b＞＜j,a|f＞ should be replaced by Σ<SUB>j</SUB>＜g|Φ<SUB>j</SUB>＞＜Φ<SUB>j</SUB>|f＞.
　To solve the problem that functional integral is not done over χ(a) and χ(b), I set χ(a)=χ(b)=y and integrate over y.
However it failed because time development yield twice.

This idea is seen in my self e-mail written on 2013/05/16 as a memo.



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4. ∫dχ(a)∫dχ(b)[∫Dχ(t≠a and t≠b){∫dχ(a)f(χ(a))<SUP>*</SUP>Φ[χ]}{∫dχ(b)g(χ(b))Φ[χ]<SUP>*</SUP>}][∫Dχ(t≠a and t≠b){∫dχ(a)f(χ(a))Φ[χ]<SUP>*</SUP>}{∫dχ(b)g(χ(b))<SUP>*</SUP>Φ[χ]}]

　This approach was to try to improve the previous approach.
　To do so, I paired complex conjugates.
　The following try belong to this approach.

・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > Biquadratic Formula of Probability

　This approach failed because the order of summation and squaring in the result of calculation is incorrect.

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5. ∫dχ(a)[∫Dχ(t≠a){∫dχ(a)f(χ(a))<SUP>*</SUP>Φ[χ]}Φ[χ]<SUP>*</SUP>][∫Dχ(t≠b){∫dχ(b)g(χ(b))Φ[χ]<SUP>*</SUP>}Φ[χ]]<SUB>χ(b)=χ(a)</SUB>

　I invented this approach to prevent the fault of the 4th approach from arising.
　The following try belong to this approach.

・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > Corrected Biquadratic Formula of Probability

　This approach failed because the result of calculation is ＜f|g＞, which does not contain any time development at all.


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6. The Stacked Daruma Game Method
　This approach is a compromise in the problem that the functional integral is not done over χ(a) and χ(b) in the 4th and 5th approach.

　The following try belong to this approach.

・Forum > Problems in Grammatical Physics > Quantum Field Theory on the Time-Axis > Quadratic Formula

　This approach was cursed with accumulation of phase shifts.
　So, I stopped pushing this approach once at this stage.



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After 6, Before 7.

　Starting with noticing that ∫dχ(a) f(χ(a))* Φ[χ] is not the quantum history in which the initial state is f, I investigated how the linear expansion of a quantum history is.

　The following try belong to this approach.

・Forum > Problems in Grammatical Physics > Incompatible Linearity > Basis and Expansion

This aspect is seen in my self e-mail written on 2013/05/18 as a memo.

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7. Antecedent of Earth Worm Deformation of the Time Axis
　χ'(t)=χ((a+ε)+(t-a)/2) (a ≦ t ＜ a + 2ε)
　χ'(t)=χ(t) (a ＜ t or t ≧ a + 2ε)
　Φ'[χ'] = [dar(f;a,a+ε)]<SUB>pre</SUB>Φ[χ]

This idea was refered to in my self e-mail written on 2013/05/20 as a memo.


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8. Contract of (Φ/＜f|Φ＞)Φ and (Φ/＜g|Φ＞)Φ.
　This approach aims at flicking a piece away off each stacked Daruma without being followed by collapse.
　However, it failed because each denominator consists of many terms.

　If Φ[χ] ＝ Π<SUB>t</SUB> φ(χ(t),t),
　each of ＜f|Φ＞ and ＜g|Φ＞ consists of a single term.
　If Φ[χ] ＝ Σ<SUB>j</SUB> Π<SUB>t</SUB> φ<SUB>j</SUB>(χ(t),t),
　each of ＜f|Φ＞ and ＜g|Φ＞ consists of many terms.

　This approach was only thought in my mind and was only taken as a memo by self e-mail, it was not posted to any web site.
This idea was refered to in my self e-mail written at 2013/05/22/18:53JST as a memo.



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9. Earth Worm Deformation of the Time Axis.

　This idea appeared at News > 2013/05/23/11:26:01.

This idea is seen in my self e-mail written at 2013/05/22/19:59JST as a memo.


　This does not work well because it changes the power index of old type wavefunctions of corresponding points in time.

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soroban shift

This idea is seen in my self e-mail written at 2013/05/24/01:29JST as a memo.

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10. Operation such that f→|φ＞ ＝ ＜f|φ＞|φ＞.

This idea is seen in my self e-mail written at 2013/05/24/01:43JST as a memo.

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11. Flick χ(b) dependence away off dar(f;a,a+ε)Φ, and
flick χ(a) dependence away off dar(f;b,b+ε)Φ.

This idea is seen in my self e-mail written at 2013/05/24/01:46JST as a memo.

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12. Try to make a formula resembling to the formula of the expectation value of the old quantum mechanics.

This idea is seen in my self e-mail written at 2013/05/24/02:10JST as a memo.

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13. δ/δχ(t) ln Φ[χ]
　When I tried to develop this method, I noticed that energy ambiguity problem prevent me to go further.

This idea is seen in my self e-mail written at 2013/05/24/08:42JST as a memo.

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Last edited at 2013/07/23/15:40JST

As reported at the article posted at 2013/05/25/17:04:45JST, I succeeded in establishing a new probability formula by inventing the twist remove normalization.

The new probability formula and the twist remove normalization are exhibited in Forum > Products of Grammatical Physics > Theory of Quantum History Entangled in Time-like Direction > Probability Formula in New Grammar.
The stacked Daruma game method is used there.

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