A=e^iθ,B=e^-iθとおくと、ファインマン・パラメータ | ||||||||||||
1/AB=1/(A-B)^2⋆[1/k-1/k+1] | ||||||||||||
∴(2iSinθ)^2=[1/k-1/k+1] オイラーの公式より | ||||||||||||
※sinh[x]=[1/k-1/k+1] (A=e^x,B=e^-xのファインマン・パラメータ) | ||||||||||||
⑦∫dx/√(x^2+1)=Log(x+√(x^2+1))=asinh(x)(九州大,矢野『微分積分学』p84) | ||||||||||||
∴d=sinh[x]=(1/2)/√k(k+1)=√ℓ(ℓ+1) | ||||||||||||
√X(X+1) | ⋆ | d=asinh(x) | % | 文献値 | ⋆ | 文/d≒16 | ⋆ | Sin(x/98) | ⋆ | √Sin(x/98) | ⋆ | X=x/98 |
0.101529 | / | 0.881374 | % | 14.13472 | ⋆ | 16.03715 | ⋆ | 0.010204 | ⋆ | 0.1010144 | ⋆ | 0.010204 |
0.144308 | / | 1.443635 | % | 21.02203 | ⋆ | 14.561869 | ⋆ | 0.020407 | ⋆ | 0.1428522 | ⋆ | 0.020408 |
0.177621 | / | 1.818446 | % | 25.01085 | ⋆ | 13.753966 | ⋆ | 0.030607 | ⋆ | 0.1749499 | ⋆ | 0.030612 |
0.206112 | / | 2.094713 | % | 30.42487 | ⋆ | 14.524604 | ⋆ | 0.040805 | ⋆ | 0.2020025 | ⋆ | 0.040816 |
0.231567 | / | 2.312438 | % | 32.93506 | ⋆ | 14.242568 | ⋆ | 0.050998 | ⋆ | 0.225828 | ⋆ | 0.05102 |
0.254898 | / | 2.49178 | % | 37.58617 | ⋆ | 15.084065 | ⋆ | 0.061186 | ⋆ | 0.2473585 | ⋆ | 0.061224 |
0.276642 | / | 2.644121 | % | 40.91871 | ⋆ | 15.475356 | ⋆ | 0.071368 | ⋆ | 0.2671476 | ⋆ | 0.071429 |
0.297147 | / | 2.776472 | % | 43.32707 | ⋆ | 15.605079 | ⋆ | 0.081542 | ⋆ | 0.2855556 | ⋆ | 0.081633 |
0.316656 | / | 2.893444 | % | 48.00515 | ⋆ | 16.591007 | ⋆ | 0.091708 | ⋆ | 0.3028328 | ⋆ | 0.091837 |
0.33534 | / | 2.998223 | % | 49.77383 | ⋆ | 16.60111 | ⋆ | 0.101864 | ⋆ | 0.3191611 | ⋆ | 0.102041 |
0.353332 | / | 3.093102 | % | 52.97 | ⋆ | 17.125202 | ⋆ | 0.112009 | ⋆ | 0.334678 | ⋆ | 0.112245 |
0.370733 | / | 3.179785 | % | 56.446 | ⋆ | 17.751512 | ⋆ | 0.122143 | ⋆ | 0.3494899 | ⋆ | 0.122449 |
0.387621 | / | 3.259573 | % | 59.347 | ⋆ | 18.206988 | ⋆ | 0.132264 | ⋆ | 0.3636817 | ⋆ | 0.132653 |
0.404061 | / | 3.333478 | % | 60.831 | ⋆ | 18.24851 | ⋆ | 0.142372 | ⋆ | 0.3773218 | ⋆ | 0.142857 |
0.420106 | / | 3.402307 | % | 65.112 | ⋆ | 19.137605 | ⋆ | 0.152464 | ⋆ | 0.3904667 | ⋆ | 0.153061 |
※SinX=X-X^3/3!+X^5/5!-X^7/7!+⋆⋆⋆ | ||||||||||||
SinhX=X+X^3/3!+X^5/5!+X^7/7!+⋆⋆⋆ |
⑦∫dx/√(x^2+1)=Log(x+√(x^2+1))=asinh(x)(九州大,矢野『微分積分学』p84) | ||||||||||||
∴d=sinh[x]=(1/2)/√k(k+1)=√ℓ(ℓ+1) | ||||||||||||
x | X | =x/98 | ⋆ | X(X+1) | ⋆ | √X(X+1) | ⋆ | d=asinh(x) | % | 文献値 | ⋆ | 文/d≒16 |
1 | ⋆ | 0.01 | ⋆ | 0.0103 | ⋆ | 0.101529 | / | 0.881374 | % | 14.13472 | ⋆ | 16.03715 |
2 | ⋆ | 0.02 | ⋆ | 0.0208 | ⋆ | 0.144308 | / | 1.443635 | % | 21.02203 | ⋆ | 14.561869 |
3 | ⋆ | 0.031 | ⋆ | 0.0315 | ⋆ | 0.177621 | / | 1.818446 | % | 25.01085 | ⋆ | 13.753966 |
4 | ⋆ | 0.041 | ⋆ | 0.0425 | ⋆ | 0.206112 | / | 2.094713 | % | 30.42487 | ⋆ | 14.524604 |
5 | ⋆ | 0.051 | ⋆ | 0.0536 | ⋆ | 0.231567 | / | 2.312438 | % | 32.93506 | ⋆ | 14.242568 |
6 | ⋆ | 0.061 | ⋆ | 0.065 | ⋆ | 0.254898 | / | 2.49178 | % | 37.58617 | ⋆ | 15.084065 |
7 | ⋆ | 0.071 | ⋆ | 0.0765 | ⋆ | 0.276642 | / | 2.644121 | % | 40.91871 | ⋆ | 15.475356 |
8 | ⋆ | 0.082 | ⋆ | 0.0883 | ⋆ | 0.297147 | / | 2.776472 | % | 43.32707 | ⋆ | 15.605079 |
9 | ⋆ | 0.092 | ⋆ | 0.1003 | ⋆ | 0.316656 | / | 2.893444 | % | 48.00515 | ⋆ | 16.591007 |
10 | ⋆ | 0.102 | ⋆ | 0.1125 | ⋆ | 0.33534 | / | 2.998223 | % | 49.77383 | ⋆ | 16.60111 |
11 | ⋆ | 0.112 | ⋆ | 0.1248 | ⋆ | 0.353332 | / | 3.093102 | % | 52.97 | ⋆ | 17.125202 |
12 | ⋆ | 0.122 | ⋆ | 0.1374 | ⋆ | 0.370733 | / | 3.179785 | % | 56.446 | ⋆ | 17.751512 |
13 | ⋆ | 0.133 | ⋆ | 0.1502 | ⋆ | 0.387621 | / | 3.259573 | % | 59.347 | ⋆ | 18.206988 |
14 | ⋆ | 0.143 | ⋆ | 0.1633 | ⋆ | 0.404061 | / | 3.333478 | % | 60.831 | ⋆ | 18.24851 |
15 | ⋆ | 0.153 | ⋆ | 0.1765 | ⋆ | 0.420106 | / | 3.402307 | % | 65.112 | ⋆ | 19.137605 |
∵1/AB=1/(A-B)2[1/k-1/k+1]ファインマン・パラメータ | ||||||||||||
A=e^x,B=e^-xとおくと |
∵1/AB=1/(A-B)2[1/k-1/k+1]ファインマン・パラメータ | ||||||||||||
A=e^x,B=e^-xと | ||||||||||||
(e^x-e^-x)^2=[1/k-1/k+1]=1/k(k+1) | ||||||||||||
(2sinh[x])^2=[1/k-1/k+1]=1/k(k+1) | ||||||||||||
∴sinh[x]=(1/2)/√k(k+1)=√ℓ(ℓ+1) | ||||||||||||
またasinh(α)=asinh(-α) | ||||||||||||
cosh(α)=1/Sinθ橋本(p117) | ||||||||||||
①双曲線関数(オイラーディスク、2点間距離) | ||||||||||||
a(sinh(ℓx))=ℓ・cosh(ℓx) | ||||||||||||
a(cosh(ℓx))=ℓ・sinh(ℓx) | ||||||||||||
Ψ(x)=sinh(ℓx)とおくと,△Ψ=ℓ^2Ψ | ||||||||||||
(△-ℓ^2)Ψ(x)=HΨ(x)シュレーディンガー | ||||||||||||
②零点(ℓ⁰)=±√[16d⋆L(L+1)]) | ||||||||||||
零点(ℓ⁰)=±√(ℓ^2+△ーiε)≒±√(ℓ^2+△)ーiε=±√[16d⋆L(L+1)]) | ||||||||||||
③Wick回転、次の典型的なloap積分 | ||||||||||||
∫d^4ℓ/(ℓ^2-△+iε)=∫d^4ℓ/(ℓ⁰^2-ℓ^2-△+iε) | ||||||||||||
この被積分関数の極値は、複素ℓ⁰平面ニおいては、 | ||||||||||||
ℓ⁰=±√(ℓ^2+△ーiε)≒±√(ℓ^2+△)ーiε=±√[16d+L(L+1)]ーiε | ||||||||||||
∵△φ=0(ベースラインor1/4) | , | ℓ^2=16d | ⋆ | 零点ℓ⁰ | ||||||||
ℓ | L | =ℓ/98 | ⋆ | L(L+1) | ⋆ | √L(L+1) | ⋆ | d=asinh(ℓ) | % | 文献値 | ⋆ | 文/d≒16 |
1 | ⋆ | 0.01 | ⋆ | 0.0103 | ⋆ | 0.101529 | / | 0.881374 | % | 14.13472 | ⋆ | 16.03715 |
2 | ⋆ | 0.02 | ⋆ | 0.0208 | ⋆ | 0.144308 | / | 1.443635 | % | 21.02203 | ⋆ | 14.561869 |
3 | ⋆ | 0.031 | ⋆ | 0.0315 | ⋆ | 0.177621 | / | 1.818446 | % | 25.01085 | ⋆ | 13.753966 |
4 | ⋆ | 0.041 | ⋆ | 0.0425 | ⋆ | 0.206112 | / | 2.094713 | % | 30.42487 | ⋆ | 14.524604 |
5 | ⋆ | 0.051 | ⋆ | 0.0536 | ⋆ | 0.231567 | / | 2.312438 | % | 32.93506 | ⋆ | 14.242568 |
6 | ⋆ | 0.061 | ⋆ | 0.065 | ⋆ | 0.254898 | / | 2.49178 | % | 37.58617 | ⋆ | 15.084065 |
7 | ⋆ | 0.071 | ⋆ | 0.0765 | ⋆ | 0.276642 | / | 2.644121 | % | 40.91871 | ⋆ | 15.475356 |
8 | ⋆ | 0.082 | ⋆ | 0.0883 | ⋆ | 0.297147 | / | 2.776472 | % | 43.32707 | ⋆ | 15.605079 |
9 | ⋆ | 0.092 | ⋆ | 0.1003 | ⋆ | 0.316656 | / | 2.893444 | % | 48.00515 | ⋆ | 16.591007 |
10 | ⋆ | 0.102 | ⋆ | 0.1125 | ⋆ | 0.33534 | / | 2.998223 | % | 49.77383 | ⋆ | 16.60111 |
11 | ⋆ | 0.112 | ⋆ | 0.1248 | ⋆ | 0.353332 | / | 3.093102 | % | 52.97 | ⋆ | 17.125202 |
12 | ⋆ | 0.122 | ⋆ | 0.1374 | ⋆ | 0.370733 | / | 3.179785 | % | 56.446 | ⋆ | 17.751512 |
13 | ⋆ | 0.133 | ⋆ | 0.1502 | ⋆ | 0.387621 | / | 3.259573 | % | 59.347 | ⋆ | 18.206988 |
14 | ⋆ | 0.143 | ⋆ | 0.1633 | ⋆ | 0.404061 | / | 3.333478 | % | 60.831 | ⋆ | 18.24851 |
15 | ⋆ | 0.153 | ⋆ | 0.1765 | ⋆ | 0.420106 | / | 3.402307 | % | 65.112 | ⋆ | 19.137605 |
∵1/AB=1/(A-B)2[1/k-1/k+1] | ||||||
A=e^x,B=e^-x | ||||||
(e^x-e^-x)^2=[1/k-1/k+1]=1/k(k+1) | ||||||
(2sinh[x])^2=[1/k-1/k+1]=1/k(k+1) | ||||||
∴sinh[x]=(1/2)/√k(k+1)=√ℓ(ℓ+1) | ||||||
またasinh(α)=asinh(-α) | ||||||
F=16⋆ASINH(n)フックの法則 | ||||||
∵オイラーディスク上の2点間の距離d=ASINH(α) | ||||||
n | ⋆ | asinh(n) | ⋆ | 文献値 | ⋆ | 文/d≒16 |
-10 | ⋆ | -2.998223 | ⋆ | |||
-9 | ⋆ | -2.893444 | ⋆ | |||
-8 | ⋆ | -2.776472 | ⋆ | |||
-7 | ⋆ | -2.644121 | ⋆ | |||
-6 | ⋆ | -2.49178 | ⋆ | |||
-5 | ⋆ | -2.312438 | ⋆ | |||
-4 | ⋆ | -2.094713 | ⋆ | |||
-3 | ⋆ | -1.818446 | ⋆ | |||
-2 | ⋆ | -1.443635 | ⋆ | |||
-1 | ⋆ | -0.881374 | ⋆ | |||
0 | ⋆ | 0 | ⋆ | |||
1 | ⋆ | 0.8813736 | ⋆ | 14.1347 | ⋆ | 16.03715 |
2 | ⋆ | 1.4436355 | ⋆ | 21.022 | ⋆ | 14.56187 |
3 | ⋆ | 1.8184465 | ⋆ | 25.0109 | ⋆ | 13.75397 |
4 | ⋆ | 2.0947125 | ⋆ | 30.4249 | ⋆ | 14.5246 |
5 | ⋆ | 2.3124383 | ⋆ | 32.9351 | ⋆ | 14.24257 |
6 | ⋆ | 2.4917799 | ⋆ | 37.5862 | ⋆ | 15.08407 |
7 | ⋆ | 2.6441208 | ⋆ | 40.9187 | ⋆ | 15.47536 |
8 | ⋆ | 2.7764723 | ⋆ | 43.3271 | ⋆ | 15.60508 |
9 | ⋆ | 2.893444 | ⋆ | 48.0052 | ⋆ | 16.59101 |
10 | ⋆ | 2.998223 | ⋆ | 49.7738 | ⋆ | 16.60111 |
11 | ⋆ | 3.0931022 | ⋆ | 52.97 | ⋆ | 17.1252 |
12 | ⋆ | 3.1797854 | ⋆ | 56.446 | ⋆ | 17.75151 |
13 | ⋆ | 3.2595726 | ⋆ | 59.347 | ⋆ | 18.20699 |
14 | ⋆ | 3.3334776 | ⋆ | 60.831 | ⋆ | 18.24851 |
15 | ⋆ | 3.4023066 | ⋆ | 65.112 | ⋆ | 19.1376 |