Feynman Encounter with Mathematics 52, 200 9 [] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull. Sci. Math. vol. 28 (2004) 97 25. [2] D. Fujiwara and N. Kumano-go, Smooth functional derivatives in Feynman path integral by time slicing approximation. Bull. Sci. Math. vol. 29 (2005) 57 79. [3] N. Kumano-go and D. Fujiwara, Feynman path integrals and semiclassical approximation. RIMS Kokyuroku Bessatsu B5 (2008) 24-263.
. Introduction 948 Feynman Schrödinger K(T, x, x 0 ) K(T, x, x 0 ) = (i T + 2 2 e i S[γ] D[γ] ) V (T, x) u(t, x) = 0 γ : [0, T ] R d γ(0) = x 0, γ(t ) = x T S[γ] = dγ 2 V (t, γ)dt γ D[γ] 2 dt 0 R. P. Feynman, Rev. Mod. Phys. 20 (948). (T, x) (0, x 0 ) 0 T
Feynman Feynman F [γ] F [γ]e i S[γ] D[γ] (DF )[γ][η] 960 Cameron e i S[γ] D[γ] R. H. Cameron, J. of Math. and Phys. Sci. 39(960). Remark limit
(DF )[γ][η] Feynman e i S[γ] F [γ]d[γ] Riemann(-Stieltjes) limit γ + η Qγ (DF )[γ][η] 0 0 D. Fujiwara
(x, x 0 ) R 2d F [γ] F Remark Schrödinger R 2d F [γ] Fujiwara(979,99) Kitada-H. Kumano-go(98) Yajima(99) N. Kumano-go(995) Fujiwara-Tsuchida(997) W. Ichinose(997) Nelson(964) Cameron-Storvick(983) Wiener Itô(967) Albeverio-Hoegh Krohn(976) Truman(972) Albeverio, Hoegh-Krohn, Mazzucchi, Mathematical theory of Feynman path integrals, 2nd, Springer, 2008. Johnson-Lapidus Gill-Zachary(2002) Operational calculus Johnson and Lapidus, The Feynman integral and Feynman s operational calculus, Oxford, 2000. T. Ichinose-Tamura(987) 2 Dirac Path Integral, 994.
2. Feynman Assumption of S[γ] = T 0 2 dγ dt 2 V (t, γ)dt. V (t, x) : R R d R x α V (t, x) α x V (t, x) C α( + x ) max(2 α,0). Example of F [γ] F ( ) () α x B(t, x) C α( + x ) m t F [γ] = B(t, γ(t)) F, F [γ] F, T Riemann(-Stieltjes) F [γ] = B(t, γ(t))dt F. T (2) x αb(t, x) C α F [γ] = e T T B(t,γ(t))dt F. (3) Z : R R d C d α x Z(t, x) + α x tz(t, x) C α ( + x ) m, t ( x Z) = ( x Z) F [γ] = T T Z(t, γ(t)) dγ(t) F.
F [γ] F F F F [γ] F Theorem Smooth algebra F [γ], G[γ] F, η : [0, T ] R d, d d P () F [γ] + G[γ] F, F [γ]g[γ] F. (2) F [γ + η] F, F [P γ] F. (3) (DF )[γ][η] F. Remark γ : [0, T ] R d η : [0, T ] R d (DF )[γ][η] = d dθ θ=0 F [γ + θη].
T,0 : T = T J+ > T J > > T > T 0 = 0 [0, T ] t j = T j T j, T,0 = max j J+ t j x = x J+ x J, x J,..., x R d γ T,0 j =, 2,..., J, J + (T j, x j ) (T j, x j ) S[γ T,0 ], F [γ T,0 ] x J+, x J,, x, x 0 S[γ T,0 ] = S T,0 (x J+, x J,..., x, x 0 ), F [γ T,0 ] = F T,0 (x J+, x J,..., x, x 0 ). (T J, x J ) (T 3, x 3 ) (T, x) (T, x ) (0, x 0 ) (T 2, x 2 ) T 0 = 0 T T 2 T 3 T J T = T J+
Feynman Theorem 2 Feynman T F [γ] F J+ ( ) e i d/2 S[γ] F [γ]d[γ] lim e i S[γ T,0 ] F [γ T,0 ] T,0 0 2πi t j= j R dj (x, x 0 ) R 2d i.e., well-defined. J j= dx j ( ) Remark F [γ] R d dx j =. J, J.
Remark S[γ T,0 ] S T,0 (x J+, x J,, x, x 0 ) Feynman J+ lim T,0 0 j= ( 2πi t j ) d/2 e i S T,0 (x J+,x J,...,x,x 0 ) R dj Nelson S T,0 Trotter v L 2 (R d ) L 2 - J+ ( ) d/2 lim J 2πi T /(J + ) j= R d R d e J+ i j= ( (x j x j ) 2 2T /(J+) E. Nelson, J. Math. Phys. 5 (964). ( J+ (xj x j ) 2 Tj S[γ T,0 ] = V (t, t T j x j + 2t j T j T j T j j= J j= V (x j ) T J+ dx j ) v(x 0 ) J j=0 T ) j t x j )dt T j T j ( ) S[γ T,0 ], F [γ T,0 ] dx j.
3. Theorem 3 T m 0, 0 T T T, f(t, x) : R R d C α x f(t, x), α x tf(t, x) x αf(t, x) + α x tf(t, x) C α ( + x ) m e i S[γ]( ) f (T, γ(t )) f (T, γ(t )) D[γ] = e i S[γ]( T T ) ( x f)(t, γ(t)) dγ(t)+ ( t f)(t, γ(t))dt D[γ]. Remark T T T ( x f)(t, γ(t)) dγ(t) γ B(t) (?) T
B(t) γ T,0 B(T j ) = x j Itô γ T,0 T T Z(t, B(t)) db(t) j Z(T j, x j ) (x j x j ). Stratonovich γ T,0 T Z(t, B(t)) db(t) ( Tj + T j Z, x ) j + x j T 2 2 j (x j x j ). γ T,0 T T Z(t, γ T,0 (t)) dγ T,0 (t).
Itô Riemann (T, x) (0, x 0 ) T 0 = 0 T T T = T J+ (T, x) (0, x 0 ) T 0 = 0 T T T = T J+
Proof of Theorem 3 Example Theorem () F 2 [γ] = F [γ] = f (T, γ(t )) f (T, γ(t )) F, T T ( x f)(t, γ(t)) dγ(t) + T γ T,0 Theorem 2 e i S[γ] F [γ]d[γ] lim T,0 0 = lim T,0 0 F [γ T,0 ] = F 2 [γ T,0 ]. ( J+ ) d/2 2πi t j= j J+ ( ) d/2 j= 2πi t j e i S[γ] F 2 [γ]d[γ]. T ( t f)(t, γ(t))dt F. R dj e i R dj e i S[γ T,0 ] F [γ T,0 ] S[γ T,0 ] F2 [γ T,0 ] J j= J j= dx j dx j
4. Riemann(-Stieltjes) lim Theorem 4 T m 0, 0 T T T, B(t, x) : [0, T ] R d C x α B(t, x) x αb(t, x) C α( + x ) m T ( ) ( T e i S[γ] B(t, γ(t))d[γ] dt = e i S[γ] T T B(t, γ(t))dt ) D[γ]. Remark limit x αb(t, x) C α e i S[γ]+ i T0 B(τ,γ(τ ))dτ D[γ] ( ) i n T τn τ2 = dτ n dτ n dτ n= 0 0 0 e i S[γ] B(τ n, γ(τ n ))B(τ n, γ(τ n )) B(τ, γ(τ ))D[γ].
Nelson x j x j (T, x) x j x j (0, x 0 ) T 0 = 0 T j t T j T = T J+ γ T,0 (t) = t T j T j T j x j + T j t T j T j x j (T, x) (0, x 0 ) T 0 = 0 T j t T j T = T J+
Proof of Theorem 4 γ T,0 (t) B(t, γ T,0 (t)) t [T, T ] x j Lebesgue T,0 J+ ( ) d/2 J e i S[γ T,0 ] B(t, γ T,0 (t)) dx j 2πi t j R dj j= t [T, T ] Theorem 2 e i S[γ] B(t, γ(t))d[γ] j= J+ ( ) d/2 lim e i S[γ T,0 ] B(t, γ T,0 (t)) T,0 0 2πi t j= j R dj t [T, T ] e i S[γ] B(t, γ(t))d[γ] t [T, T ] Riemann J j= dx j
T T T T ( ) e i S[γ] B(t, γ(t))d[γ] dt J+ ( ) d/2 lim e i S[γ T,0 ] B(t, γ T,0 (t)) T,0 0 2πi t j= j R dj T dt lim T T,0 0 T J+ ( ) d/2 = lim e i S[γ T,0 ] B(t, γ T,0 (t)) T,0 0 T 2πi t j R dj j= Fubini J+ ( ) d/2 = lim e i S[γ T,0 ] T,0 0 2πi t j= j R dj ( ) T e i S[γ] B(t, γ(t))dt D[γ]. T T T J j= J j= B(t, γ T,0 (t))dt dx j dt. dx j dt. J j= dx j
5. F [γ + η] F [Qγ] Theorem 5 T F [γ] F η : [0, T ] R d e i S[γ+η] F [γ + η]d[γ] = γ(0)=x 0,γ(T )=x γ(0)=x 0 +η(0),γ(t )=x+η(t ) e i S[γ] F [γ]d[γ]. Remark T F [γ] F d d Q e i S[Qγ] F [Qγ]D[γ] = γ(0)=x 0,γ(T )=x γ(0)=qx 0,γ(T )=Qx e i S[γ] F [γ]d[γ].
Proof of Theorem 5 Theorem 2 γ T,0 (T j ) = x j e i S[γ+η] F [γ + η]d[γ] γ(0)=x 0,γ(T )=x J+ = lim T,0 0 j= ( 2πi t j ) d/2 R dj e i S[γ T,0 +η] F [γ T,0 + η] T,0 η η(t j ) = y j, j = 0,,..., J, J + J j= dx j. (T, x) (0, x 0 ) η T 0 = 0 T = T J+
γ T,0 + η j =, 2,..., J + (T j, x j + y j ) (T j, x j + y j ) J+ ( ) d/2 = lim T,0 0 2πi t j= j e i S T,0 (x J+ +y J+,x J +y J,...,x +y,x 0 +y 0 ) R dj F T,0 (x J+ + y J+, x J + y J,..., x + y, x 0 + y 0 ) x j + y j x j, j =, 2,..., J J+ ( ) d/2 = lim e i T,0 0 2πi t j= j R dj F T,0 (x J+ + y J+, x J,..., x, x 0 + y 0 ) y J+ = η(t ) y 0 = η(0) = e i S[γ] F [γ]d[γ]. γ(0)=x 0 +η(0),γ(t )=x+η(t ) J j= dx j. S T,0 (x J+ +y J+,x J,...,x,x 0 +y 0 ) J j= dx j.
6. (DF )[γ][η] Theorem 6 T F [γ] F η(0) = η(t ) = 0 η : [0, T ] R d e i S[γ] (DF )[γ][η]d[γ] = i e i S[γ] (DS)[γ][η]F [γ]d[γ]. Remark F [γ] S[γ] = T 0 dγ 2 V (t, γ)dt 2 dt η(0) = η(t ) = 0 η : [0, T ] R d T ( ) 0 = e i dγ S[γ] dη dt dt ( xv )(t, γ(t))η(t) dtd[γ]. 0
Remark T,0 γ η γ(t j ) = x j, η(t j ) = y j, j = 0,,..., J, J + θ R γ + θη j =, 2,..., J, J + (T j, x j + θy j ) (T j, x j + θy j ) F [γ + θη] = F T,0 (x J+ + θy J+, x J + θy J,..., x + θy, x 0 + θy 0 ). γ (T, x) (0, x 0 ) η T 0 = 0 T = T J+
(DF )[γ][η] (DF )[γ][η] = d J+ dθ θ=0 F [γ + θη] = ( xj F T,0 )(x J+, x J,..., x, x 0 ) y j. j=0 Remark T F [γ] F η : [0, T ] R d e i S[γ] F [γ + η]d[γ] = + 0 ( θ) L L! L l=0 l! e i S[γ] (D l F )[γ][η] [η]d[γ] e i S[γ] (D L+ F )[γ +θη][η] [η]d[γ]dθ.
7. 0 Theorem 7 0 T F [γ] F ( ) e i d/2 ( ) S[γ] F [γ]d[γ] = e i S[γcl ] D(T, x, x 0 ) /2 F [γ cl ] + Υ (, x, x 0 ) 2πi T γ cl γ cl (0) = x 0, γ cl (T ) = x D(T, x, x 0 ) Morette-Van Vleck Υ (, x, x 0 ) C( + x + x 0 ) m. (T, x) (0, x 0 ) γ cl T 0 = 0 T = T J+
8. Theorems (),2,7 992 Fujiwara Schrödinger F [γ],, J+ ( ) d/2 J e i S T,0 (x J+,x J,...,x,x 0 ) dx j 2πi t j= j R dj j= ( ) d/2 = e i ( S T,0 (x,x 0) D T,0 2πi T (x, x 0) /2 + Υ (, x, x T,0 0)). (T J, x J ) (T 3, x 3 ) (T, x) (T, x ) (0, x 0 ) (T 2, x 2 ) T 0 = 0 T T 2 T 3 T J T = T J+
Remark Remark γ cl S[γ cl ] Hessian T,0 : T = T J+ > T J > > T > T 0 = 0, ( T,TN+, Tn,0) : T = T J+ > > T N+ > T n > > T 0 = 0 D T,0 D ( T,TN+, Tn,0) C(T N+ T n ) 2, Υ T,0 Υ ( T,TN+, Tn,0) C (T N+ T n ) 2, Υ T,0 C. T,0 0 D T,0 D C T,0 T, Υ T,0 Υ C T,0 T ( ), Υ C. 0, 999.
0 ( ) Υ T,0 Υ C T,0 T? = ( ) F [γ] Υ T,0 Υ C T,0 T ( + x + x 0 )? Remark =.054 0 34 (J s).
( Υ T,0 Υ ɛ, 0 ɛ ) ( T,TN+, Tn,0) T,0 0 ɛ ( ) d/2 ɛ = e i ( ) S T,0 D /2 2πi T T,0 + Υ T,0 ɛ ɛ = 0 ( 2πi T ( 2πi T ) d/2 e i S ɛ (D ɛ /2 + Υ ɛ ) ) d/2 e i S ( T,TN+, Tn,0 ) ( D /2 ( T,TN+, Tn,0) + Υ ( T,TN+, Tn,0) Υ T,0 Υ ( T,TN+, Tn,0) = Υ Υ 0 = 0 ( ɛυ ɛ ) dɛ ɛ Υ ɛ C(T N+ T n ) 2 ( + x + x 0 ) Υ ɛ ).
Υ ɛ? Υ ɛ D ɛ /2 D ɛ /2 S ɛ S ɛ
= = =. γ T,0 F [γ] F [γ] Cauchy T,0 : T = T J+ > T J > > T > T 0 = 0, ( T,TN+, Tn,0) : T = T J+ > > T N+ > T n > > T 0 = 0
(T, x) (0, x 0 ) T 0 = 0 T n T N+ T = T J+ Key Lemma x j = T j T n T N+ T n x N+ + T N+ T j T N+ T n x n, j = n,..., N (x N,..., x n ) = (x N,..., x n ) γ T,0 = γ ( T,TN+, Tn,0). (T, x) (0, x 0 ) T 0 = 0 T n T N+ T = T J+
J dx j. j= Key Lemma (x N,..., x n ) = (x N,..., x n ) F T,0 (x J+,..., x N+, x N,..., x n, x n,..., x 0 ) = F [γ T,0 ] = F [γ ( T,TN+, Tn,0)] = F ( T,TN+, Tn,0)(x J+,..., x N+, x n,..., x 0 ). J+ n dx j dx j. j=n+ j=
Proof of Theorems (),2,7, J. () F [γ] F = ( ) ( ) F [γ] F [γ] (2) T T B(t, γ(t))dt F, B(t, γ(t)) F. ( ) F (3) F [γ], G[γ] F = F [γ] + G[γ], F [γ]g[γ] F. + ()(3) (2)
9. F [γ] F Assumption (critical point) l L J + x L,l = (x L, x L,..., x l ) x L,l = x L,l (x L+, x l ) ( xl,l S T,0 )(x J+,L+, x L,l, x l,0) = 0 Definition F F [γ] Assumption F [γ] F
Assumption m 0 M A M, X M T,0, 0 = j 0 < j < j < j 2 < j 2 < < j K J +, j K+ = J + α jk+, α jk M () ( K k=0 α j k+ x jk+ α j k x jk )F T,0 (x J+, x J,j K +, x j K,, x js+, x j s+ 2,j s +, x j s, x js, x j s 2,j s +, x j s, x js, x j s 2,j s 2 +, x j s 2,, x j, x j 2,, x 0) A M (X M ) K+ ( + x J+ + x jk + + x js+ + x js + x js + x js + x js + x js 2 + + x j + x 0 ) m. (2), (3), (4) Remark Theorem (),2,3,4,7
Proof F T,0 (, x N+, x N,n, x n, ) = F ( T,TN+, Tn,0)(, x N+, x n, ). T n T N+ T n T N+
Assumption m 0, u j 0, J+ j= u j U < M A M, X M T,0, α j M, j = 0,,..., J + k J J+ () ( j=0 α j x j )F T,0 (x J+, x J,..., x, x 0 ) A M (X M ) J+ ( + J+ (2) ( α j x j ) xk F T,0 (x J+, x J,..., x, x 0 ) j=0 A M (X M ) J+ (u k+ + u k )( + J+ j=0 x j ) m. Remark Theorem (),2,3,4,7 J+ j=0 x j ) m,
Definition T,0 F T,0 (x J+, x J,..., x, x 0 ) C (R d(j+2) ) γ : [0, T ] R d η l : [0, T ] R d, l =, 2,..., L L L (D L L F )[γ] [η l ] = ( )F [γ + θ l η l ] θ l θ = =θ L =0 l= l= l= γ η 0,l0 η,l η 2,l2 η 3,l3 η j,lj T 0 = 0 T T 2 T 3 T j T j T j+ T = T J+
Assumption of F [γ] F m 0, ρ(t), ρ (t) M A M, X M J+ J+ () (D j=0 L j F )[γ] j=0 L j l j = J+ (2) (D + J+ j=0 L j F )[γ][η] j=0 J+ [η j,lj ] A M (X M ) J+ ( + γ ) m L j l j = [η j,lj ] A M (X M ) J+ ( + γ ) m T 0 η(t) d ρ (t) J+ j=0 L j l j = η j,lj j=0 L j l j = η j,lj, T,0, γ : [0, T ] R d, η : [0, T ] R d L j = 0,,..., M, [T j, T j+ ] η j,lj : [0, T ] R d, l j =, 2,..., L j γ = max 0 t T γ(t) Remark Theorem,2,3,4,5,6,7
0. Theorem (),2,7 J+ ( ) d/2 J ( ) lim e i S[γ T,0 ] F [γ T,0 ] dx j, T,0 0 2πi t j= j R dj j=, J F T,0 (x J+, x J,..., x, x 0 ) = F [γ T,0 ] Step. H. Kumano-go-Taniguchi ( ) J C C C Step 2. Fujiwara ( ) J C Step 3. ( ) T,0 0
. H. Kumano-go-Taniguchi H. Kumano-go, Pseudo differential operators, MIT Press, p.360 Fourier J+ j= = ( ) d/2 2πi t j ( ) d/2 e i S T,0 (x,x 0) q T,0 (, x, x 0). 2πi T R dj e i S T,0 (x J+,x J,...,x,x 0 ) F T,0 (x J+, x J,..., x, x 0 ) Lemma KT m 0 M A M, X M α j M, j =, 2,..., J, J + J+ ( j=0 α j x j )F T,0 (x J+, x J,..., x, x 0 ) A M (X M ) J+ ( + J+ J C J q T,0 (, x J+, x 0 ) C J ( + x J+ + x 0 ) m. j=0 J j= x j ) m. Remark x αb(t, x) C α F [γ] = e T B(t,γ(t))dt 0 J+ T j Tj j= T B(t,γ Tj,T F T,0 = e j j (t,x j,x j ))dt = J+ j= e T B(t,γ Tj,T j j (t,x j,x j ))dt. dx j
Proof of Lemma KT D. Fujiwara, N. Kumanogo, K. Taniguchi, Funkcial. Ekvac. 40 (997) m = 0 ω j (x j, x j ) 2πi t j+ ( = 2πi T j+ S T,0 (x J+, x J,..., x, x 0 ) = J+ j= (x j x j ) 2 2t j t j 0 (/t j ) ( ) d/2 (x i j+ x j ) 2 ( ) 2t d/2 (x i j x 0 ) 2 e j+ 2T e j 2πi T j ) d/2 (x i j+ x 0 ) 2 ( 2T e j+ 2π ) d e i S (x,x 0 ) i (x x 0 ) 2 T,0 2T q T,0 (, x, x 0 ) = R d e ( J Φ = (x j T j x j+ t j+ x 0 )ξ j T j+ T j+ j= 2π J+ j= ω j (x j, x j ). i ( t j+ T j ξ 2 2T j+ j +(x j T j x T j+ t j+ x j+ T 0 )ξ j ) j+ dξj J j= ) dj R 2dJ e i Φ F T,0 t j+ T j 2T j+ ξ 2 j J+ j= J j= dx j dξ j, ω j (x j, x j ).
M j = i( ξ j Φ) ξj + ξj Φ, N 2 j = i( x j Φ) xj, j =, 2,..., J + xj Φ 2 M j e i Φ = e i Φ, N j e i Φ = e i Φ. ( ) dj J ( ) e i dj Φ F T,0 dx j dξ j = e i Φ F 2π R 2dJ 2π R 2dJ T,0 j= J j= dx j dξ j, F T,0 = (N J )d+ (N 2 )d+ (N )d+ (M J )d+ (M 2 )d+ (M )d+ F T,0 M j, N j M j, N j
J J C J ξj Φ = x j T j x j+ t j+ x 0 t j+t j ξ j, T j+ T j+ T j+ M j = a j (x j+, ξ j, x j, x 0 ) ξj + a 0 j (x j+, ξ j, x j, x 0 ), J 4d xj+, ξj, xj M j N j+, M j, N j M j F T,0 J C J M j = C ( + ξj Φ 2 ) /2, N j = C ( + xj Φ 2 ) /2.
z j = ξj Φ, ζ j = xj Φ, j =, 2,..., J J F T,0 (C ) J ( + j= z j 2 ) (d+)/2 ( + ζ j 2 ), (d+)/2 det (x J,..., x, ξ J,..., ξ ) (z J,..., z, ζ J,..., ζ ) (C ) J. (x J,..., x, ξ J,..., ξ ) (z J,..., z, ζ J,..., ζ ) ( ) dj J e i Φ F 2π R 2dJ T,0 dx j dξ j j= ( ) dj = e i Φ F 2π R 2dJ T,0 det (x J,..., x, ξ J,..., ξ ) J (z J,..., z, ζ J,..., ζ ) dz j dζ j. (z J,..., z, ζ J,..., ζ ) J C J j=
2. Fujiwara D. Fujiwara, Nagoya Math. J. 24 (99). γ T,0 (T j, x j ) (T j, x j ) S[γ T,0 ] = S T,0 (x J+, x J,..., x, x 0 ), F [γ T,0 ] = F T,0 (x J+, x J,..., x, x 0 ). (T J, x J ) (T 3, x 3 ) (T, x) (T, x ) (0, x 0 ) (T 2, x 2 ) T 0 = 0 T T 2 T 3 T J T = T J+ Remark γ T,0 (T, x) (0, x 0 ) S[γ T,0 ] = S T,0 (x, x 0 ), F [γ T,0 ] = F T,0 (x, x 0 ).
( (xj,...,x )S T,0 )(x J+, x J,..., x, x 0) = 0 (x J,..., x ) J+ j= D T,0 (x J+, x 0 ) = t d j det( 2 (x J,...,x ) S T,0 )(x J+, x J,..., x, x 0) T J+ J+ ( ) d/2 J e i S T,0 (x J+,x J,...,x,x 0 ) F T,0 (x J+, x J,..., x, x 0 ) dx j 2πi t j= j R dj j= ( ) d/2 ( ) = e i S T,0(x,x 0 ) D T,0 (x, x 0 ) /2 F T,0 (x, x 0 ) + Υ T,0 (, x, x 0 ). 2πi T Lemma F m 0 M A M, X M T,0 α j M, j = 0,,..., J, J + J+ ( j=0 α j x j )F T,0 (x J+, x J,..., x, x 0 ) A M (X M ) J+ ( + J+ j=0 x j ) m. J C Υ T,0 (, x, x 0 ) CT ( + x J+ + x 0 ) m.
Remark J = 0,, 2 Lemma F T = T > T 0 = 0 J = 0 α x α 0 x 0 F T,0 (x, x 0 ) A M (X M ) ( + x + x 0 ) m. T = T 2 > T > T 0 = 0 J = α 2 x 2 α x α 0 x 0 F T,T,0(x 2, x, x 0 ) A M (X M ) 2 ( + x 2 + x + x 0 ) m. T = T 3 > T 2 > T > T 0 = 0 J = 2 α 3 x 3 α 2 x 2 α x α 0 x 0 F T,T2,T,0(x 3, x 2, x, x 0 ) A M (X M ) 3 ( + x 3 + x 2 + x + x 0 ) m. Remark x αb(t, x) C α F [γ] = e T B(t,γ(t))dt 0 F T,0 = F [γ T,0 ] = Tj J+ j= e T B(t,γ Tj,T j j (t,x j,x j ))dt.
Proof of Lemma F m = 0 x, x 2,..., x J x (M F T,0 ) (R F T,0 ) ( ) d/2 ( ) d/2 e i S T,0 F T,0 (..., x 2, x, x 0 )dx 2πi t 2 2πi t R ( ) d d/2 = e i S ( T,T2,0) (M F 2πi T )(..., x T,0 2, x 0 ) 2 ( ) d/2 + e i S ( T,T2,0) (R F 2πi T )(..., x T,0 2, x 0 ). 2
(M F T,0 ) x 2 (M F T,0 )(..., x 2, x 0 ) = D T2,T,0(x 2, x 0 ) /2 F T,0 (..., x 2, x, x 0) = D (x T2,0 2, x 0 ) /2 F ( T,T2,0)(..., x 2, x 0 ), ( T,T2, 0) : T = T J+ > T J > > T 2 > T 0 = 0 (R F T,0 ) x 2 (t 2 ) (R F T,0 ) C(t 2 ). (T, x) (0, x 0 ) (T 2, x 2 ) T 0 = 0 T T 2 T = T J+
(M F T,0 ) x 2 x 2 ( ) d/2 ( ) d/2 e i S ( T,T2,0) (M F T,0 )(..., x 3, x 2, x 0 )dx 2 2πi t 3 2πi T 2 R ( ) d d/2 = e i S ( T,T3,0) (M 2 M F 2πi T )(..., x T,0 3, x 0 ) 3 ( ) d/2 i + e S ( T,T3,0) (R 2 M F T,0 )(..., x 3, x 0 ). 2πi T 3
(M 2 M F T,0 ) x 3 (M 2 M F T,0 )(..., x 3, x 0 ) = D T3,T 2,0(x 3, x 0 ) /2 D T2,0 (x 2, x 0) /2 F ( T,T2,0)(..., x 3, x 2, x 0). = D (x T3,0 3, x 0 ) /2 F ( T,T3,0)(..., x 3, x 0 ), ( T,T3, 0) : T = T J+ > T J > > T 3 > T 0 = 0 (R 2 M F T,0 ) x 3 (t 3 ) (R 2 M F T,0 ) C(t 3 ). (T 3, x 3 ) (T, x) (0, x 0 ) T 0 = 0 T 2 T 3 T = T J+
(M 2 M F T,0 ) x 3 x 3 Theorem 7 (M J M J... M F T,0 ) = D T,0 (x, x 0 ) /2 F T,0 (x, x 0 ) = D T,0 (x, x 0 ) /2 F [γ T,0 ]. (T, x) (0, x 0 ) γ T,0 T 0 = 0 T = T J+
(R F T,0 ) x 2 x 2 x 3 ( ) d/2( ) d/2( ) d/2 e i S ( T,T2,0) (R F T,0 )(..., x 4, x 3, x 2, x 0 )dx 3 2πi t 4 2πi t 3 2πi T 2 R ( ) d d/2 ( ) d/2 = e i S ( T,T4,T 2,0) (M 3 R F 2πi (t 4 + t 3 ) 2πi T )(..., x T,0 4, x 2, x 0 ) 2 ( ) d/2 ( ) d/2 + e i S ( T,T4,T 2,0) (R 3 R F 2πi (t 4 + t 3 ) 2πi T )(..., x T,0 4, x 2, x 0 ). 2 (R 3 R F T,0 ) C(t 4 )C(t 2 ). (M 3 R F T,0 ) x 4 x 4 (R 3 R F T,0 ) x 4 x 4 x 5
Fujiwara x j x j+ x j+ x j+ x j+ x j+2
q T,0 (, x J+, x 0 ) = q 0 (x J+, x 0 ) + qjk,jk,...,j (x J+, x 0 ). q 0 (x J+, x 0 ) = D T,0 (x, x 0 ) /2 F T,0 (x, x 0 ) Theorem 7 0 = j 0 < j < j < j 2 < j 2 < < j K < j K J +, (j K, j K,..., j ) q jk,j K,...,j (x J+, x 0 ) ( ) d/2 e i S T,0(x J+,x 0 ) q jk,j 2πi T K,...,j (x J+, x 0 ) = K+ k= RdK e i S (T,T jk,...,t j,0) b jk,j K,...,j (x J+, x jk,..., x j, x 0 ) ( 2πi (T jk T jk ) K dx jk, k= ) d/2
b jk,j K,...,j (x J+, x jk,..., x j, x 0 ) = (Q J Q 3 Q 2 Q F T,0 )(x J+, x jk,..., x j, x 0 ). Q j = Identity if j = j K, j K,..., j R j if j = j K, j K,..., j M j otherwise (t jk ) b jk,j K,...,j (x J+, x jk,..., x j, x 0 ) C K ( K k= ) ( t jk ).
H. Kumano-go-Taniguchi q jk,j K,...,j (x J+, x 0 ) (C ) K ( K q jk,j K,...,j (x J+, x 0 ) k= ) ( t jk ). Υ T,0 (, x J+, x 0 ) = qjk,jk,...,j (x J+, x 0 ) J+ t j = T, 0 < < j= Υ T,0 (, x J+, x 0 ) K ) ((C ) K ( t jk ) ( J+ k= ) ( + C t j ) j= (C )T
3. m 0, u j 0, J+ j= u j U < M A M, X M T,0, α j M, j = 0,,..., J + k J J+ () ( j=0 α j x j )F T,0 (x J+, x J,..., x, x 0 ) A M (X M ) J+ ( + J+ (2) ( α j x j ) xk F T,0 (x J+, x J,..., x, x 0 ) j=0 A M (X M ) J+ (u k+ + u k )( + J+ j=0 x j ) m. J+ j=0 x j ) m, Remark x αb(t, x) C α F [γ] = e T B(t,γ(t))dt 0 xk F T,0 = Tj J+ j= e T B(t,γ Tj,T j j (t,x j,x j ))dt ( T k+ T k xk B(t, γ Tk+,T k (x k+, x k ))dt + T k T k xk B(t, γ Tk,T k (x k, x k ))dt).
J+ ( ) d/2 J ( ) e i S[γ T,0 ] F [γ T,0 ] dx j 2πi t j= j R dj j= ( ) d/2 = e i S T,0(x,x 0 ) q T,0 (, x, x 0 ) 2πi T ( ) d/2 ( ) = e i S T,0(x,x 0 ) D T,0 (x, x 0 ) /2 F T,0 (x, x 0 ) + Υ T,0 (, x, x 0 ) 2πi T Lemma F. Υ T,0 (, x, x 0 ) CT (T + U)( + x + x 0 ) m, q T,0 (, x, x 0 ) C ( + x + x 0 ) m. Lemma F m = 0
(T, x) (0, x 0 ) T 0 = 0 T = T J+ (T, x) (0, x 0 ) T 0 = 0 T = T J+ CT (T + U)
(T, x) (0, x 0 ) T 0 = 0 T = T J+ C T,0 ( ) Cauchy T,0 : T = T J+ > T J > > T > T 0 = 0, ( T,TN+, Tn,0) : T = T J+ > > T N+ > T n > > T 0 = 0,
J+ j= = ( ) d/2 e i S[γ T,0 ] F [γ T,0 ] 2πi t j ( ) d/2 e i S T,0(x,x 0 ) q T,0 (, x, x 0 ), 2πi T J j= dx j J+ ( j=n+2 = ( 2πi T 2πi t j ) d/2 ( 2πi (T N+ T n ) ) n j= ( 2πi t j ) d/2 i e S[γ ( T,TN+, Tn,0 )] F [γ ( T,TN+, Tn,0)] ) d/2 e i S T,0(x,x 0 ) q ( T,TN+, Tn,0)(, x, x 0 ). J j=n+ n dx j j= dx j
(T n, x n ) (T, x) (0, x 0 ) (T N+, x N+ ) T 0 = 0 T n T N+ T = T J+ (T n, x n ) (T, x) (0, x 0 ) (T N+, x N+ ) T 0 = 0 T n T N+ T = T J+ C C(T N+ T N )(T N+ T n + U N+ U n ) C
q T,0 (, x, x 0 ) q ( T,TN+, Tn,0)(, x, x 0 ) C(T N+ T n )(T N+ T n + U N+ U n )( + x + x 0 ) m. Theorem 2 Feynman q(t,, x, x 0 ) q T,0 (, x, x 0 ) q(t,, x, x 0 ) C T,0 (U + T )( + x + x 0 ) m, ( ) T,0 0 R 2d
( Fujiwara ) Fujiwara D. Fujiwara and N. Kumano-go, J. Math. Soc. Japan Vol 58, No. 3 (2006). Lemma γ T,T2,T,0 (0, x 0 ), (T, x ), (T 2, x 2 ), (T, x) ( ) q(t ) q(t, x, x 0 ) lim x D(T, x, x 0 ) /2 x F [γ 2 =γ T,T2,T T2,0] cl (T 2 ) T (x, x 0 ) R 2d x =γ cl (T ). (T, x) x (0, x 0 ) x 2 γ cl T 0 = 0 T T 2 T = T J+
Theorem 8 Lemma q(t) ( ) e i d/2 S[γ] F [γ]d[γ] = e i S[γcl ] D(T, x, x 0 ) /2 2πi T ( F [γ cl ] + i T ) D(t, γ cl (t), x 0 ) /2 q(t)dt + 2 Υ (, x, x 0 ) 2 0 Υ (, x, x 0 ) C( + x + x 0 ) m. Remark F [γ] i 2 T 0 D(t, γ cl (t), x 0 ) /2 ( y D(t, y, x 0 ) /2 ) y=γ cl (t) dt Schrödinger Birkhoff