A Feynman path integral-like method of quantization on
Riemannian manifolds and related problems
Yoshihisa Miyanishi
The graduate school of Engineering and Science
Osaka University

March 8, 2015

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and1 related/ 27

§1 Introduction. Feynman path integrals (heuristics)§1.1 Classical Mechanics

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and2 related/ 27

§1 Introduction. Feynman path integrals (heuristics)§1.1 Classical Mechanics(Hamiltonian formulation){nH(x, p) = 12 |p|2 + V (x) ∈ C ∞ (Rnx × Rp ) : Hamiltonian
dx
= ∂H
, dp
= − ∂H: Hamitlonian flow
dt
∂p
dt
∂x

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and2 related/ 27

§1 Introduction. Feynman path integrals (heuristics)§1.1 Classical Mechanics(Hamiltonian formulation){nH(x, p) = 12 |p|2 + V (x) ∈ C ∞ (Rnx × Rp ) : Hamiltonian
dx
= ∂H
, dp
= − ∂H: Hamitlonian flow
dt
∂p
dt
∂x
Legendre transform gives(Lagrangian formulation)
For X(τ ) ∈ Rn and Ẋ = dXdτ
{ ()
L X, Ẋ = 12 Ẋ 2 − V (x) : Lagrangian
d ∂L
∂L
− ∂X
= 0 : Euler-Lagrange eqdτ ∂ Ẋ

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and2 related/ 27

§1 Introduction Feynman path integrals (heuristics)
space xX ′′′X′

Action integralS(X)∫(t, x) = st 12 L(X, Ẋ)dτ

X̄
X̂

X ′′

(s, y)
O

X̄(·) classical pathX̂(·) linear pathttime

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and3 related/ 27

§1 Introduction Feynman path integrals (heuristics)
space xX ′′′X′

Action integralS(X)∫(t, x) = st 12 L(X, Ẋ)dτ
R.Feynman proposed
the quantization is given by∫i
S(0,t,x,y)hf (y)D[X]
Ωe

X̄
X̂

X ′′

−it

= e h Ĥ f (x)(s, y)
O

X̄(·) classical pathX̂(·) linear pathttime

Ω is the path spaceconnecting (0, y) and (t, x).
D[X] is the Lebesgue-likemeasure on Ω

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and3 related/ 27

§1 Introduction Feynman path integrals (heuristics)
space xX ′′′X′

Action integralS(X)∫(t, x) = st 12 L(X, Ẋ)dτ
R.Feynman proposed
the quantization is given by∫i
S(0,t,x,y)hf (y)D[X]
Ωe

X̄
X̂

X ′′

−it

= e h Ĥ f (x)(s, y)O

Remark.

X̄(·) classical pathX̂(·) linear pathttime

Ω is the path spaceconnecting (0, y) and (t, x).
D[X] is the Lebesgue-likemeasure on Ω

We can not construct Feynman path measure (Cameron)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and3 related/ 27§.2 Time slicing approximations (An alternative method of F.P.I.)
An alternative method of F.P.I. (rough sketch)
H(x, p) ∈ C ∞ (T ∗ Rn )Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and4 related/ 27§.2 Time slicing approximations (An alternative method of F.P.I.)
An alternative method of F.P.I. (rough sketch)
H(x, p) ∈ C ∞ (T ∗ Rn )Strong Assumption (Time locally unique path condition)
For small t − s and for x, y ∈ Rn ,Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and4 related/ 27§.2 Time slicing approximations (An alternative method of F.P.I.)
An alternative method of F.P.I. (rough sketch)
H(x, p) ∈ C ∞ (T ∗ Rn )Strong Assumption (Time locally unique path condition)
For small t − s{and for x, y ∈ Rn ,X̄(τ ) satisfies E-L eq.
∃1 X̄(τ ) s.t.
X̄(s) = x and X̄(t) = y

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and4 related/ 27§.2 Time slicing approximations (An alternative method of F.P.I.)
An alternative method of F.P.I. (rough sketch)
H(x, p) ∈ C ∞ (T ∗ Rn )Strong Assumption (Time locally unique path condition)
For small t − s{and for x, y ∈ Rn ,X̄(τ ) satisfies E-L eq.
∃1 X̄(τ ) s.t.
X̄(s) = x and X̄(t) = yS(s, t, x, y) =∫ts L(X̄(τ ), X̄(τ )) : phase function defined by H.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and4 related/ 27§.2 Time slicing approximations (An alternative method of F.P.I.)
An alternative method of F.P.I. (rough sketch)
H(x, p) ∈ C ∞ (T ∗ Rn )Strong Assumption (Time locally unique path condition)
For small t − s{and for x, y ∈ Rn ,X̄(τ ) satisfies E-L eq.
∃1 X̄(τ ) s.t.
X̄(s) = x and X̄(t) = yS(s, t, x, y) =∫tdefined by H.
s L((X̄(τ ), X̄(τ )) : phase function)
2
D(s, t, x, y) = det ∂ S(s, t, x, y)/∂x∂y : van-Vleck determinant

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and4 related/ 27§.2 Time slicing approximations (An alternative method of F.P.I.)
An alternative method of F.P.I. (rough sketch)
H(x, p) ∈ C ∞ (T ∗ Rn )Strong Assumption (Time locally unique path condition)
For small t − s{and for x, y ∈ Rn ,X̄(τ ) satisfies E-L eq.
∃1 X̄(τ ) s.t.
X̄(s) = x and X̄(t) = yS(s, t, x, y) =∫tdefined by H.
s L((X̄(τ ), X̄(τ )) : phase function)
2
D(s, t, x, y) = det ∂ S(s, t, x, y)/∂x∂y : van-Vleck determinant∆ : 0 = t0 < t1 < · · · < tn−1 < tn = t.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and4 related/ 27§.2 Time slicing approximations (An alternative method of F.P.I.)
An alternative method of F.P.I. (rough sketch)
H(x, p) ∈ C ∞ (T ∗ Rn )Strong Assumption (Time locally unique path condition)
For small t − s{and for x, y ∈ Rn ,X̄(τ ) satisfies E-L eq.
∃1 X̄(τ ) s.t.
X̄(s) = x and X̄(t) = yS(s, t, x, y) =∫tdefined by H.
s L((X̄(τ ), X̄(τ )) : phase function)
2
D(s, t, x, y) = det ∂ S(s, t, x, y)/∂x∂y : van-Vleck determinant∆ : 0 = t0 < t1 < · · · < tn−1 < tn = t.
∫
U (t − s)f (x) ≡(2πi)−n/2 D(s, t, x, y) exp{ hi S(s, t, x, y)}f (y)dy
Rn

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and4 related/ 27§.2 Time slicing approximations (An alternative method of F.P.I.)
An alternative method of F.P.I. (rough sketch)
H(x, p) ∈ C ∞ (T ∗ Rn )Strong Assumption (Time locally unique path condition)
For small t − s{and for x, y ∈ Rn ,X̄(τ ) satisfies E-L eq.
∃1 X̄(τ ) s.t.
X̄(s) = x and X̄(t) = yS(s, t, x, y) =∫tdefined by H.
s L((X̄(τ ), X̄(τ )) : phase function)
2
D(s, t, x, y) = det ∂ S(s, t, x, y)/∂x∂y : van-Vleck determinant∆ : 0 = t0 < t1 < · · · < tn−1 < tn = t.
∫
U (t − s)f (x) ≡(2πi)−n/2 D(s, t, x, y) exp{ hi S(s, t, x, y)}f (y)dy
n
R
(
)
∏
−it
Ĥ
lim
[U (ti − ti−1 )] = exp
|∆|→0
h
i

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and4 related/ 27§.2 Time slicing approximations (An alternative method of F.P.I.)
An alternative method of F.P.I. (rough sketch)
H(x, p) ∈ C ∞ (T ∗ Rn )Strong Assumption (Time locally unique path condition)
For small t − s{and for x, y ∈ Rn ,X̄(τ ) satisfies E-L eq.
∃1 X̄(τ ) s.t.
X̄(s) = x and X̄(t) = yS(s, t, x, y) =∫tdefined by H.
s L((X̄(τ ), X̄(τ )) : phase function)
2
D(s, t, x, y) = det ∂ S(s, t, x, y)/∂x∂y : van-Vleck determinant∆ : 0 = t0 < t1 < · · · < tn−1 < tn = t.
∫
U (t − s)f (x) ≡(2πi)−n/2 D(s, t, x, y) exp{ hi S(s, t, x, y)}f (y)dy
n
R
(
)
∏
−it
Ĥ
lim
[U (ti − ti−1 )] = exp
|∆|→0
h
i
Problem1.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and4 related/ 27§.2 Time slicing approximations (An alternative method of F.P.I.)
An alternative method of F.P.I. (rough sketch)
H(x, p) ∈ C ∞ (T ∗ Rn )Strong Assumption (Time locally unique path condition)
For small t − s{and for x, y ∈ Rn ,X̄(τ ) satisfies E-L eq.
∃1 X̄(τ ) s.t.
X̄(s) = x and X̄(t) = yS(s, t, x, y) =∫tdefined by H.
s L((X̄(τ ), X̄(τ )) : phase function)
2
D(s, t, x, y) = det ∂ S(s, t, x, y)/∂x∂y : van-Vleck determinant∆ : 0 = t0 < t1 < · · · < tn−1 < tn = t.
∫
U (t − s)f (x) ≡(2πi)−n/2 D(s, t, x, y) exp{ hi S(s, t, x, y)}f (y)dy
n
R
(
)
∏
−it
Ĥ
lim
[U (ti − ti−1 )] = exp
|∆|→0
h
i
∂
) ∈ L(L2 (Rn ))?
Problem1. Ĥ = H(x, 1i ∂x

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and4 related/ 27§.2 Time slicing approximations (Summary. Euclidean case)xX̄(t2 , x2 )(t, x)(t3 , x3 )(t1 , x1 )(0, y)
Ot
X̄(·) piecewise classical path

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and5 related/ 27§.2 Time slicing approximations (Summary. Euclidean case)xX̄(t2 , x2 )∫t˙S(s, t, x, y) ≡ s L(X̄, X̄)dτ(t, x) The action S is integrals over(t3 , x3 )piecewise classical paths(t1 , x1 )(0, y)
Ot
X̄(·) piecewise classical path

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and5 related/ 27§.2 Time slicing approximations (Summary. Euclidean case)xX̄(t2 , x2 )∫t˙S(s, t, x, y) ≡ s L(X̄, X̄)dτ(t, x) The action S is integrals over(t3 , x3 )piecewise classical paths
By using the density of paths a(t1 , x1 )(0, y)
Ot
X̄(·) piecewise classical path

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and5 related/ 27§.2 Time slicing approximations (Summary. Euclidean case)xX̄(t2 , x2 )∫t˙S(s, t, x, y) ≡ s L(X̄, X̄)dτ(t, x) The action S is integrals over(t3 , x3 )piecewise classical paths(t1 , x1 )By using the density of paths a∫
M a(tj , tj+1 , xj , xj+1 )ie h S(tj ,tj+1 ,xj ,xj+1 ) f (xj )dxj
= U (tj+1 − tj )f (xj+1 )(small time evolution op.)(0, y)
Ot
X̄(·) piecewise classical path

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and5 related/ 27§.2 Time slicing approximations (Summary. Euclidean case)xX̄(t2 , x2 )(t1 , x1 )∫t
˙S(s, t, x, y) ≡ s L(X̄, X̄)dτ(t, x) The action S is integrals over(t3 , x3 )piecewise classical paths
By using the density of paths a∫
M a(tj , tj+1 , xj , xj+1 )i(0, y)

e h S(tj ,tj+1 ,xj ,xj+1 ) f (xj )dxj
= U (tj+1 − tj )f (xj+1 )(small time evolution op.)Time slicing approximations
are defined bytX̄(·) piecewise classical path
−1
N∏
−1
∫
∫ N∏
i
a(tj , tj+1 , xj , xj+1 )e h S(tj ,tj+1 ,xj ,xj+1 ) f (y)dxj
Rn · · · Rn
j=0
j=0
N∏
−1
∫
i
=[
U (tj+1 − tj )]f (x) → Ω e h S(0,t,x,y) f (y)D[X] (N → ∞).
O

j=0

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and5 related/ 27

Known results (Fujiwara Theory) (Euclidean case)
Assumption V (x) ∈ C ∞ (Rn ), |∂ α V (x)| < Cα for |α| ≧ 2.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and6 related/ 27

Known results (Fujiwara Theory) (Euclidean case)
Assumption V (x) ∈ C ∞ (Rn ), |∂ α V (x)| < Cα for |α| ≧ 2.
1. S(t, x, y) =∫t 1 ˙2
0 [ 2 X̄(τ ) − V (X̄(τ ))]dτ(The classical path connecting (0, y) and (t, x) is time locally unique. )Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and6 related/ 27

Known results (Fujiwara Theory) (Euclidean case)
Assumption V (x) ∈ C ∞ (Rn ), |∂ α V (x)| < Cα for |α| ≧ 2.
1. S(t, x, y) =∫t 1 ˙2
0 [ 2 X̄(τ ) − V (X̄(τ ))]dτ(The classical path connecting (0, y) and (t, x) is time locally unique. )2. D(t, x, y) = det(∂ 2 S(t, x, y)/∂x∂y)(van Vleck determinant)a(t, x, y) = (2πih)−n/2 D(t, x, y)1/2 .

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and6 related/ 27

Known results (Fujiwara Theory) (Euclidean case)
Assumption V (x) ∈ C ∞ (Rn ), |∂ α V (x)| < Cα for |α| ≧ 2.
1. S(t, x, y) =∫t 1 ˙2
0 [ 2 X̄(τ ) − V (X̄(τ ))]dτ(The classical path connecting (0, y) and (t, x) is time locally unique. )2. D(t, x, y) = det(∂ 2 S(t, x, y)/∂x∂y)(van Vleck determinant)a(t, x, y) = (2πih)−n/2 D(t, x, y)1/2 .
3. U (t)f (x) ≡∫Rn a(t, x, y)eiS(t,x,y)hf (y)dy.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and6 related/ 27

Known results (Fujiwara Theory) (Euclidean case)
Assumption V (x) ∈ C ∞ (Rn ), |∂ α V (x)| < Cα for |α| ≧ 2.
1. S(t, x, y) =∫t 1 ˙2
0 [ 2 X̄(τ ) − V (X̄(τ ))]dτ(The classical path connecting (0, y) and (t, x) is time locally unique. )2. D(t, x, y) = det(∂ 2 S(t, x, y)/∂x∂y)(van Vleck determinant)a(t, x, y) = (2πih)−n/2 D(t, x, y)1/2 .
3. U (t)f (x) ≡∫Rn a(t, x, y)eiS(t,x,y)hf (y)dy.

Theorem (D. Fujiwara)
For t ̸= 0,∏
2[U (ti − ti−1 )] = exp −it[− h2 △ + V (x)] (Operator norm)limh|∆|→0 i

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and6 related/ 27

Other diﬀerent alternative definitions of Feynman path
integrals
Other alternative methods for path integrals.
1. Trotter Kato forumulas.
2. Analytic continuation of Wiener measureby using complex Planch constant h, m or t
3. An improper integral on Hilbert spaces.
（K.Ito, Albeverio ）4. Non-standard analysis (*measure of the Dirac operator
and take the limit c → ∞etc.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and7 related/ 27

Other diﬀerent alternative definitions of Feynman path
integrals
Other alternative methods for path integrals.
1. Trotter Kato forumulas.
2. Analytic continuation of Wiener measureby using complex Planch constant h, m or t
3. An improper integral on Hilbert spaces.
（K.Ito, Albeverio ）4. Non-standard analysis (*measure of the Dirac operator
and take the limit c → ∞etc.
Here, we employ the time slicing products.
to derive the curvature from action integrals.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and7 related/ 27

§3 Path integral-like method on Riemannian manifold
Settings.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and8 related/ 27

§3 Path integral-like method on Riemannian manifold
Settings.
1. (M, g) : n-dim C ∞ oriented complete Riemannian manifold

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and8 related/ 27

§3 Path integral-like method on Riemannian manifold
Settings.
1. (M, g) : n-dim C ∞ oriented complete Riemannian manifold
2. H(x, p) = 12 |p|2

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and8 related/ 27

§3 Path integral-like method on Riemannian manifold
Settings.
1. (M, g) : n-dim C ∞ oriented complete Riemannian manifold
2. H(x, p) = 12 |p|2 = 12 g ij pi pj(on local charts)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and8 related/ 27

§3 Path integral-like method on Riemannian manifold
Settings.
1. (M, g) : n-dim C ∞ oriented complete Riemannian manifold
2. H(x, p) = 12 |p|2 = 12 g ij pi pj(on local charts)

Hamiltonian gives the geodesic flow on T ∗ M (various speed)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and8 related/ 27

§3 Path integral-like method on Riemannian manifold
Settings.
1. (M, g) : n-dim C ∞ oriented complete Riemannian manifold
2. H(x, p) = 12 |p|2 = 12 g ij pi pj(on local charts)

Hamiltonian gives the geodesic flow on T ∗ M (various speed)
3. d = d(x, y)(geodesic distance)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and8 related/ 27

§3 Path integral-like method on Riemannian manifold
Settings.
1. (M, g) : n-dim C ∞ oriented complete Riemannian manifold
2. H(x, p) = 12 |p|2 = 12 g ij pi pj(on local charts)

Hamiltonian gives the geodesic flow on T ∗ M (various speed)
3. d = d(x, y)(geodesic distance)∫t 1
2
4. S(t, x, y) = 0 2 gx(t) (ẋ(t), ẋ(t)) dt = |d(x,y)|2t(The action integral over the shortest path)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and8 related/ 27

§3 Path integral-like method on Riemannian manifold
Settings.
1. (M, g) : n-dim C ∞ oriented complete Riemannian manifold
2. H(x, p) = 12 |p|2 = 12 g ij pi pj(on local charts)

Hamiltonian gives the geodesic flow on T ∗ M (various speed)
3. d = d(x, y)(geodesic distance)∫t 1
2
4. S(t, x, y) = 0 2 gx(t) (ẋ(t), ẋ(t)) dt = |d(x,y)|2t(The action integral over the shortest path)
5. van Vleck determinants on manifolds
D(t, x, y) = G−1/2 (x)G−1/2 (y) det(∂ 2 S(t, x, y)/∂x∂y)χ(d(x, y)) : cut oﬀ(bump ft. with compact support contained in d(x, y) < injrad(M ) )a(t, x, y) = χ(d(x, y))D(t, x, y)1/2

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and8 related/ 27

§3 Path integral-like method on Riemannian manifold
Settings.
1. (M, g) : n-dim C ∞ oriented complete Riemannian manifold
2. H(x, p) = 12 |p|2 = 12 g ij pi pj(on local charts)

Hamiltonian gives the geodesic flow on T ∗ M (various speed)
3. d = d(x, y)(geodesic distance)∫t 1
2
4. S(t, x, y) = 0 2 gx(t) (ẋ(t), ẋ(t)) dt = |d(x,y)|2t(The action integral over the shortest path)
5. van Vleck determinants on manifolds
D(t, x, y) = G−1/2 (x)G−1/2 (y) det(∂ 2 S(t, x, y)/∂x∂y)χ(d(x, y)) : cut oﬀ(bump ft. with compact support contained in d(x, y) < injrad(M ) )a(t, x, y) = χ(d(x, y))D(t, x, y)1/2

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and8 related/ 27

§3 Path integral-like method on Riemannian manifold
Definition (Shortest path approximations on manifolds)
U (t)f (x) ≡ (2πi)−n/2∫

M a(t, x, y)eiS(t, x, y) f (y) dy(Remark. For the simplicity, let h = 1.)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and9 related/ 27

§3 Path integral-like method on Riemannian manifold
Definition (Shortest path approximations on manifolds)
U (t)f (x) ≡ (2πi)−n/2∫

M a(t, x, y)eiS(t, x, y) f (y) dy(Remark. For the simplicity, let h = 1.)Md(x, y)yxMyxd(x, y)We don’t consider d(x, y) ≧ injrad(M )Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and9 related/ 27

§3 Path integral-like method on Riemannian manifold
Definition (Shortest path approximations on manifolds)
U (t)f (x) ≡ (2πi)−n/2∫

M a(t, x, y)eiS(t, x, y) f (y) dy(Remark. For the simplicity, let h = 1.)Md(x, y)yxMyxd(x, y)We don’t consider d(x, y) ≧ injrad(M )Problem. lim [U (t/N )]N f (x) = exp(−itĤ)f (x) ?
N →∞What is the Ĥ ?

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015 and9 related/ 27

§4 Asymptotically conic non-trapping scattering case
Assumtions for scattering case

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and10 related/ 27

§4 Asymptotically conic non-trapping scattering case
Assumtions for scattering case(A1) ∂M is n − 1 dimensional smooth Riemmanian manifold,Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and10 related/ 27

§4 Asymptotically conic non-trapping scattering case
Assumtions for scattering case(A1) ∂M is n − 1 dimensional smooth Riemmanian manifold,with metric h = hjk dy i dy k

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and10 related/ 27

§4 Asymptotically conic non-trapping scattering case
Assumtions for scattering case(A1) ∂M is n − 1 dimensional smooth Riemmanian manifold,with metric h = hjk dy i dy k(A2) ∃ compact set K ⊂ M s.t.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and10 related/ 27

§4 Asymptotically conic non-trapping scattering case
Assumtions for scattering case(A1) ∂M is n − 1 dimensional smooth Riemmanian manifold,with metric h = hjk dy i dy k(A2) ∃ compact set K ⊂ M s.t.
2

g = dx+x4

hjk (x,y)dy i dy k
= dr 2 + r 2 hjk ( 1r , y)dy j dy k
x2

on the asymptotic region
M \K = (0, ϵ) × ∂M = {(x, y) : 0 < x < ϵ, y ∈ ∂M }Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and10 related/ 27

§4 Asymptotically conic non-trapping scattering case
Assumtions for scattering case(A1) ∂M is n − 1 dimensional smooth Riemmanian manifold,with metric h = hjk dy i dy k(A2) ∃ compact set K ⊂ M s.t.
2

g = dx+x4

hjk (x,y)dy i dy k
= dr 2 + r 2 hjk ( 1r , y)dy j dy k
x2

on the asymptotic region
M \K = (0, ϵ) × ∂M = {(x, y) : 0 < x < ϵ, y ∈ ∂M }Example(The radial compactification map)Rn(z, 1)M

RC(z)RC : (z, 1) ∈ Rn
→ (√ z 2 , √ 1
1+|z|1+|z|2n) ∈ S+∂M

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and10 related/ 27

§4 Asymptotically conic non-trapping case(A3)(Non-trapping) For all compact set K ⊂ int(M ), ∃T > 0 s.t.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and11 related/ 27

§4 Asymptotically conic non-trapping case(A3)(Non-trapping) For all compact set K ⊂ int(M ), ∃T > 0 s.t.
π{exp tXH (x, p)} ∩ K = ∅ for ∀|t| > T(i.e. Every geodesic γ : R → M reaches ∂M at ±∞. )(A4) The scalar curvature R(x) on M is bounded.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and11 related/ 27

§4 Asymptotically conic non-trapping case(A3)(Non-trapping) For all compact set K ⊂ int(M ), ∃T > 0 s.t.
π{exp tXH (x, p)} ∩ K = ∅ for ∀|t| > T(i.e. Every geodesic γ : R → M reaches ∂M at ±∞. )(A4) The scalar curvature R(x) on M is bounded.

Lemma (The action integral S(t, x, y) is well defined locally)
Under (A1)∼(A4),
injrad(M ) > 0.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and11 related/ 27

§4 Asymptotically conic non-trapping case(A3)(Non-trapping) For all compact set K ⊂ int(M ), ∃T > 0 s.t.
π{exp tXH (x, p)} ∩ K = ∅ for ∀|t| > T(i.e. Every geodesic γ : R → M reaches ∂M at ±∞. )(A4) The scalar curvature R(x) on M is bounded.

Lemma (The action integral S(t, x, y) is well defined locally)
Under (A1)∼(A4),
injrad(M ) > 0.
(Notation)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and11 related/ 27

§4 Asymptotically conic non-trapping case(A3)(Non-trapping) For all compact set K ⊂ int(M ), ∃T > 0 s.t.
π{exp tXH (x, p)} ∩ K = ∅ for ∀|t| > T(i.e. Every geodesic γ : R → M reaches ∂M at ±∞. )(A4) The scalar curvature R(x) on M is bounded.

Lemma (The action integral S(t, x, y) is well defined locally)
Under (A1)∼(A4),
injrad(M ) > 0.
(Notation)−△M + R(x)=6∫

R Edρ(E) : spectral resolution

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and11 related/ 27

§4 Asymptotically conic non-trapping case(A3)(Non-trapping) For all compact set K ⊂ int(M ), ∃T > 0 s.t.
π{exp tXH (x, p)} ∩ K = ∅ for ∀|t| > T(i.e. Every geodesic γ : R → M reaches ∂M at ±∞. )(A4) The scalar curvature R(x) on M is bounded.

Lemma (The action integral S(t, x, y) is well defined locally)
Under (A1)∼(A4),
injrad(M ) > 0.
(Notation)−△M + R(x)=6∫

R Edρ(E) : spectral resolution

ρ(E) : L2 (M ) → L2 (M ) : spectral projector

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and11 related/ 27

§4 Asymptotically conic non-trapping case (Results)(in preparation)

Theorem (strong limits)
Assume (A1)∼ (A4). For t ̸= 0[()]
s- lim [U (t/N )]N ρ(N )f (x) = exp −it − 12 (△ − R)
f (x) in L2
6
N →∞Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and12 related/ 27

§4 Asymptotically conic non-trapping case (Results)(in preparation)

Theorem (strong limits)
Assume (A1)∼ (A4). For t ̸= 0[()]
s- lim [U (t/N )]N ρ(N )f (x) = exp −it − 12 (△ − R)
f (x) in L2
6
N →∞Assume (A1)∼ (A4). If R(x) ≦ 0, then injrad(M ) = ∞.
Moreover we can take χ(d(x, y)) = 1 and U ∗ (t)U (t) ∈ Ψ0sc

Theorem (strong limits without cut oﬀ and spectral projectors)
Assume (A1) ∼ (A4), If R(x) ≦[0 and (χ(d(x, y)) =)1.] Thens- lim [U (t/N )]N f (x) = exp − it − 12 (△ − R) f (x) in L2
6
N →∞Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and12 related/ 27

§5 In the case of a compact manifold (Sphere)
Setting

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and13 related/ 27

§5 In the case of a compact manifold (Sphere)
Setting
1. (M, g) = (S 2 , gst )(2dim standard sphere in R3 )Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and13 related/ 27

§5 In the case of a compact manifold (Sphere)
Setting
1. (M, g) = (S 2 , gst )(2dim standard sphere in R3 )2. d = d(x, y) = arccos(⃗x·y⃗)(geodesic distance)

x
d
y
S2

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and13 related/ 27

§5 In the case of a compact manifold (Sphere)
Setting
1. (M, g) = (S 2 , gst )(2dim standard sphere in R3 )2. d = d(x, y) = arccos(⃗x·y⃗)(geodesic distance)

x

x

x

d
y
S2

y

S2

y

S2
(We don’t consider d ≧ π.)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and13 related/ 27

§5 Path integrals on the sphereSetting.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and14 related/ 27

§5 Path integrals on the sphereSetting.
3. S(t, x, y) =∫t 1

0 2 gx(t) (ẋ(t), ẋ(t)) dt =|d(x,y)|22t(The action integral over the shortest path)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and14 related/ 27

§5 Path integrals on the sphereSetting.
3. S(t, x, y) =∫t 1

0 2 gx(t) (ẋ(t), ẋ(t)) dt =|d(x,y)|22t(The action integral over the shortest path)
4. Van Vleck determinant on manifolds
D(t, x, y) = G−1/2 (x)G−1/2 (y) det(∂ 2 S(t, x, y)/∂x∂y)χ(d(x, y)) : cut oﬀ(bump ft. with compact support contained in d(x, y) < π. )a(t, x, y) = χ(d(x, y))D(t, x, y)1/2

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and14 related/ 27

§5 Path integrals on the sphereSetting.
3. S(t, x, y) =∫t 1

0 2 gx(t) (ẋ(t), ẋ(t)) dt =|d(x,y)|22t(The action integral over the shortest path)
4. Van Vleck determinant on manifolds
D(t, x, y) = G−1/2 (x)G−1/2 (y) det(∂ 2 S(t, x, y)/∂x∂y)χ(d(x, y)) : cut oﬀ(bump ft. with compact support contained in d(x, y) < π. )a(t, x, y) = χ(d(x, y))D(t, x, y)1/2

Definition (Shortest path approximations on S 2 )U (t)f (x) ≡ (2πi)−1∫S 2 a(t, x, y)eiS(t, x, y) f (y) dy(Remark. For the simplicity, let h = 1.)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and14 related/ 27

§5 Path integrals on the sphere.

Setting

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and15 related/ 27

§5 Path integrals on the sphere.

Setting−△S 2 + R=
6∫

R Edρ(E) : spectral resolution

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and15 related/ 27

§5 Path integrals on the sphere.

Setting−△S 2 + R=
6∫

R Edρ(E) : spectral resolution

Remark.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and15 related/ 27

§5 Path integrals on the sphere.

Setting−△S 2 + R=
6∫

R Edρ(E) : spectral resolution

Remark.
{uj , Ej } : Eigenfunction expansion on S 2 .

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and15 related/ 27

§5 Path integrals on the sphere.

Setting−△S 2 + R=
6∫

R Edρ(E) : spectral resolution

Remark.
{uj , Ej } : Eigenfunction expansion on S 2 .
({uj } denotes C.O.B in L2 (S 2 ) , Ej the eigenvalue)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and15 related/ 27

§5 Path integrals on the sphere.

Setting−△S 2 + R=
6∫

R Edρ(E) : spectral resolution

Remark.
{uj , Ej } : Eigenfunction expansion on S 2 .
({uj } denotes C.O.B in L2 (S 2 ) , Ej the eigenvalue)ρ(E) : L2 (S 2 ) → L.h.{uj | Ej ≦ E} : spectral projector(Spectral projectors )Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and15 related/ 27

§5 Path integrals on the sphere. (Results)[Adv. Appl. Math. Anal. (2014)]
Theorem (operator norm)
For t ̸= 0 and small ε > 0,[()]
lim [U (t/N )]N ρ(N 1/3−ε ) = exp − it − 12 (△ − R
)
in L2
6
N →∞

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and16 related/ 27

§5 Path integrals on the sphere. (Results)[Adv. Appl. Math. Anal. (2014)]
Theorem (operator norm)
For t ̸= 0 and small ε > 0,[()]
lim [U (t/N )]N ρ(N 1/3−ε ) = exp − it − 12 (△ − R
)
in L2
6
N →∞

Theorem (strong limits)
For t ̸= 0[()]
s- lim [U (t/N )]N ρ(N )f (x) = exp −it − 12 (△ − R)
f (x) in L2
6
N →∞Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and16 related/ 27

§5 Path integrals on the sphere. (Results)[Adv. Appl. Math. Anal. (2014)]
Theorem (operator norm)
For t ̸= 0 and small ε > 0,[()]
lim [U (t/N )]N ρ(N 1/3−ε ) = exp − it − 12 (△ − R
)
in L2
6
N →∞

Theorem (strong limits)
For t ̸= 0[()]
s- lim [U (t/N )]N ρ(N )f (x) = exp −it − 12 (△ − R)
f (x) in L2
6
N →∞CorollaryLet uj be an eigenfunction of Laplacian.
[( For t ̸= 0,
s- lim [U (t/N )]N uj = exp
N →∞

− it

− 12 (△ − R)
6)]uj in L2

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and16 related/ 27

§5 Path integrals on the sphere (Results)
Remark 1.
For t ̸= 0,[()]
lim ∥[U (t/N )]N − exp − it − 12 (△ − R)∥L2 ̸=0.
6
N →∞(Time slicing products does not converge in operator norm
without spectral projector, )Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and17 related/ 27

§5 Path integrals on the sphere (Results)
Remark 1.
For t ̸= 0,[()]
lim ∥[U (t/N )]N − exp − it − 12 (△ − R)∥L2 ̸=0.
6
N →∞(Time slicing products does not converge in operator norm
without spectral projector, )Remark 2.
f (x) ∈ G1/6 (S 2 ) (Gevrey class). For t ̸= 0,[()]
s- lim [U (t/N )]N f (x) = exp − it − 12 (△ − R)
f (x) in L2 .
6
N →∞(The convergence for low energy functions)High energy functions cannot be captured by shortest pathapproximations.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and17 related/ 27

§5 Path integrals on the sphere (Related results)
Let f (x) ∈ C ∞ (S 2 ) and t = 8πm∈ Q (k and m are relativelykprime.)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and18 related/ 27

§5 Path integrals on the sphere (Related results)
Let f (x) ∈ C ∞ (S 2 ) and t = 8πm∈ Q (k and m are relativelykprime.)
s- lim {U (8πm/kN )}N ρ(N )f (x) × eiRt/12
N →∞∫ ∑∞ ( 2l+1 ) −4πi{3ml(l+1)+1}/3k 1/2= S 2 l=0 4π e
C (cos d(x, y)) f (y)dy
} l
{
2k−1
∑
= e2πim/3k
Γ(m, k, j) cos 2πj
A f (x)k
j=0

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and18 related/ 27

§5 Path integrals on the sphere (Related results)
Let f (x) ∈ C ∞ (S 2 ) and t = 8πm∈ Q (k and m are relativelykprime.)
s- lim {U (8πm/kN )}N ρ(N )f (x) × eiRt/12
N →∞∫ ∑∞ ( 2l+1 ) −4πi{3ml(l+1)+1}/3k 1/2= S 2 l=0 4π e
C (cos d(x, y)) f (y)dy
} l
{
2k−1
∑
= e2πim/3k
Γ(m, k, j) cos 2πj
A f (x)
k
j=0

2k−1
2
1 ∑
where Γ(m, k, j) = 2πeπi(l m+lj)/k is a Gaussian sum,l=0√1
A = −△ + 4 ,Cl ： Gegenbauer polynomials are defined by
∞
∑
1/2
1
Cl (x) tl .
=(1−2xt+t2 )1/2
l=0

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and18 related/ 27

§5.1 Outline of proof (We mainly discuss the case of sphere)
For the case of Sphere S 2
D(t, x, y) =We findd(x, y)t2 sin d(x, y)

for 0 ≦ d < π.

For χ(d)K(t, x, y) = χ(d)D(t, x, y)1/2 eiS , we obtain
(
)
∂
1
R
i
+ △x −(χ(d)K(t, x, y))
∂t
2
12
(
)
[
]
d2 − sin2 d
1
R
1
= χ
+
+
−
(△
χ)
K(t, x, y)x
8122
8d2 sin2 d()∂χ sin d − d cos d+K(t, x, y)∂d
2d sin d
( )
∂χ id
+
K(t, x, y).
∂dt

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and19 related/ 27

§5.1 Outline of proof (We mainly discuss the case of sphere)
By using stationary phase methods and integral equations,

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and20 related/ 27

§5.1 Outline of proof (We mainly discuss the case of sphere)
By using stationary phase methods and integral equations,

For the case of sphere S 2
For Ĥ = − 12 (△S 2 − R)
6Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and20 related/ 27

§5.1 Outline of proof (We mainly discuss the case of sphere)
By using stationary phase methods and integral equations,

For the case of sphere S 2
For Ĥ = − 12 (△S 2 − R)
6∥U (t)f ∥L2 ≦ (1 + C1 t)∥f ∥L2 + C2 t2 ∥(−△S 2 + 1)f ∥L2 · · · (3)2∥U (t)f − exp(−itĤ)f ∥L2 ≦ C32t ∥(−△S 2 + 1)3 f ∥L2· · · (4)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and20 related/ 27

§5.1 Outline of proof (We mainly discuss the case of sphere)
By using stationary phase methods and integral equations,

For the case of sphere S 2
For Ĥ = − 12 (△S 2 − R)
6∥U (t)f ∥L2 ≦ (1 + C1 t)∥f ∥L2 + C2 t2 ∥(−△S 2 + 1)f ∥L2 · · · (3)2∥U (t)f − exp(−itĤ)f ∥L2 ≦ C32t ∥(−△S 2 + 1)3 f ∥L2· · · (4)For the scattering manifold case
For Ĥ = − 12 (△M − R)
6Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and20 related/ 27

§5.1 Outline of proof (We mainly discuss the case of sphere)
By using stationary phase methods and integral equations,

For the case of sphere S 2
For Ĥ = − 12 (△S 2 − R)
6∥U (t)f ∥L2 ≦ (1 + C1 t)∥f ∥L2 + C2 t2 ∥(−△S 2 + 1)f ∥L2 · · · (3)2∥U (t)f − exp(−itĤ)f ∥L2 ≦ C32t ∥(−△S 2 + 1)3 f ∥L2· · · (4)For the scattering manifold case
For Ĥ = − 12 (△M − R)
6∥U (t)f ∥L2 ≦ (1 + C1 t)∥f ∥L2 + C2 t2 ∥f ∥Hl for l ≧ 2· · · (3)′2

∥U (t)f − exp(−itĤ)f ∥L2 ≦ C32t ∥f ∥Hm for m > [n/2] + 1· · · (4)′∩s (M ) : weighted scattering Sobolev sp.
Here Hk = ks=0 xk−s Hsc

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and20 related/ 27

§5.1 Outline of proof (We mainly discuss the case of
sphere)( )N
1
1
The binomical coeﬃcients bounds< k!
yields the following
k Nkestimates∥{e−itĤ − Uχ (t/N )n }f (x)∥L2[]= ∥ e−itĤ − {e−itĤ/N (1 + Ẽ(t/N ))}N f (x)∥L2
N ( )
∑
N
∥{e−i(N −k)tĤ/N Ẽ(t/N )k }f (x)∥L2
≦
k
≦
≦

k=1
N (
∑
k=1
N
∑

k=1

N
k

)(

C̃ )k ( t )2k
2

1 ( C̃t2 )k
k! 2N

N

∥(−△ + 1)3k f (x)∥L2∥(−△ + 1)3k f (x)∥L2 .

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and21 related/ 27

§6 Summary(Case 1) M = Rn , H(x, p) = 12 |p|2 + V (x) ∈ C ∞ (T ∗ M )Classical mechanics

Canonical quantization

Feynman quantization

V (x) = O(|x|2 )+error.

2Ĥ = − h2 △ + V (x)lim [U ( Nt )]N()= exp −itĤh(Fujiwara theory)

N →∞Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and22 related/ 27

§6 Summary(Case 1) M = Rn , H(x, p) = 12 |p|2 + V (x) ∈ C ∞ (T ∗ M )Classical mechanics

Canonical quantization

Feynman quantization

V (x) = O(|x|2 )+error.

2Ĥ = − h2 △ + V (x)lim [U ( Nt )]N()= exp −it
Ĥ
h
N →∞

(Fujiwara theory)−it(Integral kernel) e h Ĥ f (x) =∫K(t, x, y)f (y) dy.

Rn

Classical mechanics
V (x) = O(|x|2 )+error.

V (x) = C|x|n(C > 0, n ≧ 4)

Orbits of CM
time locallyglobal diﬀeo
on config. space
infinite manysmall periodic curves

integral kernel
K(t, x, y)∈ C ∞ ((0, t) × R2n )If n = 1, K(t, x, y)
is nowhere C 1

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and22 related/ 27

§6 Summary(Case 2) (M, g) : Riem. mfd., H(x, p) = 12 |p|2 ∈ C ∞ (T ∗ M )Manifolds
Asymptotically conic+ cutoﬀ
Asymptotically conic
R(x) ≦ 0 without cutoﬀ

Feynman quantization
lim [U ( t )]N ρ(N )N →∞ [ N ()]= exp − it − 21 (△ − R)
6lim [U ( Nt )]N[()]= exp − it − 21 (△ − R
)
6
N →∞

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and23 related/ 27

§6 Summary(Case 2) (M, g) : Riem. mfd., H(x, p) = 12 |p|2 ∈ C ∞ (T ∗ M )Manifolds
Asymptotically conic+ cutoﬀ
Asymptotically conic
R(x) ≦ 0 without cutoﬀ
Manifolds
S 2 (sphere)+ cutoﬀ
S2
without cutoﬀ

Feynman quantization
lim [U ( t )]N ρ(N )N →∞ [ N ()]= exp − it − 21 (△ − R)
6lim [U ( Nt )]N[()]= exp − it − 21 (△ − R
)
6
N →∞

Feynman quantization
lim [U ( t )]N ρ(N )N →∞ [ N ()]= exp − it − 12 (△ − R)
6lim [U ( Nt )]NN →∞How to define rigorously ?

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and23 related/ 27

Prospects(1) General manifolds? Potentials?

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and24 related/ 27

Prospects(1) General manifolds? Potentials?
(2) Relationships between Various quantizations?
(e.g. Geometric quantization, Feynman quantization,
Deformation quantization etc.).

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and24 related/ 27

Prospects(1) General manifolds? Potentials?
(2) Relationships between Various quantizations?
(e.g. Geometric quantization, Feynman quantization,
Deformation quantization etc.).
・In the framework of geometric quantization, we can notconstruct the real polarization of 12 |p|2 on L2 (S 2 ).

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and24 related/ 27

Prospects(1) General manifolds? Potentials?
(2) Relationships between Various quantizations?
(e.g. Geometric quantization, Feynman quantization,
Deformation quantization etc.).
・In the framework of geometric quantization, we can notconstruct the real polarization of 12 |p|2 on L2 (S 2 ).
・Quantized operators may diﬀer depending on the definitions.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and24 related/ 27

Prospects(1) General manifolds? Potentials?
(2) Relationships between Various quantizations?
(e.g. Geometric quantization, Feynman quantization,
Deformation quantization etc.).
・In the framework of geometric quantization, we can notconstruct the real polarization of 12 |p|2 on L2 (S 2 ).
・Quantized operators may diﬀer depending on the definitions.

(3) Application? Spectral geometry △ + βR・β = 0(well-known)
・β = 1/6 (geometry of spectrum clustering)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and24 related/ 27

Prospects(1) General manifolds? Potentials?
(2) Relationships between Various quantizations?
(e.g. Geometric quantization, Feynman quantization,
Deformation quantization etc.).
・In the framework of geometric quantization, we can notconstruct the real polarization of 12 |p|2 on L2 (S 2 ).
・Quantized operators may diﬀer depending on the definitions.

(3) Application? Spectral geometry △ + βR・β = 0(well-known)
・β = 1/6 (geometry of spectrum clustering)(4) Application? Gutzwiller trace formula(eg. Albeverio’s proof using Feynman path integral (Euclidean case). )Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and24 related/ 27

Prospects(1) General manifolds? Potentials?
(2) Relationships between Various quantizations?
(e.g. Geometric quantization, Feynman quantization,
Deformation quantization etc.).
・In the framework of geometric quantization, we can notconstruct the real polarization of 12 |p|2 on L2 (S 2 ).
・Quantized operators may diﬀer depending on the definitions.

(3) Application? Spectral geometry △ + βR・β = 0(well-known)
・β = 1/6 (geometry of spectrum clustering)(4) Application? Gutzwiller trace formula(eg. Albeverio’s proof using Feynman path integral (Euclidean case). )γtr(e−itH ) ∼∑heiSγ /h √ γ

γ:closed geodesic

c e

πimγ
4

det(I−Pγ )?

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and24 related/ 27

Prospects(1) General manifolds? Potentials?
(2) Relationships between Various quantizations?
(e.g. Geometric quantization, Feynman quantization,
Deformation quantization etc.).
・In the framework of geometric quantization, we can notconstruct the real polarization of 12 |p|2 on L2 (S 2 ).
・Quantized operators may diﬀer depending on the definitions.

(3) Application? Spectral geometry △ + βR・β = 0(well-known)
・β = 1/6 (geometry of spectrum clustering)(4) Application? Gutzwiller trace formula(eg. Albeverio’s proof using Feynman path integral (Euclidean case). )γtr(e−itH ) ∼∑heiSγ /h √ γ

γ:closed geodesic

c e

πimγ
4

det(I−Pγ )?

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and24 related/ 27

§3 Appendix

・Quantization (Euclidean space)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and25 related/ 27

§3 Appendix

・Quantization (Euclidean space)
Canonical quantization

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and25 related/ 27

§3 Appendix

・Quantization (Euclidean space)
Canonical quantizationn∞∗ nA, B ∈ C ∞ (Rnx × Rp ) = C (T R )Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and25 related/ 27

§3 Appendix

・Quantization (Euclidean space)
Canonical quantizationn∞∗ nA, B ∈ C ∞ (Rnx × Rp ) = C (T R )()∑ ∂A ∂B∂B ∂A{A, B} =−∂xα ∂pα∂xα ∂pα

Poisson Bracket

αYoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and25 related/ 27

§3 Appendix

・Quantization (Euclidean space)
Canonical quantizationn∞∗ nA, B ∈ C ∞ (Rnx × Rp ) = C (T R )()∑ ∂A ∂B∂B ∂A{A, B} =−∂xα ∂pα∂xα ∂pα

Poisson Bracketα⇕Â, B̂ ∈ L(L2 (Rn ))\[Â, B̂] = ih{A,
B}
(i.e.

Canonical quantization∂x̂ = x, p̂ = hi ∂x.)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and25 related/ 27

§7 Appendix
・Quantization (manifolds)
Geometric quantization

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and26 related/ 27

§7 Appendix
・Quantization (manifolds)
Geometric quantization
A, B ∈ C ∞ (T ∗ M )Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and26 related/ 27

§7 Appendix
・Quantization (manifolds)
Geometric quantization∞ (T ∗ M )A, B ∈ C∑(ω =dxα ∧ dpα : Darboux coordinates)

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and26 related/ 27

§7 Appendix
・Quantization (manifolds)
Geometric quantization∞ (T ∗ M )A, B ∈ C∑(ω =dxα ∧ dpα : Darboux coordinates))∑ ( ∂A ∂B∂B ∂A−
Poisson Bracket{A, B} =∂xα ∂pα∂xα ∂pαα

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and26 related/ 27

§7 Appendix
・Quantization (manifolds)
Geometric quantization∞ (T ∗ M )A, B ∈ C∑(ω =dxα ∧ dpα : Darboux coordinates))∑ ( ∂A ∂B∂B ∂A−
Poisson Bracket{A, B} =∂xα ∂pα∂xα ∂pα⇓αÂ, B̂ ∈ L(L2 (T ∗ M ))^[Ã, B̃] = ih{A,B}

Prequantization

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and26 related/ 27

§7 Appendix
・Quantization (manifolds)
Geometric quantization∞ (T ∗ M )A, B ∈ C∑(ω =dxα ∧ dpα : Darboux coordinates))∑ ( ∂A ∂B∂B ∂A−
Poisson Bracket{A, B} =∂xα ∂pα∂xα ∂pα⇓αÂ, B̂ ∈ L(L2 (T ∗ M ))^[Ã, B̃] = ih{A,B}

Prequantization(i.e. H̃ = −ihXH + η(XH ) + H= −ih{( ∂H) ∂ − ( ∂H) ∂ } − 12 (x ∂H+ p ∂H
)+H
∂p ∂x
∂x ∂p
∂x
∂p

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and26 related/ 27

§7 Appendix
・Quantization (manifolds)
Geometric quantization∞ (T ∗ M )A, B ∈ C∑(ω =dxα ∧ dpα : Darboux coordinates))∑ ( ∂A ∂B∂B ∂A−
Poisson Bracket{A, B} =∂xα ∂pα∂xα ∂pα⇓αÂ, B̂ ∈ L(L2 (T ∗ M ))^[Ã, B̃] = ih{A,B}

Prequantization(i.e. H̃ = −ihXH + η(XH ) + H= −ih{( ∂H) ∂ − ( ∂H) ∂ } − 12 (x ∂H+ p ∂H
)+H
∂p ∂x
∂x ∂p
∂x
∂p
⇓
1
We can’t take the real polarization of ^|p|2 . However for T ∗ S 2
2∑
1∃Lα ∈ C ∞ (S 2 ) s.t. 2 |p|2 = α L2α The prequantization L̃α satisfy

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and26 related/ 27

§7 Appendix
・Quantization (manifolds)
Geometric quantization∞ (T ∗ M )A, B ∈ C∑(ω =dxα ∧ dpα : Darboux coordinates))∑ ( ∂A ∂B∂B ∂A−
Poisson Bracket{A, B} =∂xα ∂pα∂xα ∂pα⇓αÂ, B̂ ∈ L(L2 (T ∗ M ))^[Ã, B̃] = ih{A,B}

Prequantization(i.e. H̃ = −ihXH + η(XH ) + H= −ih{( ∂H) ∂ − ( ∂H) ∂ } − 12 (x ∂H+ p ∂H
)+H
∂p ∂x
∂x ∂p
∂x
∂p
⇓
1
We can’t take the real polarization of ^|p|2 . However for T ∗ S 2
2∑
1∃Lα ∈ C ∞ (S 2 ) s.t. 2 |p|2 = α L2α The prequantization L̃α satisfy
∑
h2
1
2 ∗
∗
α L̃α π Yl,m = 2 (l(l + 1) + 8 )π Yl,mYoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and26 related/ 27

Thank you for your attention.

Yoshihisa Miyanishi (The graduate school of Engineering
A Feynman pathand Science
integral-likeOsakamethodUniversity)of quantization on RiemannianMarch 8,manifolds
2015
and27 related/ 27