量子論的粒子の波束効果は見えるのか？
Kenji Nishiwaki( जीवाKenđi Nishiwaki
니시와키 겐지जीवा
西脇 健二)
Based on works with Kenzo Ishikawa (Hokkaido)
Osamu Jinnouchi (Titech) and Kin-ya Oda (Tokyo Woman’s Christian)Eur. Phys. J.C 83 (2023) 10, 978 [arXiv:2308.09933]

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कि

शि

नि

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कें

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कें

セミナー講演 @ 岡山理科大学, 25th July 2024 [Fri]経歴
神戸

学部 (2002—2006) @ 神戸大学
修士過程 (2006—2008) @ 神戸大学
博士過程 (2008—2011) @ 神戸大学
Ph.D. 指導教員: 林青司教授 (Prof. C.S.Lim)

HRI, インド

PD (Oct .2011—Sep. 2014) @ Harish-Chandra Research Institute (India)
PD (Oct. 2014—Sep. 12 2018) @ KIAS (South Korea)
PD (Sep. 14 2018—Oct. 31 2019) @ Ruđer Bošković Institute (Croatia)
Asst. Prof. (Nov. 4 2019—) @ Shiv Nadar Institution of Eminence (India)[ザグレブ, クロアチア][ソウル, 韓国]

デリー(インドの首都)
Beyond theplane-wave transitions bywave packets: anomalous kinetic effect in quarkonium decays
Kenji Nishiwaki( जीवाKenđi Nishiwaki
ニューデリーの니시와키겐지
西脇 健二)
Here
国際空港
Based on works with Kenzo
Ishikawa (Hokkaido)
車で二時間程度
Osamu Jinnouchi (Titech) and Kin-ya Oda (Tokyo Woman’s Christian)[based on works to be summarised]

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कें

[pictures from the web]

デリー(インドの首都)
Beyond theplane-wave transitions bywave packets: anomalous kinetic effect in quarkonium decays
Kenji Nishiwaki( जीवाKenđi Nishiwaki
니시와키 겐지
西脇 健二)

Here

Based on works with Kenzo Ishikawa (Hokkaido)
Osamu Jinnouchi (Titech) and Kin-ya Oda (Tokyo Woman’s Christian)[based on works to be summarised]

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⬅︎

कि

शि

नि

कें

[pictures from the web]

インドはIT大国として有名ですが、(2010-)
技術提供(2018-)[pictures from the web]

教育にも力をいれているようです。

Aloke Kanjilal

Arindam Chatterjee

Bhaskar Kaviraj

Binson Babu

Mayukh Majumder

Priya Johari

Rana Nandi

Sajal Ghosh

Subhra Sen Gupta

Sucheta Mondal

Sujit K. Tandel

Ipsita Mandal

Kenji Nishiwaki

Samarendra Pratap Singh Santosh Kumar

Susanta Sinha Roy Syed Mohammad Kamil[current faculty members of our Department of Physics]

教育にも力をいれているようです。

Aloke Kanjilal

Arindam Chatterjee

Bhaskar Kaviraj

Binson Babu

Mayukh Majumder

Priya Johari

Rana Nandi

Sajal Ghosh

Ipsita Mandal

Kenji Nishiwaki

Samarendra Pratap Singh Santosh Kumar
Theorists

Subhra Sen Gupta

Sucheta Mondal

Sujit K. Tandel

Susanta Sinha Roy Syed Mohammad Kamil[current faculty members of our Department of Physics]

教育にも力をいれているようです。

Distinguished Visiting Professors
Aloke Kanjilal

Arindam Chatterjee

Bhaskar Kaviraj

Rohini Madhusudan Godbole
Mayukh Majumder

Priya Johari

Rana Nandi

Binson Babu

Ipsita Mandal

Kenji Nishiwaki

Swadesh M Mahajan
Sajal Ghosh

Samarendra Pratap Singh Santosh Kumar
Theorists

Subhra Sen Gupta

Sucheta Mondal

Sujit K. Tandel

Susanta Sinha Roy Syed Mohammad Kamil[current faculty members of our Department of Physics]

何故研究職を志すことにしたのか？
純粋に物理・研究が面白く、ずっと続けたいと思ったから。
(会社で普通に働けるか余り自身がなかったこともあるかもです・・・)
海外に出ることに、抵抗は殆どありませんでした。
(明確な根拠はなかった。体の強さ[これ大事]はある程度自信はあった。)

何故研究職を志すことにしたのか？
純粋に物理・研究が面白く、ずっと続けたいと思ったから。
(会社で普通に働けるか余り自身がなかったこともあるかもです・・・)
海外に出ることに、抵抗は殆どありませんでした。
(明確な根拠はなかった。体の強さ[これ大事]はある程度自信はあった。)
何故インドで教員になったか？
初めてのポスドクがインドの研究所だったので、インドではある程度名前が知られていた(と期待した)。
研究所・大学が(2010年度後半においても)新設されており、今後十数年のインド国内の学術分野の発展の見通しの良さ。
上と同様に、経済面に対する今後暫くの見通しの良さ。
インド市民でない人の就職も、少しずつ増えていたので時期的にもよかった？
(研究だけでなく)教育・アドミッションの仕事も英語で出来る。
純粋な好奇心。

第一部:
背景の理解と何を議論したいかの説明本日の話のキーワード

波束量子力学
波(の)束 (wave packet)

分子・原子といったこの世の中のミクロな構造を記述する為の言語・方法論(Quantum Theory)本日の話のキーワード

波束量子力学
波(の)束 (wave packet)

色々な波を束状に集めると、どんな形も表現できる。

分子・原子といったこの世の中のミクロな構造を記述する為の言語・方法論(Quantum Theory)

物理学(Physics)は、初めこの世のマクロな構造(身の回りの世界)を理解することから始まった。
[古典理論 (Classical Theory)]

本日の話のキーワード

波束量子力学
波(の)束 (wave packet)

分子・原子といったこの世の中のミクロな構造を記述する為の言語・方法論(Quantum Theory)

どんな形状のミクロな構造でも“正確に記述できる方法論!

本日の話のメイン・テーマ本日の話のキーワード

波束量子力学??
波(の)束 (wave packet)

色々な波を束状に集めると、どんな形も表現できる。

???
分子・原子といったこの世の中のミクロな構造を記述する為の言語・方法論(Quantum Theory)?

物理学(Physics)は、初めこの世のマクロな構造(身の回りの世界)を理解することから始まった。
[古典理論 (Classical Theory)]

本日の話のキーワード

波束量子力学、?? 背景を???
う
ょ
な
し
波(の)束 (wave packet)
分子・原子といった
的
ま
本
め
基
初
この世の中のミクロな構造をらかろ記述する為の言語・方法論
ことる(Quantum
Theory)す
色々な波を束状に集めると、
解
理
どんな形も表現できる。
?
物理学(Physics)は、初めこの世のマクロな構造(身の回りの世界)を理解することから始まった。
[古典理論 (Classical Theory)]古典物理(マクロな身の回りの世界を記述する理論)
マクロな物の動きの記述 [古典力学 (Classical Mechanics)]

古典物理(マクロな身の回りの世界を記述する理論)
マクロな物の動きの記述 [古典力学 (Classical Mechanics)]
…

例えば

太陽
地球
地面(地球)
地上で手を話すと、物体(りんご)は下に落ちる

地球は太陽の周りを周回している[pictures from the web]

古典物理(マクロな身の回りの世界を記述する理論)
マクロな物の動きの記述 [古典力学 (Classical Mechanics)]
…

例えば

太陽
地球
地面(地球)
りんごの速度 (velocity)(向きのある量[=ベクトル])<latexit sha1_base64="cadA/p9nL2vUJzh18aGjNbc5dfQ=">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</latexit>

速度が変化する度合い(向きのある量[=ベクトル])

dvv apple
mapple= F appledt

りんごの質量 (mass) 時間=(物体の動きにくさ) (time)≒(物体の重さ)

りんごにかかる力 (force)

dvv EarthmEarth= F Earthdt<latexit sha1_base64="N6i5Nt0+Ykkmzr9GGN6pAJT2ZYs=">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</latexit>

ニュートンの運動方程式が、物の動きを支配する!
[pictures from the web]

古典物理(マクロな身の回りの世界を記述する理論)
マクロな物の動きの記述 [古典力学 (Classical Mechanics)]…

F Earth<latexit sha1_base64="YAE1svMzUbX2M0FD8x/Y/bb1nu8=">AAACEXicdVBJSgNBFK12jHFqh52bwiC4Ch2Jw0oCoriMYAZIQqiu/CRFqgeqfoux6VN4Abd6A3fi1hN4Ac9hJemAEX1Q8HjvT/XcUAqNjvNpzc0vLC4tZ1ayq2vrG5v21nZVB5HiUOGBDFTdZRqk8KGCAiXUQwXMcyXU3MHFyK/dgdIi8G9xGELLYz1fdAVnaKS2vdsMPTe+StpNhHuML5nCftK2c07+2BmBOnlnSlKlkCo5kqLctr+anYBHHvjIJdO6UXBCbMVmmOASkmwz0hAyPmA9aBjqMw90Kx5fn9ADo3RoN1Dm+UjH6s+OmHlaDz3XVHoM+/q3NxL/8hoRds9asfDDCMHnk0XdSFIM6CgK2hEKOMqhIYwrYW6lvM8U42gCm9niKjYATLImmOnv6f+kepQvnOSLN8Vc6TyNKEP2yD45JAVySkrkmpRJhXDyQJ7IM3mxHq1X6816n5TOWWnPDpmB9fENBYyeLw==</latexit>

例えば

F apple<latexit sha1_base64="cP7tK/4y3QGCjxfFV2e9FOR2PWs=">AAACEXicdVBJSgNBFK2OU4xTHHZuCoPgKnQkDisJCOIyghkgCaG68pMUqe4uqn6LsckpvIBbvYE7cesJvIDnsJJ0wIg+KHi896d6npLCoOt+OqmFxaXllfRqZm19Y3Mru71TNWGkOVR4KENd95gBKQKooEAJdaWB+Z6Emje4HPu1O9BGhMEtDhW0fNYLRFdwhlZqZ/eayvfiq1G7iXCPMVNKwqidzbn5E3cM6ubdGUmUQqLkSIJyO/vV7IQ88iFALpkxjYKrsBUzjYLbeZlmZEAxPmA9aFgaMB9MK55cP6KHVunQbqjtC5BO1J8dMfONGfqerfQZ9s1vbyz+5TUi7J63YhGoCCHg00XdSFIM6TgK2hEaOMqhJYxrYW+lvM8042gDm9viaTYAHGVsMLPf0/9J9ThfOM0Xb4q50kUSUZrskwNyRArkjJTINSmTCuHkgTyRZ/LiPDqvzpvzPi1NOUnPLpmD8/ENNgueTQ==</latexit>

太陽地球
地面(地球)

物質間に引力が働くと考えると、<latexit sha1_base64="cadA/p9nL2vUJzh18aGjNbc5dfQ=">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</latexit>

dvv apple
mapple= F apple
dt

dvv EarthmEarth= F Earthdt<latexit sha1_base64="N6i5Nt0+Ykkmzr9GGN6pAJT2ZYs=">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</latexit>[pictures from the web]

古典物理(マクロな身の回りの世界を記述する理論)
マクロな物の動きの記述 [古典力学 (Classical Mechanics)]
v apple,0<latexit sha1_base64="3db5nyDHTn41cYnWHkVcPIITAf0=">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</latexit>

えば…

<latexit sha1_base64="a2I7TgUtuNTMBAByVgPmHK2tYaw=">AAACEXicdVDLSgMxFM34tr7qY+cmWARXZVJKbTdSEMGlgq1CW0omvW1DMw+SO2Id5iv8Abf6B+7ErV/gD/gdZmoFFT0QOJxzXzlepKRB131zZmbn5hcWl5ZzK6tr6xv5za2mCWMtoCFCFeorjxtQMoAGSlRwFWngvqfg0hsdZ/7lNWgjw+ACxxF0fD4IZF8Kjlbq5nfake8l12m3jXCDyQnXOEy7+YJbdF2XMUYzwg4rriW1WrXEqpRllkWBTHHWzb+3e6GIfQhQKG5Mi7kRdhI7TAoFaa4dG4i4GPEBtCwNuA+mk0yuT+m+VXq0H2r7AqQT9XtHwn1jxr5nK32OQ/Pby8S/vFaM/WonkUEUIwTic1E/VhRDmkVBe1KDQDW2hAst7a1UDLnmAm1gP7Z4mo8A05wN5uv39H/SLBVZpVg+LxfqR9OIlsgu2SMHhJFDUien5Iw0iCC35J48kEfnznlynp2Xz9IZZ9qzTX7Aef0AloCeiw==</latexit>

F Earth<latexit sha1_base64="YAE1svMzUbX2M0FD8x/Y/bb1nu8=">AAACEXicdVBJSgNBFK12jHFqh52bwiC4Ch2Jw0oCoriMYAZIQqiu/CRFqgeqfoux6VN4Abd6A3fi1hN4Ac9hJemAEX1Q8HjvT/XcUAqNjvNpzc0vLC4tZ1ayq2vrG5v21nZVB5HiUOGBDFTdZRqk8KGCAiXUQwXMcyXU3MHFyK/dgdIi8G9xGELLYz1fdAVnaKS2vdsMPTe+StpNhHuML5nCftK2c07+2BmBOnlnSlKlkCo5kqLctr+anYBHHvjIJdO6UXBCbMVmmOASkmwz0hAyPmA9aBjqMw90Kx5fn9ADo3RoN1Dm+UjH6s+OmHlaDz3XVHoM+/q3NxL/8hoRds9asfDDCMHnk0XdSFIM6CgK2hEKOMqhIYwrYW6lvM8U42gCm9niKjYATLImmOnv6f+kepQvnOSLN8Vc6TyNKEP2yD45JAVySkrkmpRJhXDyQJ7IM3mxHq1X6816n5TOWWnPDpmB9fENBYyeLw==</latexit>

F apple例<latexit sha1_base64="cP7tK/4y3QGCjxfFV2e9FOR2PWs=">AAACEXicdVBJSgNBFK2OU4xTHHZuCoPgKnQkDisJCOIyghkgCaG68pMUqe4uqn6LsckpvIBbvYE7cesJvIDnsJJ0wIg+KHi896d6npLCoOt+OqmFxaXllfRqZm19Y3Mru71TNWGkOVR4KENd95gBKQKooEAJdaWB+Z6Emje4HPu1O9BGhMEtDhW0fNYLRFdwhlZqZ/eayvfiq1G7iXCPMVNKwqidzbn5E3cM6ubdGUmUQqLkSIJyO/vV7IQ88iFALpkxjYKrsBUzjYLbeZlmZEAxPmA9aFgaMB9MK55cP6KHVunQbqjtC5BO1J8dMfONGfqerfQZ9s1vbyz+5TUi7J63YhGoCCHg00XdSFIM6TgK2hEaOMqhJYxrYW+lvM8042gDm9viaTYAHGVsMLPf0/9J9ThfOM0Xb4q50kUSUZrskwNyRArkjJTINSmTCuHkgTyRZ/LiPDqvzpvzPi1NOUnPLpmD8/ENNgueTQ==</latexit>v Earth太陽<latexit sha1_base64="xxvesNC43/g0xQthva4lLoZfwYw=">AAACEXicdVDLSgMxFM3UV62v+ti5CRbBVZkU0XYjBTcuK1gttKVk0lsNzcyE5I5Yh36FP+BW/8CduPUL/AG/w4xWsKIHAodz7isn0Epa9P03LzczOze/kF8sLC2vrK4V1zfObZwYAU0Rq9i0Am5ByQiaKFFBSxvgYaDgIhgeZ/7FNRgr4+gMRxq6Ib+M5EAKjk7qFbc6OgzS63Gvg3CDKddawbhXLPll3/cZYzQj7PDAd6RWq1ZYlbLMciiRCRq94nunH4skhAiF4ta2ma+xm3KDUrh5hU5iQXMx5JfQdjTiIdhu+nn9mO46pU8HsXEvQvqp/uxIeWjtKAxcZcjxyv72MvEvr53goNpNZaQThEh8LRokimJMsyhoXxoQqEaOcGGku5WKK264QBfY1JbA8CHguOCC+f49/Z+cV8rsoLx/ul+qH00iypNtskP2CCOHpE5OSIM0iSC35J48kEfvznvynr2Xr9KcN+nZJFPwXj8Axv+eqQ==</latexit>

地面(地球)

地球

v apple(同じ方向への速度変化)

v Earth,0<latexit sha1_base64="GxSdZJJPLImJb3n8KnvwrAPIrSU=">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</latexit>(違う方向への速度変化)

物質間に引力が働くと考えると、<latexit sha1_base64="cadA/p9nL2vUJzh18aGjNbc5dfQ=">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</latexit>

dvv apple
mapple= F apple
dt

dvv EarthmEarth= F Earthdt<latexit sha1_base64="N6i5Nt0+Ykkmzr9GGN6pAJT2ZYs=">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</latexit>

両者の運動は統一的に理解できる!

[pictures from the web]

実は、ここで以下の近似を使用した。

単

純化

りんごと
同じ質量を持つ点

地球と同じ質量を持つ点化
純
単物質の(回転運動などを無視した)、全体としての動きのみに興味があるなら、この取り扱いは良い。

[pictures from the web]

実は、ここで以下の近似を使用した。

単

純化

りんごと
同じ質量を持つ点

地球と同じ質量を持つ点化
純
単物質の(回転運動などを無視した)、全体としての動きのみに興味があるなら、この取り扱いは良い。

このような理想的に考えた物体のことを、(古典的な) 粒子(particle)と呼ぶ。

[pictures from the web]

本日の話のキーワード

波束量子力学??
波(の)束 (wave packet)

色々な波を束状に集めると、どんな形も表現できる。

???
分子・原子といったこの世の中のミクロな構造を記述する為の言語・方法論(Quantum Theory)

!
e
n
o
D

?

物理学(Physics)は、初めこの世のマクロな構造(身の回りの世界)を理解することから始まった。
[古典理論 (Classical Theory)]皆様、海で(水の)波をご覧になったことがあるかと思います。

[pictures from the web]

皆様、海で(水の)波をご覧になったことがあるかと思います。
媒質 水
この現象は、以下のように明文化できます。
波とは、広がった「物体」(媒質, 今は水)の分布パターンが、ある方向に伝搬していく現象である。

[pictures from the web]

皆様、海で(水の)波をご覧になったことがあるかと思います。
媒質 水
この現象は、以下のように明文化できます。
波とは、広がった「物体」(媒質, 今は水)の分布パターンが、ある方向に伝搬していく現象である。

音も波
媒質 空気

地震も波
媒質 地面

光も波
媒質 「真空」[pictures from the web]

伝搬方向を一方向に定めて、一番単純な場合を詳しく見てみましょう。
波の高さ三角関数(sin or cos)一つで表せる場合。
空間方向(x)0<latexit sha1_base64="WA98rcfGWqggZLS5pb3+ZFf0AGc=">AAAB/HicbVDLSgNBEOyNrxhfUY9eBoPgKexKUE8S8OIxAfOAZAmzk04yZHZ2mZkVwhJ/wKv+gTfx6r/4A36Hk2QPJrGgoajqprsriAXXxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ihRDBssEpFqB1Sj4BIbhhuB7VghDQOBrWB8P/NbT6g0j+SjmcToh3Qo+YAzaqxUd3vFklt25yDrxMtICTLUesWfbj9iSYjSMEG17nhubPyUKsOZwGmhm2iMKRvTIXYslTRE7afzQ6fkwip9MoiULWnIXP07kdJQ60kY2M6QmpFe9Wbif14nMYNbP+UyTgxKtlg0SAQxEZl9TfpcITNiYgllittbCRtRRZmx2SxtCRQdo5kWbDDeagzrpHlV9q7LlXqlVL3LIsrDGZzDJXhwA1V4gBo0gAHCC7zCm/PsvDsfzueiNedkM6ewBOfrF4wAlSE=</latexit>

時刻 t=0 での波の配位

伝搬方向を一方向に定めて、一番単純な場合を詳しく見てみましょう。
波の高さ

空間方向
(x)

0
<latexit sha1_base64="WA98rcfGWqggZLS5pb3+ZFf0AGc=">AAAB/HicbVDLSgNBEOyNrxhfUY9eBoPgKexKUE8S8OIxAfOAZAmzk04yZHZ2mZkVwhJ/wKv+gTfx6r/4A36Hk2QPJrGgoajqprsriAXXxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ihRDBssEpFqB1Sj4BIbhhuB7VghDQOBrWB8P/NbT6g0j+SjmcToh3Qo+YAzaqxUd3vFklt25yDrxMtICTLUesWfbj9iSYjSMEG17nhubPyUKsOZwGmhm2iMKRvTIXYslTRE7afzQ6fkwip9MoiULWnIXP07kdJQ60kY2M6QmpFe9Wbif14nMYNbP+UyTgxKtlg0SAQxEZl9TfpcITNiYgllittbCRtRRZmx2SxtCRQdo5kWbDDeagzrpHlV9q7LlXqlVL3LIsrDGZzDJXhwA1V4gBo0gAHCC7zCm/PsvDsfzueiNedkM6ewBOfrF4wAlSE=</latexit>

ある後の時刻 t(>0) での波の配位<latexit sha1_base64="jOUfm9KmZxwjlh3SxQydrX6NNco=">AAACJXicbVDLSgNBEJz1GeMr6tHLkCDEg2FXRD2J4MVjBKNCNsTZSW8yZHZ2mekVw7K/4G/4A171D7yJ4Mmb3+HkcTBqQUNR1U13V5BIYdB1P5yZ2bn5hcXCUnF5ZXVtvbSxeWXiVHNo8FjG+iZgBqRQ0ECBEm4SDSwKJFwH/bOhf30H2ohYXeIggVbEukqEgjO0UrtU9RHuMbu9zX0jlC8hxOo93aN36GvR7eHu2C/n7VLFrbkj0L/Em5AKmaDeLn35nZinESjkkhnT9NwEWxnTKLiEvOinBhLG+6wLTUsVi8C0stFHOd2xSoeGsbalkI7UnxMZi4wZRIHtjBj2zG9vKP7nNVMMj1uZUEmKoPh4UZhKijEdxkM7QgNHObCEcS3srZT3mGYcbYhTWwLN+oB50Qbj/Y7hL7nar3mHtYOLg8rpySSiAtkmZVIlHjkip+Sc1EmDcPJAnsgzeXEenVfnzXkft844k5ktMgXn8xsQjaYD</latexit>[t時間の間に動いた距離]=v(波の伝搬速度) × t“ sin (xvt) ”

伝搬方向を一方向に定めて、一番単純な場合を詳しく見てみましょう。
✓
◆
波の高さ
2⇡
2⇡
A ⇥ sinxt (+↵)
T
A<latexit sha1_base64="ChFV9Y7iCywrykx9X3CAFrPUnR4=">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</latexit><latexit sha1_base64="kcKoDgixDn5PQQftzwXIs4vGvAs=">AAAB/HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKqCeJePGYgHlAsoTZSW8yZHZ2mZkVQog/4FX/wJt49V/8Ab/DSbIHk1jQUFR1090VJIJr47rfTm5tfWNzK79d2Nnd2z8oHh41dJwqhnUWi1i1AqpRcIl1w43AVqKQRoHAZjC8n/rNJ1Sax/LRjBL0I9qXPOSMGivV7rrFklt2ZyCrxMtICTJUu8WfTi9maYTSMEG1bntuYvwxVYYzgZNCJ9WYUDakfWxbKmmE2h/PDp2QM6v0SBgrW9KQmfp3YkwjrUdRYDsjagZ62ZuK/3nt1IQ3/pjLJDUo2XxRmApiYjL9mvS4QmbEyBLKFLe3EjagijJjs1nYEig6RDMp2GC85RhWSeOi7F2VL2uXpcptFlEeTuAUzsGDa6jAA1ShDgwQXuAV3pxn5935cD7nrTknmzmGBThfv6cYlTI=</latexit>

空間方向
(x)

0

振幅

波長 =
[最小パターンの長さ]<latexit sha1_base64="WA98rcfGWqggZLS5pb3+ZFf0AGc=">AAAB/HicbVDLSgNBEOyNrxhfUY9eBoPgKexKUE8S8OIxAfOAZAmzk04yZHZ2mZkVwhJ/wKv+gTfx6r/4A36Hk2QPJrGgoajqprsriAXXxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ihRDBssEpFqB1Sj4BIbhhuB7VghDQOBrWB8P/NbT6g0j+SjmcToh3Qo+YAzaqxUd3vFklt25yDrxMtICTLUesWfbj9iSYjSMEG17nhubPyUKsOZwGmhm2iMKRvTIXYslTRE7afzQ6fkwip9MoiULWnIXP07kdJQ60kY2M6QmpFe9Wbif14nMYNbP+UyTgxKtlg0SAQxEZl9TfpcITNiYgllittbCRtRRZmx2SxtCRQdo5kWbDDeagzrpHlV9q7LlXqlVL3LIsrDGZzDJXhwA1V4gBo0gAHCC7zCm/PsvDsfzueiNedkM6ewBOfrF4wAlSE=</latexit><latexit sha1_base64="elhOyKEsLo0qAEPheaAnw0+QQXA=">AAACAnicbVDLSsNAFL2pr1pfVZdugkVwVRIRdSUFNy4r2Ae0oUwmk3bIZBJmboQSuvMH3OofuBO3/og/4Hc4bbOwrQcGDuecy71z/FRwjY7zbZXW1jc2t8rblZ3dvf2D6uFRWyeZoqxFE5Gork80E1yyFnIUrJsqRmJfsI4f3U39zhNTmifyEccp82IylDzklKCROn1hogEZVGtO3ZnBXiVuQWpQoDmo/vSDhGYxk0gF0brnOil6OVHIqWCTSj/TLCU0IkPWM1SSmGkvn507sc+MEthhosyTaM/UvxM5ibUex75JxgRHetmbiv95vQzDGy/nMs2QSTpfFGbCxsSe/t0OuGIUxdgQQhU3t9p0RBShaBpa2OIrEjGcVEwx7nINq6R9UXev6pcPl7XGbVFRGU7gFM7BhWtowD00oQUUIniBV3iznq1368P6nEdLVjFzDAuwvn4BgyiX6g==</latexit>

位相

周期 =
[一波長分進むのにかかる時間]<latexit sha1_base64="m8xBWHjl+5j0WFpRCkEyHn+qizo=">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</latexit>v=

T(伝搬速度)A ⇥ sin (kx2⇡◦ k :=<latexit sha1_base64="LgkYsSkYE4wC3pgZxB8zjkQy7yw=">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</latexit>!t (+↵))<latexit sha1_base64="D9XtsouQr0o72oFNh5s4cSUObwQ=">AAACFnicbVDLSsNAFJ3UV62vqCtxM1gEVyUpRUVQCm5cVrAPaEKZTCbtkMkkzEyEEoK/4Q+41T9wJ27d+gN+h9M2C9t6YOBwzj3cO8dLGJXKsr6N0srq2vpGebOytb2zu2fuH3RknApM2jhmseh5SBJGOWkrqhjpJYKgyGOk64W3E7/7SISkMX9Q44S4ERpyGlCMlJYG5lEIr66hEwiEs7qT0DxzmE77KB+YVatmTQGXiV2QKijQGpg/jh/jNCJcYYak7NtWotwMCUUxI3nFSSVJEA7RkPQ15Sgi0s2mX8jhqVZ8GMRCP67gVP2byFAk5Tjy9GSE1EguehPxP6+fquDSzShPUkU4ni0KUgZVDCd9QJ8KghUba4KwoPpWiEdI16F0a3NbPIFCovKKLsZerGGZdOo1+7zWuG9UmzdFRWVwDE7AGbDBBWiCO9ACbYDBE3gBr+DNeDbejQ/jczZaMorMIZiD8fULOy+fQA==</latexit>

波数 =[単位長さあたりのパターンの数]/(2π)

2⇡角振動数 =◦ ! :=
T [単位時間あたりに<latexit sha1_base64="+YzUu0D4kgMMbzyHSHjpCoYx4qc=">AAACFXicbZDLSsNAFIYn9VbrLepGcDNYBFclKUVFUApuXFboRWhCmUxP2qGTCzMToYT4Gr6AW30Dd+LWtS/gczhts7CtPwz8/OcczpnPizmTyrK+jcLK6tr6RnGztLW9s7tn7h+0ZZQICi0a8Ug8eEQCZyG0FFMcHmIBJPA4dLzR7aTeeQQhWRQ21TgGNyCDkPmMEqWjnnnkRAEMCL66xo4vCE2rTsyytJn1zLJVsabCy8bOTRnlavTMH6cf0SSAUFFOpOzaVqzclAjFKIes5CQSYkJHZABdbUMSgHTT6Q8yfKqTPvYjoV+o8DT9O5GSQMpx4OnOgKihXKxNwv9q3UT5l27KwjhRENLZIj/hWEV4ggP3mQCq+FgbQgXTt2I6JBqE0tDmtniCjEBlJQ3GXsSwbNrVin1eqd3XyvWbHFERHaMTdIZsdIHq6A41UAtR9IRe0Ct6M56Nd+PD+Jy1Fox85hDNyfj6BWAvnsc=</latexit>

何波長進むか]/(2π)

伝搬方向を一方向に定めて、一番単純な場合を詳しく見てみましょう。
✓
◆
波の高さ
2⇡
2⇡
A ⇥ sinxt (+↵)
T
A<latexit sha1_base64="ChFV9Y7iCywrykx9X3CAFrPUnR4=">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</latexit><latexit sha1_base64="kcKoDgixDn5PQQftzwXIs4vGvAs=">AAAB/HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKqCeJePGYgHlAsoTZSW8yZHZ2mZkVQog/4FX/wJt49V/8Ab/DSbIHk1jQUFR1090VJIJr47rfTm5tfWNzK79d2Nnd2z8oHh41dJwqhnUWi1i1AqpRcIl1w43AVqKQRoHAZjC8n/rNJ1Sax/LRjBL0I9qXPOSMGivV7rrFklt2ZyCrxMtICTJUu8WfTi9maYTSMEG1bntuYvwxVYYzgZNCJ9WYUDakfWxbKmmE2h/PDp2QM6v0SBgrW9KQmfp3YkwjrUdRYDsjagZ62ZuK/3nt1IQ3/pjLJDUo2XxRmApiYjL9mvS4QmbEyBLKFLe3EjagijJjs1nYEig6RDMp2GC85RhWSeOi7F2VL2uXpcptFlEeTuAUzsGDa6jAA1ShDgwQXuAV3pxn5935cD7nrTknmzmGBThfv6cYlTI=</latexit>

空間方向
(x)

0

振幅

波長 =
[最小パターンの長さ]<latexit sha1_base64="WA98rcfGWqggZLS5pb3+ZFf0AGc=">AAAB/HicbVDLSgNBEOyNrxhfUY9eBoPgKexKUE8S8OIxAfOAZAmzk04yZHZ2mZkVwhJ/wKv+gTfx6r/4A36Hk2QPJrGgoajqprsriAXXxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ihRDBssEpFqB1Sj4BIbhhuB7VghDQOBrWB8P/NbT6g0j+SjmcToh3Qo+YAzaqxUd3vFklt25yDrxMtICTLUesWfbj9iSYjSMEG17nhubPyUKsOZwGmhm2iMKRvTIXYslTRE7afzQ6fkwip9MoiULWnIXP07kdJQ60kY2M6QmpFe9Wbif14nMYNbP+UyTgxKtlg0SAQxEZl9TfpcITNiYgllittbCRtRRZmx2SxtCRQdo5kWbDDeagzrpHlV9q7LlXqlVL3LIsrDGZzDJXhwA1V4gBo0gAHCC7zCm/PsvDsfzueiNedkM6ewBOfrF4wAlSE=</latexit><latexit sha1_base64="elhOyKEsLo0qAEPheaAnw0+QQXA=">AAACAnicbVDLSsNAFL2pr1pfVZdugkVwVRIRdSUFNy4r2Ae0oUwmk3bIZBJmboQSuvMH3OofuBO3/og/4Hc4bbOwrQcGDuecy71z/FRwjY7zbZXW1jc2t8rblZ3dvf2D6uFRWyeZoqxFE5Gork80E1yyFnIUrJsqRmJfsI4f3U39zhNTmifyEccp82IylDzklKCROn1hogEZVGtO3ZnBXiVuQWpQoDmo/vSDhGYxk0gF0brnOil6OVHIqWCTSj/TLCU0IkPWM1SSmGkvn507sc+MEthhosyTaM/UvxM5ibUex75JxgRHetmbiv95vQzDGy/nMs2QSTpfFGbCxsSe/t0OuGIUxdgQQhU3t9p0RBShaBpa2OIrEjGcVEwx7nINq6R9UXev6pcPl7XGbVFRGU7gFM7BhWtowD00oQUUIniBV3iznq1368P6nEdLVjFzDAuwvn4BgyiX6g==</latexit>

位相

周期 =
[一波長分進むのにかかる時間]<latexit sha1_base64="m8xBWHjl+5j0WFpRCkEyHn+qizo=">AAACEnicdVDLSsNAFJ3UV62vqrhyM1gEVyGJoa0LpeDGZYW+oA1lMpm0QycPZiaFEvIX/oBb/QN34tYf8Af8DidtBSt6YOBwzj3cO8eNGRXSMD60wtr6xuZWcbu0s7u3f1A+POqIKOGYtHHEIt5zkSCMhqQtqWSkF3OCApeRrju5zf3ulHBBo7AlZzFxAjQKqU8xkkoalk+m8BoOfI5wOmAq5qEsbWXDcsXQr+pVy65CQzeMmmmZObFq9qUNTaXkqIAlmsPy58CLcBKQUGKGhOibRiydFHFJMSNZaZAIEiM8QSPSVzREARFOOj8/g+dK8aAfcfVCCefqz0SKAiFmgasmAyTH4reXi395/UT6dSelYZxIEuLFIj9hUEYw7wJ6lBMs2UwRhDlVt0I8RqoLqRpb2eJyNCEyK6livn8P/ycdSzerun1vVxo3y4qK4BScgQtgghpogDvQBG2AQQoewRN41h60F+1Ve1uMFrRl5hisQHv/Au48nhg=</latexit>v=<latexit sha1_base64="p43tEJprXkJfvhr7sYj2iIoFXWo=">AAACGnicbZDLSgMxFIYz9VbH26hLEYJFqAhlRoq6UQpuXFawF2hryaSnbWgmMyQZYRi68jV8Abf6Bu7ErRtfwOcwvSxs6w+Bj/+cwzn5/YgzpV3328osLa+srmXX7Y3Nre0dZ3evqsJYUqjQkIey7hMFnAmoaKY51CMJJPA51PzBzaheewSpWCjudRJBKyA9wbqMEm2stnMIDynDydC+sps0VPnkBJ9ihpuKCcNtJ+cW3LHwInhTyKGpym3np9kJaRyA0JQTpRqeG+lWSqRmlMPQbsYKIkIHpAcNg4IEoFrp+BtDfGycDu6G0jyh8dj9O5GSQKkk8E1nQHRfzddG5n+1Rqy7l62UiSjWIOhkUTfmWId4lAnuMAlU88QAoZKZWzHtE0moNsnNbPElGYAe2iYYbz6GRaieFbzzQvGumCtdTyPKogN0hPLIQxeohG5RGVUQRU/oBb2iN+vZerc+rM9Ja8aazuyjGVlfv2rPnxc=</latexit>eiy= cos(y) + i sin(y)オイラーの公式<latexit sha1_base64="r6iSFBT+v+cIjGbj5hKIcR7C/Ms=">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</latexit>hA ⇥ Im ei(kx !t (+↵))

i

指数関数を使って表現できる。

T(伝搬速度)A ⇥ sin (kx2⇡◦ k :=<latexit sha1_base64="LgkYsSkYE4wC3pgZxB8zjkQy7yw=">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</latexit>!t (+↵))<latexit sha1_base64="D9XtsouQr0o72oFNh5s4cSUObwQ=">AAACFnicbVDLSsNAFJ3UV62vqCtxM1gEVyUpRUVQCm5cVrAPaEKZTCbtkMkkzEyEEoK/4Q+41T9wJ27d+gN+h9M2C9t6YOBwzj3cO8dLGJXKsr6N0srq2vpGebOytb2zu2fuH3RknApM2jhmseh5SBJGOWkrqhjpJYKgyGOk64W3E7/7SISkMX9Q44S4ERpyGlCMlJYG5lEIr66hEwiEs7qT0DxzmE77KB+YVatmTQGXiV2QKijQGpg/jh/jNCJcYYak7NtWotwMCUUxI3nFSSVJEA7RkPQ15Sgi0s2mX8jhqVZ8GMRCP67gVP2byFAk5Tjy9GSE1EguehPxP6+fquDSzShPUkU4ni0KUgZVDCd9QJ8KghUba4KwoPpWiEdI16F0a3NbPIFCovKKLsZerGGZdOo1+7zWuG9UmzdFRWVwDE7AGbDBBWiCO9ACbYDBE3gBr+DNeDbejQ/jczZaMorMIZiD8fULOy+fQA==</latexit>

波数 =[単位長さあたりのパターンの数]/(2π)

2⇡角振動数 =◦ ! :=
T [単位時間あたりに<latexit sha1_base64="+YzUu0D4kgMMbzyHSHjpCoYx4qc=">AAACFXicbZDLSsNAFIYn9VbrLepGcDNYBFclKUVFUApuXFboRWhCmUxP2qGTCzMToYT4Gr6AW30Dd+LWtS/gczhts7CtPwz8/OcczpnPizmTyrK+jcLK6tr6RnGztLW9s7tn7h+0ZZQICi0a8Ug8eEQCZyG0FFMcHmIBJPA4dLzR7aTeeQQhWRQ21TgGNyCDkPmMEqWjnnnkRAEMCL66xo4vCE2rTsyytJn1zLJVsabCy8bOTRnlavTMH6cf0SSAUFFOpOzaVqzclAjFKIes5CQSYkJHZABdbUMSgHTT6Q8yfKqTPvYjoV+o8DT9O5GSQMpx4OnOgKihXKxNwv9q3UT5l27KwjhRENLZIj/hWEV4ggP3mQCq+FgbQgXTt2I6JBqE0tDmtniCjEBlJQ3GXsSwbNrVin1eqd3XyvWbHFERHaMTdIZsdIHq6A41UAtR9IRe0Ct6M56Nd+PD+Jy1Fox85hDNyfj6BWAvnsc=</latexit>

何波長進むか]/(2π)

２つ以上の波のプロファイルは、それぞれの足し算で得られる。

[pictures from the web]

フーリエ変換[https://www.yukisako.xyz/entry/fourier-transform]本日の話のキーワード

波束量子力学
!
e
n
o
D

??

波(の)束 (wave packet)

色々な波を束状に集めると、どんな形も表現できる。

???
分子・原子といったこの世の中のミクロな構造を記述する為の言語・方法論(Quantum Theory)

!
e
n
o
D

?

物理学(Physics)は、初めこの世のマクロな構造(身の回りの世界)を理解することから始まった。
[古典理論 (Classical Theory)]20世紀初め頃までに、この世の元素は原子から構成されているという考えに人類はたどり着き、以下のような模型が具体的に考えられた。
古典的な
周回運動

古典的な
電子(electron)

電子の発見 [J.J.トムソン (1897)][pictures from the web]

古典的な陽子(proton)
水素原子
α線

原子核の発見
[E.ラザフォード (1911)]

電子電荷の測定[R.A.ミリカン (1910)]

20世紀初め頃までに、この世の元素は原子から構成されているという考えに人類はたどり着き、以下のような模型が具体的に考えられた。
古典的な周回運動
クーロン引力

古典的な陽子(proton)

古典的な電子(electron)[電子電荷] =-1×(1.6×10-19[クーロン])[陽子電荷] =+1×(1.6×10-19[クーロン])

水素原子<latexit sha1_base64="a2I7TgUtuNTMBAByVgPmHK2tYaw=">AAACEXicdVDLSgMxFM34tr7qY+cmWARXZVJKbTdSEMGlgq1CW0omvW1DMw+SO2Id5iv8Abf6B+7ErV/gD/gdZmoFFT0QOJxzXzlepKRB131zZmbn5hcWl5ZzK6tr6xv5za2mCWMtoCFCFeorjxtQMoAGSlRwFWngvqfg0hsdZ/7lNWgjw+ACxxF0fD4IZF8Kjlbq5nfake8l12m3jXCDyQnXOEy7+YJbdF2XMUYzwg4rriW1WrXEqpRllkWBTHHWzb+3e6GIfQhQKG5Mi7kRdhI7TAoFaa4dG4i4GPEBtCwNuA+mk0yuT+m+VXq0H2r7AqQT9XtHwn1jxr5nK32OQ/Pby8S/vFaM/WonkUEUIwTic1E/VhRDmkVBe1KDQDW2hAst7a1UDLnmAm1gP7Z4mo8A05wN5uv39H/SLBVZpVg+LxfqR9OIlsgu2SMHhJFDUien5Iw0iCC35J48kEfnznlynp2Xz9IZZ9qzTX7Aef0AloCeiw==</latexit>

F Earth<latexit sha1_base64="YAE1svMzUbX2M0FD8x/Y/bb1nu8=">AAACEXicdVBJSgNBFK12jHFqh52bwiC4Ch2Jw0oCoriMYAZIQqiu/CRFqgeqfoux6VN4Abd6A3fi1hN4Ac9hJemAEX1Q8HjvT/XcUAqNjvNpzc0vLC4tZ1ayq2vrG5v21nZVB5HiUOGBDFTdZRqk8KGCAiXUQwXMcyXU3MHFyK/dgdIi8G9xGELLYz1fdAVnaKS2vdsMPTe+StpNhHuML5nCftK2c07+2BmBOnlnSlKlkCo5kqLctr+anYBHHvjIJdO6UXBCbMVmmOASkmwz0hAyPmA9aBjqMw90Kx5fn9ADo3RoN1Dm+UjH6s+OmHlaDz3XVHoM+/q3NxL/8hoRds9asfDDCMHnk0XdSFIM6CgK2hEKOMqhIYwrYW6lvM8U42gCm9niKjYATLImmOnv6f+kepQvnOSLN8Vc6TyNKEP2yD45JAVySkrkmpRJhXDyQJ7IM3mxHq1X6816n5TOWWnPDpmB9fENBYyeLw==</latexit>v Earth

太陽地球

v Earth,0<latexit sha1_base64="GxSdZJJPLImJb3n8KnvwrAPIrSU=">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</latexit>[picture from the web]

20世紀初め頃までに、この世の元素は原子から構成されているという考えに人類はたどり着き、以下のような模型が具体的に考えられた。
制動放射

古典的な電子(electron)
クーロン引力

古典的な陽子(proton)

しかし、
古典的な電磁気の理論によると、電磁波(=光)の放出により、電子はエネルギーを即座に失う。

原子は古典理論では不安定、記述不可!

水素原子

原子・分子といった
ミクロの世界の記述には、全く新しい方法論・言語が必要!!

20世紀初め頃までに、この世の元素は原子から構成されているという考えに人類はたどり着き、以下のような模型が具体的に考えられた。
量子化(quantisation)
制動放射

古典的な
電子(electron)
クーロン引力

古典的な陽子(proton)
水素原子

ここで、以下の大幅な思考のジャンプを行う。

[picture from the web]

ミクロな粒子の空間プロファイルは、
適切な波として表現されなければならない。
このアイディアの下に、ミクロな世界を記述する方法論・言語を
量子論/力学(quantum theory/mechanics)と呼ぶ。

20世紀初め頃までに、この世の元素は原子から構成されているという考えに人類はたどり着き、以下のような模型が具体的に考えられた。
量子化(quantisation)
制動放射

古典的な
電子(electron)
クーロン引力λr

古典的な陽子(proton)
水素原子

水素原子中の電子の適切な波はどんなプロファイルを取る?

[picture from the web]

安定に存在するためには、条件 ”n λ=2πr (n=1,2,...)” が必要[限られた波長の定常的な波のみが選ばれる。]

20世紀初め頃までに、この世の元素は原子から構成されているという考えに人類はたどり着き、以下のような模型が具体的に考えられた。
量子化(quantisation)
制動放射

古典的な
電子(electron)
クーロン斥力λr

古典的な陽子(proton)
水素原子

水素原子中の電子の適切な波はどんなプロファイルを取る?

[pictures from the web]

安定に存在するためには、条件 ”n λ=2πr (n=1,2,...)” が必要[限られた波長の定常的な波のみが選ばれる。]
この結果から、水素原子から放出される光スペクトルが説明できる!

先程は、電磁ポテンシャル内の電子を考えましたが、以下のような実験を行うと、自由な(=外から力が働いていない)
電子に対しても、量子的な粒子の驚くべき性質が見えてきます。
電子の二重スリット実験:

電子は二つの細いスリットのどちらかしか通れない

?
一発ずつ
電子を発射

スクリーンは電子の着地地点を個別に全て記録する。
Q. 多数の電子を発射した後、何が記録されているでしょうか?

[picture from the web]

先程は、電磁ポテンシャル内の電子を考えましたが、以下のような実験を行うと、自由な(=外から力が働いていない)
電子に対しても、量子的な粒子の驚くべき性質が見えてきます。
電子の二重スリット実験:

A. 干渉パターンが見える!
[picture from the web]

古典的な波の場合:◦ h1: スリット”1”からの波の、右端(absorber)での高さ◦ h2: スリット”2”からの波の、右端(absorber)での高さ

右端での波全体の強度”I12 = |h1 + h2|2”.
[The Feynman Lectures on Physics]

量子的な電子の空間プロファイル(=波動関数)の場合:◦ Φ1: スリット”1”を通る電子の波動関数の、右端(backstop)での値◦ Φ2: スリット”2”を通る電子の波動関数の、右端(backstop)での値

右端での波動関数全体の強度”P12 = |Φ1 + Φ2|2”.

電子の各地点における存在確率と理解する[The Feynman Lectures on Physics]

詳細:◦ 各電子は、点として観測される (粒子)

二
の
波

”
性
重

と
子
“粒

◦ 多数の観測結果は、干渉効果を含めた波動関数の強度( 対応する確率)に従う (波)ド・ブロイ-アインシュタインの関係式: 粒子と波の言語の対応(reduced) プランク定数◦ E =~⇥!
<latexit sha1_base64="cJ11qJcuZd2YWTHUZoAmGz8Us9o=">AAACFHicbVDLSgNBEJyNrxhfUQ8evAwGwVPYFVEvgiiCxwhGA9kl9E46yZDZBzO9QljyG/6AV/0Db+LVuz/gdziJe/BVMFBUdVM9FaZKGnLdd6c0Mzs3v1BerCwtr6yuVdc3bkySaYFNkahEt0IwqGSMTZKksJVqhChUeBsOzyf+7R1qI5P4mkYpBhH0Y9mTAshKnerWBT/h/iAEzX2SERruJxH2oVOtuXV3Cv6XeAWpsQKNTvXD7yYiizAmocCYtuemFOSgSQqF44qfGUxBDKGPbUtjsFlBPv3AmO9apct7ibYvJj5Vv2/kEBkzikI7GQENzG9vIv7ntTPqHQe5jNOMMBZfQb1McUr4pA3elRoFqZElILS0t3IxAA2CbGc/UkINQ6RxxRbj/a7hL7nZr3uH9YOrg9rpWVFRmW2zHbbHPHbETtkla7AmE2zMHtgje3LunWfnxXn9Gi05xc4m+wHn7RPEsJ3g</latexit>

エネルギー<latexit sha1_base64="N0ZGauBxH9j7G2W0dX0twOmnCFk=">AAACG3icbVDLSsNAFJ3UV62vqEtBBovgqiRS1I1QdOOygn1AE8pkOmmHTh7M3AglZOdv+ANu9Q/ciVsX/oDf4TTNwrYeGDiccy/nzvFiwRVY1rdRWlldW98ob1a2tnd298z9g7aKEklZi0Yikl2PKCZ4yFrAQbBuLBkJPME63vh26ncemVQ8Ch9gEjM3IMOQ+5wS0FLfPHbiwEvjDF9jZ+QRiR3gAVM4l8dZ36xaNSsHXiZ2QaqoQLNv/jiDiCYBC4EKolTPtmJwUyKBU8GyipMoFhM6JkPW0zQkOsxN839k+FQrA+xHUr8QcK7+3UhJoNQk8PRkQGCkFr2p+J/XS8C/clMexgmwkM6C/ERgiPC0FDzgklEQE00IlVzfiumISEJBVzeX4kkyZpBVdDH2Yg3LpH1esy9q9ft6tXFTVFRGR+gEnSEbXaIGukNN1EIUPaEX9IrejGfj3fgwPmejJaPYOURzML5+AWx2oX0=</latexit>

◦

角振動数

p = ~⇥k

空間運動量

波数

次元を合わせるために、新しい定数が必要◦ [hbar] = [エネルギー×時間] = [空間運動量×長さ][pictures from the web]

量子的な電子の空間プロファイル(=波動関数)の場合[どちらのスリットを通ったか調べる際]:◦ Φ’1: スリット”1”を通る電子の波動関数の、右端(backstop)での値◦ Φ’2: スリット”2”を通る電子の波動関数の、右端(backstop)での値

粒子の観測確率”P’12 = |Φ’1|2+|Φ’2|2 = P’1 + P’2”.
途中の微弱な操作によっても、最後の結果が大きく変化することが(干渉光の喪失!)[The Feynman Lectures on Physics]

さて、量子的な粒子の空間的なプロファイル(=波動関数)は、どんな形をしているのでしょうか?
<latexit sha1_base64="yzcRTHGmcIIoXBPb579TXd09hAM=">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</latexit>◦ [粒子のエネルギー] : Ep =◦ [粒子の空間運動量] :<latexit sha1_base64="V9v+kKG8AtP/T0vyHLUJHdpErcw=">AAACAnicbVDLSsNAFL2pr1pfVZdugkVwVRIR7bLgxmUF+4A2lMl00g6ZmQwzE6GE7PwBt/oH7sStP+IP+B1O2yxs64ELh3Pu5d57QsmoNp737ZQ2Nre2d8q7lb39g8Oj6vFJRyepwqSNE5aoXog0YVSQtqGGkZ5UBPGQkW4Y38387hNRmibi0UwlCTgaCxpRjIyVugPJw0zmw2rNq3tzuOvEL0gNCrSG1Z/BKMEpJ8JghrTu+540QYaUoZiRvDJINZEIx2hM+pYKxIkOsvm5uXthlZEbJcqWMO5c/TuRIa71lIe2kyMz0aveTPzP66cmagQZFTI1RODFoihlrknc2e/uiCqCDZtagrCi9lYXT5BC2NiElraECsXE5BUbjL8awzrpXNX9m/r1w3Wt2SgiKsMZnMMl+HALTbiHFrQBQwwv8ApvzrPz7nw4n4vWklPMnMISnK9f8UuYKg==</latexit>⇢pp2(non-relativistic), p 2 c2 + m2 c4 (relativistic)2m

p (何か固有の値を持つ)(wave) ⇠ superposition (over k ) of e i!k t+ikk ·xx<latexit sha1_base64="MkIoPwsUZv/ZYajda2crfJitGtE=">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</latexit>さて、量子的な粒子の空間的なプロファイル(=波動関数)は、どんな形をしているのでしょうか?
<latexit sha1_base64="yzcRTHGmcIIoXBPb579TXd09hAM=">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</latexit>◦ [粒子のエネルギー] : Ep =◦ [粒子の空間運動量] :<latexit sha1_base64="V9v+kKG8AtP/T0vyHLUJHdpErcw=">AAACAnicbVDLSsNAFL2pr1pfVZdugkVwVRIR7bLgxmUF+4A2lMl00g6ZmQwzE6GE7PwBt/oH7sStP+IP+B1O2yxs64ELh3Pu5d57QsmoNp737ZQ2Nre2d8q7lb39g8Oj6vFJRyepwqSNE5aoXog0YVSQtqGGkZ5UBPGQkW4Y38387hNRmibi0UwlCTgaCxpRjIyVugPJw0zmw2rNq3tzuOvEL0gNCrSG1Z/BKMEpJ8JghrTu+540QYaUoZiRvDJINZEIx2hM+pYKxIkOsvm5uXthlZEbJcqWMO5c/TuRIa71lIe2kyMz0aveTPzP66cmagQZFTI1RODFoihlrknc2e/uiCqCDZtagrCi9lYXT5BC2NiElraECsXE5BUbjL8awzrpXNX9m/r1w3Wt2SgiKsMZnMMl+HALTbiHFrQBQwwv8ApvzrPz7nw4n4vWklPMnMISnK9f8UuYKg==</latexit>⇢pp2(non-relativistic), p 2 c2 + m2 c4 (relativistic)2m

p (何か固有の値を持つ)(wave) ⇠ superposition (over k ) of e i!k t+ikk ·xx<latexit sha1_base64="MkIoPwsUZv/ZYajda2crfJitGtE=">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</latexit>

係式

の関
ン
イ
タ
ュ
シ
イ-アイン

ド・ブロ

note: (metric) = diag(-1,1,1,1);
自然単位を採用: ~ = c = 1<latexit sha1_base64="sQDJ/1mOgqtSR/dQ1Wgm9Fi4G3M=">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</latexit>

平面波(plane wave)による粒子の表現:<latexit sha1_base64="WX6IBCSNiwQbaZWLvq31TBTwgXE=">AAACAnicdVDLSgMxFM34rPVVdekmWARXZVJKO90V3LisYB/QDiWTZtrQZCYkGaEMs/MH3OofuBO3/og/4HeYaStY0QMXDufcy733BJIzbVz3w9nY3Nre2S3sFfcPDo+OSyenXR0nitAOiXms+gHWlLOIdgwznPalolgEnPaC2XXu9+6p0iyO7sxcUl/gScRCRrCxUm8oRZDKbFQquxXXdRFCMCeoUXctaTa9KvIgyi2LMlihPSp9DscxSQSNDOFY6wFypfFTrAwjnGbFYaKpxGSGJ3RgaYQF1X66ODeDl1YZwzBWtiIDF+rPiRQLrecisJ0Cm6n+7eXiX94gMaHnpyySiaERWS4KEw5NDPPf4ZgpSgyfW4KJYvZWSKZYYWJsQmtbAoVn1GRFG8z39/B/0q1WUL1Su62VW94qogI4BxfgCiDQAC1wA9qgAwiYgUfwBJ6dB+fFeXXelq0bzmrmDKzBef8CQoiYYA==</latexit>(t, x ) ⇠ e iEp t+ipp·xx

p<latexit sha1_base64="v5V60vmmDt8O3zYv6J1fN+4SHcI=">AAACCHicbVBLSgNBFOyJvxh/UZduGoPgKsyIaDZCwI3LCOYDyRDedDpJk56esfuNEIZcwAu41Ru4E7fewgt4DjvJLExiwYOi6j3qUUEshUHX/XZya+sbm1v57cLO7t7+QfHwqGGiRDNeZ5GMdCsAw6VQvI4CJW/FmkMYSN4MRrdTv/nEtRGResBxzP0QBkr0BQO0kt8ZBqDpDWV2vG6x5JbdGegq8TJSIhlq3eJPpxexJOQKmQRj2p4bo5+CRsEknxQ6ieExsBEMeNtSBSE3fjp7ekLPrNKj/UjbUUhn6t+LFEJjxmFgN0PAoVn2puJ/XjvBfsVPhYoT5IrNg/qJpBjRaQO0JzRnKMeWANPC/krZEDQwtD0tpAQaRhwnBVuMt1zDKmlclL2r8uX9ZalaySrKkxNySs6JR65JldyRGqkTRh7JC3klb86z8+58OJ/z1ZyT3RyTBThfv7JwmOo=</latexit>⇣= e+ipµ xµ⌘◦ 固有の運動量を表現◦ エネルギーの運動量依存性(分散関係)
を正しく表現しているEp<latexit sha1_base64="IrgTRZ8rOZSr+XnbQ4/nq6MZoXE=">AAACBnicdVDLSsNAFJ3UV62vqks3g0VwFZLa2mRXEMFlBfuANJTJdNoOnUnCzEQoIXt/wK3+gTtx62/4A36Hk7aCFT1w4XDOvdx7TxAzKpVlfRiFtfWNza3idmlnd2//oHx41JFRIjBp44hFohcgSRgNSVtRxUgvFgTxgJFuML3K/e49EZJG4Z2axcTnaBzSEcVIacm7HqT9mAdpnGWDcsUyXadev3CgZVqWW3UbmriuazdsaGslRwUs0RqUP/vDCCechAozJKVnW7HyUyQUxYxkpX4iSYzwFI2Jp2mIOJF+Oj85g2daGcJRJHSFCs7VnxMp4lLOeKA7OVIT+dvLxb88L1Ejx09pGCeKhHixaJQwqCKY/w+HVBCs2EwThAXVt0I8QQJhpVNa2RIINCUqK+lgvr+H/5NO1bQvzdptrdJ0lhEVwQk4BefABg3QBDegBdoAgwg8gifwbDwYL8ar8bZoLRjLmWOwAuP9C71TmlM=</latexit>( 幅広く使用されている)

ところが！
この結果を思い出そう →量子力学的粒子の配位は、空間的に局在化している筈(?)

ここまでのまとめ:[A. Tonomura, Proceedings of the National Academy of Sciences,
USA, 102, 14952 (2005)]• 平面波を利用した量子的な粒子の表現は、“粒子としての”幾つかの性質(運動量、分散関係)を正しく表現• しかし、他の重要な性質(局在性)は全く表現されていない!

ところが！
この結果を思い出そう →量子力学的粒子の配位は、空間的に局在化している筈(?)

ここまでのまとめ:[A. Tonomura, Proceedings of the National Academy of Sciences,
USA, 102, 14952 (2005)]• 平面波を利用した量子的な粒子の表現は、“粒子としての”幾つかの性質(運動量、分散関係)を正しく表現• しかし、他の重要な性質(局在性)は全く表現されていない!
波束を使用すれば、局在性も加味した完全な量子力学的粒子の表現が可能になる!

本日の話のキーワード

波束量子力学

に
算
!
!
計
e
e
n
n
o
o
D??
D
な
???
的
準
標
波(の)束 (wave packet)
分子・原子といったの。
でう
波
この世の中のミクロな構造を
ょ
面
し
平
ま
、
記述する為の言語・方法論
し
に
論
次
議
(Quantum
Theory)て色々な波を束状に集めると、し関
どんな形も表現できる。

!
e
n
o
D

?

物理学(Physics)は、初めこの世のマクロな構造(身の回りの世界)を理解することから始まった。
[古典理論 (Classical Theory)]

先程話お話した通り、量子力学/論では、諸々の議論は対応する確率量の計算になります。
物理過程一つ一つで何が起こるかは測ってみるまでわからない。しかし、確率量の性質・分布は理論から決定でき、実験でも測れる。

先程話お話した通り、量子力学/論では、諸々の議論は対応する確率量の計算になります。
物理過程一つ一つで何が起こるかは測ってみるまでわからない。しかし、確率量の性質・分布は理論から決定でき、実験でも測れる。
これの理解は、以下の身近(!?)な例の理解と基本的に同じです。

◦ 一枚300円の宝くじを一枚買って、一等賞(7億円!)が当たることは極々々々々々偶にはある。
[一つ一つの過程の極端な例][https://univ-juken.com/takarakuzi]

先程話お話した通り、量子力学/論では、諸々の議論は[賭博破戒録カイジ、沼編]対応する確率量の計算になります。
物理過程一つ一つで何が起こるかは測ってみるまでわからない。しかし、確率量の性質・分布は理論から決定でき、実験でも測れる。
これの理解は、以下の身近(!?)な例の理解と基本的に同じです。

◦ 一枚300円の宝くじを一枚買って、一等賞(7億円!)が当たることは極々々々々々偶にはある。
[一つ一つの過程の極端な例][https://univ-juken.com/takarakuzi]

先程話お話した通り、量子力学/論では、諸々の議論は[賭博破戒録カイジ、沼編]対応する確率量の計算になります。
物理過程一つ一つで何が起こるかは測ってみるまでわからない。しかし、確率量の性質・分布は理論から決定でき、実験でも測れる。
これの理解は、以下の身近(!?)な例の理解と基本的に同じです。

◦ 一枚300円の宝くじを一枚買って、一等賞(7億円!)が当たることは極々々々々々偶にはある。
[一つ一つの過程の極端な例]◦ 賞金の期待値は、理論的に計算可能。
右のデータを下にすると、約148円。
基本的には大きく損をする。
[宝くじの確率分布の性質の一例][https://univ-juken.com/takarakuzi]

この世の根本的なミクロな世界が、確率によって支配されるのは不思議な気がしますが、これは観測事実なので、認めて先に進みましょう。
更に以下では、本質に集中するため、最も単純な (1 2)-体
量子遷移に集中しましょう。
これは、重たい粒子 が相互作用により、より軽い粒子対 1 2に崩壊する過程。
<latexit sha1_base64="7c7djoFrFNVkMn8VIlDEmS3vZzc=">AAAB/3icbVDLSsNAFJ3UV62vqks3g0VwVRIR7bLgxmUF+4A2lMn0phk6mYSZG6GELvwBt/oH7sStn+IP+B0mbRa29cCFwzn3cu89XiyFQdv+tkobm1vbO+Xdyt7+weFR9fikY6JEc2jzSEa65zEDUihoo0AJvVgDCz0JXW9yl/vdJ9BGROoRpzG4IRsr4QvOMJcGrUAMqzW7bs9B14lTkBop0BpWfwajiCchKOSSGdN37BjdlGkUXMKsMkgMxIxP2Bj6GVUsBOOm81tn9CJTRtSPdFYK6Vz9O5Gy0Jhp6GWdIcPArHq5+J/XT9BvuKlQcYKg+GKRn0iKEc0fpyOhgaOcZoRxLbJbKQ+YZhyzeJa2eJpNAGeVLBhnNYZ10rmqOzf164frWrNRRFQmZ+ScXBKH3JImuSct0iacBOSFvJI369l6tz6sz0VrySpmTskSrK9fDiGWhg==</latexit><latexit sha1_base64="8OIYCSue9xhBJ+XP6PvfsDh2oMs=">AAACE3icbVDLSsNAFL2pr1pfUcGNm8EiuCpJKdplwY3LCvYBTQiT6aQZOnkwMxFK7Gf4A271D9yJWz/AH/A7nLZZ2NYDwxzOuZd77/FTzqSyrG+jtLG5tb1T3q3s7R8cHpnHJ12ZZILQDkl4Ivo+lpSzmHYUU5z2U0Fx5HPa88e3M7/3SIVkSfygJil1IzyKWcAIVlryzDOnHTLkqAQ5acg8e/HVPbNq1aw50DqxC1KFAm3P/HGGCckiGivCsZQD20qVm2OhGOF0WnEySVNMxnhEB5rGOKLSzef7T9GlVoYoSIR+sUJz9W9HjiMpJ5GvKyOsQrnqzcT/vEGmgqabszjNFI3JYlCQcaTPnYWBhkxQovhEE0wE07siEmKBidKRLU3xBR5TNa3oYOzVGNZJt16zr2uN+0a11SwiKsM5XMAV2HADLbiDNnSAwBO8wCu8Gc/Gu/FhfC5KS0bRcwpLML5+AfhanWU=</latexit>

!

初状態
(一体)

1 2

<latexit sha1_base64="xVFJb4+lUTUWnvZHg1YLj2+F4U0=">AAACCHicbVBLSgNBEO2Jvxh/UZduGoPgKsyEoFkG3LiMYD6QDKGnU5M06fnYXSOEIRfwAm71Bu7ErbfwAp7DzmQWmvigqMd7VVTxvFgKjbb9ZRU2Nre2d4q7pb39g8Oj8vFJR0eJ4tDmkYxUz2MapAihjQIl9GIFLPAkdL3pzcLvPoLSIgrvcRaDG7BxKHzBGRrJHcQTMXRo1mrDcsWu2hnoOnFyUiE5WsPy92AU8SSAELlkWvcdO0Y3ZQoFlzAvDRINMeNTNoa+oSELQLtp9vScXhhlRP1ImQqRZurvjZQFWs8Cz0wGDCd61VuI/3n9BP2Gm4owThBCvjzkJ5JiRBcJ0JFQwFHODGFcCfMr5ROmGEeT058rnmJTwHnJBOOsxrBOOrWqc1Wt39UrzUYeUZGckXNySRxyTZrklrRIm3DyQJ7JC3m1nqw36936WI4WrHznlPyB9fkDOkqZ3g==</latexit>

終状態
(二体)

p1
<latexit sha1_base64="9vrZ4UIK7lZW6sJmsMza5/IipPs=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9lV0R4LXjxWsB/QLiWbZtvYbBKSrFCW/gcvHhTx6v/x5r8xbfegrQ8GHu/NMDMvUpwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jEw1oU0iudSdCBvKmaBNyyynHaUpTiJO29H4dua3n6g2TIoHO1E0TPBQsJgRbJ3U6qkR6wf9csWv+nOgVRLkpAI5Gv3yV28gSZpQYQnHxnQDX9kww9oywum01EsNVZiM8ZB2HRU4oSbM5tdO0ZlTBiiW2pWwaK7+nshwYswkiVxngu3ILHsz8T+vm9q4FmZMqNRSQRaL4pQjK9HsdTRgmhLLJ45gopm7FZER1phYF1DJhRAsv7xKWhfV4Lp6eX9VqdfyOIpwAqdwDgHcQB3uoAFNIPAIz/AKb570Xrx372PRWvDymWP4A+/zBzrgjuE=</latexit>

1P<latexit sha1_base64="aWQnsngYA/hZu/If6dJrzT9pRL4=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0m0aI8FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq777ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmGG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU00TTKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwqafMZmkhkqyXBSmHJkY5Y+jMVOUGD6zBBPF7K2IRFhhYmw8FRuCt/ryOule1b2b+vVDo9ZqFnGU4QzO4RI8uIUW3EMbOkAggmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPiU44d</latexit>

time<latexit sha1_base64="LUu0m8UM3ApNR1oSOz9G6VCsVCY=">AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqexW0R4LXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurOd9o8La+sbmVnG7tLO7t39QPjxqG5VqylpUCaU7ITFMcMlallvBOolmJA4FewjHNzP/4Ylpw5W8t5OEBTEZSh5xSqyT2r1kxPu1frniVb058Crxc1KBHM1++as3UDSNmbRUEGO6vpfYICPacirYtNRLDUsIHZMh6zoqScxMkM2vneIzpwxwpLQrafFc/T2RkdiYSRy6zpjYkVn2ZuJ/Xje1UT3IuExSyyRdLIpSga3Cs9fxgGtGrZg4Qqjm7lZMR0QTal1AJReCv/zyKmnXqv5V9eLustKo53EU4QRO4Rx8uIYG3EITWkDhEZ7hFd6QQi/oHX0sWgsonzmGP0CfPzxkjuI=</latexit>

p2

2

この世の根本的なミクロな世界が、確率によって支配されるのは不思議な気がしますが、これは観測事実なので、認めて先に進みましょう。
更に以下では、本質に集中するため、最も単純な (1 2)-体
量子遷移に集中しましょう。
<latexit sha1_base64="/8dAKD5AcGb3Lu32MZXW5LwdsBs=">AAACFXicbVDLSsNAFL3xWesr6kZwM1hEVyUpRbssuHFZwT6gCWEynTRDJw9mJkIJ9Tf8Abf6B+7ErWt/wO9w2mZhWw8MczjnXu69x085k8qyvo219Y3Nre3STnl3b//g0Dw67sgkE4S2ScIT0fOxpJzFtK2Y4rSXCoojn9OuP7qd+t1HKiRL4gc1Tqkb4WHMAkaw0pJnnjqtkCFHJchJQ3bp2cVf88yKVbVmQKvELkgFCrQ888cZJCSLaKwIx1L2bStVbo6FYoTTSdnJJE0xGeEh7Wsa44hKN59dMEEXWhmgIBH6xQrN1L8dOY6kHEe+roywCuWyNxX/8/qZChpuzuI0UzQm80FBxpE+eBoHGjBBieJjTTARTO+KSIgFJkqHtjDFF3hE1aSsg7GXY1glnVrVvq7W7+uVZqOIqARncA5XYMMNNOEOWtAGAk/wAq/wZjwb78aH8TkvXTOKnhNYgPH1C8yQncc=</latexit><latexit sha1_base64="8OIYCSue9xhBJ+XP6PvfsDh2oMs=">AAACE3icbVDLSsNAFL2pr1pfUcGNm8EiuCpJKdplwY3LCvYBTQiT6aQZOnkwMxFK7Gf4A271D9yJWz/AH/A7nLZZ2NYDwxzOuZd77/FTzqSyrG+jtLG5tb1T3q3s7R8cHpnHJ12ZZILQDkl4Ivo+lpSzmHYUU5z2U0Fx5HPa88e3M7/3SIVkSfygJil1IzyKWcAIVlryzDOnHTLkqAQ5acg8e/HVPbNq1aw50DqxC1KFAm3P/HGGCckiGivCsZQD20qVm2OhGOF0WnEySVNMxnhEB5rGOKLSzef7T9GlVoYoSIR+sUJz9W9HjiMpJ5GvKyOsQrnqzcT/vEGmgqabszjNFI3JYlCQcaTPnYWBhkxQovhEE0wE07siEmKBidKRLU3xBR5TNa3oYOzVGNZJt16zr2uN+0a11SwiKsM5XMAV2HADLbiDNnSAwBO8wCu8Gc/Gu/FhfC5KS0bRcwpLML5+AfhanWU=</latexit>

!

初状態
(一体)

!

1 2
終状態
(二体)

初状態
(一体)

0 0
1 2
終状態
(二体)

他の相互作用が存在するなら、
0 0
別種のより軽い粒子対 1 2に崩壊する過程も存在。
<latexit sha1_base64="sgmRvidVFuMEseT+4xbEtoHTryg=">AAACDHicbVDLSsNAFL2pr1ofjbp0M1hEVyUpRbssuHFZwT6gDWEynbRDJ5MwMxFK6C/4A271D9yJW//BH/A7nLZZ2NYDl3s4517u5QQJZ0o7zrdV2Nre2d0r7pcODo+Oy/bJaUfFqSS0TWIey16AFeVM0LZmmtNeIimOAk67weRu7nefqFQsFo96mlAvwiPBQkawNpJvlwfJmF35Llr2mm9XnKqzANokbk4qkKPl2z+DYUzSiApNOFaq7zqJ9jIsNSOczkqDVNEEkwke0b6hAkdUedni8Rm6NMoQhbE0JTRaqH83MhwpNY0CMxlhPVbr3lz8z+unOmx4GRNJqqkgy0NhypGO0TwFNGSSEs2nhmAimfkVkTGWmGiT1cqVQOIJ1bOSCcZdj2GTdGpV96Zaf6hXmo08oiKcwwVcgwu30IR7aEEbCKTwAq/wZj1b79aH9bkcLVj5zhmswPr6BYJHmnE=</latexit>

p1

p1<latexit sha1_base64="9vrZ4UIK7lZW6sJmsMza5/IipPs=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9lV0R4LXjxWsB/QLiWbZtvYbBKSrFCW/gcvHhTx6v/x5r8xbfegrQ8GHu/NMDMvUpwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jEw1oU0iudSdCBvKmaBNyyynHaUpTiJO29H4dua3n6g2TIoHO1E0TPBQsJgRbJ3U6qkR6wf9csWv+nOgVRLkpAI5Gv3yV28gSZpQYQnHxnQDX9kww9oywum01EsNVZiM8ZB2HRU4oSbM5tdO0ZlTBiiW2pWwaK7+nshwYswkiVxngu3ILHsz8T+vm9q4FmZMqNRSQRaL4pQjK9HsdTRgmhLLJ45gopm7FZER1phYF1DJhRAsv7xKWhfV4Lp6eX9VqdfyOIpwAqdwDgHcQB3uoAFNIPAIz/AKb570Xrx372PRWvDymWP4A+/zBzrgjuE=</latexit><latexit sha1_base64="62SEb41BCfWrxua6uowuCorDQqU=">AAACAnicbVDLSsNAFL2pr1pfVZdugkV0VRIp2mXBjcsK9gFtKJPpTTtkMgkzE6GE7vwBt/oH7sStP+IP+B1O2yxs64ELh3Pu5d57/IQzpR3n2ypsbG5t7xR3S3v7B4dH5eOTtopTSbFFYx7Lrk8UciawpZnm2E0kksjn2PHDu5nfeUKpWCwe9SRBLyIjwQJGiTZSp5+M2eXAHZQrTtWZw14nbk4qkKM5KP/0hzFNIxSacqJUz3US7WVEakY5Tkv9VGFCaEhG2DNUkAiVl83PndoXRhnaQSxNCW3P1b8TGYmUmkS+6YyIHqtVbyb+5/VSHdS9jIkk1SjoYlGQclvH9ux3e8gkUs0nhhAqmbnVpmMiCdUmoaUtviQh6mnJBOOuxrBO2tdV96Zae6hVGvU8oiKcwTlcgQu30IB7aEILKITwAq/wZj1b79aH9bloLVj5zCkswfr6Bdmcl3s=</latexit>

1P<latexit sha1_base64="aWQnsngYA/hZu/If6dJrzT9pRL4=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0m0aI8FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq777ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmGG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU00TTKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwqafMZmkhkqyXBSmHJkY5Y+jMVOUGD6zBBPF7K2IRFhhYmw8FRuCt/ryOule1b2b+vVDo9ZqFnGU4QzO4RI8uIUW3EMbOkAggmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPiU44d</latexit>

P01<latexit sha1_base64="aWQnsngYA/hZu/If6dJrzT9pRL4=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0m0aI8FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq777ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmGG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU00TTKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwqafMZmkhkqyXBSmHJkY5Y+jMVOUGD6zBBPF7K2IRFhhYmw8FRuCt/ryOule1b2b+vVDo9ZqFnGU4QzO4RI8uIUW3EMbOkAggmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPiU44d</latexit><latexit sha1_base64="jmOMjMHxLc5N+UpY1cxYt7+oQ24=">AAACAnicbVDLSgNBEOz1GeMr6tHLYBA9hd0QNMeAF48RzAOSJcxOZpNhZx/M9Aphyc0f8Kp/4E28+iP+gN/hJNmDSSxoKKq66e7yEik02va3tbG5tb2zW9gr7h8cHh2XTk7bOk4V4y0Wy1h1Paq5FBFvoUDJu4niNPQk73jB3czvPHGlRRw94iThbkhHkfAFo2ikTj8Zi6tBdVAq2xV7DrJOnJyUIUdzUPrpD2OWhjxCJqnWPcdO0M2oQsEknxb7qeYJZQEd8Z6hEQ25drP5uVNyaZQh8WNlKkIyV/9OZDTUehJ6pjOkONar3kz8z+ul6NfdTERJijxii0V+KgnGZPY7GQrFGcqJIZQpYW4lbEwVZWgSWtriKRpwnBZNMM5qDOukXa04N5XaQ63cqOcRFeAcLuAaHLiFBtxDE1rAIIAXeIU369l6tz6sz0XrhpXPnMESrK9f2zSXfA==</latexit>

time<latexit sha1_base64="LUu0m8UM3ApNR1oSOz9G6VCsVCY=">AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqexW0R4LXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurOd9o8La+sbmVnG7tLO7t39QPjxqG5VqylpUCaU7ITFMcMlallvBOolmJA4FewjHNzP/4Ylpw5W8t5OEBTEZSh5xSqyT2r1kxPu1frniVb058Crxc1KBHM1++as3UDSNmbRUEGO6vpfYICPacirYtNRLDUsIHZMh6zoqScxMkM2vneIzpwxwpLQrafFc/T2RkdiYSRy6zpjYkVn2ZuJ/Xje1UT3IuExSyyRdLIpSga3Cs9fxgGtGrZg4Qqjm7lZMR0QTal1AJReCv/zyKmnXqv5V9eLustKo53EU4QRO4Rx8uIYG3EITWkDhEZ7hFd6QQi/oHX0sWgsonzmGP0CfPzxkjuI=</latexit>

p2

2

time

p2

0
2

この世の根本的なミクロな世界が、確率によって支配されるのは不思議な気がしますが、これは観測事実なので、認めて先に進みましょう。
更に以下では、本質に集中するため、最も単純な (1 2)-体
量子遷移に集中しましょう。
<latexit sha1_base64="/8dAKD5AcGb3Lu32MZXW5LwdsBs=">AAACFXicbVDLSsNAFL3xWesr6kZwM1hEVyUpRbssuHFZwT6gCWEynTRDJw9mJkIJ9Tf8Abf6B+7ErWt/wO9w2mZhWw8MczjnXu69x085k8qyvo219Y3Nre3STnl3b//g0Dw67sgkE4S2ScIT0fOxpJzFtK2Y4rSXCoojn9OuP7qd+t1HKiRL4gc1Tqkb4WHMAkaw0pJnnjqtkCFHJchJQ3bp2cVf88yKVbVmQKvELkgFCrQ888cZJCSLaKwIx1L2bStVbo6FYoTTSdnJJE0xGeEh7Wsa44hKN59dMEEXWhmgIBH6xQrN1L8dOY6kHEe+roywCuWyNxX/8/qZChpuzuI0UzQm80FBxpE+eBoHGjBBieJjTTARTO+KSIgFJkqHtjDFF3hE1aSsg7GXY1glnVrVvq7W7+uVZqOIqARncA5XYMMNNOEOWtAGAk/wAq/wZjwb78aH8TkvXTOKnhNYgPH1C8yQncc=</latexit><latexit sha1_base64="8OIYCSue9xhBJ+XP6PvfsDh2oMs=">AAACE3icbVDLSsNAFL2pr1pfUcGNm8EiuCpJKdplwY3LCvYBTQiT6aQZOnkwMxFK7Gf4A271D9yJWz/AH/A7nLZZ2NYDwxzOuZd77/FTzqSyrG+jtLG5tb1T3q3s7R8cHpnHJ12ZZILQDkl4Ivo+lpSzmHYUU5z2U0Fx5HPa88e3M7/3SIVkSfygJil1IzyKWcAIVlryzDOnHTLkqAQ5acg8e/HVPbNq1aw50DqxC1KFAm3P/HGGCckiGivCsZQD20qVm2OhGOF0WnEySVNMxnhEB5rGOKLSzef7T9GlVoYoSIR+sUJz9W9HjiMpJ5GvKyOsQrnqzcT/vEGmgqabszjNFI3JYlCQcaTPnYWBhkxQovhEE0wE07siEmKBidKRLU3xBR5TNa3oYOzVGNZJt16zr2uN+0a11SwiKsM5XMAV2HADLbiDNnSAwBO8wCu8Gc/Gu/FhfC5KS0bRcwpLML5+AfhanWU=</latexit>

!

初状態
(一体)

!

1 2
終状態
(二体)

初状態
(一体)

p1

0 0
1 2
終状態
(二体)

p1
<latexit sha1_base64="9vrZ4UIK7lZW6sJmsMza5/IipPs=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9lV0R4LXjxWsB/QLiWbZtvYbBKSrFCW/gcvHhTx6v/x5r8xbfegrQ8GHu/NMDMvUpwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jEw1oU0iudSdCBvKmaBNyyynHaUpTiJO29H4dua3n6g2TIoHO1E0TPBQsJgRbJ3U6qkR6wf9csWv+nOgVRLkpAI5Gv3yV28gSZpQYQnHxnQDX9kww9oywum01EsNVZiM8ZB2HRU4oSbM5tdO0ZlTBiiW2pWwaK7+nshwYswkiVxngu3ILHsz8T+vm9q4FmZMqNRSQRaL4pQjK9HsdTRgmhLLJ45gopm7FZER1phYF1DJhRAsv7xKWhfV4Lp6eX9VqdfyOIpwAqdwDgHcQB3uoAFNIPAIz/AKb570Xrx372PRWvDymWP4A+/zBzrgjuE=</latexit><latexit sha1_base64="62SEb41BCfWrxua6uowuCorDQqU=">AAACAnicbVDLSsNAFL2pr1pfVZdugkV0VRIp2mXBjcsK9gFtKJPpTTtkMgkzE6GE7vwBt/oH7sStP+IP+B1O2yxs64ELh3Pu5d57/IQzpR3n2ypsbG5t7xR3S3v7B4dH5eOTtopTSbFFYx7Lrk8UciawpZnm2E0kksjn2PHDu5nfeUKpWCwe9SRBLyIjwQJGiTZSp5+M2eXAHZQrTtWZw14nbk4qkKM5KP/0hzFNIxSacqJUz3US7WVEakY5Tkv9VGFCaEhG2DNUkAiVl83PndoXRhnaQSxNCW3P1b8TGYmUmkS+6YyIHqtVbyb+5/VSHdS9jIkk1SjoYlGQclvH9ux3e8gkUs0nhhAqmbnVpmMiCdUmoaUtviQh6mnJBOOuxrBO2tdV96Zae6hVGvU8oiKcwTlcgQu30IB7aEILKITwAq/wZj1b79aH9bloLVj5zCkswfr6Bdmcl3s=</latexit>

1P<latexit sha1_base64="aWQnsngYA/hZu/If6dJrzT9pRL4=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0m0aI8FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq777ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmGG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU00TTKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwqafMZmkhkqyXBSmHJkY5Y+jMVOUGD6zBBPF7K2IRFhhYmw8FRuCt/ryOule1b2b+vVDo9ZqFnGU4QzO4RI8uIUW3EMbOkAggmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPiU44d</latexit>

P01

後で示すように、<latexit sha1_base64="aWQnsngYA/hZu/If6dJrzT9pRL4=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0m0aI8FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq777ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmGG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU00TTKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwqafMZmkhkqyXBSmHJkY5Y+jMVOUGD6zBBPF7K2IRFhhYmw8FRuCt/ryOule1b2b+vVDo9ZqFnGU4QzO4RI8uIUW3EMbOkAggmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPiU44d</latexit>

平面波の計算では、<latexit sha1_base64="jmOMjMHxLc5N+UpY1cxYt7+oQ24=">AAACAnicbVDLSgNBEOz1GeMr6tHLYBA9hd0QNMeAF48RzAOSJcxOZpNhZx/M9Aphyc0f8Kp/4E28+iP+gN/hJNmDSSxoKKq66e7yEik02va3tbG5tb2zW9gr7h8cHh2XTk7bOk4V4y0Wy1h1Paq5FBFvoUDJu4niNPQk73jB3czvPHGlRRw94iThbkhHkfAFo2ikTj8Zi6tBdVAq2xV7DrJOnJyUIUdzUPrpD2OWhjxCJqnWPcdO0M2oQsEknxb7qeYJZQEd8Z6hEQ25drP5uVNyaZQh8WNlKkIyV/9OZDTUehJ6pjOkONar3kz8z+ul6NfdTERJijxii0V+KgnGZPY7GQrFGcqJIZQpYW4lbEwVZWgSWtriKRpwnBZNMM5qDOukXa04N5XaQ63cqOcRFeAcLuAaHLiFBtxDE1rAIIAXeIU369l6tz6sz0XrhpXPnMESrK9f2zSXfA==</latexit>

time

Γ12

<latexit sha1_base64="LUu0m8UM3ApNR1oSOz9G6VCsVCY=">AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqexW0R4LXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurOd9o8La+sbmVnG7tLO7t39QPjxqG5VqylpUCaU7ITFMcMlallvBOolmJA4FewjHNzP/4Ylpw5W8t5OEBTEZSh5xSqyT2r1kxPu1frniVb058Crxc1KBHM1++as3UDSNmbRUEGO6vpfYICPacirYtNRLDUsIHZMh6zoqScxMkM2vneIzpwxwpLQrafFc/T2RkdiYSRy6zpjYkVn2ZuJ/Xje1UT3IuExSyyRdLIpSga3Cs9fxgGtGrZg4Qqjm7lZMR0QTal1AJReCv/zyKmnXqv5V9eLustKo53EU4QRO4Rx8uIYG3EITWkDhEZ7hFd6QQi/oHX0sWgsonzmGP0CfPzxkjuI=</latexit>

p2

2

time

p2

Γ’12

0
2

二つの崩壊過程は、それぞれ対応する単位時間あたりの崩壊確率 Γ12, Γ’12 で特徴づく。

この世の根本的なミクロな世界が、確率によって支配されるのはΓtotal := Γ12 + Γ’12 を全崩壊確率と呼ぶ。
◦
不思議な気がしますが、これは観測事実なので、認めて先にこれは、単位時間辺りにどちらかに崩壊する確率。
進みましょう。
τ := 1/(Γtotal) は時間の物理単位を持ち、これは崩壊する◦
更に以下では、本質に集中するため、最も単純な(1 2)-体粒子の平均寿命を表す。
量子遷移に集中しましょう。
<latexit sha1_base64="/8dAKD5AcGb3Lu32MZXW5LwdsBs=">AAACFXicbVDLSsNAFL3xWesr6kZwM1hEVyUpRbssuHFZwT6gCWEynTRDJw9mJkIJ9Tf8Abf6B+7ErWt/wO9w2mZhWw8MczjnXu69x085k8qyvo219Y3Nre3STnl3b//g0Dw67sgkE4S2ScIT0fOxpJzFtK2Y4rSXCoojn9OuP7qd+t1HKiRL4gc1Tqkb4WHMAkaw0pJnnjqtkCFHJchJQ3bp2cVf88yKVbVmQKvELkgFCrQ888cZJCSLaKwIx1L2bStVbo6FYoTTSdnJJE0xGeEh7Wsa44hKN59dMEEXWhmgIBH6xQrN1L8dOY6kHEe+roywCuWyNxX/8/qZChpuzuI0UzQm80FBxpE+eBoHGjBBieJjTTARTO+KSIgFJkqHtjDFF3hE1aSsg7GXY1glnVrVvq7W7+uVZqOIqARncA5XYMMNNOEOWtAGAk/wAq/wZjwb78aH8TkvXTOKnhNYgPH1C8yQncc=</latexit><latexit sha1_base64="8OIYCSue9xhBJ+XP6PvfsDh2oMs=">AAACE3icbVDLSsNAFL2pr1pfUcGNm8EiuCpJKdplwY3LCvYBTQiT6aQZOnkwMxFK7Gf4A271D9yJWz/AH/A7nLZZ2NYDwxzOuZd77/FTzqSyrG+jtLG5tb1T3q3s7R8cHpnHJ12ZZILQDkl4Ivo+lpSzmHYUU5z2U0Fx5HPa88e3M7/3SIVkSfygJil1IzyKWcAIVlryzDOnHTLkqAQ5acg8e/HVPbNq1aw50DqxC1KFAm3P/HGGCckiGivCsZQD20qVm2OhGOF0WnEySVNMxnhEB5rGOKLSzef7T9GlVoYoSIR+sUJz9W9HjiMpJ5GvKyOsQrnqzcT/vEGmgqabszjNFI3JYlCQcaTPnYWBhkxQovhEE0wE07siEmKBidKRLU3xBR5TNa3oYOzVGNZJt16zr2uN+0a11SwiKsM5XMAV2HADLbiDNnSAwBO8wCu8Gc/Gu/FhfC5KS0bRcwpLML5+AfhanWU=</latexit>

!

初状態
(一体)

!

1 2
終状態
(二体)

初状態
(一体)

p1

0 0
1 2
終状態
(二体)

p1
<latexit sha1_base64="9vrZ4UIK7lZW6sJmsMza5/IipPs=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9lV0R4LXjxWsB/QLiWbZtvYbBKSrFCW/gcvHhTx6v/x5r8xbfegrQ8GHu/NMDMvUpwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jEw1oU0iudSdCBvKmaBNyyynHaUpTiJO29H4dua3n6g2TIoHO1E0TPBQsJgRbJ3U6qkR6wf9csWv+nOgVRLkpAI5Gv3yV28gSZpQYQnHxnQDX9kww9oywum01EsNVZiM8ZB2HRU4oSbM5tdO0ZlTBiiW2pWwaK7+nshwYswkiVxngu3ILHsz8T+vm9q4FmZMqNRSQRaL4pQjK9HsdTRgmhLLJ45gopm7FZER1phYF1DJhRAsv7xKWhfV4Lp6eX9VqdfyOIpwAqdwDgHcQB3uoAFNIPAIz/AKb570Xrx372PRWvDymWP4A+/zBzrgjuE=</latexit><latexit sha1_base64="62SEb41BCfWrxua6uowuCorDQqU=">AAACAnicbVDLSsNAFL2pr1pfVZdugkV0VRIp2mXBjcsK9gFtKJPpTTtkMgkzE6GE7vwBt/oH7sStP+IP+B1O2yxs64ELh3Pu5d57/IQzpR3n2ypsbG5t7xR3S3v7B4dH5eOTtopTSbFFYx7Lrk8UciawpZnm2E0kksjn2PHDu5nfeUKpWCwe9SRBLyIjwQJGiTZSp5+M2eXAHZQrTtWZw14nbk4qkKM5KP/0hzFNIxSacqJUz3US7WVEakY5Tkv9VGFCaEhG2DNUkAiVl83PndoXRhnaQSxNCW3P1b8TGYmUmkS+6YyIHqtVbyb+5/VSHdS9jIkk1SjoYlGQclvH9ux3e8gkUs0nhhAqmbnVpmMiCdUmoaUtviQh6mnJBOOuxrBO2tdV96Zae6hVGvU8oiKcwTlcgQu30IB7aEILKITwAq/wZj1b79aH9bloLVj5zCkswfr6Bdmcl3s=</latexit>

1P<latexit sha1_base64="aWQnsngYA/hZu/If6dJrzT9pRL4=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0m0aI8FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq777ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmGG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU00TTKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwqafMZmkhkqyXBSmHJkY5Y+jMVOUGD6zBBPF7K2IRFhhYmw8FRuCt/ryOule1b2b+vVDo9ZqFnGU4QzO4RI8uIUW3EMbOkAggmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPiU44d</latexit>

P01

後で示すように、<latexit sha1_base64="aWQnsngYA/hZu/If6dJrzT9pRL4=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0m0aI8FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq777ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmGG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU00TTKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwqafMZmkhkqyXBSmHJkY5Y+jMVOUGD6zBBPF7K2IRFhhYmw8FRuCt/ryOule1b2b+vVDo9ZqFnGU4QzO4RI8uIUW3EMbOkAggmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPiU44d</latexit>

平面波の計算では、<latexit sha1_base64="jmOMjMHxLc5N+UpY1cxYt7+oQ24=">AAACAnicbVDLSgNBEOz1GeMr6tHLYBA9hd0QNMeAF48RzAOSJcxOZpNhZx/M9Aphyc0f8Kp/4E28+iP+gN/hJNmDSSxoKKq66e7yEik02va3tbG5tb2zW9gr7h8cHh2XTk7bOk4V4y0Wy1h1Paq5FBFvoUDJu4niNPQk73jB3czvPHGlRRw94iThbkhHkfAFo2ikTj8Zi6tBdVAq2xV7DrJOnJyUIUdzUPrpD2OWhjxCJqnWPcdO0M2oQsEknxb7qeYJZQEd8Z6hEQ25drP5uVNyaZQh8WNlKkIyV/9OZDTUehJ6pjOkONar3kz8z+ul6NfdTERJijxii0V+KgnGZPY7GQrFGcqJIZQpYW4lbEwVZWgSWtriKRpwnBZNMM5qDOukXa04N5XaQ63cqOcRFeAcLuAaHLiFBtxDE1rAIIAXeIU369l6tz6sz0XrhpXPnMESrK9f2zSXfA==</latexit>

time

Γ12

<latexit sha1_base64="LUu0m8UM3ApNR1oSOz9G6VCsVCY=">AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqexW0R4LXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurOd9o8La+sbmVnG7tLO7t39QPjxqG5VqylpUCaU7ITFMcMlallvBOolmJA4FewjHNzP/4Ylpw5W8t5OEBTEZSh5xSqyT2r1kxPu1frniVb058Crxc1KBHM1++as3UDSNmbRUEGO6vpfYICPacirYtNRLDUsIHZMh6zoqScxMkM2vneIzpwxwpLQrafFc/T2RkdiYSRy6zpjYkVn2ZuJ/Xje1UT3IuExSyyRdLIpSga3Cs9fxgGtGrZg4Qqjm7lZMR0QTal1AJReCv/zyKmnXqv5V9eLustKo53EU4QRO4Rx8uIYG3EITWkDhEZ7hFd6QQi/oHX0sWgsonzmGP0CfPzxkjuI=</latexit>

p2

2

time

p2

Γ’12

0
2

二つの崩壊過程は、それぞれ対応する単位時間あたりの崩壊確率 Γ12, Γ’12 で特徴づく。

鉄の同位体 Fe57 の(第一励起状態からγ線(光の量子)を放出して基底状態に落ち着く)”崩壊”までにかかった時間の実験結果

t0 = 141 [nano seconds][https://www.rri.kyoto-u.ac.jp/NRP/research/research_forward.htm]

単位時間あたりの確率は、物理次元「(時間)-1」をもつ量で、”(時間)-1で平均化した” (相対的な) 確率と理解できます。
一方で、通常考える「あるイベントが発生する確率」は、物理次元を持たない (絶対的な) 確率で記述されます。
[例] 理想的なサイコロを一回降って、一の目が出る確率は 1/6 (無次元量).

単位時間あたりの確率は、物理次元「(時間)-1」をもつ量で、”(時間)-1で平均化した” (相対的な) 確率と理解できます。
一方で、通常考える「あるイベントが発生する確率」は、物理次元を持たない (絶対的な) 確率で記述されます。
[例] 理想的なサイコロを一回降って、一の目が出る確率は 1/6 (無次元量).

なぜ平面波の計算で、絶対的な確率を考えないのでしょうか？
答えは、考えないのではなく、考えられないのです。
<latexit sha1_base64="WX6IBCSNiwQbaZWLvq31TBTwgXE=">AAACAnicdVDLSgMxFM34rPVVdekmWARXZVJKO90V3LisYB/QDiWTZtrQZCYkGaEMs/MH3OofuBO3/og/4HeYaStY0QMXDufcy733BJIzbVz3w9nY3Nre2S3sFfcPDo+OSyenXR0nitAOiXms+gHWlLOIdgwznPalolgEnPaC2XXu9+6p0iyO7sxcUl/gScRCRrCxUm8oRZDKbFQquxXXdRFCMCeoUXctaTa9KvIgyi2LMlihPSp9DscxSQSNDOFY6wFypfFTrAwjnGbFYaKpxGSGJ3RgaYQF1X66ODeDl1YZwzBWtiIDF+rPiRQLrecisJ0Cm6n+7eXiX94gMaHnpyySiaERWS4KEw5NDPPf4ZgpSgyfW4KJYvZWSKZYYWJsQmtbAoVn1GRFG8z39/B/0q1WUL1Su62VW94qogI4BxfgCiDQAC1wA9qgAwiYgUfwBJ6dB+fFeXXelq0bzmrmDKzBef8CQoiYYA==</latexit><latexit sha1_base64="eTk8wBJKAQlG0VXE5iybZe96r9M=">AAACE3icbVBLSgNBFOyJvzj+ooIbN4NBiAhhRoJmGXDjMoL5QCaEns6bpElPz9D9Rgwxx/ACbvUG7sStB/ACnsPOZ2GiBQ+KqveoRwWJ4Bpd98vKrKyurW9kN+2t7Z3dvdz+QV3HqWJQY7GIVTOgGgSXUEOOApqJAhoFAhrB4HriN+5BaR7LOxwm0I5oT/KQM4pG6uSOfM2lLyDEgv1wPrR9xXt9POvk8m7RncL5S7w5yZM5qp3ct9+NWRqBRCao1i3PTbA9ogo5EzC2/VRDQtmA9qBlqKQR6PZo+v/YOTVK1wljZUaiM1V/X4xopPUwCsxmRLGvl72J+J/XSjEst0dcJimCZLOgMBUOxs6kDKfLFTAUQ0MoU9z86rA+VZShqWwhJVB0ADi2TTHecg1/Sf2i6F0WS7elfKU8ryhLjskJKRCPXJEKuSFVUiOMPJJn8kJerSfrzXq3PmarGWt+c0gWYH3+AEq0nZY=</latexit>sin (x + y)psin2 (x + y)<latexit sha1_base64="aZJTYE+Gt8jMhaoUAvhFu65BLvM=">AAACFXicbVDLSgNBEJz17fpa9SJ4WQxCRAi7IWiOAS8eFYwGsjHMTnqTIbOzy0yvGEL8DX/Aq/6BN/Hq2R/wO5w8DiaxoKGo6qa7K0wF1+h539bC4tLyyuraur2xubW94+zu3eokUwyqLBGJqoVUg+ASqshRQC1VQONQwF3YvRj6dw+gNE/kDfZSaMS0LXnEGUUjNZ2DQHN5XwwERJi3H097dqB4u4MnTSfnFbwR3HniT0iOTHDVdH6CVsKyGCQyQbWu+16KjT5VyJmAgR1kGlLKurQNdUMljUE3+qMPBu6xUVpulChTEt2R+neiT2Ote3FoOmOKHT3rDcX/vHqGUbnR5zLNECQbL4oy4WLiDuNwW1wBQ9EzhDLFza0u61BFGZrQpraEinYBB7YJxp+NYZ7cFgv+WaF0XcpVypOI1sghOSJ54pNzUiGX5IpUCSNP5IW8kjfr2Xq3PqzPceuCNZnZJ1Owvn4BiJGeOg==</latexit>yy<latexit sha1_base64="KYE9h66qfAJExYis5Th/+yjaN2Q=">AAAB/HicbVDLSsNAFJ3UV62vqks3wSK4KokU7bLgxmUL9gFtKJPpTTt0MgkzN0II9Qfc6h+4E7f+iz/gdzhts7CtBy4czrmXe+/xY8E1Os63Vdja3tndK+6XDg6Pjk/Kp2cdHSWKQZtFIlI9n2oQXEIbOQroxQpo6Avo+tP7ud99AqV5JB8xjcEL6VjygDOKRmqlw3LFqToL2JvEzUmF5GgOyz+DUcSSECQyQbXuu06MXkYVciZgVhokGmLKpnQMfUMlDUF72eLQmX1llJEdRMqURHuh/p3IaKh1GvqmM6Q40eveXPzP6ycY1L2MyzhBkGy5KEiEjZE9/9oecQUMRWoIZYqbW202oYoyNNmsbPEVnQLOSiYYdz2GTdK5qbq31VqrVmnU84iK5IJckmvikjvSIA+kSdqEESAv5JW8Wc/Wu/VhfS5bC1Y+c05WYH39Av6KlWQ=</latexit>x<latexit sha1_base64="LMMxf9qkIIBCnz6i6JjgbC8ltJc=">AAAB/HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNUY4kXjxCIo8ENmR2aGDC7OxmZtZINvgDXvUPvBmv/os/4Hc4wB4ErKSTSlV3uruCWHBtXPfbyW1sbm3v5HcLe/sHh0fF45OmjhLFsMEiEal2QDUKLrFhuBHYjhXSMBDYCsZ3M7/1iErzSD6YSYx+SIeSDzijxkr1p16x5JbdOcg68TJSggy1XvGn249YEqI0TFCtO54bGz+lynAmcFroJhpjysZ0iB1LJQ1R++n80Cm5sEqfDCJlSxoyV/9OpDTUehIGtjOkZqRXvZn4n9dJzKDip1zGiUHJFosGiSAmIrOvSZ8rZEZMLKFMcXsrYSOqKDM2m6UtgaJjNNOCDcZbjWGdNK/K3k35un5dqlayiPJwBudwCR7cQhXuoQYNYIDwAq/w5jw7786H87lozTnZzCkswfn6BfzylWM=</latexit>

Ep<latexit sha1_base64="IrgTRZ8rOZSr+XnbQ4/nq6MZoXE=">AAACBnicdVDLSsNAFJ3UV62vqks3g0VwFZLa2mRXEMFlBfuANJTJdNoOnUnCzEQoIXt/wK3+gTtx62/4A36Hk7aCFT1w4XDOvdx7TxAzKpVlfRiFtfWNza3idmlnd2//oHx41JFRIjBp44hFohcgSRgNSVtRxUgvFgTxgJFuML3K/e49EZJG4Z2axcTnaBzSEcVIacm7HqT9mAdpnGWDcsUyXadev3CgZVqWW3UbmriuazdsaGslRwUs0RqUP/vDCCechAozJKVnW7HyUyQUxYxkpX4iSYzwFI2Jp2mIOJF+Oj85g2daGcJRJHSFCs7VnxMp4lLOeKA7OVIT+dvLxb88L1Ejx09pGCeKhHixaJQwqCKY/w+HVBCs2EwThAXVt0I8QQJhpVNa2RIINCUqK+lgvr+H/5NO1bQvzdptrdJ0lhEVwQk4BefABg3QBDegBdoAgwg8gifwbDwYL8ar8bZoLRjLmWOwAuP9C71TmlM=</latexit><latexit sha1_base64="KYE9h66qfAJExYis5Th/+yjaN2Q=">AAAB/HicbVDLSsNAFJ3UV62vqks3wSK4KokU7bLgxmUL9gFtKJPpTTt0MgkzN0II9Qfc6h+4E7f+iz/gdzhts7CtBy4czrmXe+/xY8E1Os63Vdja3tndK+6XDg6Pjk/Kp2cdHSWKQZtFIlI9n2oQXEIbOQroxQpo6Avo+tP7ud99AqV5JB8xjcEL6VjygDOKRmqlw3LFqToL2JvEzUmF5GgOyz+DUcSSECQyQbXuu06MXkYVciZgVhokGmLKpnQMfUMlDUF72eLQmX1llJEdRMqURHuh/p3IaKh1GvqmM6Q40eveXPzP6ycY1L2MyzhBkGy5KEiEjZE9/9oecQUMRWoIZYqbW202oYoyNNmsbPEVnQLOSiYYdz2GTdK5qbq31VqrVmnU84iK5IJckmvikjvSIA+kSdqEESAv5JW8Wc/Wu/VhfS5bC1Y+c05WYH39Av6KlWQ=</latexit>x<latexit sha1_base64="LMMxf9qkIIBCnz6i6JjgbC8ltJc=">AAAB/HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNUY4kXjxCIo8ENmR2aGDC7OxmZtZINvgDXvUPvBmv/os/4Hc4wB4ErKSTSlV3uruCWHBtXPfbyW1sbm3v5HcLe/sHh0fF45OmjhLFsMEiEal2QDUKLrFhuBHYjhXSMBDYCsZ3M7/1iErzSD6YSYx+SIeSDzijxkr1p16x5JbdOcg68TJSggy1XvGn249YEqI0TFCtO54bGz+lynAmcFroJhpjysZ0iB1LJQ1R++n80Cm5sEqfDCJlSxoyV/9OpDTUehIGtjOkZqRXvZn4n9dJzKDip1zGiUHJFosGiSAmIrOvSZ8rZEZMLKFMcXsrYSOqKDM2m6UtgaJjNNOCDcZbjWGdNK/K3k35un5dqlayiPJwBudwCR7cQhXuoQYNYIDwAq/w5jw7786H87lozTnZzCkswfn6BfzylWM=</latexit>単位時間あたりの確率は、物理次元「(時間)-1」をもつ量で、”(時間)-1で平均化した” (相対的な) 確率と理解できます。
一方で、通常考える「あるイベントが発生する確率」は、物理次元を持たない (絶対的な) 確率で記述されます。
[例] 理想的なサイコロを一回降って、一の目が出る確率は 1/6 (無次元量).

なぜ平面波の計算で、絶対的な確率を考えないのでしょうか？
答えは、考えないのではなく、考えられないのです。
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空間全体で、積分すると答えは∞yy<latexit sha1_base64="KYE9h66qfAJExYis5Th/+yjaN2Q=">AAAB/HicbVDLSsNAFJ3UV62vqks3wSK4KokU7bLgxmUL9gFtKJPpTTt0MgkzN0II9Qfc6h+4E7f+iz/gdzhts7CtBy4czrmXe+/xY8E1Os63Vdja3tndK+6XDg6Pjk/Kp2cdHSWKQZtFIlI9n2oQXEIbOQroxQpo6Avo+tP7ud99AqV5JB8xjcEL6VjygDOKRmqlw3LFqToL2JvEzUmF5GgOyz+DUcSSECQyQbXuu06MXkYVciZgVhokGmLKpnQMfUMlDUF72eLQmX1llJEdRMqURHuh/p3IaKh1GvqmM6Q40eveXPzP6ycY1L2MyzhBkGy5KEiEjZE9/9oecQUMRWoIZYqbW202oYoyNNmsbPEVnQLOSiYYdz2GTdK5qbq31VqrVmnU84iK5IJckmvikjvSIA+kSdqEESAv5JW8Wc/Wu/VhfS5bC1Y+c05WYH39Av6KlWQ=</latexit>x<latexit sha1_base64="LMMxf9qkIIBCnz6i6JjgbC8ltJc=">AAAB/HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNUY4kXjxCIo8ENmR2aGDC7OxmZtZINvgDXvUPvBmv/os/4Hc4wB4ErKSTSlV3uruCWHBtXPfbyW1sbm3v5HcLe/sHh0fF45OmjhLFsMEiEal2QDUKLrFhuBHYjhXSMBDYCsZ3M7/1iErzSD6YSYx+SIeSDzijxkr1p16x5JbdOcg68TJSggy1XvGn249YEqI0TFCtO54bGz+lynAmcFroJhpjysZ0iB1LJQ1R++n80Cm5sEqfDCJlSxoyV/9OpDTUehIGtjOkZqRXvZn4n9dJzKDip1zGiUHJFosGiSAmIrOvSZ8rZEZMLKFMcXsrYSOqKDM2m6UtgaJjNNOCDcZbjWGdNK/K3k35un5dqlayiPJwBudwCR7cQhXuoQYNYIDwAq/w5jw7786H87lozTnZzCkswfn6BfzylWM=</latexit>

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0x<latexit sha1_base64="LMMxf9qkIIBCnz6i6JjgbC8ltJc=">AAAB/HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNUY4kXjxCIo8ENmR2aGDC7OxmZtZINvgDXvUPvBmv/os/4Hc4wB4ErKSTSlV3uruCWHBtXPfbyW1sbm3v5HcLe/sHh0fF45OmjhLFsMEiEal2QDUKLrFhuBHYjhXSMBDYCsZ3M7/1iErzSD6YSYx+SIeSDzijxkr1p16x5JbdOcg68TJSggy1XvGn249YEqI0TFCtO54bGz+lynAmcFroJhpjysZ0iB1LJQ1R++n80Cm5sEqfDCJlSxoyV/9OpDTUehIGtjOkZqRXvZn4n9dJzKDip1zGiUHJFosGiSAmIrOvSZ8rZEZMLKFMcXsrYSOqKDM2m6UtgaJjNNOCDcZbjWGdNK/K3k35un5dqlayiPJwBudwCR7cQhXuoQYNYIDwAq/w5jw7786H87lozTnZzCkswfn6BfzylWM=</latexit>

第二部:
平面波の計算の限界と、波束計算の導入。
更に、何を見るべきか?

Problem in plane-wave S-matrix
outfree state<latexit sha1_base64="ttx+XgBAqkBJlCV/gVrVRo+Qlxg=">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</latexit>

SPW = h p1 , p2 |Te

Plane-wave
S-matrix (1→2) def.:<latexit sha1_base64="fky7IDnO0/MB4yDxzP+6fdfdiuQ=">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</latexit>= (2⇡)

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far
distant

free
stateinteraction(s)

free
state

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P

1

⇡

free
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time
far future

far past

taking all possible con gurations(up to an order)fi(factorised)amplitude

momentum eigenstates(external free: also mass eigenstates;
Ei2 = pi2 + mi2)

<latexit sha1_base64="LUu0m8UM3ApNR1oSOz9G6VCsVCY=">AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqexW0R4LXjxWsLXQLiWbZtvYbLIkWaEs/Q9ePCji1f/jzX9j2u5BWx8MPN6bYWZemAhurOd9o8La+sbmVnG7tLO7t39QPjxqG5VqylpUCaU7ITFMcMlallvBOolmJA4FewjHNzP/4Ylpw5W8t5OEBTEZSh5xSqyT2r1kxPu1frniVb058Crxc1KBHM1++as3UDSNmbRUEGO6vpfYICPacirYtNRLDUsIHZMh6zoqScxMkM2vneIzpwxwpLQrafFc/T2RkdiYSRy6zpjYkVn2ZuJ/Xje1UT3IuExSyyRdLIpSga3Cs9fxgGtGrZg4Qqjm7lZMR0QTal1AJReCv/zyKmnXqv5V9eLustKo53EU4QRO4Rx8uIYG3EITWkDhEZ7hFd6QQi/oHX0sWgsonzmGP0CfPzxkjuI=</latexit>

p2

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Pin ) ⇥ (iMPW )manifest energy-momentum
conservation
position(I)[QFT textbooks]
infree stateProblem in plane-wave S-matrix
outfree state<latexit sha1_base64="ttx+XgBAqkBJlCV/gVrVRo+Qlxg=">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</latexit>

Plane-wave
S-matrix (1→2) def.:

SPW = h p1 , p2 |Te<latexit sha1_base64="fky7IDnO0/MB4yDxzP+6fdfdiuQ=">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</latexit>= (2⇡)

4 4i

R ToutTin[QFT textbooks]
infree state(I)

dt Ĥint (t)

P 0i
|P

Pin ) ⇥ (iMPW )(Poutmanifest energy-momentum
conservation(factorised)amplitude

Corresponding probability is given as |SPW|2.

(2π)-4[(Volume)(Time) → ∞]|SPW|2 is ill-de ned due to | 4(Pout-Pin)|2 =4(Pout-Pin) ×4(0).

⇒ Only the averaged (per V and T) frequencies of events is calculable.

(Tin (= Tinitial) = -∞, Tout (= T nal) = +∞)We will see soon later.

<latexit sha1_base64="WX6IBCSNiwQbaZWLvq31TBTwgXE=">AAACAnicdVDLSgMxFM34rPVVdekmWARXZVJKO90V3LisYB/QDiWTZtrQZCYkGaEMs/MH3OofuBO3/og/4HeYaStY0QMXDufcy733BJIzbVz3w9nY3Nre2S3sFfcPDo+OSyenXR0nitAOiXms+gHWlLOIdgwznPalolgEnPaC2XXu9+6p0iyO7sxcUl/gScRCRrCxUm8oRZDKbFQquxXXdRFCMCeoUXctaTa9KvIgyi2LMlihPSp9DscxSQSNDOFY6wFypfFTrAwjnGbFYaKpxGSGJ3RgaYQF1X66ODeDl1YZwzBWtiIDF+rPiRQLrecisJ0Cm6n+7eXiX94gMaHnpyySiaERWS4KEw5NDPPf4ZgpSgyfW4KJYvZWSKZYYWJsQmtbAoVn1GRFG8z39/B/0q1WUL1Su62VW94qogI4BxfgCiDQAC1wA9qgAwiYgUfwBJ6dB+fFeXXelq0bzmrmDKzBef8CQoiYYA==</latexit>

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What is calculable?
So, what can we do in the plane-wave formalism?
<latexit sha1_base64="HbSjM53jDZ3EsyDqCMc2/s5nR0A=">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</latexit>◦

1(t, x) = p
e iEp t+ipp·xx
2Ep V“literal normalisation”(V )d3p 1 (V )d3p 2◦ [(PW) phase space] =2E1 (2⇡)3 2E2 (2⇡)3<latexit sha1_base64="fY/6GjtV0KnLWLDbu/cTU3656UA=">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</latexit>

2<latexit sha1_base64="hUeKchfB5pOnLEkh05u76XTAQYU=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUJJS1KXgxmVFa4U0lsn0th06eTBzI5Y0H+Fv+ANu9Q/ciVsXfofTNgtfBwYO59zXnCCRQqPjvFtz8wuLS8ullfLq2vrGpr21fa3jVHFo8ljG6iZgGqSIoIkCJdwkClgYSGgFw7OJ37oDpUUcXeEoAT9k/Uj0BGdopI59OL7stBHuMWu08vFtjbZRhKCpNxOTgZlNdcI45H7HrjhVZwr6l7gFqZACjY792e7GPA0hQi6Z1p7rJOhnTKHgEvJyO9VgRg9ZHzxDI2Y2+9n0UzndN0qX9mJlXoR0qn7vyFio9SgMTGXIcKB/exPxP89LsXfiZyJKUoSIzxb1UkkxppOEaFco4ChHhjCuhLmV8gFTjKPJ8ceWQLEhYF42wbi/Y/hLrmtV96hav6hXTk+KiEpkl+yRA+KSY3JKzkmDNAknD+SJPJMX69F6td6s91npnFX07JAfsD6+APAdp4Y=</latexit>|SPW | ⇥ [phase space] ⇥ [flux]<latexit sha1_base64="eYrQjbU73kOKEo7k7Q1feUO85kc=">AAACFHicbVC7SgNBFJ31GeMramFhMxgECwm7EjRlwMYygnlAdgmzk7s6ZPbBzF1JWPY3/AFb/QM7sbX3B/wOJ4/CJB4YOJxzL+fO8RMpNNr2t7Wyura+sVnYKm7v7O7tlw4OWzpOFYcmj2WsOj7TIEUETRQooZMoYKEvoe0PbsZ++wmUFnF0j6MEvJA9RCIQnKGReqVjF0UImroXtOsiDDELZDrMvV6pbFfsCegycWakTGZo9Eo/bj/maQgRcsm07jp2gl7GFAouIS+6qYaE8QF7gK6hETOpXjb5QE7PjNKnQazMi5BO1L8bGQu1HoW+mQwZPupFbyz+53VTDGpeJqIkRYj4NChIJcWYjtugfaGAoxwZwrgS5lbKH5liHE1ncym+YgPAvGiKcRZrWCaty4pzVaneVcv12qyiAjkhp+ScOOSa1MktaZAm4SQnL+SVvFnP1rv1YX1OR1es2c4RmYP19QtafJ7O</latexit><latexit sha1_base64="Cf1kuwXYG6jPbO9/bgk8VyHXCoo=">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</latexit>= (2⇡)

4 4

(Pout

3
3
1
d
p
d
p2
1
2
Pin )
|MPW |
⇥T
TV
3
3
2Ein
2E1 (2⇡) 2E2 (2⇡)

fifiwell de ned(The volume is cancelled out.)<latexit sha1_base64="tivSjBjGLInD1fx6F1bPjw3gkA4=">AAAB/nicdVDLSgNBEJyNrxhfUY9eBoPgadldlyTegl48RswLkiXMTmaTIbMPZnqFEAL+gFf9A2/i1V/xB/wOZ5MIRrSgoajqprvLTwRXYFkfRm5tfWNzK79d2Nnd2z8oHh61VJxKypo0FrHs+EQxwSPWBA6CdRLJSOgL1vbH15nfvmdS8ThqwCRhXkiGEQ84JaCluwZu9Ysly7yslh23jC3Tsiq2Y2fEqbgXLra1kqGElqj3i5+9QUzTkEVABVGqa1sJeFMigVPBZoVeqlhC6JgMWVfTiIRMedP5qTN8ppUBDmKpKwI8V39OTEmo1CT0dWdIYKR+e5n4l9dNIah6Ux4lKbCILhYFqcAQ4+xvPOCSURATTQiVXN+K6YhIQkGns7LFl2TMYFbQwXx/j/8nLce0y6Z765ZqV8uI8ugEnaJzZKMKqqEbVEdNRNEQPaIn9Gw8GC/Gq/G2aM0Zy5ljtALj/Qsys5YV</latexit>

ill-de ned!
(since T, V→∞)

What is calculable?
So, what can we do in the plane-wave formalism?
<latexit sha1_base64="HbSjM53jDZ3EsyDqCMc2/s5nR0A=">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</latexit>◦

1(t, x) = p
e iEp t+ipp·xx
2Ep V“literal normalisation”(V )d3p 1 (V )d3p 2◦ [(PW) phase space] =2E1 (2⇡)3 2E2 (2⇡)3<latexit sha1_base64="fY/6GjtV0KnLWLDbu/cTU3656UA=">AAACaHicdVFNa9tAEF2pX677pbSHUnpZYgr2xUiKSXwpGEqhRxdqO2ApYrUe2YtX0rI7CjVCPzLH/IEe+gdyzdoxtHHagYXHmzfzhrepksKg71877qPHT54+az1vv3j56vUb7+jt1JSV5jDhpSz1ecoMSFHABAVKOFcaWJ5KmKXrL9v+7BK0EWXxAzcK4pwtC5EJztBSibeeRwg/se6OZz2qVnYTNYpxaOL253aUacbr7rRHFxcnNFJ5WqsmCZo6/JoEtBtGSvQuTpr/6MKdLvyjS7yO3/d3RR+CYA86ZF/jxPsVLUpe5VAgl8yYeeArjGumUXAJ1rcyYI9dsyXMLSxYDiaud6E09JNlFjQrtX0F0h3790TNcmM2eWqVOcOVOextyX/15hVmw7gWhaoQCn5nlFWSYkm3CdOF0MBRbixgXAt7K+UrZhNC+w/3XFLN1oBN2wYTHMbwEEzDfnDaH3wfdEbDfUQt8pEcky4JyBkZkW9kTCaEkyty4xDHcX67nvve/XAndZ39zDtyr9zjW84mtlc=</latexit>

2<latexit sha1_base64="hUeKchfB5pOnLEkh05u76XTAQYU=">AAACKHicbVDLSsNAFJ34rPUVdelmsAiCUJJS1KXgxmVFa4U0lsn0th06eTBzI5Y0H+Fv+ANu9Q/ciVsXfofTNgtfBwYO59zXnCCRQqPjvFtz8wuLS8ullfLq2vrGpr21fa3jVHFo8ljG6iZgGqSIoIkCJdwkClgYSGgFw7OJ37oDpUUcXeEoAT9k/Uj0BGdopI59OL7stBHuMWu08vFtjbZRhKCpNxOTgZlNdcI45H7HrjhVZwr6l7gFqZACjY792e7GPA0hQi6Z1p7rJOhnTKHgEvJyO9VgRg9ZHzxDI2Y2+9n0UzndN0qX9mJlXoR0qn7vyFio9SgMTGXIcKB/exPxP89LsXfiZyJKUoSIzxb1UkkxppOEaFco4ChHhjCuhLmV8gFTjKPJ8ceWQLEhYF42wbi/Y/hLrmtV96hav6hXTk+KiEpkl+yRA+KSY3JKzkmDNAknD+SJPJMX69F6td6s91npnFX07JAfsD6+APAdp4Y=</latexit>|SPW | ⇥ [phase space] ⇥ [flux]<latexit sha1_base64="eYrQjbU73kOKEo7k7Q1feUO85kc=">AAACFHicbVC7SgNBFJ31GeMramFhMxgECwm7EjRlwMYygnlAdgmzk7s6ZPbBzF1JWPY3/AFb/QM7sbX3B/wOJ4/CJB4YOJxzL+fO8RMpNNr2t7Wyura+sVnYKm7v7O7tlw4OWzpOFYcmj2WsOj7TIEUETRQooZMoYKEvoe0PbsZ++wmUFnF0j6MEvJA9RCIQnKGReqVjF0UImroXtOsiDDELZDrMvV6pbFfsCegycWakTGZo9Eo/bj/maQgRcsm07jp2gl7GFAouIS+6qYaE8QF7gK6hETOpXjb5QE7PjNKnQazMi5BO1L8bGQu1HoW+mQwZPupFbyz+53VTDGpeJqIkRYj4NChIJcWYjtugfaGAoxwZwrgS5lbKH5liHE1ncym+YgPAvGiKcRZrWCaty4pzVaneVcv12qyiAjkhp+ScOOSa1MktaZAm4SQnL+SVvFnP1rv1YX1OR1es2c4RmYP19QtafJ7O</latexit><latexit sha1_base64="Cf1kuwXYG6jPbO9/bgk8VyHXCoo=">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</latexit>= (2⇡)

4 4

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2

<latexit sha1_base64="20+Q8j76o+in5EYZOcoeHQJ40l0=">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</latexit>

3
3
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d
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1
2
Pin )
|MPW |
⇥T
TV
3
3
2Ein
2E1 (2⇡) 2E2 (2⇡)|SPW | ⇥ [phase space] ⇥ [flux]
T TV<latexit sha1_base64="eYrQjbU73kOKEo7k7Q1feUO85kc=">AAACFHicbVC7SgNBFJ31GeMramFhMxgECwm7EjRlwMYygnlAdgmzk7s6ZPbBzF1JWPY3/AFb/QM7sbX3B/wOJ4/CJB4YOJxzL+fO8RMpNNr2t7Wyura+sVnYKm7v7O7tlw4OWzpOFYcmj2WsOj7TIEUETRQooZMoYKEvoe0PbsZ++wmUFnF0j6MEvJA9RCIQnKGReqVjF0UImroXtOsiDDELZDrMvV6pbFfsCegycWakTGZo9Eo/bj/maQgRcsm07jp2gl7GFAouIS+6qYaE8QF7gK6hETOpXjb5QE7PjNKnQazMi5BO1L8bGQu1HoW+mQwZPupFbyz+53VTDGpeJqIkRYj4NChIJcWYjtugfaGAoxwZwrgS5lbKH5liHE1ncym+YgPAvGiKcRZrWCaty4pzVaneVcv12qyiAjkhp+ScOOSa1MktaZAm4SQnL+SVvFnP1rv1YX1OR1es2c4RmYP19QtafJ7O</latexit><latexit sha1_base64="tivSjBjGLInD1fx6F1bPjw3gkA4=">AAAB/nicdVDLSgNBEJyNrxhfUY9eBoPgadldlyTegl48RswLkiXMTmaTIbMPZnqFEAL+gFf9A2/i1V/xB/wOZ5MIRrSgoajqprvLTwRXYFkfRm5tfWNzK79d2Nnd2z8oHh61VJxKypo0FrHs+EQxwSPWBA6CdRLJSOgL1vbH15nfvmdS8ThqwCRhXkiGEQ84JaCluwZu9Ysly7yslh23jC3Tsiq2Y2fEqbgXLra1kqGElqj3i5+9QUzTkEVABVGqa1sJeFMigVPBZoVeqlhC6JgMWVfTiIRMedP5qTN8ppUBDmKpKwI8V39OTEmo1CT0dWdIYKR+e5n4l9dNIah6Ux4lKbCILhYFqcAQ4+xvPOCSURATTQiVXN+K6YhIQkGns7LFl2TMYFbQwXx/j/8nLce0y6Z765ZqV8uI8ugEnaJzZKMKqqEbVEdNRNEQPaIn9Gw8GC/Gq/G2aM0Zy5ljtALj/Qsys5YV</latexit><latexit sha1_base64="i/y49w/wGm/YsDUz3hVhlu7AB00=">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</latexit>= (2⇡)4 4 (Pout

3
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Pin )
|MPW |
2Ein
2E1 (2⇡)3 2E2 (2⇡)3well de ned!

fi

fi

The frequency per time (= Γ: decay rate)
is well de ned and calculable.

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<latexit sha1_base64="yVzdEPkJ1PE4yXbhHhcLrDQ+tg0=">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</latexit>□ In the case of Ĥint (t) =

Z

d3 x

⇣
2

the plane-wave amplitude;
taking a simple form,
easily derived via Feynman rules⌘ˆ ˆˆ ,
p1<latexit sha1_base64="fkZXuKZooORNXIrOHwTLpEDfOmA=">AAAB/3icbVBNS8NAEJ3Ur1q/qh69BIvgqSQq6rHoxWMFWwttKJvtplm62YTdiVBCD/4Br/oPvIlXf4p/wN/hps3Btj4YeLw3w8w8PxFco+N8W6WV1bX1jfJmZWt7Z3evun/Q1nGqKGvRWMSq4xPNBJeshRwF6ySKkcgX7NEf3eb+4xNTmsfyAccJ8yIylDzglGAu9ZKQ96s1p+5MYS8TtyA1KNDsV396g5imEZNIBdG66zoJehlRyKlgk0ov1SwhdESGrGuoJBHTXja9dWKfGGVgB7EyJdGeqn8nMhJpPY580xkRDPWil4v/ed0Ug2sv4zJJkUk6WxSkwsbYzh+3B1wximJsCKGKm1ttGhJFKJp45rb4iowYTiomGHcxhmXSPqu7l/Xz+4ta46aIqAxHcAyn4MIVNOAOmtACCiG8wCu8Wc/Wu/Vhfc5aS1YxcwhzsL5+AUQRlq8=</latexit><latexit sha1_base64="4VkmPSP3EuHdD2X/uznITpXxtOU=">AAAB/3icbVBNS8NAEJ3Ur1q/qh69BIvgqSQq6rHoxWMFWwttKJvtpFm62YTdjVBCD/4Br/oPvIlXf4p/wN/hps3Btj4YeLw3w8w8P+FMacf5tkorq2vrG+XNytb2zu5edf+greJUUmzRmMey4xOFnAlsaaY5dhKJJPI5Pvqj29x/fEKpWCwe9DhBLyJDwQJGic6lXjNk/WrNqTtT2MvELUgNCjT71Z/eIKZphEJTTpTquk6ivYxIzSjHSaWXKkwIHZEhdg0VJELlZdNbJ/aJUQZ2EEtTQttT9e9ERiKlxpFvOiOiQ7Xo5eJ/XjfVwbWXMZGkGgWdLQpSbuvYzh+3B0wi1XxsCKGSmVttGhJJqDbxzG3xJRmhnlRMMO5iDMukfVZ3L+vn9xe1xk0RURmO4BhOwYUraMAdNKEFFEJ4gVd4s56td+vD+py1lqxi5hDmYH39AhDRlo8=</latexit>◦ iMPW (<latexit sha1_base64="+nlQIJkqhKRKWYoATGAwE6RhaJk=">AAACIXicbVDLSgNBEJyN7/ha9ehlMArxEnZV1IsQ9OJFiGAekA1hdtJJhsw+mOkVw7I/4G/4A171D7yJN/HudziJOWi0oKGo6qa7y4+l0Og471ZuZnZufmFxKb+8srq2bm9s1nSUKA5VHslINXymQYoQqihQQiNWwAJfQt0fXIz8+i0oLaLwBocxtALWC0VXcIZGatu7gl61Uw/hDtNKPcuKXqUvqIcR9eK+GNU+PWvbBafkjEH/EndCCmSCStv+9DoRTwIIkUumddN1YmylTKHgErK8l2iIGR+wHjQNDVkAupWOv8nonlE6tBspUyHSsfpzImWB1sPAN50Bw76e9kbif14zwe5pKxVhnCCE/HtRN5HUPDuKhnaEAo5yaAjjSphbKe8zxTiaAH9t8RUbAGZ5E4w7HcNfUjsoucelw+ujQvl8EtEi2SY7pEhcckLK5JJUSJVwck8eyRN5th6sF+vVevtuzVmTmS3yC9bHFycJo1Q=</latexit>!

=<latexit sha1_base64="tG4c8yDjFBu3dkecxSuJeLlonXA=">AAACB3icbVBLSgNBFOzxG+Mv6tJNYxDcGGZU1I0QdOMygvlAZghvOj1Jk56eprtHCEMO4AXc6g3ciVuP4QU8h51kFiax4EFR9R71qFBypo3rfjtLyyura+uFjeLm1vbObmlvv6GTVBFaJwlPVCsETTkTtG6Y4bQlFYU45LQZDu7GfvOJKs0S8WiGkgYx9ASLGAFjJf8Gn2KG/QFICZ1S2a24E+BF4uWkjHLUOqUfv5uQNKbCEA5atz1XmiADZRjhdFT0U00lkAH0aNtSATHVQTb5eYSPrdLFUaLsCIMn6t+LDGKth3FoN2MwfT3vjcX/vHZqousgY0KmhgoyDYpSjk2CxwXgLlOUGD60BIhi9ldM+qCAGFvTTEqoYEDNqGiL8eZrWCSNs4p3WTl/uChXb/OKCugQHaET5KErVEX3qIbqiCCJXtArenOenXfnw/mcri45+c0BmoHz9Quof5j+</latexit>)=
P(=p1+p2)<latexit sha1_base64="UC8O7uBi4Kp1YBNA4u9df8fxOtc=">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</latexit>(for Pin = 0)

2

(

!


)=
32⇡m

<latexit sha1_base64="fkZXuKZooORNXIrOHwTLpEDfOmA=">AAAB/3icbVBNS8NAEJ3Ur1q/qh69BIvgqSQq6rHoxWMFWwttKJvtplm62YTdiVBCD/4Br/oPvIlXf4p/wN/hps3Btj4YeLw3w8w8PxFco+N8W6WV1bX1jfJmZWt7Z3evun/Q1nGqKGvRWMSq4xPNBJeshRwF6ySKkcgX7NEf3eb+4xNTmsfyAccJ8yIylDzglGAu9ZKQ96s1p+5MYS8TtyA1KNDsV396g5imEZNIBdG66zoJehlRyKlgk0ov1SwhdESGrGuoJBHTXja9dWKfGGVgB7EyJdGeqn8nMhJpPY580xkRDPWil4v/ed0Ug2sv4zJJkUk6WxSkwsbYzh+3B1wximJsCKGKm1ttGhJFKJp45rb4iowYTiomGHcxhmXSPqu7l/Xz+4ta46aIqAxHcAyn4MIVNOAOmtACCiG8wCu8Wc/Wu/Vhfc5aS1YxcwhzsL5+AUQRlq8=</latexit>

p2

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<latexit sha1_base64="yVzdEPkJ1PE4yXbhHhcLrDQ+tg0=">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</latexit>□ In the case of Ĥint (t) =

Z

d3 x

⇣
2

the plane-wave amplitude;
taking a simple form,
easily derived via Feynman rules⌘ˆ ˆˆ ,
p1<latexit sha1_base64="fkZXuKZooORNXIrOHwTLpEDfOmA=">AAAB/3icbVBNS8NAEJ3Ur1q/qh69BIvgqSQq6rHoxWMFWwttKJvtplm62YTdiVBCD/4Br/oPvIlXf4p/wN/hps3Btj4YeLw3w8w8PxFco+N8W6WV1bX1jfJmZWt7Z3evun/Q1nGqKGvRWMSq4xPNBJeshRwF6ySKkcgX7NEf3eb+4xNTmsfyAccJ8yIylDzglGAu9ZKQ96s1p+5MYS8TtyA1KNDsV396g5imEZNIBdG66zoJehlRyKlgk0ov1SwhdESGrGuoJBHTXja9dWKfGGVgB7EyJdGeqn8nMhJpPY580xkRDPWil4v/ed0Ug2sv4zJJkUk6WxSkwsbYzh+3B1wximJsCKGKm1ttGhJFKJp45rb4iowYTiomGHcxhmXSPqu7l/Xz+4ta46aIqAxHcAyn4MIVNOAOmtACCiG8wCu8Wc/Wu/Vhfc5aS1YxcwhzsL5+AUQRlq8=</latexit><latexit sha1_base64="4VkmPSP3EuHdD2X/uznITpXxtOU=">AAAB/3icbVBNS8NAEJ3Ur1q/qh69BIvgqSQq6rHoxWMFWwttKJvtpFm62YTdjVBCD/4Br/oPvIlXf4p/wN/hps3Btj4YeLw3w8w8P+FMacf5tkorq2vrG+XNytb2zu5edf+greJUUmzRmMey4xOFnAlsaaY5dhKJJPI5Pvqj29x/fEKpWCwe9DhBLyJDwQJGic6lXjNk/WrNqTtT2MvELUgNCjT71Z/eIKZphEJTTpTquk6ivYxIzSjHSaWXKkwIHZEhdg0VJELlZdNbJ/aJUQZ2EEtTQttT9e9ERiKlxpFvOiOiQ7Xo5eJ/XjfVwbWXMZGkGgWdLQpSbuvYzh+3B0wi1XxsCKGSmVttGhJJqDbxzG3xJRmhnlRMMO5iDMukfVZ3L+vn9xe1xk0RURmO4BhOwYUraMAdNKEFFEJ4gVd4s56td+vD+py1lqxi5hDmYH39AhDRlo8=</latexit>◦ iMPW (<latexit sha1_base64="+nlQIJkqhKRKWYoATGAwE6RhaJk=">AAACIXicbVDLSgNBEJyN7/ha9ehlMArxEnZV1IsQ9OJFiGAekA1hdtJJhsw+mOkVw7I/4G/4A171D7yJN/HudziJOWi0oKGo6qa7y4+l0Og471ZuZnZufmFxKb+8srq2bm9s1nSUKA5VHslINXymQYoQqihQQiNWwAJfQt0fXIz8+i0oLaLwBocxtALWC0VXcIZGatu7gl61Uw/hDtNKPcuKXqUvqIcR9eK+GNU+PWvbBafkjEH/EndCCmSCStv+9DoRTwIIkUumddN1YmylTKHgErK8l2iIGR+wHjQNDVkAupWOv8nonlE6tBspUyHSsfpzImWB1sPAN50Bw76e9kbif14zwe5pKxVhnCCE/HtRN5HUPDuKhnaEAo5yaAjjSphbKe8zxTiaAH9t8RUbAGZ5E4w7HcNfUjsoucelw+ujQvl8EtEi2SY7pEhcckLK5JJUSJVwck8eyRN5th6sF+vVevtuzVmTmS3yC9bHFycJo1Q=</latexit>!

=<latexit sha1_base64="tG4c8yDjFBu3dkecxSuJeLlonXA=">AAACB3icbVBLSgNBFOzxG+Mv6tJNYxDcGGZU1I0QdOMygvlAZghvOj1Jk56eprtHCEMO4AXc6g3ciVuP4QU8h51kFiax4EFR9R71qFBypo3rfjtLyyura+uFjeLm1vbObmlvv6GTVBFaJwlPVCsETTkTtG6Y4bQlFYU45LQZDu7GfvOJKs0S8WiGkgYx9ASLGAFjJf8Gn2KG/QFICZ1S2a24E+BF4uWkjHLUOqUfv5uQNKbCEA5atz1XmiADZRjhdFT0U00lkAH0aNtSATHVQTb5eYSPrdLFUaLsCIMn6t+LDGKth3FoN2MwfT3vjcX/vHZqousgY0KmhgoyDYpSjk2CxwXgLlOUGD60BIhi9ldM+qCAGFvTTEqoYEDNqGiL8eZrWCSNs4p3WTl/uChXb/OKCugQHaET5KErVEX3qIbqiCCJXtArenOenXfnw/mcri45+c0BmoHz9Quof5j+</latexit>)=
P(=p1+p2)<latexit sha1_base64="UC8O7uBi4Kp1YBNA4u9df8fxOtc=">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</latexit>

2

(

!

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)=
32⇡m

<latexit sha1_base64="fkZXuKZooORNXIrOHwTLpEDfOmA=">AAAB/3icbVBNS8NAEJ3Ur1q/qh69BIvgqSQq6rHoxWMFWwttKJvtplm62YTdiVBCD/4Br/oPvIlXf4p/wN/hps3Btj4YeLw3w8w8PxFco+N8W6WV1bX1jfJmZWt7Z3evun/Q1nGqKGvRWMSq4xPNBJeshRwF6ySKkcgX7NEf3eb+4xNTmsfyAccJ8yIylDzglGAu9ZKQ96s1p+5MYS8TtyA1KNDsV396g5imEZNIBdG66zoJehlRyKlgk0ov1SwhdESGrGuoJBHTXja9dWKfGGVgB7EyJdGeqn8nMhJpPY580xkRDPWil4v/ed0Ug2sv4zJJkUk6WxSkwsbYzh+3B1wximJsCKGKm1ttGhJFKJp45rb4iowYTiomGHcxhmXSPqu7l/Xz+4ta46aIqAxHcAyn4MIVNOAOmtACCiG8wCu8Wc/Wu/Vhfc5aS1YxcwhzsL5+AUQRlq8=</latexit>

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<latexit sha1_base64="yVzdEPkJ1PE4yXbhHhcLrDQ+tg0=">AAACbHicbZFNb9MwGMedbINR3jrgNiF5VKDuUiXbNHZBmuCyY5HoNqku1RP3SWPVSSz7ybQqysfkwBfgyBfggpv1wDoeydJP/+fN/jsxWjmKop9BuLW98+jx7pPO02fPX7zs7r26dGVlJY5kqUt7nYBDrQockSKN18Yi5InGq2TxZZW/ukHrVFl8o6XBSQ7zQqVKAnlp2i1FBlRfNFNBeEu1Kqjp0yH/xIVHLg7EAZ99P+bC5El923DREakFWYsFGANNfdR0hMaU+u0YMcyUr2nRbKJV84wOp91eNIja4A8hXkOPrWM47f4Ss1JWORYkNTg3jiNDkxosKanRr68cGpALmOPYYwE5ukndGtPw916Z8bS0/vjntOq/HTXkzi3zxFfmQJnbzK3E/+XGFaVnE++WqQgLebcorTSnkq9c5jNlUZJeegBplb8rlxl458j/xb0tiYUFUtPxxsSbNjyEy6NBfDo4/nrSO/+8tmiX7bN3rM9i9pGdsws2ZCMm2Q/2J9gOdoLf4ZtwP3x7VxoG657X7F6EH/4CWA+8BQ==</latexit>□ In the case of Ĥint (t) =

Z

d3 x

⇣
2

the plane-wave amplitude;
taking a simple form,
easily derived via Feynman rules⌘ˆ ˆˆ ,
p1<latexit sha1_base64="fkZXuKZooORNXIrOHwTLpEDfOmA=">AAAB/3icbVBNS8NAEJ3Ur1q/qh69BIvgqSQq6rHoxWMFWwttKJvtplm62YTdiVBCD/4Br/oPvIlXf4p/wN/hps3Btj4YeLw3w8w8PxFco+N8W6WV1bX1jfJmZWt7Z3evun/Q1nGqKGvRWMSq4xPNBJeshRwF6ySKkcgX7NEf3eb+4xNTmsfyAccJ8yIylDzglGAu9ZKQ96s1p+5MYS8TtyA1KNDsV396g5imEZNIBdG66zoJehlRyKlgk0ov1SwhdESGrGuoJBHTXja9dWKfGGVgB7EyJdGeqn8nMhJpPY580xkRDPWil4v/ed0Ug2sv4zJJkUk6WxSkwsbYzh+3B1wximJsCKGKm1ttGhJFKJp45rb4iowYTiomGHcxhmXSPqu7l/Xz+4ta46aIqAxHcAyn4MIVNOAOmtACCiG8wCu8Wc/Wu/Vhfc5aS1YxcwhzsL5+AUQRlq8=</latexit><latexit sha1_base64="4VkmPSP3EuHdD2X/uznITpXxtOU=">AAAB/3icbVBNS8NAEJ3Ur1q/qh69BIvgqSQq6rHoxWMFWwttKJvtpFm62YTdjVBCD/4Br/oPvIlXf4p/wN/hps3Btj4YeLw3w8w8P+FMacf5tkorq2vrG+XNytb2zu5edf+greJUUmzRmMey4xOFnAlsaaY5dhKJJPI5Pvqj29x/fEKpWCwe9DhBLyJDwQJGic6lXjNk/WrNqTtT2MvELUgNCjT71Z/eIKZphEJTTpTquk6ivYxIzSjHSaWXKkwIHZEhdg0VJELlZdNbJ/aJUQZ2EEtTQttT9e9ERiKlxpFvOiOiQ7Xo5eJ/XjfVwbWXMZGkGgWdLQpSbuvYzh+3B0wi1XxsCKGSmVttGhJJqDbxzG3xJRmhnlRMMO5iDMukfVZ3L+vn9xe1xk0RURmO4BhOwYUraMAdNKEFFEJ4gVd4s56td+vD+py1lqxi5hDmYH39AhDRlo8=</latexit>◦ iMPW (<latexit sha1_base64="+nlQIJkqhKRKWYoATGAwE6RhaJk=">AAACIXicbVDLSgNBEJyN7/ha9ehlMArxEnZV1IsQ9OJFiGAekA1hdtJJhsw+mOkVw7I/4G/4A171D7yJN/HudziJOWi0oKGo6qa7y4+l0Og471ZuZnZufmFxKb+8srq2bm9s1nSUKA5VHslINXymQYoQqihQQiNWwAJfQt0fXIz8+i0oLaLwBocxtALWC0VXcIZGatu7gl61Uw/hDtNKPcuKXqUvqIcR9eK+GNU+PWvbBafkjEH/EndCCmSCStv+9DoRTwIIkUumddN1YmylTKHgErK8l2iIGR+wHjQNDVkAupWOv8nonlE6tBspUyHSsfpzImWB1sPAN50Bw76e9kbif14zwe5pKxVhnCCE/HtRN5HUPDuKhnaEAo5yaAjjSphbKe8zxTiaAH9t8RUbAGZ5E4w7HcNfUjsoucelw+ujQvl8EtEi2SY7pEhcckLK5JJUSJVwck8eyRN5th6sF+vVevtuzVmTmS3yC9bHFycJo1Q=</latexit>!

=<latexit sha1_base64="tG4c8yDjFBu3dkecxSuJeLlonXA=">AAACB3icbVBLSgNBFOzxG+Mv6tJNYxDcGGZU1I0QdOMygvlAZghvOj1Jk56eprtHCEMO4AXc6g3ciVuP4QU8h51kFiax4EFR9R71qFBypo3rfjtLyyura+uFjeLm1vbObmlvv6GTVBFaJwlPVCsETTkTtG6Y4bQlFYU45LQZDu7GfvOJKs0S8WiGkgYx9ASLGAFjJf8Gn2KG/QFICZ1S2a24E+BF4uWkjHLUOqUfv5uQNKbCEA5atz1XmiADZRjhdFT0U00lkAH0aNtSATHVQTb5eYSPrdLFUaLsCIMn6t+LDGKth3FoN2MwfT3vjcX/vHZqousgY0KmhgoyDYpSjk2CxwXgLlOUGD60BIhi9ldM+qCAGFvTTEqoYEDNqGiL8eZrWCSNs4p3WTl/uChXb/OKCugQHaET5KErVEX3qIbqiCCJXtArenOenXfnw/mcri45+c0BmoHz9Quof5j+</latexit>)=
P(=p1+p2)<latexit sha1_base64="UC8O7uBi4Kp1YBNA4u9df8fxOtc=">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</latexit>

2

(

!


)=
32⇡m

<latexit sha1_base64="fkZXuKZooORNXIrOHwTLpEDfOmA=">AAAB/3icbVBNS8NAEJ3Ur1q/qh69BIvgqSQq6rHoxWMFWwttKJvtplm62YTdiVBCD/4Br/oPvIlXf4p/wN/hps3Btj4YeLw3w8w8PxFco+N8W6WV1bX1jfJmZWt7Z3evun/Q1nGqKGvRWMSq4xPNBJeshRwF6ySKkcgX7NEf3eb+4xNTmsfyAccJ8yIylDzglGAu9ZKQ96s1p+5MYS8TtyA1KNDsV396g5imEZNIBdG66zoJehlRyKlgk0ov1SwhdESGrGuoJBHTXja9dWKfGGVgB7EyJdGeqn8nMhJpPY580xkRDPWil4v/ed0Ug2sv4zJJkUk6WxSkwsbYzh+3B1wximJsCKGKm1ttGhJFKJp45rb4iowYTiomGHcxhmXSPqu7l/Xz+4ta46aIqAxHcAyn4MIVNOAOmtACCiG8wCu8Wc/Wu/Vhfc5aS1YxcwhzsL5+AUQRlq8=</latexit>

p2

s

1

4m2
m2

i ,

S-matrix in Gaussian basis
outfree state<latexit sha1_base64="Iyk7vDZ9TchrB2PdpPnA/gkwL/w=">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</latexit>

S-matrix (1→2) def.:

S := hP1 , P2 |Teh<latexit sha1_base64="rwT4HwvihsIJNrRMSUcLgqB+910=">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</latexit>[Note: as in the plane-wave basis,but by the creation/annihilation
operators for wave packets]i

R ToutTininfree state(I)
dt Ĥint (t)

|P0 i

i
Pi = { i , Xi0 (= Ti ), X i , P i }
|
{z
}
=:Xi

This describes the amplitude for the nite probability/frequencyof the event with fully-described initial & nal particle states!

fi

fi

fi

“additional”
information

Normalisability of Gaussian
can makes S itself nite!

outfree state<latexit sha1_base64="Iyk7vDZ9TchrB2PdpPnA/gkwL/w=">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</latexit>

S-matrix (1→2) def.:

S := hP1 , P2 |Teh<latexit sha1_base64="rwT4HwvihsIJNrRMSUcLgqB+910=">AAACXnicbZDPa9swFMcVr2s7b12z7TLYRSwMOgjB3spWCoHSXnbMoGkNsWeelZdURJKNJA+C8N/Xv6G3nnrbdbtWTnNYfzwQ+vJ9ej/0KSrBjY2iq07wbOP55tb2i/Dlq53Xu903b89MWWuGY1aKUicFGBRc4dhyKzCpNIIsBJ4Xi5M2f/4bteGlOrXLCjMJc8VnnIH1Vt6F9JjPJzSVYC8YCDdqck6HNHVhavhcQs77YVqrKepCA0OX/Ipyvjc8zfnnflrJwiW+oMnd8DDxd5+uvLZJmDa07Z3l3V40iFZBH4t4LXpkHaO8e5NOS1ZLVJYJMGYSR5XNHGjLmcDGr2OwAraAOU68VCDRZG6FoqGfvDOls1L7oyxduf9XOJDGLGXhX7ZfNg9zrflUblLb2UHmuKpqi4rdDZrVgtqStlzplGtkViy9AKa535WyC/DMrKd/b4oHuUDbhB5M/BDDY3H2ZRB/G3z9ud87Ol4j2iYfyEeyR2LynRyRH2RExoSRS/KH/CX/OtfBZrAT7N49DTrrmnfkXgTvbwGBbbc6</latexit>[Note: as in the plane-wave basis,but by the creation/annihilation
operators for wave packets]i

R ToutTininfree state(I)
dt Ĥint (t)

|P0 i

i
Pi = { i , Xi0 (= Ti ), X i , P i }
|
{z
}
=:Xi

This describes the amplitude for the nite probability/frequencyof the event with fully-described initial & nal particle states!

t<latexit sha1_base64="idQ+0AEvw5tEYKBNmRz+kv9HnFY=">AAAB/HicbVDLTgJBEJzFF+IL9ehlIjHxRHbVqEeiF4+QyCOBDZkdGpgwO7uZ6TUhG/wBr/oH3oxX/8Uf8DscYA8CVtJJpao73V1BLIVB1/12cmvrG5tb+e3Czu7e/kHx8KhhokRzqPNIRroVMANSKKijQAmtWAMLAwnNYHQ/9ZtPoI2I1COOY/BDNlCiLzhDK9WwWyy5ZXcGukq8jJRIhmq3+NPpRTwJQSGXzJi258bop0yj4BImhU5iIGZ8xAbQtlSxEIyfzg6d0DOr9Gg/0rYU0pn6dyJloTHjMLCdIcOhWfam4n9eO8H+rZ8KFScIis8X9RNJMaLTr2lPaOAox5YwroW9lfIh04yjzWZhS6DZCHBSsMF4yzGsksZF2bsuX9auSpW7LKI8OSGn5Jx45IZUyAOpkjrhBMgLeSVvzrPz7nw4n/PWnJPNHJMFOF+/+UKVaA==</latexit>

Normalisability of Gaussian
can makes S itself nite!

P1 X1(@t=T1) X2(@t=T2) P2

Tout<latexit sha1_base64="/bZzYnZPTFyNLl1bW1ywBx1XfZY=">AAACB3icbVDLSsNAFJ34rPVVdekmWARXJVFRl0U3Liv0BU0ok+ltO3SSCTM3Ygn5AH/Arf6BO3HrZ/gDfofTNgvbeuDC4Zx7OZcTxIJrdJxva2V1bX1js7BV3N7Z3dsvHRw2tUwUgwaTQqp2QDUIHkEDOQpoxwpoGAhoBaO7id96BKW5jOo4jsEP6SDifc4oGsmrdz2EJ0xlglm3VHYqzhT2MnFzUiY5at3Sj9eTLAkhQiao1h3XidFPqULOBGRFL9EQUzaiA+gYGtEQtJ9Of87sU6P07L5UZiK0p+rfi5SGWo/DwGyGFId60ZuI/3mdBPs3fsqjOEGI2CyonwgbpT0pwO5xBQzF2BDKFDe/2mxIFWVoappLCRQdAWZFU4y7WMMyaZ5X3KvKxcNluXqbV1Qgx+SEnBGXXJMquSc10iCMxOSFvJI369l6tz6sz9nqipXfHJE5WF+/JAqahg==</latexit>

~√σ1

~√σ2

◦ First proposal by coherent state:[Ishikawa, Shimomura (0508303)]

all possible con gurations(up to an order)

ing
e
b
Tin
)
l
t
u
a
o
r
T
(
e
n
e
g
n
~√σ0 nite i◦ Claims on various phenomena
by Ishikawa-san et. al.
e.g. [Ishikawa, Jinnouchi, Kubota,
Sloan, Tatsuishi (1901.03019)]
Experiment by the same group → (1907.01264)(22 Na ! 22 Ne⇤ e+ ⌫, e+ (e ) ! 2 )<latexit sha1_base64="QvZ8llYvZ1amfASTqR22vzSoT60=">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</latexit>

x

fi

fi

<latexit sha1_base64="S8X/Q9zh6haDgwgLDjpC2CFhBEE=">AAACAnicbVDLSgNBEJyNrxhfUY9eBoPgKeyqqMegF48RzAOSJcxOZpMhM7PLTK8Ylr35A171D7yJV3/EH/A7nCR7MIkFDUVVN91dQSy4Adf9dgorq2vrG8XN0tb2zu5eef+gaaJEU9agkYh0OyCGCa5YAzgI1o41IzIQrBWMbid+65FpwyP1AOOY+ZIMFA85JWClVjeWQfqU9coVt+pOgZeJl5MKylHvlX+6/YgmkimgghjT8dwY/JRo4FSwrNRNDIsJHZEB61iqiGTGT6fnZvjEKn0cRtqWAjxV/06kRBozloHtlASGZtGbiP95nQTCaz/lKk6AKTpbFCYCQ4Qnv+M+14yCGFtCqOb2VkyHRBMKNqG5LYEmIwZZyQbjLcawTJpnVe+yen5/Uand5BEV0RE6RqfIQ1eohu5QHTUQRSP0gl7Rm/PsvDsfzuesteDkM4doDs7XLwDSmDs=</latexit>fi

Tin<latexit sha1_base64="ZRs66c5jhnRmkQbIJ+aSy5JO8kc=">AAACBnicbVDLSgNBEJz1GeMr6tHLYBA8hV0V9Rj04jFCXpAsYXbSmwyZfTDTK4Zl7/6AV/0Db+LV3/AH/A4nyR5MYkFDUdVNd5cXS6HRtr+tldW19Y3NwlZxe2d3b790cNjUUaI4NHgkI9X2mAYpQmigQAntWAELPAktb3Q38VuPoLSIwjqOY3ADNgiFLzhDI3XqvS7CE6YizHqlsl2xp6DLxMlJmeSo9Uo/3X7EkwBC5JJp3XHsGN2UKRRcQlbsJhpixkdsAB1DQxaAdtPpyRk9NUqf+pEyFSKdqn8nUhZoPQ480xkwHOpFbyL+53US9G9c80+cIIR8tshPJMWITv6nfaGAoxwbwrgS5lbKh0wxjialuS2eYiPArGiCcRZjWCbN84pzVbl4uCxXb/OICuSYnJAz4pBrUiX3pEYahJOIvJBX8mY9W+/Wh/U5a12x8pkjMgfr6xcsnZn7</latexit>

P0
X0(@t=T0)fi

fi

S-matrix in Gaussian basis

Short Summary
For the same focused physical 1→2 process,(note: we can similarly construct those for m→n processes.)outfree state<latexit sha1_base64="ttx+XgBAqkBJlCV/gVrVRo+Qlxg=">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</latexit>

plane-wave S-matrix:◦ with partial information

SPW = h p 1 , p 2 |Te= (2⇡)4 4 (Poutmoreinformative

not equal<latexit sha1_base64="Iyk7vDZ9TchrB2PdpPnA/gkwL/w=">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</latexit>out
free state

S := hP1 , P2 |Teh<latexit sha1_base64="rwT4HwvihsIJNrRMSUcLgqB+910=">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</latexit>◦ with full information◦ normalised appropriately

Tin

(I)
dt Ĥint (t)

P 0i
|P

<latexit sha1_base64="fky7IDnO0/MB4yDxzP+6fdfdiuQ=">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</latexit>◦ not suitably normalised

Gaussian S-matrix:i

R Toutinfree stateiPin ) ⇥ (iMPW )(R ToutTin)suitable limits/
marginalisations(I)infree state

dt Ĥint (t)

|P0 i

i
Pi = { i , Xi0 (= Ti ), X i , P i }|{z}“additional”
information=:Xi

Short Summary
For the same focused physical 1→2 process,(note: we can similarly construct those for m→n processes.)|plane-wave S-matrix|2:◦ with partial information◦ not suitably normalised(dimensionful,|SPW |
relative frequency)
1 |SPW |2 d3p 1
d3 p 2d =2Ein T V 2E1 (2⇡)3 2E2 (2⇡)32<latexit sha1_base64="RqKjRRgKSQextwOHT//WQ6+E/xQ=">AAACDHicbVBLTgJBFOzBH+KHUZduJhITV2SGEHVJdOMSo3wSGElP84AOPZ90vzGSgSt4Abd6A3fGrXfwAp7DBmYhYCUvqVS9l3opLxJcoW1/G5m19Y3Nrex2bmd3bz9vHhzWVRhLBjUWilA2PapA8ABqyFFAM5JAfU9AwxteT/3GI0jFw+AeRxG4Pu0HvMcZRS11zPz4rtNGeMKk2piMH0ods2AX7RmsVeKkpEBSVDvmT7sbstiHAJmgSrUcO0I3oRI5EzDJtWMFEWVD2oeWpgH1QbnJ7PGJdaqVrtULpZ4ArZn69yKhvlIj39ObPsWBWvam4n9eK8bepZvwIIoRAjYP6sXCwtCatmB1uQSGYqQJZZLrXy02oJIy1F0tpHiSDgEnOV2Ms1zDKqmXis55sXxbLlSu0oqy5JickDPikAtSITekSmqEkZi8kFfyZjwb78aH8TlfzRjpzRFZgPH1C3NCm6w=</latexit>

itself is ill de ned.

<latexit sha1_base64="xVDJ7ySPJOTxGVcMwzPxe8C9i7g=">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</latexit>

External states are characterisedby momenta.

|Gaussian S-matrix|2:<latexit sha1_base64="I6BjYodkBEACxeHiybMq8WCUbsE=">AAACDHicbVDLTsJAFL3FF+KDqks3E4mJK9ISoi6JblxiFDCBSqbDFCZMH5mZmpDSX/AH3OofuDNu/Qd/wO9wCl0IeJKbnJxzb+7JcSPOpLKsb6Owtr6xuVXcLu3s7u2XzYPDtgxjQWiLhDwUDy6WlLOAthRTnD5EgmLf5bTjjq8zv/NEhWRhcK8mEXV8PAyYxwhWWuqb5WnPx2pEME/u0uljrW9WrKo1A1oldk4qkKPZN396g5DEPg0U4VjKrm1FykmwUIxwmpZ6saQRJmM8pF1NA+xT6SSz4Ck61coAeaHQEyg0U/9eJNiXcuK7ejNLKZe9TPzP68bKu3QSFkSxogGZP/JijlSIshbQgAlKFJ9ogolgOisiIywwUbqrhS+uwGOq0pIuxl6uYZW0a1X7vFq/rVcaV3lFRTiGEzgDGy6gATfQhBYQiOEFXuHNeDbejQ/jc75aMPKbI1iA8fULi9mbuw==</latexit>|S|

2 itself is well de ned.

(dimensionless,absolute frequency)

3
3
3
3
d
X
d
p
d
X
d
1
1
2 p2
2
dP = |S|
(2⇡)3
(2⇡)3
<latexit sha1_base64="3YE4pUFvzIFkczW6B81xabCZN6g=">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</latexit>◦ with full information◦ normalised appropriatelyfi

fi

External states are characterisedby momenta and positions of centres.

Structure of Transitions
The plane-wave S-matrix has no time boundaries (only in time bulk).
The wave-packet S-matrix has time boundaries (also in time bulk).

t<latexit sha1_base64="wNgVTtNCOGmG/W1tvW7fdaTGPeA=">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</latexit>

Wave packets

Plane waves

Out state

ar
d
n
u
bo

Tout (! 1)y<latexit sha1_base64="6Tc/RxNM5Zms7oLUrOS7w82ees4=">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</latexit>

Out-time-boundary
region⇠<latexit sha1_base64="aouuHGikgntv0IC8ZZN8IkqSV0g=">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</latexit>

p

t

⇡

p

No

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Bulk regionT<latexit 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region<latexit 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Spatially inﬁnite
overlap region<latexit 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(t)<latexit 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What’s next?
We examined the simplest 1→2 case in wave-packet formalism.
Now, we will be interested in
1. How about the 2→2 full scattering, including the production part?
No detailed discussion today
2. When does the wave-packet effect become signi cant?

fiToday’s main topic

So, the `best’ process to see a wave-packet intrinsic nature requires{<latexit sha1_base64="Lh/1HbGH6+7S9V6YJPd6KN3c3NI=">AAAB/XicbVDLSgNBEOyNrxhfUY9eBoPgKexK0BwDXjxGMQ9IQpid9CZDZmeXmVkhLMEf8Kp/4E28+i3+gN/hJNmDSSxoKKq66e7yY8G1cd1vJ7exubW9k98t7O0fHB4Vj0+aOkoUwwaLRKTaPtUouMSG4UZgO1ZIQ19gyx/fzvzWEyrNI/loJjH2QjqUPOCMGis9dNN+seSW3TnIOvEyUoIM9X7xpzuIWBKiNExQrTueG5teSpXhTOC00E00xpSN6RA7lkoaou6l80un5MIqAxJEypY0ZK7+nUhpqPUk9G1nSM1Ir3oz8T+vk5ig2ku5jBODki0WBYkgJiKzt8mAK2RGTCyhTHF7K2EjqigzNpylLb6iYzTTgg3GW41hnTSvyt51uXJfKdWqWUR5OINzuAQPbqAGd1CHBjAI4AVe4c15dt6dD+dz0ZpzsplTWILz9Qu5vZXM</latexit>◦ domination of the boundary contribution, e.g., via a narrow phase space◦ resonant production & decay+◦ experimental anomalies being reported

So, the `best’ process to see a wave-packet intrinsic nature requires{<latexit sha1_base64="Lh/1HbGH6+7S9V6YJPd6KN3c3NI=">AAAB/XicbVDLSgNBEOyNrxhfUY9eBoPgKexK0BwDXjxGMQ9IQpid9CZDZmeXmVkhLMEf8Kp/4E28+i3+gN/hJNmDSSxoKKq66e7yY8G1cd1vJ7exubW9k98t7O0fHB4Vj0+aOkoUwwaLRKTaPtUouMSG4UZgO1ZIQ19gyx/fzvzWEyrNI/loJjH2QjqUPOCMGis9dNN+seSW3TnIOvEyUoIM9X7xpzuIWBKiNExQrTueG5teSpXhTOC00E00xpSN6RA7lkoaou6l80un5MIqAxJEypY0ZK7+nUhpqPUk9G1nSM1Ir3oz8T+vk5ig2ku5jBODki0WBYkgJiKzt8mAK2RGTCyhTHF7K2EjqigzNpylLb6iYzTTgg3GW41hnTSvyt51uXJfKdWqWUR5OINzuAQPbqAGd1CHBjAI4AVe4c15dt6dD+dz0ZpzsplTWILz9Qu5vZXM</latexit>◦ domination of the boundary contribution, e.g., via a narrow phase space◦ resonant production & decay+◦ experimental anomalies being reported[https://higgstan.com/standerd-model/]We found such a process!
heavy quarkonium decays into mesons near kinetic thresholde<latexit sha1_base64="J84cYAhe0YDX2yRhLywWkuXygQQ=">AAAB/nicbVDLSgNBEJyNrxhfUY9eFoPgxbArQT1JwIvHiOYByRpmJ73JkNnZZaZXCEvAH/Cqf+BNvPor/oDf4STZg0ksaCiquunu8mPBNTrOt5VbWV1b38hvFra2d3b3ivsHDR0likGdRSJSLZ9qEFxCHTkKaMUKaOgLaPrDm4nffAKleSQfcBSDF9K+5AFnFI10D49n3WLJKTtT2MvEzUiJZKh1iz+dXsSSECQyQbVuu06MXkoVciZgXOgkGmLKhrQPbUMlDUF76fTUsX1ilJ4dRMqURHuq/p1Iaaj1KPRNZ0hxoBe9ifif104wuPJSLuMEQbLZoiARNkb25G+7xxUwFCNDKFPc3GqzAVWUoUlnbouv6BBwXDDBuIsxLJPGedm9KFfuKqXqdRZRnhyRY3JKXHJJquSW1EidMNInL+SVvFnP1rv1YX3OWnNWNnNI5mB9/QIJOJX1</latexit><latexit sha1_base64="Sloz4zdTDyvEdt+0db3mJ91nVqE=">AAACInicbVDLSsNAFJ34rPUVdelmsBRclUREXUnBjUsFq4W2lMn0pg6dZMLMjVhCvsDf8Afc6h+4E1eCa7/DSQ1i1bM6nHMP994TJFIY9Lw3Z2Z2bn5hsbJUXV5ZXVt3NzYvjUo1hxZXUul2wAxIEUMLBUpoJxpYFEi4CkYnhX91A9oIFV/gOIFexIaxCAVnaKW+W+8qaxfpLM1pF+EWM6o0tfzbGOR9t+Y1vAnoX+KXpEZKnPXdj+5A8TSCGLlkxnR8L8FexjQKLiGvdlMDCeMjNoSOpTGLwPSyyTs5rVtlQEN7RahipBP1ZyJjkTHjKLCTEcNr89srxP+8TorhUS8TcZIixPxrUZhKiooW3dCB0MBRji1hXAt7K+XXTDOOtsGpLYFmI8C8aovxf9fwl1zuNfyDxv75fq15XFZUIdtkh+wSnxySJjklZ6RFOLkjD+SRPDn3zrPz4rx+jc44ZWaLTMF5/wSGraVN</latexit>

Qheavy<latexit sha1_base64="Egdv9CGUO6SRhv11acPKKoQmgNc=">AAACCXicbVDLSsNAFJ3UV62vqks3wSK4KomIupKCG5ct2Ae0sUymN+3QyYOZm2II+QJ/wK3+gTtx61f4A36H0zYL23rgwuGcezmX40aCK7Ssb6Owtr6xuVXcLu3s7u0flA+PWiqMJYMmC0UoOy5VIHgATeQooBNJoL4roO2O76Z+ewJS8TB4wCQCx6fDgHucUdTSY6PfQ3jCdAR0kmT9csWqWjOYq8TOSYXkqPfLP71ByGIfAmSCKtW1rQidlErkTEBW6sUKIsrGdAhdTQPqg3LS2deZeaaVgemFUk+A5kz9e5FSX6nEd/WmT3Gklr2p+J/XjdG7cVIeRDFCwOZBXixMDM1pBeaAS2AoEk0ok1z/arIRlZShLmohxZV0DJiVdDH2cg2rpHVRta+ql43LSu02r6hITsgpOSc2uSY1ck/qpEkYkeSFvJI349l4Nz6Mz/lqwchvjskCjK9frHWbWQ==</latexit><latexit sha1_base64="Qho8SYVB4tbkyFMfBqchMoGj1mY=">AAACAXicbVDLSgNBEJyNrxhfUY9eBoPgKexKUE8S8OIxgnlAsoTeySQZM7O7zPQKYcnJH/Cqf+BNvPol/oDf4STZg0ksaCiquunuCmIpDLrut5NbW9/Y3MpvF3Z29/YPiodHDRMlmvE6i2SkWwEYLkXI6yhQ8lasOahA8mYwup36zSeujYjCBxzH3FcwCEVfMEArNToDUAq6xZJbdmegq8TLSIlkqHWLP51exBLFQ2QSjGl7box+ChoFk3xS6CSGx8BGMOBtS0NQ3Pjp7NoJPbNKj/YjbStEOlP/TqSgjBmrwHYqwKFZ9qbif147wf61n4owTpCHbL6on0iKEZ2+TntCc4ZybAkwLeytlA1BA0Mb0MKWQMOI46Rgg/GWY1gljYuyd1mu3FdK1Zssojw5IafknHjkilTJHamROmHkkbyQV/LmPDvvzofzOW/NOdnMMVmA8/ULxeKXgg==</latexit>

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R

AMi`Q/m+iBQM

Anomaly in heavy vector quarkonium decaysh?2`2 Bb  HQM;@biM/BM; MQKHv U/Bb+`2TM+v #2ir22M 2tT2`BK2MiH M/ i?2Q`2iB+H `2bmHibV+ D X branches
For
heavy
vectorquarkonium(V),
dominantdecay
0 iQ two
BM i?2each`iBQ Q7
i?2 /2+v
rB/i? 7Q`
(3770) ! D0 D
i?i 7Q`(3770) ! D
bBKBH`
+P-” and#mi r2F2`
MQKHv
2tBbib“V→P
7Q` i?2 0`iBQ
are“V→P
P0”.Q7 (1020) ! KG0 Ka0 iQ (1020) ! K + K X PM i?2
Qi?2` ?M/- i?2 `iBQ Q7 ⌥(4S) ! B + B iQ ⌥(4S) ! B 0 B 0 Bb +QMbBbi2Mi rBi? i?2 biM/`/+ is the EM-charged one; (anti-particle of P+) = P◦P
i?2Q`2iB+H T`2/B+iBQMbX- is the[m`F
H2p2H- i?2b2one;
T`Q+2bb2b
`2R◦iPi?2
EM-neutral(anti-particleof P0) = P0(ss) ! K + (us̄) K (sū) ,(ss) ! K 0 (ds̄) K 0 sd¯ ! K 0 K 0 ,Ga“Kaons”(cc) ! D+ cd D (dc) , ⌥ bb ! B + ub B (bū) ,(cc) ! D0 (cu) D0 (uc) ,“D-mesons”⌥ bb ! B 0 db B 0 bd ,“B-mesons”

r?B+? +M #2 bmKK`Bx2/ b V QQ ! P (Qq) + P qQ - r?2`2 V M/ P `2 p2+iQ` M/
Tb2m/Q@b+H` K2bQMb- `2bT2+iBp2Hv- M/ Q M/ q `2 ?2pv Us- c- bV M/ HB;?i Uu- dV [m`Fb`2bT2+iBp2HvX h?Bb `iBQ Q7 /2+v rB/i?b Bb i?2Q`2iB+HHv +H2M #2+mb2 KQbi Q7 i?2 [mMimK@+?`QKQ/vMKB+b UZ*.V +Q``2+iBQMb +M+2H Qmi #2ir22M i?2 MmK2`iQ` M/ i?2 /2MQKBMiQ`X
h?2b2 /2+v T`Q+2bb2b `2 pB bi`QM; BMi2`+iBQM- M/ ?2M+2 BM i?2 HBKBi Q7 2t+i BbQbTBM
bvKK2i`v u $ d- i?2 `iBQ #2+QK2b mMBivX h?2 BbQbTBM pBQHiBQM KF2b  /2pBiBQM 7`QK
mMBivX
q2 MK2 i?2 `iBQ Q7 i?2 rB/i?b bk
R :=( ! K +K ),00
! K G KaR :=( ! D+ D )⇣⌘,! D0 D0R⌥ :=(⌥ ! B + B )⇣⌘.
⌥ ! B0B0

URVh?2 2tT2`BK2MiH `2bmHib `2 +QK#BM2/ #v i?2 S`iB+H2 .i :`QmT US.:V (R),jRS.: = 1.45 ± 0.03,

RS.: = 0.798 ± 0.010,

S.:R⌥= 1.058 ± 0.024.

UkV

i?2Q`2iB+H T`2/B+iBQMbX
R
i
i?2
[m`F
H2p2Hi?2b2
T`Q+2bb2b
`2
R iAMi`Q/m+iBQM
i?2 [m`F H2p2H- i?2b2 T`Q+2bb2b `2R

Anomaly in heavy vector quarkonium decays

+
+
+
(ss)
!
K
(us̄)
K
(sū)
,
(cc)
!
D
cd
D
(dc)
,
⌥
bb
!
B
B (bū) ,
h?2`2
Bb

HQM;@biM/BM;MQKHv
U/Bb+`2TM+v#2ir22M
2tT2`BK2MiH
M/
i?2Q`2iB+H+++ ub `2bmHibV(ss)each! K heavy(us̄) K vector(sū) , quarkonium(cc) !0D
cdtwo
D (dc), ⌥ bb !
B ubbranches
B (bū) ,
+
For
(V),
dominant
decay
0
0
0
0
0
0
BM i?2
Q7 i?2rB/i?
7Q` , (3770)!
DDD(cu)iQ Di?i7Q` , (3770)!!
D BD dbX B
bBKBH`
0/2+v
0 (uc)
0 bd ,¯ !
(ss)!`iBQK (ds̄)
d
K
K
(cc)
!
⌥
bb
0 K s
0
0
0
0
G
a ,
0 (uc)
0K
0 iQ, (1020)+ K db(ss)! K MQKHv(ds̄)
sd¯ !7Q`Ki?2! D (cu)⌥ bb !!KB
B 0 i?2bd ,+ - K 0 2tBbib
0 (cc)#mi
r2F2`
Q7 (1020)! KD
X PM
G K0`iBQ

are “V→P P ” and “V→P aP ”.

G aiQ ⌥(4S) ! B 0 B 0 Bb +QMbBbi2Mi rBi? i?2 biM/`/Qi?2`i?2 `iBQ Q7 ⌥(4S)
B B! P (Qq) + P qQ - r?2`2 V M/ P `2 p2+iQ` M/
r?B+?
+M?M/#2 bmKK`Bx2/
b V !
QQ++) qQ- r?2`2 V M/ P `2 p2+iQ` M/
r?B+?
+M
#2
bmKK`Bx2/
b
V
QQ
!
P
(Qq)
+
P
◦
P
is
the
EM-charged
one;(anti-particleofP=P
i?2Q`2iB+H T`2/B+iBQMbX
Tb2m/Q@b+H`K2bQMb- `2bT2+iBp2Hv- M/ QRM/ q `2 ?2pv Us- c- bV M/ HB;?i Uu- dV [m`FbTb2m/Q@b+H`K2bQMb`2bT2+iBp2HvM/
Q M/ ofq `2- is the0) ?2pv
0 Us- c- bV M/ HB;?i Uu- dV [m`Fbi
i?2
[m`F
H2p2Hi?2b2
T`Q+2bb2b`2◦
P
EM-neutral
one;(anti-particle
P=P`2bT2+iBp2HvX h?Bb `iBQ Q7 /2+v rB/i?b Bb i?2Q`2iB+HHv +H2M #2+mb2 KQbi Q7 i?2 [mMimK@`2bT2+iBp2HvX h?Bb `iBQ Q7 /2+v rB/i?b Bb i?2Q`2iB+HHv +H2M #2+mb2 KQbi Q7 i?2 [mMimK@++?`QKQ/vMKB+b
UZ*.V
+Q``2+iBQMb+M+2H(ss) ! K + (us̄)
K (sū),(cc) Qmi!Qmi
D#2ir22M
cd D i?2(dc),MmK2`iQ`⌥ bb !M/
B + i?2
ub /2MQKBMiQ`X
B(bū) ,+?`QKQ/vMKB+b
UZ*.V
+Q``2+iBQMb+M+2H#2ir22M
i?2MmK2`iQ`
M/
i?2/2MQKBMiQ`X
h?2b2
/2+v
T`Q+2bb2b`2
pBpB0bi`QM;
BMi2`+iBQMM/?2M+2
BMBMi?2HBKBi
2t+i
BbQbTBM
0 T`Q+2bb2b
0
0
0 Q7
0
0
0
h?2b2
/2+v
`2
bi`QM;
BMi2`+iBQMM/
?2M+2
i?2
HBKBi
Q7
2t+i
¯
(ss) ! K (ds̄) K sd ! KG Ka ,(cc) ! D (cu) D (uc) , ⌥ bb ! B db B bd ,BbQbTBM
bvKK2i`v
u $d- d-i?2i?2`iBQ
bvKK2i`v
u $`iBQ#2+QK2b#2+QK2bmMBivX
mMBivX h?2
h?2BbQbTBM
BbQbTBMpBQHiBQM
pBQHiBQMKF2b
KF2b  /2pBiBQM
/2pBiBQM 7`QK
7`QK
“Kaons”“D-mesons”“B-mesons”
mMBivX
r?B+? +M #2 bmKK`Bx2/ b V QQ ! P (Qq) + P qQ - r?2`2 V M/ P `2 p2+iQ` M/
mMBivX
k
q2
MK2
i?2
`iBQ
Q7
i?2
rB/i?bb
Tb2m/Q@b+H`K2bQMbM/
q2 MK2 i?2`iBQ `2bT2+iBp2HvQ7 i?2 rB/i?bbkQ M/ q `2 ?2pv Us- c- bV M/ HB;?i Uu- dV [m`Fb+

Measuring Isospin breaking`2bT2+iBp2HvX h?Bb `iBQ
Q7 /2+v rB/i?b Bb i?2Q`2iB+HHv+H2M #2+mb2 KQbi Q7 i?2[mMimK@
++
++B
+
( (!!
K+
K
)
(
!
D
D
)
(⌥
!
B
))
K
K
)
(
!
D
D
)
(⌥
!
B
B
⇣ ⇣Qmi #2ir22M⌘i?2⇣⇣ i?2 /2MQKBMiQ`X⌘⌘..
+?`QKQ/vMKB+b
UZ*.V
+Q``2+iBQMb+M+2HM/
R R:=:=, , R R :=, , MmK2`iQ`
RR⌥⌥:=
URV
⌘
:=
:=
URV
0
0
0
0
!
K
K
0
0
0
0
! GKGaK0 D 0 ?2M+2 BM i?2 HBKBi
!!DDD
⌥⌥!
h?2b2 /2+v T`Q+2bb2b
`2
M/
Q7
2t+i
!B
B 0BB 0BbQbTBM
a pB bi`QM; BMi2`+iBQMbvKK2i`v u $ d- i?2 `iBQ #2+QK2b mMBivX h?2 BbQbTBM pBQHiBQM KF2b  /2pBiBQM 7`QK
jj
mMBivXh?2
2tT2`BK2MiH`2bmHib`2+QK#BM2/#v
i?2
S`iB+H2
.i
:`QmT
US.:V
(R),
h?2 2tT2`BK2MiH`2bmHib `2 +QK#BM2/#v i?2 S`iB+H2 .i :`QmT US.:V (R),○ PDG
values:k
q2 MK2 i?2 `iBQ Q7 i?2 rB/i?b b

RS.:
==
1.45
0.03,
+ 0.03,RS.:( 1.45
!±K±
K )

RS.:
0.010,
RS.:=(=0.798
0.798
0.010,! D±+±D)

S.:
S.:= 1.058
RR⌥
0.024.
⌥ (⌥ =!1.058B + B±±)0.024.

UkV
UkV

+QmTHBM;gV:=
BM i?2
Tb2m/Q@b+H`@K2bQM
Kbb2bmBM+ = gV 0 - i?2 /Bz2`2M+2
P + 6= mP 0 ⌘`2bmHib
⇣
⌘
⇣
R
,
R
:=
,
R
:=
.
URV
⌥
○ plane-wave
KG0mMBiv,
Ka0
0 D 0 BM _27X (R)-9 r2 Q#iBM
0B0/2pBiBQM BM RV !
7`QK
SmiiBM; i?2 KbbpHm2b!
D⌥!
B+
00P 00 Bb #b2/ QM+
h?2i?2Q`2iB+H
T`2/B+iBQM
Q7
i?2
/2+v
`i2b
7Q`
V
!
P
P
M/
V
!
P
h?2
i?2Q`2iB+H
T`2/B+iBQM
Q7
i?2
/2+v
`i2b
7Q`
V
!
P
P
M/
V
!
P
P Bb #b2/ QM
results:
2
2
2i?2i?2
THM2@rp2
7Q`KHBbK
bQ
7`X
h?2
i`22@H2p2H`2bmHiQ7
i?2+?B`H
T2`im`#iBQM
i?2Q`v
THM2@rp2
7Q`KHBbK
bQ
7`X
h?2
i`22@H2p2H`2bmHiQ7
i?2+?B`H
T2`im`#iBQM
i?2Q`v `2/b
`2/b
g
g
g
j
+
+
THM2
THM2
THM2
⌥+
2tT2`BK2MiH`2bmHib`2 +QK#BM2/#v i?2 (0.6915
S`iB+H2±.i:`QmT
US.:V=(R), (1.047 ± 0.026) .
R h?2 =
(1.5156±
0.0033),R=
0.0046)
,
R
⌥
2
2
g20
g
g
✓
◆
✓
◆
3/2
3/2
⌥0
0
2 2
2 2 4m2 2
g
g
S.:
S.:
S.:
m
m
4m
++
THM2
V=
+
V+
V±
V 0.010,
PPR= 1.45 ± 0.03, RTHM2
R
0.798
R
UkV UjV
R ==UjV, ⌥, = 1.058 ± 0.024. U9V

i?2Q`2iB+H T`2/B+iBQMbX
R
i
i?2
[m`F
H2p2Hi?2b2
T`Q+2bb2b
`2
R iAMi`Q/m+iBQM
i?2 [m`F H2p2H- i?2b2 T`Q+2bb2b `2R

Anomaly in heavy vector quarkonium decays

+
+
+
(ss)
!
K
(us̄)
K
(sū)
,
(cc)
!
D
cd
D
(dc)
,
⌥
bb
!
B
B (bū) ,
h?2`2
Bb

HQM;@biM/BM;MQKHv
U/Bb+`2TM+v#2ir22M
2tT2`BK2MiH
M/
i?2Q`2iB+H+++ ub `2bmHibV(ss)each! K heavy(us̄) K vector(sū) , quarkonium(cc) !0D
cdtwo
D (dc), ⌥ bb !
B ubbranches
B (bū) ,
+
For
(V),
dominant
decay
0
0
0
0
0
0
BM i?2
Q7 i?2rB/i?
7Q` , (3770)!
DDD(cu)iQ Di?i7Q` , (3770)!!
D BD dbX B
bBKBH`
0/2+v
0 (uc)
0 bd ,¯ !
(ss)!`iBQK (ds̄)
d
K
K
(cc)
!
⌥
bb
0 K s
0
0
0
0
G
a ,
0 (uc)
0K
0 iQ, (1020)+ K db(ss)! K MQKHv(ds̄)
sd¯ !7Q`Ki?2! D (cu)⌥ bb !!KB
B 0 i?2bd ,+ - K 0 2tBbib
0 (cc)#mi
r2F2`
Q7 (1020)! KD
X PM
G K0`iBQ

are “V→P P ” and “V→P aP ”.

G aiQ ⌥(4S) ! B 0 B 0 Bb +QMbBbi2Mi rBi? i?2 biM/`/Qi?2`i?2 `iBQ Q7 ⌥(4S)
B B! P (Qq) + P qQ - r?2`2 V M/ P `2 p2+iQ` M/
r?B+?
+M?M/#2 bmKK`Bx2/
b V !
QQ++) qQ- r?2`2 V M/ P `2 p2+iQ` M/
r?B+?
+M
#2
bmKK`Bx2/
b
V
QQ
!
P
(Qq)
+
P
◦
P
is
the
EM-charged
one;(anti-particleofP=P
i?2Q`2iB+H T`2/B+iBQMbX
Tb2m/Q@b+H`K2bQMb- `2bT2+iBp2Hv- M/ QRM/ q `2 ?2pv Us- c- bV M/ HB;?i Uu- dV [m`FbTb2m/Q@b+H`K2bQMb`2bT2+iBp2HvM/
Q M/ ofq `2- is the0) ?2pv
0 Us- c- bV M/ HB;?i Uu- dV [m`Fbi
i?2
[m`F
H2p2Hi?2b2
T`Q+2bb2b`2◦
P
EM-neutral
one;(anti-particle
P=P`2bT2+iBp2HvX h?Bb `iBQ Q7 /2+v rB/i?b Bb i?2Q`2iB+HHv +H2M #2+mb2 KQbi Q7 i?2 [mMimK@`2bT2+iBp2HvX h?Bb `iBQ Q7 /2+v rB/i?b Bb i?2Q`2iB+HHv +H2M #2+mb2 KQbi Q7 i?2 [mMimK@++?`QKQ/vMKB+b
UZ*.V
+Q``2+iBQMb+M+2H(ss) ! K + (us̄)
K (sū),(cc) Qmi!Qmi
D#2ir22M
cd D i?2(dc),MmK2`iQ`⌥ bb !M/
B + i?2
ub /2MQKBMiQ`X
B(bū) ,+?`QKQ/vMKB+b
UZ*.V
+Q``2+iBQMb+M+2H#2ir22M
i?2MmK2`iQ`
M/
i?2/2MQKBMiQ`X
h?2b2
/2+v
T`Q+2bb2b`2
pBpB0bi`QM;
BMi2`+iBQMM/?2M+2
BMBMi?2HBKBi
2t+i
BbQbTBM
0 T`Q+2bb2b
0
0
0 Q7
0
0
0
h?2b2
/2+v
`2
bi`QM;
BMi2`+iBQMM/
?2M+2
i?2
HBKBi
Q7
2t+i
¯
(ss) ! K (ds̄) K sd ! KG Ka ,(cc) ! D (cu) D (uc) , ⌥ bb ! B db B bd ,BbQbTBM
bvKK2i`v
u $d- d-i?2i?2`iBQ
bvKK2i`v
u $`iBQ#2+QK2b#2+QK2bmMBivX
mMBivX h?2
h?2BbQbTBM
BbQbTBMpBQHiBQM
pBQHiBQMKF2b
KF2b  /2pBiBQM
/2pBiBQM 7`QK
7`QK
“Kaons”“D-mesons”“B-mesons”
mMBivX
r?B+? +M #2 bmKK`Bx2/ b V QQ ! P (Qq) + P qQ - r?2`2 V M/ P `2 p2+iQ` M/
mMBivX
k
q2
MK2
i?2
`iBQ
Q7
i?2
rB/i?bb
Tb2m/Q@b+H`K2bQMbM/
q2 MK2 i?2`iBQ `2bT2+iBp2HvQ7 i?2 rB/i?bbkQ M/ q `2 ?2pv Us- c- bV M/ HB;?i Uu- dV [m`Fb+

Measuring Isospin breaking`2bT2+iBp2HvX h?Bb `iBQ
Q7 /2+v rB/i?b Bb i?2Q`2iB+HHv+H2M #2+mb2 KQbi Q7 i?2[mMimK@
++
++B
+
( (!!
K+
K
)
(
!
D
D
)
(⌥
!
B
))
K
K
)
(
!
D
D
)
(⌥
!
B
B
⇣ ⇣Qmi #2ir22M⌘i?2⇣⇣ i?2 /2MQKBMiQ`X⌘⌘..
+?`QKQ/vMKB+b
UZ*.V
+Q``2+iBQMb+M+2HM/
R R:=:=, , R R :=, , MmK2`iQ`
RR⌥⌥:=
URV
⌘
:=
:=
URV
0
0
0
0
!
K
K
0
0
0
0
! GKGaK0 D 0 ?2M+2 BM i?2 HBKBi
!!DDD
⌥⌥!
h?2b2 /2+v T`Q+2bb2b
`2
M/
Q7
2t+i
!B
B 0BB 0BbQbTBM
a pB bi`QM; BMi2`+iBQMbvKK2i`v u $ d- i?2 `iBQ #2+QK2b mMBivX h?2 BbQbTBM pBQHiBQM KF2b  /2pBiBQM 7`QK
jj
mMBivXh?2
2tT2`BK2MiH`2bmHib`2+QK#BM2/#v
i?2
S`iB+H2
.i
:`QmT
US.:V
(R),
h?2 2tT2`BK2MiH`2bmHib `2 +QK#BM2/#v i?2 S`iB+H2 .i :`QmT US.:V (R),○ PDG
values:k
q2 MK2 i?2 `iBQ Q7 i?2 rB/i?b b

RS.:
==
1.45
0.03,
+ 0.03,RS.:( 1.45
!±K±
K )

RS.:
0.010,
RS.:=(=0.798
0.798
0.010,! D±+±D)

S.:
S.:= 1.058
RR⌥
0.024.
⌥ (⌥ =!1.058B + B±±)0.024.

UkV
UkV

+QmTHBM;gV:=
BM i?2
Tb2m/Q@b+H`@K2bQM
Kbb2bmBM+ = gV 0 - i?2 /Bz2`2M+2
P + 6= mP 0 ⌘`2bmHib
⇣
⌘
⇣
R
,
R
:=
,
R
:=
.
URV
⌥
○ plane-wave
KG0mMBiv,
Ka0
0 D 0 BM _27X (R)-9 r2 Q#iBM
0B0/2pBiBQM BM RV !
7`QK
SmiiBM; i?2 KbbpHm2b!
D⌥!
B+
00P 000.32σ+
2.1σ
9.5σ!!
h?2
i?2Q`2iB+H
T`2/B+iBQM
Q7
i?2
/2+v
`i2b
7Q`
V
!
P
P
M/
V
!
P
h?2
i?2Q`2iB+H
T`2/B+iBQM
Q7
i?2
/2+v
`i2b
7Q`
V
!
P
P
M/
V
!
P
P BbBb #b2/#b2/ QM
QM
results:
2
2
2i?2i?2
THM2@rp2
7Q`KHBbK
bQ
7`X
h?2
i`22@H2p2H`2bmHiQ7
i?2+?B`H
T2`im`#iBQM
i?2Q`v
THM2@rp2
7Q`KHBbK
bQ
7`X
h?2
i`22@H2p2H`2bmHiQ7
i?2+?B`H
T2`im`#iBQM
i?2Q`v `2/b
`2/b
g
g
g
j
+
+
THM2
THM2
THM2
⌥+
2tT2`BK2MiH`2bmHib`2 +QK#BM2/#v i?2 (0.6915
S`iB+H2±.i:`QmT
US.:V=(R), (1.047 ± 0.026) .
R h?2 =
(1.5156±
0.0033),R=
0.0046)
,
R
⌥
2
2
g20
g
g
✓
◆
✓
◆
3/2
3/2
⌥0
0
2 2
2 2 4m2 2
g
g
S.:
S.:
S.:
m
m
4m
++
THM2
THM2
V
+
V
+
V
V
P
P
R
= 1.45 ± 0.03,
R
=
0.798
±
0.010,R
UkV UjV[forRisospin-symmetriccase (gP+ = g,P0⌥
R ==, )] = 1.058 ± 0.024. U9V
UjVp2HvXbiM/`/
+Q``2+iBQMb
Q7 i?2`iBQ 9 W[mMimK@2H2+i`Q/vMKB+b
H`;2`M/ BbQbTBM@#`2FBM;
`iBQ i?2g 2 +i?2Q`2iB+H/g 2 0 7m`i?2`T`2/B+iBQM
KF2 Bi ǳbQK2
T2`- i?2 bm#b+`BTibyR- M/k /2MQi2 VUZ1.V- +Q``2+iBQMb
P - M/
PiQ- i?2KF2`iBQ
W H`;2`M/
BbQbTBM@#`2FBM;
+Q``2+iBQMb
iQ i?25.2`iBQbbmKBM;
g 2 + m,/g 2 0 i?i
7m`i?2`KF2Bi-/QK@ǳbQK2
k WǴ9T+F2i(k)H`;2`-BbH2/BM;iQ  H`;2` #v
MQKHv
2``Q`
22 :mbbBM rp2+?`+i2`Bx2/
 b2iQ7Q7`Qm;?Hv
T`K2i2`b, X 0i?2
, X,
P Bb
The
mass
difference
in
the
nal
statesdeviates
the
ratio
from
unity.
kWǴ(k)H`;2`H2/BM;iQH`;2`
MQKHv
Q7`Qm;?Hv
5.2
bbmKBM;
i?i
i?2
2``Q`
Bb
/QK@
0
BMi2/
#v
i?i
BM
1[X
UkVX
AM
_27X
(j)
i?2
mi?Q`b
BMi`Q/m+2
bK2`2/
/2+v
`i2
i?i
Bb

m Bb i?2 Kbbc
Bb i?2 rB/i?@b[m`2/c M/ 0X Bb  `272`2M+2 iBK2 i r?B+? i?2
+i2`Bx2/ #v BMi2/
7mM+iBQM
b2i #vQ7Q7i?iT`K2i2`b,(j)Xi?2,i?2
X,
P BMi`Q/m+2-M/ }MH THM2@rp2
1[X UkVX
AMm,_27X
mi?Q`b bK2`2/bii2bc
/2+v i?Bb
`i2bK2`BM;
i?i Bb 
i?2BM
2M2`;v/Bz2`2M+2#2ir22M
BMBiBH+F2i iF2b
i?2:mbbBM
7Q`K
rBi?
i?2 +2Mi`Hi?2pHm2b
Q7
i?2
T2F
TQbBiBQM
X on
M/
7mM+iBQM
i?2
2M2`;v/Bbi`B#miBQM/Bz2`2M+2
BMBiBH}MHrB/i?
THM2@rp2
bK2`BM;
In M/
theplane-wave
calculation,
and
R(ψ)
only
BbX
#v0 i?2/m2
i?2 R(Φ)
BM+HmbBQM
Q7 i?2b depend
r2HH bii2bc
b #v  i?Bb
T?2MQK2MQ@[m`2/c
BbQ7GQ`2MixBM `272`2M+2
iBK2#2ir22M
iiQr?B+?
i?2M/mK P X
BbHQ;B+H#v i?27Q`K
GQ`2MixBM
iQ i?2 BM+HmbBQM
Q7 i?2 rB/i?
r2HH b #v i?2T?2MQK2MQ@
7+iQ` /Bbi`B#miBQM
Tmi #v ?M//m2
iQ `2;mH`Bx2
M mHi`pBQH2i
UloVb/Bp2`;2M+2c
MQKHv
Bi?
i?2+2Mi`HpHm2bQ7
i?2
T2F
TQbBiBQM
X
M/
in7+iQ`theisospin-symmetricg/2MbBiv0).
HQ;B+H
7Q`K#2Tmi#v?M/iQ `2;mH`Bx2
UloV
/Bp2`;2M+2ci?2?BM i?2 the+?B`Hmasses
T2`im`#iBQM
i?2Q`vi?2
Bb MQKHv
7Q` +M
2tTHBM2/rBi?
2z2+iBp2@BMi2`+iBQM@>KBHiQMBM
Kbb
T`K2i2`M
M mHi`pBQH2i'limit1.5 :2o (gBM +i?2=
T?2MQK2MQHQ;B+H
7Q`K
7Q`+MAM#2_27X
2tTHBM2/
rBi?  Kbb
T`K2i2`i?2' 1.5 :2oi?2 T?2MQK2MQHQ;B+H
7+iQ`X(9)- i?2 mi?Q`b?p2 2biBKi2/
Q7 i?2BM2H2+i`QK;M2iB+
bi`m+im`27Q`K
Q7
hM 2z2+ib
i
⇥
⇤
µ M/
+
+ +QMi`B#miBQMbµBb
0 should
0 sinceFQMb
Qi?2`
iQPgIti?2`/BiBp2+Q``2+iBQMbM/ i?2
7+iQ`X
AMP_27X(9)i?2 mi?Q`b?p2
2biBKi2/i?22z2+ibbi`m+im`2
0Q7
02H2+i`QK;M2iB+
be
good
m
~ md Q7~ O(1) MeV,b UAV 2z2+iBp2@BMi2`+iBQM@>KBHiQMBM
v-dimensionless
i?2/2MbBiv
u`2bmHiMi
coupling,
while
gKQ/2H@/2T2M/2Mi
counterpart(insteadofthei?2decay
0 is the+@)µfor
H
=
ig
V
@
P
P
@
P
+
ig
V
P
P
@
P
,
UdV
µ
µ
µ
V
+
V
0
BMi-2z+Q``2+iBQMb?p2KQ/2H@/2T2M/2Mi
im`M2/
QmigiQ #2
iBMvX AM _27X (8)irQ
UM/ M/
MQM@`2HiBpBbiB+
FQMb
M/symmetry,
Qi?2`+QMi`B#miBQMb
iQ i?2+Q``2+iBQMbi?2 `2bmHiMit of exact SU (2)flavorwe getIn`/BiBp2
the"`2Bi@qB;M2`
ordinaryms ~ O(102) MeV and mc ~ O(1) GeV.
+ = g0i(see e.g. [2]).whileh⇤ coupling,
GQ`2MixBMV
ivT2b
Q7isp2`;2//2+v
rB/i?bQp2`
BMBiBH@bii2
2M2`;v`2 BMi`Q/m+2/
rBi?
+Q``2+iBQMb?p2
im`M2/Qmicounterpart
iQ #2
iBMvX
AM(instead_27X(8)-i?2
irQgU) for"`2Bi@qB;M2`
M/
MQM@`2HiBpBbiB++µ
00
imensionless
while
g
the
of
the
decay
0
0
he
decay
width
takes
the
form,
0
+
±µ
P
+GQ`2MixBMV
ig
V`2 i?2
PivT2b@}2H/bP p2`;2/
P @/2+v
P rB/i?b, BMi2`pHb
UdVµ Q7µ2M2`;v
V 00T?2MQK2MQHQ;B+HHv- P@-P
P 0 - M/P
`2T`2b2MiBM;
i?2
p2+iQ`
i?2M/+?`;2/
Tb2m/Q@
irQ+?Qb2M
RXyRyĜRXyey:2oRXyyyĜRXRyy:2o- iQrBi?
`2@
Qp2`
i?2 K2bQMBMBiBH@bii2
2M2`;v
`2 BMi`Q/m+2/✓ 2◆
of exact SU (2) Htflavorsymmetry,
we
get
g
=
g
(see
e.g.
[2]).
In
the
ordinary
3+0
i?2Tb2m/Q@b+H`MQKHvX
2 g+ |k|
irQ+T?2MQK2MQHQ;B+HHv+?Qb2M
2M2`;v
BMi2`pHb
RXyRyĜRXyey `2bT2+iBp2Hv:2o M/ RXyyyĜRXRyy
2bQMb- i?2 !
M2mi`HK2bQMM/
Bib MiBT`iB+H2M/ :2ogV + iQ `2@
KK=,e decay width takesthe
2Q7 p2`;2/ /2+v rB/i? Bb BMi`Q/m+2/ BM _27X (e)- M/ i?2 `2bmHiMi
6Q`
Rform,- MQi?2`
ivT2
Ht
i?2
MQKHvX
3 K2bQM4⇡m[Branon,Escribano,Lucio,Pancheri,
hep-ph/0003273]
`2 i?2 p2+iQ`@K2bQM
2z2+iBp2+QmTHBM;biQ i?2Tb2m/Q@+?`;2/ Tb2m/Q@b+H`b
M/ iQ i?2`2T`2b2MiBM;
i?2
p2+iQ`
i?2+?`;2/✓◆MQKHv#2+QK2
2p2MQ7KQ`2bB;MB}+MiX
h?2`2 Bb
Bb MQ
2tTHMiBQM
7Q` i?BbMQKHvbQ
2ivT2
3p2`;2/
6Q` R ?b
- MQi?2`/2+v rB/i?
BMi`Q/m+2/
BM _27X(e)- 9.5
M/ i?2`2bmHiMig2|k|1+ i?Bb
1/2 r2 iF2+
Isospinbreaking
and7Q`QED
corrections
do notTb2m/Q@b+H`b`2bT2+iBp2HvX
AMi?2
HBKBi2 TT2`K2bQMM/Bib
MiBT`iB+H2`2bT2+iBp2HvM/
gV +BbBbQbTBM@bvKK2i`B+! MQKHv
K7`X
K |k|
= #2+QK2
,
◦
?b
KQ`2
bB;MB}+MiX
h?2`2
MQ
2tTHMiBQM
i?Bb
9.5
MQKHvbQ
m2 2p2M
4m
,
(1.5)
+
2
K
3
4⇡
m
2
h?2
#Qp2
bK2`BM;fp2`;BM;i?2 2M2`;v
bB;MB}+Mi 2z2+ib
resolvethe discrepancyof #2+mb2
R(Φ). i?2 /2@+QmTHBM;b iQ7`X
i?2 +?`;2/ Tb2m/Q@b+H`bQp2`M/
iQ i?2T`QpB/2b
gV +i?2=gV replacements(=:), 2M2`;v
U3Vi?2 /2@
0V !
P P1obtainedBbM2`i?`2b?QH/m
2m
bBimiBQMb
M2`i?2 i?`2b?QH/Bi Bb /2bB`#H2
0 1/2
V
0 are
V g'
P X AMbK2`BM;fp2`;BM;
Qp2`
i?2bB;MB}+Mi
2z2+ib #2+mb2
g?Bb
valuesfor r2!+v
Kh?2
K#Qp2after(g
g0 and m
2
2+ !T`QpB/2b
K+ !
TT2`iF2
i?2
BbQbTBM@bvKK2i`B+
HBKBi|k|Pi?2=
m KQ`2
4m
, '
(1.5)
+
iQ V
i`2i
/2+v
`B;Q`QmbHv
#v
mbBM;
rp2
T+F2ib 7Q`M2`i?2 BMBiBH
M/ }MHBibii2bX
_2@
K
+v
!
P
Bb
M2`
i?2
i?`2b?QH/m
2m
XAMbBimiBQMb
i?2
i?`2b?QH/Bb /2bB`#H2
VP
Combining them leads us to2the resultHere,the form
factor [ψ(r=0)] of the vector meson+HH+QmTHBM;
i?i i?2 S@Ki`Bt
BM i?2
THM2@rp2+QMiBMbi?2bT+2
2M2`;v@KQK2MimK@+QMb2`pBM;
B+? i?2 2z2+iBp2
g iF2bi?27Q`K BM7Q`KHBbK
i?2
KQK2MimK

2zKQ`2iQ i`2i i?2 /2+v#v mbBM; rp2 T+F2ib 7Q` i?2 BMBiBH M/ }MH bii2bX _2@✓`B;Q`QmbHv
◆
0
0
2
=
g
(=:
g
),
U3V+QKTmiBM;+/2Hi
7mM+iBQM
M/
Bbi?2Q`2iB+HHv
BHH@/2}M2/
i?2
T`Q##BHBiv`i?2` i?M i?2
V 0 for+R.
values
!+HH
KKi?2
are
obtained
after
replacements(g! g0 andm!
isr?2M
cancelledouting+ THM2@rp2(VK!
KS@Ki`Bt)+ +QMiBMb
K
i?i
BM i?2
7Q`KHBbK
i?2
2M2`;v@KQK2MimK@+QMb2`pBM;µ QMHv(1.6)⇣`i2XR"FGR2
r2HH@/2}M2/
/2+v
+M
#2 µ+H+mHi2/
 i`MbBiBQM`i?2`
7`QKUNV rp2◦ Rth :=them
2 gT`Q##BHBiv
gleads( 0us
,0 Pto00 ,⌘M/
P1 ,Bb=:Pi?2Q`2iB+HHv
(P
,
)
(P
P
) , i?2bT`Q##BHBiv
Combiningtheresult2 ) :=µ
00/2Hi
BHH@/2}M2/
r?2M+QKTmiBM;
i?M
i?2
V
2z7mM+iBQM
1
2
g
0
!
K
K
iQ  TB` thQ7 rp2bT+2
T+F2ibX h?Bb Bb i?2Q`2iB+HHv KQ`2 `2HB#H2X
2b i?2 7Q`K `i2X
BMT+F2ii?2 KQK2MimK
r2HH@/2}M2/ /2+v
T`Q##BHBiv+M #2 +H+mHi2/ QMHv b  i`MbBiBQM 7`QK  rp2
✓
◆
2
+
AMK
i?2 K
T`2pBQmb
MHvb2b(j- 9- 8)- Bi ?b #22M bbmK2/ i?i i?2 i`MbBiBQM T`Q+2bb2b `2!3/2T+F2ibX
gQ7(T+F2i!i?2) Q72 rp2◆
`2
p2+iQ`K2bQM
V - KQ`2
i?2 Tbm/Q@b+H`K2bQM P iQ 7Qm`@KQK2Mi
Bb i?2Q`2iB+HHv`2HB#H2X+ i?2h?Bb
0 - P1 - M/ P✓
2⇣
2 TB`
2
8 AM i?Bb TT2`⌘ KµUTHM2@rp2V
Rth := g+/2b+`B#2/
=:
R
(1.6)
m µ#v 4mi?2
`i2b
HQM2Xr2
T`2b2MiM
MHvbBb#b2/ QM
+
FGR2
2
AM
i?2
T`2pBQmb
MHvb2b(j98)Bi?b#22M
bbmK2/i?ii?2i`MbBiBQM
T`Q+2bb2b:= gV "=
) (P) , "µ gBb
MiBT`iB+H2
P0-,2`2bT2+iBp2Hvp2+iQ` Q7 i?2(1.7)
p2+iQ` K2bQM rBi? `2. 0 i?2 TQH`BxiBQM UNV
0 K 0 PM/
µ (P
0
1
2
!
K
2
2
8 AM i?Bb TT2`- r2 T`2b2Mi M MHvbBb #b2/ QM
g/2b+`B#2/
m #v i?2
4mthUTHM2@rp2V0`i2b
HQM2X
3
0 9 *QM+`2i2HvK
Bib ?2HB+BivX

!3/2
✓
◆
9
2
2
2 get
iTT2`Q7 i?2
p2+iQ`K2bQM
V-i`MbBiBQM
i?2
Tbm/Q@b+H`K2bQM
PBMBiBH- J2o,bii2 7Q`m± 0.016)J2o,m Qz@b?2HH= (3773.7 ±0.4)m⌥V= iQ(10579.4± 1.2) J2o,
G-Livedigits,we*QM+`2i2HvBb
r2i?27`QK MM QM@b?2HH
m = (1019.461
4m
gBMp2biB;i2
K++
2mK =p2+iQ`(987.354 ±Q7
0.032)J2o,
2mD= (3739.32± 0.10) J2o,2mB(1.7)= (10558.7 ± 0.24). p2+iQ`N J2o,e0 M/
i?2
K2bQM
rBi?
2
2Qz@b?2HH
2 ±2M2`;v
i2
7Q`i?2P=
PTQH`BxiBQM
- ?pBM;M
E
QM@b?2HH
QM2bE
E`2bT2+iBp2Hv,µ Bbm=(1019.461
0.016)
J2o,m=(3773.7
±
0.4)
J2o,
m
=
(10579.4±
1.2) J2o,1
2⌥
gR0FGR22m=Km1.51558
4mK'±0 1.516,= (995.222
0.026) J2o,2mD = (3729.68 ± 0.10) J2o, (1.8)
2mB = (10559.3 ± 0.24) J2o,
Note: q 2mK = (987.354 ± 0.032) J2o,q 2mD = (3739.32 ± 0.10) J2o, 2mB = (10558.7 ± 0.24) J2o,
mofVh?2ould
benegligibleup
to=i?2
therounded
In2m
Ref.
[2]
(asTb2m/Q@b+H`a±quotation
theBiQiH
bbmKBM;biM/`/
2``Q`digits.
T`QT;iBQM
7Q` i?2irB+2
KbbX/2+v rB/i?b`2J2o,Livedigits,
we
get
2
2
2
2m
(995.222±
0.026)
J2o,=(3729.68
0.10)
J2o,
2m
= (10559.3± 0.24)eKD
E0 :=M Qz@b?2HH
mV + P0BMBiBHi Vbii2mV =7Q` VE0iQ M
i QM@b?2HH
m
'
E
i
,
BiBQM 7`QK
0V
V
V]), the radiative correction=factor(intheordinary
QFT
formalism),⌘=
1.042,
2E(4.249 ±2``Q`
0.013)
J2o,(27.2± 1.0) J2o, KbbX 0h?2(20.5
± 2.5)
J2o. `2
⌥ =NTb2m/Q@b+H`e0 M/R
bbmKBM;i?2biM/`/T`QT;iBQM
7Q` i?2 =irB+2
iQiH
/2+v
rB/i?bq
2`;v 1E
QM@b?2HH
QM2bE
E`2bT2+iBp2Hv,=
1.51558'
1.516,
(1.8)
FGR2
1
2
+

+

0

0

+

+

+

0

0

0

fi

0

+

第三部:
波束計算の詳細と、クォーコニウム崩壊を通しての波束効果の実験的検証

Predictions for the Ratio
For+<latexit sha1_base64="LiDk8ntUhX7OdxPrrhfbCbtA4k0=">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</latexit>

!D D

and

Overall normalisation
does not contribute to the ratio R(ψ).

0! D D0<latexit sha1_base64="5B5aVYffXY5c+hjaFFu6K40R04I=">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</latexit>

Parameters: m , mD+ (= mD ), mD0 (= mD0 ), ⌧ (=Experimental data are available.

1); R0 ,

D,

; Nd

oint
p
e
l
d
d
a
within s ox.
appr

wave-packet pro leSize ofSize of
Parameter ofWave Packet Wave Packet
Form factor [Length]( nal D & Dbar) (initial ψ)[Length2][Length2]
Note:⇥<latexit sha1_base64="qVPkSh9WjQ3K1D5cUwaJUJ3ZWGE=">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</latexit>

2
=
O
10
MeV
de-Brogile<latexit sha1_base64="q12uPrmuAVTsy/F84MOc1cmUaEg=">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</latexit>

de-Brogile .

p

D

⇤
1

fi

fi

○ Tin and Tout provide overall effects (NOT to the ratio).

Renormalisationof initial-ψ’s
wavefunction(due to decaying
nature)22`7(z)Ratio:= pPredictions for the⇡For<latexit sha1_base64="LiDk8ntUhX7OdxPrrhfbCbtA4k0=">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</latexit>

z

e

x2

dx

0

Overall normalisation! D D anddoes not7mM+iBQM
contribute G(T)to the ratio
R(ψ).
#2BM; i?2 :mbb 2``Q` 7mM+iBQMX h?2 rBM/Qr#2+QK2b+0! D D01+ (=M/ x2`Q⌧mTDBM ),M/
TQmi
⌧
TX
6Q`
0;Bp2M
Q
m ,7Q`
mDT
mD07Q`
(= m
),
⌧
(=
); R
, D , +QM};m`iBQM
; Nd
Parameters:
0D<latexit sha1_base64="5B5aVYffXY5c+hjaFFu6K40R04I=">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</latexit>

oint
p
Re
e
l
d
d
a
|.
t - i?2 iBK2 `2;BQMb TBM ⌧ t ⌧ TQmi - within|ts xT. BM}t2b i?2 pHm2 Q7
approwave-packet
pro leExperimental
data are available.
`2 +HH2/ i?2#mHF- BM@iBK2@#QmM/`vM/
Qmi@iBK2@#QmM/`v`2;
T?2MQK2MQHQ;B+H MHvbBb #2HQr- r2 rBHH M2;H2+i i?2 Qmi@iBK2@#Qm
rBHH /Bb+mbbX

Size ofSize ofkXk .B{2`2MiBH /2+v
T`Q##BHBiv
Parameterof
Wave Packet Wave Packet
Form factor [Length]( nal D & Dbar) (initial ψ)
6`QK i?2 S@Ki`Bt UkRV- i?2 /Bz2`2MiBH/2+v
T`Q##BHBiv
+M
#2
/
2
2
[Length ][Length ][Reminder]Note:/ X 1 / P1 / X 2 / P2
2⇤◦⇥ /PV !P P =SV !P P
1
3
3
2
(2⇡)
(2⇡)
Renormalisation
10 MeVde-Brogile = Or✓33of initial-ψ’s 2
p 1
/
P
/
P
1
t
Non-relativistic
work2 ne.
2 D approximations
4 wavefunctiont ( !)
2.
de-Brogile= |g2z | NV(2⇡)
e
3 2E (2⇡)3 2E
2E(2⇡)(due ⇡
to decaying
0
1
2
+
P
V !P
P
WP
s
◦ RV
nature)(the Ratio◆in3 terms of transition probability)
:=
✓
T T0
t
○ Tin and Tout provide
effects(NOT
tothe
ratio).
+PV !Poverall
0 Pt 0
s
⌧0
3
3
R
2
4⌧
⇥
/
X
/
X
e
|G(T)|
e
1
2
5
⇡
0 1 2
3

3

<latexit sha1_base64="qVPkSh9WjQ3K1D5cUwaJUJ3ZWGE=">AAACW3icbVBNSxxBEO0d86GjSVaDJy9NFsFAXGZE1IsgevESVMiuws641PTU7Dbb80F3jbgM8+/8Ex5y9eBVf0B6Pw5+pKDpx6v3qKoXFUoa8rz7hrPw4eOnz4tL7vLKl6/fmqtrXZOXWmBH5CrXVxEYVDLDDklSeFVohDRSeBmNTib9yxvURubZHxoXGKYwyGQiBZCl+s3rQGFCPTdQ1hNDPyC8pSrG7WOdD6TCmh+6QQo0FKCqs3qq3uK+d11t79Q8+DXT/8ZubRnfMloOhvTTnf1hv9ny2t60+Hvgz0GLzeu833wI4lyUKWYkFBjT872Cwgo0SWG3cYPSYAFiBAPsWZhBiiaspjnUfNMyMU9ybV9GfMq+dFSQGjNOI6ucnGTe9ibk/3q9kpKDsJJZURJmYjYoKRWnnE9C5bHUKEiNLQChpd2ViyFoEGSjfzUl0jBCql0bjP82hvegu9P299q7F7uto+N5RItsg/1gW8xn++yInbJz1mGC3bFH9sSeG3+dBcd1VmZSpzH3fGevyln/B/fxtpE=</latexit><latexit sha1_base64="q12uPrmuAVTsy/F84MOc1cmUaEg=">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</latexit>

fi

fi

fi

<latexit sha1_base64="Um4SJ+19+o1motMahC3laEZssMs=">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</latexit>3

3

Bp2 THM2@rp2 `2bmHi rBi?Qmi i?2 7Q`K@7+iQ`

Predictions for the Ratio

For<latexit sha1_base64="LiDk8ntUhX7OdxPrrhfbCbtA4k0=">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</latexit>! D+ Dand! D0 D0<latexit sha1_base64="5B5aVYffXY5c+hjaFFu6K40R04I=">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</latexit>

Parameters: m , mD+ (= mD ), mD0 (= mD0 ), ⌧ (=Experimental data are available.

in

et
k
c
a
p
e
v
a
lw
l
a
m
as
( )1/2 = 0.01 [MeV -1 ]1.1

DPlane-Wave(Non-R)

Plane-Wave (R)Wave-Packet (bulk+boundary+interference)

0.8

PDG(±1)

0.7

Plane-Wave parton-level(±1)

0.01

0.02

0.03

0.04

R0 [MeV-1 ]
+
fi

R

0.9

0

0

); R0 ,

D,

; Nd

wave-packet pro le◦ Parton-level (factored form factor)
deviated from PDG (as reported).

WP-bulk

1.0

1

0.05

Bp2 THM2@rp2 `2bmHi rBi?Qmi i?2 7Q`K@7+iQ`

Predictions for the Ratio

For<latexit sha1_base64="LiDk8ntUhX7OdxPrrhfbCbtA4k0=">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</latexit>! D+ Dand! D0 D0<latexit sha1_base64="5B5aVYffXY5c+hjaFFu6K40R04I=">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</latexit>

Parameters: m , mD+ (= mD ), mD0 (= mD0 ), ⌧ (=Experimental data are available.

in

et
k
c
a
p
e
v
a
lw
l
a
m
as
( )1/2 = 0.01 [MeV -1 ]D
Plane-Wave(Non-R)

1.1

Plane-Wave (R)PDG(±1)

0.7

Plane-Wave parton-level(±1)

0.02

0.03

0.04

R0 [MeV-1 ]

fi

+

; Nd

wave-packet pro leprovides the compositeness well.
good ts!

0.8

0.01

D,

◦ The improved form factor

Wave-Packet (bulk+boundary+interference)

fi

R

0.9

); R0 ,◦ Parton-level (factored form factor)
deviated from PDG (as reported).

WP-bulk

1.0

1

0

0

0.05

Bp2 THM2@rp2 `2bmHi rBi?Qmi i?2 7Q`K@7+iQ`

Predictions for the Ratio

For<latexit sha1_base64="LiDk8ntUhX7OdxPrrhfbCbtA4k0=">AAACNnicbVDLSgMxFM34rPVVdekmWARBLDNStO4KunBZwT6grSWT3rahmcyQ3BHL0G/xN/wBt7p2407d+gmmj0W1Xrhwck5uTu7xIykMuu6bs7C4tLyymlpLr29sbm1ndnYrJow1hzIPZahrPjMghYIyCpRQizSwwJdQ9fuXI716D9qIUN3iIIJmwLpKdARnaKlW5qIRGUEbGNKru2PbJxbDAyaUqTYdpmdUlzZC+9LIKLGnYSuTdXPuuOg88KYgS6ZVamU+G+2QxwEo5JIZU/fcCJsJ0yi4BOsVG4gY77Mu1C1ULADTTMYrDumhZdq0E2rbCumYnZ1IWGDMIPDtzYBhz/zVRuR/Wj3GTqGZCBXFCIpPjDqxpHbnUV60LTRwlAMLGNfC/pXyHtOMo031l4uvWR9wmLbBeH9jmAeV05x3lsvf5LPFwjSiFNknB+SIeOScFMk1KZEy4eSRPJMX8uo8Oe/Oh/M1ubrgTGf2yK9yvn8AlMuqiA==</latexit>! D+ Dand! D0 D0<latexit sha1_base64="5B5aVYffXY5c+hjaFFu6K40R04I=">AAACh3icbVFda9RAFJ1ErW3Uuir44oODS6HFdk2k1EIRKhbsU2lLty3sbMPN7Ox22JkkzNwIS8gf8B/6B3z2JzgZg/TDCwOHc889dzg3K5W0GMc/g/DBw0dLj5dXoidPn60+7714eW6LynAx5IUqzGUGViiZiyFKVOKyNAJ0psRFNv/a9i++C2NlkZ/hohRjDbNcTiUHdFTa+6FTVlq5SRnVaX1w9b5hm+ufPdxqNv7RsadbzApn127zrJNEDKHyJtRr2DfQGq7qraTx7MaeM4lO07g1Y1bONKQHN3Cr2YuO0kna68eD2Be9D5IO9ElXx2nvF5sUvNIiR67A2lESlziuwaDkSjQRq6wogc9hJkYO5qCFHdc+s4auOWZCp4VxL0fq2ZsTNWhrFzpzSg14be/2WvJ/vVGF091xLfOyQpHzv4umlaJY0PYAdCKN4KgWDgA30v2V8mswwNGd6daWzMBcYBO5YJK7MdwH5x8Hyc5g+2S7v7/bRbRM3pB3ZJ0k5BPZJ4fkmAwJJ7+D18HbgIYr4YdwJ+y0YdDNvCK3KvzyBxkbvzU=</latexit>

Parameters: m , mD+ (= mD ), mD0 (= mD0 ), ⌧ (=Experimental data are available.

in

et
k
c
a
p
e
v
a
lw
l
a
m
as
( )1/2 = 0.01 [MeV -1 ]D
Plane-Wave(Non-R)

1.1

Plane-Wave (R)0.7

Plane-Wave parton-level(±1)

0.02

0.03

0.04

R0 [MeV-1 ]fiwave-packet pro le00◦ Wave-packet calculations explain
the PDG result.

0.8

+

; Nd

provides the compositeness well.
good ts!

PDG(±1)

0.01

D,◦ The improved form factor

Wave-Packet (bulk+boundary+interference)

fi

R

0.9

); R0 ,◦ Parton-level (factored form factor)
deviated from PDG (as reported).

WP-bulk

1.0

1

0.05

Bp2 THM2@rp2 `2bmHi rBi?Qmi i?2 7Q`K@7+iQ`

Predictions for the Ratio

For<latexit sha1_base64="LiDk8ntUhX7OdxPrrhfbCbtA4k0=">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</latexit>! D+ Dand! D0 D0<latexit sha1_base64="5B5aVYffXY5c+hjaFFu6K40R04I=">AAACh3icbVFda9RAFJ1ErW3Uuir44oODS6HFdk2k1EIRKhbsU2lLty3sbMPN7Ox22JkkzNwIS8gf8B/6B3z2JzgZg/TDCwOHc889dzg3K5W0GMc/g/DBw0dLj5dXoidPn60+7714eW6LynAx5IUqzGUGViiZiyFKVOKyNAJ0psRFNv/a9i++C2NlkZ/hohRjDbNcTiUHdFTa+6FTVlq5SRnVaX1w9b5hm+ufPdxqNv7RsadbzApn127zrJNEDKHyJtRr2DfQGq7qraTx7MaeM4lO07g1Y1bONKQHN3Cr2YuO0kna68eD2Be9D5IO9ElXx2nvF5sUvNIiR67A2lESlziuwaDkSjQRq6wogc9hJkYO5qCFHdc+s4auOWZCp4VxL0fq2ZsTNWhrFzpzSg14be/2WvJ/vVGF091xLfOyQpHzv4umlaJY0PYAdCKN4KgWDgA30v2V8mswwNGd6daWzMBcYBO5YJK7MdwH5x8Hyc5g+2S7v7/bRbRM3pB3ZJ0k5BPZJ4fkmAwJJ7+D18HbgIYr4YdwJ+y0YdDNvCK3KvzyBxkbvzU=</latexit>

Parameters: m , mD+ (= mD ), mD0 (= mD0 ), ⌧ (=Experimental data are available.

in

et
k
c
a
p
e
v
a
lw
l
a
m
as
( )1/2 = 0.01 [MeV -1 ]D
Plane-Wave(Non-R)

1.1

Plane-Wave (R)0.7

Plane-Wave parton-level(±1)

0.02

0.03

0.04

R0 [MeV-1 ]fiwave-packet pro le00◦ Wave-packet calculations explain
the PDG result.

0.8

+

; Nd

provides the compositeness well.
good ts!

PDG(±1)

0.01

D,◦ The improved form factor

Wave-Packet (bulk+boundary+interference)

fi

R

0.9

); R0 ,◦ Parton-level (factored form factor)
deviated from PDG (as reported).

WP-bulk

1.0

1◦ R0 ~ O(10-2) MeV-1 ~ (QCD scale)-1
is a reasonable choice.

0.05

(b?QrM BM 1[X UkV)- M/ i?2 7Q`KmH 7Q` i?2 MBp2 THM2@rp2 `2bmHi rBi?Qmi
2z2+i Bb b?QrM BM 1[X
UdyVX
Predictions
for the Ratio

For<latexit sha1_base64="LiDk8ntUhX7OdxPrrhfbCbtA4k0=">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</latexit>! D+ Dand! D0 D0<latexit sha1_base64="5B5aVYffXY5c+hjaFFu6K40R04I=">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</latexit>

Parameters: m , mD+ (= mD ), mD0 (= mD0 ), ⌧ (=Experimental data are available.

1); R0 ,

D,

; Nd

wave-packet pro le

t

ke
c
a
p
ve
a
w
rge

a
l
a
n
i

1.1

Plane-Wave(Non-R)1.0

is far away from PDG(outside the shown region),
also does not make sense.
Boundary part should be taken.

Plane-Wave (R)Wave-Packet (bulk+boundary+interference)

0.9
0.8

PDG(±1)

0.7

Plane-Wave parton-level(±1)

0.01

0.02

0.03

0.04

0.05

R0 [MeV-1 ]fiR◦ Bulk-only wave-packet result(D )1/2 = 1 [MeV-1 ]+0(b?QrM BM 1[X UkV)- M/ i?2 7Q`KmH 7Q` i?2 MBp2 THM2@rp2 `2bmHi rBi?Qmi
2z2+i Bb b?QrM BM 1[X
UdyVX
Predictions
for the Ratio

For<latexit sha1_base64="LiDk8ntUhX7OdxPrrhfbCbtA4k0=">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</latexit>! D+ Dand! D0 D0<latexit sha1_base64="5B5aVYffXY5c+hjaFFu6K40R04I=">AAACh3icbVFda9RAFJ1ErW3Uuir44oODS6HFdk2k1EIRKhbsU2lLty3sbMPN7Ox22JkkzNwIS8gf8B/6B3z2JzgZg/TDCwOHc889dzg3K5W0GMc/g/DBw0dLj5dXoidPn60+7714eW6LynAx5IUqzGUGViiZiyFKVOKyNAJ0psRFNv/a9i++C2NlkZ/hohRjDbNcTiUHdFTa+6FTVlq5SRnVaX1w9b5hm+ufPdxqNv7RsadbzApn127zrJNEDKHyJtRr2DfQGq7qraTx7MaeM4lO07g1Y1bONKQHN3Cr2YuO0kna68eD2Be9D5IO9ElXx2nvF5sUvNIiR67A2lESlziuwaDkSjQRq6wogc9hJkYO5qCFHdc+s4auOWZCp4VxL0fq2ZsTNWhrFzpzSg14be/2WvJ/vVGF091xLfOyQpHzv4umlaJY0PYAdCKN4KgWDgA30v2V8mswwNGd6daWzMBcYBO5YJK7MdwH5x8Hyc5g+2S7v7/bRbRM3pB3ZJ0k5BPZJ4fkmAwJJ7+D18HbgIYr4YdwJ+y0YdDNvCK3KvzyBxkbvzU=</latexit>

Parameters: m , mD+ (= mD ), mD0 (= mD0 ), ⌧ (=Experimental data are available.

1); R0 ,

D,

; Nd

wave-packet pro le

t

ke
c
a
p
ve
a
w
rge

a
l
a
n
i

1.1

Plane-Wave(Non-R)1.0

is far away from PDG(outside the shown region),
also does not make sense.
Boundary part should be taken.

Plane-Wave (R)Wave-Packet (bulk+boundary+interference)◦ R0 for explaining PDG is similar
for smaller and bigger wave packets.

0.9
0.8

PDG(±1)

0.7

Plane-Wave parton-level(±1)

0.01

0.02

0.03

0.04

0.05

R0 [MeV-1 ]fiR◦ Bulk-only wave-packet result(D )1/2 = 1 [MeV-1 ]+0Predictions for the Ratio
For6B;m`2
Kaons:
Ry, h?2 THM2@rp2 `iBQ +QKT`BM; i?2 /2+v `i2b Q7

1.2

R

R

+ D iQ0 D 0 Bb!
D!
D(K )1/2 = 1 [MeV-1 ](K )1/2 = 0.1 [MeV -1 ]/`rM b 1.6
 7mM+iBQM Q7 R0 X 6Q` +QKT`BbQM- r2 HbQ 1.6
b?Qr i?2 rp2@T+F2i `2bmHib 7Q` irQ }t2/ppPlane-Wave parton-level(±1)
Plane-Wave parton-level(±1)11rp2@T+F2i bBx2b Q7 i?2 .@K2bQMb
Q7
=
1
J2o
UH27i
TM2HV
M/
=
0.01
J2o
D
D
PDG(±1)PDG(±1)U`B;?i TM2HVX
h?2 Qi?2` +QMp2MiBQMb `2 i?2 bK2 1.4
b BM 6B;X NX
1.4
Wave-Packet (full)

1.0 Plane-Wave

1.2

Wave-Packet (full)

1.0 Plane-Wave

e.
n
k
0.8
0.8
r
o
0.01
0.02
0.03
0.04
0.05
0.01
0.02
0.03
0.04
0.05
w
o
s
l
a
R [MeV ]R [MeV ]sesa
For B-mesons:
C
n
o
s
e
+ K iQ0 K 0 Bbm
6B;m`2 N, h?2 THM2@rp2`iBQ+QKT`BM;
i?2
/2+v
`i2b
Q7
!
K
!
K
B
d
n
a
/`rM b  7mM+iBQM+TiBQMb ǳ`2HǴ M/ ǳLQM@`2HǴ
i?2 `2HiBpBbiB+M/n(Q7) R0=- 1r?2`2[MeV i?2]( ) =K2M
0.01 [MeV]oaK `2bmHib b?QrM BM 1[bX UeRV M/ UejV1.10
1.10 `2bT2+iBp2HvX 6Q` +QKT`BbQM- r2 HbQMQM@`2HiBpBbiB+(Non-rel)(Non-rel)Plane-Wave (rel)

0

B

1/2

Plane-Wave (rel)

-1

0

-1

B

-1

1/2

-1

p

R

1
=
1
J2o
K
p
PDG(±1)PDG(±1)
1
1.06 M/
UH27i TM2HVU`B;?i TM2HVX 1.06
AM #Qi? TM2Hb- i?2 THM2@rp2 `2bmHib `2
K = 0.1 J2oparton-level(±1)
1.04
i?2 bK2
bBM+2 i?2v Plane-Wave`2 BM/2T2M/2Mi
Q7 K X h?21.04
2tT2`BK2MiHPlane-Wave`2bmHi-parton-level(±1)b?QrM BM 1[X UkV- Bb
1.02
T`QpB/2/1.02#v S.: (R)XR

1.08 bBx2b Q7 i?2 EQMb Q7
b?Qr i?21.08
rp2@T+F2i `2bmHib 7Q` irQ }t2/ rp2@T+F2i

1.00
0.98

Wave-Packet (full)Plane-Wave(Non-rel)

1.00
0.98

Plane-Wave (rel)Wave-Packet (full)Plane-Wave(Non-rel)Plane-Wave (rel)fi

6Q` MQi?2` +QKT`BbQM- r2 rBHH HbQ b?Qr MHvb2b mbBM; Bib MQM@`2HiBpBbiB+ TT`QtBKi2/
0.96
0.96
0.01
0.02
0.03
0.04
0.05
0.01
0.02
0.03
0.04
0.05
7Q`K,
R0 [MeV-1 ]R0 [MeV-1 ] ⌘2
⇣
2
R
m
|V
V
|
0
0
1
2
0
P
P
1/2
+1
m (m
2m + )3/2

Constraint via Resonant shape
via e e+ !
<latexit sha1_base64="+XwdVtbWh88RoJM3PBfKSSWPizg=">AAACGHicbZDLSgMxFIYz9VbrbdRlN8EiCNIyI0W7LLgRuqlgL9AbmfRMG5rJDElGKKULX8MXcKtv4E7cuvMFfA7TaRe29YfAx3/O4eT8XsSZ0o7zbaU2Nre2d9K7mb39g8Mj+/ikrsJYUqjRkIey6REFnAmoaaY5NCMJJPA4NLzR7azeeASpWCge9DiCTkAGgvmMEm2snp2Fbh66l7itQ9yOhiyBijEq3XzPzjkFJxFeB3cBObRQtWf/tPshjQMQmnKiVMt1It2ZEKkZ5TDNtGMFEaEjMoCWQUECUJ1JcsQUnxunj/1Qmic0Tty/ExMSKDUOPNMZED1Uq7WZ+V+tFWu/1JkwEcUaBJ0v8mOOzaGzRHCfSaCajw0QKpn5K6ZDIgnVJrelLZ4kI9DTjAnGXY1hHepXBfe6ULwv5sqlRURplEVn6AK56AaV0R2qohqi6Am9oFf0Zj1b79aH9TlvTVmLmVO0JOvrFxi8nmM=</latexit>! K +K  (K )/ =[- ] =[- ]





- 


  

( -  +     +  - ) []( -  +     +  - ) [] -









  (K)/ =[- ] =[- ]



-



   - [] []+K=[ 6B;m`2 Rk, h?2}ii2/`2bQMM+2]/Bbi`B#miBQMb
Q7 BM e e+ ! !
K`2 /`rM 7Q`+
pmΦ are1 Note: σ(e e →Φ) and
i?2 irQ
b2ib Q7 i?2 }t2/ T`K2i2`bM/ R0 = 0.0015 J2o 1 (G27i
TM2H)
K = 10 J2op1determinedbyKbbstatisticalM/M/ R0 = 0.01 J2o 1 (_B;?i TM2H)r?2`2 i?2Q7 M/ts.i?2
K = 10 J2o
MQ`KHBxiBQM 7+iQ` Ne e+ ! `2 /2i2`KBM2/ i?`Qm;? Qm` biiBbiB+H MHvbBb #b2/ QM i?2
2 7mM+iBQM /2}M2/ BM 1[X Ud3VX h?2 #2bi@}i T`K2i2`b M/ i?2 2 7mM+iBQMb 7Q` i?2 H27if`B;?i
TM2H`2 b?QrM BM 1[bX UdNV M/ U3yV f BM 1[bX U3RV M/ U3kV- `2bT2+iBp2Hv- 7Q` i?2 rp2@T+F2i
UqSV M/ THM2@rp2 rBi?QmifrBi? 7Q`K 7+iQ` U"qf"q rBi? 66VX

Ç 7Q`

p




K = 10 J2o-
( 

)

1

M/ R0 = 0.0015 J2o 1 (G27i TM2H Q7 6B;X Rk),

 = 1019.8
J2o,
mqS#2bi@}i

Sq@S`iQMfi  (-+     +  - )/() -(  ) / [- ]


NeqS
e+ !

Sq@S`iQM#2bi@}i= 6.23 ⇥ 105 J2o 1 ,41

Constraint via Resonant shape
via e e+ !

! 2D (D+ D and D0 D0 )<latexit sha1_base64="Gv8747VCn9xPV3aqqjZHNZ93M3E=">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</latexit>  ( )/ =[- ] =[- ]

(+-    ) [](+-    ) [](  () 
 
  




  













-





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

  ( )/ =[- ] =[- ]

- 






(  ()) 
 
  





  









 []







 []

 =[- ]

+ 

+

+  (-+    )/() + !
6B;m`2
R9,
h?2
}ii2/
`2bQMM+2
/Bbi`B#miBQMb
Q7
BM
e
e
!
2D
/`rM
7Q`
i?2 }t2/-e+`2
Note:
σ(e
→ψ)
and
m
ψ are

p
p
1
1
T`K2i2`bM/ R0 = 0.0015 J2o 1 U7Q` i?2 H27i TM2HV- M/
D = 1 J2o
D = 1 J2o
determined by statistical ts.
1M/ R0 = 0.01 J2o U7Q` i?2 `B;?i TM2HV- r?2`2 i?2 Kbb Q7 M/ i?2 MQ`KHBxiBQM 7+iQ`
Ne e
`2 /2i2`KBM2/ i?`Qm;? Qm` biiBbiB+H MHvbBb #b2/ QM i?2 2 7mM+iBQM /2}M2/ BM+!
1[X U3NVX h?2 #2bi@}i T`K2i2`b M/ i?2 2 7mM+iBQMb 7Q` i?2 H27if`B;?i TM2H `2 b?QrM BM
1[bX 
UNyV M/ UNRV f BM 1[bX UNkV M/ UNjV- `2bT2+iBp2HvX h?2 Qi?2` +QMp2MiBQMb `2 i?2 bK2
b i?Qb2 Q7 6B;X RkX

 -( 
h?2 2 7mM+iBQMb
`2) /2}M2/ b

fi

2 ( )/ [ - ], qS :=



X



X

⇣

qS
iA

⇣

2tT-A
iA

⌘2

⌘2

,

まとめ
1. 平面波の計算は、”平均的な確率的振る舞い”しか表せない。
2. 波束で計算するなら、理論的にどんな物理過程に関しても、“平均値を超えた” 絶対確率が得られる。
3. 波束でしか見えない物理は、どうもありそうだ。

まとめ
1. 平面波の計算は、”平均的な確率的振る舞い”しか表せない。
2. 波束で計算するなら、理論的にどんな物理過程に関しても、“平均値を超えた” 絶対確率が得られる。
3. 波束でしか見えない物理は、どうもありそうだ。
より一般の物理過程(及び観測)に適応するには、更なる一般論の議論が必要 (future works)。

まとめ
1. 平面波の計算は、”平均的な確率的振る舞い”しか表せない。
2. 波束で計算するなら、理論的にどんな物理過程に関しても、“平均値を超えた” 絶対確率が得られる。
3. 波束でしか見えない物理は、どうもありそうだ。
より一般の物理過程(及び観測)に適応するには、更なる一般論の議論が必要 (future works)。

k
n
a
h
T

!
u
yo

Backup Slides

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2
2 (2⇡
p
SV !P P = ig2z NV
e 2
2⇡ t G(T)s)
2EA ⇡ A

Form of Gaussian wave-packet S-matrix
A=0

V (T
2

⇥e

T0 +i t !)+

2
V t
8

Fe (|V1

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2`7
2
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,

C 2 := C 2

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m Bb i?2 Kbbc
Bb i?2 rB/i?@b[m`2/c M/ 0X Bb t ✓`272`2M+2s`+i2`Bx2/ #v  b2i Q7 T`K2i2`b m, , X , X,
P 3
3
R
2
(T T )+
Re
⇥
/
X
/
X
e
|G(T)|e
Fe (|V1
2
5
r?2`2iF2b
i?2 rBM/Qr
7mM+iBQM
G(T)Bb/2}M2/
b
T+F2ii?2 :mbbBM
7Q`K
rBi?
i?2
+2Mi`HpHm2bQ7
i?2
T2F
TQbBiBQM
X
M/
⇡
0 1 2
?@b[m`2/c M/ X 0 Bb  `272`2M+2 iBK2 i r?B+? i?2
MimK P Xh ⇣⌘i2
Z TQmi1
V t +irBi?
i?2+2Mi`HpHm2bQ7
i?2
T2F
TQbBiBQM
X t M/
/t
t
T
!
2
2
Bi?BM i?2+?B`H
T2`im`#iBQM
i?2Q`vi?22z2+iBp2@BMi2`+iBQM@>KBHiQMBM
Bbp
G(T) :=
e t r?2`2 r2 ?p2 iF2M i?2 p2`;2 Qp2` i?2 /2MbBiv?2HB+Biv 0 - r?B+? `2bmHib BM i?2 ?2HB
2⇡ t
i
TBM ⇥
2z2+iBp2+QmTHBM;,h⇤UAV"V µ P + @ P!µBb P 0 @ P 0 P 0 @ P 0 ,!#+ /2MbBiv
b 2z2+iBp2@BMi2`+iBQM@>KBHiQMBM
Q`v- H
i?2
=
ig
P
@
P
+
ig
V
UdV
µ
µ
µ
µ
V
+
V
0
V t
V t
BMi-2z
2 X
2
g
g
T
T
+
i
!
T
T
+
i
!
1
2
µ
µ
BM
t
Qmi
t
2
V
V
2 :=
2
2
h
i
|g
|
"
(P
,
)
(P
P
)
=
(P
P
)
. Uk
µ
0
0
1
2
2z
1
2
p
p
⇤
=
2`7
2`7
,
3
3
+
µ
0
0
0@ P 0 ,2±
02
2 tTb2m/Q@
0
P
@
P
+
ig
V
P
@
P
P
UdV K2bQM- i?2 +?`;2/tµµV 0 `2 i?2 }2H/b
V- P µ- P - M/ P
`2T`2b2MiBM;
i?2 p2+iQ`

Form of Gaussian wave-packet S-matrix

2

t

0

V

2
V t
4

0

>2`2i?2 Hbi
7m`i?2` bbmK2b
i?2 pMBb?BM;
K2bQMbi?2 M2mi`H Tb2m/Q@b+H` K2bQMM/
Bib 2[mHBivMiBT`iB+H2`2bT2+iBp2HvM/ BMBiBH
gV + KQK2MimK P0 = 0X
R8
q2 /QTi i?2 7QHHQrBM; MQiiBQM g7Q`
`#Bi``v
b+H`
M/
p2+iQ`p`B#H2b
M/
C- `2bT2+iBp2Hv,
Tmi2
i?2 BMi2;`i2//2+v
T`Q##BHBiv
mM/2` Ci?Bb
bbmKTiBQM
BM a2+X kXj M/ BM
=
g
(→g)
is
suggested
via
+
0
`2
i?2
p2+iQ`@K2bQM
2z2+iBp2+QmTHBM;biQi?2+?`;2/
Tb2m/Q@b+H`b
M/
iQ
i?2
b
`2T`2b2MiBM;
i?2
p2+iQ`
K2bQMi?2+?`;2/
Tb2m/Q@V0>2`27i2`- r22 bbmK2 #Qi? i?2 7QHHQrBM; +QM/BiBQMb
2
X CA AM i?Bb
theTT2`isospind)
H Tb2m/Q@b+H`b`2bT2+iBp2HvX
r2 symmetryiF2i?2
HBKBi2
`
K2bQM- M/ Bib
MiBT`iB+H2`2bT2+iBp2HvM/X
gV C
A (u
2
+BbQbTBM@bvKK2i`B+
2

C :=

,

C := T s T,t + iA Tb2m/Q@b+H`b M/ iQ i?2A2
2 +QmTHBM;b iQ i?2 +?`;2/
A=0
gV + = gV 0 (=: gV ), A=0 p2 t
i?Bb
TT2`- r2 iF2 i?2 BbQbTBM@bvKK2i`B+ HBKBiRe
s

BM

V

t

C :=TC T

!

1,

Qmi

U3V

C .V t + i
p

2

t

!

2 t

V
AM 1[X UkjV- T QM i?2 `B;?i@?M/ bB/2 Bb S?vbB+HHv`2TH+2/ 7`QK
Q`B;BMH/2}MBiBQM
Q7 G(T)
BMH2bi(RR) QM2
b TQ7!
2+? Q7i?2i?2b2+QM/BiBQMb
Bb biBb}2/
r?2M ii?2T7QHHQrB2

r?B+? i?2 2z2+iBp2 +QmTHBM; g2z iF2b i?2 7Q`K BM i?2 KQK2MimK bT+2+ = gV 0(=: gV ),/BiBQMb Bb K2i,

U3V

p
UNV
TQmi |
BMi2`+iBQM iBK2 T Bb t V r?2M i?2pF2b i?2 7Q`K BM i?2 KQK2MimK bT+2
7`QK TBM UQ` TQmi V +QKT`2/ iQ i?2 i2KTQ`H rB/i? Q7 i?2 Qp2`HTt - r?
+Q``2bTQM/biQ i?2+b2 U6B;X K2bQM
k- H27iVc P P0 - P1 - M/ P2 `2 i?2 7Qm`@KQK2Mi Q7 i?2 p2+iQ`
K2bQM
V - ǳ#mHF@HBF2Ǵ
i?2 Tbm/Q@b+H`
µ
µ
p
1 UNV
1
)
:=
g
"
(P
,
)
(P
P
b2MiBT`iB+H2
M/
p2+iQ` Q7 i?2 p2+iQ` K2bQM rBi?
0
V µ P0- `2bT2+iBp2HvÇ
12 ) , "µ Bb i?2 TQH`BxiBQMt
V r?2M i?2 ǳK2M HB72iBK2 i?`Qm;? #mHF 2z2+iǴ V Bb Km+? b?Qµpµ
g2z ( 0 , P0 , P1 , P2 ) := gV "µÇ(P|T0 , T0BM)|(PP
N 1 t UQ`2 )|T,7i2` i?2 b[m`2 +QKTH2iBQM Q7 t M/ i?2 MHviB+ :mbbBM BMi2;`iBQM Qp2` x BM UR9VK/2 BM (RR)- r2 `2T`2b2Mi i?2 S@Ki`Bt b 7QHHQrb,
!
✓
◆
2
Y
p
1
1 3/4
t ( !)2s ( P )2 R
3/2
2
2 (2⇡
p
SV !P P = ig2z NV
e 2
2⇡ t G(T)s)
2EA ⇡ A

Form of Gaussian wave-packet S-matrix
A=0

V (T
2

⇥e

T0 +i t !)+

2
V t
8

Fe (|V1

Uk

V2 |) ,

r?2`2 i?2 rBM/Qr 7mM+iBQM G(T) Bb /2}M2/ bReh ⇣⌘i2Z TQmi1
V t +i
/t
t
T
t !
2
p
G(T) :=
e 2t
2⇡ t
TBM
" factor for decaying Φ!
rescaling
V t
T TBM+i t !
1
2
p
=
2`7
2`7
2
2 t

T

TQmi

V

p

2

t

+i t !

2 t

!#

,

Uk

overlaps of the wave packets
normalisation factors
R8q2 /QTi i?2 7QHHQrBM;
7Q` `#Bi``v b+H` M/ p2+iQ`(includingp`B#H2b C M/
C- `2bT2+iBp2Hv,
approximatedof freeMQiiBQM
Gaussians
22XX
conservation)CA
CAEnergy-Momentum
2
2
2
C :=

,

s

A=0

Re

A

C :=

,

s

A=0

C := CC .

A

AM 1[X UkjV- T QM i?2 `B;?i@?M/ bB/2 Bb `2TH+2/ 7`QK i?2 Q`B;BMH /2}MBiBQM Q7 G(T) BM (RR) b T ! T

N

V

2

7i2` i?2 b[m`2 +QKTH2iBQM Q7 t M/ i?2 MHviB+ :mbbBM BMi2;`iBQM Qp2` x BM UR9VK/2 BM (RR)- r2 `2T`2b2Mi i?2 S@Ki`Bt b 7QHHQrb,
!
✓
◆
2
Y
p
1
1 3/4
t ( !)2s ( P )2 R
3/2
2
2 (2⇡
p
SV !P P = ig2z NV
e 2
2⇡ t G(T)s)
2EA ⇡ A

Form of Gaussian wave-packet S-matrix
A=0

V (T
2

⇥e

T0 +i t !)+

2
V t
8

Fe (|V1

Uk

V2 |) ,

r?2`2 i?2 rBM/Qr 7mM+iBQM G(T) Bb /2}M2/ bRe
● Geometricalvariables characterise
S.
h ⇣
⌘i2
Z TQmi1
V t +i
/t
t
T
t !
2 t
2
p
G(T)
:=
e
( ! ⇠ E := Eout E
P in )
in , P := P out
2⇡
t
TBM
"
!
t
V t
P11
T TBM
+ i t !P2
2
p
=
2`7
2`7
2
2 t
Tout
~√σ1
~√σ2
<latexit sha1_base64="seJLsVxsNZlgnuzqCMIdpZ+LBBI=">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</latexit><latexit sha1_base64="idQ+0AEvw5tEYKBNmRz+kv9HnFY=">AAAB/HicbVDLTgJBEJzFF+IL9ehlIjHxRHbVqEeiF4+QyCOBDZkdGpgwO7uZ6TUhG/wBr/oH3oxX/8Uf8DscYA8CVtJJpao73V1BLIVB1/12cmvrG5tb+e3Czu7e/kHx8KhhokRzqPNIRroVMANSKKijQAmtWAMLAwnNYHQ/9ZtPoI2I1COOY/BDNlCiLzhDK9WwWyy5ZXcGukq8jJRIhmq3+NPpRTwJQSGXzJi258bop0yj4BImhU5iIGZ8xAbQtlSxEIyfzg6d0DOr9Gg/0rYU0pn6dyJloTHjMLCdIcOhWfam4n9eO8H+rZ8KFScIis8X9RNJMaLTr2lPaOAox5YwroW9lfIh04yjzWZhS6DZCHBSsMF4yzGsksZF2bsuX9auSpW7LKI8OSGn5Jx45IZUyAOpkjrhBMgLeSVvzrPz7nw4n/PWnJPNHJMFOF+/+UKVaA==</latexit><latexit sha1_base64="/bZzYnZPTFyNLl1bW1ywBx1XfZY=">AAACB3icbVDLSsNAFJ34rPVVdekmWARXJVFRl0U3Liv0BU0ok+ltO3SSCTM3Ygn5AH/Arf6BO3HrZ/gDfofTNgvbeuDC4Zx7OZcTxIJrdJxva2V1bX1js7BV3N7Z3dsvHRw2tUwUgwaTQqp2QDUIHkEDOQpoxwpoGAhoBaO7id96BKW5jOo4jsEP6SDifc4oGsmrdz2EJ0xlglm3VHYqzhT2MnFzUiY5at3Sj9eTLAkhQiao1h3XidFPqULOBGRFL9EQUzaiA+gYGtEQtJ9Of87sU6P07L5UZiK0p+rfi5SGWo/DwGyGFId60ZuI/3mdBPs3fsqjOEGI2CyonwgbpT0pwO5xBQzF2BDKFDe/2mxIFWVoappLCRQdAWZFU4y7WMMyaZ5X3KvKxcNluXqbV1Qgx+SEnBGXXJMquSc10iCMxOSFvJI369l6tz6sz9nqipXfHJE5WF+/JAqahg==</latexit>

R8

X1(@T1)

X2(@T2)

2
X
CA

∝R
s

A=0

~ √σCt :=

,

A

s

2
X
CA

A=0

V

p

2

t

+i t !

2 t,,

C 2 := C 2

Uk

2

C .

A

AM 1[X UkjV- T QM i?2 `B;?i@?M/ bB/2 Bb `2TH+2/ 7`QK i?2 Q`B;BMH /2}MBiBQM Q7 G(T) BM (RR) b T ! T

~ √σs

Tin
<latexit sha1_base64="ZRs66c5jhnRmkQbIJ+aSy5JO8kc=">AAACBnicbVDLSgNBEJz1GeMr6tHLYBA8hV0V9Rj04jFCXpAsYXbSmwyZfTDTK4Zl7/6AV/0Db+LV3/AH/A4nyR5MYkFDUdVNd5cXS6HRtr+tldW19Y3NwlZxe2d3b790cNjUUaI4NHgkI9X2mAYpQmigQAntWAELPAktb3Q38VuPoLSIwjqOY3ADNgiFLzhDI3XqvS7CE6YizHqlsl2xp6DLxMlJmeSo9Uo/3X7EkwBC5JJp3XHsGN2UKRRcQlbsJhpixkdsAB1DQxaAdtPpyRk9NUqf+pEyFSKdqn8nUhZoPQ480xkwHOpFbyL+53US9G9c80+cIIR8tshPJMWITv6nfaGAoxwbwrgS5lbKh0wxjialuS2eYiPArGiCcRZjWCbN84pzVbl4uCxXb/OICuSYnJAz4pBrUiX3pEYahJOIvJBX8mY9W+/Wh/U5a12x8pkjMgfr6xcsnZn7</latexit>

TQmi

q2 /QTi i?2 7QHHQrBM; MQiiBQM 7Q` `#Bi``v b+H` M/ p2+iQ` p`B#H2b C M/ C- `2bT2+iBp2Hv,
C :=

Re

T

!#

X0(@T0)

~√σ0

P0

N

x
<latexit sha1_base64="S8X/Q9zh6haDgwgLDjpC2CFhBEE=">AAACAnicbVDLSgNBEJyNrxhfUY9eBoPgKeyqqMegF48RzAOSJcxOZpMhM7PLTK8Ylr35A171D7yJV3/EH/A7nCR7MIkFDUVVN91dQSy4Adf9dgorq2vrG8XN0tb2zu5eef+gaaJEU9agkYh0OyCGCa5YAzgI1o41IzIQrBWMbid+65FpwyP1AOOY+ZIMFA85JWClVjeWQfqU9coVt+pOgZeJl5MKylHvlX+6/YgmkimgghjT8dwY/JRo4FSwrNRNDIsJHZEB61iqiGTGT6fnZvjEKn0cRtqWAjxV/06kRBozloHtlASGZtGbiP95nQTCaz/lKk6AKTpbFCYCQ4Qnv+M+14yCGFtCqOb2VkyHRBMKNqG5LYEmIwZZyQbjLcawTJpnVe+yen5/Uand5BEV0RE6RqfIQ1eohu5QHTUQRSP0gl7Rm/PsvDsfzuesteDkM4doDs7XLwDSmDs=</latexit>

They are functions ofXi, Pi, σi (i=0,1,2)

V2

7i2` i?2 b[m`2 +QKTH2iBQM Q7 t M/ i?2 MHviB+ :mbbBM BMi2;`iBQM Qp2` x BM UR9VK/2 BM (RR)- r2 `2T`2b2Mi i?2 S@Ki`Bt b 7QHHQrb,
!
✓
◆
2
Y
p
1
1 3/4
t ( !)2s ( P )2 R
3/2
2
2 (2⇡
p
SV !P P = ig2z NV
e 2
2⇡ t G(T)s)
2EA ⇡ A

Form of Gaussian wave-packet S-matrix
A=0

V (T
2

⇥e

T0 +i t !)+

2
V t
8

Fe (|V1

Uk

V2 |) ,

r?2`2 i?2 rBM/Qr 7mM+iBQM G(T) Bb /2}M2/ bRe
● Geometricalvariables characteriseS.
h ⇣⌘i2 The limit (σs→∞ and σt→∞)
Z TQmi1
V t +i
/t
t
T
t !
2 t
2
p
G(T)
:=
e
( ! ⇠ E := Eout E
P in )
in , P := P out
2⇡
Recovery of the energy-momentum
t
TBM
"
!
!#
t
V t
V t
P11
T TBM
+
i
!
T
T
+ iconservation
P
t
Qmi
t !
2
2
2
p
p
=
2`7
2`7
,
Uk
Note:
2
2 t
2 t
Tout
0
1
~√σ1
~√σ2
r
X
(@T
)
X
(@T
)
1
1
2
2
R8
(p
p0 )2 C- `2bT2+iBp2Hv, A
q2 /QTi i?2 7QHHQrBM; MQiiBQM 7Q` `#Bi``v b+H` M/ p2+iQ`CM/@ p`B#H2b2e
! (p p0 )
|{z}
2⇡
2
2
X
X
<latexit sha1_base64="seJLsVxsNZlgnuzqCMIdpZ+LBBI=">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</latexit><latexit sha1_base64="idQ+0AEvw5tEYKBNmRz+kv9HnFY=">AAAB/HicbVDLTgJBEJzFF+IL9ehlIjHxRHbVqEeiF4+QyCOBDZkdGpgwO7uZ6TUhG/wBr/oH3oxX/8Uf8DscYA8CVtJJpao73V1BLIVB1/12cmvrG5tb+e3Czu7e/kHx8KhhokRzqPNIRroVMANSKKijQAmtWAMLAwnNYHQ/9ZtPoI2I1COOY/BDNlCiLzhDK9WwWyy5ZXcGukq8jJRIhmq3+NPpRTwJQSGXzJi258bop0yj4BImhU5iIGZ8xAbQtlSxEIyfzg6d0DOr9Gg/0rYU0pn6dyJloTHjMLCdIcOhWfam4n9eO8H+rZ8KFScIis8X9RNJMaLTr2lPaOAox5YwroW9lfIh04yjzWZhS6DZCHBSsMF4yzGsksZF2bsuX9auSpW7LKI8OSGn5Jx45IZUyAOpkjrhBMgLeSVvzrPz7nw4n/PWnJPNHJMFOF+/+UKVaA==</latexit><latexit sha1_base64="/bZzYnZPTFyNLl1bW1ywBx1XfZY=">AAACB3icbVDLSsNAFJ34rPVVdekmWARXJVFRl0U3Liv0BU0ok+ltO3SSCTM3Ygn5AH/Arf6BO3HrZ/gDfofTNgvbeuDC4Zx7OZcTxIJrdJxva2V1bX1js7BV3N7Z3dsvHRw2tUwUgwaTQqp2QDUIHkEDOQpoxwpoGAhoBaO7id96BKW5jOo4jsEP6SDifc4oGsmrdz2EJ0xlglm3VHYqzhT2MnFzUiY5at3Sj9eTLAkhQiao1h3XidFPqULOBGRFL9EQUzaiA+gYGtEQtJ9Of87sU6P07L5UZiK0p+rfi5SGWo/DwGyGFId60ZuI/3mdBPs3fsqjOEGI2CyonwgbpT0pwO5xBQzF2BDKFDe/2mxIFWVoappLCRQdAWZFU4y7WMMyaZ5X3KvKxcNluXqbV1Qgx+SEnBGXXJMquSc10iCMxOSFvJI369l6tz6sz9nqipXfHJE5WF+/JAqahg==</latexit><latexit sha1_base64="bcjPdBXBSak3khVlQxPOeKi1dvM=">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</latexit>

∝R

C :=

s

A=0

Re

CA

~ √σCt :=

,

A

s

A=0

,

!1

C 2 := C 2

2

C .

A

AM 1[X UkjV- T QM i?2 `B;?i@?M/ bB/2 Bb `2TH+2/ 7`QK i?2 Q`B;BMH /2}MBiBQM Q7 G(T) BM (RR) b T ! T

~ √σs

Tin
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X0(@T0)

~√σ0

P0

N

x
<latexit sha1_base64="S8X/Q9zh6haDgwgLDjpC2CFhBEE=">AAACAnicbVDLSgNBEJyNrxhfUY9eBoPgKeyqqMegF48RzAOSJcxOZpMhM7PLTK8Ylr35A171D7yJV3/EH/A7nCR7MIkFDUVVN91dQSy4Adf9dgorq2vrG8XN0tb2zu5eef+gaaJEU9agkYh0OyCGCa5YAzgI1o41IzIQrBWMbid+65FpwyP1AOOY+ZIMFA85JWClVjeWQfqU9coVt+pOgZeJl5MKylHvlX+6/YgmkimgghjT8dwY/JRo4FSwrNRNDIsJHZEB61iqiGTGT6fnZvjEKn0cRtqWAjxV/06kRBozloHtlASGZtGbiP95nQTCaz/lKk6AKTpbFCYCQ4Qnv+M+14yCGFtCqOb2VkyHRBMKNqG5LYEmIwZZyQbjLcawTJpnVe+yen5/Uand5BEV0RE6RqfIQ1eohu5QHTUQRSP0gl7Rm/PsvDsfzuesteDkM4doDs7XLwDSmDs=</latexit>V2

7i2` i?2 b[m`2 +QKTH2iBQM Q7 t M/ i?2 MHviB+ :mbbBM BMi2;`iBQM Qp2` x BM UR9VK/2 BM (RR)- r2 `2T`2b2Mi i?2 S@Ki`Bt b 7QHHQrb,
!
✓
◆
2
Y
p
1
1 3/4
t ( !)2s ( P )2 R
3/2
2
2 (2⇡
p
SV !P P = ig2z NV
e 2
2⇡ t G(T)s)
2EA ⇡ A

Form of Gaussian wave-packet S-matrix

i x = (t, x) BM

A=0

⇥e

V (T
2

T0 +i t !)+

2
V t
8

Fe (|V1

V2 |) ,

Uk

r?2`2 i?2 rBM/Qr 7mM+iBQM G(T) Bb /2}M2/ bReh ⇣ three wave packets⌘i2
Z TQmi (around which: time of overlap
overlap).
“window function”
1
t
V
/t
t T
+i t !
2p
G(T) :=
e 2t
2⇡ t
TBM
"
!
!#
t
V t
V t
P11
T TBM
+
i
!
T
T
+i t !
P
t
Qmi
2
2
2
p
p
=
2`7
2`7
,
Uk
2
2 t
2 t
Tout
~√σ1
~√σ2

U83V

G(T) ,

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X1(@T1)

X2(@T2)q2 /QTi i?2 7QHHQrBM; MQiiBQM 7Q` `#Bi``v b+H` M/ p2+iQ` p`B#H2b C M/ C- `2bT2+iBp2Hv,
de at
∝
R
2
2
X
X
CA
CA
2
2
2
C := s
,
C
:=
,
C
:=
C
C
.
s
~ √σt
(T) ,
U83V A
A
R8

A=0

A=0

Re
not
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U8NV
N
~√σ0
(con gurations ofx.e Tatin
X0(@T0) P0
external particles)V<latexit sha1_base64="S8X/Q9zh6haDgwgLDjpC2CFhBEE=">AAACAnicbVDLSgNBEJyNrxhfUY9eBoPgKeyqqMegF48RzAOSJcxOZpMhM7PLTK8Ylr35A171D7yJV3/EH/A7nCR7MIkFDUVVN91dQSy4Adf9dgorq2vrG8XN0tb2zu5eef+gaaJEU9agkYh0OyCGCa5YAzgI1o41IzIQrBWMbid+65FpwyP1AOOY+ZIMFA85JWClVjeWQfqU9coVt+pOgZeJl5MKylHvlX+6/YgmkimgghjT8dwY/JRo4FSwrNRNDIsJHZEB61iqiGTGT6fnZvjEKn0cRtqWAjxV/06kRBozloHtlASGZtGbiP95nQTCaz/lKk6AKTpbFCYCQ4Qnv+M+14yCGFtCqOb2VkyHRBMKNqG5LYEmIwZZyQbjLcawTJpnVe+yen5/Uand5BEV0RE6RqfIQ1eohu5QHTUQRSP0gl7Rm/PsvDsfzuesteDkM4doDs7XLwDSmDs=</latexit>

fi<latexit sha1_base64="ZRs66c5jhnRmkQbIJ+aSy5JO8kc=">AAACBnicbVDLSgNBEJz1GeMr6tHLYBA8hV0V9Rj04jFCXpAsYXbSmwyZfTDTK4Zl7/6AV/0Db+LV3/AH/A4nyR5MYkFDUdVNd5cXS6HRtr+tldW19Y3NwlZxe2d3b790cNjUUaI4NHgkI9X2mAYpQmigQAntWAELPAktb3Q38VuPoLSIwjqOY3ADNgiFLzhDI3XqvS7CE6YizHqlsl2xp6DLxMlJmeSo9Uo/3X7EkwBC5JJp3XHsGN2UKRRcQlbsJhpixkdsAB1DQxaAdtPpyRk9NUqf+pEyFSKdqn8nUhZoPQ480xkwHOpFbyL+53US9G9c80+cIIR8tshPJMWITv6nfaGAoxwbwrgS5lbKh0wxjialuS2eYiPArGiCcRZjWCbN84pzVbl4uCxXb/OICuSYnJAz4pBrUiX3pEYahJOIvJBX8mY9W+/Wh/U5a12x8pkjMgfr6xcsnZn7</latexit>7i2` i?2 b[m`2 +QKTH2iBQM Q7 t M/ i?2 MHviB+ :mbbBM BMi2;`iBQM Qp2` x BM UR9VK/2 BM (RR)- r2 `2T`2b2Mi i?2 S@Ki`Bt b 7QHHQrb,
!
✓
◆
2
Y
p
1
1 3/4
t ( !)2s ( P )2 R
3/2
2
2 (2⇡
p
SV !P P = ig2z NV
e 2
2⇡ t G(T)s)
2EA ⇡ A

Form of Gaussian wave-packet S-matrix

i x = (t, x) BM

A=0“Bulk-like”
2t“Decay within wave-packet overlap”
V (T T +i
Vt !)+ 80e (|V1 V2 |) ,
2
⇥e
F
Uk
p
1 pp
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T<latexit 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A=0

A=0

Re
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X0(@T0)~√σ0P0

In stateNx⌧01<latexit sha1_base64="xJhCTyTZxUIfhZf7PHkXwQHSDCE=">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</latexit><latexit sha1_base64="MhawRdgmFZdWa/gZHZBabHL8DKw=">AAACCHicdVBLSgNBEO2Jvxh/UZduGoPgxjCTxN8u6EKXEcwHkjHUdDpJk+6ZsbtHCMNcwAu41Ru4E7fewgt4DjsfIRF9UPB4r4qqel7ImdK2/WmlFhaXllfSq5m19Y3Nrez2Tk0FkSS0SgIeyIYHinLm06pmmtNGKCkIj9O6N7gc+fUHKhUL/Fs9DKkroOezLiOgjeS2rkAIaNfu4iMnaWdzdt4eA8+Q46JdPC9gZ6rk0BSVdvar1QlIJKivCQelmo4dajcGqRnhNMm0IkVDIAPo0aahPgiq3Hh8dIIPjNLB3UCa8jUeq7MTMQilhsIznQJ0X/32RuJfXjPS3TM3Zn4YaeqTyaJuxLEO8CgB3GGSEs2HhgCRzNyKSR8kEG1ymtviSRhQnWRMMD/f4/9JrZB3TvKlm1KufDGNKI320D46RA46RWV0jSqoigi6R0/oGb1Yj9ar9Wa9T1pT1nRmF83B+vgGpkyaKw==</latexit><latexit sha1_base64="S8X/Q9zh6haDgwgLDjpC2CFhBEE=">AAACAnicbVDLSgNBEJyNrxhfUY9eBoPgKeyqqMegF48RzAOSJcxOZpMhM7PLTK8Ylr35A171D7yJV3/EH/A7nCR7MIkFDUVVN91dQSy4Adf9dgorq2vrG8XN0tb2zu5eef+gaaJEU9agkYh0OyCGCa5YAzgI1o41IzIQrBWMbid+65FpwyP1AOOY+ZIMFA85JWClVjeWQfqU9coVt+pOgZeJl5MKylHvlX+6/YgmkimgghjT8dwY/JRo4FSwrNRNDIsJHZEB61iqiGTGT6fnZvjEKn0cRtqWAjxV/06kRBozloHtlASGZtGbiP95nQTCaz/lKk6AKTpbFCYCQ4Qnv+M+14yCGFtCqOb2VkyHRBMKNqG5LYEmIwZZyQbjLcawTJpnVe+yen5/Uand5BEV0RE6RqfIQ1eohu5QHTUQRSP0gl7Rm/PsvDsfzuesteDkM4doDs7XLwDSmDs=</latexit>

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Form
ofGaussianwave-packet
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i x = (t, x) BM<latexit 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Beyond the simplest calculation
We examined the simplest 1→2 case in wave-packet formalism.
Now, we will be interested in
1. When the wave-packet effect becomes signi cant?

fi

2. How about the 2→2 full scattering, including the production part?

1. When the wave-packet effect becomes signi cant?
Let me remind you that:
The plane-wave calculation includes only the pure bulk part.

Con gurations where the boundary parts are dominant.
In the S-matrix, when we focus on momentum-non-conserving(off-shell) part, the difference between the bulk and boundary
becomes signi cant.

fi

fi

fi

A valid phase space is nallow, near kinetic threshold.

2. How about the 2→2 full scattering, including the production part?
The full format of S for the resonant process (ϕϕ ➔ Φ ➔ ϕϕ) is available.

production

decay<latexit sha1_base64="ssTLdJhthir8jRD9p5O1B6SsOYM=">AAAB/3icbVDLSsNAFL2pr1pfVZduBovgqiQi2mXBjcsK9gFtKJPppBk6mYSZG6GELvwBt/oH7sStn+IP+B0mbRa29cCFwzn3cu89XiyFQdv+tkobm1vbO+Xdyt7+weFR9fikY6JEM95mkYx0z6OGS6F4GwVK3os1p6Enedeb3OV+94lrIyL1iNOYuyEdK+ELRjGXBnEghtWaXbfnIOvEKUgNCrSG1Z/BKGJJyBUySY3pO3aMbko1Cib5rDJIDI8pm9Ax72dU0ZAbN53fOiMXmTIifqSzUkjm6t+JlIbGTEMv6wwpBmbVy8X/vH6CfsNNhYoT5IotFvmJJBiR/HEyEpozlNOMUKZFdithAdWUYRbP0hZP0wnHWSULxlmNYZ10rurOTf364brWbBQRleEMzuESHLiFJtxDC9rAIIAXeIU369l6tz6sz0VrySpmTmEJ1tcvQWGWpg==</latexit><latexit sha1_base64="ssTLdJhthir8jRD9p5O1B6SsOYM=">AAAB/3icbVDLSsNAFL2pr1pfVZduBovgqiQi2mXBjcsK9gFtKJPppBk6mYSZG6GELvwBt/oH7sStn+IP+B0mbRa29cCFwzn3cu89XiyFQdv+tkobm1vbO+Xdyt7+weFR9fikY6JEM95mkYx0z6OGS6F4GwVK3os1p6Enedeb3OV+94lrIyL1iNOYuyEdK+ELRjGXBnEghtWaXbfnIOvEKUgNCrSG1Z/BKGJJyBUySY3pO3aMbko1Cib5rDJIDI8pm9Ax72dU0ZAbN53fOiMXmTIifqSzUkjm6t+JlIbGTEMv6wwpBmbVy8X/vH6CfsNNhYoT5IotFvmJJBiR/HEyEpozlNOMUKZFdithAdWUYRbP0hZP0wnHWSULxlmNYZ10rurOTf364brWbBQRleEMzuESHLiFJtxDC9rAIIAXeIU369l6tz6sz0VrySpmTmEJ1tcvQWGWpg==</latexit>[Ishiwaka,KN,Oda(2006.14159, 2102.12032]◦ It is necessary for full analyses.

<latexit sha1_base64="aWQnsngYA/hZu/If6dJrzT9pRL4=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0m0aI8FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq777ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmGG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU00TTKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwqafMZmkhkqyXBSmHJkY5Y+jMVOUGD6zBBPF7K2IRFhhYmw8FRuCt/ryOule1b2b+vVDo9ZqFnGU4QzO4RI8uIUW3EMbOkAggmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPiU44d</latexit><latexit sha1_base64="dmNxYmNRKOTMr7IzTSLfeLRwyns=">AAACGXicdVDLSgMxFM34rPVVdamLYBF0Uyal1LoruHFZwbZCW0smvdXQzIPkjliG2fgb/oBb/QN34taVP+B3mLEVVPRA4HDOvdyc40VKGnTdN2dmdm5+YTG3lF9eWV1bL2xstkwYawFNEapQn3vcgJIBNFGigvNIA/c9BW1vdJz57WvQRobBGY4j6Pn8MpBDKThaqV/YOet3EW4wkUF6kUzo/gAEHx+kab9QdEuu6zLGaEbYYdW15OioVmY1yjLLokimaPQL791BKGIfAhSKG9NhboS9hGuUQkGa78YGIi5G/BI6lgbcB9NLPlOkdM8qAzoMtX0B0k/1+0bCfWPGvmcnfY5X5reXiX95nRiHtZ4NGMUIgZgcGsaKYkizSuhAahCoxpZwoaX9KxVXXHOBtrgfVzzNR4Bp3hbzlZ7+T1rlEquWKqeVYr0+rShHtsku2SeMHJI6OSEN0iSC3JJ78kAenTvnyXl2XiajM850Z4v8gPP6AcIPodE=</latexit>◦ More knowledge on<latexit sha1_base64="ssTLdJhthir8jRD9p5O1B6SsOYM=">AAAB/3icbVDLSsNAFL2pr1pfVZduBovgqiQi2mXBjcsK9gFtKJPppBk6mYSZG6GELvwBt/oH7sStn+IP+B0mbRa29cCFwzn3cu89XiyFQdv+tkobm1vbO+Xdyt7+weFR9fikY6JEM95mkYx0z6OGS6F4GwVK3os1p6Enedeb3OV+94lrIyL1iNOYuyEdK+ELRjGXBnEghtWaXbfnIOvEKUgNCrSG1Z/BKGJJyBUySY3pO3aMbko1Cib5rDJIDI8pm9Ax72dU0ZAbN53fOiMXmTIifqSzUkjm6t+JlIbGTEMv6wwpBmbVy8X/vH6CfsNNhYoT5IotFvmJJBiR/HEyEpozlNOMUKZFdithAdWUYRbP0hZP0wnHWSULxlmNYZ10rurOTf364brWbBQRleEMzuESHLiFJtxDC9rAIIAXeIU369l6tz6sz0VrySpmTmEJ1tcvQWGWpg==</latexit><latexit sha1_base64="ssTLdJhthir8jRD9p5O1B6SsOYM=">AAAB/3icbVDLSsNAFL2pr1pfVZduBovgqiQi2mXBjcsK9gFtKJPppBk6mYSZG6GELvwBt/oH7sStn+IP+B0mbRa29cCFwzn3cu89XiyFQdv+tkobm1vbO+Xdyt7+weFR9fikY6JEM95mkYx0z6OGS6F4GwVK3os1p6Enedeb3OV+94lrIyL1iNOYuyEdK+ELRjGXBnEghtWaXbfnIOvEKUgNCrSG1Z/BKGJJyBUySY3pO3aMbko1Cib5rDJIDI8pm9Ax72dU0ZAbN53fOiMXmTIifqSzUkjm6t+JlIbGTEMv6wwpBmbVy8X/vH6CfsNNhYoT5IotFvmJJBiR/HEyEpozlNOMUKZFdithAdWUYRbP0hZP0wnHWSULxlmNYZ10rurOTf364brWbBQRleEMzuESHLiFJtxDC9rAIIAXeIU369l6tz6sz0VrySpmTmEJ1tcvQWGWpg==</latexit>(decay)Tin(Note: within “1→2”, this isa parameter.)◦ Full analysis of P(ϕϕ ➔ Φ ➔ ϕϕ)time<latexit sha1_base64="kE16ATGP+7Splo7z5J5mPfDi5qg=">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</latexit>(decay)Tout<latexit sha1_base64="dmNxYmNRKOTMr7IzTSLfeLRwyns=">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</latexit>(decay)Tin<latexit sha1_base64="1qKEZCyqn7jrb61wFkT6p89KxcI=">AAACGHicbZDJSgNBEIZ74hbjFvWYy2AQ4iXMSNAcA148RsgGSRx6OjVJk56F7hoxDHPwNXwBr/oG3sSrN1/A57CzHExiQcPH/1dRXb8bCa7Qsr6NzMbm1vZOdje3t39weJQ/PmmpMJYMmiwUoey4VIHgATSRo4BOJIH6roC2O76Z+u0HkIqHQQMnEfR9Ogy4xxlFLTn5QsPpITxiwoP0PpljSWn3Ik2dfNEqW7My18FeQJEsqu7kf3qDkMU+BMgEVaprWxH2EyqRMwFprhcriCgb0yF0NQbUB9VPZkek5rlWBqYXSv0CNGfq34mE+kpNfFd3+hRHatWbiv953Ri9al/fF8UIAZsv8mJhYmhOEzEHXAJDMdFAmeT6ryYbUUkZ6tyWtriSjgHTnA7GXo1hHVqXZfuqXLmrFGvVRURZUiBnpERsck1q5JbUSZMw8kReyCt5M56Nd+PD+Jy3ZozFzClZKuPrF61goS4=</latexit><latexit sha1_base64="+iIrciE+ccy7fPlSByWc737WiVI=">AAACGnicbZDJSgNBEIZ7XGPcoh5FGAxCvIQZCZqLEPDiMUI2SMahp1NJmvQsdNeIYZiTr+ELeNU38CZevfgCPoed5WASCxo+/r+Kqv69SHCFlvVtrKyurW9sZray2zu7e/u5g8OGCmPJoM5CEcqWRxUIHkAdOQpoRRKo7wloesObsd98AKl4GNRwFIHj037Ae5xR1JKbO7muuR2ER0zCGNP7ZMoFpf3zNHVzeatoTcpcBnsGeTKrqpv76XRDFvsQIBNUqbZtRegkVCJnAtJsJ1YQUTakfWhrDKgPykkm30jNM610zV4o9QvQnKh/JxLqKzXyPd3pUxyoRW8s/ue1Y+yVnYQHUYwQsOmiXixMDM1xJmaXS2AoRhook1zfarIBlZShTm5uiyfpEDDN6mDsxRiWoXFRtC+LpbtSvlKeRZQhx+SUFIhNrkiF3JIqqRNGnsgLeSVvxrPxbnwYn9PWFWM2c0Tmyvj6BTpcogA=</latexit>(scat)
Tin(scat)= Toutinput parametersof scattering

is not yet due to the complicatedphase space..

2. How about the 2→2 full scattering, including the production part?
We consider the scheme of approximation for “ϕϕ ➔ Φ ➔ ϕϕ”.

production[Ishiwaka,Jinnouchi,KN,Oda(ongoing)]

decay<latexit sha1_base64="ssTLdJhthir8jRD9p5O1B6SsOYM=">AAAB/3icbVDLSsNAFL2pr1pfVZduBovgqiQi2mXBjcsK9gFtKJPppBk6mYSZG6GELvwBt/oH7sStn+IP+B0mbRa29cCFwzn3cu89XiyFQdv+tkobm1vbO+Xdyt7+weFR9fikY6JEM95mkYx0z6OGS6F4GwVK3os1p6Enedeb3OV+94lrIyL1iNOYuyEdK+ELRjGXBnEghtWaXbfnIOvEKUgNCrSG1Z/BKGJJyBUySY3pO3aMbko1Cib5rDJIDI8pm9Ax72dU0ZAbN53fOiMXmTIifqSzUkjm6t+JlIbGTEMv6wwpBmbVy8X/vH6CfsNNhYoT5IotFvmJJBiR/HEyEpozlNOMUKZFdithAdWUYRbP0hZP0wnHWSULxlmNYZ10rurOTf364brWbBQRleEMzuESHLiFJtxDC9rAIIAXeIU369l6tz6sz0VrySpmTmEJ1tcvQWGWpg==</latexit><latexit sha1_base64="ssTLdJhthir8jRD9p5O1B6SsOYM=">AAAB/3icbVDLSsNAFL2pr1pfVZduBovgqiQi2mXBjcsK9gFtKJPppBk6mYSZG6GELvwBt/oH7sStn+IP+B0mbRa29cCFwzn3cu89XiyFQdv+tkobm1vbO+Xdyt7+weFR9fikY6JEM95mkYx0z6OGS6F4GwVK3os1p6Enedeb3OV+94lrIyL1iNOYuyEdK+ELRjGXBnEghtWaXbfnIOvEKUgNCrSG1Z/BKGJJyBUySY3pO3aMbko1Cib5rDJIDI8pm9Ax72dU0ZAbN53fOiMXmTIifqSzUkjm6t+JlIbGTEMv6wwpBmbVy8X/vH6CfsNNhYoT5IotFvmJJBiR/HEyEpozlNOMUKZFdithAdWUYRbP0hZP0wnHWSULxlmNYZ10rurOTf364brWbBQRleEMzuESHLiFJtxDC9rAIIAXeIU369l6tz6sz0VrySpmTmEJ1tcvQWGWpg==</latexit><latexit sha1_base64="aWQnsngYA/hZu/If6dJrzT9pRL4=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0m0aI8FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq777ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmGG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU00TTKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwqafMZmkhkqyXBSmHJkY5Y+jMVOUGD6zBBPF7K2IRFhhYmw8FRuCt/ryOule1b2b+vVDo9ZqFnGU4QzO4RI8uIUW3EMbOkAggmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPiU44d</latexit>◦ We assume the factorisation(for a resonance, it will works.)<latexit sha1_base64="aWQnsngYA/hZu/If6dJrzT9pRL4=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0m0aI8FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq777ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmGG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU00TTKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwqafMZmkhkqyXBSmHJkY5Y+jMVOUGD6zBBPF7K2IRFhhYmw8FRuCt/ryOule1b2b+vVDo9ZqFnGU4QzO4RI8uIUW3EMbOkAggmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPiU44d</latexit>⌦◦ The nal-state pro le isdetermined in the decay part.

<latexit sha1_base64="Vol0Z6zhODbcYmv/EfI7ONMQ1L4=">AAACAnicbVDLSgMxFM3UV62vqks3wSK4KjMi2mXBjcsK9gHtUDLpnTZMZjIkd4QydOcPuNU/cCdu/RF/wO8wfSxs64HA4Zx7OTcnSKUw6LrfTmFjc2t7p7hb2ts/ODwqH5+0jMo0hyZXUulOwAxIkUATBUropBpYHEhoB9Hd1G8/gTZCJY84TsGP2TARoeAMrdTuKRQxmH654lbdGeg68RakQhZo9Ms/vYHiWQwJcsmM6Xpuin7ONAouYVLqZQZSxiM2hK6lCbMhfj47d0IvrDKgodL2JUhn6t+NnMXGjOPATsYMR2bVm4r/ed0Mw5qfiyTNEBI+DwozSVHR6d/pQGjgKMeWMK6FvZXyEdOMo21oKSXQLAKclGwx3moN66R1VfVuqtcP15V6bVFRkZyRc3JJPHJL6uSeNEiTcBKRF/JK3pxn5935cD7nowVnsXNKluB8/QLOQJgU</latexit><latexit sha1_base64="ssTLdJhthir8jRD9p5O1B6SsOYM=">AAAB/3icbVDLSsNAFL2pr1pfVZduBovgqiQi2mXBjcsK9gFtKJPppBk6mYSZG6GELvwBt/oH7sStn+IP+B0mbRa29cCFwzn3cu89XiyFQdv+tkobm1vbO+Xdyt7+weFR9fikY6JEM95mkYx0z6OGS6F4GwVK3os1p6Enedeb3OV+94lrIyL1iNOYuyEdK+ELRjGXBnEghtWaXbfnIOvEKUgNCrSG1Z/BKGJJyBUySY3pO3aMbko1Cib5rDJIDI8pm9Ax72dU0ZAbN53fOiMXmTIifqSzUkjm6t+JlIbGTEMv6wwpBmbVy8X/vH6CfsNNhYoT5IotFvmJJBiR/HEyEpozlNOMUKZFdithAdWUYRbP0hZP0wnHWSULxlmNYZ10rurOTf364brWbBQRleEMzuESHLiFJtxDC9rAIIAXeIU369l6tz6sz0VrySpmTmEJ1tcvQWGWpg==</latexit><latexit sha1_base64="ssTLdJhthir8jRD9p5O1B6SsOYM=">AAAB/3icbVDLSsNAFL2pr1pfVZduBovgqiQi2mXBjcsK9gFtKJPppBk6mYSZG6GELvwBt/oH7sStn+IP+B0mbRa29cCFwzn3cu89XiyFQdv+tkobm1vbO+Xdyt7+weFR9fikY6JEM95mkYx0z6OGS6F4GwVK3os1p6Enedeb3OV+94lrIyL1iNOYuyEdK+ELRjGXBnEghtWaXbfnIOvEKUgNCrSG1Z/BKGJJyBUySY3pO3aMbko1Cib5rDJIDI8pm9Ax72dU0ZAbN53fOiMXmTIifqSzUkjm6t+JlIbGTEMv6wwpBmbVy8X/vH6CfsNNhYoT5IotFvmJJBiR/HEyEpozlNOMUKZFdithAdWUYRbP0hZP0wnHWSULxlmNYZ10rurOTf364brWbBQRleEMzuESHLiFJtxDC9rAIIAXeIU369l6tz6sz0VrySpmTmEJ1tcvQWGWpg==</latexit>◦ The intermediate Φ state is
NOT a free asymptotic state.
We take account of this nature bytime<latexit sha1_base64="kE16ATGP+7Splo7z5J5mPfDi5qg=">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</latexit><latexit sha1_base64="dmNxYmNRKOTMr7IzTSLfeLRwyns=">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</latexit>(decay)Tin(decay)Tout<latexit sha1_base64="iaU3T+zdxmvOCphqqpubkpwrgmA=">AAACJnicbVDLSsNAFJ34rPUVdekmWIR2UxIp2mVBCi4r2Ac0IUwmN+3QyYOZiVBCvsHf8Afc6h+4E3Hnyu9w2mZhWw9cOJxzL/fe4yWMCmmaX9rG5tb2zm5pr7x/cHh0rJ+c9kSccgJdErOYDzwsgNEIupJKBoOEAw49Bn1vcjvz+4/ABY2jBzlNwAnxKKIBJVgqydVrbdes2knoZZ3cNWuGLWNVlPmQtfMly9UrZt2cw1gnVkEqqEDH1X9sPyZpCJEkDAsxtMxEOhnmkhIGedlOBSSYTPAIhopGOAThZPOXcuNSKb4RxFxVJI25+nciw6EQ09BTnSGWY7HqzcT/vGEqg6aT0ShJJURksShImaH+nuVj+JQDkWyqCCacqlsNMsYcE6lSXNricTwBmZdVMNZqDOukd1W3ruuN+0al1SwiKqFzdIGqyEI3qIXuUAd1EUFP6AW9ojftWXvXPrTPReuGVsycoSVo378Nq6S/</latexit>

P 0 ) ! Ẽ0 (P
P 0)
E0 (P
q
= P 20 + m2
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P 0) i' E0 (PP 0)
2E0 (P(Weisskopf-Wigner Approximation)fi

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X

2
1 ,
s( P
tP
Q)2:=R +i[···
. Q2
(31)
1 C := sP
Z TthePP
s
4(!)]Gaussian
2
2
A
Q
·Q
out
2
2
2
One
can
exactly
perform
theintegrals
overinteraction
point
(t,fo
x
2
2
2
1 :=
p approximation(27)
toA lower-case letters a, b, dt
A !) B x =eA=0 A=0where.
intinplane-wave
limit
A gete.g.
Abbreviate
to
.
(We
also
let
the
.
.
.
s
(t
T
i
t
Q
=
and
Q
=
Q
·
Q
=
,
which
A
A 2E
A
A
s 2 t ABp
P
P2 s A
G(T)are:= in one-to-one
ecorrespondence.
A
A B
A
A We may freely choose[Ishikawa,
Oda(1809.04285)]
eithervariable
P
or
V
,
which
A
A
A=0
!
Eq.
(27)
to
get
2⇡ t
laddle-point
states
1
and
2
such
that
:=
,
etc.)
P
P
T
in
definition
(31).
a
a=1
2
ZY
approximation(18)
inlower-casethet large
for all the✓ in◆ define,✓ for any Q
2 width2expansion
Ra, b, . . . 3/2 p
i
1Wefurthersletters
2
to
.
(We
also
let
the
Q
,
t
2
p
3/4
1
(
!)
(
P
)
A
P
P
A=0
A
1
T
T
+
i
!
T Tout +
T+i
x p
X+Vof
t the
ies interaction
P
2 )]
2
2 (2⇡
11
in
t
t !)]
P
[
(
3the
2
S=
(⇡
)
)
2⇡
G(T)
,
(58)
!
•pobtainisthespatialsize
region:
s
t
A
2
kets,
we
S-matrix
for
a
given
configuration
(⇧
,
⇧
,
⇧
):
sP
2
s t
s
p
p
d x:=
e weighted,(27)
erf e.g.
erf also letp
0 abbreviate
1 =2Hereafter,
we
to
.
(We
notes
sum
(and
not
the
complexconjugate):For
2E
such the
that
,
etc.)
2 following
A=0
A
A time-like size of theY 2t is! theP2
2 2 R region:2 t p P
2 tAa
2 tinteraction
1
t
s
2i
2• a=1
3/4
3/2
2
(
!)
(
P
)
!
run
for
12and
2 suchthat2⇡ a G(T):= a=1, etc
Q p:= Q quantities
Q, ) CA and(32)
12 states2=
2 the final
andthree-vector
Q,
respectively,
wedefine
p
•
S
(⇡
e
(2⇡
)
,
t
2
s
Y
A
s
t
A
2
2
2
X
R
i
1
s
t

Details on S(Φ ➔ ϕϕ)• T is what
wearoundcall the whichintersection
time, aroundwhich the interaction
call the intersection
time,
the interaction
occurs:V ·X V

V ·X

V ·X

VT·:=
X

V ·X

V ·X

V ·X

T :=
V ·X

V ·X

=
,
T :=
=
,= ⌅ (0) is the location(34)of the
V
where
X
A
A
( !) 2E2 in( Pconjugate):)+i[···1] VFor
which
ing
(and
not etherehave (⇡
definedthewindowfunction:
2scomplex2
2Asum
pweweighted
p
!ing:
) 3/4
A
:= X
.
(30) P Q2A
s
s
A
P
2
2
X
QA ·QB• we
The
overlinedenotesthefollowing
weightedsum
not
2Q A=0.where
2 2E
Z=
time
t
=
0:
A, respectively,
Q
and
CAA=andQt :=As (andzorquantities
CQ
define
A
nd C
=
Q
·
Q
,
which
follow
from
the
A
2
where
X
=
⌅
(0)
is
the
locationofthecenter
of
each
wave
pack
Z
A
A=0
A
A
A
s
2
2
AB
, Tout dt
QT :=
.
(31)
A B 1
V
xand Q , re
s
s2
Z T:=
Z
arbitraryscalarandthree-vector
quantitiesCperf(z):=
edx
definition
(31).
(t
i
!)
A
t
out
⌅
(0)
is
the
locationofthecenterofeach
wave
packetatourreference
A
2
2
1
2 t 1
A we
A
AP )]
p
G(T)
:=
e
[t
(T+i
!)]
P
P
where
have
definedthewindowfunction:
◦
x
X+V
t
i
t
⇡
[
(
3
time
t
=
0:
s
0
dt e A2 t weTinx e Q2A2s to A. (We also
, let the lower-case(27)
C⇥A2⇡ dt X
Hereafter,
abbreviatee.g.
lettersa,b,. . . XA := X
p
A=0
A
X
P
P
2

✓
◆
✓
◆
•whichthe time-likesize of
:=
. the
(31)
Tin
2 time, aroundC•final
TQ
is
what
we
call
intersection
interactionoctt is the
A
s 1,
+the
P, A
(A
=
0,
2),
(28)
Z
interaction
region:runforthestates
1
and
2
such
that
:=
,
etc.)
is
the
Gauss
error
function.
In
the
small
and
large
|z|
limits,
its
(asy
:=
,
Q :=
1
T TTout
T Tout
+ iC
! s
a=1
ne,
in + i At !
A for any QA ,
2 t
1 a
dt
p
p
A
=
erf
erf
,
(59)
(t T i t !)XA := XAA A V A TA .
indicate the following:
2 t
respectively,
2s
p
2
2
G(T)
:=
e
t meretnot the complex conjugate): For
overlinedenotesthefollowing
weightedsum(anda = 1, •2) The
being
their
masses.
(This
isarephrasingV·
X we
V ·will
X call
V ·theX overlap
V · X(35)
•
R
is
what
exponen
2
X
:=
X
V
T
.
2
2
.
(33)
A
A
A
A
2⇡
t :=
t
T
2
in
Q
:=
Q
Q
,
(32)
T
:=
=
,
tdefine,n-shellenergies:
2z, we defineVarbitrary scalar
and three-vector
quantities
CA andQA , forrespectively,
2
 ✓
◆the
✓
◆
3
We further
any
Q
hich
A
p
for
non-overlap
of
the
wave
packets aVerf(z)
=
+
O
z
,
s
we
will
call
the
overlap
exponent
that
gives
the
suppressionq • R is what
1
T
T
+
i
!
T
T
+
i
!
in
t
out
t
⇡• T is whatZ
we callthe interse
2
P Q2A2
P
X
X
z
2
p
p
22
✓
◆
erf
,
2 = theerf
Q
·Q
C
Q
2
2
A
B
2
A
2
sectiontime,
aroundwhichinteraction
occurs:
A
2
E
:=
m
+
P
(A
=
0,
1,
2),
(28)
2
Q
:=
Q
Q
,
(32)
A
x
•
and
Q
=
Q
·
Q
=
,
which
followfromtheeers velocities:
will
call
the
overlap
exponent
that
gives
the
suppression
factoraccountingA1
thenon-overlapofthewave Qpackets
the
A for
Care
, correspondence.
:=of sthe at. 2 intersection(31) point:QQ, z 3a2:=2eBthewhereX=s:=⌅p(0)
iscenterofeachpacketA A P A or VAB
z 2:= wave
erf(z)(60)
t dxlocationt+ Q
Asone-to-one
AA
variablein
A , whichperf(z)=
sgn(z)e+OA⇡ 0A⇡z R :=A
A
P
overlap
thewave
packetsattheintersection
point:
T :=
time
t
=
0:
A
P
VV· m
Xof
V
·
X
V
·
X
V
·
X
m2 and
:=
m
(a
=
1,
2)
being
their
masses.
(This
is
mere
a
rephrasing
2
a:=
Q
·Q
P
P
2
2
2
2
A
B
.
(29)
Q
QA
the
interaction
2
A · Q region:
2
=
=
,
(34)
A
Q
= Q
=
,
which
follow
from
the
t•
X
T
ne Gauss
which
where
Q
=
and
Q
=
Q
·
Q
=
2
s
AB
E
error
function.
In
the
small
and
large
|z|
limits,its(asymptotic)
expansionreads,sAany
B QA ,sdefine, forVregion:
A
AB
where we have defined R
a signvariable:
A function for a complex
A-like sizeWe
ofsfurther
theA interaction:=.
22 (31).
Z zXA := X8A s V A! 1
definition
whereTAt.[ XA = ⌅A (0)]is the lectively,
X
T
2
,
⇧
)
to
(⇧
,
⇧
,
⇧
).
X
2
2
2
0 group
1center
21velocities:
22
corresponding
locationof theof eachwave
packetp
at2 :=
ourQ
referencex2 the>mometum•We
defineandenergy
shift
R
:=
.
(36)
time
Q
Q
,
1
for
<z t>=0 0:
or(32)(<z= 0 and=zsperf(z)
:=
e
dx
•
<
:=
.
(30)
s
:=
.
(33)
interaction region:A t 2z 2
sizeof the interaction
region:3 s•t ist the⇡time-like0psgn(z)
:=
=
+
O
z
,
(61)
P
1
for
<z
<
0
or
(<z = 0 and fac=zV
A=0••erf(z)A
2We
definethemometumandenergyshifts,
etc:
P
P
R
is
what
we
will
call
the
overlap
exponent
that
gives
the
suppression
2
>
Q
Q
·Q
⇡
V
:=
.
(29)
2
9
A
A
B :
P
:=
P, which
E := E1
Qs2 = s A
and
Q
=
Q
·
Q
=
follow
from✓◆
10+ Pfor2 z =P0.
0 , theEs
P2Xwhere
P
A A
AB
A
s
A
B
:= X
V
TAthe
. letnon-overlap(35)
forofletters
the wavepackets
at theintersection
point:A
A. (We
Aalso
1
2
:=
.
:=
.
(33)
e.g.
to
the
lower-case
a,
b,
.
.
.
•
t
sethe
Gauss
error
function.
In
the
small
and
large
|z|
limits,its(asymptotic)
expansionrtz
3
definition
(31).
all
the
intersection
time,aroundwhich
theinteraction
occurs:A=0
A2
mometumandenergyshifts,etc:
2 erf(z)PPp=sgn(z)
+
e
+
O
z
,
(62)
V
breviate( 0 ,that⇧0 ; 1V, ⇧1 ;:=2 , ⇧22) to,P(⇧
, ⇧1 P, ⇧21).+ P 2 ⇡zP 0 ,
:=
E
:=
E
+
E
E
,
:=
0
•
R
is
what
we
will!calltheEov
1
2
0
nd
2
such
etc.)
From
Eq.
(58),
we
see
that
the
S-matrixis
exponentiallysuppresaa=1pespectively,
2
2
X
T
•
is
the
time-like
size
of
the
interaction
region:
•
“i
[·
·
·
]”
denotestheirrelevant
purefor
thecasenon-overlap
ofimagin
thewverlap exponent
that
gives
the
suppression
factoraccountingt
V ·X V ·X
V · X•is T
V is· X
nearlyconserved,
P
⇠
0.
This
is
also
the
for
the
energy
cons
what
weFor
callR
the:=
intersectiontime,which the i. P .around (37)
on1we
time,
aroundwhich
theinteraction
occurs:ollowing
weighted
sum
(and
not
the
complexconjugate):
•+
T
:=
=
,
(34)
re
have
definedasignfunctionforacomplexvariable:
Pwave
P
P
,
E
:=
E
+
E
E
,
!
:=
E
V
·
t
2
0
1
2
0
packets at the intersection
point: 2zthe
2 boundary
9
them
they
disappear
regions,
atimaginary
whichhereafterinvarianceis explsthe translationalt as thatV
sneglect
s
•
“i
[·
·
·
]”
denotestheirrelevant
pure
terms
are
indepen3-vector quantities CA and 8
QAerf(z), respectively,
wedefine:=As. above, the overlap exponent R (33)
p
=
+
z
,
tO
below.
said
gives V
another
su
2
V
·
X
V
·
X
·
X
V
2
V
V(0)
· Xis theV ·location
X X2 Vneglect
·of
X
V
·
X
>
1 themfor <zhereafter> 0⇡ or (<zas= they0 and =z> 0), T :=when we take theTdisappear
absolute<=
centerofQ
each
wavedoandpacketat ourreferencet
X•CA R := = • Wethe
X
packets
notoverlap.
A
,
(34)
2
✓
◆
.
(36)
definethemometumenergyshifts,etc:
A2Votes theirrelevant
pure
imaginary
terms
that
are
independent
of
x.
We
will
s
sgn(z) :=t QV:=
(63)
foror (<z= 0 2and< 0),(31)=z 1
10
s <z < .0time,• ,sT is whats we call
the1intersection
aroundinteraction 3occurs:>z which the• We define the mometum ansones
accordingly.)
0 1
0 2
1 2
"For any pair of three-vectors QA and#Q0A (A= 0, 1, 2), w[Ishikawa,Oda (1809.04285)]
22
2( V 2)( V 1V 2)
22 ( V 1)
0
V = s
+
+
. · Q0
Q
·
Q
Q
( Q1
1
2
1
2
2
0
0
1
1 2
Q0· Q
Q · 0Q 2= s++

Details on S(Φ ➔ ϕϕ)
0 1

n we get•where we define, for any QA ,12"# 1
1 ( V 1 )2 ( V 2 )2 ( V 1
V 2 )2
+
+
,
t =
s

0 1

0 2

0 2

Qa := Qa

1 2

Q0 .

Note that we always have Q1Q2 = Q1 Q2 . Especially
X1 · V 1
X2 · V 2 ( X1X
V 2)2) · ( V 1
"
+
+
,
• T=
s t
2
2
( Q2 )
( Q1
Q
0 1
0 2
1 12)
2
2 ( Q
Q = s
+
+
(
2
2
2
0 1
0 2
1 2
( X1 )( X2 )( X1
X2 )"• R= s
+
+
2
2
(Q
Q
)
(Q
Q
)
0 1
0 2
1 2
1
0
2
0
= s2
+
) +

0 1
0 2
X1 · V 1
X2 · V 2 ( X1
X2 ) · ( V 1
V 2)
+
+
.
s t
0 or
1 more concretely,
0 2
1 2
"
2
2
( X1 ) V a( !
X2V) a . Expres( X1
that for a parent particle at rest, P 0 = 0, we may simply
2
2 replace
X = s++rious limits are shown in Appendix C.
0 1
0 2
1
et us prove the non-negativity of R. In general, the"weighted
average2
2 for any
( V 2)( V1
2
2 ( V 1)or Q satisfies
V = s
+
+



0 1

2

2

0 2