QEDの赤外発散と漸近対称性
杉下 宗太郎 (名古屋大, 来月から京大基研)

平井隼人氏 (木更津高専) との共同研究に基づく[1901.09935], [2009.11716], [2209.00608]

KMI分野横断セミナー「時空の漸近構造、赤外発散、重力波」, 12月18日, 2022

Introduction•

重力理論における漸近対称性と低エネルギーの物理• 重力波のメモリー効果• soft graviton theorem•

電磁気学でも同様の関係• 電磁波のメモリー効果• soft photon theorem[Strominger, … ]

赤外発散
•

量子電磁気学(QED)における赤外発散•

散乱のS行列 S ,↵ = h | S |↵i<latexit sha1_base64="DxFSiWfRkbk6l2ZaFohDCH+K0xw=">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</latexit>↵<latexit sha1_base64="oZIPhSK601kQmwb8zQtSqxaZxXI=">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</latexit>

⼀般に摂動計算のループレベルで発散• 紫外発散 (⾼エネルギー)• ⾚外発散 (低エネルギー)•

最近の理解: 赤外発散は漸近対称性と密接に関連している.
• QEDの漸近対称性を保証するには⾚外発散が不可⽋.
• メモリー効果を量⼦論で実現するにはドレス状態が必要.

<latexit sha1_base64="bXeK98gfbDE4ptIR7T16q6a+i5g=">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</latexit>Outline
1. Introduction
2. Asymptotic symmetry in QED
3. IR divergences in QED and dressed states
4. Dressed states and asymptotic symmetry

Outline
1. Introduction
2. Asymptotic symmetry in QED
3. IR divergences in QED and dressed states
4. Dressed states and asymptotic symmetry

Asymptotic symmetry in QED• Gravitational theory on asympt flat spacetime has infinite-dim asympt sym(BMS symmetry).
• Electromagnetic theory also has an infinite dim sym related to asymptotic[He, Mitra, Porfyriadis, Strominger (2014), (2015)]
behaviors of fields.
• QED has a U(1) gauge symmetry:
Parts of them are physical symmetry (not gauge redundancy).
e.g., global trsf.

• This symmetry leads to the electromagnetic memory effect.
[Hirai, SS (2018)]

Infinite asymptotic symmetry• Local U(1) transformations:• Charge of local U(1) trsf:
If

for any configuration of fields, the trsf is gauge redundancy.

Not the case forwhich doesn’t decay at the asymptotic region (large trsf).

• Asymptotic symmetry is physical sym.
• D.o.f of symmetry is a function of two-sphere

Infinite dim physical symmetry<latexit sha1_base64="YhS9WfTDT1/YEAG0g1VsxUhBjTc=">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</latexit>lim ✏(x) = ✏(0) (✓, ')r!1

Conservation law for asymptotic symmetryn Symmetry

Conservation law@µ J µ = 0<latexit sha1_base64="0MuoQ0B9RixcY7YKYZsSZAzn4WM=">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</latexit>µ(J µ = F µ⌫ @⌫ ✏ + jmat✏)

Infinite number of conserved charges Qin [✏] = Qfin [✏]<latexit sha1_base64="Q0anvjZDNoqKGU53LhgNg9dumYA=">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</latexit><latexit sha1_base64="8bfDCqVjpQuqQVtWkjb9f+xu2L4=">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</latexit>

Qfin [✏]<latexit sha1_base64="iBSP3chYPsaIznbMuNTl4dv52nw=">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</latexit>

Qin [✏]

initial<latexit sha1_base64="kcK86aIxCnWPJKv75FuBPafxY3Y=">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</latexit>• ✏(0) = const.
<latexit sha1_base64="tnkDk5KrUDbTv2dXUl06gYsVvTQ=">AAANd3icnVfNbttGEJ6kzZ8aNU57zCFCDTcOoAiUIbTpwUCKuIUPbewidmwkSgKSXkmE+AeSciMTOvZhilzbd+gL9NBH6LmnAj3km9mlLcsSRZcEyd2Z+WZmZ2eGpBP7XppZ1l9Xrn708bXrN27eqn1yu/7pnZW7n71Io1Hiqn038qPk0LFT5Xuh2s+8zFeHcaLswPHVgTN8yvyDY5WkXhTuZeNYvQ7sfuj1PNfOQHq7cr+r4tTzo/BNvm49nKx3s4HK7Gaje2wn8cB7+HZl1WpZcjQuDtpmsErm2I3uXm9Ql44oIpdGFJCikDKMfbIpxfmK2mRRDNprykFLMPKEr2hCNVoDegQ5BRkb9CHufcxYtkM/Q/bISM6Xe2WoIeZsPRU7LvzxcSVANmjN+tP61frH+sN6b/1t/bdQF9scwb8BOUAPS6zmkBmDpmAhwb1X6mEGT2LapA3EIoP8CNQAZxN2IlCYHwh/Y4bvi+5MeJ0ZXoLY9OHHPC7HgG1HEhPmjEtXk4pUauhFvBOJq8IuuKIpADfEfuTgObK/Ch56IpOKFSV7PwA+x1pbuCaQDWUnGO1DzqMTyLF+bdUXK32JO2uOJUtSeOdhzJo6kj+TpRhPvNMZqD3YmEIu2vEAdPaNdTXNPEW8Aqyxecphj3q4CglHZJqS5xk0vTOYgt/DmkOJaUFhLUGpL7FkVSr1kS6pjp7shQv5AdacnEonmCvw9C6lJuo+qmCH9ukRbdN3tIfnN4hqR1CL8/a5iQLXEmfUCBJ92WVlqqMmsT+S1XLlxHLfRORRdbi+l51vGJkfoHkP2MOSGLB+WyLBUXi3JArTdb5YZ5G9RWaUVbWS9ZVbdcTHoWRieWfKsOrHEjvOTR0dJftWXo26e7I3R6WSWlu6NAJFFnOniaQKy1fIPXqIOj0vVVC55nUMuK/kUnOu+GCLTCQ+tUSrznrm6T3lqq1m+xFip6A/NH2nQBX881ztX7HX2vKm6UEtvIF0tq5VPBv0DHVygOdT1M2POL8FZUuyutpZk743v3cmZl/53ci5wbSzTM5P+avSPyeluvhNNZKMXaTtTKKKvlAinZuuHco73ZGq6OIsxx5LveoRZ0Qu8+WY+ALqQQXceAY1roSZtTWuZOtkBnVSCTNr66SSreEMargU4yI7Naaoc1e6fi70ZdidBdidCtjtBdjtCtiXC7AvDZZrfUti4EunUahCLbljvrxs6WeJaMmkA+bmWbsUNkXn0VlfjC6Ljw3+uXSe+Rp28fRMbW+ZL6cAV2aqi7+7PIlAF/diNjEd42y+Jn1Td9KG+KTkDTvdRRtzOv7lPepJDylW6U/NC6+mKf/bL/xztGf/MC4OXmy02l+1Oj91Vp88Nn8fN+kefUHr6Gpf0xPk3C6+cFz6hd7Tb/T77X/r9+tf1te16NUrBvM5nTvq7Q9C4aby</latexit>

• ✏

(0)

(✓, ')

final

Conservation of electric charges
Other conservation laws (memory effect)

Conserved quantity
Q[✏] = Qhard [✏] + Qsoft [✏]<latexit sha1_base64="gWl/HRT148yvc4QSvKvoY5ax7Dw=">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</latexit><latexit sha1_base64="dXHzNXsytxOahzgPr6elZbTQOic=">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</latexit>

Qhard
in [✏] =Z

Z<latexit sha1_base64="sZlrHyZOp1E/EiUTjq/c3XH+xrw=">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</latexit>

Qsoft
in [✏] =dS · E ✏(0)S 2 (t= 1)“electric multiple moment”
hard
[✏]
=
6
Q
For general ✏(✓, ') , Qhard
infin [✏]<latexit sha1_base64="Ia9zCgpTx4tbpsVx2XMMy1xeuu4=">AAANpnicnVdPc9tUEN8WKK3BkJZjLy6ZDD2kHiXjofSQmTINTA6lSadJk2kcgiw/OxrrH3pyiaPxB2J641PwBZiBI9w4c+XAb/c9JY5jywrSSHpvd3+7+/btrqROEvg6c5w/btx87/0Pbn14+07to4/rn3y6dPfeax0PU0/teXEQpwcdV6vAj9Re5meBOkhS5YadQO13Bs+Yv/9WpdqPo91slKij0O1Hfs/33Ayk46XnL4/bmTrNcj8af5+b4Ymbdsfjw7ZKtB/E0VE7Uj82CrleieDx0rLTdORoXB2s2cEy2WMnvnvrAbWpSzF5NKSQFEWUYRyQSxrnIa2RQwloR5SDlmLkC1/RmGq0AvQQcgoyLugD3PuYsWyLfoJs10rOlju01Ahztq7Fjgd/AlwpkA1acX5zfnb+cX513jl/O//O1cU2h/DvhDpAD0qs5pAZgaZgIcW9V+phBk8S2qB1xCKD/BDUEOcq7MSgMD8U/voUPxDdmfBaU7wUsenDj1lcjgHbjiUmzBmVrkaLlLb0It6pxFVhFzzRFIIbYT9y8Dqyvwoe+iKjxYqSvT8BPsdam7jGkI1kJxgdQM6nM8ixfmM1ECt9iTtrTiRLNLzzMWZNLcmf8UKML96ZDDQerE8g5+14CDr7xrpW7VwjXiHWuHrOYY96uAqJjsisSp5n0HRqMQW/hzVHEtOCwlrCUl8SySot9aEXVEdP9sKD/AnWnJ5Lp5gr8MwuaRv1AFWwTXv0iLboG9rF8wmi2hLU/Lx9ZaPAtcQZNYREX3ZZ2eqoSey7slqunETuG4g8qg7Xt7LzDSvzHJp3gT0oiQHrdyUSHIXTBVGYrPP5OovsLTKjrKqVrK/cakd8HEgmlnemDKv+SmLHuWmio2Tfyi1o2z/Zn26px0afXhiDIo+518RSh+UecJceoFIvSxVUrnoTBe4suVSdJz64IhOLT03RavKeeWZXuW6r2X6E6Cnoj2znKVAF/zLX+FfstrG8YbtQE+8gk68rFc8GvUCl7OP5DJXzHc6vQdmUvK521qTzze6eqd1XfjtydjDtIpfzc/6ydNBxqS5+Vw0lZ+dpu5Cooi+SSOe2b0fyVu9IXbRxlmPfSsWaEWdELvPFmOQK6osKuNEUalQJM21rVMnW2RTqrBJm2tZZJVuDKdRgIcZDdhpMUeee9P1c6Iuw23Ow2xWwW3OwWxWwb+Zg31gs1/qmxCCQTqNQhUZy2357udLPUtGSSQfM7bN2LaxG5zFZX4yui08s/pV0ntkadvD0bW1v2m+nEFdmq4u/vHyJQBv3Yja2HeNiviJ903TShvik5B072UUbMzr+9T3qSQ8pVhlMzAuvJin/2y/8daxN/2NcHbxeb6592Wy9bC0/fWz/P27TffqcHqKrPaanyLkdfON49Av9Tn/SX/WH9Rf1vfq+Eb15w2I+o0tH/Yf/AL/avGA=</latexit><latexit sha1_base64="IJHRBuFJb1go9e88QFVlvS/rml8=">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</latexit>“incoming electromagnetic waves”

dV F rA @A ✏I

• Current conservation@µ J µ = 0<latexit sha1_base64="qMjZhkP69VkCs4CBQhkAImpNZjw=">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</latexit>µ(J µ [✏] = F µ⌫ @⌫ ✏ + jmat✏)

<latexit sha1_base64="K+UqcDXPTSwQpRviUn4YXTI1dww=">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</latexit>

Qin [✏] = Qfin [✏]This conservation law is nothing but the memory effect. [Hirai & SS (2018)]

Memory effect as a conservation law
soft
hard
soft
[✏]
+
Q
[✏]
=
Q
[✏]
+
Q
• Conservation law Qhard
in
in
fin
fin [✏]<latexit sha1_base64="5H3/EbuI5M9RPWQYHJwrBCaG0YM=">AAAN43icnVdPb9xUEJ9tobQLC2lRT5VgIYpAIl050QrKIVJRA8oBSKomTdRsiGzv88Za/5Oft83G2g+AuCGufADEF+ELcOAjcOLAlQO/N+852WzWXqe2bL83M7+ZefNmxraTBL7MLOuvxo2bb7x5663bd5pvv9N6972lu/eey3iUumLPjYM4PXBsKQI/EnuZnwXiIEmFHTqB2HeGTxR//6VIpR9Hu9k4EUehPYh8z3ftDKTjpR+fHvcycZrlfjT5QY9O7LQ/OeyJRPpBHB19dlVCxl42JbFRSHiLlXilWo6Xlq2OxUf76mDNDJbJHDvx3VsfUY/6FJNLIwpJUEQZxgHZJHEe0hpZlIB2RDloKUY+8wVNqEkrQI8gJyBjgz7EfYCZku3SK8j2jeR8uUNDjTBX1iXbceFPgCsFsk0r1p/Wb9a/1h/W79bf1n+lupTNEfw7IQfoYYXVHDJj0AQspLh7lR5m8CShDVpHLDLIj0ANca7CTgyK4ofMX5/hB6w7Y153hpciNgP4MY+rYqBsxxwTxRlXrkaylDT0It4px1VgF1zWFIIbYT9y8BzeXwEPfZaRbEXw3p8An2OtHVwTyEa8EwodQM6nM8gp/dpqwFYGHHelOeEskfDOx1hp6nL+TBZifPZOZ6D2YH0KWbbjIejKN6Vr1cwl4hVijavnHOWRh6uQcFhmlfM8g6ZTgyn4HtYccUwLitISVvqScFZJrg+5oDo83gsX8idYc3ounWIuwNO7JE3UA1TBNu3RQ9qir2kXzy8R1S6jyvP2mYmCqiWVUSNIDHiXhamOJse+z6tVlZPwfQORR9Xh+oZ3vm1kvoXmXWAPKmKg9NscCRWF0wVRmK7zcp1F9haZUVXVgtdXbdVhH4ecidWdKcOqH3HsVG7q6Ajet2oL0vRP5U+/0mOtTy6MQZHHqtfEXIfVHqguPUSlXpYqqKrqdRRUZ8m56lz2wWaZmH3qsFad94qnd1XVbT3bDxE9Af2R6TwFquBf5mr/it3WljdMF+rgHaTzdaXm2abvUSn7eD5B5XyH8ytQNjmv651N7nzzu2dq9lW9HVV2KNpFLufn/GXuoJNKXepdNeKcLdN2IVFHX8SRzk3fjvit7nBd9HBWY19yxeqRyoic54sxyRXUJzVw4xnUuBZm1ta4lq2zGdRZLcysrbNatoYzqOFCjIvs1Jiizl3u+znTF2G3S7DbNbBbJditGtgXJdgXBqtqfZNjEHCnEahCLbltvr1s7mcpa8m4A+bm2bwWVqLz6KwvRtfFJwb/jDvPfA07ePqmtjfNt1OIKzPVpb68fI5AD/diNjEd42K+wn1Td9I2+yT4HTvdRdtzOv71PfK4hxSrDKbmhVfTlNf2C38da7P/GFcHz9c7a593uk+7y48fmf+P2/SAPqZP0dW+oMfIuR1847j0T+N+44PGhy3R+qn1c+sXLXqjYTDv06Wj9ev/8qXU0Q==</latexit>

In Lorenz gauge @µ Aµ = 0 , AB (u, r, ⌦) = AB (u, ⌦) + O(1/r)<latexit sha1_base64="jIwLukYtedsI8XzZlkkQgLmrn5Q=">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</latexit>(0)<latexit sha1_base64="1CSoE5yczF2AWKb/ekAkKPZ+i6k=">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</latexit>

angular component (u = t<latexit sha1_base64="llZxnh9W0XMorzDMmINCWUGb4fI=">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</latexit><latexit sha1_base64="wwS6mJHxeYN135z+wQkiOwQ7uN4=">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</latexit>

Qsoft
fin [✏] =

Z

Z

(0)

d2 ⌦ AB @ B ✏(0) ,

S2

(0)

(0)

AB = AB (u = 1)

(0)

r)

AB (u =

1)<latexit sha1_base64="8XR86i8PeuusSFOjNd8NpeIgAJs=">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</latexit>(0)d2 ⌦ AB @ B ✏(0) = Qhard
in [✏]

S2

soft
Qhard
[✏]
+
Q
fin
in [✏]

memory effect• There is a permanent shift of the final EM field, and the leading part of it
is fixed by the change of the hard charges (and the initial EM field).
• electromagnetic analogue of the gravitational memory effectOutline
1. Introduction
2. Asymptotic symmetry in QED
3. IR divergences in QED and dressed states
4. Dressed states and asymptotic symmetry

Scattering in QED• QED describes quantum dynamics of electromagnetic
waves and charged particles.
• The fundamental quantity is S-matrix.
S ,↵ = h | S |↵i<latexit sha1_base64="DxFSiWfRkbk6l2ZaFohDCH+K0xw=">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</latexit>↵<latexit sha1_base64="oZIPhSK601kQmwb8zQtSqxaZxXI=">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</latexit>• Perturbatively computed by Feynman diagrams<latexit sha1_base64="bXeK98gfbDE4ptIR7T16q6a+i5g=">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</latexit>

Infrared divergences in QED
Loop integral may diverge by virtual soft photons

𝑘~0Infrared divergences in QED
Loop integral may diverge by virtual soft photons

almost on-shell electron if the virtual photon is soft
𝑘~0
soft photon approx~ log 𝜆𝜆 : IR cutoff

Infrared divergences in QED
Loop integral may diverge by virtual soft photons

almost on-shell electron if the virtual photon is soft
𝑘~0
soft photon approx~ log 𝜆𝜆 : IR cutoff~ (log 𝜆)!

Factorization of IR divergences
We can sum up all of IR divergent parts. The divergent parts are factorized.

S-matrix element with IR cutoff 𝜆:Λ: new IR cutoff
soft

hard

The power 𝐴 depends only onof initial and final charged particles
and is independent of the details of scattering.
This IR factorization is valid as long as we can use the soft photon approx for |~k| < ⇤.
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We will say ‘soft’ for<latexit sha1_base64="ni/4quKAU+E1nCWrPGcVqCn348A=">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</latexit>< |~k| < ⇤ and ‘hard’ for |~k| > ⇤ .
<latexit sha1_base64="DGCEa4ROIAPBvbiPwM5HEq4qMO4=">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</latexit>

No nontrivial scattering?

The power 𝐴 > 0 except for trivial process.

if we remove the IR cutoff

Any S-matrix elements are zero except for trivial diagonal elements???
It seems that there is no nontrivial scattering…

This is the IR problem in QED.

Conventional treatment (inclusive formalism)[Bloch and Nordsieck (1937)]n Any detector has a resolution 𝐸" . Soft photons 𝑘 < 𝐸" can’t be detected.
• Detector can’t distinguish• Measured cross-section:
In the sum, we can take IR cutoff 𝜆 → 0.

ü We got the IR finite quantity. No realistic problem if we forget the S-matrix.
But, each S-matrix element vanishes:
Is the S-matrix meaningless?

Dressed state formalism[Chung (1965), Kibble (1968),
Kulish & Faddeev (1970)]

u Use of Fock states is probably not appropriate.
• Asymptotic states are usually assumed to be approximated by free states.
• This assumption might be not good for long-range interactions
like electromagnetic (and gravitational) force.

l Chung’s dressed states

Charged particle has a cloud of coherent photons

S-matrix for dressed states<latexit sha1_base64="LMuaczGVgfdrpSHa/2FV6cWvnTY=">AAANkXicnVdPb+NUEJ9dYFkCgS575BJRVXBoI7eKYHuoVGhBlfjTQttttU2JbPclteJ/sp1lU8ufhTPicyBxghMHPgJnrhz4zbznNk0Tx8WW7fdm5jczb97M2HZi30szy/rrwcPXXn/j0ZuP32q8/U7z3feWnrz/PI1GiauO3ciPklPHTpXvheo48zJfncaJsgPHVyfOcIf5Jy9VknpReJSNY3Ue2IPQ63uunYHUW9rsZipMo+SsZ53nXSex867tx5d2UeSF+iFf+763Uxy2MOJBd6gyCKnMLnpWb2nZaltytO4O1s1gmcxxED151KIuXVBELo0oIEUhZRj7ZFOK84zWyaIYtHPKQUsw8oSvqKAGrQA9gpyCjA36EPcBZizboR8he2EkZ8udGWqIOVtPxY4Lf3xcCZAtWrH+tH62/rF+s36x/rb+nauLbY7g3yU5QA8rrOaQGYOmYCHBvV/pYQZPYtqiDcQig/wI1ADnKuxEoDA/EP7GFN8X3ZnwOlO8BLEZwI9ZXI4B244kJswZV64mFanU0Mt4JxJXhV1wRVMAboj9yMFzZH8VPPREJhUrSvb+Evgca23jKiAbyk4w2oecR1eQY/3aqi9WBhJ31hxLlqTwzsOYNXUkf4qFGE+80xmoPdiYQM7b8QB09o11rZp5ingFWOPqNYc96uMqJRyRWZU8z6DplcGU/D7WHEpMSwprCSp9iSWrUqmPdEF19GUvXMhfYs3JtXSCuQJP71Jqou6jCvbpmNZoj76gIzw3EdWOoObn7aGJAtcSZ9QIEgPZZWWqoyGxv5DVcuXEct9C5FF1uL6UnW8Zma+h+QjY04oYsH5bIsFReLUgCpN1Pl9nmb1lZlRVtZL1VVt1xMehZGJ1Z8qw6mcSO85NHR0l+1Zdjbp7sjcXlZJaW7owAmUWc6eJpAqrV8g9eog6vS1VUrnmdQy4r+RSc674YItMJD61RavOeubpPeWqrWd7DbFT0B+avlOiSv5trvav3Gttecv0oDbeQDpbV2qeLfoWdXKC5w7q5hucn4GyK1ld72xI35vdOxOzr/xu5Nxg2k0m59f8ZemfRaUuflONJGPnabuRqKMvlEjnpmuH8k53pCq6OKuxL6Ve9YgzIpf5Ykx8B/VRDdx4CjWuhZm2Na5l62oKdVULM23rqpat4RRquBDjIjs1pqxzV7p+LvRF2P052P0a2L052L0a2BdzsC8Mlmt9V2LgS6dRqEItuW++vGzpZ4loyaQD5ubZuBc2RefRWV+O7ouPDf5QOs9sDQd4eqa2d82XU4ArM9XF312eRKCLezkrTMe4ma9I39SdtCU+KXnDTnbR1oyOf3+P+tJDylX6E/PSq0nK//YL/xzr038YdwfPN9rrn7Q733WWt5+Zv4/H9AF9SB+jq31K28i5A3zhuPQT/Uq/0x/Np83N5nbzcy368IHBPKVbR/Or/wBYB7IL</latexit>

RC
RC
h↵|
e
Se
| i0
0

[Chung (1965)]

Cloud of photons also interacts.

New diagrams in addition to usual ones.
These diagrams have IR divs
which cancel those in usual diagrams.

Totally IR finite!
Fig from Hirai’s phD thesis• S-matrix elements for dressed states are IR finite unlike the Fock states.

A reason of the IR problem
n Fock space is incomplete
some states orthogonal to all Fock states w/ finite particles
Coherent statesFock• orthogonal to any finite Fock state in the limit removing the IR cutoff

l time-evolved state

in QED is roughly this coherent state.
Amplitudes to Fock states vanish.

Outline
1. Introduction
2. Asymptotic symmetry in QED
3. IR divergences in QED and dressed states
4. Dressed states and asymptotic symmetry

Dressed states and asymptotic symmetry• Dressed states were old subjects. [Chung (1965), Kibble (1968), Kulish & Faddeev (1970)]• They have been reconsidered in the context of asymptotic sym.
[Mirbabayi & Porrati (2016), Gabai & Sever (2016),
Kapec, Perry, Raclariu & Strominger (2017),
Carney, Chaurette, Neuenfeld & Semenoff (2018),…]

Fock states• Asymptotic symmetry requires

soft
hard
soft
Qhard
[✏]
+
Q
[✏]
=
Q
[✏]
+
Q
in
in
fin
fin [✏]<latexit sha1_base64="5H3/EbuI5M9RPWQYHJwrBCaG0YM=">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</latexit>• However, usual Fock states Qsoft [✏] = 0<latexit sha1_base64="biiQU+KcBExrVAU0gt/nDV8QO5o=">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</latexit>

hard• For non-trivial scattering, general hard charges change Qhard
in [✏] 6= Qfin [✏]<latexit sha1_base64="1LyiJx3QrWpo1OSpOS/jiZqMsd8=">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</latexit>

except for the global part• Thus, the scattering between Fock states cannot satisfy the conservation law.
Or, the symmetry does not allow such scattering.
S ,↵ = 0 .
• This is consistent with the result lim!0<latexit sha1_base64="/61Uu4+3BUhmbNqgPVzlzd5ER48=">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</latexit>[Kapec, Perry, Raclariu & Strominger (2017)]

IR divergence is necessary<latexit sha1_base64="7TK3vKknN5HQuDFewGvfWbZ/DoM=">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</latexit>

lim S ,↵ = 0!0• The amplitude for the prohibited process by asymptotic sym
vanishes thanks to the IR divergence.
• IR divergence is necessary to maintain asympt sym.
and the IR problem is not a problem in this sense.

Necessity of dressed states• Fock states are not enough to maintain the asympt sym
because they have vanishing soft charges Qsoft [✏] = 0<latexit sha1_base64="biiQU+KcBExrVAU0gt/nDV8QO5o=">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</latexit>• Dressed states (coherent state of soft photons) can have non-vanishingsoft charges Qsoft [✏] 6= 0
<latexit sha1_base64="MAOeAQF2+KeEgdYpwdRNQ9+UddQ=">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</latexit>• Our questions:• What dressed states lead to non-vanishing amplitude lim S ,↵ 6= 0 ?
<latexit sha1_base64="63t+qUCznvm68lwYgp3R9JiLwvg=">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</latexit>• Is it consistent with the asymptotic symmetry?

!0IR factorization[Hirai & SS (2022)]• General dressed asymptotic states<latexit sha1_base64="bJYf0tVkpV+LhPa6f6iwIaiDB/A=">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</latexit>|↵iin = eC↵ |↵i0 ,<latexit sha1_base64="fm0SaLQsAFSxLVnbSxn5r90ouN4=">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</latexit>

C↵ =

Z ⇤s

| iout = eD | i0

h
d3 k
A ~
~k)
C
(
k)a
(
A
↵
(2⇡)3 (2!)

C↵A⇤ (~k)a†A (~k)

isoft photons

n We can evaluate IR factor for general dressed states.
hard
S ,↵ = S soft,↵ (⇤s , )S ,↵ (⇤s )<latexit sha1_base64="X+3RlNFe7DZ94qoXKh3uxlz733c=">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</latexit>

This is generalization of IR factorization formula for Fock states<latexit sha1_base64="apqvcMGaSSAIsaZG2oIaJpELwyk=">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</latexit>

S ,↵ =

✓

⇤s

◆#

s
S ⇤,↵

General IR factor[Hirai & SS (2022)]

hard
S ,↵ = S soft
(⇤
,
)S
s
,↵
,↵ (⇤s )<latexit sha1_base64="X+3RlNFe7DZ94qoXKh3uxlz733c=">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</latexit>

1
2

Z ⇤s

RA,↵ = RAA
R↵,<latexit sha1_base64="mZHcIbdqmqUBmOPg6aHTb0j12+Y=">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</latexit><latexit sha1_base64="jb8MGLPPftfQKplVIJOLPywCWVc=">AAANdHicnVdPj9tUEJ8WKCU0sC03yiEQRXBII+8qauGwUqtd0B6gu6i73VWbNrK9L1kr/ifbaZu1IvXTICEO8DH4Ahz4CJyROHHgN/Oed7PZxPFiy/Z7M/ObmTdvZmw7se+lmWX9ee36O+++d+P9mx/UPrxV/+jjtdt3nqbROHHVgRv5UXLk2KnyvVAdZF7mq6M4UXbg+OrQGW0x//CVSlIvCvezSaxeBPYw9Aaea2cg9dc+3VQv88f9fLvfc1Rmtxtb/Z7txyf2dNpfa1odS47G5cG6GTTJHHvR7RufU4+OKSKXxhSQopAyjH2yKcX5nNbJohi0F5SDlmDkCV/RlGrUAnoMOQUZG/QR7kPMWLZLryF7bCQXyz031BBztp6KHRf++LgSIBvUsv6wfrb+tn63frH+sv5dqottjuHfCTlAj0qs5pCZgKZgIcF9UOphBk9i2qQNxCKD/BjUAGcbdiJQmB8If2OO74vuTHjdOV6C2AzhxyIux4BtRxIT5kxKV5OKVGroRbwTiavCLriiKQA3xH7k4DmyvwoeeiKTihUle38CfI61dnBNIRvKTjDah5xHp5Bj/dqqL1aGEnfWHEuWpPDOw5g1dSV/pisxnninM1B7sDGDXLbjAejsG+tqm3mKeAVYY/uMwx4NcBUSjsi0Jc8zaHpjMAV/gDWHEtOCwlqCUl9iyapU6iNdUR0D2QsX8idYc3ImnWCuwNO7lJqo+6iCXTqge7RD39I+nt8gql1BLc/bJyYKXEucUWNIDGWXlamOmsT+WFbLlRPLfRORR9Xh+k52vmFkvofmfWCPSmLA+m2JBEfhzYoozNb5cp1F9haZUVbVStZXbtURH0eSieWdKcOqv5bYcW7q6CjZt3ILqemf7M9xqcdaX7oyBkUec6+JpA7LPeAuPUKlXpQqqFz1OgrcWXKpOld8sEUmEp86olXnPfP0rnLdVrN9D9FT0B+azlOgCv5Frvav2G1tedN0oQ7eQTpfWxXPBj1GpRziuYXK+QHnI1C2Ja+rnTXpfIu7Z2L2ld+OnB1MO8/l/IzflA46LdXF76qx5OwybecSVfSFEunc9O1Q3uqO1EUPZzn2lVSsHnFG5DJfjYkvob6sgJvMoSaVMPO2JpVsnc6hTith5m2dVrI1mkONVmJcZKfGFHXuSt/Phb4Ku7sEu1sBu7MEu1MB+2wJ9pnBcq1vSwx86TQKVagld823ly39LBEtmXTA3DxrV8Km6Dw664vRVfGxwT+RzrNYwx6enqntbfPtFODKTHXxl5cnEejhXsympmOcz1vSN3UnbYhPSt6xs120saDjX92jgfSQYpX+zLzwapbyv/3CX8f6/D/G5cHTjc76/U73x27z4X3z/3GT7tIX9BW62gN6iJzbwzeOS2/pJ/qVfrv1T/2zerPe0qLXrxnMJ3ThqHf+A/KnpVQ=</latexit>=e

ND ,C↵

where

Re(ND ,C↵ ) =

d3 k
A
|R
,↵
(2⇡)3 (2!)<latexit sha1_base64="mmKk4iC6HlpdAZiJ6pl/1+sNK1c=">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</latexit>

A

R :=

X

n2

C↵A + DA |2  0

pn · ✏ Aen
pn · k( = ↵, )The integrand generally has 1/𝑘 ! terms and thus the integral gives log 𝜆 div.
It means the IR problem<latexit sha1_base64="0PQENxHv/WA2Zh4xFzmyDntd/+4=">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</latexit>

lim S ,↵ = 0!0We have to choose the dress so that they cancel the divergences
to obtain non-vanishing S-matrix elements.

Dress code

[Hirai & SS (2022)]<latexit sha1_base64="mZHcIbdqmqUBmOPg6aHTb0j12+Y=">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</latexit><latexit sha1_base64="ZtCNNun60yTHef7frVRSZjd/aGE=">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</latexit>

S ,↵ = e

ND ,C↵

S hard,↵ (⇤s )Re(ND ,C↵ ) =

1
2

Z ⇤s

d3 k
A
|R
,↵
(2⇡)3 (2!)

C↵A + DA |2  0l Dress code for non-vanishing S-matrix elements:<latexit sha1_base64="hGBEHFj+7jEGJ4o+nARAVEqnn6I=">AAANhnicnVdPb+NUEJ9dYCmBQBeOXAJVxSKSyKkCuxwqFbWgHoAWtt1Wu1ki231JrfifbGfZ1MqX4FsgvgBXOPEFOPAROHPlwG/mPbdpmjgutmy/NzO/mXnzZsa2E/temlnWX3fuvvLqa/deX3uj9uZb9bffWb//7pM0GieuOnYjP0pOHTtVvheq48zLfHUaJ8oOHF+dOKNd5p+8UEnqReFRNonV88Aeht7Ac+0MpP56c6/fc1Rmt3b7PduPz+3t1vf9XEhNTZh+Ej0Y/ZC3OtOP++sbVtuSo3Fz0DGDDTLHYXT/3gfUozOKyKUxBaQopAxjn2xKcT6jDlkUg/acctASjDzhK5pSjTaBHkNOQcYGfYT7EDOW7dKPkD0zkovlnhlqiDlbT8WOC398XAmQDdq0/rR+tv6x/rB+sf62/l2qi22O4d85OUCPSqzmkJmApmAhwX1Q6mEGT2Lapi3EIoP8GNQAZxN2IlCYHwh/a47vi+5MeN05XoLYDOHHIi7HgG1HEhPmTEpXk4pUauhFvBOJq8IuuKIpADfEfuTgObK/Ch56IpOKFSV7fw58jrW2cU0hG8pOMNqHnEcXkGP92qovVoYSd9YcS5ak8M7DmDV1JX+mKzGeeKczUHuwNYNctuMB6Owb62qaeYp4BVhj85LDHg1wFRKOyDQlzzNoemkwBX+ANYcS04LCWoJSX2LJqlTqI11RHQPZCxfy51hzcimdYK7A07uUmqj7qIIDOqYW7dOXdITn54hqV1DL8/axiQLXEmfUGBJD2WVlqqMmsT+T1XLlxHLfRuRRdbi+kp1vGJmvofkI2NOSGLB+WyLBUXi5Igqzdb5cZ5G9RWaUVbWS9ZVbdcTHkWRieWfKsOpHEjvOTR0dJftWbiE1/ZP9OSv1WOtLV8agyGPuNZHUYbkH3KVHqNTrUgWVq15HgTtLLlXnig+2yETiU1u06rxnnt5VrttqtluInoL+0HSeAlXwr3O1f8Vua8vbpgu18Q7S+bpZ8WzQt6iUEzx3UTnf4PwClD3J62pnTTrf4u6ZmH3ltyNnB9Oucjm/5G9IB52W6uJ31Vhydpm2K4kq+kKJdG76dihvdUfqooezHPtCKlaPOCNyma/GxDdQH1XATeZQk0qYeVuTSrYu5lAXlTDzti4q2RrNoUYrMS6yU2OKOnel7+dCX4U9WII9qIDdX4Ldr4B9ugT71GC51vckBr50GoUq1JIH5tvLln6WiJZMOmBunrVbYVN0Hp31xei2+NjgH0vnWazhEE/P1Pae+XYKcGWmuvjLy5MI9HAvZlPTMa7mm9I3dSdtiE9K3rGzXbSxoOPf3qOB9JBilf7MvPBqlvK//cJfR2f+H+Pm4MlWu/NZu/tdd2Pnkfn/WKP36UN6gK72kHaQc4f4xnHpJ/qVfqPf62v1dv3T+kMteveOwbxH1476zn+UzKry</latexit>DX

pn · ✏ A
en
,
p
·
k
n
n2↵

<latexit sha1_base64="Z88urlP5YvogiAsnKalkTW1HIp4=">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</latexit>• A choice is

C↵ =

C↵A =

R ,↵ + o(kD =S ,↵ = S hard,↵ (⇤s )<latexit sha1_base64="iltIo7Kndf0Q97hQLSBkMO9Lb4c=">AAANl3icnVdPb9xUEJ8WKO3CQgon1IshiijSduVEqwKHFa0aqhwKSZWkidptI9v7dtda/5Oft+3G2k/DCfFF4AriwEfgzJUDvzfvOdk4u14HW7bfm5nfzLx5M2PbTQJfZrb917Xr77z73o33b95qfPBh86OP125/8kzGk9QTh14cxOmx60gR+JE4zPwsEMdJKpzQDcSRO36k+EevRSr9ODrIpol4GTrDyB/4npOBdLLW3T/Je67InJbVc4Jk5MwalmV1cVn7r/KRk/ZnZYG7vSfQ33dO5Fcna+t22+bDujzYNIN1MsdefPvG59SjPsXk0YRCEhRRhnFADkmcL2iTbEpAe0k5aClGPvMFzahBG0BPICcg44A+xn2ImZLt0BvI9o3kYrkXhhphrqxLtuPBnwBXCqRFG/af9s/2P/av9i/23/a/S3UpmxP4NyIX6HGF1RwyU9AELKS4Dyo9zOBJQl3aQiwyyE9ADXG2YCcGRfFD5m+V+AHrzpjXKfFSxGYIPxZxVQyU7ZhjojjTytVIlpKGXsQ75bgK7ILHmkJwI+xHDp7L+yvgoc8ykq0I3vsR8DnW2sY1g2zEO6HQAeR8OoWc0q+tBmxlyHFXmhPOEgnvfIyVpg7nz2wlxmfvdAZqD7bmkMt2PARd+aZ0tcxcIl4h1tg64yiPBrgKCZdlWpznGTS9NZiCP8CaI45pQVFawkpfEs4qyfUhV1THgPfCg/wIa07PpFPMBXh6l6SJeoAq2KVDukc79D0d4Pktotph1PK83TdRULWkMmoCiSHvsjDV0eDY93m1qnISvncReVQdrse885aReQLNB8AeV8RA6Xc4EioKb1dEYb7Ol+sssrfIjKqqFry+aqsu+zjmTKzuTBlW/Q3HTuWmjo7gfau2IE3/VP70Kz3W+uTKGBR5rHpNzHVY7YHq0mNU6kWpgqqqXkdBdZacq85jHxyWidmnNmvVea94eldV3dazfQ/RE9Afmc5ToAr+Ra72r9htbblrulAb7yCdrxs1T4t+RKUc4fkIlfMDzoegbHNe1zsb3PkWd8/U7Kt6O6rsULTzXM7P+OvcQWeVutS7asI5u0zbuUQdfRFHOjd9O+K3ust10cNZjX3NFatHKiNynq/GJJdQX9bATUuoaS1M2da0lq3TEuq0FqZs67SWrXEJNV6J8ZCdGlPUucd9P2f6KuzuEuxuDezOEuxODezzJdjnBqtqfZtjEHCnEahCLblrvr0c7mcpa8m4A+bm2bgSVqLz6KwvRlfFJwa/z51nsYY9PH1T29vm2ynElZnqUl9ePkegh3sxm5mOcT7f4L6pO6nFPgl+x853UWtBx7+6RwPuIcUqg7l54dU85X/7hb+OzfI/xuXBs6325v1252ln/cF98/9xk+7QF3QXXe1reoCc28M3jkc/0W/0O/3R/Kz5XfNxc0eLXr9mMJ/ShaP59D9PxbCn</latexit>

A

X

n2

1

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pn · ✏ Aen
pn · kIR finite!

• Other choices than Chung’s are possible.

lim k o(k 1 ) = 0

k!0(Chung’s dress)

Quantum memory effect• Dress code

D

C↵ =

R ,↵ + o(k 1 )final

initialchange of hard info

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Qsoft
fin [✏]<latexit sha1_base64="iuRgch2ABOV8iPJrzF61Xr9li0I=">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</latexit>

Qsoft
in [✏] =[Hirai & SS (2022)]

hard
Qhard
[✏]
+
Q
fin
in [✏]

Hard charges are the same as those for classical charged point particles.

• Dress code is the electromagnetic memory effect!
The result is consistent with the asymptotic symmetry.
• Amplitudes for prohibited transition btw dressed states vanishes
thanks to the IR divergence.

Superselection rule|ini = f↵ |↵i + f↵0 |↵0 i<latexit sha1_base64="yfkpSh3tSjmQPNM2XRe7Glxlz4Y=">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</latexit>• Two states cannot interfere if they belong to different sectorsof the asymptotic charges.
• We can take a genuine superposition only when the two states
have the same asymptotic charges.
p i, |p~0 i have
• Undressed momentum states for charged particles |~different asymptotic charges.
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Z<latexit sha1_base64="FcJg+nPrTVkSugFrEMnExJAYusM=">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</latexit>• Wave packet is a superposition of momentum eigenstateZ<latexit sha1_base64="kg8HJTJxYi+5PB32Q4GK+6QqhAg=">AAANmHicnVdPb+NUEJ9dYFkCgS7c4GKoKhYpG7lVBMuhqKgFigS0sO222s1S2c5LasX/ZDtlUyvfhgviiyCuwIGPwJkrB34z77lN08Rx15bt92bmNzNv3szYdpPAz3Lb/vvW7ZdefuXOq3dfa7z+RvPNt1buvf04i0eppw69OIjTY9fJVOBH6jD380AdJ6lyQjdQR+5wm/lHZyrN/Dg6yMeJehY6g8jv+56Tg3Sy8lnXj3Kr1z1TXpFMui2re+akyal/31A+stSPxfbFbNIdqrwws8mJfbKyardtOazrg3UzWCVz7Mf37rxPXepRTB6NKCRFEeUYB+RQhvMprZNNCWjPqAAtxcgXvqIJNWgN6BHkFGQc0Ie4DzBj2Q79BNmekZwv99RQI8zZeiZ2PPgT4EqBtGjN/sv+xf7X/s3+1f7H/m+hLrY5gn+n5AI9rLBaQGYMmoKFFPd+pYc5PElokzYQixzyI1BDnC3YiUFhfij8jRl+ILpz4XVmeCliM4Af87gcA7YdS0yYM65cTSZSmaGX8U4lrgq74ImmENwI+1GA58r+Knjoi0wmVpTs/SnwBdbaxjWBbCQ7wegAcj6dQ471a6uBWBlI3FlzIlmSwTsfY9bUkfyZLMX44p3OQO3BxhRy0Y6HoLNvrKtl5hniFWKNrQsOe9THVUq4ItOSPM+h6bnBlPw+1hxJTEsKawkrfUkkqzKpj2xJdfRlLzzIn2LN6YV0irkCT+9SZqIeoAr26JAe0C59QQd4foqodgS1OG8fmShwLXFGjSAxkF1WpjoaEvuerJYrJ5H7JiKPqsP1pey8ZWS+geYDYI8rYsD6HYkER+H5kihM1/linWX2lplRVdVK1ldt1RUfh5KJ1Z0px6ofSuw4N3V0lOxbtYXM9E/2p1fpsdaXLY1Bmcfca2Kpw2oPuEsPUalXpUoqV72OAneWQqrOEx8ckYnFp7Zo1XnPPL2rXLf1bD9A9BT0R6bzlKiSf5Wr/St3W1veNF2ojXeQzte1mqdF36FSjvDcRuV8i/NzUHYkr+udDel887tnavaV346cHUy7zOXigr8qHXRSqYvfVSPJ2UXaLiXq6Isk0oXp25G81V2piy7OauyZVKwecUYUMl+OSa6hPqyBG8+gxrUws7bGtWydz6DOa2FmbZ3XsjWcQQ2XYjxkp8aUde5J3y+Evgy7twC7VwO7uwC7WwP7ZAH2icFyre9IDALpNApVqCX3zLeXI/0sFS25dMDCPBs3wmboPDrry9FN8YnBP5LOM1/DPp6+qe0d8+0U4spNdfGXly8R6OJeziamY1zO16Rv6k5qiU9K3rHTXdSa0/Fv7lFfeki5ymBqXno1TXlhv/DXsT77j3F98Hijvf5xu/N9Z3Xrofn/uEvv0Qd0H13tE9pCzu3jG8ejn+l3+oP+bL7b3Gp+1fxai96+ZTDv0JWj+cP/Cnuz3w==</latexit>• Need dress to make wave packets.

d~
p '(~
p)eC(~p) |~
pi 0

d~
p '(~
p) |~
pi

Allowed dress
X<latexit sha1_base64="HbtbcjeIysDkAhL9dMqnVbRndwY=">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</latexit>ifi eCi |↵i i• To have the same asympt charges, the dress should be
R↵i + C + o(k 1 )<latexit sha1_base64="EM12Ta6zV8C97i2meuUiPj4HRdY=">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</latexit>

Ci =Chung’s dress<latexit sha1_base64="tOpuDRaLZM6u1OQPAQX9tnwDtaQ=">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</latexit>

A
R↵
=
i

X

Common to all 𝑖pn · ✏ Aen
pn · kn2↵iUndressed vs Dressed (1)The standard inclusive formalism cannot compute S-matrix elements
but can obtain the finite total cross-section.
In any realistic setup, we cannot detect soft photons.
So the inclusive computations with conventional Fock states seem to be enough...

Can we distinguish undress and dress when we can observe only inclusive quantities?
Carney, Chaurette, Neuenfeld & Semenoff (2018) claimed that the answer is yes.

Fock states and Chung states give different results for superposed initial states.

Undressed vs Dressed (2)Consider superposition of different initial states |ini = f↵ |↵i + f↵0 |↵0 i .
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Compute inclusive cross section for the hard final states | i.
<latexit sha1_base64="jEHYgCw8GhhfNOmgjLlugnQRgZs=">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</latexit><latexit sha1_base64="UocNtsRLCqdD8jNkkQ/SsYD5BO0=">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</latexit>

hard(in ! ) = (in ! ) + (in !

+ ) + (in !

+ 2 ) + ···In general, we expect that the cross section contains the interference termof the initial phase f↵ f↵⇤0 .
<latexit sha1_base64="wn4CZOQY1JK8+OmDesVLqUcyxiE=">AAANc3icnVdPb9xUEJ8WKO3ShRROVS9LQ1QOycqJVgUOkYoaUA5Agpo0UbttZDtvN9b6n/y8pRtr1U/TA0KCr8EX4MBH4Iy4ceA3856TzWbX62DL9nsz85uZN29mbHtpGOjccf68dv2dd9+78f7NW40Pbjc//GjpzsdPdTLMfLXvJ2GSHXquVmEQq/08yEN1mGbKjbxQHXiDx8w/eKUyHSTxXj5K1YvI7cdBL/DdHKSjpbu9o64bpiduq3dUmNGD8cuuq/OjpWWn7cjRujxYt4NlssducufGp9SlY0rIpyFFpCimHOOQXNI4n9M6OZSC9oIK0DKMAuErGlODVoAeQk5BxgV9gHsfM5bt0E+QPbaSs+WeW2qMOVvXYseHPyGuDMgWrTh/OD87fzu/O784fzn/ztXFNofw74Q8oAcVVgvIjEBTsJDh3qv0MIcnKW3SBmKRQ34IaoRzFXYSUJgfCX9jih+K7lx4nSlehtj04ccsLseAbScSE+aMKlejRUpbehnvTOKqsAu+aIrAjbEfBXie7K+Ch4HIaLGiZO9PgC+w1jauMWRj2QlGh5AL6BRyrN9YDcVKX+LOmlPJEg3vAoxZU0fyZ7wQE4h3JgONBxsTyHk7HoHOvrGuVTvXiFeENa6ecdijHq5SwhOZVcnzHJpeW0zJ72HNscS0pLCWqNKXVLJKS33oBdXRk73wIX+CNWdn0hnmCjyzS9pGPUQV7NA+rdE2fUN7eH6FqHYENT9vn9gocC1xRg0h0ZddVrY6GhL7Y1ktV04q901EHlWH61vZ+ZaV+Q6a94A9rIgB63clEhyF1wuiMFnn83WW2VtmRlVVK1lftVVPfBxIJlZ3phyr/lJix7lpoqNk36otaNs/2Z/jSo+NPr0wBmUec69JpA6rPeAuPUClXpQqqVz1JgrcWQqpOl98cEUmEZ/aotXkPfPMrnLd1rO9hugp6I9t5ylRJf8i1/hX7raxvGm7UBvvIJOvKzXPFv2ASjnA8zEq53ucX4OyJXld72xI55vdPTO7r/x25Oxg2nkuF2f8Zemg40pd/K4aSs7O03YuUUdfLJEubN+O5a3uSV10cVZjX0nFmhFnRCHzxZj0EupBDdxoCjWqhZm2Napl63QKdVoLM23rtJatwRRqsBDjIzsNpqxzX/p+IfRF2J052J0a2O052O0a2GdzsM8slmt9S2IQSqdRqEIjuWO/vVzpZ5loyaUDFvbZuBJWo/OYrC9HV8WnFv9EOs9sDbt4Bra2t+y3U4Qrt9XFX16BRKCLezkb245xPl+Rvmk6aUt8UvKOneyirRkd/+oe9aSHlKsMJ+alV5OU/+0X/jrWp/8xLg+ebrTXH7Y7P3aWHz20/x836R7dp8/R1b6gR8i5XXzj+PSG3tKv9Nvtf5r3mvebnxnR69cs5hO6cDTX/gMayqVR</latexit>

However, Carney, Chaurette, Neuenfeld & Semenoff show that
• If we use Fock states (undress), no interference term. Decoherence!
• If we use Chung states (dress), we have the interference term.
The decoherence represents just the superselection rule.

Superselection rule|ini = f↵ |↵i + f↵0 |↵0 i<latexit sha1_base64="ltPejYeZ8e669IZRFg/AxyU2Gb4=">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</latexit>

Undressed states have only hard charges, and they are different for different configs.

Qas,↵ 6= Qas,↵0<latexit sha1_base64="rD+/JXUq57K9ZlcCd+Ug7DxmH54=">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</latexit>

always decohered

Two states can interfere only when they have the same asymptotic charges. Qas,↵ = Qas,↵0<latexit sha1_base64="CePDXqLPjQz399kTdpMrJMx0/Ec=">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</latexit>

They should have appropriate soft dresses.

Inclusive cross section for general dress
Consider |ini =<latexit sha1_base64="qCZ6tfrfXPx2bdf2GOtlTxL43uA=">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</latexit>

X↵f↵ eC↵ |↵i0<latexit sha1_base64="i47dYgQ4PB9Jak2KRkdUWUC2KkM=">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</latexit>

hard[Hirai & SS (2022)](in ! ) =X

Independent of final dress

hard hard†f↵ f↵⇤0 S↵soft
0 ,↵ S ,↵ S 0
↵,

↵,↵0

soft factor for ↵ ! ↵0<latexit sha1_base64="0WG9+Ru7Wx3v+vo2Zr2aSAESer0=">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</latexit>• If Qas,↵ 6= Qas,↵0 as undressed states, we have S↵soft
0 ,↵ = ↵0 ,↵
X
hard†
hard
(in ! ) =|f↵ |2 S hard
S
Decoherence,↵↵,<latexit sha1_base64="WLqDIxAUgTVUKIQ/QCrKy6Cfng0=">AAANkHicnVdPbxtVEJ8WKMVgSNsjF0MUlYNrbSKrLYeorRpQhIAEJWmi1m20Xj87K+8/7VuXOCt/E74A4ltwQoIrBz4CZ64c+M28t4nj2OsNu9rd92bmNzNv3szsbjcJfJ05zl83br7z7nu33r/9Qe3Dj+off7Jy5+4LHY9STx14cRCnR11Xq8CP1EHmZ4E6SlLlht1AHXaHz5l/+Fal2o+j/WycqNehO4j8vu+5GUjHKw/33nQydZrlOu5nk+O84wbJiXu/aZ6TzU5PBZl7hX68suq0HDkaVwfrdrBK9tiN79z6jDrUo5g8GlFIiiLKMA7IJY3zFa2TQwlorykHLcXIF76iCdVoDegR5BRkXNCHuA8wY9k2/QjZnpWcL/fKUiPM2boWOx78CXClQDZozfnT+dn5x/nN+cX52/l3oS62OYJ/J9QFelhiNYfMGDQFCynu/VIPM3iS0CZtIBYZ5EeghjibsBODwvxQ+Bsz/EB0Z8Jrz/BSxGYAP+ZxOQZsO5aYMGdcuhotUtrSi3inEleFXfBEUwhuhP3IwevK/ip46IuMFitK9v4E+BxrbeGaQDaSnWB0ADmfziDH+o3VQKwMJO6sOZEs0fDOx5g1tSV/JksxvnhnMtB4sDGFXLTjIejsG+tq2rlGvEKssXnOYY/6uAqJrsg0Jc8zaDq1mILfx5ojiWlBYS1hqS+JZJWW+tBLqqMve+FB/gRrTs+lU8wVeGaXtI16gCrYoQN6QNv0Fe3j+SWi2hbU4rzds1HgWuKMGkFiILusbHXUJPY9WS1XTiL3TUQeVYfra9n5hpX5Fpr3gT0qiQHrdyUSHIXTJVGYrvPFOovsLTKjrKqVrK/cald8HEomlnemDKt+LLHj3DTRUbJv5Ra07Z/sT6/UY6NPL41Bkcfca2Kpw3IPuEsPUamXpQoqV72JAneWXKrOEx9ckYnFp5ZoNXnPPLOrXLfVbD9A9BT0R7bzFKiCf5lr/Ct221jetF2ohXeQyde1imeDvkelHOL5HJXzHc5noGxJXlc7a9L55nfP1O4rvx05O5h2kcv5OX9VOuikVBe/q0aSs4u0XUhU0RdJpHPbtyN5q3elLjo4y7FvpWLNiDMil/lyTHIFdb8CbjyDGlfCzNoaV7J1NoM6q4SZtXVWydZwBjVcivGQnQZT1LknfT8X+jLszgLsTgXs9gLsdgXsywXYlxbLtb4lMQik0yhUoZHcsd9ervSzVLRk0gFz+6xdC6vReUzWF6Pr4hOL35POM1/DLp6+re0t++0U4spsdfGXly8R6OBezCa2Y1zM16Rvmk7aEJ+UvGOnu2hjTse/vkd96SHFKoOpeeHVNOV/+4W/jvXZf4yrgxcbrfWHrfYP7dWnj+z/x236lD6nL9DVHtFT5NwuvnE8+ol+pd/pj/rd+uP6k/ozI3rzhsXco0tH/Zv/AMgUsOs=</latexit><latexit sha1_base64="vzOfIEl6F2H5JEXT3hN+AR3QK0Q=">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</latexit><latexit sha1_base64="2qyNco5mbgnVsYdfITFP7ho9UtU=">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</latexit>↵• Chung’s dress. We have S↵soft
0 ,↵ = 1.
<latexit sha1_base64="z9NAJagAY3t5KE6FoZPe7dhSkZc=">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</latexit>

No decoherence• Other appropriate dresses S↵soft
0 ,↵ 6= 0 . We can have interference terms.
<latexit sha1_base64="RyRO3FGSFc89g1BifP/V0GwI/As=">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</latexit>

Allowed dress to have all interference terms
0 ,↵ 6= 0• Only when ↵, ↵ have the same asymptotic charges, S↵soft<latexit sha1_base64="Peh0BhwupThHNB42nbMF2VE5iKQ=">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</latexit>

0<latexit sha1_base64="RyRO3FGSFc89g1BifP/V0GwI/As=">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</latexit>• The decoherence is just the superselection rule.
X<latexit sha1_base64="HbtbcjeIysDkAhL9dMqnVbRndwY=">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</latexit>ifi eCi |↵i i• To have the same asympt charges, the dress should be
R↵i + C + o(k 1 )<latexit sha1_base64="EM12Ta6zV8C97i2meuUiPj4HRdY=">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</latexit>

Ci =Chung’s dress<latexit sha1_base64="tOpuDRaLZM6u1OQPAQX9tnwDtaQ=">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</latexit>

A
R↵
=
i

X

Common to all 𝑖pn · ✏ Aen
pn · kn2↵iSoft graviton
l Gravity also has the IR problem.
(soft graviton theorem is similar to soft photon theorem)l We should use states dressed by soft gravitons to obtain IR-safe S-matrix.
l We expect that there is a decoherence problem similar to QED.
l The decoherence is due to the IR divergence (or superselection rule of gravity),and so we have to be careful when we discuss decoherence in quantum gravity.

Conclusion

Future problemsl Extension to gravity & non-Abelian gauge theoriesl Development of formalism using dressed states (e.g., dressed wave-packet)l Subleading computations (work in progress)

Summaryl The asympt sym plays an important role in the soft sector of QED.
l Don’t use the Fock states if you want finite S-matrix elements.
l IR problem is not a problem, and IR divergences are necessary for symmetry.
l Transition occurs only when Qas,↵ = Qas,<latexit sha1_base64="tQPIUFggwuaCoF7uwp3xbRk9/pA=">AAANgHicnVfNbttGEJ44bZqoVeokx16EGkZ9UBTKEPIHGEgRt/AhrR3Ejo1EgUHSK5kQ/0BSiWVCr5D3KHrttT33BXLoI/Tcaw/9ZnZpy7JE0SVBcndmvpnZ2Zkh6cS+l2aW9de1peuffX7ji5u3al9+Vb/99fKdu6/TaJi4as+N/Cg5cOxU+V6o9jIv89VBnCg7cHy17wyeM3//vUpSLwp3s1Gs3gV2P/R6nmtnIB0ur708zLuZOslyOx03u7YfH9vjjYtER2X2+HB5xWpZcjQuD9pmsELm2Inu3GhQl44oIpeGFJCikDKMfbIpxfmW2mRRDNo7ykFLMPKEr2hMNVoFegg5BRkb9AHufcxYtkMfIHtkJGfLvTXUEHO2noodF/74uBIgG7RqfbJ+sf6x/rR+tf62/p2ri20O4d8xOUAPSqzmkBmBpmAhwb1X6mEGT2LaoHXEIoP8ENQAZxN2IlCYHwh/fYrvi+5MeJ0pXoLY9OHHLC7HgG1HEhPmjEpXk4pUauhFvBOJq8IuuKIpADfEfuTgObK/Ch56IpOKFSV7fwx8jrW2cI0hG8pOMNqHnEenkGP92qovVvoSd9YcS5ak8M7DmDV1JH/GCzGeeKczUHuwPoGct+MB6Owb62qaeYp4BVhj84zDHvVwFRKOyDQlzzNoOjGYgt/DmkOJaUFhLUGpL7FkVSr1kS6ojp7shQv5Y6w5OZNOMFfg6V1KTdR9VME27dF92qIfaBfPJ4hqR1Dz8/aViQLXEmfUEBJ92WVlqqMmsT+S1XLlxHLfQORRdbh+lJ1vGJkX0LwL7EFJDFi/LZHgKJwsiMJknc/XWWRvkRllVa1kfeVWHfFxIJlY3pkyrPqxxI5zU0dHyb6VV6PunuzNUamk1pYujECRxdxpIqnC8hVyjx6gTi9KFVSueR0D7iu51JwrPtgiE4lPLdGqs555ek+5aqvZvo/YKegPTd8pUAX/Ilf7V+y1trxhelALbyCdrasVzwb9jDrZx/M56uYnnN+DsilZXe2sSd+b3TsTs6/8buTcYNp5Judn/BXpn+NSXfymGkrGztN2LlFFXyiRzk3XDuWd7khVdHGWY99LveoRZ0Qu88WY+BLquwq40RRqVAkzbWtUydbpFOq0Emba1mklW4Mp1GAhxkV2akxR5650/Vzoi7Dbc7DbFbBbc7BbFbBv5mDfGCzX+qbEwJdOo1CFWnLbfHnZ0s8S0ZJJB8zNs3YlbIrOo7O+GF0VHxv8K+k8szXs4OmZ2t40X04BrsxUF393eRKBLu7FbGw6xvl8Vfqm7qQN8UnJG3ayizZmdPyre9STHlKs0p+YF15NUv63X/jnaE//YVwevF5vtR+2Oi87K88em7+Pm/QNfUtr6GqP6BlybgdfOC59pN/od/qjvlRfqz+ot7Xo0jWDuUcXjvrT/wCu96tE</latexit>

Choose your dresses fitting with this conservation law.

THANK YOU