Secularly growing loop corrections in strong background fjeld - - PowerPoint PPT Presentation

Secularly growing loop corrections in strong background fjeld

❆❦❤♠❡❞♦✈ ❊✳ ❚✳✱ ❇✉r❞❛ P✳✱ P♦♣♦✈ ❋✳✱ ❙❛❞♦❢②❡✈ ❆✳ ❛♥❞ ❙❧❡♣✉❦❤✐♥ ❱✳ ▼■❚P✱ ▼❛✐♥③ ✭❏✉♥❡ ✷✵✶✺✮

Motivation

❙❡❝✉❧❛r ❣r♦✇t❤ ♦❢ ❧♦♦♣ ❝♦rr❡❝t✐♦♥s ✐s ♣r❛❝t✐❝❛❧❧② ✐♥❡✈✐t❛❜❧❡ ✐♥ ♥♦♥✕st❛t✐♦♥❛r② s✐t✉❛t✐♦♥s ✭▲❛♥❞❛✉ ❛♥❞ ▲✐❢s❤✐t③✱ ❳✲t❤ ✈♦❧✉♠❡✮ ❚❤✐s ❣r♦✇t❤ ✐s t❤❡ ■❘ ❡✛❡❝t✳ ◆♦ ♠♦❞✐✜❝❛t✐♦♥s ♦❢ ❯❱ ♣❤②s✐❝s✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❛r❡ ♦❢ t❤❡ s❛♠❡ ♦r❞❡r ❛s ❝❧❛ss✐❝❛❧ ❝♦♥tr✐❜✉t✐♦♥s✱ ✐❢ ♦♥❡ ✇❡✐❣❤ts ❧♦♥❣ ❡♥♦✉❣❤✳ ❞❡ ❙✐tt❡r s♣❛❝❡ ✐♥t❡r❛❝t✐♥❣ ◗❋❚ ✭r❡✈✐❡✇ ❛r❳✐✈✿✶✸✵✾✳✷✺✺✼✮✳ ◗❊❉ ♦♥ str♦♥❣ ❡❧❡❝tr✐❝ ✜❡❧❞ ❜❛❝❦❣r♦✉♥❞ ❜❡②♦♥❞ t❤❡ ❜❛❝❦❣r♦✉♥❞ ✜❡❧❞ ❛♣♣r♦①✐♠❛t✐♦♥ ✭❛r❳✐✈✿✶✹✵✺✳✺✷✷✺✮✳ ▲♦♦♣ ❝♦rr❡❝t✐♦♥ t♦ ❍❛✇❦✐♥❣ r❛❞✐❛t✐♦♥ ✭❛r❳✐✈✿✶✺✵✾✳ ✳✳✳✮✳

2 / 21

Motivation

❙❡❝✉❧❛r ❣r♦✇t❤ ♦❢ ❧♦♦♣ ❝♦rr❡❝t✐♦♥s ✐s ♣r❛❝t✐❝❛❧❧② ✐♥❡✈✐t❛❜❧❡ ✐♥ ♥♦♥✕st❛t✐♦♥❛r② s✐t✉❛t✐♦♥s ✭▲❛♥❞❛✉ ❛♥❞ ▲✐❢s❤✐t③✱ ❳✲t❤ ✈♦❧✉♠❡✮ ❚❤✐s ❣r♦✇t❤ ✐s t❤❡ ■❘ ❡✛❡❝t✳ ◆♦ ♠♦❞✐✜❝❛t✐♦♥s ♦❢ ❯❱ ♣❤②s✐❝s✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❛r❡ ♦❢ t❤❡ s❛♠❡ ♦r❞❡r ❛s ❝❧❛ss✐❝❛❧ ❝♦♥tr✐❜✉t✐♦♥s✱ ✐❢ ♦♥❡ ✇❡✐❣❤ts ❧♦♥❣ ❡♥♦✉❣❤✳ ❞❡ ❙✐tt❡r s♣❛❝❡ ✐♥t❡r❛❝t✐♥❣ ◗❋❚ ✭r❡✈✐❡✇ ❛r❳✐✈✿✶✸✵✾✳✷✺✺✼✮✳ ◗❊❉ ♦♥ str♦♥❣ ❡❧❡❝tr✐❝ ✜❡❧❞ ❜❛❝❦❣r♦✉♥❞ ❜❡②♦♥❞ t❤❡ ❜❛❝❦❣r♦✉♥❞ ✜❡❧❞ ❛♣♣r♦①✐♠❛t✐♦♥ ✭❛r❳✐✈✿✶✹✵✺✳✺✷✷✺✮✳ ▲♦♦♣ ❝♦rr❡❝t✐♦♥ t♦ ❍❛✇❦✐♥❣ r❛❞✐❛t✐♦♥ ✭❛r❳✐✈✿✶✺✵✾✳ ✳✳✳✮✳

2 / 21

Motivation

❙❡❝✉❧❛r ❣r♦✇t❤ ♦❢ ❧♦♦♣ ❝♦rr❡❝t✐♦♥s ✐s ♣r❛❝t✐❝❛❧❧② ✐♥❡✈✐t❛❜❧❡ ✐♥ ♥♦♥✕st❛t✐♦♥❛r② s✐t✉❛t✐♦♥s ✭▲❛♥❞❛✉ ❛♥❞ ▲✐❢s❤✐t③✱ ❳✲t❤ ✈♦❧✉♠❡✮ ❚❤✐s ❣r♦✇t❤ ✐s t❤❡ ■❘ ❡✛❡❝t✳ ◆♦ ♠♦❞✐✜❝❛t✐♦♥s ♦❢ ❯❱ ♣❤②s✐❝s✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❛r❡ ♦❢ t❤❡ s❛♠❡ ♦r❞❡r ❛s ❝❧❛ss✐❝❛❧ ❝♦♥tr✐❜✉t✐♦♥s✱ ✐❢ ♦♥❡ ✇❡✐❣❤ts ❧♦♥❣ ❡♥♦✉❣❤✳ ❞❡ ❙✐tt❡r s♣❛❝❡ ✐♥t❡r❛❝t✐♥❣ ◗❋❚ ✭r❡✈✐❡✇ ❛r❳✐✈✿✶✸✵✾✳✷✺✺✼✮✳ ◗❊❉ ♦♥ str♦♥❣ ❡❧❡❝tr✐❝ ✜❡❧❞ ❜❛❝❦❣r♦✉♥❞ ❜❡②♦♥❞ t❤❡ ❜❛❝❦❣r♦✉♥❞ ✜❡❧❞ ❛♣♣r♦①✐♠❛t✐♦♥ ✭❛r❳✐✈✿✶✹✵✺✳✺✷✷✺✮✳ ▲♦♦♣ ❝♦rr❡❝t✐♦♥ t♦ ❍❛✇❦✐♥❣ r❛❞✐❛t✐♦♥ ✭❛r❳✐✈✿✶✺✵✾✳ ✳✳✳✮✳

2 / 21

Motivation

❙❡❝✉❧❛r ❣r♦✇t❤ ♦❢ ❧♦♦♣ ❝♦rr❡❝t✐♦♥s ✐s ♣r❛❝t✐❝❛❧❧② ✐♥❡✈✐t❛❜❧❡ ✐♥ ♥♦♥✕st❛t✐♦♥❛r② s✐t✉❛t✐♦♥s ✭▲❛♥❞❛✉ ❛♥❞ ▲✐❢s❤✐t③✱ ❳✲t❤ ✈♦❧✉♠❡✮ ❚❤✐s ❣r♦✇t❤ ✐s t❤❡ ■❘ ❡✛❡❝t✳ ◆♦ ♠♦❞✐✜❝❛t✐♦♥s ♦❢ ❯❱ ♣❤②s✐❝s✳ ◗✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❛r❡ ♦❢ t❤❡ s❛♠❡ ♦r❞❡r ❛s ❝❧❛ss✐❝❛❧ ❝♦♥tr✐❜✉t✐♦♥s✱ ✐❢ ♦♥❡ ✇❡✐❣❤ts ❧♦♥❣ ❡♥♦✉❣❤✳ ❞❡ ❙✐tt❡r s♣❛❝❡ ✐♥t❡r❛❝t✐♥❣ ◗❋❚ ✭r❡✈✐❡✇ ❛r❳✐✈✿✶✸✵✾✳✷✺✺✼✮✳ ◗❊❉ ♦♥ str♦♥❣ ❡❧❡❝tr✐❝ ✜❡❧❞ ❜❛❝❦❣r♦✉♥❞ ❜❡②♦♥❞ t❤❡ ❜❛❝❦❣r♦✉♥❞ ✜❡❧❞ ❛♣♣r♦①✐♠❛t✐♦♥ ✭❛r❳✐✈✿✶✹✵✺✳✺✷✷✺✮✳ ▲♦♦♣ ❝♦rr❡❝t✐♦♥ t♦ ❍❛✇❦✐♥❣ r❛❞✐❛t✐♦♥ ✭❛r❳✐✈✿✶✺✵✾✳ ✳✳✳✮✳

2 / 21

Adiabatic catastrophe

❙✉♣♣♦s❡ ♦♥❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✜♥❞✿ ⟨𝒫⟩t0 (t) = ⟨ Ψ ⃒ ⃒ ⃒T ei

∫︁ t

t0 dt′H(t′) 𝒫 T e−i

∫︁ t

t0 dt′H(t′)⃒

⃒ ⃒ Ψ ⟩ , ✭✶✮ ❡✳❣✳ ⟨T𝜈𝜉⟩ ♦r ⟨J𝜈⟩✳ ❍❡r❡ H(t) = H0(t) + V (t)✳ T ✖ t✐♠❡✕♦r❞❡r✐♥❣✱ T ✖ ❛♥t✐✕t✐♠❡✕♦r❞❡r✐♥❣✳ t0 ✖ ✐♥✐t✐❛❧ ♠♦♠❡♥t ♦❢ t✐♠❡✱ |Ψ⟩ ✖ ✐♥✐t✐❛❧ st❛t❡✱ ⟨Ψ |𝒫| Ψ⟩ (t0) ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ❣✐✈❡♥✳

3 / 21

Adiabatic catastrophe

❚r❛♥s❢❡rr✐♥❣ t♦ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♣✐❝t✉r❡✿ ⟨𝒫⟩t0 (t) = ⟨︁ Ψ ⃒ ⃒S+(+∞, t0) T [𝒫0(t) S(+∞, t0)] ⃒ ⃒ Ψ ⟩︁ . ✭✷✮ ❍❡r❡ S(t2, t1) = T e−i

∫︁ t2

t1 dt′V0(t′)❀ 𝒫0(t) ❛♥❞ V0(t) ❛r❡

♦♣❡r❛t♦rs ✐♥ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♣✐❝t✉r❡✳ ❙❧✐❣❤t❧② ❝❤❛♥❣✐♥❣ t❤❡ ♣r♦❜❧❡♠✿ ⟨𝒫⟩t0 (t) = ⟨︁ Ψ ⃒ ⃒S+

t0(+∞, −∞) T [𝒫0(t) St0(+∞, −∞)]

⃒ ⃒ Ψ ⟩︁ . ✭✸✮ ❍❡r❡ t0 ✐s t❤❡ t✐♠❡ ♠♦♠❡♥t ❛❢t❡r ✇❤✐❝❤ t❤❡ ✐♥t❡r❛❝t✐♦♥s✱ V (t)✱ ❛r❡ ❛❞✐❛❜❛t✐❝❛❧❧② t✉r♥❡❞ ♦♥✳

4 / 21

Discussion

❲❤❡♥ ❞♦❡s t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t0 ❞✐s❛♣♣❡❛r❄ ❖t❤❡r✇✐s❡ ✇❡ ❤❛✈❡ ❛❞✐❛❜❛t✐❝ ❝❛t❛str♦♣❤❡ ❛♥❞ ❜r❡❛❦✐♥❣ ♦❢ ✈❛r✐♦✉s s②♠♠❡tr✐❡s✿ ❊✳❣✳ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s st♦♣ t♦ ❞❡♣❡♥❞ ♦♥❧② ♦♥ |t1 − t2|✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t0 ❞✐s❛♣♣❡❛rs ✇❤❡♥ t❤❡ s✐t✉❛t✐♦♥ ✐s ♦r ❜❡❝♦♠❡s st❛t✐♦♥❛r②✳

5 / 21

Discussion

❚❤❡ s❡♠✐♥❛❧ ❡①❛♠♣❧❡ ♦❢ t❤❡ st❛t✐♦♥❛r② s✐t✉❛t✐♦♥ ✐s ✇❤❡♥ t❤❡ ❢r❡❡ ❍❛♠✐❧t♦♥✐❛♥ H0 ✐s t✐♠❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ❤❛s ❛ s♣❡❝tr✉♠ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇✿ H0 |vac⟩ = ✵ ❛♥❞ |𝜔⟩ = |vac⟩✳ ■♥ ❢❛❝t✱ ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡ ❜② ❛❞✐❛❜❛t✐❝ t✉r♥✐♥❣ ♦♥ ❛♥❞ t❤❡♥ s✇✐t❝❤✐♥❣ ♦✛ V (t) ✇❡ ❞♦ ♥♦t ❞✐st✉r❜ t❤❡ ❣r♦✉♥❞ st❛t❡✿ ⟨︁ vac ⃒ ⃒S+(+∞, −∞) ⃒ ⃒ excited state ⟩︁ = ✵, ✇❤✐❧❡ ⃒ ⃒⟨︁ vac ⃒ ⃒S+(+∞, −∞) ⃒ ⃒ vac ⟩︁⃒ ⃒ = ✶. ■t ❞♦❡s ♥♦t ♠❛tt❡r ✇❤❡♥ ♦♥❡ t✉r♥s ♦♥ ✐♥t❡r❛❝t✐♦♥s✳ ❚❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t0 ❞✐s❛♣♣❡❛r❡❞✦

6 / 21

Stationary case

❋✉rt❤❡r♠♦r❡✱ ✐♥ t❤❡ ❧❛tt❡r ❝❛s❡ ✇❡ ♦❜t❛✐♥✿ ⟨𝒫⟩ (t) = ∑︂

sta

⟨︁ vac ⃒ ⃒S+(+∞, −∞) ⃒ ⃒ sta ⟩︁ ⟨sta |T [𝒫0(t) S(+∞, −∞)]| vac⟩ = = ⟨︁ vac ⃒ ⃒S+(+∞, −∞) ⃒ ⃒ vac ⟩︁ ⟨vac |T [𝒫0(t) S(+∞, −∞)]| vac⟩ = = ⟨vac |T [𝒫0(t) S(+∞, −∞)]| vac⟩ ⟨vac |S(+∞, −∞)| vac⟩ . ❚❤✐s ✇❛② ✇❡ ❛rr✐✈❡ ❛t ❤❛✈✐♥❣ ♦♥❧② t❤❡ T✕♦r❞❡r❡❞ ❡①♣r❡ss✐♦♥s ❛♥❞ t❤❡♥ ❝❛♥ ✉s❡ ❋❡②♥♠❛♥ t❡❝❤♥✐q✉❡✳ ❖t❤❡r s✐t✉❛t✐♦♥ ✇❤❡♥ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t0 ❞✐s❛♣♣❡❛rs ✐❢ t❤❡r❡ ✐s ❛ st❛t✐♦♥❛r② st❛t❡ ✭❡✳❣✳ t❤❡r♠❛❧ ❞❡♥s✐t② ♠❛tr✐① ✐♥ ✢❛t s♣❛❝❡✕t✐♠❡✮✳

7 / 21

Discussion

■s t❤❡r❡ ❛ st❛t✐♦♥❛r② st❛t❡ ✐❢ ❛ ❜❛❝❦❣r♦✉♥❞ ✜❡❧❞ ✐s ♥❡✈❡r s✇✐t❝❤❡❞ ♦✛❄ ❲❤❛t ✐s t❤❛t st❛t❡✱ ✐❢ ✐t ✐s ♣r❡s❡♥t❄ ❲❤❛t ✐❢ t❤❡r❡ ✐s ♥♦ st❛t✐♦♥❛r② st❛t❡❄ ❍♦✇ ❞♦❡s t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t0 r❡✈❡❛❧s ✐ts❡❧❢❄ t0 ❞♦❡s ♥♦t ❛♣♣❡❛r ✐♥ ❯❱ r❡♥♦r♠❛❧✐③❛t✐♦♥✦ ■♥ ❯❱ ❧✐♠✐t ♦♥❡ ❛❧✇❛②s ❝❛♥ ✉s❡ t❤❡ ❋❡②♥♠❛♥ t❡❝❤♥✐q✉❡✱ ❜❡❝❛✉s❡ ❤✐❣❤ ❢r❡q✉❡♥❝② ♠♦❞❡s ❛r❡ ♥♦t s❡♥s✐t✐✈❡ t♦ ❜❛❝❦❣r♦✉♥❞ ✜❡❧❞s✳ ❚♦ ❛♥s✇❡r t❤❡ ❛❜♦✈❡ q✉❡st✐♦♥s ♦♥❡ ❤❛s t♦ ❝❛❧❝✉❧❛t❡ ❞✐r❡❝t❧②✿ ⟨𝒫⟩t0 (t) = ⟨︁ Ψ ⃒ ⃒S+

t0(+∞, −∞) T [𝒫0(t) St0(+∞, −∞)]

⃒ ⃒ Ψ ⟩︁ ✭✹✮ ❢♦r ✈❛r✐♦✉s ❝❤♦✐❝❡s ♦❢ 𝒫✳

8 / 21

Non–stationary case

❙❝❤✇✐♥❣❡r ♥♦t❛t✐♦♥s✿ S ✖ ✏+✑ ✈❡rt❡①❡s✱ S+ ✖ ✏−✑ ✈❡rt❡①❡s✿ D++(✶, ✷) = ⟨ Ψ ⃒ ⃒ ⃒T (︂ 𝜒(✶) 𝜒(✷) )︂⃒ ⃒ ⃒ Ψ ⟩ , D−−(✶, ✷) = ⟨ Ψ ⃒ ⃒ ⃒T (︂ 𝜒(✶) 𝜒(✷) )︂⃒ ⃒ ⃒ Ψ ⟩ , D+−(✶, ✷) = ⟨Ψ |𝜒(✶) 𝜒(✷)| Ψ⟩ , D−+(✶, ✷) = ⟨Ψ |𝜒(✷) 𝜒(✶)| Ψ⟩ . ✭✺✮ ❊✈❡r② ✜❡❧❞ ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❛ ♠❛tr✐① ♦❢ ♣r♦♣❛❣❛t♦rs✳

9 / 21

Non–stationary case

❆❢t❡r ❑❡❧❞②s❤✬s r♦t❛t✐♦♥ ♦❢ 𝜒+ ❛♥❞ 𝜒−✱ ✇❡ ♦❜t❛✐♥✿ DR,A(✶, ✷) = 𝜄 (±∆t1,2) (︂ D+−(✶, ✷) − D−+(✶, ✷) )︂ = = 𝜄 (±∆t1,2) [︂ 𝜒(✶) , 𝜒(✷) ]︂ ✭✻✮ ✖ st❛t❡ ✐♥❞❡♣❡♥❞❡♥t ❘❡t❛r❞❡❞ ❛♥❞ ❆❞✈❛♥❝❡❞ ♣r♦♣❛❣❛t♦rs✳ ❚❤❡② ❝❤❛r❛❝t❡r✐③❡ ♦♥❧② t❤❡ s♣❡❝tr✉♠ ♦❢ ❡①❝✐t❛t✐♦♥s✳ ❚❤❡ ❑❡❧❞②s❤ ♣r♦♣❛❣❛t♦r✿ DK(✶, ✷) = ✶ ✷ (︂ D+−(✶, ✷) + D−+(✶, ✷) )︂ = = ✶ ✷ ⟨ Ψ ⃒ ⃒ ⃒ {︂ 𝜒(✶) , 𝜒(✷) }︂⃒ ⃒ ⃒ Ψ ⟩ . ✭✼✮

10 / 21

Discussion

■❢ ✇❡ ❤❛✈❡ s♣❛t✐❛❧❧② ❤♦♠♦❣❡♥❡♦✉s ♥♦♥✕st❛t✐♦♥❛r② st❛t❡✿ 𝜒(t,⃗ x) = ∫︁ dD−1⃗ p (︁ a⃗

p ei ⃗ p ⃗ x gp(t) + h.c.

)︁ ✱ ❢♦r t❤❡ ❝❛s❡ ♦❢ r❡❛❧ s❝❛❧❛r ✜❡❧❞✱ t❤❡♥ ∫︂ dD−1⃗ p e−i ⃗

p (⃗ x1−⃗ x2) DK (t1, t2, |⃗

x1 − ⃗ x2|) ≡ DK

p (t1, t2) =

= (︃✶ ✷ + ⟨ a+

⃗ p a⃗ p

⟩)︃ gp(t1) g∗

p(t2) +

⟨︁ a⃗

p a−⃗ p

⟩︁ gp(t1) gp(t2) + c.c. ✖ ❝❛rr✐❡s ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❜❛❝❦❣r♦✉♥❞ st❛t❡✦ ■♥ ◗❊❉✱ ❣❧♦❜❛❧ ❞❡ ❙✐tt❡r ❛♥❞ ❜❧❛❝❦ ❤♦❧❡ ❝♦❧❧❛♣s❡ ❝❛s❡ t❤❡ ❢♦r♠✉❧❛s ❛r❡ ❛ ❜✐t ❞✐✛❡r❡♥t✱ ❜✉t t❤❡ s✐t✉❛t✐♦♥ ✐s ❝♦♥❝❡♣t✉❛❧❧② t❤❡ s❛♠❡✳

11 / 21

Discussion

■♥ ❛ ❢r❡❡ t❤❡♦r② ⟨ a+

⃗ p a⃗ p

⟩ = const, ⟨︁ a⃗

p a−⃗ p

⟩︁ = const✳ ❆❧❧ t✐♠❡ ❞❡♣❡♥❞❡♥❝❡ ✐s ❣♦♥❡ ✐♥t♦ ❤❛r♠♦♥✐❝ ❢✉♥❝t✐♦♥s ✖ gp(t)✳ ■❢ t❤❡ ✐♥✐t✐❛❧ st❛t❡ ✐s t❤❡ ❣r♦✉♥❞ ♦♥❡✿ |Ψ⟩ = |ground⟩ ❛♥❞ ap |ground⟩ = ✵✱ ✇❡ ♦❜✈✐♦✉s❧② ❤❛✈❡ t❤❛t ⟨ a+

⃗ p a⃗ p

⟩ = ⟨︁ a⃗

p a−⃗ p

⟩︁ = ✵. ❆❧❧ q✉❛s✐✕❝❧❛ss✐❝❛❧ r❡s✉❧ts ✭♥♦♥✕✐♥t❡r❛❝t✐♥❣ ✜❡❧❞s✱ ❜❛❝❦❣r♦✉♥❞ ✜❡❧❞ ❛♣♣r♦①✐♠❛t✐♦♥✮ ❢♦❧❧♦✇ ❢r♦♠ t❤❡ tr❡❡✕❧❡✈❡❧ ♣r♦♣❛❣❛t♦r✿ DK

p (t1, t2) = ✶

✷ (︂ gp(t1) g∗

p(t2)

+ g∗

p(t1) gp(t2)

)︂ . ✭✽✮ ❊✳❣✳ ⟨T𝜈𝜉⟩0 ✐♥ ❞❡ ❙✐tt❡r s♣❛❝❡ ❛♥❞ ❜❧❛❝❦ ❤♦❧❡ ❝♦❧❧❛♣s❡✱ ❛♥❞ ⟨J𝜈⟩0 ✐♥ ◗❊❉✳

12 / 21

Secular growth of loop corrections

❍♦✇❡✈❡r✱ ✐❢ ♦♥❡ t✉r♥s ♦♥ ✐♥t❡r❛❝t✐♦♥s✱ t❤❡♥ ⟨ a+

⃗ p a⃗ p

⟩ ❛♥❞ ⟨︁ a⃗

p a−⃗ p

⟩︁ st❛rt t♦ ❞❡♣❡♥❞ ♦♥ t✐♠❡✳ ❙❛② ❢♦r 𝜇𝜒3 ✭♦r 𝜇𝜒4✮ t❤❡♦r② ❛t ❧♦♦♣ ❧❡✈❡❧✱ ❛s t = t1+t2

2

→ +∞✱ ✇❡ ♦❜t❛✐♥ t❤❛t DK

p (t1, t2) =

(︃✶ ✷ + np(t) )︃ gp(t1) g∗

p(t2) + 𝜆p(t) gp(t1) gp(t2) + c.c..

❆t ♦♥❡ ❧♦♦♣ ❧❡✈❡❧ np(t) ∝ 𝜇2 ∫︂ dD−1⃗ q1 ∫︂ dD−1⃗ q2 ∫︂∫︂ t

t0

dt3 dt4 𝜀 (⃗ p + ⃗ q1 + ⃗ q2) × ×g∗

p (t3) gp (t4) g∗ q1(t3) gq1(t4) g∗ q2(t3) gq2(t4) + O (t1 − t2),

𝜆p(t) ∝ −𝜇2 ∫︂ dD−1⃗ q1 ∫︂ dD−1⃗ q2 ∫︂∫︂ t

t0

dt3 dt4 𝜀 (⃗ p + ⃗ q1 + ⃗ q2) × ×g∗

p (t3) g∗ p (t4) g∗ q1(t3) gq1(t4) g∗ q2(t3) gq2(t4) + O (t1 − t2).

13 / 21

Secular growth of loop corrections

■❢ t❤❡r❡ ✐s ♥♦ ❜❛❝❦❣r♦✉♥❞ ✜❡❧❞✱ t❤❡♥ gp ∝ e−i 𝜗(p) t √

𝜗(p) ❛♥❞

np(t) ∝ 𝜇2 (t − t0) ∫︂ dD−1⃗ q1 ∫︂ dD−1⃗ q2𝜀 (⃗ p + ⃗ q1 + ⃗ q2) × ×𝜀 (︂ 𝜗(p) + 𝜗(q1) + 𝜗(q2) )︂ . ✭✾✮ ❍❡♥❝❡✱ np(t) = ✵ ❞✉❡ t♦ ❡♥❡r❣② ❝♦♥s❡r✈❛t✐♦♥✳ ❚❤❡r❡ ✐s ♥♦ ❡♥❡r❣② ❝♦♥s❡r✈❛t✐♦♥ ✐♥ t✐♠❡✕❞❡♣❡♥❞❡♥t ❜❛❝❦❣r♦✉♥❞ ✜❡❧❞s ✭♦r ❡♥❡r❣② ✐s ♥♦t ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇✮✱ t❤❡♥ ✇❡ ❣❡♥❡r✐❝❛❧❧② ♦❜t❛✐♥✿ np(t) ∝ 𝜇2 (t − t0) × (production rate), 𝜆p(t) ∝ −𝜇2 (t − t0) × (backreaction rate). ✭✶✵✮ ❚❤❡ ❘❍❙ ✐s t❤❡ ❝♦❧❧✐s✐♦♥ ✐♥t❡❣r❛❧✳

14 / 21

Explicit examples (QED)

■♥ ◗❊❉ ✇✐t❤ ⃗ E = const ❢♦r♠✉❧❛s ❛ ❜✐t ❞✐✛❡r❡♥t✳ ❍❛r♠♦♥✐❝s ❛r❡ gp(t) = g(p + eEt)✳ ❆❧❧ ❡①♣r❡ss✐♦♥s ❛r❡ ✐♥✈❛r✐❛♥t ✉♥❞❡r p → p + a ❛♥❞ t → t − a/eE✳ ❆s t❤❡ r❡s✉❧t✱ ❜❡②♦♥❞ t❤❡ ❜❛❝❦❣r♦✉♥❞ ✜❡❧❞ ❛♣♣r♦①✐♠❛t✐♦♥✱ ❢♦r ♣❤♦t♦♥s ✇❡ ♦❜t❛✐♥ t❤❛t✿ np(t) ∝ e2 (t − t0) × (production rate), 𝜆p(t) = ✵. ✭✶✶✮ ❇❡❝❛✉s❡ ♦❢ t❤❛t t0 ❝❛♥♥♦t ❜❡ t❛❦❡♥ t♦ ♣❛st ✐♥✜♥✐t②✳ ❍❡♥❝❡✱ ✇❡ ❤❛✈❡ ❛❞✐❛❜❛t✐❝ ❝❛t❛str♦♣❤❡✳ ❋♦r ❝❤❛r❣❡❞ ✜❡❧❞s np ❛♥❞ 𝜆p ❛r❡ t✐♠❡✕❞❡♣❡♥❞❡♥t✱ ❜✉t ❞♦ ♥♦t ❣r♦✇ ❛s t − t0 → ∞✳

15 / 21

Explicit examples (de Sitter, expanding patch)

■♥ ❡①♣❛♥❞✐♥❣ P♦✐♥❝❛r❡ ♣❛t❝❤✿ gp(t) = 𝜃

D−1 2

h(p𝜃)✱ ✇❤❡r❡ 𝜃 = e−t ❛♥❞ h(p𝜃) ✐s ❛ ❇❡ss❡❧ ❢✉♥❝t✐♦♥✳ ❚❤❡r❡ ✐s ✐♥✈❛r✐❛♥❝❡ ✉♥❞❡r p → p a ❛♥❞ 𝜃 → 𝜃/a✳ ❋♦r t❤❡ ❝❛s❡ ♦❢ ♠❛ss✐✈❡ s❝❛❧❛rs✱ m > D−1

2 ✱ ✐♥ t❤❡ ❧✐♠✐t p𝜃 → ✵✱

✇❡ ♦❜t❛✐♥ t❤❛t np(𝜃) ∝ 𝜇2 ❧♦❣ (︃ m p𝜃 )︃ × (production rate), 𝜆p(𝜃) ∝ −𝜇2 ❧♦❣ (︃ m p𝜃 )︃ × (backreaction rate). ✭✶✷✮ ◆♦ ❞✐✈❡r❣❡♥❝❡✱ ❜✉t t❤❡r❡ ✐s s❡❝✉❧❛r ❣r♦✇t❤✳

16 / 21

Explicit examples (de Sitter, contracting patch)

❈♦♥tr❛❝t✐♥❣ P♦✐♥❝❛r❡ ♣❛t❝❤ ✐s ❥✉st t✐♠❡✕r❡✈❡rs❛❧ ♦❢ t❤❡ ❡①♣❛♥❞✐♥❣ ♦♥❡✳ ❋♦r t❤❡ ❝❛s❡ ♦❢ ♠❛ss✐✈❡ s❝❛❧❛rs✱ m > D−1

2 ✱ ✐♥ t❤❡ ❧✐♠✐t

p𝜃0 → ✵ ❛♥❞ p𝜃 → +∞✱ ✇❡ ♦❜t❛✐♥ t❤❛t np(𝜃) ∝ 𝜇2 ❧♦❣ (︃ m p𝜃0 )︃ × (production rate), 𝜆p(𝜃) ∝ −𝜇2 ❧♦❣ (︃ m p𝜃0 )︃ × (backreaction rate). ✭✶✸✮ ❍❡r❡ 𝜃0 = et0 ✐s t❤❡ t✐♠❡ ❛❢t❡r ✇❤✐❝❤ ✐♥t❡r❛❝t✐♦♥s ❛r❡ ❛❞✐❛❜❛t✐❝❛❧❧② t✉r♥❡❞ ♦♥✳ ■♥ t❤✐s ❝❛s❡ ✖ ■❘ ❞✐✈❡r❣❡♥❝❡ ❛♥❞✱ ❤❡♥❝❡✱ ❛❞✐❛❜❛t✐❝ ❝❛t❛str♦♣❤❡✳ ■♥ ❣❧♦❜❛❧ ❞❡ ❙✐tt❡r t❤❡r❡ ✐s ❛❧s♦ ❛❞✐❛❜❛t✐❝ ❝❛t❛str♦♣❤❡✳

17 / 21

Explicit examples (black hole collapse)

❍❛r♠♦♥✐❝s ❛r❡ ♠✉❝❤ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞✱ ❜✉t ❛t ❢✉t✉r❡ ✐♥✜♥✐t② t❤❡② ❞❡♣❡♥❞ ♦♥ 𝜕 e−t/2rg ✳ ■♥✈❛r✐❛♥❝❡ ✉♥❞❡r 𝜕 → 𝜕 a ❛♥❞ t → t + ✷rg ❧♦❣ a✳ ❆s t❤❡ r❡s✉❧t✱ ✐❢ t❤❡ ❝♦❧❧❛♣s❡ ❤❛❞ st❛rt❡❞ ❛t t = ✵✱ t❤❡♥ ✇❡ ♦❜t❛✐♥ np(t) ∝ 𝜇2 t × (production rate), 𝜆p(t) ∝ −𝜇2 t × (backreaction rate). ✭✶✹✮ ❈❤❛♥❣❡ ♦❢ t❤❡ ❍❛✇❦✐♥❣✬s t❤❡r♠❛❧ s♣❡❝tr✉♠❄ ■♥❢♦r♠❛t✐♦♥ ♣❛r❛❞♦①❄

18 / 21

Discussion

❲❤❛t s❤♦✉❧❞ ♦♥❡ ❞♦ ✇✐t❤ t❤❡s❡ ❣r♦✇✐♥❣ ✇✐t❤ t✐♠❡ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s❄ ◆♦t❡ t❤❛t ✐❢ ❜❛❝❦❣r♦✉♥❞ ✜❡❧❞ ✐s ♦♥ ❢♦r ❧♦♥❣ ❡♥♦✉❣❤✱ t❤❡♥ 𝜇2(t − t0) ∼ ✶ ❛♥❞ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ❛r❡ ♦❢ t❤❡ s❛♠❡ ♦r❞❡r ❛s ❝❧❛ss✐❝❛❧ ❝♦♥tr✐❜✉t✐♦♥s❀ np ∼ ✶ ✖ ❝❧❛ss✐❝❛❧ ❡✛❡❝ts✳ ❲❡ ♥❡❡❞ t♦ s✉♠ ✉♣ ❧❡❛❞✐♥❣ ❝♦rr❡❝t✐♦♥s ❢r♦♠ ❛❧❧ ❧♦♦♣s✿ s✉♠ (︁ 𝜇2(t − t0) )︁n ❛♥❞ ❞r♦♣ ♦✛ ❡✳❣✳ 𝜇4(t − t0) ≪ 𝜇2(t − t0)✳ ❉♦❡s t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t0 ❞✐s❛♣♣❡❛r ❛❢t❡r t❤❡ s✉♠♠❛t✐♦♥❄ ❲❡ ❞✐❞ t❤✐s s✉♠♠❛t✐♦♥ ✐♥ ❞❡ ❙✐tt❡r s♣❛❝❡ ✭❡①♣❛♥❞✐♥❣ ❛♥❞ ❝♦♥tr❛❝t✐♥❣ P♦✐♥❝❛r❡ ♣❛t❝❤❡s✮ ❛♥❞ ✐♥ ◗❊❉ ✇✐t❤ ❝♦♥st❛♥t ✜❡❧❞ ❜❛❝❦❣r♦✉♥❞✳

19 / 21

Summation of leading loop corrections

❚♦ ❞♦ t❤❡ s✉♠♠❛t✐♦♥ ♦♥❡ ❤❛s t♦ s♦❧✈❡ t❤❡ s②st❡♠ ♦❢ t❤❡ ❉②s♦♥✕❙❝❤✇✐♥❣❡r ❡q✉❛t✐♦♥s ❢♦r ♣r♦♣❛❣❛t♦rs ❛♥❞ ✈❡rt❡①❡s ✐♥ t❤❡ ■❘ ❧✐♠✐t✳ ■♥ ❛❧❧ t❤❡ ❛❜♦✈❡ ❧✐st❡❞ ❝❛s❡s ✈❡rt❡①❡s ❞♦ ♥♦t r❡❝❡✐✈❡ ❣r♦✇✐♥❣ ✇✐t❤ t✐♠❡ ❝♦rr❡❝t✐♦♥s✳ ❆❧s♦ r❡t❛r❞❡❞ ❛♥❞ ❛❞✈❛♥❝❡❞ ♣r♦♣❛❣❛t♦rs ❞♦ ♥♦t s❡❝✉❧❛r❧② ❣r♦✇✐♥❣ ❝♦rr❡❝t✐♦♥✳ ❍❡♥❝❡✱ t♦ s✉♠ ✉♣ ❧❡❛❞✐♥❣ ❝♦rr❡❝t✐♦♥s ✇❡ ♣✉t t❤❡♠ t♦ ❜❡ ♦❢ tr❡❡✕❧❡✈❡❧ ❢♦r♠✳ ❆♥s❛t③ ❢♦r t❤❡ ❑❡❧❞②s❤ ♣r♦♣❛❣❛t♦r✿ DK

p (t1, t2) =

(︃✶ ✷ + np(t) )︃ gp(t1) g∗

p(t2) + 𝜆p(t) gp(t1) gp(t2) + c.c..

❆s t❤❡ r❡s✉❧t ✇❡ ♦❜t❛✐♥ ❛ s②st❡♠ ♦❢ ❇♦❧t③♠❛♥♥ t②♣❡ ♦❢ ❡q✉❛t✐♦♥s ❢♦r np ❛♥❞ 𝜆p✳ ❙♦❧✉t✐♦♥ ♦❢ t❤❡s❡ ❡q✉❛t✐♦♥s✱ ✇✐t❤ s♣❡❝✐✜❡❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✱ s♦❧✈❡s t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ s✉♠♠❛t✐♦♥ ♦❢ s✉❝❤ ❝♦rr❡❝t✐♦♥s✳

20 / 21

Summation of leading loop corrections

❉②s♦♥✕❙❝❤✇✐♥❣❡r ❡q✉❛t✐♦♥s ❛r❡ ❝♦✈❛r✐❛♥t ✉♥❞❡r s✐♠✉❧t❛♥❡♦✉s ❇♦❣♦❧②✉❜♦✈ r♦t❛t✐♦♥s ♦❢ ❤❛r♠♦♥✐❝s ❛♥❞ np ❛♥❞ 𝜆p✳ ❍❡♥❝❡✱ t♦ s✉♠ ✉♣ ❧❡❛❞✐♥❣ ■❘ ❝♦rr❡❝t✐♦♥s ✇❡ ❤❛✈❡ t♦ ✜♥❞ ❤❛r♠♦♥✐❝s ❢♦r ✇❤✐❝❤ t❤❡r❡ ✐s s✉❝❤ ❛ s♦❧✉t✐♦♥ t❤❛t 𝜆p = ✵✳ ❖t❤❡r✇✐s❡ t❤❡r❡ ✐s ♥♦ ❤♦♣❡ ❢♦r st❛t✐♦♥❛r② st❛t❡✦ ■♥s♣✐r❛t✐♦♥ ❢r♦♠ t❤❡ ♥♦♥✕st❛t✐♦♥❛r② t❤❡♦r② ❢♦r s✉♣❡r❝♦♥❞✉❝t♦rs✳

21 / 21