Group Theoretic Approach to Theory of Fermion Production Minho Son - - PowerPoint PPT Presentation

KEK-KIAS-NCTS Theory Workshop, Dec 4 - 7, 2018

Minho Son

Korea Advanced Institute of Science and Technology (KAIST)

Based on Min, SON, Suh 1808.00939

Group Theoretic Approach to Theory of Fermion Production

Particle Production

• Preheating via parametric resonance or

excitation in post-inflationary era

• Gravitational waves from preheating
• Axion-inflation via gauge boson (𝜚𝐺𝐺

#) or fermion (𝜖

%𝜚 𝑘%() production

Anbor, Sorbo 10’ Adshead, Pearce, Peloso, Roberts, Sorbo 18’

Many literature (hard to list all here)

Kofman, Linde, Starobinsky 97’

List goes on …..

Relaxation with particle production

• Relaxation with particle production

Hook, Marques-Tavares 15’ SON, Ye, You 18’ Fonseca, Morgante, Servant 18’ …

+ Leptogenesis

SON, Ye, You 18’

Particle Production

Relaxation with particle production + Leptogenesis

SON, Ye, You 18’

We are going to ‘Reformulate’ of theory of fermion production in a completely new manner

Traditional Approach To Theory of Fermion Production

called technique of ‘Bogoliubov’ coefficient

𝒯 = + 𝑒-𝑦 −𝑕 𝜔 2 𝑗 𝑓 5

% 𝛿5𝐸 % − 𝑛 + 𝑕 𝜚

𝜔 + 1 2 𝜖

%𝜚 < − 𝑊(𝜚)

𝑒𝑡< = 𝑒𝑢< − 𝑏 𝑢 <𝑒𝐲< = 𝑏 𝑢 <(𝑒𝜐< − 𝑒𝐲<) Under rescaling 𝜔 → 𝑏FG/<𝜔 ℒ = 𝜔 2 𝑗 𝛿%𝜖

% − 𝑛𝑏 + 𝑕 𝜚

𝜔 + 1 2 𝑏<𝜃%K𝜖

%𝜚𝜖K𝜚 − 𝑏-𝑊(𝜚)

𝑕 𝜚 = : Yukawa-type coupling : derivative coupling

Common Interaction type in literature

The model

On the metric: L ℎ𝜚 1 𝑔 𝛿%𝛿(𝜖

%𝜚

We will assume spatially homogenous scalar field : 𝜖

%𝜚 = 𝜚̇ We will not distinguish 𝑢 and 𝜐 unless it is necessary

ℒ = 𝜔 2 𝑗 𝛿%𝜖

% − 𝑛𝑏 − 1

𝑔 𝛿P𝛿(𝜚̇ 𝜔 + 1 2 𝑏<𝜃%K𝜖

%𝜚𝜖K𝜚 − 𝑏-𝑊(𝜚)

A subtlety with derivative coupling

ΠR = 𝜀ℒ 𝜀𝜔̇ = 𝑗𝜔T ΠU = 𝜀ℒ 𝜀𝜚̇ = 𝑏<𝜚̇ − 1 𝑔 𝜔 2𝛿P𝛿(𝜔 ℋ = ΠR𝜔̇ + ΠU𝜚̇ − ℒ = 𝜔 2 −𝑗 𝛿W𝜖W + 𝑛𝑏 + 1 𝑔 𝛿P𝛿(𝜚̇ 𝜔 − 1 2𝑏< 𝜔 2𝛿P𝛿(𝜔 < 𝑔 < + 1 2𝑏< ΠU

< + 𝑏(𝑊(𝜚)

Definition of particle number is ambiguous Massless limit is not manifest

Fermion Production is formulated in Hamiltonian formalism

Adshead, Sfakianakis 15’

ℒ = 𝜔 2 𝑗 𝛿%𝜖

% − 𝑛𝑏 − 1

𝑔 𝛿P𝛿(𝜚̇ 𝜔 + 1 2 𝑏<𝜃%K𝜖

%𝜚𝜖K𝜚 − 𝑏-𝑊(𝜚)

Hamiltonian formalism ΠR = 𝜀ℒ 𝜀𝜔̇ = 𝑗𝜔T ΠU = 𝜀ℒ 𝜀𝜚̇ = 𝑏<𝜚̇ ℋ = 𝜔 2 −𝑗 𝛿W𝜖W + 𝑛X − 𝑗 𝑛Y𝛿( 𝜔 + 1 2𝑏< ΠU

< + 𝑏-𝑊(𝜚)

A way out: field redefinition

𝜔 → 𝑓FWZ[U/\𝜔

ℒ = 𝜔 2 𝑗 𝛿%𝜖

% − 𝑛𝑏 cos2𝜚

𝑔 + 𝑗 𝑛𝑏 sin 2𝜚 𝑔 𝛿( 𝜔 + 1 2 𝑏<𝜃%K𝜖

%𝜚𝜖K𝜚 − 𝑏-𝑊(𝜚)

No 𝜔 - dependence in conjugate momentum ΠU Entire fermion sector is quadratic in 𝜔

= 𝑛X = 𝑛Y

Massless limit is manifest

: particle number is unambiguously defined Adshead, Pearce, Peloso, Roberts, Sorbo 18’ Adshead, Sfakianakis 15’

Fermion production

ℋ = 𝜔 2 −𝑗 𝛿W𝜖W + 𝑛X − 𝑗 𝑛Y𝛿( 𝜔 + 1 2𝑏< ΠU

< + 𝑏-𝑊(𝜚)

Quantum field 𝜔

𝜔 = + 𝑒G𝑙 2𝜌 G/< 𝑓W𝐥⋅𝐲 f 𝑉h 𝐥,𝑢 𝑏h 𝐥 + 𝑊

h −𝐥,𝑢 𝑐h T −𝐥 hk±

To estimate Fermion Production, we quantize 𝜔 while keeping pseudo-scalar as a classical field

𝑉h = 𝑣h 𝐥,𝑢 𝜓h(𝐥) 𝑠 𝑤h 𝐥,𝑢 𝜓h(𝐥) 𝜓h 𝐥 = 𝑙 + 𝑠 𝜏 ⃗ ⋅ 𝐥 2𝑙 𝑙 + 𝑙G 𝜓̅h where 𝜓̅T = 1 0 , 𝜓̅F = 0 1

Garbrecht, Prokopec, Schmidt 02’ for generic complex mass

** helicity basis for an arbitrary 𝐥

We follow notation and convention in Adshead, Pearce, Peloso, Roberts, Sorbo 18’

ℋR = f + 𝑒𝑙G

hk±

𝑏h

T 𝐥 , 𝑐h −𝐥

𝐵h 𝐶h

∗

𝐶h −𝐵h 𝑏h(𝐥) 𝑐h

T(−𝐥)

𝑜h,y = 0 𝑏h

T 𝐥;𝑢 𝑏h(𝐥; 𝑢) 0

𝐵h = 1 2 − 𝑛X 4𝜕 𝑣h

< − 𝑤h < − 𝑙

2𝜕 𝑆𝑓 𝑣h

∗𝑤 h − 𝑠𝑛Y

2𝜕 𝐽𝑛(𝑣h

∗𝑤 h)

𝐶h = 𝑠 𝑓Wh•€ 2 2 𝑛X𝑣h𝑤

h − 𝑙 𝑣h < − 𝑤h < − 𝑗𝑠𝑛Y(𝑣h < + 𝑤h <)

Fermion number density for a particle with helicity 𝑠 𝑏h(𝐥) 0 = 0 𝑏h 𝐥; 𝑢 0 ≠ 0 𝑏h 𝐥 , 𝑏h

T 𝐥

↔ one-particle state due to 𝐶h = 0 At 𝑢 = 0 At 𝑢 ≠ 0

w/ 𝑏h(𝐥; 𝑢), 𝑏h

T(𝐥;𝑢) are diagonalized 𝑏h(𝐥), 𝑏h T(𝐥)at 𝑢 ≠ 0

𝑏h 𝐥 , 𝑏h

T 𝐥

↮ one-particle state anymore due to 𝐶h ≠ 0

ℋR = f + 𝑒𝑙G

hk±

𝑏h

T 𝐥 , 𝑐h −𝐥

𝐵h 𝐶h

∗

𝐶h −𝐵h 𝑏h(𝐥) 𝑐h

T(−𝐥)

= 1 2 − 𝑛X 4𝜕 𝑣h

< − 𝑤h < − 𝑙

2𝜕 𝑆𝑓 𝑣h

∗𝑤h − 𝑠𝑛Y

2𝜕 𝐽𝑛(𝑣h

∗𝑤h)

𝑜h,y = 0 𝑏h

T 𝐥; 𝑢 𝑏h(𝐥; 𝑢) 0 = 𝛾h <

𝐵h = 1 2 − 𝑛X 4𝜕 𝑣h

< − 𝑤h < − 𝑙

2𝜕 𝑆𝑓 𝑣h

∗𝑤 h − 𝑠𝑛Y

2𝜕 𝐽𝑛(𝑣h

∗𝑤 h)

𝐶h = 𝑠 𝑓Wh•€ 2 2 𝑛X𝑣h𝑤

h − 𝑙 𝑣h < − 𝑤h < − 𝑗𝑠𝑛Y(𝑣h < + 𝑤h <)

Fermion number density for a particle with helicity 𝑠

𝑏h(𝐥; 𝑢) = 𝛽h 𝑏h(𝐥) − 𝛾h

∗ 𝑐h T(𝐥)

w/ 𝑏h(𝐥; 𝑢), 𝑏h

T(𝐥;𝑢) are diagonalized 𝑏h(𝐥), 𝑏h T(𝐥)at 𝑢 ≠ 0

𝑐h

T(𝐥;𝑢) = 𝛾h 𝑏h(𝐥) + 𝛽h ∗ 𝑐h T(𝐥)

Bogoliubov coeff.

• Diag. ops

at 𝑢 ≠ 0 In terms of diag.

• ps at 𝑢 = 0

= 1 2 − 𝑛X 4𝜕 𝑣h

< − 𝑤h < − 𝑙

2𝜕 𝑆𝑓 𝑣h

∗𝑤h − 𝑠 𝑛Y

2𝜕 𝐽𝑛(𝑣h

∗𝑤h)

𝑜h,y = 0 𝑏h

T 𝐥;𝑢 𝑏h(𝐥; 𝑢) 0

looks too technical … Any simplication?

Solving EOM of 𝑣h, 𝑤h with correct initial condition is another source of confusion

= 1 2 − 𝑛X 4𝜕 𝑣h

< − 𝑤h < − 𝑙

2𝜕 𝑆𝑓 𝑣h

∗𝑤h − 𝑠 𝑛Y

2𝜕 𝐽𝑛(𝑣h

∗𝑤h)

𝑜h,y = 0 𝑏h

T 𝐥;𝑢 𝑏h(𝐥; 𝑢) 0

𝜔 ∼ 𝑉h 𝐥, 𝑢 𝑏h 𝐥 + 𝑊

h −𝐥,𝑢 𝑐h T(−𝐥)

𝑉h = 𝑣h 𝐥, 𝑢 𝜓h(𝐥) 𝑠 𝑤h 𝐥, 𝑢 𝜓h(𝐥) = 𝑣h 𝑠𝑤h ⊗ 𝜓h ≡ 𝜊h ⊗ 𝜓h

Recall a Fourier mode in ‘helicity’ basis

Solving EOM of 𝑣h, 𝑤h with correct initial condition is another source of confusion

looks too technical … Any simplication?

= 1 2 − 𝑛X 4𝜕 𝑣h

< − 𝑤h < − 𝑙

2𝜕 𝑆𝑓 𝑣h

∗𝑤h − 𝑠 𝑛Y

2𝜕 𝐽𝑛(𝑣h

∗𝑤h)

𝑜h,y = 0 𝑏h

T 𝐥;𝑢 𝑏h(𝐥; 𝑢) 0

𝜂 ⃗

h = 𝜊h T𝜏

⃗ 𝜊h

𝜂h ‹ = 1 2 𝑠(𝑣h

∗𝑤h + 𝑣h𝑤h ∗) = 𝑠 𝑆𝑓(𝑣h ∗𝑤h)

𝜂h < = − 𝑗 2𝑠(𝑣h

∗𝑤h − 𝑣h𝑤h ∗) = 𝑠 𝐽𝑛(𝑣h ∗𝑤h)

𝜂h G = 1 2 𝑣h

< − 𝑤h <

𝜔 ∼ 𝑉h 𝐥, 𝑢 𝑏h 𝐥 + 𝑊

h −𝐥,𝑢 𝑐h T(−𝐥)

𝑉h = 𝑣h 𝐥, 𝑢 𝜓h(𝐥) 𝑠 𝑤h 𝐥, 𝑢 𝜓h(𝐥) = 𝑣h 𝑠𝑤h ⊗ 𝜓h ≡ 𝜊h ⊗ 𝜓h

Then we realize that Recall a Fourier mode in ‘helicity’ basis

collapses into one vector

Solving EOM of 𝑣h, 𝑤h with correct initial condition is another source of confusion

looks too technical … Any simplication?

= 1 2 − 𝑛X 4𝜕 𝑣h

< − 𝑤h < − 𝑙

2𝜕 𝑆𝑓 𝑣h

∗𝑤h − 𝑠 𝑛Y

2𝜕 𝐽𝑛(𝑣h

∗𝑤h)

𝑜h,y = 0 𝑏h

T 𝐥;𝑢 𝑏h(𝐥; 𝑢) 0

𝜂 ⃗

h = 𝜊h T𝜏

⃗ 𝜊h

𝜂h ‹ = 1 2 𝑠(𝑣h

∗𝑤h + 𝑣h𝑤h ∗) = 𝑠 𝑆𝑓(𝑣h ∗𝑤h)

𝜂h < = − 𝑗 2𝑠(𝑣h

∗𝑤h − 𝑣h𝑤h ∗) = 𝑠 𝐽𝑛(𝑣h ∗𝑤h)

𝜂h G = 1 2 𝑣h

< − 𝑤h <

𝐫 = 𝑠𝑙 𝑦

• ‹ + 𝑛Y 𝑦
• < + 𝑛X 𝑦
• G

w/ 𝜊h ≡ 𝑣h 𝑠𝑤h

* We will see the origin

• f this vector later

= 1 2 − 𝑛X 4𝜕 𝑣h

< − 𝑤h < − 𝑙

2𝜕 𝑆𝑓 𝑣h

∗𝑤h − 𝑠 𝑛Y

2𝜕 𝐽𝑛(𝑣h

∗𝑤h)

𝑜h,y = 0 𝑏h

T 𝐥;𝑢 𝑏h(𝐥; 𝑢) 0

𝜂 ⃗

h = 𝜊h T𝜏

⃗ 𝜊h

𝜂h ‹ = 1 2 𝑠(𝑣h

∗𝑤h + 𝑣h𝑤h ∗) = 𝑠 𝑆𝑓(𝑣h ∗𝑤h)

𝜂h < = − 𝑗 2𝑠(𝑣h

∗𝑤h − 𝑣h𝑤h ∗) = 𝑠 𝐽𝑛(𝑣h ∗𝑤h)

𝜂h G = 1 2 𝑣h

< − 𝑤h <

𝐫 = 𝑠𝑙 𝑦

• ‹ + 𝑛Y 𝑦
• < + 𝑛X 𝑦
• G

𝜂𝑠,𝐫 behave like vector reps of SO(3) w/ 𝜊h ≡ 𝑣h 𝑠𝑤h

* We will see the origin

• f this vector later

What is this mysterious SO(3)?

Group Theoretic Approach

‘Reparametrization’ Group

𝛿%,𝛿K = 2 𝜃%K1- 𝛿 % → 𝑉𝛿 %𝑉F‹ : GL(4,C) ℒ = 𝜔T𝛿P 𝑗𝛿 %𝜖% − 𝑛 𝜔 → ℒ = 𝜔T 𝑉T𝑉𝛿 P𝑉F‹ 𝑗𝑉𝛿%𝑉F‹𝜖% − 𝑛 𝑉𝜔 𝑉T𝑉 = 𝑉𝑉T = 1 : U(4)

Clifford Algebra Dirac Theory

We assign the transformation of 𝜔, 𝜔 → 𝑉𝜔

While 𝛿% is fixed and only 𝜔 transforms in the Lorentz group,

𝛿 % → 𝛿 %, 𝜔 → Λ‹/<𝜔 ,

there is a freedom in choosing a representation of the gamma matrices. This freedom is totally unphysical.

We consider the following subgroup of 𝑉(4) 𝑇𝑉 2 ‹×𝑇𝑉 2 <×𝑉(1) ⊂ 𝑉(4)

The rep of subgroup is constructed as a ‘tensor product’ of two 𝑇𝑉(2)’s and phase rotation, e.g. Under 𝑇𝑉 2 ‹ ⊗ 𝑇𝑉 2 < transformation (we associate 𝑉(1) with 𝜊h) 𝑏‹‹ 𝑏‹< 𝑏<‹ 𝑏<< ⊗ 𝑉< = 𝑏‹‹𝑉< 𝑏‹<𝑉< 𝑏<‹𝑉< 𝑏<<𝑉< = 𝑉‹ 𝜔 ∼ 𝜊h ⊗ 𝜓h → 𝑉‹ ⊗ 𝑉< 𝜊h ⊗ 𝜓h = 𝑉‹𝜊h ⊗ (𝑉<𝜓h) This is what we are looking for

We consider the following subgroup of 𝑉(4) 𝑇𝑉 2 ‹×𝑇𝑉 2 <×𝑉(1) ⊂ 𝑉(4)

The rep of subgroup is constructed as a ‘tensor product’ of two 𝑇𝑉(2)’s and phase rotation, e.g. Under 𝑇𝑉 2 ‹ ⊗ 𝑇𝑉 2 < transformation (we associate 𝑉(1) with 𝜊h) 𝑏‹‹ 𝑏‹< 𝑏<‹ 𝑏<< ⊗ 𝑉< = 𝑏‹‹𝑉< 𝑏‹<𝑉< 𝑏<‹𝑉< 𝑏<<𝑉< = 𝑉‹ 𝜔 ∼ 𝜊h ⊗ 𝜓h → 𝑉‹ ⊗ 𝑉< 𝜊h ⊗ 𝜓h = 𝑉‹𝜊h ⊗ (𝑉<𝜓h)

Looks similar to space rotation of Lorentz group.

This is what we are looking for

But it can not be identified with SU(2) space rotation 𝜔 2𝛿%𝜔 → 𝜔T𝑉T𝑉𝛿P𝑉F‹𝑉𝛿%𝑉F‹𝑉𝜔 = 𝜔 2𝛿%𝜔 𝜔 2𝛿%𝜔 → 𝜔 2 Λ‹/<

F‹ 𝛿%Λ‹/<𝜔 = Λ K % 𝜔

’ 𝛿%𝜔 E.g.

𝛿P = 𝐽< −𝐽< = 𝜏G ⊗ 𝐽< 𝛿W = 𝜏W −𝜏W = 𝑗 𝜏< ⊗ 𝜏W 𝛿( = 𝐽< 𝐽< = 𝜏‹ ⊗𝐽< 𝛿P = 𝐽< 𝐽< = 𝜏‹ ⊗ 𝐽< 𝛿W = 𝜏W −𝜏W = 𝑗 𝜏< ⊗ 𝜏W 𝛿( = −𝐽< 𝐽< = −𝜏G ⊗ 𝐽<

Weyl Representation Dirac Representation

𝜔“”•– = 𝜔— 𝜔X 𝜔˜™š›œ = 1 2 𝜔— + 𝜔X −𝜔— + 𝜔X

w/ 𝑉‹(𝜌/2) = 𝑓W •

ž Ÿ ž =

‹ <

1 1 −1 1

𝜔“”•– → 𝑉‹𝜔“”•– = 𝜔˜™š›œ 𝛿“”•–

%

→ 𝑉‹𝛿“”•–

%

𝑉‹

F‹ = 𝛿˜™š›œ %

Two representations are related via a similarity transformation

A well-known example of 𝑇𝑉(2)‹

is what our group theoretic approach is based on

we will drop subscript from now on

𝑇𝑉(2)< does not play any important role.

𝑇𝑉(2)‹×𝑉(1)

w/ 𝐫 = 𝑠𝑙 𝑦

• ‹ + 𝑛Y 𝑦
• < + 𝑛X 𝑦
• G

𝜖¡𝜊h = −𝑗 𝐫 ⋅ 𝜏 ⃗ 𝜊h

Gives rise to EOM of fundamental rep.

SU(2) embedding

• f SO(3) vector 𝐫

𝑗 𝛿 %𝜖% − 𝑛X + 𝑗 𝑛Y𝛿( 𝜔 = 0 𝑗 𝜏G𝜖¡ − 𝑗𝑠𝑙𝜏< − 𝑛X𝐽< + 𝑗𝑛Y𝜏‹ ⊗ 𝐽< (𝜊h ⊗ 𝜓h) = 0

Dirac equation in inertial frame EOM in tensor form for a Fourier mode can be written as (using 𝜏 ⃗ ⋅ 𝐥 𝜓h = 𝑠𝑙𝜓h)

SU(2) fundamental

: it is called Weyl equation in condensed matter physics

Group Theoretic Approach

w/ 𝐫 = 𝑠𝑙 𝑦

• ‹ + 𝑛Y 𝑦
• < + 𝑛X 𝑦
• G

𝜊h ≡ 𝑣h 𝑠𝑤h 𝜖¡𝜊h = −𝑗 𝐫 ⋅ 𝜏 ⃗ 𝜊h ü Fundamental rep. of SU(2)

• EOM of fundamental rep.

SU(2) embedding

• f SO(3) vector

Group Theoretic Approach

𝜂 ⃗

h = 𝜊T𝜏

⃗ 𝜊 : vector 1 2 𝜖¡𝜂 ⃗

h = 𝐫×𝜂

⃗

h

ü In terms of SO(3) ∼ SU(2) reps 𝜖¡𝜂h W = 1 2 𝜊h

T 𝑗𝐫 ⋅ 𝜏

⃗, 𝜏W 𝜊h = 2𝜗W£y𝑟£𝜂h y Bilinear of 𝜊h : 𝜊h

T𝐵 𝜊h

𝜊T 𝜊 (= 1) : scalar

w/ 𝐵 = arbitrary 2×2 complex matrix

• EOM of vector rep.

the only non-trivial rep.

Group Theoretic Approach

Analog to classical precession motion

1 2 𝑒𝜂 ⃗

h

𝑒𝑢 = 𝐫×𝜂 ⃗

h Classical precession of a vector 𝑠 ⃗ with angular velocity 𝜕

𝜕 𝑠 ⃗ 𝑒𝑠 ⃗ 𝑒𝑢 = 𝜕×𝑠 ⃗

torque

Quantum mechanical fermion production 𝑠 ⃗ = 𝐍 (magnetization), 𝜕 = 𝜕𝐂 = −𝛿𝐂 : called block eq.

𝐹 = 𝜕𝐂 ⋅ 𝐍

𝑒𝐍 𝑒𝑢 = 𝜕𝐂×𝐍 E.g. when

? = 𝒓 ⋅ 𝜂 ⃗

h

𝐫 as angular velocity

ℋR = f + 𝑒𝑙G

hk±

𝑏h

T 𝐥 , 𝑐h −𝐥

𝐵h 𝐶h

∗

𝐶h −𝐵h 𝑏h(𝐥) 𝑐h

T(−𝐥)

𝐵h = 1 2 − 𝑛X 4𝜕 𝑣h

< − 𝑤h < − 𝑙

2𝜕 𝑆𝑓 𝑣h

∗𝑤h − 𝑠𝑛Y

2𝜕 𝐽𝑛(𝑣h

∗𝑤h)

𝐶h = 𝑠 𝑓Wh•€ 2 2 𝑛X𝑣h𝑤h − 𝑙 𝑣h

< − 𝑤h < − 𝑗𝑠𝑛Y(𝑣h < + 𝑤h <)

𝐵h = 𝐫 ⋅ 𝜂 ⃗

h

= 𝜕 cos𝜄

Now it is clear that each matrix element should be a function of 𝐫 and 𝜂 ⃗

h in our

group theoretic approach

𝐶h = 𝐫×𝜂 ⃗

h Diagonal element Off-diagonal element

One can easily see why eigenvalues are ±𝜕 = ±|𝐫|

Particle number density

= 𝜕 sin 𝜄

𝐫 = 𝜕 = 𝑙< + 𝑛<

Particle number density

ℋR = f +𝑒𝑙G

hk±

𝑏h

T 𝐥 ,𝑐h −𝐥

𝐵h 𝐶h

∗

𝐶h −𝐵h 𝑏h(𝐥) 𝑐h

T(−𝐥)

𝑜h,y = 0 𝑏h

T 𝐥; 𝑢 𝑏h(𝐥;𝑢) 0 = 𝛾h <= 𝑔(𝐫 ⋅ 𝜂

⃗

h, |𝐫|)

𝑜h,y = 𝐵 ± 𝐶 𝐫 ⋅ 𝜂 ⃗

h

|𝐫|

𝐵h = 𝐫 ⋅ 𝜂 ⃗

h ,

𝐶h = 𝐫×𝜂 ⃗

h

• 1. It should be at most linear in 𝜂

⃗

h (note 𝜂

⃗

h = 1)

𝐵 − 𝐶 ≤ 𝑜h,y ≤ 𝐵 + 𝐶

which gives rise to inequality,

𝑜h,y = 1 2 1 − 𝐫 ⋅ 𝜂 ⃗

h

|𝐫|

• 2. Pauli-blocking

0 ≤ 𝑜h,y ≤ 1

′ − ′ sign chosen for the consistency with the form of 𝐵h

(** agrees with our explicit computation)

In our approach, a few group properties can uniquely determine fermion number density

w/ 𝐫 = 𝑠𝑙 𝑦

• ‹ + 𝑛Y 𝑦
• < + 𝑛X 𝑦
• G

Solution of EOM

Closed form of solution is available

1 2 𝜖¡𝜂 ⃗

h = 𝐫×𝜂

⃗

h = 𝐫 ⋅ 𝐌 𝜂

⃗

h

𝑜h,y = 1 2 1 − 𝐫 ⋅ 𝜂 ⃗

h

|𝐫|

• Initial condition (↔ zero particle number) at 𝑢 = 𝑢P is straightforward than other

approach

𝜂 ⃗

h(𝑢P,𝑢P) = 𝐫(𝑢P)

|𝐫(𝑢P)| 𝜂 ⃗

h 𝑢, 𝑢P = 𝑈 exp + 𝑒𝑢³ (𝐫 ⋅ 𝐌)(𝑢′) ¡ ¡´

𝐫(𝑢P) |𝐫(𝑢P)|

• Just like solving Schrödinger eq. for the unitary op., EOM can be iteratively solved

Expanding involves commutators of 𝐫 ⋅ 𝐌 WKB solution might be the case with vanishing commutators

Numerical example

𝜚 𝑢 = 𝜚P sin(𝑢) for chaotic potential, 𝑊 𝜚 ∼ 𝑛<𝜚< 𝑛 = 1,

U´ \ = 10

chosen for all plots 𝑙 = 1 𝑙 = 10 𝑙 = 12

Region where fermion production happens, and WKB

• approx. is not valid

Region where WKB

• approx. is valid

Case where WKB approx. is not valid

1 2𝜖¡𝜂 ⃗

h = 𝐫×𝜂

⃗

h

with 𝜂 ⃗

h(0) = 𝐫(P) |𝐫(P)|

ℒ = 𝜔 2 𝑗 𝛿%𝜖

% − 𝑛𝑏 − 1

𝑔 𝛿P𝛿(𝜚̇ 𝜔 + ⋯

To, via 𝜔 → 𝑓TWZ[U/\𝜔,

𝜂 ⃗

h → 𝑆 𝑢 𝜂

⃗

h ,

where 𝑆 𝑢 = 1 cos2𝜚 𝑔 ⁄ −sin 2𝜚 𝑔 ⁄ sin2𝜚 𝑔 ⁄ cos2𝜚 𝑔 ⁄

is equivalent to, in terms of 𝜂 ⃗

h,

ℒ = 𝜔 2 𝑗 𝛿 %𝜖% − 𝑛X + 𝑗 𝑛Y𝛿( 𝜔 + ⋯ Transformation from `Inertial Frame’ vs `Rotating Frame’

in `Rotating Frame’

ℒ = 𝜔 2 𝑗 𝛿%𝜖

% − 𝑛𝑏 − 1

𝑔 𝛿P𝛿(𝜚̇ 𝜔 + ⋯

To, via 𝜔 → 𝑓TWZ[U/\𝜔,

𝜂 ⃗

h → 𝑆 𝑢 𝜂

⃗

h ,

where 𝑆 𝑢 = 1 cos2𝜚 𝑔 ⁄ −sin 2𝜚 𝑔 ⁄ sin2𝜚 𝑔 ⁄ cos2𝜚 𝑔 ⁄

is equivalent to, in terms of 𝜂 ⃗

h,

ℒ = 𝜔 2 𝑗 𝛿 %𝜖% − 𝑛X + 𝑗 𝑛Y𝛿( 𝜔 + ⋯ Transformation from

in `Inertial Frame’

`Inertial Frame’ vs `Rotating Frame’

This rotating frame is non-inertial frame Needs to supplement extra terms, e.g. Coriolis , centrifugal forces etc, to keep physics independent

Under 𝜂 ⃗

h → 𝑆 𝑢 𝜂

⃗

h , 1 2 𝜖¡𝜂 ⃗

h = 𝐫×𝜂

⃗

h = 𝐫 ⋅ 𝐌 𝜂

⃗

h

→ 1 2 𝜖¡(𝑆𝜂 ⃗

h) = 𝐫 ⋅ 𝐌 (𝑆𝜂

⃗

h)

1 2 𝜖¡𝜂 ⃗

h = 𝑆¸ 𝐫 ⋅ 𝐌 𝑆 𝜂

⃗

h − 1

2 𝑆¸𝑆̇𝜂 ⃗

h

w/ 𝑆¸𝑆̇

W£ ≡ 𝜗W£y𝜕¹º y

Similarly to the classical mechanics, EOM transforms like

EOM in `Rotating Frame’

Under 𝜂 ⃗

h → 𝑆 𝑢 𝜂

⃗

h , 1 2 𝜖¡𝜂 ⃗

h = 𝐫×𝜂

⃗

h = 𝐫 ⋅ 𝐌 𝜂

⃗

h

→ 1 2 𝜖¡(𝑆𝜂 ⃗

h) = 𝐫 ⋅ 𝐌 (𝑆𝜂

⃗

h)

1 2 𝜖¡𝜂 ⃗

h = 𝑆𝐫×𝜂

⃗

h + 1

2 𝜕¹º×𝜂 ⃗

h = 𝑆𝐫 + 𝜕¹º ×𝜂

⃗

h = 𝐫′×𝜂

⃗

h

1 2 𝜖¡𝜂 ⃗

h = 𝑆¸ 𝐫 ⋅ 𝐌 𝑆 𝜂

⃗

h − 1

2 𝑆¸𝑆̇𝜂 ⃗

h

w/ 𝑆¸𝑆̇

W£ ≡ 𝜗W£y𝜕¹º y

Similarly to the classical mechanics, EOM transforms like 𝐫³ = 𝑠𝑙 + 𝜚̇ 𝑔 𝑦

• ‹ + 𝑛𝑏 𝑦
• G

: different basis amounts to choose different angular velocity

EOM in `Rotating Frame’

EOM can be brought back to the universal form

Particle number density in `Rotating (non-inertial) Frame’

Particle number density in rotating frame 𝑜h,y = 0 𝑏h

T 𝐥; 𝑢 𝑏h(𝐥; 𝑢) 0 = 𝑔(𝐫³ ⋅ 𝜂

⃗

h,|𝐫′|)

It should be at most linear in 𝜂 ⃗

h.

Higher order terms should vanish to match to the one in inertial frame in 𝜚̇ → 0 limit 𝑜h,y = 1 2 1 − 𝐫′ ⋅ 𝜂 ⃗

h

|𝐫′| ℋR = 𝜔 2 −𝑗 𝛿W𝜖W + 𝑛𝑏 + 1 𝑔 𝛿P𝛿(𝜚̇ 𝜔 − 1 2𝑏< 𝜔 2𝛿P𝛿(𝜔 < 𝑔 <

: matches to the quadratic term

• 1. It looks like particle numbers are different in two different frames.
• 2. Establishing the ‘final’ particle number as a basis-independent quantity seems very

non-trivial, e.g. Inertial frame vs. Non-inertial frame

See Adshead, Sfakianakis 15’ for a related discussion

* does not take into account of quartic coupling etc..

Summary

We proposed a new group theoretic approach to theory of fermion production

• 3. This approach applies to any fermion system
• 1. Based on the ‘Reparametrization’ group of gamma matrcies

a. Possible extension is gravitino production, fermion production from gravitational background, fermion production in extra-dim. Spacetime b. Application to relaxation scenario c. Group theoretic approach for both fermion- and gauge boson production

• 2. Insightful visualization of quantum mechanical fermion production dynamics.

a. Totally unphysical symmetry (that we never cared) provides us with totally different viewpoint of a very complicated process such as fermion production a. Dynamics is analogous to the classical precession. b. Crystal clear initial condition unlike the traditional approach. c. Systematic comparison between Exact solution vs WKB solution.

Backup slides

Lorentz Group

𝛿P = 𝐽< 𝐽< = 𝜏‹ ⊗ 𝐽< 𝛿W = 𝜏W −𝜏W = 𝑗 𝜏< ⊗ 𝜏W 𝛿( = −𝐽< 𝐽< = −𝜏G ⊗ 𝐽<

𝑇%K = 𝑗 4[𝛿 %,𝛿 K]

Weyl Representation

𝐾W ≡ 1 2𝜗W£y𝑇£y = 1 2 𝐽< ⊗ 𝜏W 𝐿W ≡ 𝑇WP = 𝑗 2 𝜏G ⊗ 𝜏W 𝐾—, X W = 𝐾W ∓ 𝑗 𝐿W 2 = 1 2 𝐽< ± 𝜏G ⊗ 𝜏W 2 𝜔 = 𝜔— 𝜔X 1 2 , 0 ⊕ 0, 1 2

Spinor rep. satisfying Lorentz algebra (space rotation) , (boost) : 𝑇𝑉 2 —×𝑇𝑉 2 X

: Rep. of 𝑇𝑉 2 —×𝑇𝑉 2 X is constructed as a ‘tensor sum’

𝜔 ∼ 𝜊h ⊗ 𝜓h → 𝑓FWÁ⋅Â

⃗𝜔 = 𝜊 ⊗ 𝑓FWÁ⋅Ã < 𝜓h

On the other hand