Fermions in LQC Mercedes Martn-Benito (with B. Elizaga Navascus and - - PowerPoint PPT Presentation

Fermions in LQC

(with B. Elizaga Navascués and G.A. Mena Marugán)

Mercedes Martín-Benito

01/27

INTRODUCTION

• Standard Cosmology: Primordial cosmological perturbations
• ver flat FLRW supplemented with inflation
• Quantum vacuum fluctuations of the inflaton and the metric
• Matter is fermionic: Could the evolution of primordial fermion

fields affect the scalar and tensor perturbations?

• Framework: hybrid LQC
• LQG inspired treatment for the global mode of the geometry
• Fock quantization for the inhomogeneities
• In LQC, the presence of the big bounce might affect drastically

the behavior of fermion fields at the early Universe

Fermions in Loop Quantum Cosmology

02/27

• For simplicity, in this talk we analyze only fermion perturbations
• We include a scalar field subject to a potential
• We truncate the action at quadratic perturbative order

[D’Eath and Halliwell, ‘87]

• Flat FLRW cosmology, with toroidal spatial sections
• We couple the system to a Dirac field, treated as a perturbation
• The fermionic part is quadratic in the Dirac field. At our order
• f truncation, the Dirac field couples only to the homogeneous

tetrad, and decouples from the rest of possible perturbations

CLASSICAL SYSTEM

03/27

• FLRW with fermion perturbations
• Hybrid quantization (global mode Hamiltonian constraint)
• Unperturbed FLRW: loop quantization
• Classical system (annihilation/creation variables)

OUTLINE

• Born-Oppenheimer approx. Schrödinder Eq.
• Quantum dynamics of fermion perturbations
• Evolution operator and unitarity
• Production of particles/antiparticles and backreaction

LOOP QUANTIZATION OF HOMOGENEOUS FLRW

04/27

05/27

• Flat FLRW cosmology, with toroidal spatial sections

Euclidean metric cte equal to angular coordinate with period 2π/l0

4π/(3l3

0)

• Homogeneous scalar field subject to a potential (e.g. mass term)

φ

• Hamiltonian

H|0 = e−3α 2 ✓ 3 4π π2

φ − π2 α + 4πσ2

3 e6αm2φ2 ◆ ds2 = σ2(−N 2

0 (t)dt2 + e2α(t) 0hijdθidθj)

• LQC variables:

constraint

{b, v} = 2 , V ∝ |v| , eα = ✓ 3V 4πσ ◆ 1

3

, πα = −3 2vb

HOMOGENEOUS MODEL

= 0

ˆ v|vi = v|vi hv|v0i = δv,v0 ; v 2 R L2(Rd, dµd) ⊗ L2(R, dφ)

• Kinematical Hilbert space and elementary quantum operators:

[Bojowald, Ashtekar, Lewandowski, Pawlowski, Singh,….]

• Hamiltonian constraint:

(vb)2

represents

06/27

d eiλb|vi = |v + 2λi

ˆ C0 = 1 2 ⇣ ˆ π2

φ − ˆ

H(2) ⌘ , ˆ H(2) = 3 4πγ2 ˆ Ω2

0 − ˆ

V 2m2 ˆ φ2

b → \ sin(b) = c eib − d e−ib 2i (C0 = 4πe3αH|0/3)

Immirzi parameter

POLYMER QUANTIZATION

07/27

is the square root of

• For massless scalar is time independent
• Evolution picture: Scalar plays the role of time

ˆ H(2)

χ0(v) = Z ∞ dλe✏

(v)ψ(λ)ei!()0 ,

ω(λ) = s 3λ 4πl2

P~γ2

(generalized) eigenfunctions of . Non-degenerate and absolutely continuous spectrum

ˆ Ω2

L2(R, dλ)

χ(v, φ) = ˆ U(v, φ)χ0(v) , ˆ U(v, φ) = P " exp i Z φ

φ0

d˜ φˆ h0(v, ˜ φ) !# d˜ φ ˆ H0

PHYSICAL STATES

ˆ H0 ∝ q Ω2

˜ χ

[Ashtekar, Pawlowski, Singh]

08/27

• Expectation value of the volume on semiclassical states

φ v LQC classical

[by courtesy of J. Olmedo]

DYNAMICS

H2 = 8πG 3 ρφ , H > 0 H2 = 8πG 3 ρφ , H < 0 h ˆ Vφiχ

[Ashtekar, Pawlowski, Singh]

08/27

• Expectation value of the volume on semiclassical states

φ v LQC classical

[by courtesy of J. Olmedo]

quantum bounce

h ˆ Vφiχ

DYNAMICS

H2 = 8πG 3 ρφ ✓ 1 − ρφ ρmax ◆

• The spectrum of the energy density in the Physical Hilbert space

is bounded from above:

ρmax

[Ashtekar, Corichi, Singh]

09/27

FLAT FLRW WITH FERMION PERTURBATIONS

10/27

• Dirac field: cross-sections of the resulting spinor bundle
• Partial internal gauge fixing (time gauge):
• Using Weyl representation, pair of two-component spinors of

definite chirality: (left-handed) and (right-handed)

¯ χA0 ej

0 = 0

• and defined on parametrized by the time coordinate

¯ χA0

• Expansion in the eigenspinors of the Dirac operator on
• Discrete spectrum labeled by ( ) :

±!k = ±2⇡ l0 |~ k + ~ ⌧| T 3 T 3

• Spin structure characterized by ~

⌧

DIRAC FIELD WITH MASS M

ϕA ϕA

~ k ∈ Z3

k

(DΨ = MΨ)

11/27

ωk ωk −ωk −ωk

(x~

k, ¯

y~

k) = (m~ k, ¯

s~

k) or (t~ k, ¯

r~

k)

• Let us denote

~ k ∈ Z3

• Grasman variables
• For simplicity: there are no zero-modes,

~ ⌧ 6= ~ ~ k 6= ~ ⌧

MODE EXPANSION

¯ A0(x) = e−3↵(⌘)/2 3/2l3/2 X

~ k,(±)

⇥ ¯ s~

k(⌘) ¯

w

~ k,(+) A0

(~ ✓) + t~

k(⌘) ¯

w

~ k,(−) A0

(~ ✓) ⇤ 'A(x) = e−3↵(⌘)/2 3/2l3/2 X

~ k,(±)

⇥ m~

k(⌘)w ~ k,(+) A

(~ ✓) + ¯ r~

k(⌘)w ~ k,(−) A

(~ ✓) ⇤ ,

12/27

• Canonical system:

{α, π↵} = 1 , {φ, π} = 1 , {x~

k, ¯

x~

k}D = −i

, {y~

k, ¯

y~

k}D = −i

• Subject to a global Hamiltonian constraint coupling both sectors

H~

k = σM

• s−~

k−2~ ⌧m~ k + ¯

m~

k¯

s−~

k−2~ ⌧ + r~ kt−~ k−2~ ⌧ + ¯

t−~

k−2~ ⌧ ¯

r~

k

• −e−↵ωk
• ¯

m~

km~ k + ¯

t~

kt~ k − r~ k¯

r~

k − s~ k¯

s~

k

• H|0 + HD = 0

, HD = X

~ k6=⌧

H~

k

HAMILTONIAN CONSTRAINT

~ ⌧

13/27

• We ask them to respect the linearity of the equations of motion

and the dynamical decoupling between modes

• Infinite ambiguity in their definition
• The coefficients extract part of the dynamics of the field, as

attributed to the homogeneous geometry

ANNIHILATION AND CREATION VAR. 1

a(x,y)

~ k

= f

~ k,(x,y) 1

x~

k + f ~ k,(x,y) 2

¯ y−~

k−2~ ⌧

¯ b(x,y)

~ k

= g

~ k,(x,y) 1

x~

k + g ~ k,(x,y) 2

¯ y−~

k−2~ ⌧

{a(x,y)

~ k

, ¯ a(x,y)

~ k

} = −i

{b(x,y)

~ k

,¯ b(x,y)

~ k

} = −i

(α, πα)-dependent in general

14/27

• Variables for instantaneous diagonalization

[D’Eath and Halliwell, ‘87]

• If we were regarding the Dirac field as a test field, this choice would

lead to a unique Fock quantization (up to unitary equivalence):

• Vacuum invariant under the symmetries of the e.o.m
• Unitary implementation of the quantum dynamics
• Usual convention for particles/antiparticles in the massless limit

[Cortez, Elizaga Navascués, Martín-Benito, Mena Marugán, Velhinho, ‘16]

f

~ k,(x,y) 1

= −g

~ k,(x,y) 2

= s ξk − ωk 2ξk

f

~ k,(x,y) 2

= g

~ k,(x,y) 1

= s ξk + ωk 2ξk

ANNIHILATION AND CREATION VAR. 2

ξk = q ω2

k + (σMeα)2

Restricts the asymptotic behavior in the ultraviolet limit ωk → ∞

14/27

• Variables for instantaneous diagonalization

[D’Eath and Halliwell, ‘87]

• If we were regarding the Dirac field as a test field, this choice would

lead to a unique Fock quantization (up to unitary equivalence):

• Vacuum invariant under the symmetries of the e.o.m
• Unitary implementation of the quantum dynamics
• Usual convention for particles/antiparticles in the massless limit

[Cortez, Elizaga Navascués, Martín-Benito, Mena Marugán, Velhinho, ‘16]

Restricts the asymptotic behavior in the ultraviolet limit ωk → ∞

ANNIHILATION AND CREATION VAR. 2

ξk = q ω2

k + (σMeα)2

f

~ k,(x,y) 1

= −g

~ k,(x,y) 2

= Me↵ 2ωk + o(ω−1

k )

f

~ k,(x,y) 2

= g

~ k,(x,y) 1

= 1 + O(ω−2

k )

15/27

• Inhomogeneous sector

(x~

k, ¯

x~

k, y~ k, ¯

y~

k)

→ (a(x,y)

~ k

, ¯ a(x,y)

~ k

, b(x,y)

~ k

,¯ b(x,y)

~ k

) α- dependent canonical transformation

• We need to correct the homogeneous sector

generates the dynamical evolution of the creation and annihilation variables

π↵ → ˘ π↵ = π↵ + iσMωk 2ξ2

k

e↵ X

~ k,(x,y)

✓ a(x,y)

~ k

b(x,y)

~ k

+ ¯ a(x,y)

~ k

¯ b(x,y)

~ k

◆

−

CANONICAL TRANSFORMATION

+ α-dependent term quadratic in the fermion variables

˘ HD H|0(α, ˘ πα)−(˘ πα − πα)∂˘

παH|0(α, ˘

πα) + HD[a, b]

16/27

• Main quantum gravity effects are those affecting the global

degrees of freedom of the geometry. By quantizing the homogeneous geometry sector with QG techniques, we hope to retain the main quantum effects

• Inhomogeneities, even though are also quantum, can be

treated in a more conventional way. Assumption:

HYBRID QUANTIZATION

16/27

QG QFT/CS

• Main quantum gravity effects are those affecting the global

degrees of freedom of the geometry. By quantizing the homogeneous geometry sector with QG techniques, we hope to retain the main quantum effects

• Inhomogeneities, even though are also quantum, can be

treated in a more conventional way. Assumption:

HYBRID QUANTIZATION

• Main quantum gravity effects are those affecting the global

degrees of freedom of the geometry. By quantizing the homogeneous geometry sector with QG techniques, we hope to retain the main quantum effects

16/27

• Inhomogeneities, even though are also quantum, can be

treated in a more conventional way.

QG Hybrid LQC

Assumption:

QFT/CS

HYBRID QUANTIZATION

17/27

ˆ C = ˆ C0 + 1 2 ⇥ˆ ΥF + ˆ ΥI ⇤

ˆ ΥI = i X

~ k6=~ ⌧,(x,y)

Mωk l0γ ˆ ϑI

k

✓ ˆ a(x,y)

~ k

ˆ b(x,y)

~ k

+ ˆ a(x,y)†

~ k

ˆ b(x,y)†

~ k

◆

Operators defined on the homogeneous sector. They depend on ωk

• Kinematical Hilbert space Hhom ⊗ FD
• Hamiltonian Constraint operator

ˆ ΥF = X

~ k6=~ ⌧,(x,y)

2l0 ˆ ϑF

k

✓ ˆ a(x,y)†

~ k

ˆ a(x,y)

~ k

+ ˆ b(x,y)†

~ k

ˆ b(x,y)

~ k

◆

QUANTUM CONSTRAINT

18/27

• Born-Oppenheimer ansatz: Consider states whose dependence
• n the FLRW geometry and the inhomogeneities ( ) split

N

Ψ = Γ(v, φ)ψ(N, φ)

Its evolution is generated by a positive hamiltonian such that is negligible:

ˆ ˜ H0

( ˆ ˜ H0)2 − ˆ H(2)

Γ ≈ χ ˆ π2

φψ

• We can disregard transitions from to other FLRW states

Γ

• Born-Oppenheimer approximation:
• negligible in comparison to h ˆ

˜ H0iΓˆ πφψ

• Taking expectation values on and under those approximations:

Γ

BORN-OPPENHEIMER APPROX.

19/27

• explores the quantum nature of the homogeneous geometry,

as it depends on the expectation values and , rather than on classical functions

ˆ HD hˆ ϑF

k iΓ

hˆ ϑI

kiΓ

C(Γ)

D (φ) = h( ˆ

˜ H0)2 ˆ H(2)

0 iΓ Backreaction onto the homogeneous geometry

idφψ = ˆ HDψ , ˆ HD = " hˆ ΥF + ˆ ΥIiΓ 2h ˆ ˜ H0iΓ + C(Γ)

D (φ)

# ψ

• Infinite number of expectation values: the fermion perturbations

do not feel a dressed metric determined by a finite number of dressed quantities, unlike for scalar and tensor perturbations

SCHRÖDINGER EQUATION

19/27

• explores the quantum nature of the homogeneous geometry,

as it depends on the expectation values and , rather than on classical functions

ˆ HD hˆ ϑF

k iΓ

hˆ ϑI

kiΓ

Hybrid LQC QFT/CS

C(Γ)

D (φ) = h( ˆ

˜ H0)2 ˆ H(2)

0 iΓ Backreaction onto the homogeneous geometry

idφψ = ˆ HDψ , ˆ HD = " hˆ ΥF + ˆ ΥIiΓ 2h ˆ ˜ H0iΓ + C(Γ)

D (φ)

# ψ

SCHRÖDINGER EQUATION

19/27

• explores the quantum nature of the homogeneous geometry,

as it depends on the expectation values and , rather than on classical functions

ˆ HD hˆ ϑF

k iΓ

hˆ ϑI

kiΓ

QFT/ QFT/CS Hybrid LQC quantum spacetime

C(Γ)

D (φ) = h( ˆ

˜ H0)2 ˆ H(2)

0 iΓ Backreaction onto the homogeneous geometry

idφψ = ˆ HDψ , ˆ HD = " hˆ ΥF + ˆ ΥIiΓ 2h ˆ ˜ H0iΓ + C(Γ)

D (φ)

# ψ

SCHRÖDINGER EQUATION

20/27

QUANTUM DYNAMICS OF THE FERMION PERTURBATIONS

21/27

State-dependent conformal time

• Heisenberg picture: ,

ˆ O ∈ FD

dφ ˆ O = i[ ˆ HD, ˆ O] + ∂φ ˆ O

• To make contact with QFT/CS results:

dηΓ = l0h ˆ V 2/3iΓ h ˆ ˜ H0iΓ dφ (dη = e−αdt)

( in LQC !! )

h ˆ V 2/3iΓ > 0

• Annihilation and creation operators:

d⌘Γˆ a(x,y)

~ k

(η, η0) = −iF (Γ)

k

ˆ a(x,y)

~ k

(η, η0) + G(Γ)

k ˆ

b(x,y)†

~ k

(η, η0) d⌘Γˆ b(x,y)†

~ k

(η, η0) = iF (Γ)

k

ˆ b(x,y)†

~ k

(η, η0) − G(Γ)

k ˆ

a(x,y)

~ k

(η, η0) F (Γ)

k

= hˆ ϑF

k iΓ

h ˆ V 2/3iΓ G(Γ)

k

= Mωk 2l2 hˆ ϑI

kiΓ

γh ˆ V 2/3iΓ

EVOLUTION OF FERMION OPERATORS

22/27

• In the classical limit

f (Γ)

1,k =

v u u tF (Γ)

k

− ωk 2F (Γ)

k

, f (Γ)

2,k =

v u u tF (Γ)

k

+ ωk 2F (Γ)

k

ˆ y†

−~ k−2~ ⌧(η, η0) = f (Γ) 2,k ˆ

a(x,y)

~ k

(η, η0) − f (Γ)

1,k ˆ

b(x,y)†

~ k

(η, η0) ˆ x~

k(η, η0) = f (Γ) 1,k ˆ

a(x,y)

~ k

(η, η0) + f (Γ)

2,k ˆ

b(x,y)†

~ k

(η, η0)

F (Γ)

k

→ ξk , f (Γ)

l,k → fl

• Asymptotic analysis in the limit of the e.o.m for ,

ˆ x~

k

ˆ y~

k

• To obtain the explicit expression, first we define

ωk → ∞ ˆ b(x,y)†

~ k

(η, η0) = −¯ βk(η, η0)ˆ a(x,y)

~ k

+ ¯ αk(η, η0)ˆ b(x,y)†

~ k

ˆ a(x,y)

~ k

(η, η0) = αk(η, η0)ˆ a(x,y)

~ k

+ βk(η, η0)ˆ b(x,y)†

~ k

EVOLUTION AS A BOGOLIUBOV TRANSF.

23/27

αk = e−iωk(η−η0) + O(ω−2

k )

• Asymptotic behavior

βk = i M 4l2

0ω2 k

h λ(Γ),0 e−iωk(η−η0) − λ(Γ)

0 eiωk(η−η0)i

+ O(ω−3

k )

Expectation value on of the quantum Hubble parameter

Γ

Vanishes in LQC if the initial time is chosen at the bounce

ˆ b(x,y)†

~ k

(η, η0) = −¯ βk(η, η0)ˆ a(x,y)

~ k

+ ¯ αk(η, η0)ˆ b(x,y)†

~ k

ˆ a(x,y)

~ k

(η, η0) = αk(η, η0)ˆ a(x,y)

~ k

+ βk(η, η0)ˆ b(x,y)†

~ k

UNITARITY OF THE EVOLUTION

X

~ k

|β~

k|2 < ∞

23/27

• There exists unitary operator

ˆ b(x,y)†

~ k

(η, η0) = −¯ βk(η, η0)ˆ a(x,y)

~ k

+ ¯ αk(η, η0)ˆ b(x,y)†

~ k

ˆ a(x,y)

~ k

(η, η0) = αk(η, η0)ˆ a(x,y)

~ k

+ βk(η, η0)ˆ b(x,y)†

~ k

αk = e−iωk(η−η0) + O(ω−2

k )

• Asymptotic behavior

βk = i M 4l2

0ω2 k

h λ(Γ),0 e−iωk(η−η0) − λ(Γ)

0 eiωk(η−η0)i

+ O(ω−3

k )

Expectation value on of the quantum Hubble parameter

Γ

Vanishes in LQC if the initial time is chosen at the bounce

X

~ k

|β~

k|2 < ∞

UNITARITY OF THE EVOLUTION

ˆ U = ˆ U −1ˆ a(x,y)

~ k

ˆ U = ˆ U −1ˆ b(x,y)†

~ k

ˆ U

24/27

• Fock vacuum : annihilated by all operators ,

ˆ a(x,y)

~ k

ˆ b(x,y)

~ k

• Evolved vacuum

EVOLUTION OPERATOR EVOLUTION OF THE FOCK VACUUM

|0i

ˆ U|0i

idφ ˆ U|0i = ˆ HD ˆ U|0i

• Solution of the Schrödinger eq.

24/27

• Fock vacuum : annihilated by all operators ,

ˆ a(x,y)

~ k

ˆ b(x,y)

~ k

• Evolved vacuum

EVOLUTION OPERATOR EVOLUTION OF THE FOCK VACUUM

|0i

ˆ U|0i

idφ ˆ U|0i = ˆ HD ˆ U|0i

• Solution of the Schrödinger eq.

for a specific backreaction term

ˆ HD = " hˆ ΥF + ˆ ΥIiΓ 2h ˆ ˜ H0iΓ + C(Γ)

D (φ)

#

24/27

• Fock vacuum : annihilated by all operators ,

ˆ a(x,y)

~ k

ˆ b(x,y)

~ k

• Evolved vacuum

EVOLUTION OPERATOR EVOLUTION OF THE FOCK VACUUM

|0i

ˆ U|0i

idφ ˆ U|0i = ˆ HD ˆ U|0i

• Solution of the Schrödinger eq.

is indeed the generator of the evolution

ˆ HD ˆ U

24/27

• Fock vacuum : annihilated by all operators ,

ˆ a(x,y)

~ k

ˆ b(x,y)

~ k

• Evolved vacuum

EVOLUTION OPERATOR EVOLUTION OF THE FOCK VACUUM

|0i

ˆ U|0i

idφ ˆ U|0i = ˆ HD ˆ U|0i

• Solution of the Schrödinger eq.

QFT/ QFT/CS Hybrid LQC quantum spacetime

is indeed the generator of the evolution

ˆ HD ˆ U

25/27

• The number of particles (or antiparticles) produced out of the

vacuum in its evolution (as perceived by the original vacuum) is

X

~ k

|β~

k|2 < ∞

• Finite production with our choice for creation and annihilation
• perators (same conclusion in geometrodynamics )

[D’Eath and Halliwell]

• Contribution of modes with large : proportional to

ωk M 2

really small for known fermion fields

• In LQC, initial conditions taken at the bounce minimize even

more this production of particles

PRODUCTION OF PARTICLES

26/27

• Convergence asymptotic behavior

G(Γ)

k =(∆k) =

M 2 8l4

0ω3 k

λ(Γ) n λ(Γ) λ(Γ),0 cos [2ωk(η η0)]

• + O(ω−4

k )

• Divergence absorbed with the renormalization term

c(x,y)

k

= − M 2 8l4

0ω3 k

Z η

η0

⇣ λ(Γ) ⌘2 + O(ω−4

k ) Vanishes in LQC if the initial time is chosen at the bounce

• Dominant term instead of thanks to our Fock rep.

O(ωk)

[D’Eath and Halliwell]

O(ω−3

k )

C(Γ)

D (φ) = l0h ˆ

V 2/3iΓ h ˆ ˜ H0iΓ X

~ k6=~ ⌧,(x,y)

h G(Γ)

k =(∆k) + d⌘Γc(x,y) k

i

BACKREACTION

26/27

• Convergence asymptotic behavior

G(Γ)

k =(∆k) =

M 2 8l4

0ω3 k

λ(Γ) n λ(Γ) λ(Γ),0 cos [2ωk(η η0)]

• + O(ω−4

k )

• Divergence absorbed with

c(x,y)

k

= − M 2 8l4

0ω3 k

Z η

η0

⇣ λ(Γ) ⌘2 + O(ω−4

k )

Vanishes in LQC if the initial time is chosen at the bounce

• Dominant term instead of thanks to our Fock rep.

O(ωk)

[D’Eath and Halliwell]

O(ω−3

k )

C(Γ)

D (φ) = l0h ˆ

V 2/3iΓ h ˆ ˜ H0iΓ X

~ k6=~ ⌧,(x,y)

h G(Γ)

k =(∆k) + d⌘Γc(x,y) k

i

• could be made finite with a different choice of Fock rep.

C(Γ)

D (φ)

BACKREACTION

27/27

• Hybrid LQC quantization of a flat FRW universe with an

inflaton and with fermion perturbations

• Born-Oppenheimer ansatz for physical states and Schrödinger

equation for fermion pert. under certain approximations

• The resulting Hamiltonian explores the quantum nature of the

geometry: QFT/quantum spacetime

• It generates a unitary evolution analog to that of QFT/CS

thanks to our Fock representation for perturbations

• Evolved Fock vacuum exact solution of the Schrödinger eq.
• Well behaved production of particles and backreaction

CONCLUSIONS

THANK YOU FOR YOUR ATTENTION