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

Fermions in LQC Mercedes Martín-Benito (with B. Elizaga Navascués and G.A. Mena Marugán)

INTRODUCTION - Standard Cosmology: Primordial cosmological perturbations over 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? - 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 - Framework: hybrid LQC LQG inspired treatment for the global mode of the geometry - Fock quantization for the inhomogeneities - 01/27

CLASSICAL SYSTEM [D’Eath and Halliwell, ‘87] - Flat FLRW cosmology, with toroidal spatial sections - We include a scalar field subject to a potential - We couple the system to a Dirac field , treated as a perturbation - We truncate the action at quadratic perturbative order - The fermionic part is quadratic in the Dirac field. At our order of truncation, the Dirac field couples only to the homogeneous tetrad, and decouples from the rest of possible perturbations - For simplicity, in this talk we analyze only fermion perturbations 02/27

OUTLINE - Unperturbed FLRW: loop quantization - FLRW with fermion perturbations - Classical system (annihilation/creation variables) - Hybrid quantization (global mode Hamiltonian constraint) - Born-Oppenheimer approx. Schrödinder Eq. - Quantum dynamics of fermion perturbations - Evolution operator and unitarity - Production of particles/antiparticles and backreaction 03/27

LOOP QUANTIZATION OF HOMOGENEOUS FLRW 04/27

HOMOGENEOUS MODEL - Flat FLRW cosmology, with toroidal spatial sections ds 2 = σ 2 ( − N 2 0 ( t ) dt 2 + e 2 α ( t ) 0 h ij d θ i d θ j ) Euclidean angular coordinate cte equal to 4 π / (3 l 3 0 ) metric with period 2 π /l 0 - Homogeneous scalar field subject to a potential (e.g. mass term) φ ✓ 3 H | 0 = e − 3 α - Hamiltonian α + 4 πσ 2 ◆ e 6 α m 2 φ 2 4 π π 2 φ − π 2 = 0 constraint 2 3 - LQC variables: ✓ 3 V ◆ 1 , π α = − 3 3 , V ∝ | v | , e α = { b, v } = 2 2 vb 4 πσ 05/27

POLYMER QUANTIZATION [Bojowald, Ashtekar, Lewandowski, Pawlowski, Singh,….] - Kinematical Hilbert space and elementary quantum operators: L 2 ( R d , dµ d ) ⊗ L 2 ( R , d φ ) v | v i = v | v i ˆ d h v | v 0 i = δ v,v 0 v 2 R e i λ b | v i = | v + 2 λ i ; e ib − d c e − ib b → \ sin( b ) = 2 i - Hamiltonian constraint: ( C 0 = 4 π e 3 α H | 0 / 3) C 0 = 1 3 ⇣ ⌘ V 2 m 2 ˆ H (2) H (2) π 2 Ω 2 φ 2 ˆ φ − ˆ ˆ 4 πγ 2 ˆ 0 − ˆ ˆ , = 0 0 2 Immirzi parameter represents ( vb ) 2 06/27

PHYSICAL STATES - Evolution picture: Scalar plays the role of time Z φ " !# d ˜ φ ˆ h 0 ( v, ˜ χ ( v, φ ) = ˆ ˆ d ˜ φ ˆ U ( v, φ ) χ 0 ( v ) U ( v, φ ) = P exp φ ) , i H 0 φ 0 H (2) ˆ is the square root of 0 q - For massless scalar is time independent ˆ Ω 2 H 0 ∝ 0 L 2 ( R , d λ ) Z ∞ s 3 λ � ( v ) ψ ( λ ) e i ! ( � ) � 0 , d λ e ✏ ˜ χ 0 ( v ) = χ ω ( λ ) = 4 π l 2 P ~ γ 2 0 Ω 2 ˆ (generalized) eigenfunctions of . Non-degenerate and absolutely continuous spectrum 0 07/27

DYNAMICS [Ashtekar, Pawlowski, Singh] - Expectation value of the volume on semiclassical states LQC classical H 2 = 8 π G , H > 0 ρ φ 3 h ˆ V φ i χ v H 2 = 8 π G , H < 0 ρ φ 3 φ [by courtesy of J. Olmedo] 08/27

DYNAMICS [Ashtekar, Pawlowski, Singh] - Expectation value of the volume on semiclassical states LQC classical ✓ ◆ H 2 = 8 π G ρ φ 1 − ρ φ 3 ρ max h ˆ V φ i χ v quantum bounce φ [by courtesy of J. Olmedo] - The spectrum of the energy density in the Physical Hilbert space is bounded from above: ρ max [Ashtekar, Corichi, Singh] 08/27

FLAT FLRW WITH FERMION PERTURBATIONS 09/27

DIRAC FIELD WITH MASS M - Dirac field: cross-sections of the resulting spinor bundle ( D Ψ = M Ψ ) - Using Weyl representation, pair of two-component spinors of definite chirality: (left-handed) and (right-handed) ϕ A ¯ χ A 0 - Partial internal gauge fixing (time gauge): e j 0 = 0 - and defined on parametrized by the time coordinate ϕ A T 3 ¯ χ A 0 - Expansion in the eigenspinors of the Dirac operator on T 3 ± ! k = ± 2 ⇡ | ~ - Discrete spectrum labeled by ( ) : ⌧ | k + ~ ~ k ∈ Z 3 k l 0 - Spin structure characterized by ~ ⌧ 10/27

MODE EXPANSION ω k − ω k ' A ( x ) = e − 3 ↵ ( ⌘ ) / 2 ~ ~ k, (+) k, ( − ) X ( ~ ( ~ ⇥ ⇤ k ( ⌘ ) w ✓ ) + ¯ k ( ⌘ ) w ✓ ) m ~ r ~ , A A � 3 / 2 l 3 / 2 0 ~ k, ( ± ) ~ k ∈ Z 3 � A 0 ( x ) = e − 3 ↵ ( ⌘ ) / 2 ~ ~ k, (+) k, ( − ) X ( ~ ( ~ ⇥ ⇤ ¯ ¯ k ( ⌘ ) ¯ ✓ ) + t ~ k ( ⌘ ) ¯ ✓ ) s ~ w w A 0 A 0 � 3 / 2 l 3 / 2 0 ~ k, ( ± ) ω k − ω k - Let us denote ( x ~ k , ¯ k ) = ( m ~ k , ¯ k ) or ( t ~ k , ¯ k ) y ~ s ~ r ~ - Grasman variables ~ - For simplicity: there are no zero-modes, ⌧ 6 = ~ ~ k 6 = ~ ⌧ 0 11/27

HAMILTONIAN CONSTRAINT - Canonical system: { α , π ↵ } = 1 { φ , π � } = 1 { x ~ k } D = − i { y ~ k } D = − i k , ¯ k , ¯ , , x ~ , y ~ - Subject to a global Hamiltonian constraint coupling both sectors X H | 0 + H D = 0 , H D = H ~ k ~ k 6 = ⌧ ~ ⌧ � � ⌧ + ¯ H ~ k = σ M s − ~ ⌧ m ~ m ~ s − ~ ⌧ + r ~ k t − ~ t − ~ r ~ k + ¯ k ¯ ⌧ ¯ k − 2 ~ k − 2 ~ k − 2 ~ k − 2 ~ k − e − ↵ ω k � � k + ¯ m ~ k m ~ t ~ k t ~ k − r ~ r ~ k − s ~ s ~ ¯ k ¯ k ¯ k 12/27

ANNIHILATION AND CREATION VAR. 1 - We ask them to respect the linearity of the equations of motion and the dynamical decoupling between modes ~ ~ a ( x,y ) k, ( x,y ) k, ( x,y ) = f x ~ k + f y − ~ ¯ 1 2 { a ( x,y ) a ( x,y ) ~ k − 2 ~ ⌧ k } = − i , ¯ ~ ~ k k { b ( x,y ) b ( x,y ) ~ ~ , ¯ b ( x,y ) k, ( x,y ) k, ( x,y ) } = − i ¯ = g x ~ k + g y − ~ ¯ ~ ~ k k 1 2 ~ k − 2 ~ ⌧ k ( α , π α ) -dependent in general - Infinite ambiguity in their definition - The coefficients extract part of the dynamics of the field, as attributed to the homogeneous geometry 13/27

ANNIHILATION AND CREATION VAR. 2 - Variables for instantaneous diagonalization [D’Eath and Halliwell, ‘87] s ξ k + ω k ~ ~ k, ( x,y ) k, ( x,y ) f = g = 2 1 2 ξ k q ω 2 k + ( σ Me α ) 2 ξ k = s ξ k − ω k ~ ~ k, ( x,y ) k, ( x,y ) f = − g = 1 2 2 ξ k - 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 - Usual convention for particles/antiparticles in the massless limit - Unitary implementation of the quantum dynamics [Cortez, Elizaga Navascués, Martín-Benito, Mena Marugán, Velhinho, ‘16] Restricts the asymptotic behavior in the ultraviolet limit ω k → ∞ 14/27

ANNIHILATION AND CREATION VAR. 2 - Variables for instantaneous diagonalization [D’Eath and Halliwell, ‘87] ~ ~ k, ( x,y ) k, ( x,y ) = 1 + O ( ω − 2 = g k ) f 2 1 q ω 2 k + ( σ Me α ) 2 ξ k = = Me ↵ ~ ~ k, ( x,y ) k, ( x,y ) + o ( ω − 1 = − g k ) f 1 2 2 ω k - 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 - Usual convention for particles/antiparticles in the massless limit - Unitary implementation of the quantum dynamics [Cortez, Elizaga Navascués, Martín-Benito, Mena Marugán, Velhinho, ‘16] Restricts the asymptotic behavior in the ultraviolet limit ω k → ∞ 14/27

CANONICAL TRANSFORMATION - Inhomogeneous sector ( a ( x,y ) a ( x,y ) , b ( x,y ) b ( x,y ) , ¯ ( x ~ k , ¯ k , ¯ k ) , ¯ ) x ~ k , y ~ y ~ → ~ ~ ~ ~ k k k k α - dependent canonical transformation - We need to correct the homogeneous sector ✓ ◆ π ↵ = π ↵ + i σ M ω k a ( x,y ) b ( x,y ) a ( x,y ) b ( x,y ) e ↵ X ¯ + α -dependent term quadratic in the fermion variables π ↵ → ˘ + ¯ − 2 ξ 2 ~ ~ ~ ~ k k k k k ~ k, ( x,y ) H | 0 ( α , ˘ π α ) − (˘ π α − π α ) ∂ ˘ π α H | 0 ( α , ˘ π α ) + H D [ a, b ] ˘ H D generates the dynamical evolution of the creation and annihilation variables 15/27

HYBRID QUANTIZATION Assumption: - Main quantum gravity effects are those affecting the global degrees of freedom of the geometry. - Inhomogeneities, even though are also quantum, can be treated in a more conventional way. By quantizing the homogeneous geometry sector with QG techniques, we hope to retain the main quantum effects 16/27

HYBRID QUANTIZATION Assumption: - Main quantum gravity effects are those affecting the global degrees of freedom of the geometry. - Inhomogeneities, even though are also quantum, can be treated in a more conventional way. By quantizing the homogeneous geometry sector with QG techniques, we hope to retain the main quantum effects QG QFT/CS 16/27

HYBRID QUANTIZATION Assumption: - Main quantum gravity effects are those affecting the global degrees of freedom of the geometry. - Inhomogeneities, even though are also quantum, can be treated in a more conventional way. By quantizing the homogeneous geometry sector with QG techniques, we hope to retain the main quantum effects Hybrid LQC QG QFT/CS 16/27