Unstable KleinGordon modes in an accelerating universe Dark Energy does not behave like particles or radiation Quantised unstable modes no particle or radiation interpretation Accelerating universe ID: 497773
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Slide1
Unstable Klein-Gordon Modes in an Accelerating UniverseSlide2
Unstable Klein-Gordon modes in an accelerating universe
Dark Energy
-does not behave like particles or radiation
Quantised unstable modes
-no particle or radiation interpretation
Accelerating universe
-produces unstable Klein-Gordon modesSlide3
Plan
Solve K-G coupled to exponentially accelerating space background
Canonical quantisation
->Hamiltonian partitioned into stable and unstable components
Fundamental units of unstable component have no Fock representationFinite no. of unstable modes + Stone von Neumann theorem -> Theory makes senseSlide4
BASICS
CM QM QFT -
Qm
Harmonic
-Fock Space Oscillator Slide5
Classical Mechanics
Lagrangian
Euler-Lagrange equations
Conjugate momentum
Hamiltonian (energy)
Slide6
Quantum Mechanics
Dynamical variables → non-commuting operators
Most commonly used
Expectation value
Slide7
Quantum Harmonic Oscillator
Hamiltonian – energy operato
r
Eigenstates
with eigenvalue
Creation and annihilation operators
=
Number operator
Slide8
Quantum Field Theory
Euler-Lagrange equations
→ Klein-Gordon equation
Conjugate
field
Commutation relations
Hamiltonian density
Slide9
Fock
Space
Basis
where are e’vectors with energy e’value Vectors
Vacuum
state
Creation
and
annihilation operators
Number operator
Commutation relations
Slide10
Klein-Gordon
Unstable when
requires
Change to time
coordinate
K-G becomesSlide11
Canonical Quantisation
Commutation relations for creation and annihilation operators
Hamiltonian density
Slide12
Hamiltonian
Sum of quadratic terms
Bogoliubov
transformationSlide13
Bogoliubov
t
ransformation preserves Canonical Commutation RelationsSlide14
Bogoliubov Transformation
Preserves eigenvalues
of
Real when
Purely imaginary when
Slide15
Energy PartitioningSlide16
Slide17
Slide18
Existence of Preferred Physical Representation
Stone-von Neumann Theorem guarantees a preferred representation for H
D
H
L has usual Fock representationThere is a preferred representation for the whole systemSlide19
Cosmological Consequences
Modes become unstable when
First mode k=2.2
t ≈
now
Modes of wavelength 1.07
μ
m
t ≈ 100×current age of universe
Slide20
Current/Future work
This theory is semi-classical
Dark energy at really long wavelengths
A quantum gravity theory
D
ark energy at short wavelengths (we hope!)Slide21
Horava
Gravity (Horava Phys. Rev. D 2009)
C
andidate for a UV completion General Relativity
H
igher
derivative corrections to the
Lagrangian
Dispersion
relation for scalar fields
(Visser Phys. Rev. D 2009)Slide22
Development of unstable modes