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Unstable Klein-Gordon Modes in an Accelerating Universe. 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

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unstable klein gordon modes in an accelerating universe1
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 modes

slide3
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 Fockrepresentation
  • Finite no. of unstable modes + Stone von Neumann theorem
  • -> Theory makes sense
basics
BASICS
  • CM
  • QMQFT
  • -Qm Harmonic -FockSpace Oscillator
classical mechanics
Classical Mechanics
  • Lagrangian
  • Euler-Lagrange equations
  • Conjugate momentum
  • Hamiltonian (energy)
quantum mechanics
Quantum Mechanics
  • Dynamical variables → non-commuting operators
  • Most commonly used
  • Expectation value
quantum harmonic oscillator
Quantum Harmonic Oscillator
  • Hamiltonian – energy operator
  • Eigenstates with eigenvalue
  • Creation and annihilation operators
  • =
  • Number operator
quantum field theory
Quantum Field Theory
  • Euler-Lagrange equations
  • → Klein-Gordon equation
  • Conjugate field
  • Commutation relations
  • Hamiltonian density
fock space
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

  • Change to time coordinate
  • K-G becomes
  • Unstable when requires
canonical quantisation
Canonical Quantisation
  • Commutation relations for creation and annihilation operators
  • Hamiltonian density
hamiltonian
Hamiltonian
  • Sum of quadratic terms
  • Bogoliubov transformation
bogoliubov transformation
Bogoliubov Transformation
  • Preserves eigenvalues of
  • Real when
  • Purely imaginary when
existence of preferred physical representation
Existence of Preferred Physical Representation

Stone-von Neumann Theorem guarantees a preferred representation for HD

HL has usual Fock representation

There is a preferred representation for the whole system

cosmological consequences
Cosmological Consequences
  • Modes become unstable when
  • First mode k=2.2

t ≈ now

  • Modes of wavelength 1.07μm

t ≈ 100×current age of universe

current future work
Current/Future work
  • This theory is semi-classical
  • Dark energy at really long wavelengths
  • A quantum gravity theory
  • Dark energy at short wavelengths (we hope!)
horava gravity horava phys rev d 2009
Horava Gravity (HoravaPhys. Rev. D 2009)
  • Candidate for a UV completion General Relativity
  • Higher derivative corrections to the Lagrangian
  • Dispersion relation for scalar fields (VisserPhys. Rev. D 2009)