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
-produces unstable Klein-Gordon modes
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
representation Finite no. of unstable modes + Stone von Neumann theorem -> Theory makes sense
BASICS
CM
QM QFT
-Qm Harmonic -Fock Space Oscillator
Classical Mechanics Lagrangian
Euler-Lagrange equations
Conjugate momentum
Hamiltonian (energy)
Quantum Mechanics Dynamical variables → non-commuting operators
Most commonly used
Expectation value
Quantum Harmonic Oscillator Hamiltonian – energy operator Eigenstates with eigenvalue
Creation and annihilation operators=
Number operator
Quantum Field Theory Euler-Lagrange equations
→ Klein-Gordon equation Conjugate field
Commutation relations
Hamiltonian density
0)( 2 Rmg
Fock Space Basis where are e’vectors with energy e’value Vectors Vacuum state Creation and annihilation operators Number operator Commutation relations
Klein-Gordon
222 1)16(23
m
Unstable when requires
02
te 1 Change to time coordinate
K-G becomes
0)( 2 Rmg
Canonical Quantisation)}()({),(
)(),,,( *†
0
21
kklmkk lm
l
lmlm
l
lSk
fafaYkr
krJr
'''†
'''
†'''
†'''
],[
0],[
0],[
mmllkkmlkklm
mlkklm
mlkklm
iaa
aa
aa
𝜋=𝜕ℒ𝜕 ��
=𝜕𝜙𝜕𝜂=∑𝑘∈𝑆∑𝑙=0
∞
∑𝑚=−𝑙
𝑙 𝐽𝑙+1
2
(𝑘𝑟 )
√𝑘𝑟𝑌 𝑙𝑚(𝜃 ,𝜑) {𝑎𝑘𝑙𝑚 𝑓 ′
𝑘(𝜂)+𝑎𝑘𝑙𝑚† 𝑓 𝑘′ ∗(𝜂) }
Commutation relations for creation and annihilation operators
Hamiltonian density
Hamiltonian Sum of quadratic terms
Bogoliubov transformation
†
††† ][
21
mkl
klm
mkl
klm
klmmklk lmmklklmklm
aa
aa
DaaaaH
),'(00),'()1(0),'(),'()1(00),'()1(),'(0
),'()1(00),'(
*
*
***
***
kkkkm
kkkkm
kkm
kk
kkm
kk
klklm
ffWffWffWffW
ffWffWffWffW
AD
Bogoliubov transformation preserves Canonical Commutation Relations
†
††† ][
21
mkl
k lm
mkl
k lm
klmmklk lmmklk lmklm
bb
bb
DbbbbH
TDTD klmklm†
†
†
†
†
mkl
klm
mkl
klm
mkl
klm
mkl
klm
bb
bb
T
aa
aa
Bogoliubov Transformation Preserves eigenvalues of
Real when
Purely imaginary when
k lmDI
2
2
00ˆI
II
22 k
klmklm i 22 k
Energy Partitioning
}{2
)( ††††
1022
mklmklk lmklmklmklmmklmklk
l
mlk
SkL bbbbbbbbH
DL HHH
}{2
)( ††††
1022
mklmklklmklmklmklmmklmklk
l
mlk
SkD bbbbbbbbiH
𝑆= {𝑘∈ℝ : (∃ ℓ∈ℕ∪0 ) 𝑗ℓ′ (𝑘 )=0 }
𝜕𝜙𝜕𝑟 =0 ,𝑟=1
𝑆= {𝑘∈ℝ : (∃ ℓ∈ℕ∪0 ) 𝑗ℓ′ (𝑘 )=0 }
𝜕𝜙𝜕𝑟 =0 ,𝑟=1
22 k
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 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 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)
𝜔2=𝑚2+𝑘2+𝑎1𝑘4+𝑎2𝑘6
Candidate for a UV completion General Relativity Higher derivative corrections to the Lagrangian Dispersion relation for scalar fields (Visser Phys. Rev. D 2009)
Development of unstable modes
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