Download Unstable Modes of a Boson Field in an Accelerating

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
Oscillator
-Fock Space
Classical Mechanics

Lagrangian
𝐿 𝑞, 𝑞 = 𝑇 − 𝑉 =
1
1
𝑚𝑞2 − 𝑘𝑞2
2
2

Euler-Lagrange equations
𝜕𝐿
𝜕 𝜕𝐿
−
=0
𝜕𝑞𝑖 𝜕𝑡 𝜕𝑞𝑖

Conjugate momentum
𝜕𝐿
𝑝𝑖 =
= 𝑚𝑞𝑖
𝜕𝑞𝑖

Hamiltonian (energy)
1 𝑝2
𝐻 𝑞, 𝑝 = 𝑇 + 𝑉 = 𝑝𝑞 − L =
+ 𝑘𝑞 2
2 𝑚
Quantum Mechanics

Dynamical variables → non-commuting operators
[𝑥𝑖 , 𝑥𝑗 ] = [𝑝𝑖 , 𝑝𝑗 ]=0
𝑥𝑖 , 𝑝𝑗 = 𝑖δ𝑖𝑗

Most commonly used
𝑥𝑖 Ψ 𝐱, 𝑡 = 𝑥𝑖 Ψ(𝐱, 𝑡)
𝜕Ψ
𝑝𝑖 Ψ = −𝑖
𝜕𝑥𝑖

Expectation value
𝑂 = ∫ Ψ ∗ 𝑂Ψ 𝑑 3 𝑥
Quantum Harmonic Oscillator

Hamiltonian – energy operator

Eigenstates 𝜓𝑛 with eigenvalue 𝐸𝑛 = 𝑛 +

Creation and annihilation operators
𝑎=
1
√2
𝑚𝜔
𝑥
ℏ
+𝑖
ℏ
𝑝
𝑚𝜔
𝑎† =
,
𝐻 =𝑇+𝑉 =
1
√2
𝑚𝜔
𝑥
ℏ
Number operator
−𝑖
𝑎† 𝑎 𝜓𝑛 = 𝑛 𝜓𝑛
𝐻=
ℏ𝜔
ℏ
𝑝
𝑚𝜔
𝑎† 𝜓𝑛 = 𝜓𝑛+1
𝑎𝜓𝑛 = 𝜓𝑛−1

1
2
𝑎† 𝑎
1
+
ℏ𝜔
2
1 𝑝2
2 𝑚
+ 𝑚𝜔2 𝑥 2
Quantum Field Theory

Euler-Lagrange equations
𝜕ℒ
𝜕 𝜕ℒ
− 𝜇
=0
𝜕𝜙 𝜕𝑥 𝜕𝜙,𝜇
→ Klein-Gordon equation

g       (m 2  R)  0
Conjugate field
𝜕ℒ
𝜋(𝐱, 𝑡) =
𝜕𝜙,0

Commutation relations
𝜙 𝐱, 𝑡 , 𝜙 𝐲, 𝑡 = [𝜋 𝐱, 𝑡 , 𝜋(𝐲, 𝑡)]=0
𝜙 𝐱, 𝑡 , 𝜋 𝐲, 𝑡 = 𝑖δ(𝐱 − 𝐲)

Hamiltonian density
ℋ 𝜋, 𝜙 = 𝜋𝜙 − ℒ
Fock Space

Basis
|ℏ𝜔𝑘
where |ℏ𝜔𝑘 are e’vectors with energy e’value ℏ𝜔𝑘
|𝑛1 , 𝑛2 , … , 𝑛𝑘 , …

Vectors

Vacuum state

Creation
|0
𝑎𝑘† … , 𝑛𝑘 , … =
and annihilation operators

Number operator 𝑁𝑖 = 𝑎𝑖† 𝑎𝑖

Commutation relations
𝑛𝑘 + 1 … , 𝑛𝑘 + 1, …
𝑎𝑘 |… , 𝑛𝑘 , … = 𝑛𝑘 |… , 𝑛𝑘 − 1, …
𝑁𝑖 |𝑛1 , 𝑛2 , … , 𝑛𝑖 , … = 𝑛𝑖 |𝑛1 , 𝑛2 , … , 𝑛𝑖 , …
𝑎𝑗 , 𝑎𝑘 = 𝑎𝑗† , 𝑎𝑘† = 0
𝑎𝑗 , 𝑎𝑘† = 𝑖𝛿𝑗𝑘
Klein-Gordon

g      (m  R )  0

Change to time coordinate

K-G becomes
2
  1 e
  0
2
3 2

   m  2(6  1) 1    2


2

Unstable when 𝜇2 < 0, requires
𝜉<
t
1
6
𝑚=0
Canonical Quantisation
 (r , ,  , )  
kS
𝜕ℒ
𝜕𝜙
𝜋=
=
=
𝜕𝜂
𝜕𝜙

l
 
l 0
ml
∞
𝑙
J l  1 (kr )
2
kr
𝐽
𝑘∈𝑆 𝑙=0 𝑚=−𝑙
𝑙+
1
2
†
Ylm ( ,  ){aklm f k ( )  aklm
f k* ( )}
𝑘𝑟
𝑘𝑟
Commutation relations for creation and annihilation operators
†
𝑎𝑘𝑙𝑚 , 𝑎𝑘 ′ 𝑙′ 𝑚′ = 𝑎𝑘𝑙𝑚
, 𝑎𝑘†′ 𝑙′ 𝑚′ = 0
[aklm , ak 'l 'm ' ]  0
†
[aklm
, ak†'l 'm ' ]  0
[aklm , ak†'l 'm ' ]  i kk ' ll ' mm '
∗
†
𝑌𝑙𝑚 (𝜃, 𝜑) 𝑎𝑘𝑙𝑚 𝑓 ′ 𝑘 (𝜂) + 𝑎𝑘𝑙𝑚
𝑓𝑘′ (𝜂)
𝑎𝑘𝑙𝑚 , 𝑎𝑘†′ 𝑙′ 𝑚′ = 𝑖𝛿𝑘𝑘 ′ 𝛿𝑙𝑙′ 𝛿𝑚𝑚′
Hamiltonian density
ℋ 𝜋, 𝜙 = 𝜋𝜙 − ℒ
Hamiltonian

Sum of quadratic terms
H klm
1 †
 [aklm
2
 W ( f k ' , f k* )
0
0
(1) m W ( f k* ' , f k* )


0
W ( f k ' , f k* ) (1) m W ( f k* ' , f k* )
0


Dklm  Akl
m
*


0
(1) W ( f k ' , f k ) W ( f k ' , f k )
0
 m

0
0
W ( f k ' , f k* ) 
(1) W ( f k ' , f k )

Bogoliubov transformation
akl†  m
aklm
 aklm 
a 
kl  m 

akl  m ]Dklm †
 aklm 
 † 
akl  m 
Bogoliubov transformation preserves
Canonical Commutation Relations
H klm
1 †
 [bklm
2
 aklm 
 bklm 
a 
b 
 kl  m   T  kl  m 
†
†
 aklm

 bklm

 † 
 † 
a
 kl  m 
bkl  m 
bkl†  m
bklm
 bklm 
b 
bkl  m ]D klm  kl† m 
 bklm 
 † 
bkl  m 
D klm  T † DklmT
Bogoliubov Transformation

Preserves eigenvalues 𝜔𝑘 of
𝜔𝑘𝑙𝑚 = ±𝐴𝑘𝑙 𝑊
𝑓𝑘′ , 𝑓𝑘∗ 2
−𝑊
IˆDklm
𝑓𝑘′ , 𝑓𝑘
I
Iˆ   2
0
′∗ ∗
𝑊(𝑓𝑘 , 𝑓𝑘 )

Real when k 2   2

Purely imaginary klm  i klm when k 2   2
0 

 I 2 
Energy Partitioning
H  HL  HD
HL 
HD 

l
k ( )
m 1
2

kS
l 0
2
2
k 

l

kS
l 0
2
2
k 
m 1
†
†
{bkl  m bkl†  m  bklmbklm
 bklm
bklm  bkl†  mbkl  m }
i k ( )
†
†
{bkl  m bkl  m  bklmbklm  bklm
bklm
 bkl†  mbkl†  m }
2
𝜕𝜙
= 0,
𝜕𝑟
𝑟=1
𝑆 = 𝑘 ∈ ℝ: ∃ℓ ∈ ℕ ∪ 0
′
𝑗ℓ
𝑘 =0
𝜕𝜙
= 0,
𝜕𝑟
𝑟=1
𝑆 = 𝑘 ∈ ℝ: ∃ℓ ∈ ℕ ∪ 0
k 
2
2
′
𝑗ℓ
𝑘 =0
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
𝑘 2 = 𝜇2
𝑡 = log 𝑘/√2

First mode k=2.2
t ≈ now

Modes of wavelength 1.07μm
t ≈ 100×current age of universe
Current/Future work



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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)



Candidate for a UV completion General Relativity
Higher derivative corrections to the Lagrangian
Dispersion relation for scalar fields (Visser Phys. Rev. D 2009)
𝜔2 = 𝑚2 + 𝑘 2 + 𝑎1 𝑘 4 + 𝑎2 𝑘 6
Development of unstable modes