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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 -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 , , , ) kS 𝜕ℒ 𝜕𝜙 𝜋= = = 𝜕𝜂 𝜕𝜙 l l 0 ml ∞ 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 kS l 0 2 2 k l kS 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 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