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Path Integral Formulation of Loop Quantum Cosmology

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Path Integral Formulation of Loop Quantum Cosmology Adam D. Henderson Institute for Gravitation and the Cosmos Pennsylvania State University International Loop Quantum Gravity Seminar March 17 2008 Motivations 1. There have been arguments against the singularity resolution in LQC using path integrals. The canonical quantization is well understood and the singularity resolution is robust, so where do the path integral arguments break down? 2. Recent developements have allowed for the construction of new spin foam models that are more closely related to the canonical theory. It is important to make connections to spin-foams by building path integrals from the canonical theory when possible. 3. A path integral representation of Loop Quantum Cosmology will aid in studying the physics of more complicated LQC models (k = 1, Λ = 0, Bianchi). Using standard path integral techniques one can 6 argue why the effective equations are so surprisingly accurate. Introduction We construct a path integral for the exactly soluble Loop Quantum • Cosmology starting with the canonical quantum theory. The construction defines each component of the path integral. Each • has non-trivial changes from the standard path integral. We see the origin of singularity resolution in the path integral • representation of LQC. The structure of the path integral features similarities to spin foam • models. The path integral can give an argument for the surprising accuracy • of the effective equations used in more complicated models. Outline Standard Construction Exactly Soluble LQC LQC Path Integral Other Forms of Path Integral Singularity Resolution/Effective Equations Conclusion Path Integrals - Overview Path Integrals: Covariant formulation of quantum theory expressed • as a sum over paths Formally the path integral can be written as • qeiS[q]/~ or q peiS[q,p]/~ (1) D D D Z Z To define a path integral directly involves defining the components • of these formal expressions: 1. What is the space of paths q(t) over which we integrate? 2. : What is the measure on this space of paths? 3. SD[q]: What is the phase associated with each path. Each of these components can be determined by constructing the • path integral from the canonical theory (when possible). Standard Construction Outline the standard construction of a path integral representation • for Non-Relativistic Quantum Mechanics with a polynomial Hamiltonian, H(q, p). We want path integral representation of the propagator • iHˆ ∆t/~ x′ e− x = x′, t′ x, t (2) h | | i h | i Split the exponential into a product of N identical terms. • N iHˆ ǫ/~ ∆t x′ e− x where ǫ = (3) h | | i N nY=1 Insert a complete basis in x between each exponential. • iHˆǫ/~ iHˆ ǫ/~ x′ e− dxn 1 xn 1 xn 1 e− ... x (4) h | − | − ih − | | i Z Step 2 - Expand in ǫ and Evaluate By defining x = x & x = x this can be written in a simple form. • N ′ 0 N 1 N iHˆ ∆t/~ − iHˆ ǫ/~ x′ e− x = dxm xn e− xn 1 (5) h | | i h | | − i mY=1 Z nY=1 In the limit N (ǫ 0) we can expand each term in the 2nd • →∞ → product of eqn (5) in ǫ. iHˆ ǫ/~ 2 xn e− xn 1 = xn 1 iHˆ ǫ/~ xn 1 + (ǫ ) (6) h | | − i h | − | − i O This can be written as an integral over momentum p . • n 1 iHˆ ǫ/~ ipn(xn xn−1)/~ xn e− xn 1 = dpne − (7) h | | − i 2π~ Z 2 [1 iǫH(pn, xn 1, xn)/~ + (ǫ )] × − − O Step 3 Inserting the matrix elements x e iHˆ ǫ/~ x into the expression • n − n 1 for the full matrix element andh simplifying.| | − i N 1 N iHˆ∆t/~ − 1 x′ e− x = Lim dxm dpn (8) h | | i N 2π~ →∞ mY=1 Z nY=1 Z N N iǫ P (pn(xn xn−1)/ǫ~) 2 e n=1 − [1 iǫH(pn, xn, xn 1)/~ + (ǫ )] − − O n 1 Y= The limit N defines the measure of the phase space path • integral integral→∞ as the integral over the position at each time between t & t’ and the integral over all momenta. Almost in path integral form, except the final product. • Infinite Product: Re-Exponentiation The infinite product has the form • N 2 Lim (1 + anǫ + (ǫ )) where ǫ =∆t/N (9) N O →∞ nY=1 If an = a n: Recall from intro calculus • ∀ Lim (1 + aǫ + (ǫ2))N = ea∆T (10) N O →∞ If the terms are not the same for all n this can be generalized. • N N P anǫ Lim (1 + anǫ)= Lim e n=1 (11) N N →∞ →∞ nY=1 Extending to the product in the previous slide N N 2 Lim [1 iǫH(pn, xn 1, xn)~+ (ǫ )] = exp i/~ Lim ǫH(pn, xn 1, xn) N − − O − N − →∞ n=1 " →∞ n=1 # Y X (12) Final Result Combining these results: • N 1 N iHˆ ∆t/~ − 1 x′ e− x = Lim dxm dpn (13) h | | i N 2π~ →∞ mY=1 Z nY=1 Z exp i/~ S[x, p, N] Where we recognize S as the discretized action. • N S[x, p, N]= ǫ (pn(xn xn 1)/ǫ H[xn, xn 1, pn]) (14) − − − − Xn=1 In the limit N this is the action • →∞ S[x, p]= dt(px˙ H(x, p)) (15) − Z Overview of Path Integral Construction Strategy to construct path integral 1. Start from the propagator x , t x, t h ′ ′| i i∆t/~H iǫH/~ 2. Split e− b into N copies e− b iǫH/~ 3. Insert a complete basis between each factor of e− b iHˆ ǫ/~ 4. Evaluate each xn e− xn 1 to first order in ǫ h | | − i 5. Re-exponentiate resulting expression. SLQC - Classical We work with k=0 FRW model in the b, ν variables with a scalar • field. 3o a V 4πγG pa ν = ε 2 , b = ε o 2 (16) 2πℓPlγ − 3 V a With Poisson bracket • b,ν = 2/~ (17) { } Classically they have range ( , ). • −∞ ∞ To simplify the constraint the lapse is chosen to be N = a3N • ′ The phase space action is then • ~ N S = dt[b˙ν + p φ˙ ′ (p2 3π~2Gb2ν2)] (18) 2 φ − 2 φ − Z Argument Against Singularity Resolution We can try to construct a path integral directly from this classical • theory. The path integral will be something of the form, • iS/~ ν′, t′ ν, t = b ν...e (19) h | i D D Z The action evaluated along the classical path between ν and ν is • ′ pφ S = ln(ν/ν′) (20) √12πG Quantum corrections are negligible when S >> ~ • cl No quantum corrections to classical singular paths No bounce! • → SLQC - Physical Hilbert Space We will construct the path integral starting from the physical Hilbert • space of SLQC. The physical states are solutions to the quantum Hamiltonian • constraint. ∂2ψ(ν, φ)= Θψ(ν, φ) (21) φ − The physical Hilbert space can be obtained by group averaging • b procedure. Find that physical states satisfy a Schrodinger like equation. • i∂ ψ(ν, φ)= √Θψ(ν, φ) (22) − φ The physical inner product is. d • λ 1 ψ1 ψ2 = ψ¯1(ν, φo)ψ2(ν, φo) (23) h | i π ν νX=4nλ | | Schrodinger Eqn Path Integral Construction → We have SLQC written in the form of a Schrodinger equation with φ • as time, so we apply the construction above to obtain a path integral. Similar to the construction in non-relativistic quantum mechanics • we want a path integral representation of the propagator. i√Θ∆φ ν′ e d ν = ν′, φ′ ν, φ (24) h | | i h | i More generally we could construct the path integral from the • definition of the physical inner product in terms of group averaging. SLQC Propagator - Exact We have some knowledge of the exact propagator. • One can show that • ν′, φ′ ν, φ = 0 if ν′ < 0 & ν > 0 (25) h | i This allows us to simplify the calculations by restricting to positive • or negative ν The propagator can be written as an integral. • π/λ λ i[ bν 1 tan−1(e∆φ tan(λb/2))ν′] ν′, φ′ ν, φ = db e 2 − λ (26) h | i 2πν Z0 +(∆φ ∆φ) →− Step 1 - Split Exponential/Insert Complete Basis As before we split the exponential into N copies and insert a • complete basis of ν between each. π 1 = ν ν ν (27) λ | | | ih | νX=4nλ Giving • N 1 N − π iǫ√Θ ν′, φ′ ν, φ = νn νn e d νn 1 (28) h | i λ | | h | | − i " ν # nY=1 Xn nY=1 Important difference: Instead of continuous integrals there are • discrete sums over ν at each φ. The next step is to compute each term of the product: • iǫ√dΘ νn e νn 1 (29) h | | − i Step 2 - Evaluate Each Term in Product Want to evaluate each term to first order in ǫ • iǫ√Θ νn e d νn 1 (30) h | | − i Problem: Even computing to first order in epsilon requires knowing • the spectrum of Θ The resolution is to rewrite each term as • b ǫ iǫ√dΘ ∞ ∞ νn e νn 1 = dpφn pφn Θ(pφn ) dNn (31) h | | − i | | 2π~ Z−∞ Z−∞ Nn 2 2 ipφ ǫ/~ iǫ (pφ ~ Θ) x e n νn e− 2~ n − b νn 1 h | | − i We then only need to evaluate the the following to first order in • epsilon. ~ iǫ Nn Θ) νn e 2 b νn 1 (32) h | | − i Step 2b - Try Again Evaluating this term is simple given the action of Θ • ~ ~ iǫ Nn Θ λ Nn 3πG νn + νn 1 2 b b − νn e νn 1 = δνn,νn−1 iǫ 2 νn (33) h | | − i πνn 1 − 2 4λ 2 − 2 [δνn,νn−1+4λ + δνn,νn−1 4λ 2δνn,νn−1 ] + (ǫ ) × − − O Expressed as an integral by writing the delta functions as integrals • over b.
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