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Scattering Using Minkowski Path Integrals AMCS Scattering using Minkowski path integrals AMCS - April 22, 2016 W. N. Polyzou - The University of Iowa Work motivated by Katya Nathanson (UI thesis) and Palle Jorgensen Journal of Mathematical Physics 56, 092102 (2015) Research supported by the US DOE Office of Science Physics Background - Quantum field theory • Quantum field theory = microscopic theory of strong, weak, and electromagnetic interactions. • Solution requires solving an infinite number of coupled non-linear integral equations in an infinite number of variables. • Mathematical existence of solutions is in question (Millennium problem). • Weak coupling case - perturbation theory exists after subtracting infinite quantities, does not converge, but gives the most accurate prediction in physics (electron magnetic moment). g=2 = 1:15965218178(77)(th) g=2 = 1:15965218073(28)(ex) • Some fundamental truth - many questions • Strong force - perturbation theory not applicable. • Feynman - reduced solution to a formal limit of 1-dimensional integrals - interpreted as an integral over classical paths. • No positive countably additive candidate for the Feynman measure on the space of paths. • Rigorous tool for generating a well-defined perturbation series. • Provides a computational algorithm that is inherently non-perturbative. Convergence not proven. • Analytic continuation to imaginary time recovers a positive measure on spaces of functions. Conventional wisdom / observations • Only a small subset of path integrals can be computed exactly (typically those that can be reduced to an infinite product of Gaussian integrals) Z 2 Y −i(ai x +bi xi +ci ) e i dxi • Numerical treatments require analytic continuationsp and Monte Carlo integration (converges like 1= N independent of dimension). • Testable in quantum mechanics - implications for quantum field theory? Nathanson and Jorgensen • Recast the path integral as an expectation value of a random variable over a complex probability on a space of paths: X K(x0; t0; x; t) = E[e−iV [γ]] = P[γ]e−iV [γ] γ • Result satisfies Schrodinger equation. • Method is not limited to Gaussian random variables. • Method does not require analytic continuation to imaginary time. • May be applicable to scattering? • Computational methods may extend to field theory case? Quantum mechanics: • Mathematical setting: Complex Hilbert space inner product ( ; φ) • Physical prediction: Probability for measuring state in system prepared in state φ ( ; φ)(φ, ) P = = j( ^; φ^)j2 ( ; )(φ, φ) • Origin of dynamics: Equivalent experiments at different times give identical probabilities ! time translations given by unitary one-parameter group, U(t) = e−iHt : U(t)U(s) = U(t + s) U(0) = IUy(t) = U(−t) U(t): U(t) (0) = (t)( ^(t); φ^(t)) = ( ^(0); φ^(0)) Scattering theory • Main tool for studying properties of particles that are too small to see. • Same principle as a microscope; light reflected (scattered) from target viewed far away from target - (i.e. in eye) provides image of target. • Resolution limited by wavelength - in quantum theory the relevant wavelength is ~=(momentum) where ~ is Planck's constant=(2π). Scattering probabilities 2 2 P = j( ^+(t); ^−(t))j = j( ^+(0); ^−(0))j : Scattering initial conditions lim kU(t) ^±(0) − U0(t) ^±0(0)k = 0 t→±∞ m lim k ^±(0) − U(−t)U0(t) ^±0(0)k = 0: t→±∞ Scattering states (vectors) expressed in terms of free particle asymptotes Relation of scattering probabilities to dynamics 2 P = lim j( ^+0(0); U0(−t)U(t)U(t)U0(−t) ^−0(0))j = t!1 2 lim j( ^+0(−t); U(2t) ^−0(−t))j ≈ t!1 2 j( ^+0(−T ); U(2T ) ^−0(−T ))j T sufficiently large • This form is natural for a path integral formulation. • U(−t)U0(t) adiabatically switches on interaction - state remains localized near point of scattering. What is the path integral? U(t) = e−iHt = e−i(K+V )t [K; V ] 6= 0 x ex = lim (1 + )N : N!1 N Operator version = Trotter product formula U(t) = e−i(K+V )t = [e−i(K+V )t=N ]N = t t N −iK t −iV t N lim [(1 − iK )(1 − iV )] = lim [e N e N ] N!1 N N N!1 insert spectral (eigenfunction) expansions of K and V so they become multiplication operators. Result: scattering probability amplitude has the form ( ^+(0); ^−(0)) ≈ N 2 Z pi Y dpi dxi ipi (xi−1−xi )−i ∆t−iV (xi )∆t lim +0(−T ; x0)dx0 e 2µ −0(xN ; −T ) N!1 2π i=1 where ∆t := 2T =N: and x0 = xf : The integrals over pi are Fresnel integrals - they can be computed exactly (Henstock integration / Cauchy's theorem). The elementary integrals over pi have the general structure 1 r Z 2 π 2 e−iap +ibpdp = eib =(4a): −∞ ia After integrating the scattering probability amplitude becomes the limit of finite dimensional integrals −2iTH ( f 0(−T ); e i0(−T )) = Z N µ N=2 Y ∗ i µ (x −x )2−iV (x )∆t ^ 2∆t i−1 i i ^ lim ( ) dxi f 0(x0; −T )e i0(xN ; −T ): N!1 2πi∆t i=0 Feynman interpreted this as an integral over paths −2iTH ( ^f 0(−T ); e ^i0(−T )) = Z PN µ xi−1−xi 2 N µ N=2 i i=1 2 ( ∆t ) −V (xi ) ∆t Y lim ( ) ^f 0(x0; −T )dx0e dxj ^i0(xN ; −T ) N!1 2πi∆t j=1 N X µ xi−1 − xi 2 A [γ] := ( ) − V (x ) ∆t ! N 2 ∆t i i=1 Z tf µ γ_ 2(t) − V (γ(t)) dt 2 ti N µ N=2 Y D [γ] ! ( ) dx N 2πi∆t j j=1 Z Z iA[γ] iAN [γ] ( ^f 0(−T ); D[γ]e ^i0(−T )) := lim ( ^f 0(−T ); DN [γ]e ^i0(−T )) N!1 • In this form this form the quantum evolution between (xi ; ti ) and (xf ; tf ) is the \weighted average" over all classical paths starting at xi at time ti and finishing at xf at time tf . • The weight is eiA[γ] where A[γ] is the classical action functional. • This interpretation has proved to be very important for understanding properties of perturbative quantum field theory. • It has also led to non-perturbative methods (with imaginary time) that are consistent with experiment. • Quantum field theory - same structure except action is replaced by action functional for a classical field. Practical considerations • The dimension of integrals is infinite. • The integrand is oscillatory • Field theory case - the number of paths is also infinite. • Current applications - (1) imaginary time replaces oscillating integrand by positive integrand (2) integrals over the real line are replaced by integrals over a compact interval (3) large dimensional integrals evaluated by Monte Carlo integration, which converges like p1 independent of dimension. N • Methods not suitable for scattering (large times, large volumes, and oscillation are important for scattering). Method of Nathanson, Jorgenson, Muldowney • Replace integrals on real line by sums of integrals over sub-intervals. • Choose sub-intervals small enough so potential is approximately constant on each sub-interval. • Factor out the constant potential on each sub-interval and do the remaining integral over the sub interval exactly. • Result: complex probability on cylinder sets. Potential contribution becomes a random variable defined on the cylinder sets. • Limit as the number of intervals and time slices ! 1 converges to a solution to the Schr¨odingerequation. Replace integral over each \time-slice" by M Z Xn Z dxn = dxn: m=0 Imn The sub-intervals are chosen to be disjoint and cover the real line. The scattering probability amplitude becomes −2iTH ( ^f 0(−T ); e ^i0(−T )) = µ lim ( )N=2× N!1 2πi∆t N Z µ 2 X Y ^∗ i (xn−1−xn) −iV (xn)∆t ^ dxn f 0(x0; −T )e 2∆t i0(xN ; −T ): Im m0···mN N=0 n Complex probability on cylinder sets (Nathanson and Jorgensen) P(x0; Im1 ··· ImN ) := N Z µ N=2 Y i µ (x −x )2 ( ) dx e 2∆t n−1 n 2πi∆t n n=1 Imn These are interpreted as complex probabilities because X P(x0; Im1; ··· ; ImN ) = 1: m1···mN The approximate integral can be expressed as the following expectation value with respect to this compled probability distribution −2iHT X h ^f 0(T )je j ^i0(−T )i = lim lim × N!1 Vol(Imn)!0 m0;m1···mN PN ^∗ −i i=1 V (ymn)∆t ^ f 0(−T ; Im0)P(ym0; Im1; ··· ; ImN )e (ymN ; −T ): where Z ^∗ ^∗ ymn 2 Imn f 0(−T ; Im0) = f 0(−T ; x)dx Im0 • Gives a new interpretation - integral over paths replaced by expectation value over paths. • Can be used with any V (x) (not limited to Gaussian interactions). R • Weight e−i V (γ(t))dt non-trivial on support of V . • Part of the oscillations can be treated analytically. • Limit gives complex probability. Too many cylinder sets. Complex probability difficult to compute; too numerous to compute (one for each cylinder set). • Fix: compute the complex probability to proceed by a single time step. • Approximate the N-step complex probability by a product of N single-step complex probabilities. • Including the potential term in each of the one-step probabilities reduces the problem to the computation of the Nth power of the one-step propagation. • Advantage 1: The single-step probability can be computed exactly. • Advantage 2: The calculation reduces to computing the N − th power of an (M + 1) × (M + 1), which is much more efficient that summing over all cylinder sets. P(ym0; Im1; ··· ; ImN ) ≈ P1(ym0; Im1)P1(ym1; Im2) ··· P1(ymN−1; ImN ) where ymk 2 Imn PN −i V (ymn)∆t P(ym0; Im1; ··· ; ImN )e i=1 ≈ N Y −iV (ymk )∆t P1(ymk−1; Imk )e k=1 Result is Nth power of −iV (yn)∆t Mmn = P1(ym; In)e Structure of one-step matrix −iV (yk )∆t Mm;k = P(ym; Ik )e 1 r m r m P(y ; I ) = [(C ( (x[k+1]−y[n]))−C ( (x[k]−y[n]))+ n k 2 c 2∆t c 2∆t r m r m S ( (x[k + 1] − y[n])) − S ( (x[k] − y[n]))) c 2∆t c 2∆t r m r m +i(S ( (x[k + 1] − y[n])) − S ( (x[k] − y[n])) c 2∆t c 2∆t r m r m −C ( (x[k + 1] − y[n])) + C ( (x[k] − y[n])))] c 2∆t c 2∆t N yk 2 Ik = [xk ; xk+1] Need M Computing powers of matrix easy • Diagonalize MN = WDN W y • Use Krylov methods N−1 X Vn = MV^n−2 − V^k (V^k ; MV^n−1) k=1 1=2 Vn = Vn=(Vn; Vn) approximates matrix by lower dimensional matrix M; M2; (M2)2; ((M2)2)2 ··· Advantages • Sum over cylinder sets replaced by large power of a fixed matrix.
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