Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Variational Methods for Path Integral Scattering J. Carron Paul-Scherrer Institute, Villigen April 9, 2009 ETH Master thesis Supervisors : R. Rosenfelder, J. Fr¨ ohlich arXiv:0903.0273
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Outline Path Integrals for Scattering The Feynman-Jensen Variational Principle A Stationary Expression for the Path Integral. Our Ansatz Analytical Results The Approximation in the Eikonal Representation The Approximation in the Ray Representation Numerical Results
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary The Stage. • Non-relativistic quantum mechanics. • Elastic scattering at a potential V ( x ) , vanishing at infinity. • Incoming and outgoing momenta k i and k f . • Mean momentum and momentum transfer K = 1 2 ( k i + k f ) , q = k f − k i .
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Path Integrals for Scattering. Main Features. • A phase e iS , S an action, is functionally integrated over two different velocities v ( t ) , w ( t ) . • w : phantom degree of freedom. Removes all seemingly divergent quantities. ( → The kinetic term of w in the action has the wrong sign).
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Path Integrals for Scattering. Main Features. • A phase e iS , S an action, is functionally integrated over two different velocities v ( t ) , w ( t ) . • w : phantom degree of freedom. Removes all seemingly divergent quantities. ( → The kinetic term of w in the action has the wrong sign). • Interacting part of S : values of the potential are integrated along a one-particle trajectory ξ ( t , v , w ) . • The path integral describes the quantum fluctuations around a reference trajectory. → ξ ( t , v , w ) = ξ ref ( t ) + ξ quant ( t , v , w ) .
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary The Formulae. � � � � T i → f = i K e i S int − 1 d 2 b e − i q · b D v D w e iS free . m � � � S free = m v 2 ( t ) − w 2 ( t ) dt , 2 � S int = − dt V ( ξ ( t )) , ξ ( t ) = ξ ref ( t ) + ξ quant ( t , v , w ) .
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary The Formulae. Two Representations. • Eikonal representation : ξ ref ( t ) = b + K w 3-dimensional , mt . • Ray representation : ξ ref ( t ) = b + K mt + q w � K , 2 m | t | . In both cases, ξ quant ( v , w ) is linear in the velocities.
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Reference Trajectory.
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Outline Path Integrals for Scattering The Feynman-Jensen Variational Principle A Stationary Expression for the Path Integral. Our Ansatz Analytical Results The Approximation in the Eikonal Representation The Approximation in the Ray Representation Numerical Results
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary • Imagine you want to solve a path integral for an action S , knowing its value for another action S t . You may write � � � e i ( S − S t ) � � � D x e i ( S − S t ) e iS t D xe iS = D xe iS t := D xe iS t . � D x e iS t
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary • Imagine you want to solve a path integral for an action S , knowing its value for another action S t . You may write � � � e i ( S − S t ) � � � D x e i ( S − S t ) e iS t D xe iS = D xe iS t := D xe iS t . � D x e iS t • Consider in place of the above expression the following functional: � D x e iS t . F [ S t ] = e i � S − S t �
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary • Imagine you want to solve a path integral for an action S , knowing its value for another action S t . You may write � � � e i ( S − S t ) � � � D x e i ( S − S t ) e iS t D xe iS = D xe iS t := D xe iS t . � D x e iS t • Consider in place of the above expression the following functional: � D x e iS t . F [ S t ] = e i � S − S t � • It holds that � D x e iS F [ S ] = δ F | S = S t = 0 . and
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary • Imagine you want to solve a path integral for an action S , knowing its value for another action S t . You may write � � � e i ( S − S t ) � � � D x e i ( S − S t ) e iS t D xe iS = D xe iS t := D xe iS t . � D x e iS t • Consider in place of the above expression the following functional: � D x e iS t . F [ S t ] = e i � S − S t � • It holds that � D x e iS F [ S ] = δ F | S = S t = 0 . and • We have thus found a stationary expression for the path integral, which we can solve for a nearby action S t .
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Corrections. � e it ∆ S � Two ways to expand : • Expansion in moments : � e it ∆ S � ∞ � (∆ S ) k � � ( it ) k = . k ! k = 0 • Expansion in cumulants λ k : � ∞ � � e it ∆ S � � ( it ) k := exp k ! λ k . k = 1 � (∆ S ) 2 � − � ∆ S � 2 ⇒ λ 1 = � ∆ S � , λ 2 =
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Corrections. � e it ∆ S � Two ways to expand : • Expansion in moments : � e it ∆ S � ∞ � (∆ S ) k � � ( it ) k = . k ! k = 0 • Expansion in cumulants λ k : � ∞ � � e it ∆ S � � ( it ) k := exp k ! λ k . k = 1 � (∆ S ) 2 � − � ∆ S � 2 ⇒ λ 1 = � ∆ S � , λ 2 = • Our variational approximation is the first term of the cumulant expansion. • The first correction term is given by � � − 1 F [ S t ] → F [ S t ] exp 2 λ 2 .
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Outline Path Integrals for Scattering The Feynman-Jensen Variational Principle A Stationary Expression for the Path Integral. Our Ansatz Analytical Results The Approximation in the Eikonal Representation The Approximation in the Ray Representation Numerical Results
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Which Trial Action ? The trial action S t has to satisfy two criteria: 1. It must have a physical motivation. 2. It must be simple enough to allow analytical calculations. (very restrictive !)
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Our Ansatz I. Motivation: • In a high-energy approach to our path integral, one would expand the interacting part of the action in V ( ξ ref + ξ quant ( v , w )) ≈ V ( ξ ref ) + ∇ V ( ξ ref ) · ξ quant ( v , w ) . • This makes the interacting part of the action linear in the velocities ( → leads to eikonal-like expansions).
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Our Ansatz II. This suggests: • Our Ansatz will be to add to the free action a linear term in the velocities. • The variational procedure will pick up for us the best linear term possible, while emulating the structure of the high-energy expansion.
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary What we do. In our path integral formulae for the T -Matrix, instead of � D v D w e iS , we will therefore consider � F [ S t ] = e i � S − S t � D v D w e iS t , where the trial action is linear in the velocities, � � → S t = S free + dt B ( t ) · v ( t ) + dt C ( t ) · w ( t )
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Expectations. The problem is reduced to: 1. The computation of the needed expectation values. 2. The solution to the variational equations for B ( t ) and C ( t ) arising from the stationarity condition.
Path Integrals for Scattering The Feynman-Jensen Variational Principle Analytical Results Numerical Results Summary Expectations. The problem is reduced to: 1. The computation of the needed expectation values. 2. The solution to the variational equations for B ( t ) and C ( t ) arising from the stationarity condition. We expect: 1. To recover in the high-energy limit (at least) the leading and next-to-leading term of the eikonal expansion. 2. That the approximation should also be valid for lower energies or larger scattering angles.
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