The Universe as an Anharmonic Ocillator and other Unexplicable Mysteries Talk at QMCD 09 Fred Cooper NSF, SFI, LANL-CNLS March 20, 2009 1 Introduction The world according to Carl (as influenced by my being at Los Alamos). Find a new non-perturbative approximation scheme for Field Theory See how well it works on the energy eigenvalues etc of the Anharmonic Oscillator Use it in various quantum field Theory applications Use whatever new techniques are invented to solve real problems relevant to the lab boundary value problems, Stochastic processes, Solitary Waves 1
Figure 1: The World According to PTolemy My Life with Carl (in terms of Research) Bound state (Mean Field) perturbation Theory (aka the large-n aprroximation for φ 4 field Theory) Strong Coupling expansion Ultra Local Expansions Gaussian Integrals are not the only ones you can do!- Spinoffs Boundary Layer Theory Langevin Equation, Post-Gaussian Trial Functions to study Solitary wave Blow-up Discrete Time Quantum Mechanics Real Rime Tunneling in the double well anharmonic oscillator Delta Expansion: Anharmonic Oscillator SUSY anharmonic Oscillator Quantum Field theories PTolemaic Phyiscs Explaining why Ptolemy (actually PT) was right after all How to live with a Punster and survive! - Eat a lot of Green Chile 2
Figure 2: New Mexico s Finest from Hatch 2 Path Integral Formulation of Mean Field Perturbation theory (large-n) expansion One summer Carl, Gerry and I got together and wondered whether the mean-field approximation that was understood in terms of Functional Differential equations could be put into Path Integral form. In a Pioneering paper using Carl s abilities in Steepest Descent and Stationary Phase approximations, we were able to convert quartic self interacting theories to Yukawa and QED like theories and also do all order renormalization of the new Dyson Equations. F. Cooper, C. Bender and G. Guralnik, Path Integral Formulation of Mean-Field Perturbation Theory, Ann. Phys. 109, 165 (1977). This was the starting point for all future works on phase transitions both at Finite Temperature and in non-equilibrium situations that has led to many applications. 3
Figure 3: A MEAN Field? 3 Strong Coupling or UltraLocal epansions Carl realized that if we start with a local potential energy, one could treat the potential energy part of the path integral exactly even if the integrals were exponentials to arbitrary powers. This led to a Kinetic Energy expansion which is a Lattice Strong Coupling Expansion. F. Cooper, C. Bender, G. Guralnik and Sharp, D., Strong Coupling Expansion in Quantum Field Theory, Phys. Rev. D 19, 1865 (1979).. This led to a lot of achievements: For the Anharmonic Oscillator the renormalized coupling constant at infinite bare coupling was calculated to be 6! We found Strong Numerical Evidence that at d=4 scalar selfinteracting field theory is a trivial theory in the continuum limit We found new way of solving Boundary Layer Problems F. Cooper, C. Bender, G. Guralnik, E. Mjolsness, H. Rose and 4
D. Sharp, A Novel Approach to the Solution of Boundary Layer Problems, Adv. Appl. Math.1, 22 (1980). New way of understanding Supersymmetry breaking F. Cooper, C. Bender and A. Das, A New Test for Spontaneous Breakdown of Supersymme- try, Phys. Rev. D 50, 28, 1473 (1983). C. Bender, F. Cooper and A. Das, Continuum Limit of Supersymmetric Field Theories on a Lattice, Phys. Rev. Letts. 50, 397 (1983) New Way of Solving the Langevin Equation! F. Cooper, C. Bender and B. Freedman, A New Strong Coupling Expansion for Quantum Field Theory Based on the Langevin Equation, Nucl. Phys. B 219, 61 (1983) 5
Figure 4: Who ordered this? 4 Finite Elephant (element) Theory) AKA Discrete time Quantum Mechanics Carl Invented a new way of doing time dependent quantum theory by finding a discrete time operator update equation that preserved the Canonical Commutation relations. This led to some exact and approximate analytic calculations. F. Cooper, K. A. Milton and L. M. Simmons Jr., The Quantum Roll: A Study in the Long Time Behavior of the Finite Element Method, Phys. Rev. D32, 2056 (1985). as well as a new numerical method for looking at quantum tunneling Quantum Tunneling Using Discrete Time Operator Difference Equations. Carl M. Bender, Fred Cooper, James E. O Dell, L.M. Simmons, Jr. Phys.Rev.Lett.55:901,1985. 6
Figure 5: The Rosetta Stone, found in the Nile Delta in 1799 5 The δ expansion Carl asked the question can you do perturbation theory in the NONlinearity of the equation! x 4 x 2+δ This expansion when coupled with Variational Improvements led to excellent results for the energy Eigenstates of various Anharmonic Oscillators and their SUSY extensions and better results than the leading order in large N approximation for Scalar Field Theory. C. Bender, F. Cooper, and K. A. Milton, The δ Expansion for Stochastic Quantization, Phys. Rev. D39, 3684 (1989). C. Bender F. Cooper, and K. A. Milton, The δ Expansion and Local Gauge Invariance, Phys. Rev. D40, 1354 (1989). F. Cooper and P. Roy, δ Expansion for the Superpotential, Phys. Letts. A143, 202 (1990). F.Cooper and M. Moshe, Scaling Relations and the δ Expansion, Phys. Lett. B 259, 101 (1991). 7
Figure 6: More of Ptolemy? 6 Ptolemaic Physics Figure 7: Which Universe is ours? Carl decided to challenge the whole foundation of Quantum Mechanics! Hermiticity was NOT necessary for Real Eigenvalues only PT Symmetry. Motivation: Is it possible to make an asymptotically free non-trivial scalar Field Theory in a PTolemaic world. Deep Question: Is the PT Symmetric World one universe in our Multiverse... Carl M. Bender, Fred Cooper, Peter Meisinger and Van M. Savage Variational Ansatz for PT Symmetric Quantum Mechanics Phys. Lett. A 259 (1999) 8
u x,t 1.0 0.8 0.6 0.4 0.2 4 2 2 4 x Figure 8: Evolution of Unstable Solitary wave Solutions Compactons In PT symmetric generalized K-dV Equations Carl Bender, Fred Cooper, Avinash Khare, Bogdan Mihaila and Avadh Saxena, arxiv:0810.3460v1 [math-ph] Shramana-to be published. Presently we are investigating the breakup of unstable PT compactons. 9
7 Working with Carl Figure 9: Rapunzel, do we have an LPU? Carl is a great collaborator to work with... great food... great jokes... great interchange of ideas. However there was this overall desire for an LPU (least publishable unit). This meant being put up in the 3rd floor of his house until we finished a paper! (This often only took 2-3 days!)... So Rapunzel... what do you say??? 10