STOCHASTIC VARIATIONAL METHOD AND QUANTIZATION TAKESHI KODAMA

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Transcript of STOCHASTIC VARIATIONAL METHOD AND QUANTIZATION TAKESHI KODAMA

  • Slide 1
  • STOCHASTIC VARIATIONAL METHOD AND QUANTIZATION TAKESHI KODAMA
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  • STOCHASTIC VARIATIONAL METHOD AND QUANTIZATION TAKESHI KODAMA AND TOMOI KOIDE
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  • STOCHASTIC VARIATIONAL METHOD AND QUANTIZATION TAKESHI KODAMA AND TOMOI KOIDE Federal University of Rio de Janeiro
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  • VARIATIONAL METHOD
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  • Consider Physical Laws comes from an Optimization procedure of a Scalar Quantity
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  • VARIATIONAL METHOD Useful for physical insight Formal development of the theory Approximation Methods Consider Physical Laws comes from an Optimization procedure of a Scalar Quantity
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  • FIRST STEPS IN PHYSICS Pierre-Louis Moreau de Maupertuis July17,1698 Jully 27,1759 Analogy of optics to mechanics with concept of minimization of Ation
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  • VARIATIONAL FORMLATION OF CLASSICAL MECHANICS ou
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  • q(t) t
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  • IMPORTANT ! q(t) t Fixed
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  • IMPORTANT ! q(t) t Fixed Although the variational approach assumes the future information as fixed, the resultant equation reduces to the problem of initial condition !!
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  • ANOTHER ASPECTS OF VARIATIONAL APPROACH SYMMETRY AND CONSERVATION LAWS Amalie Emmy Noether March, 23, 1882March, 23, 1882 - April, 14, 1935April, 14, 1935
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  • ANOTHER ASPECTS OF VARIATIONAL APPROACH SYMMETRY AND CONSERVATION LAWS Amalie Emmy Noether March, 23, 1882March, 23, 1882 - abril, 14, 1935abril, 14, 1935 Action is scalar !
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  • ANOTHER ASPECTS OF VARIATIONAL APPROACH SYMMETRY AND CONSERVATION LAWS In Quantum Mechanics, this role of Variational Approach is replaced by the representation of operators in Hilbert space of physical states.
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  • ONCE THE VARIATIONAL APPROACH IS ESTABLISHED FOR A PROBLEM.. Use as Approximation method
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  • ONCE THE VARIATIONAL APPROACH IS ESTABLISHED FOR A PROBLEM.. Use as Approximation method or Model Construction
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  • EXAMPLES IN QM Hartree-Fock Approx. QMD Model
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  • As we know Variational Principle in Classical Mechanics Can be understood as Stationary Path in Feynmans Path Integral Representation of Quantum Mechanis.
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  • As we dont know Inversely, Quantum Mechanics can be derived from the Classical Mechanics in terms of Variational Principle...?
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  • Quantum Mechanics is so beautifully established as a linear representation in Hilbert Space for Physical states. So, why do you want to think something different...?
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  • GENEALOGY OF NON CONVENTIONAL FORMULATION OF QUANTUM MECHANICS Bohm-Vigier Hidden variables Edward Nelson - Stochastic Method Parisi-Wu Stochastic Quantization 5 th Dim (time) Kunio Yasue Stochastic Variational Method de Broglie A. Eddington E. Madelung
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  • Conferncia Solvay Bruxela 1911 THE BIGINING Solvay Conf.-1911
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  • Conferncia Solvay Bruxela 1911 THE BIGINING Solvay Conf.-1911
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  • SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION Ensemble of dust particle under the potential V E. Madelung
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  • SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION Ensemble of dust particle under the potential V For interacting adiabatic fluid, add the internal energy to V,
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  • SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION Ensemble of dust particle under the potential V
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  • SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION Ensemble of dust particle under the potential V Continuity condition NEW
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  • SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION Ensemble of dust particle under the potential V A constant to make adimensional by partial integration.
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  • SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION Still continues Variation with respect to the velocity field,
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  • SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION Still continues Variation with respect to the velocity field, Eliminate the velocity
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  • SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION Still continues Change of variables:
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  • SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION Still continues This would be the action for Schroedinger Equation, if we dont have the term, choosing.
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  • SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION Still continues Can we have some mechanism to generate the internal energy U to compensate?
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  • ANOTHER CONNECTION BETWEEN CLASSICAL PHYSICS AND QUANTUM MECHANICS LAWS Smilarity in Partition function in Statistial Physics and the Green function, with Wick Rotation But, then the diffusion equation is then Schroedinger Equation Is this internal energy is associated with the diffution current due to a presence of noises such as thermal internal energy?
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  • HOW TO DEAL WITH THE PRESENCE OF NOISE?
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  • 38 STOCHASTIC PROCESS The effect of microscopic degrees of freedom can be treated as noise, with Stochastic Differential Equation (SDE) Classical case Stochastic
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  • HOW TO INCORPORATE IN VARIATIONAL SCHEME? q(t) t Fixed
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  • NECESSITY OF TWO KINDS OF SDE 41 with white noise B ACKWARD SDE F ORWARD SDE
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  • OTHERWISE, STARTING FROM THE INITIAL CONDITION . q(t) t
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  • OTHERWISE, STARTING FROM THE INITIAL CONDITION . q(t) t Fokker-Planck Equation for Browinian Motion
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  • THEN, HOW ABOUT STARTING FROM THE FINAL CONDITION ? . q(t) t
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  • THEN, HOW ABOUT STARTING FROM THE FINAL CONDITION ? . q(t) t Fokker-Planck Equation for Browinian Motion
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  • HOW TO DO? RECONNECTION !
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  • Number of the ways to reconnect at is proportional to
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  • MAXIMUM ENTROPY ASSUMPTION NEW We live in the world when the entropy associated to this recombination process is maximum (i.e., the two flows get in equilibrium). WE REQUIRE: with
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  • MAXIMUM ENTROPY ASSUMPTION NEW We live in the world when the entropy associated to this recombination process is maximum (i.e., the two flows get in equilibrium). We get We get
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  • MAXIMUM ENTROPY ASSUMPTION NEW We live in the world when the entropy associated to this recombination process is maximum (i.e., the two flows get in equilibrium). We get We get
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  • ONE DENSITY AND TWO FLOW VELOCITIES
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  • : Translational (CM) velocity : Velocity of internal motion
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  • NOW, WHAT IS THE OPTIMAL PATH?
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  • VARIATIONAL PRINCIPLE FOR STOCHASTIC TRAJECTORIES 55 Define the Action for Stochastic Variables. We are talking necessarily about the distribution of trajectories and not a particular trajectory Yasue, J. Funct. Anal, 41, 327 (81), Guerra&Morato, Phys. Rev. D27, 1774 (83), Nelson, Quantum Fluctuations (85).
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  • VARIATIONAL PRINCIPLE FOR STOCHASTIC TRAJECTORIES 56 Define the Action for Stochastic Variables. We are talking necessarily about the distribution of trajectories and not a particular trajectory Yasue, J. Funct. Anal, 41, 327 (81), Guerra&Morato, Phys. Rev. D27, 1774 (83), Nelson, Quantum Fluctuations (85).
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  • VARIATIONAL PRINCIPLE FOR STOCHASTIC TRAJECTORIES 57 Define the Action for Stochastic Variables. We are talking necessarily about the distribution of trajectories and not a particular trajectory Yasue, J. Funct. Anal, 41, 327 (81), Guerra&Morato, Phys. Rev. D27, 1774 (83), Nelson, Quantum Fluctuations (85).
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  • VARIATIONAL PRINCIPLE FOR STOCHASTIC TRAJECTORIES 58 Define the Action for Stochastic Variables. We are talking necessarily about the distribution of trajectories and not a particular trajectory Yasue, J. Funct. Anal, 41, 327 (81), Guerra&Morato, Phys. Rev. D27, 1774 (83), Nelson, Quantum Fluctuations (85).
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  • OUR FORMULATION NEW FLUID WITH THE INTERNAL FLOW VELOCITY
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  • OUR FORMULATION NEW FLUID WITH THE INTERNAL FLOW VELOCITY The fluid carries the internal energy
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  • ACTION SHOULD BE
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  • IN SHORT, -Quantum Mechanical action for the Schroedinger equation, is equivalent to that of the fluid with two components, one obeys FSDE, other BSDE with a simple variable transformation,
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  • RE