Tomoi Koide (IF,UFRJ) - PDF Free Download

International Conference of Physics (2016) Tomoi Koide (IF,UFRJ) Pedagogical introduction, Koide et. al., J. Phys. Conf. 626, 012055 (`15)

Variational Principle in Class. Mech. b a 2 t F m dx() t I( x) dt V ( x( t)) t I 2 dt OPTIMIZATION Newton equation

Variational Principle in Quan. Mech. b a V * F 4 * I(, ) d x i H V I t OPTIMIZATION Schrödinger equation Note that Variation of WF, not trajectory * (, ) I T V

Path Integral Approach b a b [ Dx]expiI a All paths contribute! b a

For this, we need to extend the formulation of the variational method.

A Optimized path? B

A Optimized path! B

We cannot follow the optimized path!! 1. Zig-zag by fluctuation 2. Each times, different paths A B

Optimal control Effect of variables which we cannot control. b a 2 t F m dx() t I( x) dt V ( x( t)) t I 2 dt Newton equation

Yasue, J. Funct. Anal. 41 327 (1981) Nelson, Quantum Fluctuations (Princeton, NJ: Princeton University Press, 1985) Guerra&Morato, Phys. Rev. D27 1774 (1983) Pavon, J. Math. Phys. 36 6774 (1995) Nagasawa, Stochastic Process in Quantum Physics (Bassel:Birkhaeuser, 2000) Cresson and Darses, J. Math. Phys. 48 072703 (2007) Holm, arxiv:1410.8311 [math-ph] Chen,Cruzeiro&Ratiu, arxiv:1506.05024 [math-ph] Stochastic Variational method is one approach to calculate optimization including such a fluctuation.

KEY POINT 5 important steps

CLASSICAL TRAJECTORY Velocioty dr ( t) r( t dt) r( t) udt Zig-zag motion?

Zig-zag in Brownian motion molecure Brownian particle

Zig-zag in Brownian motion molecure Brownian particle dr ( t) 2 dw ( t)

Zig-zag in Brownian motion molecure Brownian particle Noise intensity Gaussian white noise (Wiener process) dr ( t) 2 dw ( t) E[ dw ] 0 E dw 2 [( i) ] dt Direction is random Mean magnitude is dt

CLASSICAL TRAJECTORY Velocioty dr ( t) r( t dt) r( t) udt Zig-zag motion?

CLASSICAL TRAJECTORY Velocioty dr ( t) r( t dt) r( t) udt STOCHASTIC TRAJECTORY dr ( t) u( r( t), t) dt 2 dw ( t) 0 E[ dw ] 0 E dw 2 [( i) ] dt

How define the velocity at A? r x C A B t dt t t dt

How define the velocity at A? r x C A B t dt t t dt lim dt0 r( t dt) r( t) dt lim dt0 r( t dt) r( t) dt

Bernstein Process Forward stochastic differential equation dr ( t) u( r( t), t) dt 2 dw ( t) dt 0 2 E ( dwi ( t)) d t We employ the variational procedure to determine these unknown functions.

Bernstein Process Forward stochastic differential equation dr ( t) u( r( t), t) dt 2 dw ( t) dt 0 2 E ( dwi ( t)) d t Backward stochastic differential equation dr ( t) u( r( t), t) dt 2 dw ( t) dt 0 2 E ( dwi ( t)) dt We employ the variational procedure to determine these unknown functions.

Consistency Condition Probability density ( x, t) E ( x r( t)) The Fokker-Plank equation (forward) u The Fokker-Plank equation (backward) t t u These two should be equivalent u u 2 ln Also related to Bayesian statistics Caticha, JPA44, 225303(`11)

Time Drivative Operations Because of the two different definitions of velocities, we can introduce the two different time derivatives. (Nelson) Mean forward derivative Mean backward derivative Dr Dr lim dt0 lim dt0 dr dt dr dt u u See, Koide, Kodama&Tsushima, JP Conf. 626, 012055 (`15)

Zheng&Meyer, S`eminarie de Probabilit`es XVIII, Lecture Notes in Mathematics Vol 1059, P223, (Springer-Verlag, Berlin, 1984). Partial Integration Formula CLASSICAL b a dx b dt Y X ( b) Y( b) X ( a) Y( a) dtx dt a dy dt STOCHASTIC b a dte ( DX ) Y EX ( b) Y( b) X ( a) Y( a) dte X ( DY ) b a See, Koide, Kodama&Tsushima, JP Conf. 626, 012055 (`15)

Ito Formula (Ito s lemma) This is a kind of Tayler expansion for stochastic variables. Taylor dr udt 2 df ( r( t), t) dt u f ( r( t), t) O( dt ) t Ito dr udt 2 dw 2 df ( r( t), t) dt t u f ( r( t), t) 2 f dw 2 O( dt )

Let s apply!!

Stochastic Representation of Action Classical action We consider 3) b 2 m dr () t Icla dt V ( r ( t)) 2 dt a For example 1) Dr Dr 2 dr 2) Dr Dr dt Dr Dr Dr Dr 3) 2 Stochastic action I sto b a dte m ( Dr ) ( Dr ) 2 2 2 2 V ( r)

Stochastic Variation for Kinetic Term r r r r( a) r( b) 0 b a m dt E r r mdt r 2 b ( D ) ( D ) E( D ) ( Dr) a b m dte a b a u( D r) m dte Du r Ito formula du( r, t) Du( r, t) lim t u u dt0 dt

Variation of Action I 0 2 t um um V ( r) m 2 1/ 2 2 1 um ( u u) / 2

Variation of Action I 0 2 t um um V ( r) m 2 1/ 2 2 1 um ( u u) / 2 when 0

Variation of Action I 0 2 t um um V ( r) m 2 1/ 2 2 1 um ( u u) / 2 u t m d dt when 0 The Newton equation d 1 u m ( r ( t ), t ) V ( r ( t )) dt m

Variation of Action I 0 2 t um um V ( r) m 2 1/ 2 2 1 um ( u u) / 2 u t m d dt when 0 The Newton equation d 1 u m ( r ( t ), t ) V ( r ( t )) dt m The dynamics of t is given by the FP equation. u u m

Derivation of Schrödinger Equation Introduction of phase u m /(2 ) Eq. of variation 2 1/ 2 2 1 t V m 0

Derivation of Schrödinger Equation Introduction of phase u m /(2 ) Eq. of variation 2 1/ 2 2 1 t V m 0 Introduction of wave function i e Yasue, JFA 41, 327 (`81) 1 it V 2 m 2 2 m i V t The Schrödinger equation 2m

Optimization in macro. scale Optimization in micro. scale Newton equation NOISE Schrödinger equation GLASSES

Canonical Quantization and SVM L( r, r) optimization d L L dt r r 0 LEGENDRE TRANSFORM H( r, p) p L / r optimization r p r, H p, H PB PB

Canonical Quantization and SVM L( r, r) optimization d L L dt r r 0 LEGENDRE TRANSFORM H( r, p) p L / r optimization canonical quantization r p r, H PB 1 [ rh ˆ, ˆ ] i p, HPB 1 [ ph ˆ, ˆ ] i

Canonical Quantization and SVM SVM L( r, r) optimization d L L 0 dt r r D L D L L ( Dr) ( Dr) r 0 LEGENDRE TRANSFORM H( r, p) p L / r optimization canonical quantization r p r, H PB 1 [ rh ˆ, ˆ ] i p, HPB 1 [ ph ˆ, ˆ ] i

Invariance for spatial translation The change of the action t rt () r() t f f I dte L ( r A, Dr, Dr ) dte L ( r, Dr, Dr ) t t i t i f d m m dt E Dr Dr A dt 2 2 Misawa, J. Math. Phys. 29 2178 (`88) t t i A

Invariance for spatial translation The change of the action t rt () r() t f f I dte L ( r A, Dr, Dr ) dte L ( r, Dr, Dr ) t t i t i f d m m dt E Dr Dr A dt 2 2 If the action is invariant for the spatial translation, Misawa, J. Math. Phys. 29 2178 (`88) t t i A conserved momentum operator! m E 3 Dr Dr d x * ( x, t )( i x) ( x, t ) 2 Comment : we do not need the normal ordering product. Koide&Kodama, PTEP 093A03 ( 15)

Classical Many-Body Dynamics Positions and velocities of all particles Mass density and velocity of fluid classical variation SVM classical variation SVM N-body Newton s eq. Ideal fluid eq. (Euler) Loffredo&Morato, JPA40,8709(`07), Koide, PLA379,2007(`15)

Classical Many-Body Dynamics Positions and velocities of all particles Mass density and velocity of fluid classical variation SVM classical variation SVM N-body Newton s eq. N-body Scrödinger eq. Ideal fluid eq. (Euler) Loffredo&Morato, JPA40,8709(`07), Koide, PLA379,2007(`15) Gross -Ptaevskii eq.

SVM is a useful method of for quantization of nonrelativistic particles and bosonic fileds. SVM is applicable as a method for coase-grainings of micro. dynamics (Navier-Stokes-Fourier, Gross-Pitaevskii). The stochastic Noether theorem The uncertainty relations Quantum-Classical hybrids

SVM Analytical Mechanics stochastic particle classical field classical particle curved space-time Noether theorem stochasic field dissipative systems rigid body fermion field brachistochrone curve canonical transform anomaly

Stochastic Analytical Mechanics SVM Analytical Mechanics stochastic particle classical field classical particle curved space-time Noether theorem stochasic field dissipative systems rigid body fermion field brachistochrone curve canonical transform anomaly

Pedagogical introduction, Koide et. al., J. Phys. Conf. 626, 012055 (`15)

Classical Many-Body Dynamics Positions and velocities of all particles Mass density and velocity of fluid

Classical Many-Body Dynamics classical variation Positions and velocities of all particles N-body Newton s eq. classical variation Mass density and velocity of fluid Ideal fluid eq. (Euler)

Classical Many-Body Dynamics Positions and velocities of all particles Mass density and velocity of fluid classical variation SVM classical variation SVM N-body Newton s eq. N-body Scrödinger eq. Loffredo&Morato, JPA40,8709(`07), Koide, PLA379,2007(`15) Ideal fluid eq. (Euler)?

Classical variation of fluid Action of (ideal) fluid ( xt, ) I v dtd x v x t I 2 tf 3 M 2 ( M, ) (, ) t M Classical variation Internal energy density

Classical variation of fluid Action of (ideal) fluid Fluid velocity ( xt, ) I v dtd x v x t I 2 tf 3 M 2 ( M, ) (, ) t M Mass density Classical variation Internal energy density

Classical variation of fluid Action of (ideal) fluid Fluid velocity ( xt, ) I v dtd x v x t I 2 tf 3 M 2 ( M, ) (, ) t M Mass density Classical variation Internal energy density Euler equation 1 t v v P M P d d 1/ M M Pressure TdS de PdV P de dv / adiabatic ( ds 0) S

Application of SVM Applying SVM to the same action of (ideal) fluid, i When we choose 1 d 2 dm t Noise intensity Koide&Kodama, JPA45, 255204 (`12) 1 2 um i e 2M 1 ( M ) V ( x) U M 2 M M M 0 2 quantum fluctuation external force two-body interaction

Application of SVM Koide&Kodama, JPA45, 255204 (`12) 2 2 i t V U0 2M Gross-Pitaevskii equation BEC Nucleation

Koide, JP Conf. 410, 012025 (`13) Replacement of kinetic terms The general expression of dr ( R, t) / dt 2 is given by 2 2 A Dr B Dr C Dr Dr For noise=0, the classical kinetic term should be reproduced. dr ( R, t) / dt 2 1 1 1 1 2 1 2 2 2 2 2 2 Dr 1 Dr 2 Dr Dr The calculation so far corresponds to ( 1, 2) ( 0,1/ 2).

Forward B A Direction of time

backward B A Direction of time

B A Direction of time See also the Bernstein process.

Quantum-Classical hybrids

Zurek, Physics Today 44, 36 (`91) Near the boundary, we can expect a coexistence of Class. and Quan. degrees of freedom.

Quantum measurement Quantum-to-classical transition in early universe Einstein gravity interacting with quantum objects Simplification of complex simulation of quantum chemistry Models of quantum-classical hybrids should satisfy, Energy conservation Positivity of probability Newton equation + Schrödinger equation in no int. V=0. Most of proposed models cannot satisfy these conditions.

Quantum Two Particles Lagrangian 2 2 2 2 M ( Drq) ( Drq) m ( Drq ) ( Drq ) L E V ( rq, rq ) 2 2 2 2 drq vdt dw M Stochastic variables drq udt dw m '

Hybrid Lagrangian L 2 2 2 2 M ( ) ( ) ( ) ( ) c Drc Dr mq Dr c q Dr q E V( rc, rq ) 2 2 2 2 Classical variable Stochastic variable dr v dt c c dr q udt dw m

Classical Variation d 1 q( rc, xq, t) q vc ( rc, xq, t) cv ( rc, xq ) dt m M t r ( x, t) x ds v ( r ( x, s), x, s) c q c0 c c q q t 0 Stochastic Variation d M i t V t dt 2m 2 2 2 q( rc, xq, ) q ( rc, xq ) vc ( rc, xq, t) q( rc, xq, ) i ( r, x, t) c q ( r, x, t) ( x, t) e q c q q Koide, PLA379 2007 (2015)

It was confirmed Check list by Caro, Diósi, Elze et al. O.K. Energy conservation Stochastic Noether theorem O.K. Positivity of probability O.K. Newton equation + Schrödinger equation in no int. V=0. O.K. Generalized Ehrenfest theorem Other successful models, Hall&Reginatto, PRA72, 062109 (`05) Elze, PRA85, 052109 (`12);Lampo et al., PRA90, 042120 (`14), Radonji et al., PRA85, 064101 (`12)

Local, realism Non-local, realism Takabayasi Hydrodynamic Representation EPR paradox Madelung de Broglie Bohm Vigier Nelson-Newton Equation SVM Schödinger Eddington Nelson Yasue Stochastic Quantization Parisi Wu Namiki AdS/CFT Micro-canonical Quantization Chaotic Quantization

Nelson s Stochastic Quantization SVM is the reformulation of Nelson s approach in the framework of a variational principle. In the original Nelson s approach, quantization indicates m 2 d x ( ) 2 xv x dt REPLACE m Nelson-Newton equation DDx DDx 2 x V ( x)

Svaiter, Menezes, Alcalde, Tópicos em teoria quântica dos campos Parisi-Wu s Stochastic Quan.(I) Wick rotation S[ ] Damgaard&Hueffel, PR. 152, 227 (`87) Krein et al., IJMP A29, 1450030 (`14) S [ ] E Intro. of a virtual time 4 D 4+1 D ( x ) E ( x, ) E We consider the stochastic motion in this virtual time, SDE SE[ ] d( xe, ) d 2 dw ( xe, ) ( x, ) E Wiener Process

Parisi-Wu s Stochastic Qua.(II) For the free case, 1 S d k k k m k 2 4 2 2 E[ ] E( E, ) E ( E, ) we can solve the SDE as 2 2 2 kem kem 2 ( s) E 0 ( k, ) e ( k,0) ds 2e E dw ( ke, s) ds Propagator G( k, p ) lim E ( k, ) ( p, ') E E E E ' 4 (4) (2 ) ( ke pe) k 1 m 2 2 E

These successes are just accidental? As a pedagogical review, Koide, Kodama&Tsushima, JP Conf. 626, 012055 (`15) http://iopscience.iop.org/1742-6596/626/1