STOCHASTIC VARIATIONAL METHOD AND

QUANTIZATION

TA K E S H I KO DA M A

STOCHASTIC VARIATIONAL METHOD AND

QUANTIZATION

TA K E S H I KO DA M A

A N D

T O M O I KO I D E

STOCHASTIC VARIATIONAL METHOD AND

QUANTIZATION

TA K E S H I KO DA M A

A N D

T O M O I KO I D E

Federal University of Rio de Janeiro

VARIATIONAL METHOD

VARIATIONAL METHOD

Consider Physical Laws comes from

an Optimization procedure of

a Scalar Quantity

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

FIRST STEPS IN PHYSICS …

Pierre-Louis Moreau de MaupertuisJuly17,1698 –Jully 27,1759

Analogy of optics to mechanics with conceptof minimization of “Açtion”

VARIATIONAL FORMLATION OF CLASSICAL MECHANICS

[ ( )] ( , )dq

I q t dt L qdt

[ ( ), ( )] 0, ( ), ( )I q t p t q t p t

ou

[ ( ), ( )] ( , )dq

I q t p t dt p H q pdt

[ ( )] 0, ( )I q t q t

q(t)

t

[ ( )] 0, ( )I q t q t

IMPORTANT !

q(t)

t

Fixed

[ ( )] 0, ( )I q t q t

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 !!

[ ( )] 0, ( )I q t q t

0d L L

dt qq

ANOTHER ASPECTS OF VARIATIONAL APPROACH

SYMMETRY AND CONSERVATION LAWS

Amalie Emmy NoetherMarch, 23, 1882 - April, 14, 1935

ANOTHER ASPECTS OF VARIATIONAL APPROACH

SYMMETRY AND CONSERVATION LAWS

Amalie Emmy NoetherMarch, 23, 1882 - abril, 14, 1935

Action is scalar !

ANOTHER ASPECTS OF VARIATIONAL APPROACH

SYMMETRY AND CONSERVATION LAWS

QM tI dt t i H t

In Quantum Mechanics, this role of Variational Approach is replaced by the representation of operators in Hilbert space of physical states.

ONCE THE VARIATIONAL APPROACH IS ESTABLISHED FOR A PROBLEM…..

,0,

True True

True

I I q t

I

Use as Approximation method

1

, 1,.., ,

0,

, 1,..,

N

i ïi

True App i

App

i

q t C t n

I I C t i N

I

for C t i N

ONCE THE VARIATIONAL APPROACH IS ESTABLISHED FOR A PROBLEM…..

,0,

True True

True

I I q t

I

Use as Approximation method or Model Construction

1

, 1,.., ,

0,

, 1,..,

N

i ïi

True App i

App

i

q t C t n

I I C t i N

I

for C t i N

,

0,

M

True Model M

Model

M

q t

I I

I

for

EXAMPLES IN QM

• Hartree-Fock Approx.• QMD Model• …

QM tI dt t i H t

As we know…

Variational Principle in Classical Mechanics Can be understood as Stationary Path in Feynman’s Path Integral Representation of Quantum Mechanis.

As we don’t know…

Inversely, Quantum Mechanics can be derived from the Classical Mechanics in terms of Variational Principle...?

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...?

GENEALOGY OF NON CONVENTIONAL FORMULATION OF QUANTUM MECHANICS

Bohm-VigierHidden variables

Edward Nelson- Stochastic Method

Parisi-WuStochastic Quantization5th Dim (time)

Kunio YasueStochastic Variational Method

de Broglie

A. Eddington

E. Madelung

Conferência Solvay Bruxela 1911THE BIGININGSolvay Conf.-1911

Conferência Solvay Bruxela 1911THE BIGININGSolvay Conf.-1911

SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION

Ensemble of dust particle under the potential V

3 21[ , ] ,

2

f

i

t

tI v d r mv V

( , ) ( ( ))x t x x t

( )( , )

x

dx tv x t

dt

E. Madelung

SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION

Ensemble of dust particle under the potential V

3 21[ , ] ,

2

f

i

t

tI v d r mv V

( , ) ( ( ))x t x x t

( )( , )

x

dx tv x t

dt

For interacting adiabatic fluid, add the internal energy to V,

, /V V U U

SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION

Ensemble of dust particle under the potential V

3 21[ , ] ,

2

f

i

t

tI v dt d r mv V

( , ) ( ( ))x t x x t

( )( , )

x

dx tv x t

dt

3 2

[ , , ]

1( , ) ( )

2

f

i

t

t

I v

d r mv V r t v

SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION

Ensemble of dust particle under the potential V

3 21[ , ] ,

2

f

i

t

tI v dt d r mv V

( , ) ( ( ))x t x x t

( )( , )

x

dx tv x t

dt

3 2

[ , , ]

1( , ) ( )

2

f

i

t

t

I v

d r mv V r t v

Continuitycondition

NEW

SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION

Ensemble of dust particle under the potential V

3 2

[ , , ]

1( , ) ( )

2

f

i

t

t

I v

d r mv V r t v

A constant to make adimensional

3 21( )

2

f

i

t

tdt d r mv V v

by partial integration.

SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION

Still continues…

3 21( )

2

f

i

t

tI dt d r mv V v

Variation with respect to the velocity field,

( ) 0,I

mvv

.vm

SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION

Still continues…

3 21( )

2

f

i

t

tI dt d r mv V v

Variation with respect to the velocity field,

( ) 0,I

mvv

.vm

Eliminate the velocity

SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION

Still continues…

23 2[ , ] ( )

2

f

i

t

tI dt d r V

m

Change of variables: ( , ) ,ie

3 *[ , ] [ , ]f

i

t

ttI I d r i H

2 2 2

2 ln2 2

H Vm m

SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION

Still continues…

3 *[ , ] [ , ]f

i

t

ttI I d r i H

2 2 2

2 ln2 2

H Vm m

This would be the action for Schroedinger Equation, if we don’t have the term , choosing .

SIMILARITY OF CLASSICAL FLUID TO SCHROEDINGER EQUATION

Still continues…

3 *[ , ] [ , ]f

i

t

ttI I d r i H

2 2 2

2 ln2 2

H V Um m

Can we have some mechanism to generate the internal energy U to compensate?

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?

3 *[ , ] [ , ]f

i

t

ttI I d r i H

2 2 2

2 ln2 2

H V Um m

HOW TO DEAL WITH THE PRESENCE OF NOISE?

38

STOCHASTIC PROCESS

The effect of microscopic degrees of freedom can be treated as noise, with Stochastic Differential Equation (SDE)

Classical case

Stochastic ( , )dr u r t dt

( , )dr V r t dt

( , )dr V r t dt

( , )dr u r t dt

HOW TO INCORPORATE IN VARIATIONAL SCHEME?

q(t)

t

Fixed

( ) ( )dq t V t

dt

NECESSITY OF TWO KINDS OF SDE

41

( , )FFdr u r t dt 0dt

2 2/ 2

2

1( ) ,

2P e

2 dt

with white noise

BACKWARD SDE

0dt ( , )BBdr u r t dt

FORWARD SDE

OTHERWISE, STARTING FROM THE INITIAL CONDITION ….

q(t)

t

( ) ( , ) ( )F Fdr t u r t dt t

OTHERWISE, STARTING FROM THE INITIAL CONDITION ….

q(t)

t

( ) ( , ) ( )F Fdr t u r t dt t

t F F Fu

Fokker-Planck Equation for Browinian Motion

THEN, HOW ABOUT STARTING FROM THE FINAL CONDITION ? ….

q(t)

t

( ) ( , ) ( )B Bdr t u r t dt t

THEN, HOW ABOUT STARTING FROM THE FINAL CONDITION ? ….

q(t)

t

t B B Bu

( ) ( , ) ( )B Bdr t u r t dt t

Fokker-Planck Equation for Browinian Motion

HOW TO DO?

RECONNECTION !

t B B Bu

t F F Fu ( , )r t

Number of the ways to reconnect atis proportional to

( , )r t

( , ) ( , )F Br t r t

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).

( , ) ( , ) ( , )F BN r t r t r t

3( ,[ , ]) ( , ) ln ,F BS t d r N r t N r t

WE REQUIRE: ( ,[ , ]) 0,F BS t

3 31 ( , ) ( , )F Bd r r t d r r t

with

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

( , ) ( , ) ( , )F Br t r t r t

t B B Bu t F F Fu

0, ,2

Ft F m m

Buuu u

2 lnF Bu Au

We get

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

( , ) ( , ) ( , )F Br t r t r t

t B B Bu t F F Fu

0, ,2

Ft F m m

Buuu u

2 lnF Bu Au

We get

ONE DENSITY AND TWO FLOW VELOCITIES

0, ,2

Ft

Bm m

u uu u

Re 2 ln .l BFuu u

ONE DENSITY AND TWO FLOW VELOCITIES

0, ,2

Ft

Bm m

u uu u

Re 2 ln .l BFuu u

ONE DENSITY AND TWO FLOW VELOCITIES

0, ,2

Ft

Bm m

u uu u

Re 2 ln .l BFuu u

2F

mBu

u u

: Translational (CM) velocity

Re F Blu uu

: Velocity of internal motion

NOW, WHAT IS THE OPTIMAL PATH?

0, ,2

Ft

Bm m

u uu u

Re 2 ln .l BFuu u

VARIATIONAL PRINCIPLE FOR STOCHASTIC TRAJECTORIES…

55

( , )b

a

I dtL X DX

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).

VARIATIONAL PRINCIPLE FOR STOCHASTIC TRAJECTORIES…

56

( , )b

a

I dtL X DX

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).

VARIATIONAL PRINCIPLE FOR STOCHASTIC TRAJECTORIES…

57

( , )b

a

I dtL X DX

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).

VARIATIONAL PRINCIPLE FOR STOCHASTIC TRAJECTORIES…

58

( , )b

a

I dtL X DX

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).

OUR FORMULATION NEW

FLUID WITH THE INTERNAL FLOW VELOCITY

0, ,2

Ft

Bm m

u uu u

Re 2 ln .l BFuu u

OUR FORMULATION NEW

FLUID WITH THE INTERNAL FLOW VELOCITY

0, ,2

Ft

Bm m

u uu u

Re 2 ln .l BFuu u

2 22 ln .

2 2Beff

F

m mU u u

The fluid carries the internal energy

2 .effm m

ACTION SHOULD BE3

21( )

2

f

i

t

t

m m

I dt d r

mu V U u

3 *f

i

t

ttd r i H

2 2 2

2 ln2 2

H V Um m

224 ln .

2 2Beff

F

m mU u u

ACTION SHOULD BE3

21( )

2

f

i

t

t

m m

I dt d r

mu V U u

3 *f

i

t

ttd r i H

2 2 2

2 ln2 2

H V Um m

4 .m

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,

QM tI dt t i H t

( , ) ( , ) ,ie

RECONSTRUCTION OF QUANTUM MECHANICS

, , , ,O t dt r t O

From the fluid form:

* ˆ, ,dt r t O r t

For example, from the Noether’s theorem, the generator for the spatial translation is

,P dt r t

* , ,dt r t r ti

The generator for the time translation is

22 2 2, ln2 m

mE dt r t u V

* , ,dt r t H r t

CRUCIAL QUESTION:

It seems that the use of the wavefunction is just a mathematical convenient representation of the stochastic motivated two component fluid dynamics !

Then, the concept of amplitude is not essential ?? Do we not need the Hilbert space for state vectors?

TAKABAYASI-WALLSRTOM CRITICS

( , ) ( , ) ,ie

,( , ) , , ,imnr t r e

2m

C

mu dl n

If l is single-valued function with no singularity,

But inversly we should allow y , for example

0m

C

mu dl

Or, n does not necessarily to be 0, otherwise not quantize the angular momentum….

1. Our formalism is easy to extend to quantize the sytem of a field.

2. Shroedinger Equation should be the special case of one particle sector of non-relativistic case for the Klein-Gordon or Dirac Equation

3. In this case, the velocity is refer to the flow in the functional space

4. Takabayashi-Wallstrom criticism may not be the deffect, but may be even better, (to reduce the linear space for the functional states)

1. Space-Time manifold was born together with the known fields as effective theory. The Gaussian noise indicate the central limit theorem, which seems to be consistent with the maximum entropy….. In this case,

2. The origin of the noises is universal (that’s why the Planck constant and c enter together. In fact, this is the case. Noise=hc )

3. No need for the multiple valued phase. Maybe useful to reduce the huge quantum field state vector space.

4. If exist, it can be attributed to the difference between boson and fermions….

OUR RESPONSE AND SPECULATIONS

1. Our formalism is easy to extend to quantize the sytem of a field.

2. Shroedinger Equation should be the special case of one particle sector of non-relativistic case for the Klein-Gordon or Dirac Equation

3. In this case, the velocity is refer to the flow in the functional space

4. Takabayashi-Wallstrom criticism may not be the deffect, but may be even better, (to reduce the linear space for the functional states)

1. Space-Time manifold was born together with the known fields as effective theory. The Gaussian noise indicate the central limit theorem, which seems to be consistent with the maximum entropy….. In this case,

2. The origin of the noises is universal (that’s why the Planck constant and c enter together. In fact, this is the case. Noise=hc )

3. No need for the multiple valued phase. Maybe useful to reduce the huge quantum field state vector space.

4. If exist, it can be attributed to the difference between boson and fermions….

OUR RESPONSE AND SPECULATIONS

Spec

ulat

ion

witho

ut n

o

resp

onsi

bilit

y

REFERENCES

- P.R. Holstein, The Quantum Theory of Motion, Cambridge Univeristy

- E. Nelson, Phys. Rev. 150, 1079 (1966); Quantum Fluctuations, (Princeton Univ. Press, Prinston, NJ, 1985).

- K. Yasue, J. Funct. Anal. 41, 327 (1981); - F. Guerra and L. M. Morato, Phys. Rev. D27,

1774 (1983); - M. Pavon, J. Math. Phys. 36, 6774 (1995); - M. Nagasawa, Stochastic Process in Quantum

Physics, (Birkhäuser, 2000). - T. Koide and T. Kodama, J. Phys. A: Math. Theor.

45, 255204 (2012) and references therein. KK2 : - T. Koide and T. Kodama, arXiv:1306.6922; T.

Koide ,T. Kodama, and K. Tsushina, arXiv:1406.6295. Space : See for example, H.S. Snyder, Phys. Rev. 71, 38 (1947); P. Jizba and F. Scardigli, Phys.Rev.D86,025029 (2012).

THANK YOU FOR YOUR ATTENTION,