STOCHASTIC VARIATIONAL METHOD AND QUANTIZATION TAKESHI KODAMA.
-
Upload
shauna-griffin -
Category
Documents
-
view
224 -
download
0
Transcript of STOCHASTIC VARIATIONAL METHOD AND QUANTIZATION TAKESHI KODAMA.
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,