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Variational Approach in
Quantum Field Theories
-- to Dynamical Chiral Phase Transition --
Yasuhiko TSUE
Physica Division, Faculty of Science,
Kochi University, Japan
Introduction and Motivation
Chiral Phase Transitions
Relativistic Heavy Ion Collision
Some posibilities of dynamical process
T  TC

T0





Chiral symmetric phase
Symmetry broken phase
Dynamical Chiral Phase
Transition
Introduce a possible method to describe it
including higher order quantum effects
Some of ways the process is classically represented
① Coherent displacement of chiral
condensate
We ivestigate ・・・
② Isospin rotation of chiral condensate
③ Roll down to the sigma direction
Treat the chiral condensate and fluctuation
modes around it self-consistently
R. Jackiw, A. Kerman (Phys.Lett. ,1979)
Time dependent variational approach with a squeezed state
or a Gaussian wavefunctional
.
Disoriented Chiral Condensate （DCC)

Production or Decay of DCC
⇒Time evolution of chiral cndensate
in quatum fluctuations
⇒amplitudes of quantum fluctuation
modes are not so small
・・・amplification of
quantum meson modes
It is necessary to treat the time evolution
of chiral condensate (mean field) and
quantum meson modes (fluctuations)
appropriately (not perturbatively)
Our Method
―Dynamical Chiral Phase Transition
・・・How to describe the time evolution of chiral
condensate (mean field) in quantum meson
modes self-consistently ?
―Nuclear Many-Body Problems
・・・Is it possible to apply the methods developed
in microscopic theories of collective motion in
nucleus to quantum field theories ?
↓↓↓↓
Time-Dependent Variational Method in
Quantum Field Theories
Y.T.and Y.Fujiwara, Prog.Theor.Phys.86 (1991) 469
Table of Contents




Time-Dependent Variational Approach (TDVA) with
a Squeezed State in Quantum Mechanics
--- Equivalent to Gaussian approximation in the
functional Schrödinger picture --TDVA with a Squeezed State in Quantum Scalar
Field Theory
Application to dynamical chiral phase transition
・isospin rotation
・late time of chiral phase transition
Summary
Time-dependent variational approach with
a squeezed state in quantum mechanics
Functional Schrödinger Picture in
Quantum Mechanics
Coherent state
Vacuum of shifted operator
bˆ   (aˆ   )   0
  exp  aˆ    *aˆ  0


Quantum mechanical system

1 ˆ2
ˆ
H  P  V (Qˆ )
2
Coherent State
・・・Classical image for QM system



ˆ
ˆ
Q
(aˆ  aˆ ) , P  i
(aˆ  aˆ  )
2
2


(t )  exp aˆ    *aˆ 0

i
 exp  p(t )Qˆ  q(t ) Pˆ

 
 1

Q 0    exp    Q 2 
 
 2

1/ 4
vacuum


0

aˆ 0  0
Coherent State

Expectation values
(t ) Qˆ (t )  q(t ) ,


2
2
ˆ
(t ) Q (t )  q(t ) 
,
2
Uncertainty relation


2
ˆ
q   (t ) Q  q(t ) (t )  1 ,
2
2
(t ) Pˆ (t )  p(t )

2
2
ˆ
(t ) P (t )  p(t ) 
2


2
ˆ
p   (t ) P  p(t ) (t )  1 
2
2

q p  
2
・・・Fixed quantum fluctuation・・・
Squeezed state
vacuum of Bogoliubov transformed operator


B

*
cˆ   (aˆ   ) cosh | B | (aˆ   )
sinh | B |   0
|B|


1
2
* 2 
  exp aˆ   aˆ exp  Baˆ  B aˆ  0
2


*
Squeezed State

To include “Quantum effects” appropriately
・・・・extended coherent state ⇒ Squeezed State




1

(t )  exp aˆ   aˆ exp  Baˆ  2  B*aˆ 2  0
2


 2G 
1/ 4
*

i
exp  p(t )Qˆ  q(t ) Pˆ

Coherent state

 1 
 ˆ2
1

 i 2(t ) Q  0
 exp    
2G (t )

 
 2 
Squeezed state
Squeezed State

Expectation values
(t ) Qˆ (t )  q(t ) ,
(t ) Pˆ (t )  p(t )
 1

(t ) Qˆ 2 (t )  q(t ) 2  G (t ) , (t ) Pˆ 2 (t )  p(t ) 2  
 4G (t )(t ) 2 
 4G


Uncertainty relation


 (t ) Pˆ  p(t ) 
2
ˆ
q   (t ) Q  q(t ) (t )  G (t ),
2
p 
2

2
 1

(t )  
 4G (t )(t ) 2 
 4G


2
q p   1  4G(t )(t )
2
quantum fluctuations are included through G(t) and Σ(t)
Squeezed State
⇒ is equivalent to Gaussian wave function

Wave function representation
Q  sq  (Q)
 2G 
1/ 4 ipq / 2 
e
i

1 1
2
exp  p(t )Q  q(t )   
 i(t ) Q  q(t )  
  4G (t )



Gaussian wave function
・・・center : q(t)
; its velocity : p(t)
・・・width of Gaussian : G(t) ; its velocity : ４GΣ
Y.T.and Y.Fujiwara, Prog.Theor.Phys.86 (1991) 443
Equations of motion derived by
Time-Dependent Variational Principle

Time-dependent variational principle (TDVP)
 S    (t ) i t  Hˆ (t ) dt  0
⇒
H
H
H
H


q 
, p  
, G 
,   
p
q

G
－G and Σ appear with 
・・・describe the dynamics of quantum fluctuations
Y.T.and Y.Fujiwara, Prog.Theor.Phys.86 (1991) 443
Time-Dependent Variational Approach in
quantum scalar field theory
Time-Dependent Variational Approach in
Quantum Scalar Field Theory

We extend the variational method with a
squeezed state in Quantum Mechanical System
to Quantum Field Theory
TDVP + Squeezed State
↓↓↓↓↓
Time-dependent variational approach with
a Gaussian wavefunctional
based on the functional Schrödinger picture
x   ( x)


p  i   ( x)  i
x
( x)
x, p  i,
( x),  ( y)  i ( x  y)
Squeezed state
and
Gaussian wavefunctional
in quantum field theory
Squeezed State in Quantum Scalar Field Theory

Squeezed State
 (t )  exp{ ( ka aka   ka* aka )} exp{
ka
1
(aka Bka ,k 'b ak'b  aka Bka ,k 'b ak 'b )} 0

2 kk ',ab
 exp S (t )  exp T (t )  0




S (t )  i  d 3 x  a ( x , t )ˆa ( x )   a ( x , t )ˆ a ( x )




  1 ( 0 ) 1  
   
1  
T (t )    d 3 x d 3 y ˆa ( x )  Gab
( x , y )  Gab
( x , y, t )  i ab ( x , y, t )ˆb ( y )
4



( 0)  
(k:momentum ; a,b:isospin)
Gab
( x, y)  0 ˆa ( x )ˆb ( y) 0

cf.) squeezed state in Quantum Mechanics




1

(t )  exp aˆ    *aˆ exp  Baˆ  2  B*aˆ 2  0
2

 2G 
1/ 4


 1 
 ˆ2
1
i

ˆ
ˆ

exp  p(t )Q  q(t ) P  exp    
 i 2(t ) Q  0
2G (t )


 
 2 

From Squeezed State
to Gaussian wave functional

Functional Schrödinger Picture

[ ( x )]    (t )





 N exp i  dx a ( x , t )ˆ a ( x , t )

 
  1 1  
    
 exp   dxdyˆ a ( x , t ) Gab
( x , y, t )  i ab ( x , y, t ) ˆb ( y, t )
4






ˆ a ( x , t )  a ( x )   a ( x , t )
Gaussian wave functional・・・dynamical “variables’’


・・・center  ( x , t ,)
its conjugate momentum  ( x , t )
 
 
・・・Gaussian width G ( x , y, t ) , its velocity 2G  G ( x, y, t )
Gaussian wavefunctional

Expectation values




(t )  a ( x ) (t )   a ( x , t ) , (t )  a ( x ) (t )   a ( x , t )

 

(t ) ˆ a ( x , t )ˆb ( y, t ) (t )  Gab ( x , y, t )

 
1 1  

(t ) ˆ a ( x , t )ˆ b ( y, t ) (t )  Gab ( x , y, t )  4G ab ( x , y, t )
4






ˆa ( x, t )  a ( x )  a ( x, t ) , ˆ a ( x, t )   a ( x )   a ( x, t )
Time-Dependent Variational Approach with
a Gaussian Wave Functional

TDVP

Trial functions in a Gaussian wave functional
 S    (t ) i t  Hˆ (t ) dt  0

 ( x, t )

 ( x, t )
 
G ( x , y, t )
 
( x , y , t )
Application to Dynamical Chiral Phase
Transition,
especially,
Disoriented Chiral Condensate (DCC)
problems
DCC Formation
Nonequilibrium chiral dynamics
and
two-particle correlations
Dr. Ikezi’s talk
DCC
as a collective isospin rotation
DCC
as a collective Isospin Rotation
effects of collective rotation of chiral condensate
in isospin space
⇔
・Phase diagram in isospin rotation ?
・Damping mechanism of collective
isospin rotation ?
・Damping time ?
・Number of emitted mesons ?
Investigate them in O(4) linear
sigma model in time-dependent
variational method
Variational Approach in Gaussian wave functional
Y.Tsue, D.Vautherin & T.Matsui, PTP 102 (1999) 313
・Hamiltonian density
H
 2 m02 2  
 22

1 2  1
  a ( x )   a ( x )  
a ( x ) 
a ( x )  c a ( x ) a 0
2
2
2
24


・Gaussian wave functional




[ ( x )]  N exp i  dx a ( x , t )ˆ a ( x , t )



 
  1 1  
    
ˆ
 exp   dxdy a ( x , t ) Gab ( x , y, t )  i ab ( x , y, t ) ˆb ( y, t )
4






ˆ a ( x , t )  a ( x )   a ( x , t )
・Dynamical variables

a ( x, t )

 a ( x, t )
 
Gab ( x, y, t )
 
 ab ( x, y, t )
Dynamical Variables

Mean filed (chiral condensate)


(t ) a ( x ) (t )  a ( x, t ) ,
 : chiral condensate



(t )  a ( x ) (t )   a ( x, t )
Center and its momentum
Quantum fluctuations around the mean field


 


(t )  a ( x )b ( y ) (t )   a ( x , t )b ( y, t )  Gab ( x , y, t )
 
Gaussian Width and
 ab ( x , y, t )
its momentum
Both should be determined self-consistently
Eqs. of motion for condensate

TDVP
  dt (t ) i t  Hˆ (t )  0

Eq. of motion for condensate・・・Klein-Gordon type

      
 2 


2
     m0   ( x , t )   Tr G( x , x )  G( x , x )   ( x , t )  c1
6
3
6


 


Gab ( x, y, t )  ˆa ( x, t )ˆb ( y, t )
: Thermal average
Eq. of motion for fluctuations


Reduced density matrix--- like TDHB theory





ˆ
ˆ
ˆ
ˆ
 i  a ( x , t ) b ( y, t )
 a ( x , t )b ( y, t )
 

M ab ( x , y; t ) 


 ˆ a ( x, t )ˆ b ( y, t )
i ˆ a ( x , t )ˆb ( y, t )

2iG
G 


  1
 G / 4  4G  2iG 
 1  
   ( x  y)
 2

Eq. of Motion ・・・Liouville von-Neumann equation
iM  H , M

 0 1

H  
 0





ab     2  m02   2  ˆcˆc   ab   ab   ˆ aˆb  
6
6
3


  2  M a2 ab
Reformulation for fluctuations

Mode functions
 un 
 un 
i t    H  
 vn 
 vn 

 un  1  un 
    
M
,
 vn  2  vn 
Eq. of Motion ・・・manifestly covariant form
□ M  u
2
a

a
n
0
Feynman propagator



* 
x S y   (t x  t y ) u ( x , t x )un ( y, t y )  (t y  t x ) un ( x , t x )un ( y, t y )
n 0
and
*
n
 
G( x , x , t )  x S x
n 0
Finite Temperature
・Density operator

 exp   W (, q ) 



D(, q ) 
, Z  Tr exp   W (, q ) 
2
Z

 



q
W ( , q )  W   I ab  q  I ab   dx ˆc ( x )ˆc ( x )
2
mean field Hamiltonian
・Annihilation operator




 
3 


b(k , a)   d x vk ( x , a)ˆa ( x )  uk ( x , a)
 
ˆa ( x ) 

・Averaged value
 of particle
 number


・Thus,

1

na (k )  Tr Db (k , a)b(k , a) 
exp Ea (k )  1







ˆa ( x )ˆb ( y )  Tr Dˆ a ( x )ˆb ( y )


* 

 ( y , b)  c.c.
  ab 
n
(
k
)

1
/
2

u
(
x
,
a
)
u
a
k
k


k


Collective isospin rotation
・ effects of isospin rotation where isospin components 0 and 1 are
mixed → isospin rotating frame
0 
 0  i 0 
 


0
 i 0 0 
 ( x)  U ( x)  ,  y  

0 0 0 
 


0
    
 




U ( x)  exp iqx y  , q  ( , q )



 
 2  N 1

 2
2
ˆ
ˆ
ˆ
ˆ
  q  m0  0   c ( x )c ( x )  0 ( x )0 ( x ) 0  c
6
6 c 0
3


 
 

 
M r ( x , y, t )  U ( x , t )M ( x , y, t )U ( y, t )
1 
  y
 


H (q)  U ( x)  i  H U ( x)   


(
q
)

 t

y



Effects of isospin rotation of chiral condensate (c=0)
・Time-like
Phase diagram
isospin rotation :
・Space-like isospin rotation :
|q| vs. condensate
ω↑
q↓
←T
T vs. condensate
Y.Tsue, D.Vautherin & T.Matsui, Prog.Theor.Phys. 102 (1999) 313
Brief Summary
q2＞0…enhancement of chiral symmetry breaking
cf.) centrifugal force
 q2＜0…existence of critical q
⇒ restoration of chiral symmetry
Quantum effects lead to more rapid change of chiral condensate

T (MeV)
0
20
40
60
80
|qc|(MeV)
50.0
49.9
49.7
47.2
37.7

Cf.) Classical case
6
2
    q  m02
2
0
2
 2 M 2
2
qc 
 354 MeV  at T  0
2
・Quantum fluctuations smear out the effective potential
・Quantum fluctuations make symmetry breaking more difficult
to reach
Quantum effects are important
Decay of collective isospin rotation of
chiral condensate
---Decay of DCC---
Lifetime of collective isospin rotation
H
 2 m02 2  
 22

1 2  1
  a ( x )   a ( x )  
a ( x ) 
a ( x )  c a ( x ) a 0
2
2
2
24

- c≠0 ・・・ explicit chiral symmetry breaking
Consider the linear response with respect to c
- Chiral condensate   exp iqx   ( 0)

y

↑
Isospin rotation

 ( x)
↑
↑
ｃ＝０
ｃ≠０
()
c  0  ( x)   Hext
 exp  i t    M
Reduced density matrix --- linearization ---
M  M c 0  M
()

 exp  i t 
Explicit chiral symmetry breaking : c≠0
↓↓
iM  H , M
``External source term” for quantum fluctuation


i M  Hc0 , M   Hext , M c0   Hind , M c0 
Energy of collective isospin rotation of chiral condensate
leads to deacy of collective isospin rotation and
leads to two-pion emissions
two meson emission
Damping time & number of emitted pions
Y.Tsue, D.Vautherin & T.Matsui, Phys. Rev. D61 (2000) 076006
・Damping time
 (q) 



1
1
1
2
E0   a2   a   02  2  q 2
2
2
2
E0 Energy density of collective rotating condensate
 E  E  1 Tr HM   Tr H 0M 0    (q) t
 
V
 t Energy density of two meson
M
⇒
N 
⇒
・Number of emitted pions ,
E if classical field configuration occupies volume V
V
4
3
for
E

(
160
MeV)
and
V

(
10
fm)
0
t
     40 fm /c
N  15 mesons per fm / c
Larger than the collision time ～a few fm/c
1 2 2
0   (160 MeV) 4
2
for   2 2M 
③ Amplification of quantum meson modes in
role-down of chiral condensate
K.Watanabe, Y.T. and S.Nishiyama, Prog.Theor.Phys.113 (2005) 369
The set of basic equations of motion again


      


      m02   2 ( x , t )   Tr G( x , x )  G( x , x )   ( x , t )  c1
6
3
6


Chiral Condensate
Quantum Meson Fields
Y. Tsue, D. Vautherin and T. Matsui (Prog.Theor.Phys. ,1999)
Numerical results without spatial expansion
Late time of Chiral phase transition
Small deviation around static configurations
cf. Mathieu equation
cf. Forced oscillation
Dimensionless variables
The unstable regions for quantum pion modes
⇐ Mathieu equation
Summary
We have presented the time-dependent variational
method with a squeezed state or a Gaussian
wavefunctional in quantum scalar field theories.
We have applied our method to the problems of
dynamical process of chiral phase transition.
Further,・・・
Nonequilibrium chiral dynamics and
two-particle correlations by using the squeezed state

Functional Schrödinger picture in quantum theory
R.Jackiw and A.Kerman, Phys.Lett.71A (1979) 158
R.Balian and M.Vènéroni, Phys.Rev.Lett. 47 (1981) 1353, 1765
O.Eboli, R.Jackiw and S.-Y.Pi, Phys.Rev. D37 (1988) 3557
R.Jackiw, Physica A158 (1989) 269

Coherent state and squeezed state
W.-M.Zhang, D.H.Feng and R.Gilmore, Rev.Mod.Phys.62 (1990) 867



Our references














TDVA with squeezed state in qantum mechanics
Y.T., Y.Fujiwara, A.Kuriyama and M.Yamamura, Prog.Theor.Phys.85 (1991) 693
Y.T.and Y.Fujiwara, Prog.Theor.Phys.86 (1991) 443
Y.T. Prog.Theor.Phys.88 (1992) 911
Fermionic squeezed state in Quantum many-fermion systems
Y.T., A.Kuriyama and M.Yamamura, Prog.Theor.Phys.92 (1994) 545
Y.T., N.Azuma, A.Kuriyama and M.Yamamura, Prog.Theor.Phys.96 (1996) 729
Y.T. and H.Akaike, Prog.Theor.Phys. 113 (2005) 105
H.Akaike, Y.T. and S.Nishiyama, Prog.Theor.Phys. 112 (2004) 583
TDVA with squeezed state in scalar field theory
Y.T.and Y.Fujiwara, Prog.Theor.Phys.86 (1991) 469
Application to DCC physics
Y.T., D.Vautherin and T.Matsui, Prog.Theor.Phys.102 (1999) 313
Y.T., D.Vautherin and T.Matsui, Phys.Rev. D61 (2000) 076006
N.Ikezi, M.Asakawa and Y.T., Phys.Rev. C69 (2004) 032202(R)
Application to dynamical chiral phase transition
Y.T., A.Koike and N.Ikezi, Prog.Theor.Phys.106 (2001) 807
Y.T., Prog.Theor.Phys.107 (2002) 1285
K.Watanabe, Y.T. and S.Nishiyama, Prog.Theor.Phys.113 (2005) 369