Variational Approach in Quantum Field Theories

-- to Dynamical Chiral Phase Transition --

Yasuhiko TSUEPhysica Division, Faculty of Science,

Kochi University, Japan

Introduction and Motivation

Chiral Phase Transitions

Chiral symmetric phase

Dynamical Chiral Phase Transition

Relativistic Heavy Ion Collision

CTT

0T

Symmetry broken phase

Introduce a possible method to describe it including higher order quantum effects

Some posibilities of dynamical process

Some of ways the process is classically represented .

③ 　 Roll down to the sigma direction

Treat the chiral condensate and fluctuation modes around it self-consistently

Time dependent variational approach with a squeezed state or a Gaussian wavefunctional

R. Jackiw, A. Kerman (Phys.Lett. ,1979)

① Coherent displacement of chiral condensate

② Isospin rotation of chiral condensateWe ivestigate ・・・

Disoriented Chiral Condensate （ DCC)

Production or Decay of DCC　　⇒ Time evolution of chiral cndensat

e 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 c

hiral condensate (mean field) in quantum meson modes self-consistently ?

―Nuclear 　 Many-Body Problems　　・・・ Is it possible to apply the methods deve

loped in microscopic theories of collective motion in nucleus to quantum field theories ?

↓↓↓↓

Time-Dependent Variational Method in Quantum Field TheoriesY.T.and Y.Fujiwara, Prog.Theor.Phys.86 (1991) 469

Table of Contents Time-Dependent Variational Approach 　 (TDVA) wit

h a Squeezed State in Quantum Mechanics --- Equivalent to Gaussian approximation in the functional Schrödinger picture --- TDVA with a Squeezed State in Quantum Scalar Fie

ld 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

0)ˆ(ˆ ab

0ˆ*ˆexp

aa

Quantum mechanical system

Coherent State

　・・・ Classical image for QM system

vacuum

)ˆ(ˆ2

1ˆ 2 QVPH

)ˆˆ(2

ˆ,)ˆˆ(2

ˆ aaiPaaQ

0ˆ)(ˆ)(exp

0ˆˆexp)( *

PtqQtpi

aat

2

4/1

2

1exp0 QQ

00ˆ a

Coherent State

Expectation values

Uncertainty relation

・・・ Fixed quantum fluctuation ・・・

)()(ˆ)(,)()(ˆ)( tptPttqtQt

2

1)()(ˆ)(,

2

1)()(ˆ)(

2222 ttpPtpttqQtq

2

pq

2)()(ˆ)(,

2)()(ˆ)( 2222

tptPttqtQt

Squeezed state

vacuum of Bogoliubov transformed operator

0||sinh||

)ˆ(||cosh)ˆ(ˆ *

B

B

BaBac

0ˆˆ2

1expˆˆexp 2*2*

aBaBaa

Squeezed State To include “Quantum effects” appropriately

　　・・・・ extended coherent state 　⇒ Squeezed State

0ˆ)(2)(2

1

2

1expˆ)(ˆ)(exp2

0ˆˆ2

1expˆˆexp)(

24/1

2*2*

QtitG

PtqQtpi

G

aBaBaat

Coherent stateCoherent state Squeezed stateSqueezed state

222

22

)()(44

1)()(ˆ)(

),()()(ˆ)(

ttGG

ttpPtp

tGttqQtq

Squeezed State Expectation values

Uncertainty relation

quantum fluctuations are included through G(t) and Σ(t)

)()(ˆ)(,)()(ˆ)( tptPttqtQt

2)()(412

ttGpq

22222 )()(4

4

1)()(ˆ)(,)()()(ˆ)( ttG

GtptPttGtqtQt

22/4/1 )()()(4

11)()(exp2

)(

tqQtitG

tqQtpi

eG

QQ

ipq

sq

Squeezed State⇒ is equivalent to Gaussian wave

function Wave function representation

　Gaussian wave function

　　　・・・ center : q(t) ; its velocity : p(t) 　　　　　・・・ width of Gaussian : G(t) ; 　 its velocity : ４ G

Σ　　　　　　　　　　　　　 Y.T.and Y.Fujiwara, Prog.Theor.Phys.86 (19

91) 443

Equations of motion derived by Time-Dependent Variational Principle

Time-dependent variational principle (TDVP)

　 ⇒

　　－ G 　 and Σ appear with

　　・・・ describe the dynamics of quantum fluctuations

Y.T.and Y.Fujiwara, Prog.Theor.Phys.86 (1991) 443

G

HHG

q

Hp

p

Hq

,,,

0)(ˆ)( dttHitS t

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

　 )()()(

xix

xipxx

)()(),(,, yxiyxipx

Squeezed state and

Gaussian wavefunctional in quantum field theory

Squeezed State in Quantum Scalar Field Theory

Squeezed State

(k:momentum ; a,b:isospin)

cf.) squeezed state in Quantum Mechanics

)(ˆ),,(),,(),(4

1)(ˆ)(

)(ˆ),()(ˆ),()(

0)(exp)(exp

0)}(2

1exp{)}(exp{)(

11)0(33

3

'',','

',*

ytyxityxGyxGxydxdtT

xtxxtxxditS

tTtS

aBaaBaaat

babababa

aaaa

bkbkkakabkabkk

bkkakakakakaka

ka

0ˆ)(2)(2

1

2

1expˆ)(ˆ)(exp2

0ˆˆ2

1expˆˆexp)(

24/1

2*2*

QtitG

PtqQtpi

G

aBaBaat

0)(ˆ)(ˆ0),()0( yxyxG baab

),()(),(ˆ

),(ˆ),,(),,(4

1),(ˆexp

),(ˆ),(exp

)()]([

1

txxtx

tytyxityxGtxydxd

txtxxdiN

tx

aaa

bababa

aa

From Squeezed State to Gaussian wave functional

Functional Schrödinger Picture

　

Gaussian wave functional ・・・ dynamical “variables’’

・・・ center 　　　　 , its conjugate momentum 　　　

・・・ Gaussian width 　　　　　　 , its velocity 　　　　　　　　　　　　　　　　

),( tx ),( tx

),,( tyxG

),,(2 tyxGG

Gaussian wavefunctional

Expectation values

),()(),(ˆ,),()(),(ˆ

),,(4),,(4

1)(),(ˆ),(ˆ)(

),,()(),(ˆ),(ˆ)(

),()()()(,),()()()(

1

txxtxtxxtx

tyxGtyxGttytxt

tyxGttytxt

txtxttxtxt

aaaaaa

ababba

abba

aaaa

Time-Dependent Variational Approach with a Gaussian Wave Functional

TDVP 　

Trial functions in a Gaussian wave functional 　

　),( tx

0)(ˆ)( dttHitS t

),( tx ),,( tyxG

),,( tyx

Application to Dynamical Chiral Phase Transition, especially,

Disoriented Chiral Condensate (DCC) problems

DCC FormationNonequilibrium chiral dynamics

and two-particle correlations

Dr. Ikezi’s talk

DCC as a collective isospin rotation

・ Phase diagram in isospin rotation ?・ Damping mechanism of collective isospin rotation ?・ Damping time ?・ Number of emitted mesons ?

DCC as a collective Isospin Rotation

effects of collective rotation of chiral condensate in isospin space

　　　　　　　　　　　　　　　　　　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

・ Gaussian wave functional

・ Dynamical variables

),()(),(ˆ

),(ˆ),,(),,(4

1),(ˆexp

),(ˆ),(exp)]([

1

txxtx

tytyxityxGtxydxd

txtxxdiNx

aaa

bababa

aa

),( txa

),( txa

),,( tyxGab

),,( tyxab

0

2222022 )()(

24)(

2)(

2

1)(

2

1aaaaaa xcxx

mxx

H

Mean filed (chiral condensate)

Quantum fluctuations around the mean field

Both should be determined self-consistently

: chiral condensate : chiral condensate

Dynamical Variables

Center and its momentumCenter and its momentum

Gaussian Width and its momentum

Gaussian Width and its momentum

),()()()(,),()()()( txtxttxtxt aaaa

),,(

),,(),(),()()()()(

tyx

tyxGtytxtyxt

ab

abbaba

Eqs. of motion for condensate

TDVP

Eq. of motion for condensate ・・・ Klein-Gordon type

1),(),(3

),(Tr6

),(6

220 ctxxxGxxGtxm

averageThermal:),(ˆ),(ˆ),,( tytxtyxG baab

0)(ˆ)( tHitdt t

Eq. of motion for fluctuations Reduced density matrix--- like TDHB theory

Eq. of Motion ・・・ Liouville von-Neumann equation

　　　　

　　　　　　　　　　　　　　　　　　　　　　　　

GiGG

GiG

yxtytxitytx

tytxtytxityx

baba

babaab

244/

2

)(2

1

),(ˆ),(ˆ),(ˆ),(ˆ

),(ˆ),(ˆ),(ˆ),(ˆ);,(

1

M

aba

babaabccab

M

m

i

22

220

2 ˆˆ3

ˆˆ66

0

10

,

H

H MM

Reformulation for fluctuations Mode functions

　　　　　　　　　　　　　　　　　　　　　　　 , Eq. of Motion ・・・ manifestly covariant form

　 Feynman propagator

　　　　　 and 　　　　　　　　　　　　　　　　　　　

xSxtxxG ),,(

02 ana uM□

n

n

n

nt v

u

v

ui H

0

*

0

* ),(),()(),(),()(n

ynxnxyn

ynxnyx tyutxutttyutxuttySx

n

n

n

n

v

u

v

u

2

1M

mean field Hamiltonianmean field Hamiltonian

Finite Temperature・ Density operator

・ Annihilation operator

・ Averaged value of particle number

・ Thus,

),(expTr,),(exp

),( qWZZ

qWqD

)(ˆ)(ˆ2

),(2

xxxdq

IqIWqW ccabab

)(ˆ),()(ˆ),(),( 3

xaxuxaxvxdakb

akak

1)(exp

1),(),(Tr)(

kEakbakDbkn

a

a

kkkaab

baba

ccbyuaxukn

yxDyx

..),(),(2/1)(

)(ˆ)(ˆTr)(ˆ)(ˆ

*

・ effects of isospin rotation where isospin components 0 and 1 are mixed → isospin rotating frame

Collective isospin rotation

cxxxxmqN

ccc

000

1

0

20

20

2 )(ˆ)(ˆ3

)(ˆ)(ˆ66

y

y

r

qxU

tixUq

tyUtyxtxUtyx

)(

1)()()(

),(),,(),(),,(

HH

MM

),(,exp)(

000

00

00

,

0

0)()(

0

qqiqxxU

i

i

xUx

y

y

Effects of isospin rotation of chiral condensate (c=0) Phase diagram

|q| vs. condensate

T vs. condensate

Y.Tsue, D.Vautherin & T.Matsui, Prog.Theor.Phys. 102 (1999) 313

・・ Time-like isospin rotation :Time-like isospin rotation :・・ Space-like isospin rotation : Space-like isospin rotation :

ω↑ω↑

q ↓q ↓

←T←T

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 　　　

Cf.) 　 Classical case

・ Quantum fluctuations smear out the effective potential　　・ Quantum fluctuations make symmetry breaking more difficult 　　　　 to reach 　　　　　　　　　　　　　　 Quantum effects are

important

0atMeV3542

6

22

2

20

2220

TM

q

mq

c

T (MeV) 0 20 40 60 80

|qc|(MeV) 50.0 49.9 49.7 47.2 37.7

Decay of collective isospin rotation of chiral condensate

---Decay of DCC---

Lifetime of collective isospin rotation

- c≠0 　・・・ explicit chiral symmetry breakingConsider the linear response with respect to c- Chiral condensate

Reduced density matrix --- linearization ---

　　 ↑

Isospin rotation

　　 ↑

Isospin rotation

　↑　

ｃ＝０

　↑　

ｃ＝０

　↑　

ｃ≠０

　↑　

ｃ≠０

)(exp )0( xiqx y

titixc expexp)(0 )()(ext MH

MMM 0c

0

2222022 )()(

24)(

2)(

2

1)(

2

1aaaaaa xcxx

mxx

H

　Explicit chiral symmetry breaking : c≠0　　　　　　　　　　　　↓↓``External source term” for quantum fluctuation

MHM ,i

0ind0ext0 ,,, ccci MHMHMHM

Energy of collective isospin rotation of chiral condensate leads to deacy of collective isospin rotation and leads to two-pion emissions

two meson two meson emissionemission

Damping time & number of emitted pionsY.Tsue, D.Vautherin & T.Matsui, Phys. Rev. D61 (2000) 076006

・ Damping time Energy density of collective rotating condensate

　　　 Energy density of two meson

・ Number of emitted pions , if classical field configuration occupies volume V

Larger than the collision time ～ a few fm/c

t

qE

E0)(

t

qE

E0)(

⇒ ⇒

M

VtN

E

M

VtN

E

2220

220 2

1

2

1

2

1qaa

E

tqV

)(TrTr1

00 MHHME

c/fm40

340 fm)10(andMeV)160(for VE

M22forMeV)160(2

1 4220

cN /fmpermesons15

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

Chiral Condensate

Quantum Meson Fields

The set of basic equations of motion again

Y. Tsue, D. Vautherin and T. Matsui (Prog.Theor.Phys. ,1999)

1),(),(3

),(Tr6

),(6

220 ctxxxGxxGtxm

Numerical results without spatial expansion

Late time of Chiral phase transition

cf. Mathieu equation

cf. Forced oscillation

Dimensionless variables

Small deviation around static configurations

The unstable regions for quantum pion modes

⇐ 　 Mathieu equation

Summary We have presented the time-dependent variational me

thod with a squeezed state or a Gaussian wavefunctional in quantum scalar field theories.

We have applied our method to the problems of dynami

cal 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) 86

7

Our references

TDVA with squeezed state in qantum mechanics Y.T., Y.Fujiwara, 　 A.Kuriyama and M.Yamamura, Prog.Theor.Phys.85 (1991) 6

93 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