Post on 17-Jan-2016
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
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