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 Chiral symmetric phase Dynamical Chiral Phase Transition Relativistic Heavy Ion Collision 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 condensate We ivestigate ・・・

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

Quantum mechanical system Coherent State ・・・ Classical image for QM system vacuum

Coherent State Expectation values Uncertainty relation ・・・ Fixed quantum fluctuation ・・・

Squeezed state vacuum of Bogoliubov transformed operator

Squeezed State To include “Quantum effects” appropriately ・・・・ extended coherent state ⇒ Squeezed State Coherent state Squeezed state

Squeezed State Expectation values Uncertainty relation quantum fluctuations are included through G(t) and Σ(t)

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 GΣ 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) ⇒ － 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

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

From Squeezed State to Gaussian wave functional Functional Schrödinger Picture Gaussian wave functional ・・・ dynamical “variables’’ ・・・ center, its conjugate momentum ・・・ Gaussian width, its velocity

Gaussian wavefunctional Expectation values

Time-Dependent Variational Approach with a Gaussian Wave Functional TDVP Trial functions in a Gaussian wave functional

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

・ 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

Mean filed (chiral condensate) Quantum fluctuations around the mean field Both should be determined self-consistently : chiral condensate Dynamical Variables Center and its momentum Gaussian Width and its momentum

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

Eq. of motion for fluctuations Reduced density matrix--- like TDHB theory Eq. of Motion ・・・ Liouville von-Neumann equation

Reformulation for fluctuations Mode functions, Eq. of Motion ・・・ manifestly covariant form Feynman propagator and

mean field Hamiltonian Finite Temperature ・ Density operator ・ Annihilation operator ・ Averaged value of particle number ・ Thus,

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

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 : ・ Space-like isospin rotation : ω↑ q ↓ ←T

Brief Summary q 2 ＞ 0…enhancement of chiral symmetry breaking cf.) centrifugal force q 2 ＜ 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 T (MeV) |q c |(MeV)

Decay of collective isospin rotation of chiral condensate ---Decay of DCC---

Lifetime of collective isospin rotation - c≠0 ・・・ explicit chiral symmetry breaking Consider 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 ↑ ｃ≠０ ↑ ｃ≠０

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

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) ・ 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 ⇒ ⇒

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)

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 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) N.Ikezi, M.Asakawa and Y.T., Phys.Rev. C69 (2004) (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