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Variational Approach in Quantum Field Theories -- to Dynamical Chiral Phase Transition --

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.

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Variational Approach in Quantum Field Theories -- to Dynamical Chiral Phase Transition --

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  1. Variational Approach in Quantum Field Theories-- to Dynamical Chiral Phase Transition -- Yasuhiko TSUE Physica Division, Faculty of Science, Kochi University, Japan

  2. Introduction and Motivation Chiral Phase Transitions Relativistic Heavy Ion Collision Some posibilities of dynamical process Chiral symmetric phase Symmetry broken phase Dynamical Chiral Phase Transition Introduce a possible method to describe it including higher order quantum effects

  3. Some of ways the process is classically represented. ① Coherent displacement of chiral condensate We ivestigate ・・・ ② Isospin rotation of chiral condensate ③Roll 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

  4. 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)

  5. Our Method ―Dynamical Chiral Phase Transition 　　・・・How to describe the time evolution of chiral condensate (mean field) in quantum meson modes self-consistently ? ―NuclearMany-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

  6. 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

  7. Time-dependent variational approach with a squeezed state in quantum mechanicsFunctional Schrödinger Picture in Quantum Mechanics

  8. Coherent stateVacuum of shifted operator

  9. Quantum mechanical system • Coherent State 　・・・Classical image for QM system vacuum

  10. Coherent State • Expectation values • Uncertainty relation ・・・Fixed quantum fluctuation・・・

  11. Squeezed statevacuum of Bogoliubov transformed operator

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

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

  14. Squeezed State⇒ is equivalent toGaussian wave function • Wave function representation 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

  15. Equations of motion derived by Time-Dependent Variational Principle • Time-dependent variational principle (TDVP) 　 ⇒ －Gand Σ appear with 　　・・・describe the dynamics of quantum fluctuations Y.T.and Y.Fujiwara, Prog.Theor.Phys.86 (1991) 443

  16. Time-Dependent Variational Approach in quantum scalar field theory

  17. 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 TheoryTDVP + Squeezed State ↓↓↓↓↓ Time-dependent variational approach with a Gaussian wavefunctional based on the functional Schrödinger picture

  18. Squeezed state and Gaussian wavefunctional in quantum field theory

  19. Squeezed State in Quantum Scalar Field Theory • Squeezed State (k:momentum ; a,b:isospin) cf.) squeezed state in Quantum Mechanics

  20. From Squeezed Stateto Gaussian wave functional • Functional Schrödinger Picture Gaussian wave functional・・・dynamical “variables’’ ・・・center , its conjugate momentum ・・・Gaussian width, its velocity

  21. Gaussian wavefunctional • Expectation values

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

  23. Application to Dynamical Chiral Phase Transition, especially,Disoriented Chiral Condensate (DCC) problems

  24. DCC FormationNonequilibrium chiral dynamics and two-particle correlationsDr.Ikezi’s talk

  25. DCC as a collective isospin rotation

  26. DCC as a collective Isospin Rotationeffects 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

  27. Variational Approach in Gaussian wave functionalY.Tsue, D.Vautherin & T.Matsui, PTP 102 (1999) 313 ・Hamiltonian density ・Gaussian wave functional ・Dynamical variables

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

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

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

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

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

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

  34. Effects of isospin rotation of chiral condensate (c=0) Phase diagram ・Time-like isospin rotation : ・Space-like isospin rotation : |q| vs. condensate T vs. condensate Y.Tsue, D.Vautherin & T.Matsui, Prog.Theor.Phys. 102 (1999) 313 ω↑ q ↓ ←T

  35. 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 reachQuantum effects are important

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

  37. 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 --- 　↑　 ｃ≠０ 　↑　 ｃ＝０ 　　 ↑ Isospin rotation

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

  39. Energy of collective isospin rotation of chiral condensate leads todeacy of collective isospin rotation and leads to two-pion emissions two meson emission

  40. Damping time & number of emitted pions Y.Tsue, D.Vautherin & T.Matsui, Phys. Rev. D61 (2000) 076006 ・Damping time Energy density of collective rotating condensate Energy density oftwo meson ・Number of emitted pions , if classical field configuration occupies volume V Larger than the collision time ～a few fm/c ⇒ ⇒

  41. ③Amplification of quantum meson modes in role-down of chiral condensate K.Watanabe, Y.T. and S.Nishiyama, Prog.Theor.Phys.113 (2005) 369

  42. The set of basic equations of motion again Chiral Condensate Quantum Meson Fields Y. Tsue, D. Vautherin and T. Matsui (Prog.Theor.Phys. ,1999)

  43. Numerical results without spatial expansion

  44. Late time of Chiral phase transition Small deviation around static configurations cf. Mathieu equation Dimensionless variables cf. Forced oscillation

  45. The unstable regions for quantum pion modes ⇐Mathieu equation

  46. 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

  47. 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

  48. 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

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