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6. 有限温度系への応用. 6.1. 有限 温度場の理論の 簡単な紹介. 6.1.1 . Very Brief Review of Quantum Statistical Mechanics. ◎ micro canonical ensemble ・・・ isolated system. E (energy), N (particle number), V (volume) ・・・ fixed. ◎ canonical ensemble

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6. 有限温度系への応用


6.1. 有限温度場の理論の

簡単な紹介


6.1.1. Very Brief Review of

Quantum Statistical Mechanics


◎ micro canonical ensemble ・・・ isolated system

E (energy), N (particle number), V (volume) ・・・ fixed

◎ canonical ensemble

・・・ a system in contact with a heat reservoir at temperature T

T , N , V・・・ fixed

6.1.1.1. Ensemble

◎ grand canonical ensemble

・・・ system can exchange particles as well as energy with a reservoir

T , V , μ (chemical potential) ・・・ fixed


Partition function

: density matrix

Ex) baryon number in QCD

(number of baryons) – (number of antibaryons)


☆ Thermodynamic properties

◎ pressure

◎ particle number

◎ entropy

◎ energy


6.1.1.2. One bosonic degree of freedom

・ time-independent single-particle quantum mechanical state of bosons

(Each boson has the same energy ω)

・ commutation relation ・・・

◎ Hamiltonian and number operator

Ignore the zero-point energy


The states are simultaneously number and energy eigenstates.

→ We can assign a chemical potential μ to the particles.

☆ Partition function

◎ Mean numeber

◎ Mean energy


6.1.1.3 eigenstates.. Free (identical) bosons in a box (cube)

◎ boundary condition ・・・Wave functions vanish at the surface of the box.

・ momenta


◎ Hamiltonian and number operator eigenstates.

◎ Partition function


◎ partition function eigenstates.

◎ pressure

◎ particle number

◎ energy


☆ massless limit eigenstates.(μ= 0)


6.1.2 eigenstates.. Matsubara formalism


6.1.2.1 eigenstates.. Path integral in the quantum field theory

◎ Operators in the Hisenberg picture

・ Suppose the operators in two pictures agree with each other at t = 0, then

;

◎ Eigenstates

;



imaginary time eigenstates.

6.1.2.2. Partition function for bosons

in quantum statistical mechanics


6.1.2.3 eigenstates.. Neutral scalar field (μ= 0)

;

・・・ periodicity

◎ Lagrangian

◎ Fourier transformation of f

◎ Action

;



zero-point energy eigenstates.

same as the one obtained in the

quantum statistical mechanics


6.1.3 eigenstates.. Interactions and

Diagramatic Technique


We can use the methods used in the ordinary QFT eigenstates.

to calculate and .

6.1.3.1. Thermal Green’s function

and generating functional

◎ Thermal Green’s function

◎ Generating functional

・ perturbative expansion

・ Feynman diagrams


6.1.3.2 eigenstates.. Neutral scalar field(μ= 0)

;

☆ Feynman rules

QFT

FTFT

◎ propagator

◎ vertex

◎ integration


= eigenstates.

=

6.1.3.3. 1-loop correction to propagator


☆ Evaluation of Matsubara frequency sum eigenstates.

;

◎ contour C

deformation


☆ 1-loop correction eigenstates.

;

=

・ same as the quantum correction at T=0

・ includes the UV divergence

・ correction only for T>0

・ does not include any UV divergences


☆ renormalization eigenstates.

;

・・・ mass counter term

☆ effective mass

Mass is changed at non-zero T !


6.2. eigenstates.HLS in Hot Matter

  • M.H. and C.Sasaki, Phys. Lett. B 537, 280 (2002)

  • M.H., Y.Kim, M.Rho and C.Sasaki, Nucl. Phys. A 727, 437 (2003)

  • M.H. and C.Sasaki, hep-ph/0304282


+ eigenstates.

+

+

+

+

+

+

☆ vector meson mass

(propagator)-1 = (tree propagator)-1 +

◎ low temperature region

r中間子はp中間子による遮蔽効果で重くなる


◎ low temperature region eigenstates.

☆ Temporal and spatial pion decay constants

parametric pion

decay constant

hadronic thermal correction

consistent with

low-temperature theorem

difference appears already at one loop


eigenstates.pion velocity

= 0

dispersion relation for p

pion velocity

◎ low temperature region

Pion velocity is smaller than the speed of light already at one loop


eigenstates. Parameter a and r meson dominance

◎Pion EM form factor (tree level at T = 0)

rDominance

◎ low temperature region

rdominance is well satisfied in the low temperature region.


The End eigenstates.


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