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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

6.1. 有限温度場の理論の

簡単な紹介


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6.1.1. Very Brief Review of

Quantum Statistical Mechanics


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◎ 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


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Partition function

: density matrix

Ex) baryon number in QCD

(number of baryons) – (number of antibaryons)


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☆ Thermodynamic properties

◎ pressure

◎ particle number

◎ entropy

◎ energy


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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


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The states are simultaneously number and energy eigenstates.

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

☆ Partition function

◎ Mean numeber

◎ Mean energy


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

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

・ momenta


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◎ Hamiltonian and number operator eigenstates.

◎ Partition function


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◎ partition function eigenstates.

◎ pressure

◎ particle number

◎ energy


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☆ massless limit eigenstates.(μ= 0)


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6.1.2 eigenstates.. Matsubara formalism


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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

;



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imaginary time eigenstates.

6.1.2.2. Partition function for bosons

in quantum statistical mechanics


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

;

・・・ periodicity

◎ Lagrangian

◎ Fourier transformation of f

◎ Action

;



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zero-point energy eigenstates.

same as the one obtained in the

quantum statistical mechanics


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6.1.3 eigenstates.. Interactions and

Diagramatic Technique


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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


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

;

☆ Feynman rules

QFT

FTFT

◎ propagator

◎ vertex

◎ integration


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= eigenstates.

=

6.1.3.3. 1-loop correction to propagator


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☆ Evaluation of Matsubara frequency sum eigenstates.

;

◎ contour C

deformation


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☆ 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


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☆ renormalization eigenstates.

;

・・・ mass counter term

☆ effective mass

Mass is changed at non-zero T !


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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


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+ 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


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◎ 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


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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


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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.


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The End eigenstates.


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