# 6. 有限温度系への応用 - PowerPoint PPT Presentation

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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

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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

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

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

・ momenta

◎ Hamiltonian and number operator

◎ Partition function

◎ partition function

◎ pressure

◎ particle number

◎ energy

☆ massless limit (μ= 0)

6.1.2. Matsubara formalism

6.1.2.1. 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

;

☆ Transition amplitude for bosons (T=0)

imaginary time

6.1.2.2. Partition function for bosons

in quantum statistical mechanics

6.1.2.3. Neutral scalar field (μ= 0)

;

・・・ periodicity

◎ Lagrangian

◎ Fourier transformation of f

◎ Action

;

☆ Partition function

zero-point energy

same as the one obtained in the

quantum statistical mechanics

6.1.3. Interactions and

Diagramatic Technique

We can use the methods used in the ordinary QFT

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. Neutral scalar field(μ= 0)

;

☆ Feynman rules

QFT

FTFT

◎ propagator

◎ vertex

◎ integration

=

=

6.1.3.3. 1-loop correction to propagator

☆ Evaluation of Matsubara frequency sum

;

◎ contour C

deformation

☆ 1-loop correction

;

=

・ same as the quantum correction at T=0

・ includes the UV divergence

・ correction only for T>0

・ does not include any UV divergences

☆ renormalization

;

・・・ mass counter term

☆ effective mass

Mass is changed at non-zero T !

6.2. 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

+

+

+

+

+

+

+

☆ 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

☆ Temporal and spatial pion decay constants

parametric pion

decay constant

consistent with

low-temperature theorem

difference appears already at one loop

☆ 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

☆ 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