tts t Ps - - PowerPoint PPT Presentation
tts t Ps - - PowerPoint PPT Presentation
tts t Ps r t ts s
▼♦t✐✈❛t✐♦♥
0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T/ m
π
µB / m
π
Deconfined Chiral Symmetric Confined Chiral Broken Confined Chiral Symmetric Quark Gluon Plasma Hadronic Matter Quarkyonic Superfluid
[Schaefer Delta Meeting ’10] [Brauner, Fukushima & Hidaka ’09]
◗❈✷❉ ♥♦ s✐❣♥ ♣r♦❜❧❡♠ ❝♦♠♣❛r❡ ✇✐t❤ ❧❛tt✐❝❡ r❡s✉❧ts ❝♦❧♦✉r ♥❡✉tr❛❧ ❜♦✉♥❞ st❛t❡s ♦❢ t✇♦ q✉❛r❦s ♣❤❛s❡s ♦❢ s✉♣❡r✢✉✐❞✐t② ❡❛s✐❧② ❛❝❡ss✐❜❧❡ ✐♠♣❛❝t ♦❢ ❜❛r②♦♥s ♦♥ t❤❡ ♣❤❛s❡ ❞✐❛❣r❛♠
✷ ✴ ✷✾
▼♦t✐✈❛t✐♦♥
0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T/ m
π
µB / m
π
Deconfined Chiral Symmetric Confined Chiral Broken Confined Chiral Symmetric Quark Gluon Plasma Hadronic Matter Quarkyonic Superfluid
[Schaefer Delta Meeting ’10] [Brauner, Fukushima & Hidaka ’09]
◗❈✷❉ ♥♦ s✐❣♥ ♣r♦❜❧❡♠ → ❝♦♠♣❛r❡ ✇✐t❤ ❧❛tt✐❝❡ r❡s✉❧ts ❝♦❧♦✉r ♥❡✉tr❛❧ ❜♦✉♥❞ st❛t❡s ♦❢ t✇♦ q✉❛r❦s ♣❤❛s❡s ♦❢ s✉♣❡r✢✉✐❞✐t② ❡❛s✐❧② ❛❝❡ss✐❜❧❡ ✐♠♣❛❝t ♦❢ ❜❛r②♦♥s ♦♥ t❤❡ ♣❤❛s❡ ❞✐❛❣r❛♠
✷ ✴ ✷✾
▼♦t✐✈❛t✐♦♥
0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T/ m
π
µB / m
π
Deconfined Chiral Symmetric Confined Chiral Broken Confined Chiral Symmetric Quark Gluon Plasma Hadronic Matter Quarkyonic Superfluid
[Schaefer Delta Meeting ’10] [Brauner, Fukushima & Hidaka ’09]
◗❈✷❉ ♥♦ s✐❣♥ ♣r♦❜❧❡♠ → ❝♦♠♣❛r❡ ✇✐t❤ ❧❛tt✐❝❡ r❡s✉❧ts ❝♦❧♦✉r ♥❡✉tr❛❧ ❜♦✉♥❞ st❛t❡s ♦❢ t✇♦ q✉❛r❦s ♣❤❛s❡s ♦❢ s✉♣❡r✢✉✐❞✐t② ❡❛s✐❧② ❛❝❡ss✐❜❧❡ ✐♠♣❛❝t ♦❢ ❜❛r②♦♥s ♦♥ t❤❡ ♣❤❛s❡ ❞✐❛❣r❛♠
✷ ✴ ✷✾
▼♦t✐✈❛t✐♦♥
0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T/ m
π
µB / m
π
Deconfined Chiral Symmetric Confined Chiral Broken Confined Chiral Symmetric Quark Gluon Plasma Hadronic Matter Quarkyonic Superfluid
[Schaefer Delta Meeting ’10] [Brauner, Fukushima & Hidaka ’09]
◗❈✷❉ ♥♦ s✐❣♥ ♣r♦❜❧❡♠ → ❝♦♠♣❛r❡ ✇✐t❤ ❧❛tt✐❝❡ r❡s✉❧ts ❝♦❧♦✉r ♥❡✉tr❛❧ ❜♦✉♥❞ st❛t❡s ♦❢ t✇♦ q✉❛r❦s → ♣❤❛s❡s ♦❢ s✉♣❡r✢✉✐❞✐t② ❡❛s✐❧② ❛❝❡ss✐❜❧❡ ✐♠♣❛❝t ♦❢ ❜❛r②♦♥s ♦♥ t❤❡ ♣❤❛s❡ ❞✐❛❣r❛♠
✷ ✴ ✷✾
▼♦t✐✈❛t✐♦♥
0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T/ m
π
µB / m
π
Deconfined Chiral Symmetric Confined Chiral Broken Confined Chiral Symmetric Quark Gluon Plasma Hadronic Matter Quarkyonic Superfluid
[Schaefer Delta Meeting ’10] [Brauner, Fukushima & Hidaka ’09]
◗❈✷❉ ♥♦ s✐❣♥ ♣r♦❜❧❡♠ → ❝♦♠♣❛r❡ ✇✐t❤ ❧❛tt✐❝❡ r❡s✉❧ts ❝♦❧♦✉r ♥❡✉tr❛❧ ❜♦✉♥❞ st❛t❡s ♦❢ t✇♦ q✉❛r❦s → ♣❤❛s❡s ♦❢ s✉♣❡r✢✉✐❞✐t② ❡❛s✐❧② ❛❝❡ss✐❜❧❡ ✐♠♣❛❝t ♦❢ ❜❛r②♦♥s ♦♥ t❤❡ ♣❤❛s❡ ❞✐❛❣r❛♠
✷ ✴ ✷✾
▼♦t✐✈❛t✐♦♥
0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 T/ m
π
µB / m
π
Deconfined Chiral Symmetric Confined Chiral Broken Confined Chiral Symmetric Quark Gluon Plasma Hadronic Matter Quarkyonic Superfluid
[Schaefer Delta Meeting ’10] [Brauner, Fukushima & Hidaka ’09]
◗❈✷❉ ♥♦ s✐❣♥ ♣r♦❜❧❡♠ → ❝♦♠♣❛r❡ ✇✐t❤ ❧❛tt✐❝❡ r❡s✉❧ts ❝♦❧♦✉r ♥❡✉tr❛❧ ❜♦✉♥❞ st❛t❡s ♦❢ t✇♦ q✉❛r❦s → ♣❤❛s❡s ♦❢ s✉♣❡r✢✉✐❞✐t② ❡❛s✐❧② ❛❝❡ss✐❜❧❡ ✐♠♣❛❝t ♦❢ ❜❛r②♦♥s ♦♥ t❤❡ ♣❤❛s❡ ❞✐❛❣r❛♠
✷ ✴ ✷✾
▼♦t✐✈❛t✐♦♥
❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠
✸ ✴ ✷✾
▼♦t✐✈❛t✐♦♥
❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠
✹ ✴ ✷✾
▼♦t✐✈❛t✐♦♥
❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠
✺ ✴ ✷✾
❋❡❛t✉r❡s ♦❢ ◗❈✷❉
♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯ ✷ ❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t❛ t❚
❛
t✷t❛t✷ ❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉ t✷❈
✺❉ ✺❈t✷
❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯ ◆❢
▲
❙❯ ◆❢
❘
❯ ✶ ❇ ❙❯ ✷◆❢
◗❈✷❉ ▲✐
❉
▲ ❘✐
❉
❘
✐ ❉
▲ ❘
✉▲ ❞▲ ✉❘ ❞❘
❘ ✷t✷ ❘
❛❧❧♦✇s ✉s t♦ r♦t❛t❡
▲ ❘✱ s✐♠✐❧❛r❧②
✻ ✴ ✷✾
❋❡❛t✉r❡s ♦❢ ◗❈✷❉
♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗
❛ = t❚ ❛ = −t✷t❛t✷
❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉ t✷❈
✺❉ ✺❈t✷
❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯ ◆❢
▲
❙❯ ◆❢
❘
❯ ✶ ❇ ❙❯ ✷◆❢
◗❈✷❉ ▲✐
❉
▲ ❘✐
❉
❘
✐ ❉
▲ ❘
✉▲ ❞▲ ✉❘ ❞❘
❘ ✷t✷ ❘
❛❧❧♦✇s ✉s t♦ r♦t❛t❡
▲ ❘✱ s✐♠✐❧❛r❧②
✻ ✴ ✷✾
❋❡❛t✉r❡s ♦❢ ◗❈✷❉
♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗
❛ = t❚ ❛ = −t✷t❛t✷
❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯ ◆❢
▲
❙❯ ◆❢
❘
❯ ✶ ❇ ❙❯ ✷◆❢
◗❈✷❉ ▲✐
❉
▲ ❘✐
❉
❘
✐ ❉
▲ ❘
✉▲ ❞▲ ✉❘ ❞❘
❘ ✷t✷ ❘
❛❧❧♦✇s ✉s t♦ r♦t❛t❡
▲ ❘✱ s✐♠✐❧❛r❧②
✻ ✴ ✷✾
❋❡❛t✉r❡s ♦❢ ◗❈✷❉
♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗
❛ = t❚ ❛ = −t✷t❛t✷
❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯(◆❢ )▲ × ❙❯(◆❢ )❘ × ❯(✶)❇ → ❙❯(✷◆❢ )
◗❈✷❉ ▲✐
❉
▲ ❘✐
❉
❘
✐ ❉
▲ ❘
✉▲ ❞▲ ✉❘ ❞❘
❘ ✷t✷ ❘
❛❧❧♦✇s ✉s t♦ r♦t❛t❡
▲ ❘✱ s✐♠✐❧❛r❧②
✻ ✴ ✷✾
❋❡❛t✉r❡s ♦❢ ◗❈✷❉
♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗
❛ = t❚ ❛ = −t✷t❛t✷
❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯(◆❢ )▲ × ❙❯(◆❢ )❘ × ❯(✶)❇ → ❙❯(✷◆❢ )
◗❈✷❉ ▲✐
❉
▲ ❘✐
❉
❘
✐ ❉
▲ ❘
✉▲ ❞▲ ✉❘ ❞❘
❘ ✷t✷ ❘
❛❧❧♦✇s ✉s t♦ r♦t❛t❡
▲ ❘✱ s✐♠✐❧❛r❧②
✻ ✴ ✷✾
❋❡❛t✉r❡s ♦❢ ◗❈✷❉
♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗
❛ = t❚ ❛ = −t✷t❛t✷
❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯(◆❢ )▲ × ❙❯(◆❢ )❘ × ❯(✶)❇ → ❙❯(✷◆❢ ) L◗❈✷❉ = ψ†
▲✐σµ❉µψ▲ + ψ† ❘✐σ† µ❉µψ❘
✐ ❉
▲ ❘
✉▲ ❞▲ ✉❘ ❞❘
❘ ✷t✷ ❘
❛❧❧♦✇s ✉s t♦ r♦t❛t❡
▲ ❘✱ s✐♠✐❧❛r❧②
✻ ✴ ✷✾
❋❡❛t✉r❡s ♦❢ ◗❈✷❉
♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗
❛ = t❚ ❛ = −t✷t❛t✷
❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯(◆❢ )▲ × ❙❯(◆❢ )❘ × ❯(✶)❇ → ❙❯(✷◆❢ ) L◗❈✷❉ = ψ†
▲✐σµ❉µψ▲ + ψ† ❘✐σ† µ❉µψ❘ = Ψ†✐σµ❉µΨ ▲ ❘
✉▲ ❞▲ ✉❘ ❞❘
❘ ✷t✷ ❘
❛❧❧♦✇s ✉s t♦ r♦t❛t❡
▲ ❘✱ s✐♠✐❧❛r❧②
✻ ✴ ✷✾
❋❡❛t✉r❡s ♦❢ ◗❈✷❉
♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗
❛ = t❚ ❛ = −t✷t❛t✷
❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯(◆❢ )▲ × ❙❯(◆❢ )❘ × ❯(✶)❇ → ❙❯(✷◆❢ ) L◗❈✷❉ = ψ†
▲✐σµ❉µψ▲ + ψ† ❘✐σ† µ❉µψ❘ = Ψ†✐σµ❉µΨ
Ψ = ψ▲ ˜ ψ❘
- =
✉▲ ❞▲ ˜ ✉❘ ˜ ❞❘ , ˜ ψ❘ = σ✷t✷ψ∗
❘
❛❧❧♦✇s ✉s t♦ r♦t❛t❡
▲ ❘✱ s✐♠✐❧❛r❧②
✻ ✴ ✷✾
❋❡❛t✉r❡s ♦❢ ◗❈✷❉
♣s❡✉❞♦r❡❛❧✐t② ♦❢ ❙❯(✷)❝ ❣❛✉❣❡ ❣r♦✉♣ ❣❡♥❡r❛t♦rs✿ t∗
❛ = t❚ ❛ = −t✷t❛t✷
❛♥t✐✉♥✐t❛r✐t② ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r✿ ❉∗ = −t✷❈γ✺❉γ✺❈t✷ ❡①t❡♥❞❡❞ ✢❛✈♦✉r s②♠♠❡tr②✿ ❙❯(◆❢ )▲ × ❙❯(◆❢ )❘ × ❯(✶)❇ → ❙❯(✷◆❢ ) L◗❈✷❉ = ψ†
▲✐σµ❉µψ▲ + ψ† ❘✐σ† µ❉µψ❘ = Ψ†✐σµ❉µΨ
Ψ = ψ▲ ˜ ψ❘
- =
✉▲ ❞▲ ˜ ✉❘ ˜ ❞❘ , ˜ ψ❘ = σ✷t✷ψ∗
❘
❛❧❧♦✇s ✉s t♦ r♦t❛t❡ ψ▲ → ˜ ψ❘✱ s✐♠✐❧❛r❧② ¯ ψψ → ψψ
✻ ✴ ✷✾
❋❡❛t✉r❡s ♦❢ ◗❈✷❉
❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪
❙❯(✹) ❙❯ ✷ ▲ ❙❯ ✷ ❘ ❯ ✶ ❇ ✵ ❙❯ ✷ ▲ ❙❯ ✷ ❘ ♠ ✵ ❙♣ ✹ ♥♦ ❙❙❇ ❢♦r ✵ ❙❯ ✷ ❱ ❯ ✶ ❇ ✵ ❙❯ ✷ ❱
✼ ✴ ✷✾
❋❡❛t✉r❡s ♦❢ ◗❈✷❉
❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪
❙❯(✹)
❄
µ ❙❯(✷)▲ × ❙❯(✷)❘ × ❯(✶)❇
❄
ψψ = ✵ ❙❯(✷)▲ × ❙❯(✷)❘
✲
♠ψ → ¯ ψψ = ✵ ❙♣(✹)
✲ ♥♦ ❙❙❇ ❢♦r µ = ✵ ❄
µ ❙❯(✷)❱ × ❯(✶)❇
✲ ❄
ψψ = ✵ ❙❯(✷)❱
✲
✼ ✴ ✷✾
❋❡❛t✉r❡s ♦❢ ◗❈✷❉
❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪
❙❯(✹)
✲
♠ψ → ¯ ψψ = ✵ ❙♣(✹)
❄
µ ❙②♠♠❡tr② ❜❡t✇❡❡♥ ¯ ψψ ❛♥❞ ψψ ✐s ❜r♦❦❡♥ ❙❯(✷)❱ × ❯(✶)❇
❄
❙❙❇ ψψ = ✵ ❙❯(✷)❱
✽ ✴ ✷✾
❋✉♥❝t✐♦♥❛❧ ❘●
❊①❛❝t ❘● ❋❧♦✇ ❊q✉❛t✐♦♥
∂❦Γ❦[Φ] = ✶ ✷ ❙❚r ✶ Γ(✷)
❦ [Φ] + ❘❦
∂❦❘❦
❬❲❡tt❡r✐❝❤ ✬✾✶❪ ❦ ✐♥t❡r♣♦❧❛t❡s ❜❡t✇❡❡♥ ♠✐r❝♦s❝♦♣✐❝ ❛❝t✐♦♥ ❙ ❛♥❞ ❢✉❧❧ q✉❛♥t✉♠
❡✛❡❝t✐✈❡ ❛❝t✐♦♥
❦
❙❜❛r❡
❦ ✵
✾ ✴ ✷✾
❋✉♥❝t✐♦♥❛❧ ❘●
❊①❛❝t ❘● ❋❧♦✇ ❊q✉❛t✐♦♥
∂❦Γ❦[Φ] = ✶ ✷ ❙❚r ✶ Γ(✷)
❦ [Φ] + ❘❦
∂❦❘❦
❬❲❡tt❡r✐❝❤ ✬✾✶❪
Γ❦ ✐♥t❡r♣♦❧❛t❡s ❜❡t✇❡❡♥ ♠✐r❝♦s❝♦♣✐❝ ❛❝t✐♦♥ ❙ ❛♥❞ ❢✉❧❧ q✉❛♥t✉♠ ❡✛❡❝t✐✈❡ ❛❝t✐♦♥ Γ
Γ❦=Λ = ❙❜❛r❡ Γ❦=✵ ≡ Γ
[Gies ’06] ✾ ✴ ✷✾
❋✉♥❝t✐♦♥❛❧ ❘●
❊①❛❝t ❘● ❋❧♦✇ ❊q✉❛t✐♦♥
✐♥t❡❣r❛t❡ ♦✉t ✢✉❝t✉❛t✐♦♥s✱ Φ =
- ϕ, ψ, ¯
ψ, ❆, ❝, ¯ ❝
- ∂kΓk[Φ] = 1
2 + 1 2 − + 1 2
✶✵ ✴ ✷✾
❋✉♥❝t✐♦♥❛❧ ❘●
❊①❛❝t ❘● ❋❧♦✇ ❊q✉❛t✐♦♥
✐♥t❡❣r❛t❡ ♦✉t ✢✉❝t✉❛t✐♦♥s✱ Φ =
- ϕ, ψ, ¯
ψ
- ∂kΓk[Φ] = 1
2 + 1 2 − + 1 2
T=μ=0
✶ ●❡❱ [Diehl ❡t al ’10] ✶✶ ✴ ✷✾
❋✉♥❝t✐♦♥❛❧ ❘●
❊✛❡❝t✐✈❡ P♦t❡♥t✐❛❧
❙❯(✹) ≃ ❙❖(✻) → ❖(✻)✲♦r❞❡r ♣❛r❛♠❡t❡r ♣♦t❡♥t✐❛❧ ✰ ❡①♣❧✐❝✐t ❜r❡❛❦✐♥❣ t❡r♠s ❯❦ = ❱❦( π✷ + σ✷ + ∆✷
✶ + ∆✷ ✷) − ❝σ − µ✷|∆|✷
- ❧♦❜❛❧ ♠✐♥✐♠✉♠ ❞❡t❡r♠✐♥❡s t❤❡ ❝♦♥❞❡♥s❛t❡s
✱ ✿ ◆♦r♠❛❧ ♣❤❛s❡✿ ♠✷
✷
✵
❝ ♠✷
✵ ❙✉♣❡r✢✉✐❞ ♣❤❛s❡✿ ♠✷
✷
✵
❝
✷
✵ ❋❧♦✇
✶✷ ✴ ✷✾
❋✉♥❝t✐♦♥❛❧ ❘●
❊✛❡❝t✐✈❡ P♦t❡♥t✐❛❧
❙❯(✹) ≃ ❙❖(✻) → ❖(✻)✲♦r❞❡r ♣❛r❛♠❡t❡r ♣♦t❡♥t✐❛❧ ✰ ❡①♣❧✐❝✐t ❜r❡❛❦✐♥❣ t❡r♠s ❯❦ = ❱❦( π✷ + σ✷ + ∆✷
✶ + ∆✷ ✷) − ❝σ − µ✷|∆|✷
- ❧♦❜❛❧ ♠✐♥✐♠✉♠ ❞❡t❡r♠✐♥❡s t❤❡ ❝♦♥❞❡♥s❛t❡s σ✱ |∆|✿
◆♦r♠❛❧ ♣❤❛s❡✿ ♠✷ − µ✷ > ✵ , σ =
❝ ♠✷ ,
|∆| = ✵ ❙✉♣❡r✢✉✐❞ ♣❤❛s❡✿ ♠✷ − µ✷ < ✵ , σ =
❝ µ✷ ,
|∆| = ✵ ❋❧♦✇
✶✷ ✴ ✷✾
❋✉♥❝t✐♦♥❛❧ ❘●
❊✛❡❝t✐✈❡ P♦t❡♥t✐❛❧
❙❯(✹) ≃ ❙❖(✻) → ❖(✻)✲♦r❞❡r ♣❛r❛♠❡t❡r ♣♦t❡♥t✐❛❧ ✰ ❡①♣❧✐❝✐t ❜r❡❛❦✐♥❣ t❡r♠s ❯❦ = ❱❦( π✷ + σ✷ + ∆✷
✶ + ∆✷ ✷) − ❝σ − µ✷|∆|✷
- ❧♦❜❛❧ ♠✐♥✐♠✉♠ ❞❡t❡r♠✐♥❡s t❤❡ ❝♦♥❞❡♥s❛t❡s σ✱ |∆|✿
◆♦r♠❛❧ ♣❤❛s❡✿ ♠✷ − µ✷ > ✵ , σ =
❝ ♠✷ ,
|∆| = ✵ ❙✉♣❡r✢✉✐❞ ♣❤❛s❡✿ ♠✷ − µ✷ < ✵ , σ =
❝ µ✷ ,
|∆| = ✵ ❋❧♦✇
✶✷ ✴ ✷✾
❋✉♥❝t✐♦♥❛❧ ❘●
❊✛❡❝t✐✈❡ P♦t❡♥t✐❛❧
❙❯(✹) ≃ ❙❖(✻) → ❖(✻)✲♦r❞❡r ♣❛r❛♠❡t❡r ♣♦t❡♥t✐❛❧ ✰ ❡①♣❧✐❝✐t ❜r❡❛❦✐♥❣ t❡r♠s ❯❦ = ❱❦( π✷ + σ✷ + ∆✷
✶ + ∆✷ ✷) − ❝σ − µ✷|∆|✷
- ❧♦❜❛❧ ♠✐♥✐♠✉♠ ❞❡t❡r♠✐♥❡s t❤❡ ❝♦♥❞❡♥s❛t❡s σ✱ |∆|✿
◆♦r♠❛❧ ♣❤❛s❡✿ ♠✷ − µ✷ > ✵ , σ =
❝ ♠✷ ,
|∆| = ✵ ❙✉♣❡r✢✉✐❞ ♣❤❛s❡✿ ♠✷ − µ✷ < ✵ , σ =
❝ µ✷ ,
|∆| = ✵ ❋❧♦✇
∂tUk = 1 2 −
✶✷ ✴ ✷✾
❋✉♥❝t✐♦♥❛❧ ❘●
■♠♣r♦✈✐♥❣ t❤❡ tr✉♥❝❛t✐♦♥
❋❧✉❝t✉❛t✐♦♥s ♦❢ t❤❡ ♣r♦♣❛❣❛t♦rs → ✇❛✈❡ ❢✉♥❝t✐♦♥ r❡♥♦r♠❛❧✐③❛t✐♦♥s ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦
∂t =
−1
+ ∂t =
−1
+
❘✉♥♥✐♥❣ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣✱ ♠◗✉❛r❦ ❤❦ ❤❦
✶✸ ✴ ✷✾
❋✉♥❝t✐♦♥❛❧ ❘●
■♠♣r♦✈✐♥❣ t❤❡ tr✉♥❝❛t✐♦♥
❋❧✉❝t✉❛t✐♦♥s ♦❢ t❤❡ ♣r♦♣❛❣❛t♦rs → ✇❛✈❡ ❢✉♥❝t✐♦♥ r❡♥♦r♠❛❧✐③❛t✐♦♥s ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦
∂t =
−1
+ ∂t =
−1
+
❘✉♥♥✐♥❣ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣✱ ♠◗✉❛r❦ = ❤❦ ¯ ψψ = ❤❦σ
✶✸ ✴ ✷✾
❋✉♥❝t✐♦♥❛❧ ❘●
■♠♣r♦✈✐♥❣ t❤❡ tr✉♥❝❛t✐♦♥
❋❧✉❝t✉❛t✐♦♥s ♦❢ t❤❡ ♣r♦♣❛❣❛t♦rs → ✇❛✈❡ ❢✉♥❝t✐♦♥ r❡♥♦r♠❛❧✐③❛t✐♦♥s ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦
∂t =
−1
+ ∂t =
−1
+
❘✉♥♥✐♥❣ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣✱ ♠◗✉❛r❦ = ❤❦ ¯ ψψ = ❤❦σ
✶✸ ✴ ✷✾
❋✉♥❝t✐♦♥❛❧ ❘●
■♠♣r♦✈✐♥❣ t❤❡ tr✉♥❝❛t✐♦♥
❋❧✉❝t✉❛t✐♦♥s ♦❢ t❤❡ ♣r♦♣❛❣❛t♦rs → ✇❛✈❡ ❢✉♥❝t✐♦♥ r❡♥♦r♠❛❧✐③❛t✐♦♥s ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦
∂t =
−1
+ ∂t =
−1
+
❘✉♥♥✐♥❣ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣✱ ♠◗✉❛r❦ = ❤❦ ¯ ψψ = ❤❦σ
✶✸ ✴ ✷✾
❘❡s✉❧ts
❚❤❡ ◗❈✷❉ P❤❛s❡ ❉✐❛❣r❛♠
✶✹ ✴ ✷✾
❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮
µ ❉❡♣❡♥❞❡♥❝❡
U
- k 0
U
- k , Z
- k 0
U
- k , h
- k 0
U
- k , Z
- k, h
- k 0
Σ T 0
50 100 150 200 250 300ΜMeV 20 40 60 80 100 MeV
✶✺ ✴ ✷✾
❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮
❚❡♠♣❡r❛t✉r❡ ❉❡♣❡♥❞❡♥❝❡
U
- k 0
U
- k , Z
- k 0
U
- k , h
- k 0
U
- k , Z
- k, h
- k 0
Μ 0
50 100 150 200 250 300TMeV 10 20 30 40 50 60 70 ΣMeV
U
- k 0
U
- k , Z
- k 0
U
- k , h
- k 0
U
- k , Z
- k, h
- k 0
Μ 200 MeV
50 100 150 200TMeV 20 40 60 80 MeV
✶✻ ✴ ✷✾
❘❡s✉❧ts
Pr❡❝♦♥❞❡♥s❛t✐♦♥
40 80 120 160 200 400 600 800 1000 [MeV] k [MeV] m - µ <∆> <∆> T = 220 MeV T = 205 MeV T = 190 MeV T = 175 MeV 20 40 60 1 10 100 1000
← µ = ✷✸✵ ▼❡❱
12 10 8 6 4 2 0.05 0.10 0.15
t ρ0,k T < Tc Tc < T < Tpc
❬❇♦❡tt❝❤❡r ❡t ❛❧ ✷✵✶✷❪ ✶✼ ✴ ✷✾
❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮
❚❤❡ ◗❈✷❉ P❤❛s❡ ❉✐❛❣r❛♠
❬❘❛♥❞❡r✐❛✱ ◆❛t✉r❡ P❤②s✐❝s ✻ ✭✷✵✶✵✮❪ ✶✽ ✴ ✷✾
❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮
❚❤❡ ◗❈✷❉ P❤❛s❡ ❉✐❛❣r❛♠
✶✾ ✴ ✷✾
❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮
❚❤❡ ◗❈✷❉ P❤❛s❡ ❉✐❛❣r❛♠
✶✾ ✴ ✷✾
❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮
❚❤❡ ◗❈✷❉ P❤❛s❡ ❉✐❛❣r❛♠
✶✾ ✴ ✷✾
❘❡s✉❧ts ✭♣r❡❧✐♠✐♥❛r②✮
❚❤❡ ◗❈✷❉ P❤❛s❡ ❉✐❛❣r❛♠
Ψ Ψ 0 Ψ Ψ 0 U
- k 0
U
- k , Z
- k 0
U
- k , h
- k 0
U
- k , Z
- k, h
- k 0
50 100 150 200 250 300ΜMeV 50 100 150 200 250 TMeV
ΨΨ 0 BEC BCS ΨΨ 0 U
- k 0
U
- k , Z
- k 0
U
- k , h
- k 0
U
- k , Z
- k, h
- k 0
50 100 150 200 250 300 ΜMeV 50 100 150 200 250 TMeV
✷✵ ✴ ✷✾
❘❡s✉❧ts
❈❤✐r❛❧ ▲✐♠✐t ❝ = ✵✱ ♠π = ✵
✷✶ ✴ ✷✾
❘❡s✉❧ts
❲❛✈❡ ❋✉♥❝t✐♦♥ ❘❡♥♦r♠❛❧✐③❛t✐♦♥
❩∆ ❩∆,❱❛❝ ❩φ ❩φ,❱❛❝
✷✷ ✴ ✷✾
❘❡s✉❧ts
▼❛ss ❙♣❡❝tr✉♠
100 200 300 400 500 50 100 150 200 250 300 [MeV] T [MeV] mσ mψ mπ , m∆
← µ = ✵
100 200 300 400 500 600 700 50 100 150 200 250 300 350 400 [MeV] µ [MeV] mσ m∆* m∆ mπ mσ
~
m∆
~*
m∆
~
❞❡t●❇♦s♦♥(♣✵, ♣ = ✵) = ✵ ❚ = ✵ →
✷✸ ✴ ✷✾
❙✉♠♠❛r②✴❖✉t❧♦♦❦
❙✉♠♠❛r②
♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩
❦✱ ❩ ❦✱ ❩ ❦
❖✉t❧♦♦❦
▲P❆ ❜❡②♦♥❞
✹
❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉
✷✹ ✴ ✷✾
❙✉♠♠❛r②✴❖✉t❧♦♦❦
❙✉♠♠❛r②
♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩
❦✱ ❩ ❦✱ ❩ ❦
❖✉t❧♦♦❦
▲P❆ ❜❡②♦♥❞
✹
❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉
✷✹ ✴ ✷✾
❙✉♠♠❛r②✴❖✉t❧♦♦❦
❙✉♠♠❛r②
♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩
❦✱ ❩ ❦✱ ❩ ❦
❖✉t❧♦♦❦
▲P❆ ❜❡②♦♥❞
✹
❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉
✷✹ ✴ ✷✾
❙✉♠♠❛r②✴❖✉t❧♦♦❦
❙✉♠♠❛r②
♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦
❖✉t❧♦♦❦
▲P❆ ❜❡②♦♥❞
✹
❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉
✷✹ ✴ ✷✾
❙✉♠♠❛r②✴❖✉t❧♦♦❦
❙✉♠♠❛r②
♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦
❖✉t❧♦♦❦
▲P❆ ❜❡②♦♥❞
✹
❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉
✷✹ ✴ ✷✾
❙✉♠♠❛r②✴❖✉t❧♦♦❦
❙✉♠♠❛r②
♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦
❖✉t❧♦♦❦
▲P❆ ❜❡②♦♥❞ φ✹ ❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉
✷✹ ✴ ✷✾
❙✉♠♠❛r②✴❖✉t❧♦♦❦
❙✉♠♠❛r②
♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦
❖✉t❧♦♦❦
▲P❆ ❜❡②♦♥❞ φ✹ ❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉
✷✹ ✴ ✷✾
❙✉♠♠❛r②✴❖✉t❧♦♦❦
❙✉♠♠❛r②
♣❤❛s❡ ❞✐❛❣r❛♠ ♦❢ ✷✲❝♦❧♦✉r ✷✲✢❛✈♦✉r ◗❈❉✱ ❜❛r②♦♥✐❝ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ ❋❘● tr❡❛t♠❡♥t ✐♥ ②✐❡❧❞s ❛ ♣r❡✲❝♦♥❞❡♥s❛t✐♦♥ ♣❤❛s❡ ✐♠♣❛❝t ♦❢ r✉♥♥✐♥❣ ❤❦ ❛♥❞ ❩∆,❦✱ ❩φ,❦✱ ❩ψ,❦
❖✉t❧♦♦❦
▲P❆ ❜❡②♦♥❞ φ✹ ❝♦♥✜♥❡♠❡♥t✴❞❡❝♦♥✜♥❡♠❡♥t ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❜❛r②♦♥s ✐♥ ✸✲❝♦❧♦✉r ◗❈❉
✷✹ ✴ ✷✾
❇❛❝❦✉♣ ❙❧✐❞❡s
✷✺ ✴ ✷✾
❇❛❝❦✉♣ ❙❧✐❞❡s
❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪
❙②♠♠❡tr② ❣r♦✉♣
- ❡♥❡r❛t♦rs
Ps❡✉❞♦✲✴●♦❧❞st♦♥❡s ❙❯(✹) ✶✺ ✲
✶
❙♣ ✹ ✶✵ ✺ ● ✭ ✱ ✱
✷✮
❙❯ ✷ ▲ ❙❯ ✷ ❘ ✻ ✹ P● ✭ ✱ ✮✱ ✶ ● ✭
✷✮
♠ ❙❯ ✷ ❱ ✸ ✸ PP● ✭ ✮✱ ✶ P● ✭ ✮✱ ✶● ✭
✷✮
✷✻ ✴ ✷✾
❇❛❝❦✉♣ ❙❧✐❞❡s
❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪
❙②♠♠❡tr② ❣r♦✉♣
- ❡♥❡r❛t♦rs
Ps❡✉❞♦✲✴●♦❧❞st♦♥❡s ❙❯(✹) ✶✺ ✲
ψψ = ∆✶ ↓
❙♣(✹) ✶✵ ✺ ● ✭ π✱ σ✱ ∆✷✮ ❙❯ ✷ ▲ ❙❯ ✷ ❘ ✻ ✹ P● ✭ ✱ ✮✱ ✶ ● ✭
✷✮
♠ ❙❯ ✷ ❱ ✸ ✸ PP● ✭ ✮✱ ✶ P● ✭ ✮✱ ✶● ✭
✷✮
✷✻ ✴ ✷✾
❇❛❝❦✉♣ ❙❧✐❞❡s
❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪
❙②♠♠❡tr② ❣r♦✉♣
- ❡♥❡r❛t♦rs
Ps❡✉❞♦✲✴●♦❧❞st♦♥❡s ❙❯(✹) ✶✺ ✲
ψψ = ∆✶ ↓
❙♣(✹) ✶✵ ✺ ● ✭ π✱ σ✱ ∆✷✮ µ ↓ ❙❯(✷)▲ × ❙❯(✷)❘ ✻ ✹ P● ✭ π✱ σ✮✱ ✶ ● ✭∆✷✮ ♠ ❙❯ ✷ ❱ ✸ ✸ PP● ✭ ✮✱ ✶ P● ✭ ✮✱ ✶● ✭
✷✮
✷✻ ✴ ✷✾
❇❛❝❦✉♣ ❙❧✐❞❡s
❙②♠♠❡tr② ❇r❡❛❦✐♥❣ P❛tt❡r♥ ◆❢ = ✷ ❬❑♦❣✉t ❡t ❛❧ ✬✾✾❪
❙②♠♠❡tr② ❣r♦✉♣
- ❡♥❡r❛t♦rs
Ps❡✉❞♦✲✴●♦❧❞st♦♥❡s ❙❯(✹) ✶✺ ✲
ψψ = ∆✶ ↓
❙♣(✹) ✶✵ ✺ ● ✭ π✱ σ✱ ∆✷✮ µ ↓ ❙❯(✷)▲ × ❙❯(✷)❘ ✻ ✹ P● ✭ π✱ σ✮✱ ✶ ● ✭∆✷✮ ♠ψ ↓ ❙❯(✷)❱ ✸ ✸ PP● ✭ π✮✱ ✶ P● ✭σ✮✱ ✶● ✭∆✷✮
✷✻ ✴ ✷✾
❇❛❝❦✉♣ ❙❧✐❞❡s
❍❛❞r♦♥✐③❛t✐♦♥
◆❏▲ ♠♦❞❡❧
⇒
λψ g g g g
= ⇒ λψ
- ( ¯
ψψ)✷ − ( ¯ ψγ✺ τψ)✷ −
- ψ❚ǫψ
- ✷
❍✉❜❜❛r❞✲❙tr❛t♦♥♦✈✐❝❤ tr❛♥s❢♦r♠❛t✐♦♥
✷
❤ ✶ ✷♠✷
✷
✇✐t❤ ❤✷ ✷♠✷ ❛♥❞ ❊♦▼
✷✼ ✴ ✷✾
❇❛❝❦✉♣ ❙❧✐❞❡s
❍❛❞r♦♥✐③❛t✐♦♥
◆❏▲ ♠♦❞❡❧
⇒
λψ g g g g
= ⇒ λψ
- ( ¯
ψψ)✷ − ( ¯ ψγ✺ τψ)✷ −
- ψ❚ǫψ
- ✷
❍✉❜❜❛r❞✲❙tr❛t♦♥♦✈✐❝❤ tr❛♥s❢♦r♠❛t✐♦♥ λψ( ¯ ψψ)✷ = ❤σ ¯ ψψ + ✶ ✷♠✷σ✷ ✇✐t❤ λψ = − ❤✷ ✷♠✷ ❛♥❞ ❊♦▼(σ) → σ = ¯ ψψ
✷✼ ✴ ✷✾
❇❛❝❦✉♣ ❙❧✐❞❡s
❍❛❞r♦♥✐③❛t✐♦♥
◆❏▲ ♠♦❞❡❧
⇒
λψ g g g g
= ⇒ λψ
- ( ¯
ψψ)✷ − ( ¯ ψγ✺ τψ)✷ −
- ψ❚ǫψ
- ✷
❍✉❜❜❛r❞✲❙tr❛t♦♥♦✈✐❝❤ tr❛♥s❢♦r♠❛t✐♦♥ λψ( ¯ ψψ)✷ = ❤σ ¯ ψψ + ✶ ✷♠✷σ✷ ✇✐t❤ λψ = − ❤✷ ✷♠✷ ❛♥❞ ❊♦▼(σ) → σ = ¯ ψψ = ✵ → ♠❛sst❡r♠ ♠q = ❤σ
✷✽ ✴ ✷✾
❇❛❝❦✉♣ ❙❧✐❞❡s
❍❛❞r♦♥✐③❛t✐♦♥
▲❛r❣❡ ❢♦✉r✲❢❡r♠✐♦♥ ❝♦✉♣❧✐♥❣ ❧✐♠✐t
λψ→∞
= ⇒ ...
▲❛r❣❡ ❤❛❞r♦♥ ♠❛ss ❧✐♠✐t ❉②♥❛♠✐❝❛❧ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ◗✉❛r❦s ✱ ●❧✉♦♥s ❆ ✱ ♠❡s♦♥s ✱ ❜❛r②♦♥s ✱ ❆
✷✾ ✴ ✷✾
❇❛❝❦✉♣ ❙❧✐❞❡s
❍❛❞r♦♥✐③❛t✐♦♥
▲❛r❣❡ ❢♦✉r✲❢❡r♠✐♦♥ ❝♦✉♣❧✐♥❣ ❧✐♠✐t
λψ→∞
= ⇒ ...
▲❛r❣❡ ❤❛❞r♦♥ ♠❛ss ❧✐♠✐t
m→∞
= ⇒
❉②♥❛♠✐❝❛❧ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ◗✉❛r❦s ✱ ●❧✉♦♥s ❆ ✱ ♠❡s♦♥s ✱ ❜❛r②♦♥s ✱ ❆
✷✾ ✴ ✷✾
❇❛❝❦✉♣ ❙❧✐❞❡s
❍❛❞r♦♥✐③❛t✐♦♥
▲❛r❣❡ ❢♦✉r✲❢❡r♠✐♦♥ ❝♦✉♣❧✐♥❣ ❧✐♠✐t
λψ→∞
= ⇒ ...
▲❛r❣❡ ❤❛❞r♦♥ ♠❛ss ❧✐♠✐t
m→∞
= ⇒
❉②♥❛♠✐❝❛❧ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ◗✉❛r❦s ψ✱ ●❧✉♦♥s ❆ = ⇒ ψ✱ ♠❡s♦♥s φ ∼ ¯ ψψ✱ ❜❛r②♦♥s ∆ ∼ ψψ✱ ❆
✷✾ ✴ ✷✾