솔러 모형

솔러 모델은 3개의 공간 및 1개의 시간 차원으로 4개의 페르미온 상호작용을 통해 상호작용하는 디락 페르미온의 양자장 이론 모델입니다.1938년 드미트리 이바넨코에 의해 도입되었고 1970년^[2] 마리오 솔러에 의해 자기 상호작용 전자의 장난감 모델로 재도입되어 조사되었다.

이 모델은 라그랑주 밀도로 설명된다.

(\displaystyle\mathcal {L}}=오버라인(\psi }}\left(i\displaystyle\flash \!!)\!/-m\right)\psi +{\frac {g}{2}}\왼쪽 오버라인(\psi }}\psi \right)^{2}

$g$ 서g {\ $displaystyle$ g $}$ 는 $g$ 커플링 상수입니다. $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ / $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ μ μ $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ ${\$ $\!/=\sum$ _ ${\mu$ $=$ $0}^{\$ mu }\ $sum ^{\mu$ }{\ $frac$ {\frac $}{\$ flac $x^{\$ $mu$ $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ }}: 파인만 $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ 슬래시 ${\overline {\psi }}=\psi ^{*}\gamma ^{0}$ 에서 ${\overline {\psi }}=\psi ^{*}\gamma ^{0}$ 0 {\style } {\ $display$ { $psi$ } {\fline $} ^{\frac$ } } $。$ 여기서 $\gamma ^{\mu }$ μ $(\displaystyle$ \ $gamma$ ^{\ $mu$ $0\leq \mu \leq 3$ $0\leq \mu \leq 3$ $(\displaystyle$ 0 $\leq \leq$ 3)은 $0\leq \mu \leq 3$ Dirac 감마 매트릭스입니다.

대응하는 방정식은 다음과 같이 쓸 수 있다.

i

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

-

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

j =

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

j

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

∂ ∂ ∂ x

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

+

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

β

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

β

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

-

g

ψ

¯

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

）

ψ

display

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

、 \

display style

i

fr

frac

=

{ j

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

=

1

}{

3

} \

alpha

^{

frac

} { }

$\alpha ^{j}$ 서 $\alpha ^{j}$ j ${\displaystyle$ \alpha $^{j$ $1\leq j\leq 3$ $1\leq j\leq 3$ 3 ${\displaystyle$ 1 $\leq$ j $\leq$ 3) $1\leq j\leq 3$ $\beta$ β {\ $displaystyle \beta}$ 는 $\beta$ Dirac 매트릭스입니다.한 차원에서는 이 모델을 ^[3]매시브 Gross-Neveu 모델이라고 합니다.^[4]

일반화

일반적으로 고려되는 일반화는 다음과 같습니다.

(\displaystyle\mathcal {L}}=오버라인(\psi }}\left(i\displaystyle\flash \!!)\!/-m\오른쪽)\psi +gfrac(왼쪽 오버라인)\psi\오른쪽)^{k+1}:{k+1}}

$k>0$ > $k>0$ { $displaystyle$ k > $0$ } 또는 짝수일 경우

{\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +F\left({\overline {\psi }}\psi \right)

\!/-m\오른쪽)\psi

+

F\left({\오버라인\psi }}\psi

\

오른쪽

$F$ 서 F $(\displaystyle$ F $)$ 는 $F$ 부드러운 기능입니다.

특징들

내부 대칭

이 방정식은 유니터리 대칭 U(1) 외에 치수 1, 2, 3에서 SU(1,1)의 글로벌 내부 ^[5]대칭을 가진다.

재규격화 가능성

Soler 모델은 k $= 1$ 에 $k=1$ 대한 $k=1$ 카운트와 1차원에서만 재규격화가 가능하며 $,$ k(\ $displaystyle$ k $})$ $k$ 의 $k$ 더 높은 차원에 대한 재규격화가 불가능합니다.

단독파 솔루션

Soler 모델은 $\phi (x)e^{-i\omega t},$ ( $\phi (x)e^{-i\omega t},$ ) $\phi (x)e^{-i\omega t},$ - $\phi (x)e^{-i\omega t},$ $\phi (x)e^{-i\omega t},$ t $,$ {\ $displaystyle$ $\phi$ $(x$ $)$ e $^{-i\obmega$ t $}$ 형식의 단독 파형 솔루션을 허용하며, 여기서 $δ$ {\ $displaystyle$ x $}$ 는 $\phi$ 국소화(x {\ $displaystyle$ x}가 $x$ 클 때 작아짐)되고 $\omega$ ^[6] $\omega$ {\ $displaystyle \ope }$ 은 $\phi (x)e^{-i\omega t},$ $\omega$ 실수이다.

대형 티링 모델로 축소

공간 차원 2에서 솔러 모델은 ${\bar {\psi }}\psi =\psi ^{*}\sigma _{3}\psi$ 티링 모델과 일치합니다. 관계 $({\bar {\psi }}\psi )^{2}=J_{\mu }J^{\mu }$ $({\bar {\psi }}\psi )^{2}=J_{\mu }J^{\mu }$ $({\bar {\psi }}\psi )^{2}=J_{\mu }J^{\mu }$ J $({\bar {\psi }}\psi )^{2}=J_{\mu }J^{\mu }$ μ $display {\bar$ { $psi$ } $= J$ ^{\ $mu$ = J $=$ J = J ${\bar {\psi }}\psi =\psi ^{*}\sigma _{3}\psi$ ${\bar {\psi }}\psi =\psi ^{*}\sigma _{3}\psi$ J display ${\bar {\psi }}\psi =\psi ^{*}\sigma _{3}\psi$ display display bar bar displaybardisplaydisplaydisplay $display$ displaydisplaydisplaybar bardisplaydisplaydisplay $displaydisplaydisplaydisplaydisplaydisplay$ displaydisplaydisplaydisplaybardisplaydisplaybardisplaydisplaydisplaydisplaydisplaybardisplaydisplaydisplaydisplaydisplaybardisplaybardisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplaydisplay $J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )$ μ $J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )$ ( $ψ$ $J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )$ 1 $J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )$ 、 $J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )$ 2 $J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )$ ) $=$ ( \ $displaystyle$ J^ { * } \ $psi$ , \ $psi$ ^ { * } \ $psi$ 、 \ psi ^ { * } \ $psi$ 、 \ $psi$ ^ { * } \ psi 、 \ psi ^ { * } \ psi 、 \ $psi$ 、 \ $psi$ 、 \ $psi$ } \ psi 、 \ psi ^ { \ $psi$ 、 \ psi $J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )$ 이 관계는 아이덴티티 $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ $ψ$ ) $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ 2 $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ + ( $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ 2 $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ + ( $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ ) 2 $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ ( $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ ） $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ = ( ψ $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ $displaystyle$ ( \ psi $^{$ * } \ $psi$ _ { $1$ } \ psi $） ^2$ + ( \ psi $^{$ \ $psi$ + $2$ } } $b {C}^{2$ ^[7]

「」를 참조해 주세요.

레퍼런스

^ Dmitri Ivanenko (1938). "Notes to the theory of interaction via particles" (PDF). Zh. Eksp. Teor. Fiz. 8: 260–266.
^ Mario Soler (1970). "Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy". Phys. Rev. D. 1 (10): 2766–2769. Bibcode:1970PhRvD...1.2766S. doi:10.1103/PhysRevD.1.2766.
^ Gross, David J. and Neveu, André (1974). "Dynamical symmetry breaking in asymptotically free field theories". Phys. Rev. D. 10 (10): 3235–3253. Bibcode:1974PhRvD..10.3235G. doi:10.1103/PhysRevD.10.3235.{{cite journal}}: CS1 maint: 여러 이름: 작성자 목록(링크)
^ S.Y. Lee & A. Gavrielides (1975). "Quantization of the localized solutions in two-dimensional field theories of massive fermions". Phys. Rev. D. 12 (12): 3880–3886. Bibcode:1975PhRvD..12.3880L. doi:10.1103/PhysRevD.12.3880.
^ Galindo, A. (1977). "A remarkable invariance of classical Dirac Lagrangians". Lettere al Nuovo Cimento. 20 (6): 210–212. doi:10.1007/BF02785129.
^ Thierry Cazenave & Luis Vàzquez (1986). "Existence of localized solutions for a classical nonlinear Dirac field". Comm. Math. Phys. 105 (1): 35–47. Bibcode:1986CMaPh.105...35C. doi:10.1007/BF01212340.
^ J. Cuevas-Maraver; P.G. Kevrekidis; A. Saxena; A. Comech & R. Lan (2016). "Stability of solitary waves and vortices in a 2D nonlinear Dirac model". Phys. Rev. Lett. 116 (21): 214101. arXiv:1512.03973. Bibcode:2016PhRvL.116u4101C. doi:10.1103/PhysRevLett.116.214101.

[1] Dmitri Ivanenko (1938). "Notes to the theory of interaction via particles" (PDF). Zh. Eksp. Teor. Fiz. 8: 260–266.

[2] Mario Soler (1970). "Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy". Phys. Rev. D. 1 (10): 2766–2769. Bibcode:1970PhRvD...1.2766S. doi:10.1103/PhysRevD.1.2766.

[3] Gross, David J. and Neveu, André (1974). "Dynamical symmetry breaking in asymptotically free field theories". Phys. Rev. D. 10 (10): 3235–3253. Bibcode:1974PhRvD..10.3235G. doi:10.1103/PhysRevD.10.3235.{{cite journal}}: CS1 maint: 여러 이름: 작성자 목록(링크)

[4] S.Y. Lee & A. Gavrielides (1975). "Quantization of the localized solutions in two-dimensional field theories of massive fermions". Phys. Rev. D. 12 (12): 3880–3886. Bibcode:1975PhRvD..12.3880L. doi:10.1103/PhysRevD.12.3880.

[5] Galindo, A. (1977). "A remarkable invariance of classical Dirac Lagrangians". Lettere al Nuovo Cimento. 20 (6): 210–212. doi:10.1007/BF02785129.

[6] Thierry Cazenave & Luis Vàzquez (1986). "Existence of localized solutions for a classical nonlinear Dirac field". Comm. Math. Phys. 105 (1): 35–47. Bibcode:1986CMaPh.105...35C. doi:10.1007/BF01212340.

[7] J. Cuevas-Maraver; P.G. Kevrekidis; A. Saxena; A. Comech & R. Lan (2016). "Stability of solitary waves and vortices in a 2D nonlinear Dirac model". Phys. Rev. Lett. 116 (21): 214101. arXiv:1512.03973. Bibcode:2016PhRvL.116u4101C. doi:10.1103/PhysRevLett.116.214101.

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솔러 모형

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