"Free fermion"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
| − | * | + | * <math>c=1/2</math> (for <math>\psi</math> real) |
| − | * | + | * <math>c=1</math> (for \psi complex) |
| 12번째 줄: | 12번째 줄: | ||
==OPE of fermionic fields== | ==OPE of fermionic fields== | ||
| − | * | + | * <math>\psi(z)\psi(w) \sim \frac{1}{(z-w)}</math> |
| − | * | + | * <math>\partial \psi(z) \psi(w) \sim -\frac{1}{(z-w)^2}</math> |
| − | * | + | * <math>\partial \psi(z) \partial \psi(w) \sim -\frac{2}{(z-w)^3}</math> |
==energy-momentum tensor== | ==energy-momentum tensor== | ||
| − | * | + | * <math>T(z)=-\frac{1}{2}:\psi(z)\partial \psi(z):=-\frac{1}{2}\left(\lim_{w\to z}\psi(z)\partial \psi(z)+\frac{1}{(z-w)^2}\right)</math> |
| − | * | + | * <math>T(z)\psi(w) \sim \frac{\psi(w)}{2(z-w)^2}+\frac{\partial \psi(w)}{(z-w)}</math> |
| − | * | + | * <math>T(z)\partial \psi(w) \sim \frac{\psi(w)}{2(z-w)^3}+\frac{3\partial \psi(w)}{2(z-w)^2}+\frac{\partial^2 \psi(w)}{(z-w)}</math> |
| − | * | + | * <math>T(z)T(w) \sim \frac{1}{4(z-w)^4}+\frac{2T(w)}{(z-w)^2}+\frac{\partial T(w)}{(z-w)}</math> |
2020년 11월 16일 (월) 04:28 판
introduction
- \(c=1/2\) (for \(\psi\) real)
- \(c=1\) (for \psi complex)
action
\(S= \int\!d^2x\, \psi^\dagger \gamma^0 \gamma^\mu \partial_\mu \psi= \int\!d^2z\, \psi^\dagger_R \bar\partial \psi_R + \psi_L^\dagger \bar\partial \psi_L\,\)
OPE of fermionic fields
- \(\psi(z)\psi(w) \sim \frac{1}{(z-w)}\)
- \(\partial \psi(z) \psi(w) \sim -\frac{1}{(z-w)^2}\)
- \(\partial \psi(z) \partial \psi(w) \sim -\frac{2}{(z-w)^3}\)
energy-momentum tensor
- \(T(z)=-\frac{1}{2}:\psi(z)\partial \psi(z):=-\frac{1}{2}\left(\lim_{w\to z}\psi(z)\partial \psi(z)+\frac{1}{(z-w)^2}\right)\)
- \(T(z)\psi(w) \sim \frac{\psi(w)}{2(z-w)^2}+\frac{\partial \psi(w)}{(z-w)}\)
- \(T(z)\partial \psi(w) \sim \frac{\psi(w)}{2(z-w)^3}+\frac{3\partial \psi(w)}{2(z-w)^2}+\frac{\partial^2 \psi(w)}{(z-w)}\)
- \(T(z)T(w) \sim \frac{1}{4(z-w)^4}+\frac{2T(w)}{(z-w)^2}+\frac{\partial T(w)}{(z-w)}\)
computational resource