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I now had some time to think about this again and I think I came up with an explanation. I would gratelygreatly appreciate Ifit if somebody had some comments wheterwhether this is reasonable:

As we know from 2-dimensional CFT where we usually like to work with the variables $z = t+ix$ resp. $\overline{z} = t-ix$ hence holomorphic/antiholomorphic expressions we have: $$T_{++} \leftrightarrow T_{zz} = T(z) \quad , \quad T_{--} \leftrightarrow T_{--} = T(\overline{z})$$ Sine we can express $T(z)$ resp. $T(\overline{z})$ ad Laurent-series as $$T(z) = \sum z^{-n-2}L_n $$ where the operators $L_n$ form a Virasoro algebra and generate the conformal transfromations. Since the physical transformations like dilations/rotations are actually generatetgenerated by the sum/difference of such generators the statement that $(f(x^+)T_{++} - f(x^-)T_{--})$ generates the conformal transformations of our theory seems reasonable. Furthermore since at point for which x=0 we have $$x^+ = t + 0 = t- 0 = x^-$$ and therefore $$(f(x^+)T_{++} - f(x^-)T_{--}) = f(t)T(t) -f(t)T(t) = 0$$ ,$$(f(x^+)T_{++} - f(x^-)T_{--}) = f(t)T(t) -f(t)T(t) = 0,$$
hence the above operator leaves invariant the line $x=0$. Where the implicit statement $T_{++} = T(x^+)$ follows from the one for $T_{zz}$ by Wick rotation

I now had some time to think about this again and I think I came up with an explanation. I would grately appreciate If somebody had some comments wheter this is reasonable:

As we know from 2-dimensional CFT where we usually like to work with the variables $z = t+ix$ resp. $\overline{z} = t-ix$ hence holomorphic/antiholomorphic expressions we have: $$T_{++} \leftrightarrow T_{zz} = T(z) \quad , \quad T_{--} \leftrightarrow T_{--} = T(\overline{z})$$ Sine we can express $T(z)$ resp. $T(\overline{z})$ ad Laurent-series as $$T(z) = \sum z^{-n-2}L_n $$ where the operators $L_n$ form a Virasoro algebra and generate the conformal transfromations. Since the physical transformations like dilations/rotations are actually generatet by the sum/difference of such generators the statement that $(f(x^+)T_{++} - f(x^-)T_{--})$ generates the conformal transformations of our theory seems reasonable. Furthermore since at point for which x=0 we have $$x^+ = t + 0 = t- 0 = x^-$$ and therefore $$(f(x^+)T_{++} - f(x^-)T_{--}) = f(t)T(t) -f(t)T(t) = 0$$ , hence the above operator leaves invariant the line $x=0$. Where the implicit statement $T_{++} = T(x^+)$ follows from the one for $T_{zz}$ by Wick rotation

I now had some time to think about this again and I think I came up with an explanation. I would greatly appreciate it if somebody had some comments whether this is reasonable:

As we know from 2-dimensional CFT where we usually like to work with the variables $z = t+ix$ resp. $\overline{z} = t-ix$ hence holomorphic/antiholomorphic expressions we have: $$T_{++} \leftrightarrow T_{zz} = T(z) \quad , \quad T_{--} \leftrightarrow T_{--} = T(\overline{z})$$ Sine we can express $T(z)$ resp. $T(\overline{z})$ ad Laurent-series as $$T(z) = \sum z^{-n-2}L_n $$ where the operators $L_n$ form a Virasoro algebra and generate the conformal transfromations. Since the physical transformations like dilations/rotations are actually generated by the sum/difference of such generators the statement that $(f(x^+)T_{++} - f(x^-)T_{--})$ generates the conformal transformations of our theory seems reasonable. Furthermore since at point for which x=0 we have $$x^+ = t + 0 = t- 0 = x^-$$ and therefore $$(f(x^+)T_{++} - f(x^-)T_{--}) = f(t)T(t) -f(t)T(t) = 0,$$
hence the above operator leaves invariant the line $x=0$. Where the implicit statement $T_{++} = T(x^+)$ follows from the one for $T_{zz}$ by Wick rotation

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I now had some time to think about this again and I think I came up with an explanation. I would grately appreciate If somebody had some comments wheter this is reasonable:

As we know from 2-dimensional CFT where we usually like to work with the variables $z = t+ix$ resp. $\overline{z} = t-ix$ hence holomorphic/antiholomorphic expressions we have: $$T_{++} \leftrightarrow T_{zz} = T(z) \quad , \quad T_{--} \leftrightarrow T_{--} = T(\overline{z})$$ Sine we can express $T(z)$ resp. $T(\overline{z})$ ad Laurent-series as $$T(z) = \sum z^{-n-2}L_n $$ where the operators $L_n$ form a Virasoro algebra and generate the conformal transfromations. Since the physical transformations like dilations/rotations are actually generatet by the sum/difference of such generators the statement that $(f(x^+)T_{++} - f(x^-)T_{--})$ generates the conformal transformations of our theory seems reasonable. Furthermore since at point for which x=0 we have $$x^+ = t + 0 = t- 0 = x^-$$ and therefore $$(f(x^+)T_{++} - f(x^-)T_{--}) = f(t)T(t) -f(t)T(t) = 0$$ , hence the above operator leaves invariant the line $x=0$. Where the implicit statement $T_{++} = T(x^+)$ follows from the one for $T_{zz}$ by Wick rotation