As we know, for 1+1D CFT, we have "radial quantization" to define the Virasoro algebra on the manifold with time or space to be periodic, like cylinder. But can we do that for both coordinates goes from \((-\infty,\infty)\)? I think if we do mode expansion as cylinder case, we would get the continuum Fourier integral.

I mean that on cylinder we can expand the energy-momentum tensor as\[T(w)=\sum_{n\in Z} L_n^{cyl}e^{-nw}\] where \(w=\tau-ix\). On cylinder, spacial coordinate \(x\) is periodic. But if we consider the case without this periodicity that both \(x\) and \(\tau\) does not perform as periodic coordinate and without any boundary, my naive expansion of the energy-momentum tensor would be something like\[T(w)=\int dk L(k)e^{-kw}\] And what do these expansion mode represent?