radial quantization

From Physics wiki

We may suppose that our 2-dimensional quantum field theory lives on a spacetime that has the topology of a cylinder, with $x^1\,$ having period $2\pi\,$ so that $x^1 \sim x^1 + 2\pi\,$ . Strictly speaking, the radius of our cylinder is determined by the metric, and we may at a later time take the limit as $R\to\infty\,$ if we wish. Otherwise we may pretend that we are doing string theory. Let the metric be

$ds^2 = (dx^0)^2 + (dx^1)^2\,$ ,

(assuming we have done a Wick rotation), and introduce the complex coordinates

$w = x^1 + i x^0\,$ (left-moving^[1]),

$\bar w = x^1 - i x^0\,$ (right-moving),

which are the analytic continuations of $x^1 - x^0_M\,$ (left-moving) and $x^1+x^0_M\,$ (right-moving) respectively. Then

$x^0 = \tfrac{1}{2}\left(w + \bar w\right)$ ,

$x^1 = -\tfrac{i}{2}\left(w - \bar w\right)$ ,

so that $ds^2 = dw d\bar w\,$ , or

$g_{\mu\nu} = \begin{pmatrix}0 & \tfrac{1}{2} \\ \tfrac{1}{2} & 0\end{pmatrix}\,$ ,

$g^{\mu\nu} = \begin{pmatrix}0 & 2 \\ 2 & 0\end{pmatrix}\,$ .

Furthermore, the volume form becomes

$dx^0 \wedge dx^1 = \begin{vmatrix} \frac{1}{2} & \frac{1}{2} \\ -\frac{i}{2} & +\frac{i}{2} \end{vmatrix} dw d\bar w = \tfrac{i}{2} dw \wedge d\bar w\,$ .

Any transformation of the form

$w = f(w)\,$

$\bar w = g(\bar w)\,$

preserves the metric up to a conformal factor, which we may drop, provided our field theory has Weyl invariance (something we should confirm quantum mechanically). Let

$z = e^{i w}\,$ ,

$\bar z = e^{-i \bar w}\,$ ,

which maps the cylinder onto the plane so that "time" runs radially outward from the origin.

References

↑ This is to conform to literature. Assume $x^1\,$ increases to the left

Retrieved from "http://www.physics.thetangentbundle.net/wiki/Conformal_field_theory/radial_quantization"

radial quantization

From Physics wiki

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