Wick Rotations

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A Euclidean metric is simply a metric that resembles the distance measure from ordinary geometry. Let s try to clarify this point considering the simpli ed case of one-time dimension and one-space dimension. In special relativity, we distinguish between space and time with the use of a change of sign so that if we are using the signature ( , + ) ds 2 = dt 2 + dx 2 . So the two-dimensional Minkowski metric would be 1 0 = 0 1 What we re after with a Euclidean metric is describing things in a way that we could using ordinary geometry. In the x-y plane, the in nitesimal measure of distance is given by dr 2 = dx 2 + dy 2. This tells us that a Euclidean metric is one for which all quantities have the same sign. We can rewrite the Minkowski metric in this way by using what is known as a Wick rotation. Simply put, we make a transformation on the time coordinate by letting t it. Then dt idt and it follows that ds 2 = ( idt )2 + dx 2 = dt 2 + dx 2, which is exactly what we want. In order to describe the worldsheet with coordinates ( , ) using a Euclidean metric, we make a Wick rotation i . In terms of the worldsheet coordinates ( , ), the metric is ds 2 = d 2 + d 2 So we see that making a Wick rotation i changes this to ds 2 = d 2 + d 2 which is a Euclidean metric. Utilizing the Euclidean metric enables us to use conformal eld theory on the string.

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A consequence of a Wick rotation is that the light-cone coordinates ( +, ) are replaced with complex coordinates ( z, z ). The description of the worldsheet is transformed into complex variables by de ning complex coordinates ( z, z ) which are functions of the real variables ( , ). One way this can be done is as follows: z = + i z = i (5.4)

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Let s use this de nition to work out a few basic quantities and show how this simpli es analysis. Keep the Polyakov action in the back of your mind. Using the Euclidean metric the Polyakov action is written as SP = 1 d d ( X X + X X ) 4 (5.5)

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We re going to nd out that going to complex variables will simplify the form of Eq. (5.5). To transform coordinates we need to know how to compute derivatives with respect to the coordinates z and z . This is easy enough. First we invert the coordinates Eq. (5.4):

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It follows that 1 = = z z 2 and so

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z+z 2

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z z 2i

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(5.6)

1 = z 2i

1 = z 2i

1 1 1 = + = + = i z z z 2 2i 2

(5.7)

The shorthand notation z = = 1 / 2( i ) is usually used. It is also easy to see that

z = z = = z + z = 1 2 1 2i = 1 2 +i 1 = ( + i ) 2

(5.8)

CHAPTER 5 Conformal Field Theory Part I

where we ve introduced the abbreviation z = . Now given that after a Wick rotation the metric for the ( , ) coordinate system is written as 1 0 g = 0 1 We can write down the metric in the new complex coordinates using Eq. (5.1). We have gzz = z z g + z z g + z z g + z z g = z z + z z 1 1 1 1 1 1 = + = = 0 2 2 2i 2i 4 4 Similarly, gzz = 0. On the other hand gzz = z z g + z z g + z z g + z z g = z z + z z 1 1 1 1 1 1 1 = + = + = = gzz 2 2 2i 2i 4 4 2 In matrix form 0 1 / 2 g = 1 / 2 0 The inverse metric, written with raised indices has components given by g zz = g zz = 0 The corresponding matrix is 0 2 g = 2 0 g zz = g zz = 2 (5.12) (5.11) (5.10) (5.9)