By C. Sieg

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**Extra resources for Noncommutativity and Holography in Field, String Theory [thesis]**

**Sample text**

The first [second] limit is to let the time coordinate of the second field approach the time coordinate of the first one from below [above]. 14) X µ (s), X ν (s) = lim R X µ (s)X ν (s − ε) − X µ (s)X ν (s + ε) ε→0 = i µν θ − θνµ = iθµν . 16) The parameter θµν can be interpreted as the noncommutativity parameter in a space where the embedding coordinates on the Dp-brane describe the noncommutative coordinates. 3) on string scattering amplitudes. As we have mentioned above, an element of the string S-matrix is a correlator of vertex operators that describe the asymptotic states.

Remember that different values for Φ describe the same theory but regularized in different ways. The value of Φ is connected to the parameters of a regularization scheme that interpolates smoothly between Pauli-Villars (Φµν = Bµν ) and the point splitting regularization (Φµν = 0). Schemes that show the existence of Φ were found in [6]. There should thus exist a map that relates the gauge fields AΦ µ for different Φ. 44) that relates the field aµ in the first ordinary effective description to the field Aµ in the second noncommutative one.

The gauge group. 5 for a short introduction of this formalism). Here we will work in ordinary spacetime where noncommutativity is described by the ∗-product and the operator trace becomes a spacetime integral. 69). 52)). In the next Section we will analyze the ∗-commutator in more detail and discuss which influence noncommutativity exerts on the choice of the gauge group. 2 to noncommutative YM theories. This leads to a relation between the noncommutative and ordinary sets of ghost fields. 3 the Feynman rules for noncommutative U(N) YM theories will be worked out.