In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

withthetwo-bodyS-matrixof(58)

and

theadditiveenergyexpressionE2=M i=14sin2(pi

2

2+i=eip i eip(L 1)=1

p=2πnL→∞

L 1

dimension (1)=λ→8π8π22n2

L2

J p=0yieldstheone-loopscaling)ofthetwo-magnonoperators[71,19] 2p+1cosπn(J,2)On=

1+n2λ/J2[5].

Hencefromtheviewpointofthespinchaintheplane-wavelimitcorrespondstoachainofdiverginglengthL>>1carryinga nitenumberofmagnonsM,whicharenothingbutthegaugedualsoftheoscillatorexcitationsoftheplane-wavesuperstring.

4.2Thethermodynamiclimitofthespin-chain

Inordertomakecontactwiththespinningstringsolutiondiscussedinsection3wewillnowconsiderthethermodynamiclimitofthespinchaininwhichthelengthLandthenumberofmagnonsMbecomelarge.ThisisnecessaryastheclassicalstringsolutionsonlylimittothetruequantumresultintheBMNtype