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2.1 The rotating point-particle

It is instructive to sketch this procedure by considering the perhaps simplest solution to the equations of motion of (4View Equation): the rotating point-particle on S5, which is a degenerated string configuration.
t = k t, r = 0, g = p-, f = k t, f = f = y = 0. (7) 2 1 2 3

One easily shows that this configurations satisfies the equations of motion and the Virasoro constraints.

A glance at Equations (4View Equation, 5View Equation) reveals that the cyclic coordinates of the action integral I = dt L are (t,f ,f ;f ,f ,f ) 1 2 1 2 3 leading to the conserved charges (E, S ,S ;J ,J ,J ) 1 2 1 2 3, corresponding to the energy E and two spins (S1,S2) on AdS5 as well as the three angular momenta (J1,J2,J3) on the five sphere respectively. The energy E and the first angular momentum J1 are the only non vanishing conserved quantities of the above point-particle configuration and take the values (we also spell out J2 for later use)

integral @L V~ -- 2pds 2 V~ -- E := --- = c 2p cosh rt = c k, @t 0 integral 2p @L-- V~ -- ds 2 2 V~ -- J1 := - @f = c 0 2p sin g cos y f1 = c k, (8) 1 integral 2p J := - @L--= V~ c- ds sin2 g sin2 y f = 0, 2 @f2 0 2p 2 J := J + J . 1 2
Hence classically E = J. One may now consider quantum fluctuations around this solution, i.e. Xm = Xmsolution(t) + -11/4-xm(t,s) c, and integrate out the quantum field xm(t,s) in a perturbative fashion. This will result in “quantum” corrections to the classical energy E0 in terms of an expansion in V~ -- 1/ c
V~ -- 1 E = ck + E2(k) + V~ --E4(k) + ... (9) c
They key idea of Berenstein, Maldacena and Nastase [33] was to consider the limit J --> oo with the parameter V ~ -- k = J/ c held fixed. This limit of a large quantum number suppresses all the higher loop contributions beyond one-loop, i.e. 
-1-- ~ J-->o o E - J = E2(k) + V~ --E4(k) + ...- --> E2(k) . (10) J
Hence, the quadratic approximation becomes exact! This quadratic fluctuation action (including the fermions) is nothing but the IIB superstring in a plane wave background [88], which arises from the AdS5 × S5 geometry through a so-called Penrose limit (see [3839] for this construction). The quantization of this string model is straightforward in the light-cone gauge [8890] and leads to a free, massive two-dimensional theory for the transverse degrees of freedom (i = 1,...,8)
integral 1 i i k2 i i I2 = dtds( -@ax @ax - ---x x + fermions) (11) 2 2
with a compact expression for the spectrum
oo --1- sum V~ -----'-2 ' -1-- c-- E2 = V~ c' 1 + c n Nn c := V~ k = J2 (12) n=- oo
where Nn := a †niain is the excitation number operator for transverse string excitations a †ni|0> with [ai ,a†j] = d d m n nm ij. The Virasoro constraints (6View Equation) reduce to the level matching condition sum n N = 0 n n known from string theory in flat Minkowski space-time. Hence, the first stringy excitation is † † an a- n|0> with V~ -- V~ -------- c'E2 = 2 1 + c'n2. For a more detailed treatment of the plane wave superstring see [9497104103].
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