arXiv:hep-th/9304011 v1 Apr 5 1993

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An incoming or outgoing boson of energy ω > 0 has wavefunction ψ L ω (t, τ) = e −iω(t+τ) , ψ R ω (t, τ) = e −iω(t−τ) , respectively. Therefore we define the S-matrix according to Def 2: Consider the asymptotic behavior of the time-ordered, connected, Minkowskian collective field Green’s function: G(t 1 , τ 1 , . . . t n , τ n ) ≡ 〈0|T ∏ ∂ τ χ(t i , τ i )|0〉 , (13.5) as τ i → +∞, t i → −∞ (1 ≤ i ≤ k), t i → +∞ (k + 1 ≤ i ≤ n = k + l). Then we define the connected S-matrix element for the process |ω 1 , . . . ω k 〉 → |ω 1, ′ . . . ω l ′ 〉 to be the function S CF c in the asymptotic formula: G ∼ ∫ ∞ 0 k∏ dω i i=1 l∏ dω i ′ δ( ∑ ω i − ∑ ω i) ′ i=1 · √ k! l! S CF c ( k∑ ω i → i=1 l∑ i=1 ω ′ i k∏ l∏ (ψω L i ) ∗ i=1 i=1 ψ R ω ′ i ) + off-shell terms . (13.6) The plane wave states are normalized such that 〈ω|ω ′ 〉 = ω δ(ω −ω ′ ). An equivalent definition has been used in [96,97] to compute the S-matrix from standard Feynman perturbation theory applied to collective field theory. While this definition is physically satisfying, it is not the best mathematical definition. An equivalent definition is obtained by continuing the Minkowskian Green’s functions to Euclidean space ∆t → −i∆X. Fourier transforming the Euclidean Green’s functions with respect to X i , we obtain mixed Green’s functions ∫ ∏ G E (q 1 , τ 1 , . . . q n , τ n ) ≡ i dX i e iq iX i G Euclidean (X 1 , τ 1 , . . . X n , τ n ) , (13.7) in terms of which we may define the S-matrix via: Def2 ′ : The large τ i asymptotics G E (q 1 , τ 1 , . . . q n , τ n ) ∼ δ( ∑ q i ) ∏ i e −|q i|τ i R(q 1 , . . . q n ) ( 1 + O(e −τ i ) ) (13.8) defines a function R n (q 1 , . . . q n ) from which we may obtain the connected S CF -matrix elements via analytic continuation |q| → −iω, where q < 0 corresponds to the incomers, and q > 0 corresponds to the outgoers. Specifically, S c → − ik+l+1 √ k! l! R. Remark: The non-obvious property that the function R n is independent of the order in which the τ i are taken to ∞ was demonstrated in [132]. 153
The equivalence of the collective field theory S-matrix and the correlators defined by shrinking macroscopic loops is demonstrated using the relation between τ-space and φ- space explained in sec. 12.4 above. In particular, transforming asymptotic wavefunctions according to (12.13), we relate l → 0 and τ → ∞ asymptotics via the integral ∫ ∞ dτ e −l2√ µ cosh τ e −|q|τ ∼ (l √ µ) |q| Γ(−|q|) , (13.9) plus terms regular in l. Notice that the two prescriptions only make complete sense when q is nonintegral. Otherwise we must use the full identity ∫ ∞ A dτ e −2l√ µ cosh τ e −|q|τ = − π sin π|q| I |q|(2 √ µl) − ∑ (−1) n (l √ µ) n n≥0 n ∑ m=0 1 e A(m−n−|q|) (13.10) m!(n − m)! m − n − |q| . The pole in the Γ–function in (13.9) is a warning that we cannot unambiguously separate the two terms in (13.10) via nonanalyticity in l. Leg Factors According to the arguments of chapt. 11, we expect that the Euclidean S-matrices S ST and S CF should agree up to an overall normalization f(q) of the vertex operators V q . This is because, for q /∈ Z, the BRST cohomology with the relevant quantum numbers is one-dimensional. Indeed, comparison with vertex operator calculations in Liouville theory, which will be described in sec. 14.2 below, shows that The factors A 0,n (V q1 , . . . V qn ) = (−i) n+1 n ∏ i=1 f(q) = Γ(−|q|) Γ(|q|) Γ(−|q i |) Γ(|q i |) R n(q 1 , . . . q n ) . (13.11) (13.12) are called “leg factors.” Notice that for the Minkowskian S-matrix they are pure phases, but the phases for incomers and outgoers are not complex conjugated. Comparison with the first quantized wavefunction for the spacetime boson, described by the Wheeler–DeWitt equation (5.21), indicates that neither normalization in (13.11) is the correct physical normalization since standard first-quantized scattering theory predicts that the genus zero 1 → 1 S-matrix is Γ(iE) Γ(−iE) . This suggests that the correct normalization of the vertex operators is obtained by taking the squareroot of (13.12). All this needs to be clarified! 154
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YCTP-P23-92 LA-UR-92-3479 hep-th/93
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7.1. Orthogonal polynomials . . . .
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contexts. The aim of these lectures
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1) As Quantum Gravity In this guise
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semiclassical, and quantum Liouvill
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The Lagrangian and Hamiltonian form
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mation, on an annulus. This is give
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Quantum mechanically, we try to und
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about the domain of integration. Th
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diffeomorphism invariance, (2.11) s
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quantum gravity [31], so it would b
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To determine ∆, we employ the sam
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The stress-energy tensor following
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Note that the uniformizing map (3.1
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In particular, passing from the pla
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Exercise. The wall analogy Show tha
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Here ν is a real number. An import
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5) As will be seen in sec. 5.4 belo
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where ∆ i is the conformal weight
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where j k ≡ −α k /γ. Strictly
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where |Σ〉 is a state created by
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equation with sources, (3.42). We s
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Exercise. Show that (3.70) is an id
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In view of these observations, the
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3.11. Surfaces with boundaries The
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4. 2D Euclidean Quantum Gravity II:
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4.3. KPZ states in 2D Quantum Gravi
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In the KP formalism of the matrix m
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In fact, the c = 1 model has much m
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We may plot the quantum numbers of
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On the RHS we have one, rather than
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5.1. Particles in D Dimensions: QFT
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where H is the Wheeler-DeWitt opera
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Euclidean signature spacetimes. The
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5.4. 2D String Theory: Minkowskian
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2) The only difference in degrees o
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Those who look for special states i
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Fig. 10: The case of the exploding
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Fig. 11: A piece of a random triang
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Since ∂ /2 eJ2 ∂J = Je J2 /2 ,
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(6.8). As familiar from field theor
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continuum, as reviewed in preceding
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7. Matrix Model Technology I: Metho
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with r n a scalar coefficient indep
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gravity. It is natural that pure gr
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with α = 1 10 . The solution to (7
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parameters to result in a (2l−1)
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and the recursion relations for thi
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The formal expansion of Q l−1/2 =
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If we consider the higher operators
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of the symmetry factor for the Feyn
Page 103 and 104: 1) The expansions in V and ζ −1
Page 105 and 106: V′ ( ) + + + . . . ∂ ∂L L =
Page 107 and 108: and the main observation is 〈∏
Page 109 and 110: 2 λ Fig. 17: The Wigner semicircle
Page 111 and 112: Example. The Gaussian potential We
Page 113 and 114: Exercise. Using the asymptotics of
Page 115 and 116: The formulae (10.1) and (10.2), whi
Page 117 and 118: 10.2. Loops to Local Operators By s
Page 119 and 120: This suggests that instead of the l
Page 121 and 122: Fig. 20: Two loops on a continuum s
Page 123 and 124: Using Feynman diagrams to obtain th
Page 125 and 126: continuous and particle states are
Page 127 and 128: Here λ 1,2 are the two turning poi
Page 129 and 130: As we have seen from the tree-level
Page 131 and 132: Laplace transform the eigenvalue de
Page 133 and 134: The perturbative expansion of the H
Page 135 and 136: 11.5. Wavefunctions and Wheeler-DeW
Page 137 and 138: x Fig. 24: The function x 2 K 0 (x)
Page 139 and 140: Remarks: 1) The definition (11.59)
Page 141 and 142: p λ Fig. 25: A generic initial con
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Page 145 and 146: It follows that if we define a nonl
Page 147 and 148: 12.5. The w ∞ Symmetry of the Har
Page 149 and 150: The w ∞ symmetry of the inverted
Page 151 and 152: coadjoint orbit quantization for a
Page 153: egarded as space, the time coordina
Page 157 and 158: Kondo effect, the Callan-Rubakov ef
Page 159 and 160: 13.4. Tree-Level Collective Field T
Page 161 and 162: presence of a wall. The function R
Page 163 and 164: ω q R(µ + ω ; V ) R q q > 0 −
Page 165 and 166: Property (i) follows from the integ
Page 167 and 168: The key observation is that the com
Page 169 and 170: (13.44) is given by F = F 0 + 1 µ
Page 171 and 172: and in (13.51) we only keep terms t
Page 173 and 174: e) The c = 1 model is equivalent to
Page 175 and 176: The expansion is only convergent fo
Page 177 and 178: Example. Let us consider the most g
Page 179 and 180: free-field techniques. But this cal
Page 181 and 182: Quite generally, the operator produ
Page 183 and 184: x p−1 ∼ y q−1 ∼ 1 (where C(
Page 185 and 186: spin j even at the self-dual radius
Page 187 and 188: 15.2. Disappointments From the quan
Page 189 and 190: We thank especially N. Seiberg for
Page 191 and 192: References [1] M. B. Green and J. S
Page 193 and 194: [36] G. Moore, N. Seiberg, and M. S
Page 195 and 196: [76] S.R. Das and A. Jevicki, “St
Page 197 and 198: Phys. B368 (1992) 671; S. Jain and
Page 199 and 200: [145] E. Martinec and S. Shatashvil
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arXiv:hep-th/9304011 v1 Apr 5 1993

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