arXiv:hep-th/9304011 v1 Apr 5 1993

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since the leading term in amplitude goes as |q| which multiplies the “operator” φ e √ 2φ . This fits in well with the Seiberg bound (3.40). By a limiting process, we may interpret e √ 2φ and φ e √ 2φ as the two KPZ dressings of the unit operator. We choose the root of the KPZ equation (2.19) so that the exponential grows at φ → −∞, this being the root we expect to correspond to a local operator. In the present case we must choose the root φ e √ 2φ , as anticipated in the paragraph following (4.11), and in accord with the argument given at the end of sec. 11.6. Exercise. Spacetime interpretation of the bounce factor Apply the low energy theorem, property (v), to the two-point function to show that the “bounce factor” is the one-point function of the tachyon zeromode [141]: 〈T 0 〉 = i log R(µ; V ) . (13.39) We regard property (vi) as intriguing: it strongly hints at a topological field theory interpretation of c = 1. 13.7. Unitarity of the S-Matrix One immediate application of the algorithm of sec. 13.5 is that we can give a very simple and conceptual discussion of the unitarity of the S-matrix [132]. 3. Bosonization 2. Free Fermion Scattering, S FF 1. Fermionization Fig. 32: Composition of three maps: fermionization, free-fermion potential scattering, and rebosonization. 165
The key observation is that the combinatorics of connecting lines according to the diagrammatic rules of the previous section is identical to the combinatorics of bosonization. We can then describe the algorithm as a three-step process: fermionization, then freefermion potential scattering, then rebosonization, as shown in fig. 32. To be more precise, we describe in/out bosonic Fock spaces F in/out made from the Heisenberg algebra of in/out massless bosons: α(η) in/out where η ∈ IR, [α(η), α(η ′ )] = η ′ δ(η + η ′ ) and the in/out vacua are defined by α(η)|0〉 = 0 for η < 0. Now, the Hilbert space of the theory may also be described in terms of the Fermionic Fock space H FF defined by the oscillators a(E) of sec. 11.3 (see (11.31, 11.32)). 60 As is well-known, the fermionization map ι b→f : α(η) → ∫ ∞ −∞ dξ a(µ + ξ) a † (µ − (η − ξ)) (13.40) defines an isometry H 0 FF ∼ = F in/out , where the superscript 0 indicates restriction to the sector with the difference #particles− #holes = 0. Thus, the prescription of sec. 13.5 may be summarized by writing the collective field S CF as a composition of three maps: S CF = ι f→b ◦ S FF ◦ ι b→f , (13.41) where ι b→f is the fermionization map, ι f→b is the inverse bosonization map, and S FF is the free-fermion potential scattering S-matrix defined by (13.32). Although standard bosonization is definitely not exact for the nonrelativistic fermion systems, the asymptotic bosonization is exact for fermions in a potential approaching V (λ) ∼ −λ 2 at infinity, and this suffices for computation of the S-matrix. From (13.41), we immediately deduce that the S-matrix is nonperturbatively unitary if and only if S FF is unitary. There are two immediate consequences of this remark. 1) In theories with no infinite wall where the reflection factors have absolute value smaller than one, the theory will fail to be nonperturbatively unitary. This is not because bosons can tunnel, but because a single fermion in a particle–hole pair can tunnel, thus leaving a nontrivial soliton sector on either side of the world. Put another way, if we insist that the (left and right) Hilbert space of the theory be H 0 FF (again the sector with #particles − #holes = 0) then the model will be non-unitary. If we allow nonzero # 60 We are describing only one world so we drop the ɛ label. 166
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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
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1) The expansions in V and ζ −1
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V′ ( ) + + + . . . ∂ ∂L L =
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and the main observation is 〈∏
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2 λ Fig. 17: The Wigner semicircle
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Example. The Gaussian potential We
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Exercise. Using the asymptotics of
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Page 117 and 118: 10.2. Loops to Local Operators By s
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Page 123 and 124: Using Feynman diagrams to obtain th
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Page 129 and 130: As we have seen from the tree-level
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Page 133 and 134: The perturbative expansion of the H
Page 135 and 136: 11.5. Wavefunctions and Wheeler-DeW
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Page 139 and 140: Remarks: 1) The definition (11.59)
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Page 149 and 150: The w ∞ symmetry of the inverted
Page 151 and 152: coadjoint orbit quantization for a
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Page 155 and 156: The equivalence of the collective f
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Page 159 and 160: 13.4. Tree-Level Collective Field T
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Page 165: Property (i) follows from the integ
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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