Quantum Field Theory and Gravity: Conceptual and Mathematical ...

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Discrete and Continuum Third Quantization of Gravity 49 The corresponding field configurations represent extended chunks of space; a generic polynomial observable is associated with a particular quantum space. We now define a classical dynamics for the GFT field. In the interaction term four geometric triangles should be glued along common edges to form a tetrahedron. The kinetic term should encode the gluing of two tetrahedra along common triangles. With ϕ 123 := ϕ(x 1 ,x 2 ,x 3 ), we define the action S = 1 ∫ [dx] 3 ϕ 123 ⋆ϕ 123 − λ ∫ [dx] 6 ϕ 123 ⋆ϕ 345 ⋆ϕ 526 ⋆ϕ 641 , (3.4) 2 4! where ⋆ relates repeated indices as φ i ⋆φ i :=(φ⋆φ − )(x i ), with φ − (x)=φ(−x). One can generalize the GFT field to a function of n arguments which when satisfying (3.1) or (3.2) can be taken to represent a general n-gon (dual to an n-valent vertex) and glued to other fields, giving a polygonized quantum space just as for triangles. There is also no difficulty in considering a more general action in which, for given ϕ(x i ), one adds other interaction terms corresponding to the gluing of triangles (polygons) to form general polyhedra or more pathological configurations (e.g. with multiple identifications among triangles). The only restriction may come from the symmetries of the action. The projection (3.2) takes into account parallel transport between different frames; from (3.2) and (3.4) we can identify a propagator and a vertex: ∫ 3∏ K(x i ,y i )= dh t (δ −xi ⋆e ht )(y i ), i=1 ∫ ∏ V(x i ,y i )= dh t t 6∏ (δ −xi ⋆e htt ′ )(y i ); i=1 (3.5) the variables h t and h tτ arising from (3.2) are interpreted as parallel transports through the triangle t and from the center of the tetrahedron τ to triangle t, respectively. We may represent the propagator and vertex as follows: t d (3.6) y 2 y 3 y 1 y 1 y 4 y 6 t y 3 x 4 x 5 t b x 1 t a x 2 τ x 6 y 2 t c x 3 y 5 x 1 x 2 x 3 The integrands in (3.5) factorize into products of functions associated to strands (one for each field argument), with a geometrical meaning: The variables (x i ,y i ) associated to the edge i correspond to the edge vectors in the frames associated to the triangles t, t ′ sharing it; opposite edge orientations in different triangles and a mismatch between reference frames associated to the same triangle in two different tetrahedra are taken into account.
50 S. Gielen and D. Oriti Using the group Fourier transform we obtain the form of Boulatov [8], S[φ] = 1 ∫ [dg] 3 ϕ(g 1 ,g 2 ,g 3 )ϕ(g 3 ,g 2 ,g 1 ) 2 − λ ∫ [dg] 6 ϕ(g 12 ,g 13 ,g 14 )ϕ(g 14 ,g 24 ,g 34 )ϕ(g 34 ,g 13 ,g 23 )ϕ(g 23 ,g 24 ,g 12 ). 4! (3.7) In this group representation, the kinetic and vertex functions are ∫ 3∏ K(g i , ˜g i )= dh δ(g k h˜g −1 k ), k=1 ∫ 4∏ (3.8) ∏ V(g ij ,g ji )= dh i δ(g ij h i h −1 j g −1 ji ). i=1 i
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Quantum Field Theory and Gravity Co
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Contents Preface ..................
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Preface The present volume arose fr
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Preface ix 6. (How) can we test qua
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Preface xi Felix Finster, Andreas G
Page 14 and 15: Preface xiii it is related to other
Page 16 and 17: Quantum Gravity: Whence, Whither? C
Page 18 and 19: Quantum Gravity: Whence, Whither? 3
Page 28: Quantum Gravity: Whence, Whither? 1
Page 31 and 32: 16 K. Fredenhagen and K. Rejzner fi
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Page 37 and 38: 22 K. Fredenhagen and K. Rejzner 5.
Page 40 and 41: The “Big Wave” Theory for Dark
Page 56 and 57: Discrete and Continuum Third Quanti
Page 80 and 81: Unsharp Values, Domains and Topoi A
Page 82 and 83: Unsharp Values, Domains and Topoi 6
Page 112 and 113: Causal Boundary of Spacetimes: Revi
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Causal Boundaries 99 to the Gromov
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Causal Boundaries 101 I + (ρ) ρ P
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Causal Boundaries 103 Figure 2. The
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Causal Boundaries 105 more pairings
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Causal Boundaries 107 point of the
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Causal Boundaries 109 (0, 1) x n x
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Causal Boundaries 111 to Busemann f
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Causal Boundaries 113 respectively.
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Causal Boundaries 115 6.3. Structur
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Causal Boundaries 117 Qualitative F
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Causal Boundaries 119 [26] E. Mingu
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122 D. Häfner Let us give an idea
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124 D. Häfner Figure 1. The collap
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126 D. Häfner Figure 2. The manifo
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128 D. Häfner and the angular mome
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130 D. Häfner Lemma 3.5. There exi
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132 D. Häfner Proposition 4.4. The
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134 D. Häfner 6. Strategy of the p
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136 D. Häfner [3] B. Carter, Black
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138 R. Oeckl it is less compelling
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140 R. Oeckl Depending on the theor
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142 R. Oeckl 2.3. Recovery of stand
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144 R. Oeckl where the condition A
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146 R. Oeckl Before we proceed to i
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148 R. Oeckl the complex classical
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150 R. Oeckl The corresponding anni
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152 R. Oeckl Proposition 4.1 (Coher
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154 R. Oeckl much larger than the s
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156 R. Oeckl Setting F equal to F
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158 F. Finster, A. Grotz and D. Sch
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184 C. Bär and N. Ginoux treat the
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186 C. Bär and N. Ginoux the categ
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188 C. Bär and N. Ginoux Example 2
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190 C. Bär and N. Ginoux 3.2. Diff
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192 C. Bär and N. Ginoux In other
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194 C. Bär and N. Ginoux Here (e j
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196 C. Bär and N. Ginoux Remark 3.
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198 C. Bär and N. Ginoux Proof. Th
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200 C. Bär and N. Ginoux To prove
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202 C. Bär and N. Ginoux We refer
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204 C. Bär and N. Ginoux where Glo
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206 C. Bär and N. Ginoux [24] R. M
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208 C. J. Fewster well-described by
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210 C. J. Fewster and J + M (p) ∩
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212 C. J. Fewster commute. As funct
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214 C. J. Fewster ι + ι − M +
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216 C. J. Fewster ϕ(ψ) M M ϕ(M)(
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218 C. J. Fewster theory. The resul
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220 C. J. Fewster Following [7], a
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222 C. J. Fewster Sketch of proof.
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224 C. J. Fewster The relative Cauc
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226 C. J. Fewster [2] Bär, C., Gin
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Local Covariance, Renormalization A
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Local Thermal Equilibrium and Cosmo
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Shape Dynamics. An Introduction Jul
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Shape Dynamics. An Introduction 259
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where dx bm a Shape Dynamics. An In
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300 M. Kiessling In the following,
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302 M. Kiessling where “δ( · )
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304 M. Kiessling the point charge,
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306 M. Kiessling (2.17) and (2.18)
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308 M. Kiessling Maxwell’s “law
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310 M. Kiessling argued that E f (
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312 M. Kiessling the Maxwell-Born-I
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314 M. Kiessling removal. For insta
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316 M. Kiessling ∂ 4πc that for
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318 M. Kiessling d dt Dirac, on the
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320 M. Kiessling The Lorenz-Lorentz
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322 M. Kiessling ( 1 c Lorentz and
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324 M. Kiessling potential fields
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326 M. Kiessling as I have done her
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328 M. Kiessling just the Hamilton-
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330 M. Kiessling 7. Closing remarks
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332 M. Kiessling [Bor1937] Born, M.
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334 M. Kiessling [Lié1898] Liénar
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How Unique Are Higher-dimensional B
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Equivalence Principle, Quantum Mech
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Atom Interferometry 347 The second
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Atom Interferometry 349 α CR , aim
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Atom Interferometry 351 of this mas
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Atom Interferometry 353 height B A
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Atom Interferometry 355 iff ψ = (
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Atom Interferometry 357 have be rea
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Atom Interferometry 359 where F (t
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Atom Interferometry 361 5.2. Interf
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Atom Interferometry 363 Evaluating
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Atom Interferometry 365 where the l
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Atom Interferometry 367 Then they s
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Atom Interferometry 369 [8] T.M. Fo
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Index n-point function, 51, 146, 23
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Index 373 conformal field theory, 1
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Index 375 gravitational redshift, 2
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Index 377 operator product, 138 par
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Index 379 standard model of particl
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