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

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Unsharp Values, Domains and Topoi 73 Clearly, dcpos with Scott-continuous functions form a category. The definition of a Scott-continuous function can be extended to posets which are not dcpos, by carefully modifying the second condition to say ‘for any directed set S that has a supremum’. The reference to ‘continuity’ is not fortuitous, asthereistheso-calledScott topology, with respect to which these (and only these) arrows are continuous. The Scott topology will be defined below. Approximation - continuous posets. The other central notion is sometimes called approximation. This is captured by the following relation on elements of P . Definition 3.4. We say that x approximates y or x is way below y, andwrite x
74 A. Döring and R. Soares Barbosa Recall that ⊔ ↑ ↡y = y means that ↡y is directed and has supremum y. Continuity basically says that the elements ‘much simpler’ than y carry all the information about y, when taken together. Algebraicity further says that the ‘primitive’ (i.e., compact) elements are enough. Bases and domains. The continuity and algebraicity requirements are often expressed in terms of the notion of a basis. The definition is slightly more involved, but the concept of a basis is useful in its own right. Definition 3.7. A subset B ⊆ P is a basis for P if, for all x ∈ X, ⊔ ↑ (B∩↡x)=x. It is immediate that continuity implies that P itself is a basis. Conversely, the existence of a basis implies continuity. Definition 3.8. A domain (or continuous domain) ⟨D,⊑⟩ isadcpowhichis continuous. Equivalently, a domain is a dcpo that has a basis. ⟨D,⊑⟩ is an algebraic domain if it is a domain and algebraic, that is, if the set K(D) of compact elements is a basis for ⟨D,⊑⟩. Anω-continuous (resp. ω-algebraic) domain is a continuous (resp. algebraic) domain with a countable basis. Note that a domain always captures the notions of convergence and of approximation as explained above. Bounded complete posets. Later on, we will need another completeness property that a poset P may or may not have. Definition 3.9. A poset is bounded complete (or a bc-poset) if all subsets S with an upper bound have a supremum. It is finitely bounded complete if all finite subsets with an upper bound have a supremum. It is almost (finitely) bounded complete if all nonempty (finite) subsets with an upper bound have a supremum. 3 We state the following result without proof. Proposition 3.10. A(n almost) finitely bounded complete dcpo is (almost) bounded complete. 4 Another property, stronger than bounded completeness, will be needed later on: Definition 3.11. An L-domain is a domain D in which, for each x ∈ D, the principal ideal ↓x is a complete lattice. The Scott topology. We now define the appropriate topology on dcpos and domains, called the Scott topology, and present some useful results. In fact, the Scott topology can be defined on any poset, but we are mostly interested in dcpos and domains. Definition 3.12. Let ⟨P,≤⟩ be a poset. A subset G of P is said to be Scott-open if 3 Note that the ‘almost’ versions don’t require a least element ⊥. 4 Bounded-complete dcpos are the same as complete semilattices (see [16]).
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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
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Preface xiii it is related to other
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Quantum Gravity: Whence, Whither? C
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Quantum Gravity: Whence, Whither? 3
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16 K. Fredenhagen and K. Rejzner fi
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18 K. Fredenhagen and K. Rejzner Th
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20 K. Fredenhagen and K. Rejzner 4.
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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
Page 114 and 115: Causal Boundaries 99 to the Gromov
Page 116 and 117: Causal Boundaries 101 I + (ρ) ρ P
Page 118 and 119: Causal Boundaries 103 Figure 2. The
Page 120 and 121: Causal Boundaries 105 more pairings
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Page 126 and 127: Causal Boundaries 111 to Busemann f
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Page 130 and 131: Causal Boundaries 115 6.3. Structur
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Page 134: Causal Boundaries 119 [26] E. Mingu
Page 137 and 138: 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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