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Causal Dynamical Triangulations and the Quest for Quantum Gravity? Appendices: Mathematical Methods for Basic and Foundational Quantum Gravity Unstarred Appendices support Part I’s basic account. Starred Appendices support Parts II and III on interferences between Problem of Time facets. Double starred ones support the Epilogues on global aspects and deeper levels of mathematical structure being contemplated as Background Independent. If an Appendix is starred, the default is that all of its sections are starred likewise; a few are marked with double stars. Appendix A Basic Algebra and Discrete Mathematics A.1 Sets and Relations For the purposes of this book, take a set X to just be a collection of distinguishable objects termed elements. Write x ∈ X if x is an element of X and Y ⊂ X for Y a subset of X, ∩ for intersection, ∪ for union and Yc = X\Y for the complement of Y in X. Subsets Y1 and Y2 are mutually exclusive alias disjoint if Y1 ∩ Y2 =∅: the empty set. In this case, write Y1 ∪ Y2 as Y1 Y2: disjoint union.Apartition of a set X is a splitting of its elements into subsets pP that are mutually exclusive = and collectively exhaustive: P pP X. Finally, the direct alias Cartesian product of sets X and Z, denoted X × Z, is the set of all ordered pairs (x, z) for x ∈ X, z ∈ Z. For sets X and Z,afunction alias map ϕ : X → Z is an assignation to each x ∈ X of a unique image ϕ(x) = z ∈ Z. Such a ϕ is injective alias 1to1if ϕ(x1) = ϕ(x2) ⇒ x1 = x2, surjective alias onto if given z ∈ Z there is an x ∈ X such that ϕ(x) = z, and bijective if it is both injective and surjective. For Y ⊂ X, the corresponding inclusion map is the injection j : Y → X with j(y)= y ∀ y ∈ Y. X is countable if it is finite or admits a bijection ϕ : X → N: the natural numbers. On the other hand the set of real numbers R is uncountable. |X| is the number of elements in the set X in the finite case, or the extension of this to the notion of cardinality of the set more generally. A binary relation R onasetX is a property that each pair of elements of X may or may not possess. We use aRbto denote ‘a and b ∈ X are related by R’. Simple examples include =,<,≤, ⊂ and ⊆. Some basic properties that an R on X might possess are as follows (∀ a,b,c ∈ X). Reflexivity: aRa. Symmetry: aRb⇒ bRa. Antisymmetry: aRb and bRa⇒ a = b. Transitivity: aRb and bRc⇒ aRc. To- tality: that one or both of aRb or bRa holds, i.e. all pairs are related. Commonly useful combinations of these include the following. 1) Equivalence relation, ∼,ifR is reflexive, symmetric and transitive. 2) Partial ordering, ,ifR is reflexive, antisymmetric and transitive, e.g. ≤ or ⊂: © Springer International Publishing AG 2017 647 E. Anderson, The Problem of Time, Fundamental Theories of Physics 190, DOI 10.1007/978-3-319-58848-3 648 A Basic Algebra and Discrete Mathematics Fig. A.1 Equipping sets with a variety of structures. The main structures considered in this book are presented in straight font, and we use italic font to denote some of the associated morphisms. Reversing the arrows gives examples of forgetting structures. τ is a topology, Dist is a metric space metric, diff are differentiable structures, γ and are connections, and U and m are Riemannian metrics; γ and U are conformally invariant versions ordering by ‘is a subset of’. 3) Total ordering, alias a chain,ifR is both a partial order and total. A fairly standard Foundational System for Mathematics involves equipping sets X with further layers of structure ς; we denote this by X,ς. For example, a set equipped with a partial order is a poset X, . One can furthermore equip an al- ready established equipped space s (rather than just a set) with extra layers of struc- ture s,ς. Such equipping is what is meant in Chap. 10 and Epilogue II.C by ‘levels of mathematical structure’, within the traditional (if in some ways restrictive) con- text that the base level consists of Set Theory. See Fig. A.1 for further examples of equipping. A (homo)morphism is a map μ : s1 → s2 that is structure-preserving. In partic- ular, if a such is bijective (equivalently invertible) it is an isomorphism,ifs1 = s2 it is an endomorphism, and if both of these apply, it is an automorphism. A forgetful map is one that ‘forgets’ (or ‘strips off’) some of the layers of struc- ture: φ :s,ς→s. There is corresponding loss of structure preservation in the maps associated with the latter equipped space. The obvious reversal of each of the equipping maps in Fig. A.1 readily provides an example of forgetful map. Let us end by noting that whereas many further mathematical entities are often thought of as arising by imposing further layers of structure on a set as per above, this is not the only way of doing mathematics. E.g. Category Theory [611, 612, 631] and Topos Theory [126, 260] offer extensions and alternatives in this regard. While these are very briefly outlined in Appendix W, all detailed applications in this book can be taken to be based upon sets. A.2 Groups 649 A.2 Groups A group g, ◦ is a set g equipped with an operation ◦ , such that ∀ g1,g2,g3 ∈ g, i) g1 ◦ g2 ∈ g (closure), ii) {g1 ◦ g2}◦g3 = g1 ◦{g2 ◦ g3} (associativity), iii) ∃ e ∈ g such that e ◦ g1 = g1 = g1 ◦ e (identity), and ∃ −1 ∈ ◦ −1 = = −1 ◦ iv) g1 g for each g1 such that g1 g1 e g1 g1 (inverse). Groups (see [69, 213] for further reading) are one of the most widely useful math- ematical structures in Physics [526]. They encode the mathematics of transforma- tions and symmetries. This book often uses g as a shorthand for g, ◦. |g| denotes the order of g, i.e. the number of elements of g. If additionally g1 ◦ g2 = g2 ◦ g1 (commutativity), g is said to be Abelian (after mathematician Niels Abel). H is a subgroup of g, denoted H ≤ g,ifH ⊆ g, and H is closed with respect to the same group operation ◦ that g possesses while containing the identity and all its elements’ inverses. If g, ◦ and K, ∗ are groups, the (direct) product group g × K is defined by having (g, k) as its elements: the Cartesian product of the sets of g and K, with group operation (g1,k1) (g2,k2) = (g1 ◦ g2,k1 ∗ k2). Two groups g, ◦ and K, ∗ are isomorphic, denoted by =∼, if there is a bijection ϕ that preserves the group structure: ϕ(g ◦ k) = ϕ(g) ∗ ϕ(k) ∀ g ∈ g,k∈ K.1 Examples of groups include 1) R, +,2)R/{0}, ×. 3) The cyclic groups Zn, which are Abelian and with the sole relation gn = e, and 4) the permutations of a finite set X form a group, Perm(X). 5) The automorphisms for each s form a group, Aut(s); Perm(X) is the simplest case, corresponding to s = X finite. 6) Lie groups (named after 19th century mathematician Sophus Lie) such as Appendix E’s standard examples and Appendix V’s larger ones. 7) Given a group g, its centre Z(g) is another group: the subgroup of g formed by the elements of g which commute with all the other elements: {c ∈ g | cg = gc ∀ g ∈ g}. Exercise Set IV is dedicated to groups. Groups are of yet further interest through acting on other mathematical or physi- cal objects (sets of points, figures, physical matter...) A group action on a set X is a → : × → map α g X X (often denoted by gg in this book) such that i) {g1 ◦ g2}x = g1 ◦{g2x} (compatibility) and ii) ex = x ∀ x ∈ X (identity). Subcases include left action gx, right action xg, and conjugate action gxg−1. By a natural group action, we just mean one which does not require a choice as to how to relate X and g due to there being one ‘obvious way’ in which it acts. E.g. Perm(X) acts naturally on X,orann × n matrix group acts naturally on the 1This is the well-known group isomorphism subcase of Appendix A.1’s more general notion of isomorphism, and is useful in classifying groups. 650 A Basic Algebra and Discrete Mathematics corresponding n-vectors. An action is faithful if g1 = g2 ⇒ g1x = g2x for some x ∈ X, whereas it is free if this is so for all x ∈ X. So free ⇒ faithful, but not vice versa. Finally, an action is transitive if for every x,y ∈ X there is a g such that gx = y. For g a group acting on a set X, and x ∈ X,thegroup orbit Orb(x) := {gx | g ∈ g}: the set of images of x, and the stabilizer alias isotropy group alias little group Stab(x) := {g ∈ g | gx = x}:thesetofg ∈ g that fix x. := denotes ‘is defined by’. A subgroup N that is invariant under conjugate action—g N g−1 = N for every g ∈ g—is called a normal subgroup of g, denoted by N g. The quotient of one group by another, g/H, only makes sense if H g. In this case, g/H := {gH | g ∈ g} is itself a group: the quotient group . Finally, the semidirect product group g = N H is given by = ∗ ◦ (n1,h1) (n2,h2) (n1 ϕh1 (n2), h1 h2) for N, ∗ g, H, ◦ a subgroup of g and ϕ : H → Aut(N) a group homomor- phism.
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