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The Propagator and Diffeomorphisms of an Interacting Field Theory The propagator and diffeomorphisms of an interacting field theory Master thesis submitted to the Institut f¨urPhysik Mathematisch-Naturwissenschaftliche Fakult¨at Humboldt-Universtit¨atzu Berlin by Paul-Hermann Balduf in partial fulfillment of the requirements for the degree Master of Science. Adlershof, July 5, 2018 Referees: Prof. Dr. Dirk Kreimer (supervisor) Dr. Christian Bogner We consider a scalar quantum field theory where the field φ is replaced by a diffeomor- 2 3 phism φ = ρ+a1ρ +a2ρ +:::. The field ρ constitutes a modified quantum field theory defined implicitly by the diffeomorphism coefficients fajgj and the Lagrange density of the underlying field φ. For a generic diffeomorphism, ρ is a non-renormalizable quan- tum field with infinitely many interaction vertices, even if φ itself is a free field. In the case that φ is an interacting field itself, ρ obtains additional vertices proportional to the coupling parameters in the Lagrangian density of φ. We examine the general Lagrangian density of a scalar field with local interactions, 1 1 µ 1 2 2 X λs s L = @µφ∂ φ − m φ − φ ; 2 2 s! s=3 and show that the S-matrix elements of ρ coincide with the ones of φ. This implies the fields φ and ρ are indistinguishable in experiments. In this sense diffeomorphisms form equivalence classes of interacting scalar quantum field theories. On the other hand, n-point functions differ between φ and ρ if their momenta are offshell. Tuning the diffeomorphism coefficients fajgj allows to change the behaviour of the offshell n-point functions of ρ. We use this freedom to eliminate all loop-corrections to the propagator of ρ for a fixed offshell four-momentum p. If φ has φs-type interaction, 1 µ 1 2 2 λ s L = @µφ∂ φ − m φ − φ ; 2 2 s! and all tadpole graphs vanish, then the field formally given by λ ρ(x) = φ(x) − φs−1(x) (s − 1)!(p2 − m2) has a free propagator to all orders in perturbation theory, i.e. ρ(p)ρ(−p) = −ip2 − m2: Contents 1 Introduction5 1.1 Minkowski spacetime.............................5 1.2 Fuss-Catalan numbers............................6 1.2.1 Generating function of A(2; 1)....................7 1.2.2 Generating function of A(3; 1)....................8 1.3 Bell polynomials............................... 10 1.4 Axiomatic quantum field theory....................... 11 1.4.1 Wightman axioms.......................... 11 1.4.2 Wightman distributions....................... 12 1.4.3 Haag`s theorem............................ 13 1.5 Perturbative quantum field theory..................... 14 1.5.1 Scattering theory in the interaction picture............ 14 1.5.2 Feynman graphs........................... 17 1.5.3 Renormalization........................... 18 1.6 Diffeomorphisms............................... 21 1.7 Motivation for this work........................... 21 1.8 Organization of the text........................... 23 2 Free theory 25 2.1 Diffeomorphism of a free scalar theory................... 25 2.1.1 Lagrangian density.......................... 25 2.1.2 Feynman rules............................ 27 2.1.3 Renormalizability........................... 29 2.2 Tree sums................................... 30 2.3 Multiple external offshell edges....................... 33 2.4 Loop amplitudes............................... 37 3 Phi3-theory 39 3.1 Diffeomorphism vertices of Phi3-theory................... 39 3.2 Cancellation of higher interaction vertices................. 41 3.2.1 Four external edges.......................... 41 3.2.2 General structure........................... 42 3.2.3 Formal definition of Sn........................ 43 3.2.4 Vanishing of Sn............................ 46 3.3 Tree sums with interaction vertices..................... 47 3.3.1 General form............................. 47 3.3.2 Example: Tree sum with four external edges........... 48 3.3.3 Example: Tree sum with five external edges............ 51 3.4 Explicit cancellation of the 2-point function for one loop......... 57 3 3.5 Explicit calculation for two loops...................... 60 3.5.1 Topology A.............................. 61 3.5.2 Topology B.............................. 62 3.5.3 Topology C.............................. 64 3.5.4 Topology D.............................. 68 3.5.5 Total result.............................. 72 3.6 All-orders cancellation of corrections.................... 72 3.6.1 Decomposition of non-vanishing graphs.............. 73 3.6.2 Factorization of tree sums...................... 74 3.6.3 Parameters of the adiabatic diffeomorphism............ 79 4 Higher order interaction 83 4.1 Feynman rules................................ 83 4.2 Tree sums and higher interaction vertices................. 85 4.3 Cancellation of corrections to the 2-point function............ 87 4.4 Multiple interaction monomials....................... 94 4.4.1 Feynman rules............................ 94 4.4.2 Tree sums............................... 95 5 Properties of the adiabatic diffeomorphism 97 5.1 Rho as a free field.............................. 97 5.1.1 Rho in momentum space....................... 97 5.1.2 Classical fields............................ 98 5.1.3 Cancellation of higher correlation functions............ 99 5.2 An identity for PhiS-theory......................... 99 5.3 Conclusion and outlook........................... 101 Bibliography 103 A Lemmas 107 A.1 Sums of partitions of momenta....................... 107 A.2 Compatibility of symmetry factors..................... 109 A.3 Combinatoric Lemmas............................ 111 B Umgangssprachliche Erl¨auterungder Arbeit 123 C Danksagung 127 D Selbstst¨andigkeitserkl¨arung 129 4 1 Introduction 1.1 Minkowski spacetime Four-dimensional Minkowski spacetime M with speed of light c = 1 is used throughout. Four-vectors are underlined and three-vectors are bold such that the four-momentum is 0 1 2 3 p = p ; p ; p ; p = (E; px; py; pz) = (E; p): The metric is flat and has negative sign, 01 0 0 0 1 B0 −1 0 0 C η = B C: @0 0 −1 0 A 0 0 0 −1 Four-dimensional spacetime indices are greek and summation is implicit, 2 µ µ ν 2 2 p = pµp = ηµνp p = E − p : (1.1) Other indices are not summed over unless explicitly denoted. The spacetime derivative @ is @µ = @xµ , especially the second derivative µ 2 @µ@ = @0@0 − @1@1 − @2@2 − @3@3 = @t − 4 = : Physical particles (i.e. \really existent" as opposed to virtual particles used in interme- diate steps in computations) obey the relativistic energy-momentum-relation E2 = p2 + m2: (1.2) By eq. (1.1) the square of the four-momentum of a physical particle is its mass squared. In four-dimensional momentum space, eq. (1.2) forms a surface called the mass shell. Hence if a particle fulfills eq. (1.2) it is called onshell, otherwise offshell. Definition 1. For a four-momentum p assigned to a particle with mass m, the offshell parameter is defined as 2 2 xp = p − m : This parameter is zero if the momentum p belongs to a physical particle with mass m. The notation here follows slightly nontrivial rules: If the index is a letter like xp, this letter indicates the name of the corresponding four-momentum. If the index is a number, it is the running number of some (usually canonically) numbered momenta like x p2 − m2: 1 = 1 (1.3) 5 If sums and differences appear in the index, these indicate the corresponding operations with momenta, not indices themselves. This is, 2 x p p − m2; 1+2 = 1 + 2 (1.4) but generally x1+2 =6 x3. Note that xp only depends on the magnitude of the momen- tum, for any four-momentum p it is xp = x−p. The equation xp = xq does not imply p = q. 1.2 Fuss-Catalan numbers Definition 2. For fixed a 2 N0 and b 2 N the Fuss-Catalan numbers are defined as b ma + b (ma + b − 1)! Am(a; b) = = b : ma + b m (ma + b − m)!m! Some useful values are A0(a; b) = 1 Am(1; 1) = 1 Am(1; 2) = m b A (0; b) = : m m Example 1: Fuss-Catalan numbers used in subsequent examples The (ordinary) Catalan numbers fCmgm2N = f1; 1; 2; 5; 14; 42; 132; 429;:::g are 1 2m (2m)! Am(2; 1) = Cm = = : (1.5) m + 1 m m!(m + 1)! In the context of φ4-theory we encounter the choice a = 3; b = 1, 1 3m fAm(3; 1)gm2N = = f1; 1; 3; 12; 55; 273; 1428; 7752 :::g: 2m + 1 m m2N Countless interpretations of the Fuss-Catalan numbers are known, see for example their entries A000108 (Am(2; 1)) or A001764 (Am(3; 1)) in the OEIS [Slo18]. Noteworthy, the Catalan number Cm is the number of different planar trees with n + 1 leaves built of 3-valent vertices. Similarly, Am(s; 1) counts the number of such trees if they are made of s-valent vertices. The generating function of the Fuss-Catalan numbers is defined by 1 X m Ca;b(t) = t Am(a; b): (1.6) m=0 6 Lemma 1.1. Iff for q 2 R a function R(t) obeys R(t) − 1 = tRq(t) then for b 2 R 1 X b mq + b Rb(t) = tm : mq + b m m=0 Proof. This is [MSV06, theorem 2.1] with the replacement R(t) = 1 + w. By lemma 1.1 the generating function eq. (1.6) of the Fuss-Catalan-numbers is defined equivalently via its functional equation a b b Ca;b(t) = tCa;b(t) + 1 : (1.7) From eq. (1.6) and A0(a; b) = 1 the boundary condition is Ca;b(0) = 1: (1.8) For b = 1 and a = 2 or a = 3 the functional equation can be solved explicitly. 1.2.1 Generating function of A(2; 1) Set a = 2; b = 1, this gives the generating function of ordinary Catalan numbers defined implicitly via eq. (1.7) 2 C2;1(t) ≡ C(t) = tC (t) + 1: (1.9) Solving the quadratic equation, there are (for any given t) two solutions p 1 ± 1 − 4t C (t) = : (1.10) 1=2 2t As can be seen in fig. 1.1 or computed explicitly, only one of them fulfills the boundary condition eq.
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