Notes on the Renormalization Group Chris Ripken Institute for Mathematics, Astrophysics and Particle Physics (IMAPP) Radboud University, Nijmegen, the Netherlands [email protected] version 6.1418, November 4, 2016 Glossary << ê ⋅ ; ⋅ ëF<< Inner product on F 9 < ê ⋅ ; ⋅ ëF< Inner product on F 12 cl.J/ Classical field generated by source J 13 d Spacetime dimension 25 v.f/ Variation of f with respect to v 9 F<< Classical field space 9 F Field space 11 F< Source space 12 f̄(Δ) Kinetic function 35 f̃.s/ Laplace transform of kinetic function 35 Γ Christoffel symbol 33 G Dual map F<< → F< 13 Γk Effective average action 16 Γ Effective action 14 G Metric on finite space 22 G Supermetric on metric fluctuations 33 G*1 Dual map F< → F<< 13 ̄g Background metric 35 g Dynamical metric 33 Inclusion map of F into F<< 13 k Scale parameter 15 M Spacetime manifold 33 Quantum measure on field space 11 NS Number of scalar fields 25 NV Polynomial order of scalar potential 28 $ Gauge parameter in supermetric 33 Chris Ripken, [email protected] 3 Notes on the Renormalization Group Qj [W ] Mellin transform 27 Rk(Δ) Cutoff function 16 R.0/.z/ Cutoff shape function 26 S Action functional 11 ΔSk Cutoff action 16 Σ -algebra on field space 11 ./ Source associated to the classical field 14 Θ(z/ Heaviside Theta-function 27 ̄ ⃗2 Vk. / Dimensionful scalar potential 25 ⃗2 Vk. / Dimensionless scalar potential 27 .n/ vk Taylor coefficients dimensionless scalar potential 28 W Generating functional of connected correlation 10 functions Faddeev-Popov ghost 33 Z Partition function 12 4 November 4, 2016 Acronyms EAA Effective Average Action 16 EH Einstein-Hilbert 35 FRGE Functional Renormalization Group Equation 15 GFP Gaussian fixed point 28 IR infrared 16 NGFP Non-Gaussian fixed point 28 QCD Quantum Chromodynamics 15 QFT Quantum Field Theory 7 RG Renormalization Group 15 RRT Riesz Representation Theorem 9 UV ultraviolet 16 Chris Ripken, [email protected] 5 Notes on the Renormalization Group 6 November 4, 2016 Chapter 2 The renormalization group equation 2.1 Fields and the path integral Indeterminacy and randomness play a fundamental role in physics. For the experimentalist, this may mean that measurements may not be 100~ precise, due to imperfections in measuring apparatuses. However, indeterminacy also plays a role in the shortcoming of our theories about nature. For instance, the sheer size of a physical system may be the cause for us being unable to predict the exact state of a system. This is where statistical physics come into play. Using statistical techniques, we may be able to make predictions on averages of systems, expectation values, and uncertainties thereof. Furthermore, randomness is an intrinsic property of nature. In quantum mechanics, two observables that are not commensurable cannot be measured at the same time with absolute precision. Quantum uncertainty can be described using the same techniques as in statistical physics. This leads to a probalistic formulation of physics. The path integral formalism is especially adapted to the indeterministic nature of quantum mechanics. 2.1.1 Field space In this section, we will review the basics of the path integral formalism. We will try to be as mathematically rigourous as possible; however, well defining a path integral is still an open question in mathematics. The path integral formalism provides a more or less well-behaved description of a so-called quantum field theory. The dynamical objects in such a theory are called fields. We denote the set of fields by F; typically, an element of F is denoted by ' Ë F. In general, we do not assume a topology or vector space structure on F. We do require that F is equipped with a -algebra. This is denoted by Σ. Integration over field space is then defined by requiring the existence of a measure on .F; Σ). The measure gives rise to a volume element d. We assume that the volume element can be written in the following form: *S['] d.'/ = D' e , (2.1.1) where the functional S : F → R is called the action functional. The symbol D' denotes typically the Lebesgue measure on F. Note however that, if F is infinite dimensional, this measure does Chris Ripken, [email protected] 11 Notes on the Renormalization Group not exist. In that case, we interpret this formula as a purely formal expression. In the following, we will not refer to the action functional again. 2.1.2 Path integrals The measure enables us to take integrals over field space F. In particular, we may integrate over the entire field space. This gives rise to the partition function Z, defined by Z = .F/ = d.'/. (2.1.2) ÊF In the case where the partition function is normalized to 1, the measure is a probality measure. If the partition function is not normalized, but finite, we can always obtain a probability measure by defining d¨ = Z*1d. In general, F has no differentiable or linear structure. In order to probe the probability measure, we define a source current to be an element of the dual of F. Definition 2.1.1 (Source space). Let F be a field space with measure . We define source space F< as the space of square integrable functions F → C. The space of sources can be equipped with a Hilbert space structure. An example of an inner product is given by the L2 inner product: êJ;KëF< = d.'/J[']K['], (2.1.3) Ê for J;K Ë F<. Furthermore, we assume that F< is separable. This means that there exists a n countable basis ^d `nËN. A nice property of F< is that it is a Banach space; thus, we have a differentiable structure as discussed in section 1.1.3. A source current J Ë F< modifies the probability distribution of the theory. We define the generating functional of the n-point correlators by Z[J] = d.'/eJ[']. (2.1.4) Ê For A; J Ë F<, we define the expectation value of A in the presence of J by *1 J['] êAëJ = Z[J] d.'/e A[']. (2.1.5) Ê We now define the generating functional W [J] as the logarithm of Z[J]: W [J] = log Z[J]. (2.1.6) Equivalently, W [J] is specified by the relation eW [J] = Z[J]. (2.1.7) In the following, we assume that W is twice continuously differentiable. 12 November 4, 2016 Chapter 2. The renormalization group equation Definition 2.1.2 (Classical field space). Let F< be a Hilbert space. We define classical field space F<< as the continuous dual of F<. In other words, F<< is given by the set of continuous linear functionals F< → C. F<< is again a Hilbert space, with inner product defined by ê, ëF<< = sup ð[J]ð (2.1.8) JËF< for Ë F<<. By the RRT, classical field space is isometrically isomorphic to source space. We will denote the isomorphism F< → F<< by G*1. Explicitly, for J;K Ë F<, we define *1 ̄ . G .J/[K] = J;K F< (2.1.9) Note that the complex conjugation of J ensures that G*1 is linear. The inverse of this map is denoted by G; this is given by the defining relation ( ) [J] = G./;J , (2.1.10) F< for J Ë F<, Ë F<<. As a result, we can rewrite [J] as *1 . [J] = , G .J/ F<< (2.1.11) Field space has a natural inclusion in classical field space; we denote this inclusion map by : F → F<<; .'/[J] = J[']. (2.1.12) The generating functional W [J] gives rise to the so-called classical field. This is a map cl: F< → F<<, given by the equivalent definitions cl.J/[K] = êKëJ = K W [J]. (2.1.13) The equivalence is proven in the following lemma. Lemma 2.1.3. The two definitions in (2.1.13) are equivalent. Proof. Let J;K Ë F<. The functional derivative of W is then given by 1 1 Z[J + "K] K W [J] = lim .W [J + "K]* W [J]/ = lim log . (2.1.14) "→0 " "→0 " Z[J] Expanding Z[J + "K] up to first order in " gives J['] 2 Z[J + "K] = Z[J] + " d.'/K[']e + O " ; (2.1.15) Ê inserting this into the expression for K W [J] gives 1 2 K W [J] = lim log 1 + " K J + O " . (2.1.16) "→0 " ê ë Expanding the logarithm up to first order in " allows us to read off the limit " → 0. This results in K W [J] = êKëJ , (2.1.17) as was to be shown. Chris Ripken, [email protected] 13 Notes on the Renormalization Group Conversely, a classical field may be uniquely associated to a source. If we assume that W is strictly convex, then a classical field Ë F<< uniquely determines a source ./Ë F<. The source ./ is then given by ./ = arg sup ^[J]* W [J]` , (2.1.18) JËF< where arg sup denotes the argument J Ë F< such that the expression attains its supremum. The effective action Γ: F<< → R is now defined by the Legendre transform of W [J]: Γ[] = sup ^[J]* W [J]` < JËF (2.1.19) = [./] * W [./]. The effective action obeys a quantum equation of motion that makes the role of J as a source explicit. Proposition 2.1.4. Let the effective action Γ[] be defined as in (2.1.19). Then for all , Ë F<<, the effective action obeys the relation Γ[] = [./]. (2.1.20) < Proof.