The Renormalization Group
Notes on the Renormalization Group
Chris Ripken
Institute for Mathematics, Astrophysics and Particle Physics (IMAPP)
Radboud University, Nijmegen, the Netherlands
version 6.1418, November 4, 2016
Glossary
∗∗ ⟨ ⋅ , ⋅ ⟩F∗∗ Inner product on F 9 ∗ ⟨ ⋅ , ⋅ ⟩F∗ 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 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
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
k(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 ' ∈ 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(') = ' e , (2.1.1) where the functional S ∶ F → ℝ is called the action functional. The symbol ' 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:
⟨J,K⟩F∗ = d(')J[']K['], (2.1.3) Ê for J,K ∈ F∗. Furthermore, we assume that F∗ is separable. This means that there exists a n countable basis {d }n∈ℕ. 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 ∈ 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 ∈ F∗, we define the expectation value of A in the presence of J by
−1 J['] ⟨A⟩J = 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.
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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
⟨ , ⟩F∗∗ = sup ð [J]ð (2.1.8) J∈F∗ for ∈ 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 ∈ 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 ∈ F∗, ∈ 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] = ⟨K⟩J = 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 ∈ 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 + " ; (2.1.15) Ê inserting this into the expression for K W [J] gives
1 2 K W [J] = lim log 1 + " K J + " . (2.1.16) "→0 " ⟨ ⟩ Expanding the logarithm up to first order in " allows us to read off the limit " → 0. This results in K W [J] = ⟨K⟩J , (2.1.17) as was to be shown.
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Conversely, a classical field may be uniquely associated to a source. If we assume that W is strictly convex, then a classical field ∈ F∗∗ uniquely determines a source ( ) ∈ F∗. The source ( ) is then given by
( ) = arg sup { [J] − W [J]} , (2.1.18) J∈F∗ where arg sup denotes the argument J ∈ 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]} ∗ J∈F (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 , ∈ F∗∗, the effective action obeys the relation
Γ[ ] = [( )]. (2.1.20)
∗ Proof. First, we note that K ( − W )[( )] = 0 for any K ∈ F , since the functional − W attains its supremum at ( ). Furthermore, convexity and differentiability of W ensure that ( ) is unique and is given by the maximizing argument. Thus, we have
K W [( )] = K [( )] = [K], (2.1.21) since is linear. Then, by the chain rule, we have W ◦[ ] = W [( )] = ( ) . (2.1.22) ( ) For the other term in the effective action, we have 1 ( ⋅ ( ⋅ )) [ ] = lim (( + " )[( + " )] − [( )]) "→0 " 1 = lim ( [( + " )] + " [( + " )] − [( )]) "→0 " 1 (2.1.23) = lim [( )] + " [( )] + " [( )] "→0 " − [( )] + (")
= [( )] + [( )]. Putting the terms together gives Γ[ ] = [( )], (2.1.24) as was to be proved.
This equation takes on a particularly remarkable form if we insert a field ' ∈ F and a source J ∈ F∗. We then obtain an equation of motion on field space that reads
(')Γ[cl(J)] = J[']. (2.1.25)
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2.2 The FRGE
We will now introduce the functional renormalization group equation. First, we discuss the reasons for introducing renormalization; from this, the concept of the renormalization group flow follows naturally. The Functional Renormalization Group Equation (FRGE) is the integro-differential equation that governs this flow.
2.2.1 Ordering of field space
In the last section, we constructed the path integral as an integration over field space F. This required the existence of a measure . However, such a measure poses several problems. First of all, although we assume that such a measure exists, it is in general extremely difficult to construct a nontrivial measure . For instance, in General Relativity, it is not clear how to construct a measure on the space of metrics of Lorentz signature on a general curved manifold. Secondly, the measure poses a philosophical question. Since the path integral is over the entire field space F, performing the integration requires knowledge about the entire field space. A physical experiment, however, may not depend on the whole of field space. For instance, a scattering experiment is sensitive to details up to the length scale, or the corresponding energy scale, at which the experiment takes place. This does not mean that the result of such an experiment is independent of higher energy scales. Rather, it means that the separate contributions of higher energy scales cannot be resolved. We say that these contributions are integrated out: the precise outcome of the experiment is determined by the average of the high-energy modes. An example of such a theory is Quantum Chromodynamics (QCD), where fundamental quarks and gluons are integrated out to form protons, neutrons and mesons. The occurence of such a typical energy scale in a model is usually a clue that a theory is only an effective theory. A theory may break down at this energy scale. This is an undesirable property of a fundamental theory describing nature; thus, the scaling of a theory be used to probe the fundamentality of the model. The introduction of an energy scale also offers technical advantages. One of these is that it may be possible to rephrase certain questions about the integration measure in a differental language. The most prominent example is the FRGE, which we will encounter in a moment. Furthermore, defining theories at different energy scales may provide a tool to calculate for instance correlation functions.
Filtration of field space
The concept of scaling in a field theoretical sense is intricately linked to the Wilsonian approach of renormalization []. In a lattice model, fields are integrated out by block-spin transformations. In a path integral approach, this is implemented by a filtration of field space. Let Uk be a filtration of F, that is, for every k ∈ ℝ+ a subset Uk ∈ Σ of F, such that k∈ℝ+ ¨ Uk¨ ⊆ Uk if k < k and U0 = F. The Wilsonian Renormalization Group (RG) is then constructed by modifying each path integral by integrating over Uk instead of over F. As an example, the block-spin transformation can then be described by choosing for Uk the space of Fourier-transformable functions on a lattice whose Fourier modes are cut off at a certain scale. In this case, the intepretation of the RG scale as an energy or length scale is rather clear, since the cutoff scale corresponds to the wavelength of the functions on the lattice.
Chris Ripken, [email protected] 15 Notes on the Renormalization Group
Cutoff renormalization The cutoff action approach generalizes the filtration of field space to cutoffs which are not sharp. This can be made clear by reformulating the filtration approach in terms of characteristic functions on the filtration. The domain of the path integral remains the entire field space F; however, the integration measure is modified. This can be generalized easily by smoothing out the characteristic functions. The energy scale introduces an ordering on field space. The ordering is parameterized by the energy scale. In general, an ordering is not necessarily to be interpreted as an energy scale. However, we will refer to a parameterization of an ordering as the scale parameter k ∈ [0, ∞). If p denotes the scale of a certain region of field space, we call the regime p ≪ k the infrared (IR) regime and p ≫ k the ultraviolet ultraviolet (UV) regime. The ordering of field space is implemented by suppressing the contributions of IR modes to the path integral. This is done by introducing the cutoff action. The cutoff action is a functional on F∗∗, given by 1 ΔS [ ] = , (Δ) . (2.2.1) k 2⟨ k ⟩F∗∗ In this action, the spectrum of the positive, selfadjoint operator Δ provides the ordering of the fields. As we shall see in subsequent chapters, an often-used choice for Δ is the covariant Laplacian Δ = −∇ ∇. 2 The cutoff function k(p ) is a monotonically increasing function in k and is monotonically 2 2 2 2 decreasing in p . In order to integrate out UV modes, we require that k(p ∕k ) → 0 as p → ∞. 2 2 2 On the other hand, we require that k(p ∕k ) → k as k → 0. The reason to introduce these conditions on the cutoff action is two-fold. First of all, these conditions ensure that k-modified generating functional W k, which we will define in a moment, remains strictly convex. This allows us to define an invertible expectation value map, which gives a well-defined effective theory at each scale k. Secondly, the cutoff regularizes the Hessian of W k. In fact, we assume that this operator is trace-class. We now use the cutoff action to regulate the path integral. We let the cutoff function regulate the path integral; we define the k-dependent path integral measure by
−ΔSk[(')] dk(') = d(')e . (2.2.2) Using this measure, we define the k-dependent partition function by
W k[J] +J['] Zk[J] =e = dk(')e ; (2.2.3) Ê using these functionals, we define k-dependent expectation values. The classical field map is also modified; we write clk(J)[K] = K W k[J]. (2.2.4) Note that this is still equal to taking the expectation value of K with respect to the source J since ΔSk['] is independent of J. We can also define the source associated to a classical field:
k( ) = arg sup [J] − W k[J] . (2.2.5) J∈F∗
We now define the Effective Average Action (EAA) by the Legendre transform of W k minus the cutoff action:
Γk[ ] = sup [J] − W k[J] − ΔSk[ ] ∗ J∈F (2.2.6) = [k( )] − W k[k( )] − ΔSk[ ].
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2.2.2 Derivation of the FRGE The flow of the EAA is governed by the FRGE. This integro-differential equation depends only on the EAA and the cutoff function k. One of the nice things about the FRGE is that it is exact; it does not rely on perturbation theory, or on any truncation of the action. This enables us to study non-perturbative properties of the quantum theory. The rest of this section is devoted to the statement of the FRGE and its proof.
Theorem 2.2.1 (Functional Renormalization Group Equation). Let W k be such that its Hessian is trace-class. Then the EAA obeys the following equation:
4 −1 5 dΓk[ ] 1 dk = Tr Hess£ Γ [ ] + (Δ) , (2.2.7) dt 2 k k dt
£ ∗∗ ∗∗ where t = log k. The operator Hess Γk[ ] ∶ F → F denotes the Hessian of the EAA; the trace Tr in this equation is then the trace over F∗∗.
The structure of the FRGE reveals the two-fold role of the cutoff function. In the path integral, it explicitly suppresses IR modes; this can be seen in the dressed propagator −1 £ Hess Γk[ ] + k(Δ) . The cutoff acts here as a mass term for small momenta, thereby regularizing the IR pole in the propagator. Secondly, the cutoff also regulates the UV behaviour 2 2 of the theory. Since the derivative of k(p ) is sharply peaked at k , the UV modes in the trace are suppressed as well. Before we can prove the theorem, we first need some lemmas.
∗ Lemma 2.2.2 (Hessian of W k). Let J,K,L ∈ F . The Hessian of the generating functional W k[J] is given by ( ) £ ̄ ̄ K, Hess W k[J] L = K̄ LW k[J] = KL − K L k,J . (2.2.8) F∗ k,J k,J ⟨ ⟩
Proof. The proof consists of calculating the derivative of LW k. We have
1 K̄ LW k[J] = lim L k,J+"K̄ − L k,J . (2.2.9) "→0 " ⟨ ⟩ ⟨ ⟩ Writing out the path integrals, we obtain
H (J+"K̄ )['] J['] I 1 ∫ dk(')e L['] ∫ dk(')e L['] K̄ LW k[J] = lim − . (2.2.10) "→0 " (J+"K̄ )['] J['] ∫ dk(')e ∫ dk(')e
Expanding the first denominator up to second order in ", we obtain 0 1 (J+"K̄ )['] K̄ LW k[J] = lim dk(')e L[']× "→0 " Ê T U ' J[']K̄ ' 1 ∫ d k( )e [ ] 2 − " + " (2.2.11) d (')eJ['] J[']2 ∫ k ∫ dk(')e J['] 1 ∫ dk(')e L['] − . J['] ∫ dk(')e
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This gives for the expansion of the total expression in " 0 J['] ̄ 1 ∫ dk(')e K[']L['] K̄ LW k[J] = lim " " J['] →0 " ∫ dk(')e ' J[']L ' ' J[']K̄ ' 1 ∫ d k( )e [ ] ∫ d k( )e [ ] 2 (2.2.12) − " + " J[']2 ∫ dk(')e ̄ ̄ = KL k,J − K k,J ⟨L⟩k,J .
Thus, the Hessian of W k is given by ( ) £ ̄ ̄ K, Hess W k[J] L = KL − K L k,J . (2.2.13) F∗ k,J k,J ⟨ ⟩
£ Furthermore, we express Hess W k in terms of the Hessian of the EAA. £ Lemma 2.2.3 (Inverse of Γk). The Hessian of Γk[ ] is related to Hess W k by −1 £ £ −1 (2.2.14) Hess W k[k( )] = G◦ Hess Γk[ ] + k(Δ) ◦G .
Proof. We start by relating the Hessian of W k to the derivative of the map clk. In fact, we have , J cl (K) [L] = G J clk (K) ,L F∗ = ⟨G◦cl (K) J,L⟩F∗ (2.2.15) from which we infer £ Hess W k[K] = G◦kcl(K). (2.2.16)
Since the map k and clk are inverses, we can write by the chain rule ◦cl (K) = cl (K) J, (2.2.17) J k k J clk(K) k k ≡ from which we derive k clk(K) ◦clk(K) = Id, (2.2.18) or equivalently −1 clk(K) = k clk(K) . (2.2.19)
Inserting this into the expression for the Hessian of W k gives £ −1 −1 Hess W k[K] = G ◦ k clk(K) . (2.2.20) ̃ We now express k to the Γk. By the classical equation of motion (2.1.20), we have ( ) £ ̃ ̃ −1 , Hess Γ [ ] = Γ [ ] = ( ) = , G ◦ ( ) ∗∗ k ∗∗ k k k F F (2.2.21) −1 , = , G ◦k( ) F∗∗ where we have used that the Hessian operator is symmetric. This gives the relation £ ̃ £ −1 Hess Γk[ ] = Hess Γk[ ] + k(Δ) = G ◦k( ). (2.2.22)
Inserting this into (2.2.20) and using the fact that clk◦k = Id then gives the desired result.
18 November 4, 2016 Chapter 2. The renormalization group equation
We can now prove the theorem.
Proof of theorem 2.2.1. The proof consists of two parts. First, we derive an expression for )tΓk[ ] in terms of inner products on F∗∗. We then use the lemmas to show that the right hand side of the FRGE is actually equal to this expression. We start by rewriting the left hand side of the FRGE. We have 1 ) Γ [ ] = ) Γ̃ [ ] − ) ΔS [ ] = [) ( )] − ) W ◦ [ ] − , ) (Δ) ∗∗ . (2.2.23) t k t k t k t k t k k 2⟨ tk ⟩F The second term in this expression can be written by the chain rule as ) W ◦ [ ] = ) W [ ( )] + W [ ( )] = ) W [ ( )] + [) ( )], (2.2.24) t k k t k k )tk( ) k k t k k t k from which we derive 1 ) Γ [ ] = −) W [ ( )] − , ) (Δ) . (2.2.25) t k t k k 2⟨ tk ⟩F∗∗
For the t-derivative of W k, we have the following expansion: 0 1 H −ΔS [(')]+J['] I 1 ∫ de k+" ) W [J] = ) log d(')e−ΔSk[(')]+J['] = lim log k k k −ΔS [(')]+J['] ÊF "→0 " ∫ de k 1 2 (2.2.26) = lim log 1 − " )kΔSk◦ k,J + " = − )kΔSk◦ k,J "→0 " ⟨ ⟩ ⟨ ⟩ 1 ( ) = − ( ⋅ ),)kk(Δ)( ⋅ ) F∗∗ . 2 ⟨ ⟩ k,J This gives the expression 1 ( ) 1 )tΓk[ ] = ⟨( ⋅ ),)tk(Δ)( ⋅ )⟩F∗∗ − ⟨ , )tk(Δ) ⟩F∗∗ 2 k, ( ) 2 k (2.2.27) 1 = ( ⋅ ),) (Δ)( ⋅ ) ∗∗ − , ) (Δ) ∗∗ . 2 ⟨ tk ⟩F ⟨ tk ⟩F This completes the first stage of the proof. ∗ ∗∗ The next step consists of using the Hilbert space structure of F and F . Let {ej}j∈ℕ be an ∗∗ ∑ orthonormal basis for F . The operator j ðej⟩⟨ejð then equals the identity operator, where we have used the conventional notation of writing ðej⟩⟨ejð for the projector onto the subspace spanned by ej. The left hand side of the FRGE can then be written as 1 É ) , e e ) , e e ) tΓk[ ] = ( ⋅ ) ð j⟩⟨ jð tk(Δ) ( ⋅ ) F∗∗ − ð j⟩⟨ jð tk(Δ) F∗∗ 2 j 1 É , e e ,) , e e ,) = ( ⋅ ) j F∗∗ j tk(Δ) ( ⋅ ) F∗∗ − j F∗∗ j tk(Δ) F∗∗ (2.2.28) 2 j 1 É = G ej G )t k(Δ)ej − G ej G ej k(Δ) . k,k( ) k,k( ) k,k( ) 2 j By lemma 2.2.2, this can be rewritten as 1 É ( ) ) Γ [ ] = G e , Hess£ W [ ( )] G ) (Δ)e t k j k k tk j ∗ 2 j F (2.2.29) 1 É ( ) = e , G−1◦Hess£ W [ ( )] ◦G ) (Δ)e . j k k tk j ∗∗ 2 j F
Chris Ripken, [email protected] 19 Notes on the Renormalization Group
By lemma 2.2.3, this can be written in terms of Γk:
@ −1 A 1 É £ )tΓk[ ] = ej, Hess Γk[ ] + k(Δ) )tk(Δ)ej , (2.2.30) 2 j F∗∗ in which we recognize the trace over F∗∗: 4 5 1 −1 ) Γ [ ] = Tr ∗∗ Hess£ Γ [ ] + (Δ) ) (Δ) , (2.2.31) t k 2 F k k tk which completes the proof.
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