Quantum Field Theory II University of Cambridge Part III Mathematical Tripos

David Skinner Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA United Kingdom

[email protected] http://www.damtp.cam.ac.uk/people/dbs26/

Abstract: These are the lecture notes for the Advanced Quantum Field Theory course given to students taking Part III Maths in Cambridge during Lent Term of 2015. The main aim is to introduce the Renormalization Group and Eﬀective Field Theories from a path integral perspective. Contents

1 Preliminaries iii 1.1 Books & Other Resources iii

2 QFT in zero dimensions 1 2.1 The partition function 1 2.2 Correlation functions 2 2.3 The Schwinger–Dyson equations 3 2.4 Perturbation theory 4 2.4.1 Graph combinatorics 5 2.5 Supersymmetry and localization 8 2.6 Eﬀective theories: a toy model 11

3 QFT in one dimension (= QM) 16 3.1 Quantum Mechanics 17 3.1.1 The partition function 19 3.1.2 Operators and correlation functions 20 3.2 The continuum limit 22 3.2.1 The path integral measure 23 3.2.2 Discretization and non–commutativity 24 3.2.3 Non–trivial measures? 26 3.3 Locality and Eﬀective Quantum Mechanics 27 3.4 The worldline approach to perturbative QFT 30

4 The Renormalization Group 34 4.1 Running couplings 34 4.1.1 Wavefunction renormalization and anomalous dimensions 36 4.2 Integrating out degrees of freedom 38 4.2.1 Polchinski’s equation 41 4.3 Renormalization group ﬂow 41 4.4 Critical phenomena 45 4.5 The local potential approximation 46 4.5.1 The Gaussian critical point 49 4.5.2 The Wilson–Fisher critical point 50

– i – Acknowledgments

Nothing in these lecture notes is original. In particular, my treatment is heavily inﬂuenced by several of the textbooks listed below, especially Vafa et al. and the excellent lecture notes of Neitzke in the early stages, then Schwartz and Weinberg’s textbooks and Hollowood’s lecture notes later in the course. I am supported by the European Union under an FP7 Marie Curie Career Integration Grant.

– ii – 1 Preliminaries

This course is the second course on Quantum Field Theory offered in Part III of the Maths Tripos, so I’ll feel free to assume you’ve already taken the first course in Michaelmas Term (or else an equivalent course elsewhere). You will also find it helpful to know about groups and representation theory, say at the level of the Symmetries, Fields and Particles course last term. There may be some overlap between this course and certain other Part III courses this term. In particular, I’d expect the material here to complement the courses on The Standard Model and on Applications of Differential Geometry to Physics very well. In turn, I’d also expect this course to be useful for courses on Supersymmetry and String Theory.

1.1 Books & Other Resources There are many (too many!) textbooks and reference books available on Quantum Field Theory. Different ones emphasize different aspects of the theory, or applications to different branches of physics or mathematics – indeed, QFT is such a huge subject nowadays that it is probably impossible for a single textbook to give an encyclopedic treatment (and absolutely impossible for a course of 24 lectures to do so). Here are some of the ones I’ve found useful while preparing these notes; you might prefer different ones to me. These lists are alphabetical by author. This first list contains the main books for the course; you will certainly want to consult (at least) one of Peskin & Schroeder, or Srednicki, or Schwartz repeatedly during the course. They will also be very helpful for people taking the Standard Model course.

Peskin, M. and Schroeder, D., An Introduction to Quantum Field Theory, • Addison–Wesley (1996). An excellent QFT textbook, containing extensive discussions of both gauge theories and renormalization. Many examples worked through in detail, with a particular emphasis on applications to particle physics.

Schwartz, M., Quantum Field Theory and the Standard Model, CUP (2014). • The new kid on the block, published just last year after being honed during the author’s lecture courses at Harvard. I really like this book – it strikes an excellent balance between formalism and applications (mostly to high energy physics), with fresh and clear explanations throughout.

Srednicki, M., Quantum Field Theory, CUP (2007). • This is also an excellent, very clearly written and very pedagogical textbook. Along with Peskin & Schroeder and Schwartz, it is perhaps the single most appropriate book for the course, although none of these books emphasize the more geometric aspects of QFT.

Zee, A., Quantum Field Theory in a Nutshell,2nd edition, PUP (2010). • An great book if you want to keep the big picture of what QFT is all about ﬁrmly

– iii – in sight. QFT is notorious for containing many technical details, and its easy to get lost. This book will put you joyfully back on track and remind you why you wanted to learn the subject in the ﬁrst place. It’s not the best place to learn how to do speciﬁc calculations, but that’s not the point.

There are also a large number of books that are more specialized. Many of these are rather advanced, so I do not recommend you use them as a primary text. However, you may well wish to dip into them occasionally to get a deeper perspective on topics you particularly enjoy. This list is particularly biased towards my (often geometric) interests:

Banks, T. Modern Quantum Field Theory: A Concise Introduction, CUP (2008). • I particularly enjoyed its discussion of the renormalization group and eﬀective ﬁeld theories. As it says, this book is probably too concise to be a main text.

Cardy, J., Scaling and Renormalization in Statistical Physics, CUP (1996). • A wonderful treatment of the Renormalization Group in the context in which it was ﬁrst developed: calculating critical exponents for phase transitions in statistical systems. The presentation is extremely clear, and this book should help to balance the ‘high energy’ perspective of many of the other textbooks.

Coleman, S., Aspects of Symmetry, CUP (1988). • Legendary lectures from one of the most insightful masters of QFT. Contains much material that is beyond the scope of this course, but so engagingly written that I couldn’t resist including it here!

Costello, K., Renormalization and Effective Field Theory, AMS (2011). • A pure mathematician’s view of QFT. The main aim of this book is to give a rigorous definition of (perturbative) QFT via path integrals and Wilsonian effective field theory. Another major achievement is to implement this for gauge theories by combining BV quantization with the ERG. Repays the hard work you’ll need to read it – for serious mathematicians only.

Deligne, P., et al., Quantum Fields and Strings: A Course for Mathematicians vols. • 1 & 2, AMS (1999). Aimed at professional mathematicians wanting an introduction to QFT. They thus require considerable mathematical maturity to read, but most certainly repay the eﬀort. For this course, I particularly recommend the lectures of Deligne & Freed on Classical Field Theory (vol. 1), Gross on the Renormalization Group (vol. 1), and especially Witten on Dynamics of QFT (vol. 2).

Nair, V.P., Quantum Field Theory: A Modern Perspective, Springer (2005). • Contains excellent discussions of anomalies, the configuration space of field theories, ambiguities in quantization and QFT at finite temperature.

Polyakov, A., Gauge Fields and Strings, Harwood Academic (1987). • A very original and often very deep perspective on QFT. Several of the most impor-

– iv – tant developments in theoretical physics over the past couple of decades have been (indirectly) inspired by ideas in this book.

Schweber, S., QED and the Men Who Made It: Dyson, Feynman, Schwinger and • Tomonaga, Princeton (1994). Not a textbook, but a tale of the times in which QFT was born, and the people who made it happen. It doesn’t aim to dazzle you with how very great these heroes were1, but rather shows you how puzzled they were, how human their misunderstandings, and how tenaciously they had to ﬁght to make progress. Inspirational stuﬀ.

Vafa, C., and Zaslow, E., (eds.), Mirror Symmetry, AMS (2003). • A huge book comprising chapters written by diﬀerent mathematicians and physicists with the aim of understanding Mirror Symmetry in the context of string theory. Chapters 8 – 11 give an introduction to QFT in low dimensions from a perspective close to the one we will start with in this course. The following chapters could well be useful if you’re taking the String Theory Part III course.

Weinberg, S., The Quantum Theory of Fields, vols. 1 & 2, CUP (1996). • Weinberg’s thesis is that QFT is the inevitable consequence of marrying Quantum Mechanics, Relativity and the Cluster Decomposition Principle (that distant experiments yield uncorrelated results). Penetrating insight into everything it covers and packed with many detailed examples. The perspective is always deep, but it requires strong concentration to follow a story that sometimes plays out over several chapters. Particles play a primary role, with ﬁelds coming later; for me, this is backwards.

Zinn–Justin, J. Quantum Field Theory and Critical Phenomena, • 4th edition, OUP (2002). Contains a very insightful discussion of the Renormalization Group and also a lot of information on Gauge Theories. Most of its examples are drawn from either Statistical or Condensed Matter Physics.

Textbooks are expensive. Fortunately, there are lots of excellent resources available freely online. I like these:

Dijkgraaf, R., Les Houches Lectures on Fields, Strings and Duality, • http://arXiv.org/pdf/hep-th/9703136.pdf An modern perspective on what QFT is all about, and its relation to string theory. For the most part, the emphasis is on more mathematical topics (e.g. TFT, dualities) than we will cover in the lectures, but the ﬁrst few sections are good for orientation.

Hollowood, T., Six Lectures on QFT, RG and SUSY, • http://arxiv.org/pdf/0909.0859v1.pdf An excellent mini–series of lectures on QFT, given at a summer school aimed at end– of–ﬁrst–year graduate students from around the UK. They put renormalization and

1I should say ‘are’; Freeman Dyson still works at the IAS almost every day.

– v – Wilsonian Effective Theories centre stage. While the final two lectures on SUSY go beyond this course, I found the first three very helpful when preparing the current notes.

Neitzke, A., Applications of Quantum Field Theory to Geometry, • https://www.ma.utexas.edu/users/neitzke/teaching/392C-applied-qft/ Lectures aimed at introducing mathematicians to Quantum Field Theory techniques that are used in computing Seiberg–Witten invariants (and a much wider range of related “Theories of class ”). I very much like the perspective of these lectures, and S we’ll follow a similar path for at least the ﬁrst part of the course.

Osborn, H., Advanced Quantum Field Theory, • http://www.damtp.cam.ac.uk/user/ho/Notes.pdf The lecture notes for a previous incarnation of this course, delivered by Prof. Hugh Osborn. They cover similar material to the current ones, but from a rather diﬀerent perspective. If you don’t like the way I’m doing things, or for extra practice, take a look here!

Polchinski, J., Renormalization and Eﬀective Lagrangians, • http://www.sciencedirect.com/science/article/pii/0550321384902876

Polchinksi, J., Dualities of Fields and Strings, • http://arxiv.org/abs/1412.5704 The ﬁrst paper gives a very clear description of the ‘exact renormalization group’ and its application to scalar ﬁeld theory. The second is a recent survey of the idea of ‘duality’ in QFT and beyond. We’ll explore this if we get time.

Segal, G., Quantum Field Theory lectures, • YouTube lectures Recorded lectures aiming at an axiomatization of QFT by one of the deepest thinkers around. I particularly recommend the lectures “What is Quantum Field Theory?” from Austin, TX, and “Three Roles of Quantum Field Theory” from Bonn (though the blackboards are atrocious!).

Tong, D., Quantum Field Theory, • http://www.damtp.cam.ac.uk/user/tong/qft.html The lecture notes for a previous incarnation of the Michaelmas QFT course in Part III. If you feel you’re missing some background from last term, this is an excellent place to look. There are also some video lectures from when the course was given at Perimeter Institute.

Weinberg, S., What Is Quantum Field Theory, and What Did We Think It Is?, • http://arXiv.org/pdf/hep-th/9702027.pdf

Weinberg, S., Eﬀective Field Theory, Past and Future, • http://arXiv.org/pdf/0908.1964.pdf

– vi – These two papers provide a fascinating account of the origins of eﬀective ﬁeld theories in current algebras for soft pion physics, and how the Wilsonian picture of Renormalization gradually changed our whole perspective of what QFT is about.

Wilson, K., and Kogut, J. The Renormalization Group and the !-Expansion, • Phys. Rep. 12 2 (1974), http://www.sciencedirect.com/science/article/pii/0370157374900234 One of the ﬁrst, and still one of the best, introductions to the renormalization group as it is understood today. Written by somone who changed the way we think about QFT. Contains lots of examples from both statistical physics and ﬁeld theory.

That’s a huge list, and only a real expert in QFT would have mastered everything on it. I provide it here so you can pick and choose to go into more depth on the topics you find most interesting, and in the hope that you can fill in any background you find you are missing.

– vii – 2 QFT in zero dimensions

Quantum Field Theory is, to begin with, exactly what it says it is: the quantum version of a field theory. But this simple statement hardly does justice to what is the most profound description of Nature we currently possess. As well as being the basic theoretical framework for describing elementary particles and their interactions (excluding gravity), QFT also plays a major role in areas of physics and mathematics as diverse as string theory, condensed matter physics, topology, geometry, combinatorics, astrophysics and cosmology. It’s also extremely closely related to statistical field theory, probability and from there even to (quasi–)stochastic systems such as finance. This all–encompassing remit means that QFT is a very powerful tool and you can do yourself damage by trying to operate it without due care. So that we don’t get lost in technical minutiæ of particular calculations in specific theories, I want to begin slowly by studying the simplest possible QFT: that of a single, real–valued field in zero dimensional space–time. That’s of course a very drastic simplification, and much of the richness of QFT will be absent here. However, you shouldn’t sneer. We’ll see that even this simple case contains baby versions of ideas we’ll study more generally later in the course, and it will provide us with a safe playground in which to check we understand what’s going on. Furthermore, it has been seriously conjectured that full, non-perturbative string theory is itself a zero–dimensional QFT (though admittedly with infinitely many fields). I expect that many of the ideas in this chapter will be things you’ve met before either in last term’s QFT course or sometimes even earlier, but my perspective here may well be somewhat different.

2.1 The partition function If our space–time M is zero–dimensional and connected, then it must be just a single point. In the simplest case, a ‘field’ on M is then just a map φ : pt R, or in other words { }→ just a real variable. Notice that in zero dimensions, the Lorentz group is trivial, so in particular there’s no notion of the fields’ spin. Even more obviously, there are no space– time directions along which we could differentiate our ‘field’, so there can be no kinetic terms. The action is just a function S(φ) and the path integral becomes just an ordinary integral. The partition function becomes

S(φ) = dφ e− (2.1) Z !R where we’ll assume that S(φ) grows suﬃciently rapidly as φ so that this integral | | → ∞ exists. Typically, we’ll take S(φ) to be a polynomial (with highest term of even degree), such as m2 λ S(φ)= φ2 + φ4 + (2.2) 2 4! ··· The partition function then depends on the values of the coupling constants, so

= (m2, λ, ) . (2.3) Z Z ···

– 1 – As a piece of notation, I’ll often write for the partition function in the free theory, where Z0 the couplings of all but the term quadratic in the field(s) are set to zero. We should think of the set of couplings as being coordinates on the infinite dimensional ‘space of theories’ in the sense that, at least for our single field φ, the theory is specified once we choose values for all possible monomials φp.

2.2 Correlation functions Beyond the partition function, the most important object we wish to compute in any QFT are (normalized) correlation functions. These are just weighted integrals

1 S(φ) f := dφf(φ)e− (2.4) # $ Z !R S where along with e− we’ve inserted some f(φ) into the integral. Here, f should be thought of as a function (or perhaps a distribution, such as δ(φ φ ) or similar) on the space of − 0 fields. We’ll assume the operators we insert are chosen so as not to disturb the rapid decay of the integrand at large values of the field, and are sufficiently well-behaved at finite φ that the integral (2.4) actually exists. The usual way to think about correlation functions comes from probability. So long S 1 S as the action S(φ) is R-valued, e− 0 so we can view e− as a probability density on ≥ the space of fields. The correlation function (2.4) is thenZ just the expectation value f of # $ f(φ) averaged over the space of fields with this measure, with the factor of 1/ ensuring Z that the probability measure is normalized. On a higher dimensional space–time M we’ll be able to insert functions at different points in space–time into the path integral, and the correlation functions will probe whether there is any statistical relation between, say, two functions f(φ(x)) and g(φ(y)) at points x, y M. ∈ In the context of QFT, we’ll often choose the functions we insert to correspond to some quantity of physical interest that we wish to measure; perhaps the energy of the quantum field in some region, or the total angular momentum carried by some electrons, or perhaps temperature fluctuations in the CMB at different angles on the night sky. For reasons that will become apparent, we’ll often call these functions ‘operators’, though the terminology is somewhat inaccurate (particularly in zero dimensions). Alternatively, recalling that the general partition function (m2, λ, ) depends on the Z ··· values of all possible couplings, we see that at least for operators that are polynomial in the fields, correlation functions describe the change in this general as we infinitesimally vary Z some combination of the couplings, evaluated at the point in theory space corresponding to our original model. For example, in the simplest case that f(φ)=φp is monomial, we have formally 1 1 ∂ φp = (m2,λ ) (2.5) p!# $ − ∂λ Z i p " Z "∗ where λ is the coupling to φp/p! in the general action, and" is the point in theory space p " ∗ where the couplings are set to their values in the specific action that appears in (2.4).

– 2 – 2.3 The Schwinger–Dyson equations Let’s consider the eﬀect of relabelling φ φ + ! in the path integral (2.1), where ! is a real → constant. Since we integrate over the entire real line, trivially

S(φ) S(φ+") = dφ e− = d(φ + !)e− (2.6) Z !R !R and furthermore d(φ + !)=dφ by translation invariance of the measure on R. Taking ! to be infinitesimal and expanding the action to first order we find

S(φ+") S(φ) dS = dφ e− = dφ e− 1 ! + Z − dφ ··· !R !R # $ (2.7) S(φ) dS = ! dφ e− Z− dφ !R to first order in !. Comparing both sides of (2.7) we see that inserting dS/dφ – the first order variation of the action wrt the ‘field’ φ – into the path integral for the partition function gives zero. In our zero dimensional context, the statement dS/dφ = 0 in the path integral is the quantum equation of motion for the field φ. Alternatively, we can obtain this result directly by integrating by parts:

S dS d S dφ e− = dφ e− = 0 (2.8) − dφ dφ !R !R % & where there is no contribution from φ since we assumed e S(φ) decays rapidly. By | |→∞ − the way, the derivative d/dφ here is really best thought of as a derivative on the space of ﬁelds; it’s just that this space of ﬁelds is nothing but R in our zero dimensional example. For example, suppose we consider the action 1 λ S = m2φ2 + φ4 (2.9) 2 4! with m2 > 0 and λ 0. (Again: there can’t be any derivative terms because we’re in zero ≥ dimensions. For the same reason, there’s no integral.) The classical equation of motion is

m2φ = λφ3/6 , (2.10) − and eq (2.7) says that this relation holds inside the path integral for the partition function. However, while the classical equations of motion hold in the path integral for the partition function, the situation is different for more general correlation functions. We now find 1 S(φ+") f = dφf(φ + !)e− # $ Z !R (2.11) ! S dS df = f dφ e− f(φ) + # $− dφ − dφ ··· Z !R # $ to lowest order in ! (we’ll assume f is at least once differentiable). Therefore we find

dS df f = (2.12) dφ dφ ' ( ' (

– 3 – which you can again derive directly by integration by parts. This equation illustrates an important difference between quantum and classical theories. Classically, the equation of motion dS/dφ =0always holds; the motion always extremizes the action. Noting that if f is some generic smooth function then so too is df/dφ, there is no reason for the rhs of (2.12) to vanish. Thus it is not true that the classical eom dS/dφ = 0 holds in the quantum system, and we can detect this failure through its effect on correlation functions. n For instance, later on we’ll often set f(φ)= i=1 pi(φ) where each of the pi(φ) are polynomials. Then we find ) n n dS dp p (φ) = j p (φ) (2.13) dφ n dφ i * i=1 , j=1 * i=j , + - +# so the effect of inserting the classical equation of motion into the path integral is to modify each of the operators pi(φ) in our correlation function in turn. This equation (and its cousins in higher dimensional QFT we’ll meet later) is known as the Schwinger–Dyson equation.

2.4 Perturbation theory Let’s return to the example of S(φ)=m2φ2/2+λφ4/4!. The partition function (m, λ) Z is easy to evaluate at λ = 0 where we find (m, 0) = √2π/m. For λ> 0 it is clear that Z the integral exists, but it looks hard to evaluate. We can try to treat it perturbatively by expanding in λ: n 4n 2 ∞ λ φ m φ2 (m, λ)= dφ − e− 2 Z 4! n! R n=0 ! - . / n 2 ∞ λ 1 4n m φ2 − dφφ e− 2 (2.14) ∼ 4! n! n=0 R - . / ! √2π 1 λ 35 λ2 = 1 + + . m × − 8 m4 384 m8 ··· # $ where each term in this result follows from the standard result for a Gaussian integral using integration by parts. In the second line we’ve exchanged the order of the integral and sum. This is a very dangerous step: infinite series are convergent iff they converge for some open disc in the complex plane (here for λ). But it’s clear that the original integral would have diverged had Re(λ) been negative, so whatever (m, λ) actually is, it can’t have a convergent power Z series around λ = 0. In fact, what we have computed is an asymptotic series for (m, λ) + n + Z as λ 0 . Recall that ∞ a λ is an asymptotic series for (λ) as λ 0 if, for all → n=0 n Z → N N ∈ 0 N n Z(λ) n=0 anλ lim − =0. λ 0+ " λN " → " 0 " This definition says that for any natural" number N, and" any !> 0, for sufficiently small N λ R 0 the first N terms of the series differ from the exact answer by less than !λ . ∈ ≥

– 4 – However, since our series actually diverges, if we instead ﬁx λ and include more and more terms in the sum, we will eventually get worse and worse approximations to the answer. In fact, we can see this divergence of the perturbation series directly. Writing φ˜ = mφ we see that the coeﬃcient of λn/m4n+1 in is Z

1 φ˜2/2 4n a := dφ˜ e− φ˜ n (4!)n n! !R (2.15) 1 ˜ ˜2 ˜ = n dφ exp φ /2 + 4n ln φ (4!) n! R − ! 1 2 The integrand has maxima when φ˜ = √4n and as n it drops oﬀ rapidly away from ± →∞ these stationary points. Thus for large n the integral is dominated from the contribution near these maxima and we have

1 2n 2n n ln n a (4n) e− e (2.16) n ≈ (4!)n n! ≈ where the second approximation follows from Stirling’s approximation n! en ln n for n ≈ → . Thus these coeﬃcients asymptotically grow faster than exponentially with n, and the ∞ series (2.14) has zero radius of convergence. Perturbation theory thus tells us important, but not complete, information about our QFT.

2.4.1 Graph combinatorics Working inductively, one can show that the coeﬃcient of the general term ( λ/m4)n − in (2.14) is 1 (4n)! . The ﬁrst factor of this comes straightforwardly from expanding (4!)nn! × 4n(2n)! the λφ4/4! vertex in the exponential, while the second factor comes from the resulting φ integral. Now, (4n!)/4n(2n)! is the number of ways of joining 4n elements into distinct pairs, suggesting that this second numerical factor should have a combinatorial interpretation that saves us having to actually perform the integral. This is what the Feynman rules provide. With the action S(φ)=m2φ2/2+λφ4/4! the Feynman rules are simply

1 λ m2 − where the propagator is constant since we are in zero dimensions. The minus sign in the S vertex comes from the fact that we are expanding e− . To compute perturbation series in QFT, Feynman tells us to construct all possible graphs (not necessarily connected) using this propagator and vertex. In the case of the partition function (m2,λ), we want Z vacuum graphs, i.e., those with no external edges. In constructing all possible such graphs, we imagine the individual vertices carry their own unique ‘labels’, so that we can tell them apart, and that likewise each of the four φ ﬁelds present in a given vertex carries its own label. Thus, the term proportional to λ receives contributions from three individual graphs

– 5 – φ1 φ3

⇒ φ2 φ4 corresponding to the three possible ways to join up the four φ ﬁelds into pairs. The partition function itself is the given by the sum of graphs

+ + + + + ∅ ···

λ λ2 λ2 λ2 Z = 1 + − + + + + 8m4 48m8 16m8 128m8 ··· where we include both connected and disconnected graphs, with the contribution of a disconnected graph being the product of the contributions of the two connected graphs. Notice that this requires that we assign a factor 1 to the trivial graph (no vertices or ∅ edges), which is also included as the zeroth–order term in the sum.

To work out the numerical factors, let Dn be the set of such graphs that contain precisely n vertices; since each vertex comes with a power of the coupling λ these diagrams will each contribute to the coefficient of λn in the expansion of Z(m2,λ). Suppose there are D graphs in this set. Now, because Feynman instructed us to join up our labelled | n| vertices in every possible way, every graph in Dn contains several copies that are identical as topological graphs but differ in the labeling of their vertices. We need to remove this n overcounting. Dn is naturally acted on by the group Gn =(S4) ! Sn that permutes each of the four fields in a given vertex (n copies of the permutation group S4 on 4 elements) n and also permutes the labels of each of the n vertices. This group has order Gn = (4!) n!, S | | which is the same factor we saw before from expanding e− in powers of λ. Thus the asymptotic series may be rewritten as

n ∞ λ D Z − | n| . (2.17) ∼ m4 G 0 n=0 n Z - . / | | In detail, the power ( λ)n is the contribution of the coupling constants in each graph, − the power of (1/m2)2n comes from the fact that any vacuum diagram with exactly n 4– valent vertices must have precisely 2n edges, each of which contributes a factor of 1/m2. Finally, the factor D is the number of diagrams that contribute at this order and the | n| factor 1/ G is the coefficient of this graph in expanding the exponential of the action | n| perturbatively in the interactions. This way of working out the numerical coefficient requires that we draw all possible graphs obtained by joining up all the fields in all possible ways, as with the single λ vertex above. That’s in principle straightforward, but in practice can be very laborious if there are many vertices, or vertices containing many powers of a field. There’s another way to

– 6 – think of D / G that sometimes makes life easier. An orbit Γ of G in D is a set of | n| | n| n n labeled graphs in Dn that are identical up to a relabeling of their ﬁelds and vertices, so that we can move from one labelled graph to another in the orbit using an element of Gn. Thus an orbit Γ is a topologically distinct graph in Dn. Let On be the set of such orbits. The orbit stabilizer theorem says that2

D 1 | n| = (2.18) Gn Aut Γ Γ On | | -∈ | | where Aut Γ is the stabilizer of any element in Γ in Gn, i.e., the elements of the permutation group Gn that don’t alter the labeled graph. For example, if a graph in Dn involves a propagator joining two fields at the same vertex, then exchanging the labeling of those fields won’t change the labeled graph. Finally then, we can rewrite our asymptotic series (2.14) as n ∞ λ 1 Z − 4 0 ∼ m Aut Γ n=0 3 . / Γ On 4 Z - -∈ | | (2.19) 1 ( λ) v(Γ) = | | , − 2 e(Γ) Aut Γ (m )| | -Γ | | where v(Γ) and e(Γ) are respectively the number of vertices and edges of the graph Γ. | | | | In zero dimensions, we’ve rederived the Feynman rule that we should weight each topologically distinct graph by v(Γ) powers of (minus) the coupling constant λ and | | − e(Γ) powers of the propagator 1/m2, then divide by the symmetry factor Aut Γ of the | | | | graph. More generally, if we have i different types of field, each with propagators 1/Pi and interacting via a set of vertice with couplings λ , then a graph Γ containing e (Γ) edges α | i | of the field of type i and v (Γ) vertices of type α contributes a factor | α |

1 1 vα(Γ) ( λα)| | (2.20) Aut Γ ei(Γ) − P | | α | | +i i + to the perturbative series. To obtain the perturbative series for the partition function we sum this expression over both connected and disconnnected vacuum graphs, including the trivial graph with no vertices. It’s often convenient to just include the connected graphs. We then have

vα(Γ) 1 ( λα)| | W +W0 Z = exp − =: e− (2.21)  Aut Γ ei(Γ)  0 conn P | | Z - | | +i,α i   where the sum in the exponential is only over connected, non–trivial graphs. Particularly in applications to statistical ﬁeld theory, W = ln is known as the free energy, while Z W = ln . The identity (2.21) easily visualized by writing the power series expansion of 0 Z0 the rhs, deﬁning the product of two connected graphs to be the disconnected graph whose two connected components are the original graphs.

2If you don’t know this already, you can ﬁnd a nicely explained proof on Gowers’s Weblog.

– 7 – In practice, it is often just as quick to think through the possible ways a given topo- S logical graph Γ may be obtained by expanding out the vertices in e− and joining pairs of ﬁelds by propagators, as to work out the symmetry factor Aut Γ . I’ll leave it to your | | taste.

2.5 Supersymmetry and localization Since the perturbative series is only an asymptotic series, it’s worth asking if we can ever do better, even for interacting theories. Generically we cannot, but in special circumstances it is possible to evaluate the partition function and even certain correlation functions exactly. There are many mechanisms by which this might happen; this section gives a toy model of one of them, known as localization, which we’ll meet again later (though it’ll be in disguise). Let’s take a theory where that in addition to our bosonic field φ, we have two fermionic 1 2 fields ψ1 and ψ2. With a zero–dimensional space–time, the space of fields is just R | . Given an action S(φ, ψi) the partition function is, as usual,

dφ dψ1 dψ2 S(φ,ψ ) = e− i (2.22) Z √2π ! where I’ve thrown a factor of 1/√2π into the measure for later convenience. Generically, we’d have to be content with a perturbative evaluation of , using Feynman diagrams Z formed from edges for the φ and ψi fields, together with vertices from all the different vertices that appear in our action. For a complicated action, even low orders of the perturbative expansion might be difficult to compute in general. However, let’s suppose the action takes the special form 1 S(φ, ψ ,ψ )= ∂h(φ)2 ψ ψ ∂2h(φ) (2.23) 1 2 2 − 1 2 where h(φ) is some (R-valued) polynomial in φ. Note that there can’t be any terms in S involving only one of the fermion fields since this term would itself be fermionic. There also 2 can’t be higher order terms in the fermion fields since ψi = 0 for a Grassmann variable, so the only thing special about this action is the relation between the purely bosonic piece and the second term involving ψ1ψ2. Now consider the transformations

δφ = ! ψ + ! ψ , δψ = ! ∂h , δψ = ! ∂h (2.24) 1 1 2 2 1 2 2 − 1 where !i are fermionic parameters. These are supersymmetry transformations in this zero– dimensional context; take the Part III Supersymmetry course to meet supersymmetry in higher dimensions. The most important property of these transformations is that they are nilpotent3. Under (2.24) the action (2.23) transforms as

δS = ∂h ∂2h(! ψ + ! ψ ) (! ∂h)ψ ∂2h ψ ( ! ∂h)∂2h = 0 (2.25) 1 1 2 2 − 2 2 − 1 − 1 3 2 2 That is, δ1 = 0, δ2 = 0 and [δ1,δ2] = 0, where δ1 is the transformation with parameter "2 = 0, etc.. You should check this from (2.24) as an exercise!

– 8 – and is thus invariant — this is what the special relation between the bosonic and fermionic terms in S buys us. (To obtain this result we used the fact that Grassmann variables anticommute.) It’s also true that the integral measure dφ d2ψ is likewise invariant; I’ll leave this too as an exercise. The action being supersymmetric will drastically simplify this QFT. Let δ be the O supersymmetry variation of some operator (φ, ψ ) and consider the correlation function O i δ . Since δS = 0 we have # O$

1 2 S 1 2 S δ = dφ d ψ e− δ = dφ d ψδ e− . (2.26) # O$ Z O O 0 ! Z0 ! % & The supersymmetry variation here acts on both φ and the fermions ψ in e S . But if it i − O acts on a fermion ψi then the resulting term does not contain that ψi and hence cannot contribute to the integral because dψ 1 = 0 for Grassmann variables. On the other hand, if it acts on φ then while the resulting term may survive the Grassmann integral, it is a 9 total derivative in the φ field space. Thus, provided does not disturb the decay of e S O − as φ , any such correlation function must vanish, δ = 0. | | → ∞ # O$ In particular, if we choose = ∂g ψ for some g(φ), then setting the parameters Og 1 ! = ! = ! we have 1 − 2 0= δ = ! ∂g ∂h ∂2gψ ψ . (2.27) # Og$ # − 1 2$ The significance of this is that the quantity ∂g ∂h ∂2gψ ψ is the first–order change in − 1 2 the action under the deformation h h+g, again so long as g does not alter the behaviour → of h as φ . The fact that δ = 0 tells that the partition function [h], which | | → ∞ # Og$ Z we might think depends on all the couplings in the vertices in the polynomial h, is in fact largely insensitive to the detailed form of h because we can deform it by any other polynomial of the same degree or lower. The most important case is if we choose g to be proportional to h, then our deformation just rescales h (1 + λ)h and so we see that → [h] is independent of the overall scale of h. By iterating this procedure, we can imagine Z rescaling h by a large factor so that the bosonic part of the action (∂h)2/2 Λ2(∂h)2/2. → As Λ , the factor e S exponentially suppresses any contribution to Z except from an →∞ − infinitesimal neighbourhood of the critical points of h where ∂h = 0. This phenomenon is known as localization of the path integral. It’s now straightforward to work out the partition function. Near any such critical point φ we have ∗ c 2 h(φ)=h(φ )+ ∗ (φ φ ) + (2.28) ∗ 2 − ∗ ··· where c = ∂2h(φ ), so the action (2.23) becomes ∗ ∗ 2 c 2 S(φ, ψi)= ∗ (φ φ ) + c ψ1ψ2 + . (2.29) 2 − ∗ ∗ ··· The higher order terms will be negligible as we focus on an infinitesimal neighbourhood of φ . Expanding the exponential in Grassmann variables the contribution of this critical ∗

– 9 – h(φ) 1 − 1 −

+1 φ

Figure 1: The supersymmetric path integral receives contributions just from inﬁnitesimal neighbourhoods of the critical points of h(φ). These alternately contribute 1 according to ± whether they are minima or maxima. point to the partition function is

1 2 c (φ φ )2/2 c c (φ φ )2 dφ d ψ e− ∗ − ∗ [1 c ψ1ψ2]= ∗ dφ e− ∗ − ∗ √2π − ∗ √2π ! ! (2.30) c 2 = ∗ = sgn ∂ h . c2 |∗ ∗ % & Summing over all the critical points, the full partition function: thus becomes

2 [h]= sgn ∂ h φ (2.31) Z | ∗ φ : ∂h φ =0 ∗ -| ∗ % & and, as expected, is largely independent of the detailed form of h. In fact, if h is a polynomial of odd degree, then ∂h = 0 must have an even number of roots with ∂2h being alternately > 0 and < 0 at each. Thus their contributions to (2.31) cancel pairwise and [h ] = 0 identically. On the other hand, if h has even degree then it has an odd number Z odd of critical points and we obtain [h ]= 1, with the sign depending on whether h Z ev ± → ±∞ as φ . (See figure 1.) | | → ∞ The fact that the partition function is so simple in this class of theories is a really remarkable result! To reiterate, we’ve found that for any form of polynomial h(φ), the partition function [h] is always either 0 or 1. If we imagined trying to compute [h] Z ± Z perturbatively, then for a non–quadratic h we’d still have to sum infinitely diagrams using the vertices in the action. In particular, we could certainly draw Feynman graphs Γ with arbitrarily high numbers of loops involving both φ and ψi fields, and these graphs would each contribute to the coefficient of some power of the coupling constants in the perturbative expansion. However, by an apparent miracle, we’d find that these graphs always cancel themselves out; the net coefficient of each such loop graph would be zero with the contributions from graphs where either φ or ψ1ψ2 run around the loop contributing with opposite sign. The reason for this apparent miracle is the localization property of the supersymmetric integral.

– 10 – In supersymmetric theories in higher dimensions, complications such as spin mean the cancellation can be less powerful, but it is nonetheless still present and is responsible for making supersymmetric quantum theories ‘tamer’ than non–supersymmetric ones. As an important example, diagrams where the Higgs particle of the Standard Model runs around a loop can have the eﬀect of destabilizing the mass of the Higgs, sending it up to a very high scale. (We’ll understand this later on.) This Spring, CERN will resume its search for a hypothesized supersymmetric partner to the Higgs that many people postulated should be present so as to cancel these dangerous loop diagrams; the ultimate mechanism is the one we’ve seen above, though it’s power is ﬁltered through the lens of a much more complicated theory. I also want to point out that localization is useful for calculating much more than just the partition function. For each a 1, 2, 3,... suppose that (φ, ψ ) is an operator that ∈{ } Oa i obeys δ = 0, i.e. each operator is invariant under supersymmetry transformations (2.24). Oa Then the (unnormalized) correlation function

2 dφ d ψ S = e− (2.32) Oa √ Oa * a , 2π a + ! + again localizes to the critical points of h. Once again, this is because deforming h h + g → leaves the correlator invariant since the deformation aﬀects the correlation function as

h h+g → δ = δ = 0 (2.33) Oa −→ Og Oa Og Oa * a , * a , * ; a <, + + + which vanishes by the same arguments as before. Here, we used the fact that δ = 0 to Oa write the operator on the rhs as a total derivative. Of course, if any of the are already of the form δ , so that this is itself the Oa O( Oa supersymmetry transformation of some , then = 0 which is not very interesting. O( # Oa$ The interesting operators are those which are δ-closed (δ = 0) but not δ-exact ( = δ ). ) O O. O( These operators describe the cohomology of the nilpotent operator δ. This is the starting– point for much of the mathematical interest in QFT: we can build supersymmetric QFTs that compute the cohomology of interesting spaces. For example, Donaldson’s theory of invariants of 4–manifolds that are homeomorphic but not diﬀeomorphic, and the Gromov– Witten generalization of intersection theory can both be understood as examples of (higher– dimensional) supersymmetric QFTs where the localization / cancellation is precise. We’ll also meet essentially the same idea again in a slightly diﬀerent context later in this course when we study BRST quantization of gauge theories.

2.6 Effective theories: a toy model Now I want to introduce a very important idea which will be central to our understanding of QFT in higher dimensions. Suppose we have two real–valued fields φ and χ, so that the space of fields is R2, and let the action be m2 M 2 λ S(φ, χ)= φ2 + χ2 + φ2χ2 (2.34) 2 2 4

– 11 – so that λ provides a coupling between the ﬁelds. We have the Feynman rules

φ χ

1/m2 1/M 2 λ − which may be used to compute perturbative expressions for correlation functions such as

1 S(φ,χ) f = dφ dχ e− f(φ, χ) # $ 2 Z !R in the usual way. For example, we have Z ln = + + + Z ! 0 " λ λ2 λ2 λ2 = + + + −4m2M 2 16m4M 4 16m4M 4 8m4M 4 as the sum of connected vacuum diagrams, and also

1 φ2 = + + + + 2! "

1 λ λ2 λ2 λ2 = + + + m2 −2m4M 2 4m6M 4 2m6M 4 4m6M 4 where the blue dots represent the insertions of the two powers of φ. I want to arrive at this result in a different way. Suppose we’re interested in correlation functions of operators that depend only on φ; for example, we might imagine that the field χ has a very high mass such that our experiment isn’t powerful enough to observe real χ production — we can only measure properties of the φ field. Having no idea what χ is doing suggests that we should perform its path integral first, i.e., we average over χ configurations at each fixed φ.

We define the effective action Seff (φ) for the φ field to be the result of carrying out this χ integral. Thus S(φ,χ)/" Seff (φ) := " log dχ e− (2.35) − #!R $ where I’ve restored the powers of ". Once we have found this effective action, we can use it to compute f for any observable that depends only on φ. Of course there’s nothing # $ mysterious here, we’re simply choosing in which order to do our integrals.

In general computing Seﬀ can be diﬃcult, but in our toy example it’s straightforward because χ appears only quadratically in S(φ, χ). We have

S(φ,χ)/" m2φ2/2" 2π" dχ e− =e− (2.36) M 2 + λφ2/2 !R =

– 12 – where the ﬁrst factor is the χ-independent part of the original action and the square root comes from the Gaussian integral over χ. Hence the eﬀective action (2.37) is

m2 " λ " M 2 S (φ)= φ2 + ln 1+ φ2 + ln eff 2 2 2M 2 2 2π # $ " m2 "λ "λ2 "λ3 = + φ2 φ4 + φ6 + (2.37) 2 4M 2 − 16M 4 48M 6 ··· . / m2 λ λ =: eff φ2 + 4 φ4 + 6 φ6 + . 2 4! 6! ··· The important point is that the effect of integrating out the ‘high energy field’ χ has changed the structure of the action. In particular, the mass term of the φ field has been shifted "λ m2 m2 = m2 + . (2.38) → eff 2M 2 Even more strikingly, we’ve generated an infinite series of new coupling terms

2 3 k 3" λ λ k+1 (2k)! λ λ4 = ,λ6 = 15" ,λ2k =( 1) " (2.39) − 2 M 4 M 6 − 2k+1k M 2k describing self–interactions of φ. It’s important to observe that the φ mass shift and new φ self–interactions all vanish as " 0; they are quantum eﬀects. Notice also that they’re each suppressed by powers of → the (high) mass M. It’s useful to think a little more detail about how these new couplings arise. We can perform the χ path integral using Feynman graphs, provided we remember that the φ ﬁeld is not propagating, but may appear on an external leg. The χ propagator and vertices are

1/M 2 φ m2 λ − − where the blue 1–valent vertex represents an insertion of the φ field, coming from expanding S = + + + + the action in powers− ofeff the vertices. These ingredients lead to the following perturbative construction of Seff :

m2 λ λ2 λ3 2 φ2 + φ4 φ6 S = = + φ +2 + 4 + 6 − eﬀ − 2 −4M 16M −48M

2 2 3 m λ 2 λ 4 λ 6 1= 2 φ2 φ + φ φ φ = 4M+2 +4 6 2! −" 2 − 16M −48M

1 λ4 where we note that since Seff is the logarithm of the χ integral, only connected diagrams − = 2 6 + appear. meff −2meff

Thursday, January 15, 2015

– 13 – The diagrammatic expansion shows that the new interactions of φ have actually been generated by loops of the χ field. In our effective description that knows only about the behaviour of the φ field, we can no longer ‘see’ the χ field ‘circulating’ around the loop. Instead, we perceive this just as a new interaction vertex for φ. (The fact that the new terms in Seff come just from single loops of χ, and hence come with just a single power of ", is special to the fact that χ appears in the original action (2.34) only quadratically. Generically there would be higher–order corrections.) Using this effective action, we now find

1 φ2 = + + 2! "

1 λ4 = 2 6 + meff −2meff where the propagator and vertices here are the ones appropriate for the effective action

Seff . Using the definition of the new couplings in terms of the original λ and M, this unsurprisingly agrees with our answer before, correct to order λ2. However, once we had the effective action, we arrived at this answer using just two diagrams, whereas above it required five. If we only care about a single correlation function then the work involved in first computing Seff and then using the new set of Feynman rules to compute the low– energy correlator is roughly the same as just using the original action to compute this correlator directly. On the other hand, if we wish to compute many low–energy correlators then we’re clearly better off investing a little time to work out Seff first. However, the real point I wish to make is this: the way we experience the world is always through Seff . Naively at least, we have no idea what new physics may be lurking just out of reach of our most powerful accelerators; there may be any number of new, hitherto undiscovered species of particle, or new dimensions of space–time, or even wilder new phenomena. More importantly, when describing low–energy physics you should only seek to describe the behaviour of the degrees of freedom (fields) that are relevant and accessible at the energy scale at which you’re conducting your experiments, even when you know what the more fundamental description is. We know that a glass of water consists of very many H2O molecules, that these molecules are bound states of atoms which each consist of many electrons orbiting around a central nucleus, that this nucleus comprises of protons and neutrons stuck together by a strong force mediated by pions, and that all these hadrons are themselves seething masses of quarks and gluons. But it would be very foolish to imagine we should describe the properties of water that are relevant in everyday life by starting from the Lagrangian for QCD. Let me make one final comment. In the example above, we started from a very simple action in equation (2.34) and obtained a more complicated effective action (2.37) after integrating out the unobserved degree of freedom χ. A more generic case would start from

– 14 – a general action (invariant under φ φ and χ χ for simplicity) →− →−

λi,j 2i 2j S((φ, χ)= φ χ (2.40) (2i)! (2j)! -i,j in which all possible even monomials in φ and χ are allowed. For example, we may have arrived at this action by integrating out some other field that was unknown in our above considerations. In this generic case, the effect of integrating out χ will not generate new interactions for φ — all possible even self–interactions are included anyway — but rather the values of the coupling constants λi,0 will get shifted, just as for the mass shift we saw above. The lesson to remember is that the effect of integrating out degrees of freedom is to change the values of the coupling constants in the effective action.

– 15 – 3 QFT in one dimension (= QM)

In one dimension there are two possible compact (connected) manifolds M: the circle S1 and the interval I. We will parametrize the interval by t [0,T] so that t = 0 and t = T ∈ are the two point–like boundaries, while we will parametrize the circle by t [0,T) with ∈ the identification t ∼= t + T . The most important example of a field on M is a map x : M N to a Riemannian → manifold (N, g) which we will take to have dimension n. That is, for each point t on our ‘space–time’ M, x(t) is a point in N. It’s often convenient to describe N using coordinates. If an open patch U N has local co-ordinates xa for a =1, . . . , n, then we let xa(t) denote ⊂ the coordinates of the image point x(t). More precisely, xa(t) are the pullbacks to M of coordinates on U by the map x. With these fields, the standard choice of action is

1 S[φ]= g (x)˙xax˙ b + V (x) dt, (3.1) 2 ab !M " # a a where gab(x) is the pullback to M of the Riemannian metric on N andx ˙ = dx /dt. We have also included in the action a choice of function V : N R, or more precisely → the pullback of this function to M, which is independent of worldline derivatives of x. In writing this action we have chosen one–dimensional metric on M to be just the ﬂat Euclidean metric δtt = 1. Under a small variation δx of x we have

b 1 ∂g (x) ∂V (x) δS = g (x)˙xaδ˙x + ab δxc x˙ ax˙ b + δxc dt ab 2 ∂xc ∂xc !M " # (3.2) d 1 ∂g (x) ∂V (x) = (g (x)˙xa)+ ab x˙ ax˙ b + δxc dt −dt ac 2 ∂xc ∂xc !M " # and requiring that this vanishes for arbitrary δφ(t) gives the Euler–Lagrange equations

d2xa dV +Γa x˙ bx˙ c = gab(x) (3.3) dt2 bc dxb where Γa = 1 gad (∂ g + ∂ g ∂ g ) is the Levi–Civita connection on N, again pulled bc 2 b cd c bd − d bc back to the worldline. The standard interpretation of all this is to image an arbitrary map x(t) describes a possible trajectory a particle might in principle take as it travels through the space N. (See ﬁgure 2.) In this context, N is called the target space of the theory, while M (or its image x(M) N) is known as the worldline of the particle. The ﬁeld equation (3.3) says ⊂ that when V = 0, classically the particle travels along a geodesic in (N, g). V itself is then interpreted as a (non–gravitational) potential4 through which this particle moves.

4The absence of a minus sign on the rhs of (3.3) is probably surprising, but follows from the action (3.1). This is actually the correct sign with a Euclidean worldsheet, because under the Wick rotation t it back → to a Minkowski signature worldline, the lhs of (3.3) acquires a minus sign. In other words, in Euclidean time F = ma! −

– 16 – x(t)

(N, g) 0 T

Figure 2: The theory (3.1) describes a map from an abstract worldline into the Rieman- nian target space (N, g). The corresponding one–dimensional QFT can be interpreted as single particle Quantum Mechanics on N.

From this perspective, it’s natural to think of the target space N as being the world in which we live, and computing the path integral for this action will lead us to single particle Quantum Mechanics, as we’ll see below. However, we’re really using this theory as a further warm–up towards QFT in higher dimensions, so I want you to also keep in mind the idea that the worldline M is actually ‘our space–time’ in a one–dimensional context, and the target space N can be some abstract Riemannian manifold unrelated to the space we see around us. For example, at physics of low–energy pions is described by a theory of this general kind, where M is our Universe and N is the group manifold SU(3).

3.1 Quantum Mechanics The usual way to do Quantum Mechanics is to pick a Hilbert space and a Hamiltonian H H, which is a Hermitian operator H : . In the case relevant above, the Hilbert H → H space would be L2(N), the space of square–integrable functions on N, and the Hamiltonian would usually be

1 1 ∂ ∂ H = ∆+ V, where ∆ := √ggab (3.4) 2 √g ∂xa ∂xb $ % is the Laplacian acting on functions in L2(N). The amplitude for the particle to travel from an initial point y N to a final point y N in Euclidean time T is given by 0 ∈ 1 ∈ HT K (y ,y )= y e− y , (3.5) T 0 1 ' 1| | 0( which is known as the heat kernel. (Here I’ve written the rhs in the Heisenberg picture, which I’ll use below. In the Schrödinger picture where states depend on time we would instead write K (y ,y )= y ,T y , 0 .) T 0 1 ' 1 | 0 ( The heat kernel is a function on I N which may be defined to be the solution of the × pde ∂ K (x, y)+HK (x, y) = 0 (3.6) ∂t t t subject to the initial condition that K (x, y)=δ(x y). I remind you that we’re in 0 − Euclidean worldline time here, and in units where ! = 1 here. Rotating to Minkowski

– 17 – signature by sending t it and restoring the ! gives instead → ∂ i! Kit(x, y)=HKit(x, y) (3.7) ∂t n that we recognize as Schr¨odinger’s equation. In the simplest example where N ∼= R with ﬂat metric gab = δab and vanishing potential V = 0, the heat kernel

1 x y 2 KT (x, y)= exp | − | (3.8) (2πT )n/2 − 2T $ % where x y is the Euclidean distance between x and y. | − | As you learned last term, Feynman showed that this heat kernel could also be represented as a path integral. The usual idea is to break the time interval T into N chunks, each of duration ∆t = T/N. We can then write

HT H∆t H∆t H∆t y e− y = y e− e− e− y ' 1| | 0( ' 1| ··· | 0( n n H∆t HT H∆t = d x1 d xN 1 y1 e− xN 1 x2 e− x1 x1 e− y0 ··· − ' | | − ( ··· ' | | (' | | ( ! (3.9) N 1 − n = d xi K∆t(y1,xN 1) K∆t(x2,x1) K∆t(x1,y0) . − ··· ! &i=1 In the second line here we have inserted the identity operator dnx x x on in i | i(' i| H between each evolution operator; in the present context this can be understood as the ' concatentation identity

n Kt1+t2 (x3,x1)= d x2 Kt2 (x3,x2) Kt1 (x2,x1) (3.10) ! obeyed by convolutions of the heat kernel.

Now, while the ﬂat space expression (3.8) for the heat kernel does not hold when gab is a more general Riemannian metric on N, in fact it is (almost) correct in the limit of small times. More precisely, it can be shown that the heat kernel always has the asymptotic form

1 d(x, y)2 lim K∆t(x, y) a(x) exp (3.11) ∆t 0 ∼ (2π∆t)n/2 − 2∆t → $ % for small t, where d(x, y) is the geodesic distance between x and y measured using the metric g, and where a(x) is some polynomial in the Riemann curvature tensor that we won’t need to be speciﬁc about. Therefore, splitting our original time interval [0,T] into very many pieces of very short duration ∆t = T/N gives

nN N 1 2 2 HT 1 − n ∆t d(xi+1,xi) y1 e− y0 = lim d xi a(xi) exp (3.12) ' | | ( N 2π∆t − 2 ∆t →∞ ( ) $ % ! &i=1 $ % as an expression for the heat kernel.

– 18 – This more or less takes us to the path integral. If it is sensible to take the limits, then we can take nN N 1 2 ? 1 − n x := lim d xi a(xi) (3.13) D N 2π∆t →∞ $ % &i=1 to be the path integral measure. Similarly, if the trajectory is at least once diﬀerentiable 2 a b then (d(xi+1,xi)/∆t) converges to gabx˙ x˙ and we can write

N 1 2 T − ∆t d(xi+1,xi) 1 a b lim exp = exp gab x˙ x˙ dt (3.14) N − 2 δt −2 →∞ ( ) 0 &i=1 $ % " ! # which recovers the action (3.1), with V = 0. (A more general heat kernel can be used to incorporate a non–zero potential.) We’ll investigate these limits further below. Accepting them for now, combining (3.13) &(3.14) we obtain the path integral expression T HT 1 a b y e− y = x exp g x˙ x˙ dt , (3.15a) ' 1| | 0( D −2 ab !CT [y0,y1] " !0 # or in other words, the heat kernel can formally be written as

S K (y ,y )= x e− . (3.15b) T 0 1 D !CT [y0,y1] The integrals in these expressions are to be taken over the space CT [y0,y1] of all continuous maps x : I N that are constrained to obey the boundary conditions x(0) = y and → 0 x(T )=y1.

3.1.1 The partition function The partition function on the circle can likewise be given and interpretation in the operator approach to Quantum Mechanics. Tracing over the Hilbert space gives

TH n HT n S Tr (e− )= d y y e− y = d y x e− (3.16) H ' | | ( D ! !N !CT [y,y] using the path integral expression (3.15b) for the heat kernel. The path integral here is (formally) taken over all continuous maps x : [0,T] N such that the endpoints are both → mapped to the same point y N. We then integrate y everywhere over N 5, erasing the ∈ memory of the particular point y. This is just the same thing as considering all continuous maps x : S1 N where the worldline has become a circle of circumference T . This shows → that TH S Tr (e− )= x e− = S1 [N, g, V ] , (3.17) H C D Z ! S1 which is nothing but the partition function on S1. In higher dimensions this formula will be the basis of the relation between QFT and Statisical Field Theory, and is really the origin of the name ‘partition function’ for in physics. Z 5 n In flat space, the heat kernel (3.8) obeys KT (y, y)=KT (0, 0) so is independent of y. Thus if N ∼= R with a flat metric, this final y integral does not converge. It will converge if N is compact, say by imposing that we live in a large box, or on a torus etc..

– 19 – 3.1.2 Operators and correlation functions As in zero dimensions, we can also use the path integral to compute correlation functions of operators. A local operator is one which depends on the field only at one point of the worldline. The simplest types of local operators come from functions on the target space. If : N R O → is a real–valued function on N, let ˆ denote the corresponding operator on . Then for O H any fixed time t (0,T) we have ∈ H(T t) Ht y ˆ(t) y = y e− − ˆ e− y (3.18) ' 1|O | 0( ' 1| O | 0( in the Heisenberg picture. Inserting a complete set of ˆ(x) eigenstates x , this is O {| (} n H(T t) Ht n H(T t) Ht d x y e− − ˆ(x) x x e− y = d x (x) y e− − x x e− y ' 1| O | (' | | 0( O ' 1| | (' | | 0( ! ! (3.19) n = d x (x) KT t(y1,x) Kt(x, y0) , O − ! where we note that in the final two expressions (x) is just a number; the eigenvalue of ˆ O O in the state x . | ( Using (3.15b), everything on the rhs of this equation can now be written in terms of path integrals. We have

H(T t) Ht n S S y1 e− − ˆ e− y0 = d xt e− (xt) e− ' | O | ( ( CT t[y1,xt] ×O × Ct[x,y0] ) ! ! − ! (3.20) S = x e− (x(t)) , D O !CT [y1,y0] where to we again note that integrating over all maps x : [0,t] N with endpoint x(t)=x , → t then over all maps x :[t, T ] N with initial point x(t) again ﬁxed to x and ﬁnally inte- → t grating over all points x N is the same thing as integrating over all maps x : [0,T] N t ∈ → with endpoints y0 and y1.

More generally, we can insert several such operators. If 0

H(T tn) H(t2 t1) Ht1 y ˆ (t ) ˆ (t ) ˆ (t ) y = y e− − ˆ (x) ˆ (x)e− − ˆ (x)e− y ' 1|On n ··· O1 2 O1 1 | 0( ' 1| On ··· O2 O1 | 0( n S = x e− (x(t )) D Oi i !CT [y0,y1] i=1 & (3.21) for the n–point correlation function. The hats on the ˆ remind us that the lhs involves Oi operators acting on the Hilbert space . The objects inside the path integral are just H Oi ordinary functions, evaluated at the point x(t ) N 6. i ∈ Notice that in order to run our argument, it was very important that the insertion times ti obeyed ti

6 A more precise statement would be that they are functions on the space of ﬁelds CT [y0,y1] obtained by pullback from a function on N by the evaluation map at time ti.

– 20 – picture had this not been the case7. On the other hand, the insertions (x(t )) in the Oi i path integral are just functions and have no notion of ordering. Thus the expression on the right doesn’t have any way to know which insertion times was earliest. For this to be consistent, for a general set of times t (0,T) we must actually have { i}∈ n S x e− i(x(ti)) = y1 î y0 (3.22) CT [y0,y1D] * O + ' |T { O }| ( ! &i=1 &i where the symbol on the rhs is defined by T ˆ (t ) := (t ) , T O1 1 O1 1 ˆ (t ) ˆ (t ) := Θ(t t ) ˆ (t ) ˆ (t ) + Θ(t t ) ˆ (t ) ˆ (t ) , T{O1 1 O2 2 } 2 − 1 O2 2 O1 1 1 − 2 O1 1 O2 2 (3.23) . . . . and so on, where Θ(t) is the Heaviside step function. By construction, these step functions mean that the rhs is now completely symmetric with respect to a permutation of the orderings. However, for any given choice of times ti, only one term on the rhs can be non–zero. In other words, insertions in the path integral correspond to the time–ordered product of the corresponding operators in the Heisenberg picture. The derivative terms in the action play an important role in evaluating these correlation functions. For suppose we had chosen our action to be just a potential term V (x(t)) dt, independent of derivativesx ˙(t). Then, regularizing the path integral by dividing M into ' many small intervals as before, we would find that neighbouring points on the worldline 2 completely decouple: unlike in (3.12) where the geodesic distance d(xi+1,xi) in the heat kernel provides cross–terms linking neighbouring points together, we would obtain simply a product of independent integrals at each time step. Inserting functions (x(t )) that Oi i are likewise independent of derivatives of x into such a path integral would not change this conclusion. Thus, without the derivative terms in the action, we would have

(t ) (t ) = (t ) (t ) (3.24) 'O1 1 O2 2 ( 'O1 1 ('O2 2 ( for all such insertions. In other words, there would be no possible non–trivial correlations between objects at diﬀerent points of our (one–dimensional) Universe. This would be a very boring world: without derivatives, the number of people sitting in the lecture theatre would have nothing at all to do with whether or not a lecture was actually going on, and what you’re thinking about right now would have nothing to do with what’s written on this page. This conclusion is a familiar result in perturbation theory. The kinetic terms in the action allow us to construct a propagator, and using this in Feynman diagrams enables us to join together interaction vertices at diﬀerent points in space–time. As the name suggests, we interpret this propagator as a particle traveling between these two space–time interactions and the ability for particles to move is what allows for non–trivial correlation functions. Here we’ve obtained the same result directly from the path integral.

7 +TH Exercise: explain what goes wrong if we try to compute y1 e y1 with T>0. $ | | %

– 21 – A wider class of local path integral insertions depend not just on x but also on its worldline derivativesx, ˙ x¨ etc.. In the canonical framework, with Lagrangian L we have δL p = = g x˙ b (3.25) a δx˙ a ab where the last equality is for our action (3.1). Thus we might imagine replacing the function (xa, x˙ a) of x and its derivative in the path integral by the operator (ˆxa,gab(ˆx)ˆp ) in the O O b canonical framework. Now, probably the ﬁrst thing you learned in Quantum Mechanics was that [ˆxa, pˆ ] = 0, b * so at least for generic functions the replacement

(xa, x˙ a) (ˆxa, gâbpˆ ) O →O b is plagued by ordering ambiguities. For example, if we represent p by8 ∂/∂xa, then a − should we replace ∂ g xax˙ b xa ab →− ∂xa or should we take ∂ ∂ g xax˙ b xa = n xa ab →−∂xa − − ∂xa or perhaps something else? Even in free theory, we need to make a normal ordering prescription among the x’s and p’s to define what a composite operator means9. From the path integral perspective, however, something smells fishy here. I’ve been emphasizing that path integral insertions (x, x˙) are just ordinary functions, not operators. O How can two ordinary functions fail to commute? To understand what’s going on, we’ll need to look into the definition of our path integral in more detail.

3.2 The continuum limit In writing down the basic path integral (3.15b), we assumed it made sense to take the limit

N ? 1 n x = lim d xi a(xi) (3.26a) D N (2π∆t)n/2 →∞ &i=1 " # to construct a measure on the space of ﬁelds. We also assumed it made sense to write

N 1 2 ? − 1 x x S[x] = lim ∆t n+1 − n (3.26b) N 2 ∆t →∞ n=1 , $ % as the continuum action (here for a free particle). Alternatively, instead of splitting the interval [0,T] into increasingly many pieces, another possible way to deﬁne a regularized path integral starts by expanding each component of the ﬁeld x(t) as a Fourier series

a a 2πit/T x (t)= x˜k e . k Z ,∈ 8The absence of a factor of i on the rhs here is again a consequence of having a Euclidean worldline. 9And even there we may not be able to make a consistent choice. Read about the Groenewald–Van Hove theorem if you want sleepless nights.

– 22 – We now regularize by truncating this to a finite sum with k N. The (free) action for | |≤ the truncated field is 2π S = k2 δ xã x˜b (3.27) T ab k k k N | ,|≤ and depends only on the Fourier coefficients. We might now try to define the path integral measure as the limit N n ? d x˜ x = lim k (3.28a) D N (2π)n/2 →∞ k= N &− as an integral over more and more of these Fourier modes with higher and higher frequencies. The continuum action would then be taken to be the infinite series

? 2π 2 a b S[x] = lim k δab x˜k x˜k (3.28b) N T →∞ k N | ,|≤ which we hope converges. The obvious question to ask is whether the limits in (3.26a) & (3.26b) or in (3.28a) &(3.28b) actually exist. Perhaps the single most important fact in QFT is that the answer to this question is “No!”.

3.2.1 The path integral measure n To prove this, let’s keep things simple and work just with the case that N ∼= R with a flat metric, so that the space of fields is naturally an infinite dimensional vector space, where addition is given by pointwise addition of the fields at each t on the worldline. We’ll start with the measure. In fact, it’s easy to prove that there is no non–trivial Lebesgue measure on an infinite dimensional vector space. To see this, first recall that for finite dimension D,dµ is a Lebesgue measure on RD if it assigns a strictly positive D volume vol(U)= U dµ>0 to every non–empty open set U R , if vol(U &) = vol(U) ⊂ D whenever U & may be obtained from U by translation, and finally if for every x R there ' ∈ exists at least one open neighbourhood Ux containing x for which vol(Ux) < . The D ∞ standard example is of course dµ =d x. Now let Cx(L) denote the open (hyper)cube D centered on x and of side length L. This cube contains 2 smaller cubes Cxn (L/2) of side length L/2, all of which are disjoint. Then

2D vol(C (L)) vol(C (L/2)) = 2D vol(C (L/2)) (3.29) x ≥ xn x n=1 , where the first inequality uses the fact that the measure is positive–definite, and the final equality uses translational invariance. We see that as D , the only way the rhs can →∞ remain finite is if vol(C (L/2)) 0 for any finite L. So the measure must assign zero x → volume to any infinite dimensional hypercube. Finally, provided our vector space V is countably infinite (which both the Fourier series and discretized path integral make plain), we can cover any open U V using at most countably many such cubes C(L/2), so ⊂ vol(U) = 0 for any U and the measure must be identically zero.

– 23 – 3.2.2 Discretization and non–commutativity The question of whether the discretized action itself converges will shed light on the puzzle of how x and p might not commute in the path integral. It suﬃces to consider the simplest case of a free particle in one dimension, so choose N = R and V = 0. Then if 0

S H(T t+) H(t+ t) Ht x e− x(t)˙x(t )= y1 e− − xˆ e− − pˆe− y0 , (3.30a) D − ' | | ( !CT [y0,y1] when the insertion of x is later than that ofx ˙, and

S H(T t) H(t t ) Ht x e− x(t)˙x(t )= y e− − pê− − − xˆ e− − y (3.30b) D + ' 1| | 0( !CT [y0,y1] when x is inserted at an earlier time thanx ˙. Taking the limits t t from above and + → t t from below, the difference between the rhs of (3.30a) & (3.30b) is − → H(T t) Ht HT y e− − [ˆx, pˆ]e− y = y e− y (3.31) ' 1| | 0( ' 1| | 0( which does not vanish. By contrast, the difference of the lhs seems to be automatically zero. What have we missed? In handling the lhs of (3.30a)-(3.30b) we need to be careful. If we regularize the path integral by discretizing [0,T], chopping it into chunks of width ∆t, then we cannot pretend we are bringing the x andx ˙ insertions any closer to each other than ∆t without also taking account of the discretization of the whole path integral. Thus we replace

lim [x(t)˙x(t )] lim [x(t)˙x(t+)] t t − − t+ t −↑ ↓ by the discretized version

xt xt ∆t xt+∆t xt x − − x − (3.32) t ∆t − t ∆t where we stop the limiting procedure as soon as x coincides with any part of the discretized derivative. The order of the factors of xt and xt ∆t here doesn’t matter; they’re just ± ordinary integration variables. Now consider the integral over xt. Apart from the insertion of (3.32), the only dependence of the discretized path integral on this variable is in the heat kernels K∆t(xt+∆t,xt) and K∆t(xt,xt ∆t). Using the explicit form of these kernels in ﬂat space we have − xt xt ∆t xt+∆t xt dxt K∆t(xt+∆t,xt) xt − − xt − K∆t(xt,xt ∆t) ∆t − ∆t − ! $ % ∂ = dxt xt K∆t(xt+∆t,xt) K∆t(xt,xt ∆t) (3.33) − ∂x − ! t $ % = dxt K∆t(xt+∆t,xt) K∆t(xt,xt ∆t)=K2∆t(xt+∆t,xt ∆t) − − ! where the second step is a simple integration by parts and the ﬁnal step uses the concatenation property (3.10). The integration over xt thus removes all the insertions from the

– 24 – Figure 3: Stimulated by work of Einstein and Smoluchowski, Jean–Baptiste Perron made many careful plots of the locations of hundreds of tiny particles as they underwent Brownian motion. Understanding their behaviour played a key role in confirming the existence of atoms. A particle undergoing Brownian motion moves an average (rms) distance of √t in time t, a fact that is responsible for non–trivial commutation relations in the (Euclidean) path integral approach to Quantum Mechanics. path integral, and the remaining integrals can be done using concatenation as before. We are thus left with K (y ,y )= y e HT y in agreement with the operator approach. T 1 0 ' 1| − | 0( There’s an important point to notice about this calculation. Had we assumed the path integral included only maps x : [0,T] N that are everywhere differentiable, rather → than merely continuous, then the limiting value of (3.32) would necessarily vanish when ∆t 0, contradicting the operator calculation. Non–commutativity arises in the path → integral approach to Quantum Mechanics precisely because we’re forced to include non– differentiable paths, i.e. our map x C0(M,N) but x/C1(M,N). But because our ∈ ∈ path integral includes non–differentiable maps we cannot assign any sensible meaning to lim∆t 0 (xt+∆t xt)/∆t and the continuum action also fails to exist. → − This non–differentiability is the familiar stochastic (‘jittering’) behaviour of a particle undergoing Brownian motion. It’s closely related to a very famous property of random walks: that after a times interval t, one has moved through a net distance proportional to √t rather than t itself. More specifically, averaging with respect to the one–dimensional ∝ heat kernel 1 (x y)2/2t Kt(x, y)= e− − , √2πt in time t, the mean squared displacement is

(x y)2 = ∞ K (x, y)(x y)2 dx = ∞ K (u, 0) u2 du = t (3.34) ' − ( t − t !−∞ !−∞ so that the rms average displacement from the starting point after time t is √t. Similarly,

– 25 – our regularized path integrals yield a ﬁnite result because the average value of

x x x x x x 2 x t+∆t − t x t+∆t − t =∆t t+∆t − t , t+∆t ∆t − t ∆t ∆t $ % which for a diﬀerentiable path would vanish as ∆t 0, here remains ﬁnite. → 3.2.3 Non–trivial measures? The requirement that the measure be translationally invariant played an important role in the proof that the naive path integral measure x doesn’t exist. Do we really need this D requirement? In fact, in one dimension, while neither x nor S[x] themselves have any D continuum meaning, the limit

N n 2 d xti ∆t xti+1 xti dµW := lim exp − (3.35) N (2π∆t)n/2 − 2 ∆t →∞ ( ( )) &i=1 $ %

n n Si of the standard measures d xti on R at each time–step together with the factor e− does exist. The limit dµ rmW is known as the Wiener measure and, as you might imagine | from our discussion above, it plays a central role in the mathematical theory of Brownian S motion. Notice that the presence of the factor e− i means that this measure is certainly not translationally invariant in the fields, avoiding the no–go theorem. For Bryce de Wit, S the competition between the efforts of e− to damp out the contribution of wild field configurations and x to concentrate on such fields was poetically “The eternal struggle D between energy and entropy.”. Wiener’s result means that in one dimension the contest is beautifully balanced. In higher dimensions the situation is less clear. Certainly, the naive path integral measure does not exist. It is believed that Quantum Field Theories that are asymptotically free do have a sensible continuum limit, for reasons we’ll see later in the course. The most important example of such a QFT is Yang–Mills theory in four dimensions: every physicist believes this exists, but you can still pick up $1,000,000 from the Clay Institute for actually proving10 it. Perhaps more surprisingly, there are plenty of very important field theories for which a continuum path integral measure, of any sort, almost certainly does not exist. The most famous example is General Relativity, but it is also true of both Quantum Electrodynamics (QED) and very likely even the Standard Model. Yet planets orbit around the Sun and satellites orbit around the Earth in exquisite agreement with the predictions of General Relativity, QED is the arena for the most accurate scientific measurements ever carried out, and the Standard Model is the Crown Jewel in our understanding of Nature at the subatomic level. Clearly, not having a well–defined continuum limit does not mean these theories are so hopelessly ill–defined as to be useless. On the contrary, we can define effective quantum theories in all these cases that make perfect sense: we just restrict ourselves to taking the path integral only over low–energy modes, or over some discretized

10Terms and conditions apply; see here for details.

– 26 – version (such as putting the theory on a lattice). So long as we probe these theories within their domain of validity, they make powerful, accurate predictions. What lies beyond may not even be a QFT at all, but something else, perhaps String Theory.

3.3 Locality and Effective Quantum Mechanics To appreciate the notion of an effective field theory in a simple setting, let’s consider what happens in one dimension. We imagine we have two different fields x and y on the same worldline, which we’ll take to be a circle to avoid complications with end–points. I’ll choose the action to be

1 1 S[x, y]= x˙ 2 + y˙2 + V (x, y) dt (3.36) 1 2 2 !S " # where the potential 1 λ V (x, y)= (m2x2 + M 2y2)+ x2y2 (3.37) 2 4 allows the two fields to interact. In terms of the one–dimensional QFT, x and y look like interacting fields with masses m and M, while from the point of view of the target space R2 you should think of them as two harmonic oscillators with frequencies m and M, coupled together in a particular way. Of course, this coupling has been chosen to mimic what we did in section 2.6 in zero dimensions. If we are interested in perturbatively computing correlation functions of (local) operators that are independent of y(t), for example x(t )x(t ) , then we could proceed by ' 2 1 ( directly using (3.36) to construct Feynman diagrams. We’d find ingredients

x y

1/(k2 + m2) 1/(k2 + M 2) λ − where k is the one–dimensional worldline momentum (which would be quantized in units of the inverse circumference of the circle). On the other hand, we learned in section 2.6 that for such a class of observables, it is expendient to first construct an effective action by integrating out the y field directly. We expect this effective action to contain infinitely many new self–interactions of x which together take into account the effect of the unobserved y field. Let’s repeat that calculation here. As far as the path integral over y(t) is concerned, x is just a fixed background field so we have formally

1 T d2 λ d2 λ y exp y + M 2 + x2 y2 = det + M 2 + x2 (3.38) D −2 −dt2 2 −dt2 2 ! " !0 $ % # $ % where I’ve imposed the boundary conditions yy˙ = 0 on y(t). Accordingly, the eﬀective |t=0,T action for x is T 1 m2 d2 λ S [x]= x˙ 2 + x2 dt tr ln + M 2 + x2 (3.39) eﬀ 2 2 − −dt2 2 !0 " # $ %

– 27 – where we’ve used the identity ln det A = tr ln A (which holds provided A is a trace–class operator; don’t worry if you don’t know what this means).

Remarkably, the eﬀective action for x is non–local! To see this, suppose G(t, t&) is the worldline Green’s function that obeys

2 d 2 M G(t, t&)=δ(t t&) (3.40) dt2 − − $ % and so is the inverse of the operator d2/dt2 M 2 on the circle. Explicitly one has −

1 M t t +βk G(t, t&)= e− | − " | (3.41) 2M k Z ,∈ where β =1/T is the inverse circumference and k represents the momentum modes. Now, using tr ln(AB) = tr (ln A + ln B) = tr ln A + tr ln B we have

1 d2 x2 d2 d2 − x2 tr ln + M 2 + tr ln + M 2 = tr ln 1 λ M 2 −dt2 2 − −dt2 − dt2 − 2 $ % $ % * $ % + 2 λ 2 λ 2 2 = dtG(t, t) x (t) dt dt& G(t&,t) x (t) G(t, t&) x (t&)+ − 2 S1 − 8 S1 S1 ··· ! ! × n ∞ λ 2 2 2 = n dt1 dtn G(tn,t1) x (t1) G(t1,t2) x (t2) G(tn 1,tn) x (tn) − 2 n 1 n ··· ··· − n=1 !(S ) , (3.42) where the second term on the lhs is a divergent, but x(t) independent constant. Integrating out y has indeed generated an inﬁnite series of new interactions for x(t), but except for the (λ) term, these interactions are now non–local! O Again, it’s instructive to see why this non–locality has arisen. The ﬁrst two terms in the series (3.42) represent the Feynman diagrams

x(t) x(t) x(t!) G(t!,t) G(t, t)

G(t, t!) x(t) x(t) x(t!) that arise in the perturbative evaluation of the y path integral. Unlike the trivial case of zero dimensions, here the y field is dynamical; in particular it has its own worldline propagator G(t, t&) that allows it to move around on the worldline. (Note that in these diagrams, I’ve drawn the external blue vertices at different places on the page just for clarity. Each external x field in the diagram on the left resides at the point t S1, while ∈ the four xs on the right live pairwise at points t and t&.) Non–locality is generally bad news in physics: the equations of motion we’d obtain from Seff [x] would be integro–differential equations stating that in order to work out the behaviour of the field x here, we first have to add up what it’s doing everywhere else in the (one–dimensional) Universe. But we don’t want the results of our experiment in CERN to

– 28 – depend on what Ming the Merciless may or may not be having for breakfast over on the far side of the Galaxy. So how bad is it here? From the explicit form (3.41) of the Green’s function we see that G(t, t&) decays exponentially quickly when t = t , with a scale set by the inverse mass M 1 of y. This suggests * & − that the effects of non–locality will be small provided we restrict attention to fields whose derivatives vary slowly on timescales M 1. More specifically, expanding x(t) we have ∼ − 2 2 2 dt dt& G(t, t&) x (t) x (t&) ! 2 2 2 2 1 2 = dt dt& G(t, t&) x (t) x (t)+2x(t)˙x(t)(t t&)+ x˙ (t)+ x(t)¨x(t) (t t&) + − 2 − ··· ! " $ % # α β 1 γ = dt x4(t)+ x2x˙ 2 + x2x¨ + (four derivative terms) + . M M 3 2 M 5 ··· ! " $ % # (3.43) In going to the last line we have performed the t& integral. To do so, note that the Green’s function G(t, t&) depends on t& only through the dimensionless combination u = M(t t&). p th − Thus we replace the factor (t t&) in the p order term in the Taylor expansion by p − (u/M) and change variables dt& =du/M to integrate over the dimensionless quantity u. In particular, the infinite series of dimensionless constants α, β, γ, are just some ··· dimensionless numbers – their precise values don’t matter for the present discussion. The important point is that every new derivative of x in these vertices is suppressed ... by a further power of the mass M of the y field. Thus, so long asx, ˙ x,¨ x ,... are all small 1 in units of M − , we should have a controllable expansion. In particular, if we truncate the infinite Taylor expansion and the infinite expansion (3.42) at any finite order, we will regain an apparently local effective action. This truncation is justified provided we restrict to processes where the momentum of the x field is M. / However, once we start to probe energies M something will go badly wrong with our ∼ truncated theory. Assuming the original action (3.36) defined a unitary theory (at least in Minkowski signature), simply performing the exact path integral over y must preserve unitary. This is because we haven’t yet made any approximations, just taken the first step to performing the full x y path integral. All the possible states of the y field are D D still secretly there, encoded in the infinite series of non–local interactions for x. However, the approximation to keep just the first few terms in Seff can’t be unitary, because we’re rejecting by hand various pieces of Feynman diagrams: we’re throwing away some of the things y might have been doing. The weak interactions are responsible for many important things, from the formation of light elements such as deuterium in the early Universe, to powering stars such as our Sun, to the radioactive β-decay of 14C used in radiocarbon dating. Since the 1960s physicists have known that these weak interactions are mediated by a field called the W–boson and in 1983, the UA1 experiment at CERN discovered this field and measured its mass to be M 80 GeV. Typically, β-decay takes place at much lower energies, so to describe them W 0 it makes sense to integrate out the dynamics of the W boson leaving us with an effective action for the proton, neutron, electron and neutrino that participate in the interaction.

– 29 – This effective action contains an infinite series of terms, suppressed by higher and higher powers of the large mass MW. Truncating this infinite effective action to its first few terms leads to Fermi’s theory of β-decay which gives excellent results at low energies. However, if ones extrapolates the results obtained using this truncated action to high energies, one finds a violation of unitarity. The non–unitarity in Fermi’s theory is what lead physicists to suspect the existence of the W–boson in the first place.

3.4 The worldline approach to perturbative QFT In this chapter, we’ve been studying the case of QFT in 1 space–time dimension as a warm–up for the higher–dimensional QFTs we’ll meet later on. Before proceeding, I’d like to point out an alternative approach to perturbative QFT that was invented by Feynman.

Let’s start by considering maps x : [0,T] Rn with the free action → T 1 m2 m2 T 1 S[x]= dt δ x˙ ax˙ b + = T + dt x˙ 2 , (3.44) 2 ab 2 2 2 !0 " # !0 where I’ve included a constant term m2/2 in the Lagrangian. This may seem like a strange step; the constant term does not aﬀect the dynamics of the ﬁeld x in any way. Indeed, the path integral over x becomes

S Tm2/2 HT x e− =e− y e− x (3.45) D ' | | ( !CI [x,y] with the constant term in the action providing just an overall factor. Its true purpose will be revealed below. a a With this action, the momentum conjugate to the ﬁeld x is pa = ∂L/∂x˙ =˙xa, so the a a n Hamiltonian is H = pax˙ L = pap /2, as expected for a free particle on R . Therefore, − by inserting complete sets of momentum eigenstates, we have

HT n n HT y e− x = d p d q y p p e− q q x ' | | ( ' | (' | | (' | ( ! n (3.46) d p ip (x y) Tp2/2 = e · − e− (2π)n ! and so the path integral becomes n S d p ip (x y) T (p2+m2)/2 x e− = e · − e− . (3.47) D (2π)n !CI [x,y] ! (An alternative way to obtain the same result is to write the ﬂat space heat kernel (3.8) as its inverse Fourier transform.) Feynman noticed that if we integrate this expression over all possible lengths T (0, ) of our worldline, then we obtain ∈ ∞ n ip (x y) ∞ S d p e · − dT x e− =2 (3.48) D (2π)n p2 + m2 !0 !CI [x,y] ! which is the Fourier transform of the momentum space propagator 1/(p2 + m2) for a scalar ﬁeld Φ(x) of mass m on the target space Rn.

– 30 – Feynman now realized that one could describe several such particles interacting with one another if one replaced the worldline I by a worldgraph Γ. For example, to obtain a perturbative evaluation of the r–point correlation function

Φ(x )Φ(x ) ... Φ(x ) ' 1 2 r ( of a massive scalar ﬁeld Φ(x) in λΦ4 theory on Rn, one could start by considering a 1– dimensional QFT living on a 4–valent graph Γ with r end–points. This QFT is described by the action (3.44), where x is constrained to map each end–point of the graph to a n diﬀerent one of the Φ insertion points xi R . We assign a length Te to each edge e of ∈ the graph, which in this context is often known as a Schwinger parameter of the graph. We now take the path integral over all maps x :Γ Rn and integrate over the Schwinger → parameters of each edge. Part of what is meant by an ‘integral over all maps x :Γ Rn’ includes an integral → over the location in Rn to which each vertex of Γ is mapped. When we perform this ip (x y) integral, the factors of e · − in the path integral (3.47) for each edge lead a to target space momentum conserving δ–function at each vertex. As in (3.48), integrating over the Schwinger parameters generates a propagator 1/(p2 +m2) for each edge of the graph. Thus, after including a factor of ( λ) v(Γ) and dividing by the symmetry factor of the graph, our − | | 1–dimensional QFT has generated the same expression as we would have obtained from Feynman rules for λΦ4 on Rn. For example, the 4–valent graph with two end–points shown here:

T T x 1 2 y z corresponds to the path integral expression

λ ∞ S ∞ S ∞ S − dT1 x e− dT2 x e− dT3 x e− 4 0 C [x,zD] × 0 C [y,zD] × 0 C [z,zD] ! ! T1 ! ! T2 ! ! T3 λ dnp dnq dn) eip (x z) eiq (y z) ei# (z z) = − dnz · − · − · − (3.49) 4 (2π)n (2π)n (2π)n p2 + m2 q2 + m2 )2 + m2 ! λ dnp dn) eip (x y) = − · − . 4 (2π)n (2π)n (p2 + m2)2 ()2 + m2) ! This is the same order λ contribution to the 2–point function Φ(x)Φ(y) that we’d obtain ' ( from (Fourier transforming) the momentum space Feynman rules for λΦ4 theory, with the graph treated as a Feynman graph in Rn rather than a one–dimensional Universe. To obtain the full perturbative expansion of Φ(x )Φ(x ) Φ(x ) we now sum over ' 1 2 ··· n ( all graph topologies appropriate to our 4–valent interaction. Thus

v(Γ) ( λ)| | ∞ e(Γ) SΓ[φ] Φ(x1)Φ(x2) Φ(xn) = − d| |T φ e− , (3.50) ' ··· ( Aut Γ 0 CΓ[x1,x2,...,xn] D ,Γ | | ! !

– 31 – where e(Γ) and v(Γ) are respectively the number of edges and vertices of Γ. | | | | This worldline approach to perturbative QFT is close to the way Feynman originally thought about the subject, presenting his diagrams at the Pocono Conference of 1948. The relation of this approach to higher (four) dimensional QFT as we usually think about it was worked out by Dyson a year later, long before people used path integrals to compute anything in higher dimensions. Above, we’ve described just the simplest version of this picture, relevant for a scalar theory on the target space. There are more elaborate D =1 QFTs that would allow us to obtain target space Quantum Mechanics for particles with spin, and we could also allow for more interesting things to happen at the interaction vertices of our worldgraphs. In this way, one can build up worldline approaches to many perturbative QFTs. This way of thinking can still be useful in practical calculations today, and still occasionally throws up conceptual surprises, but we won’t pursue it further in this course. Finally, I can’t resist mentioning that what we’ve really been studying in this section is not merely one–dimensional QFT, but one–dimensional Quantum Gravity. In one dimension a Riemannian metric is just a 1 1 matrix g (t) with positive eigenvalues; × tt in other words, a positive number at each point t of the worldline. General coordinate transformations act on this 1 1 matrix as × dt dt dt 2 g g = g = g , (3.51) tt → t"t" dt dt tt dt tt & & $ & % and so can be used to rescale the value of the metric to anything we like. Throughout this chapter, we’ve implicitly been using a coordinate system t on the worldline in which the worldline metric had been ﬁxed to 1, which we’re always free to do. The proper length T of the worldline interval I can be written

T = dt √gtt = dt& √gt"t" (3.52) !I !I and is invariant under the general coordinate transform (3.51). In fact, in one dimension the length T is the only diffeomorphism invariant of a Riemannian metric, essentially because there is no ‘room’ for any sort of curvature. Thus, the integral over the lengths of all the edges of our graph in (3.50) is best thought of as an integral over the space of all possible Riemannian metrics on Γ, up to diffeomorphism invariance. Rather grandly, this is known as the moduli space of Riemannian metrics on Γ. Furthermore, in summing over graphs Γ we were really summing over the topological type of our one dimensional Universe. Notice that the vertices of our graphs are singularities of the one–dimensional Riemannian manifold, so we’re allowing our Universe to have such wild (even non–Hausdorff) behaviour. So for fixed lengths Te the path integral over x(t) is the ‘matter’ QFT on a fixed background space Γ, while the integral over the lengths of edges in Γ together with the sum over graph topologies is Quantum Gravity. This picture is also very close to perturbative String Theory. There, as you’ll learn if you’re taking the Part III String Theory course, the worldgraph Γ is replaced by a two dimensional worldsheet (Riemann surface) Σ, the D = 1 worldline QFT replaced by a

– 32 – D = 2 worldsheet CFT11. Likewise, the integral over the moduli space of Riemannian metrics on Γ becomes an integral over the moduli space of Riemann surfaces, and ﬁnally the sum over graphs is replaced by a sum over the topology of the Riemann surface. We know that the worldgraph approach to QFT only captures some aspects of perturbation theory, and in the following chapters we’ll see that deeper insight is provided by QFT proper. Asking whether there’s a similarly deeper approach to String Theory will take you to the mystic shores of String Field Theory, about which very little is known.

11CFT = Conformal Field Theory.

– 33 – 4 The Renormalization Group

“The most incomprehensible thing about the Universe is that it is comprehensible.” So said Einstein, as quoted by an early biographer. Indeed, a humble glass of pure water consists of countless H2O molecules, which are made from atoms that involve many electrons perpetually executing complicated orbits around a dense nucleus, the nucleus itself is a seething mass of protons and neutrons glued together by pion exchange, these hadrons are made from the complicated and still poorly understood quarks and gluons which themselves maybe all we can make out of tiny vibrations of some string, or modes of a theory yet undreamed of. How then is it possible to understand anything about water without first solving all the deep mysteries of Quantum Gravity? In classical physics the explanation is really an aspect of the Principle of Least Action: if it costs a great deal of energy to excite a degree of freedom of some system, either by raising it up its potential or by allowing it to vary rapidly over space–time, then the least action configuration will be when that degree of freedom is in its ground state. The corresponding field will be constant and at a minimum of the potential. This constant is the zero mode of the field, and plays the role of a Lagrange multiplier for the remaining low– energy degrees of freedom. You used Lagrange multipliers in mechanics to confine wooden beads to steel hoops. This is a good description at low energies, but my sledgehammer can excite degrees of freedom in the hoop that your Lagrange multiplier doesn’t reach. We must re-examine this question in QFT because we’re no longer constrained to sit at an extremum of the action. The danger is already apparent in perturbation theory, for even in a process where all external momenta are small, momentum conservation at each vertex still allows for very high momenta to circulate around the loop and the value of these loop integrals would seem to depend on all the details of the high–energy theory. The Renormalization Group (RG), via the concept of universality, will emerge as our quantum understanding of why it is possible to understand physics at all.

4.1 Running couplings Suppose we have some QFT with cut–oﬀ Λ, governed by the eﬀective action

eff d 1 µ d di SΛ [ϕ; gi]= d x ∂ ϕ∂µϕ + Λ − gi i(x) . (4.1) "2 O $ ! #i Here we have allowed arbitrary local operators (x) of dimension d > 0 to appear in the Oi i action; for example could any Lorentz–invariant term of the schematic form (∂ϕ)rϕs, Oi including both fields and their derivatives. For later convenience, I’ve included explicit factors of Λ in the couplings so as to ensure that the coupling constants gi themselves are dimensionless. Now suppose we want to use this QFT to predict some physical quantity M, such as the mass or lifetime of a some physical state. The value we compute for this quantity will be a function of the couplings gi and of the cut–off Λ. It cannot depend on the fields themselves as these are integrated out in performing the path integral. Thus M = MΛ(gi).

– 34 – Dependence on the cut–off violates our intuition that the details of ultra–high–energy physics do not affect everyday life. The renormalization group equation enshrines this into a principle: physical quantities must not depend on how we regularize our theory at high energy. Since the path integral certainly knows about Λ, both through the action (4.1) and as a cut–off on the modes we consider in the path integral measure, the only way that MΛ(gi) can remain constant as Λ is varied is if the couplings gi themselves also vary with Λ in such a way to cancel out any explicit dependence in MΛ(gi). We can write formalize the notion that couplings vary to ensure physical quantities are independent of the cut–off as the equation

MΛ! (gi(Λ")) = MΛ(gi(Λ)) , (4.2) or equivalently

dMΛ(g) ∂ ∂gi(Λ) ∂ Λ = Λ +Λ MΛ(g) = 0 (4.3) dΛ ∂Λ ∂Λ ∂gi % &gi &Λ' & & & & for an infinitesimal change. Equation& (4.3) is known& as the renormalization group equation for M, or often the Callan–Symanzik equation for M in honour of its inven- tors. I remark that if a physical quantity contains (apparent) dependence on the cut–off through more things than just the couplings gi(Λ), then we’ll need to generalize the Callan– Symanzik equation to account for this new dependence; we’ll meet these more general versions later. The fact that couplings vary, or ‘run’, as we change the cut–off is an important notion. Lowering the cut–off from Λ to Λ" means that we are integrating out field modes with energies Λ < p Λ. As we saw in our toy examples in zero and one dimensions, it’s quite " | |≤ natural to expect the couplings to change under such an operation. However, it seems strange: you’ve learned that the electromagnetic coupling

e2 1 α = . 4π%0!c ≈ 137 What can it mean for the fine structure constant to depend on the cut–off? We’ll understand the answer to such questions later. Since the running of couplings is so important, we give it a special name and define the beta–function βi of the coupling gi to be its derivative with respect to the logarithm of the cut–off scale: ∂g β := Λ i . (4.4) i ∂Λ Because we included explicit powers of Λ in the action (4.1) β-functions for dimensionless couplings take the form

β (g (Λ)) = (d d) g (Λ) + βquant(g ) (4.5) i j i − i i j where the ﬁrst term just compensates the variation of the explicit power of Λ in front of quant the coupling in (4.1). The second term βi represents the quantum eﬀect of integrating

– 35 – out the high–energy modes. To actually compute this term requires us to perform the path integral and so will generically introduce dependence on all the other couplings in (4.1), so that the β-function for gi is a function of all the couplings βi(gj).

4.1.1 Wavefunction renormalization and anomalous dimensions Of all the possible couplings in (4.1), one plays a rather distinguished role. As we saw in one dimension, quantum corrections will generate new contributions to all terms in the action, including those involving derivatives. In particular, it’s perfectly possible for the kinetic term (∂ϕ)2 to receive corrections. We can generalize the form of our eﬀective action to allow for this possibility, writing

eff 1/2 d 1 2 d di (r+s)/2 SΛ [ZΛ φ, gi(Λ)] = d x ZΛ(∂φ) + Λ − gi(Λ) ZΛ i(x) (4.6) "2 O $ ! #i where ZΛ is known as the wavefunction renormalization for φ (and is not to be con- fused with the partition function !). Notice that in this definition, the wavefunction Z renormalization also appears in the interactions, scaling each one according to the number 1/2 r + s of fields it involves. At any given scale Λ, we can of course define a field ϕ := ZΛ φ to aborb this factor (and we’ve been implicitly doing this up to now). However, moving to a new scale Λ" will generically cause a new wavefunction renormalization factor to emerge. Wavefunction renormalization plays an important role in correlation functions. Sup- pose we wish to compute the n–point correlator

eff 1/2 1 S [Z φ; gi(Λ)] φ(x1) φ(xn) := φ e− Λ Λ φ(x1) φn(xn) (4.7) $ ··· % C∞(M) ΛD ··· Z ! ≤ of fields inserted at points x1, . . . , xn M using the action (4.6). Changing variables to 1/2 ∈ ϕ := ZΛ φ this is n/2 φ(x ) φ(x ) = Z− ϕ(x ) ϕ(x ) (4.8) $ 1 ··· n % Λ $ 1 ··· n % since the change in the measure φ ϕ cancels in the normalization. Upon performing D →D the ϕ path integral we will (in principle!) evaluate the remaining ϕ correlator as some (n) function ΓΛ (x1, . . . , xn; gi(Λ)) that depends on the couplings, the scale Λ and also on the fixed points x . Now, if the field insertions involve just the low–energy modes then we { i} should also be able to compute the same correlator using the low–energy theory. Accounting for the field renormalization gives

n/2 (n) n/2 (n) Z− Γ (x1, . . . , xn; gi(Λ)) = Z− Γ (x1, . . . , xn; gi(Λ")) , (4.9) Λ Λ Λ! Λ! or equivalently d ∂ ∂ Λ Γ(n)(x , . . . , x ; g (Λ)) = Λ + β + nγ Γ(n)(x , . . . , x ; g (Λ)) (4.10) dΛ Λ 1 n i ∂Λ i ∂g φ Λ 1 n i ( i ) for an inﬁnitesimal shift. Here we’ve denoted the running of the wavefunction renormalization factor by 1 ∂ ln Z γ := Λ Λ . (4.11) φ −2 ∂Λ

– 36 – γφ is known as the anomalous dimension of the field φ for reasons that will be apparent momentarily. Equation (4.10) is the generalized Callan–Syman equation appropriate for correlation functions. Once again, it simply says that a physical quantity such as a correlation function cannot depend on the cut–off. Thinking about correlation functions allows us to have an alternative interpretation of renormalization that is often useful. Correlation functions depend on scales — the typical separation between insertion points — quite apart from the cut–off. We pick some s>1 and lower our cut–off from Λ to Λ/s, but then rescale the space–time metric as g s 2g. This metric rescaling reduces lengths by a factor of s, and so restores the → − cut–off scale Λ/s to Λ. It’s important to realize that the rescaling has nothing to do with integrating out degrees of freedom in the path integral; it’s simply a rescaling of the metric. With dimensionless couplings, the action is invariant under the rescaling provided we set (d 2)/2 φ(x/s)=s − φ(x). Under the combined operator of lowering the cut–off and then rescaling, (4.9) can be written n/2 (n) d 2 ZΛ (n) Γ (sx , . . . , sx ; g (Λ)) = s − Γ (x , . . . , x ; g (Λ/s)) (4.12) Λ 1 n i Z Λ/s 1 n i * Λ/s + n(d 2)/2 where the factor of s − arises from the field insertions in (4.8), and where I’ve started with insertions at points sxi rather than xi. Notice that the couplings gi and wavefunction renormalization on the rhs are evaluated at the point Λ/s appropriate for the low–energy theory: the values of these dimensionless quantities are not affected by our subsequent metric rescaling. Equation (4.12) has an important interpretation. On the left stands a correlation function where the separations between operators are proportional to s. Thus, as s increases we are probing the long distance, infra–red properties of the theory. We see from the rhs that this may be obtained by studying a correlation function where all separations are held constant, but the cut–off is lowered by a factor of s. This makes perfect sense: the IR properties of the theory are governed by the low–energy modes that survive as we integrate out more and more high–energy degrees of freedom. This equation also allows us to gain insight into the meaning of the anomalous di- n(d 2)/2 mension γφ. The power of s − on the rhs of (4.12) is the classical scaling behaviour we’d expect for an object of mass dimension n(d 2)/2. The wavefunction renormalization − factors (and different values of the couplings) are non–trivial, arising from integrating out degrees of freedom. Nonetheless, (4.12) shows that the net effect of integrating out high– energy modes is to modify the expected classical scaling by a simple factor. To quantify this modification, set s = 1 + δs with δs 1. The prefactor in (4.12) becomes ( 1/2 d 2 ZΛ d 2 s − = 1 + − + γ δs + (4.13) Z 2 φ ··· * Λ/s + * + with γφ as in (4.11). We see that the correlation function behaves as if the field scaled with mass dimension (d 2)/2+γ rather than the classical value (d 2)/2, which is where γ − φ − φ

– 37 – gets its name. The anomalous dimension γφ can be viewed as a β-function for the kinetic term. Like any β-function, it depends on the values of all the couplings in the theory.

4.2 Integrating out degrees of freedom It’s time to start to think about how couplings in the eﬀective action actually change as the cut–oﬀ scale Λ is lowered. In this section I want to explain this in a way that I think is conceptually clear, and the natural generalization of what we have already seen in zero and one dimension. However, I’ll warn you in advance that the techniques here are not the most convenient way to calculate β-functions. We’ll consider a simpler, but conceptually murkier, technique in chapter 6.

eff As before, given an effective action SΛ defined at a cut–off at scale Λ, the partition function is Seff [ϕ]/! Λ(gi(Λ)) = ϕ e− Λ (4.14) Z C∞(M) ΛD ! ≤ where the integral is taken over the space C∞(M) Λ of smooth functions on M whose eff ≤ energy is at most Λ. As in (4.1), SΛ is the effective action allowing for couplings to all operators in the theory. The first thing to note about this integral is that it makes sense: we’ve explicitly regularized the theory by declaring that we are only allowing momentum modes up to scale Λ. For example, there can be no UV divergences12 in any perturbative loop integral following from (4.14), because the UV region is simply absent. For the partition function to be independent of the cut–off scale Λ we must have

eff eff S [φ]/! SΛ [ϕ]/! φ e− Λ! = ϕ e− (4.15) D D C∞(M) Λ C∞(M) Λ ! ≤ ! ! ≤ eff so that the effective action S at scale Λ" < Λ compensates for the change in the degrees Λ! of freedom over which we take the path integral. The space C∞(M) Λ is naturally a vector ≤ space with addition just being pointwise addition on M. Therefore we can split a general field ϕ(x) as

d d p ip x ϕ(x)= d e · ϕ˜(p) p Λ (2π) !| |≤ d d d p ip x d p ip x (4.16) = d e · ϕ˜(p)+ d e · ϕ˜(p) p Λ (2π) Λ < p Λ (2π) !| |≤ ! ! ! | |≤ =: φ(x)+χ(x) , where φ C∞(M) Λ is the low–energy part of the field, while χ C∞(M)(Λ ,Λ] has high ∈ ≤ ! ∈ ! energy. The path integral measure on C∞(M) Λ likewise factorizes as ≤ ϕ = φ χ D D D 12On a non–compact space–time manifold M there can be IR divergences. This is a separate issue, unrelated to renormalization, that we’ll handle later if I get time. If you’re worried, think of the theory as living in a large box of side L with either periodic or reflecting boundary conditions on all fields. Momentum is then quantized in units of 2π/L, so the space C∞(M) Λ is finite–dimensional. ≤

– 38 – into a product of measures over the low– and high–energy modes. Using this decomposition, the renormalization group equation (4.15) says that the eﬀective action at lower scale Λ" is deﬁned by the path integral

eff eff S [φ] := ! log χ exp S [φ + χ]/! (4.17) Λ! − D − Λ " C∞(M) $ ! (Λ!,Λ] , - over the high–energy modes only. Equation (4.17) is the renormalization group equation for the effective action. Separating out the kinetic part, we write

eﬀ 0 0 int SΛ [φ + χ]=S [φ]+S [χ]+SΛ [φ, χ] (4.18) where S0[χ] is the kinetic term

1 1 S0[χ]= ddx (∂χ)2 + m2χ2 (4.19) 2 2 ! * + for χ and S0[φ] is similar. (We can always normalize the fields in the scale Λ effective action so that Z = 1 at this starting scale.) Notice that the quadratic terms can contain no cross–terms φχ, because these modes have different support in momentum space. For ∼ int the same reason, the terms in the effective interaction SΛ [φ, χ] must be at least cubic in the fields. Since φ is non–dynamical as far as the χ path integral goes, we can bring S0[φ] out of the rhs of (4.17). Observing that the same φ kinetic action already appears on the lhs, we obtain (! = 1)

Sint[φ]= log χ exp S0[χ] Sint[φ, χ] (4.20) Λ! − D − − Λ " C∞(M) $ ! (Λ!,Λ] . / which is the renormalization group equation for the interactions. In perturbation theory, the rhs of (4.20) may be expanded as an inﬁnite series of connected Feynman diagrams. If we wish to compute the eﬀective interaction Sint[φ] as an Λ! integral over space–time in the usual way, then we should use the position space Feynman rules. As in section 3.4, the position space propagator D(χ)(x, y) for χ is

d ip (x y) (χ) d p e · − D (x, y)= d 2 2 (4.21) Λ < p Λ (2π) p + m ! ! | |≤ where we note the restriction to momenta in the range Λ" < p Λ. As usual, vertices int d | |≤ from SΛ [φ, χ] come with an integration d x over their location that imposes momentum conservation at the vertex. Now, diagrams that exclusively involve vertices which are 0 independent of φ contribute just to a field–independent term on the lhs of (4.20). This term represents the shift in vacuum energy due to integrating out the χ field; we will henceforth ignore it13. The remaining diagrams use vertices including at least one φ field,

13This is harmless in a non–gravitational theory, but is really the start of the cosmological constant problem.

– 39 –

. . n 2 . . . .

d . − . . .

= . +

Λ . .

. dΛ . r=2 ! gn gr+1 gn r+1 gn+2 − Figure 4: A schematic representation of the renormalization group equation for the effective interactions when the scale is lowered infinitesimally. Here the dashed line is a propagator of the mode with energy Λ that is being integrated out, while the external lines represent the number of low–energy fields at each vertex. All these external fields are evaluated at the same point x The total number of fields attached to a vertex is indicated by the subscripts on the couplings gi. treated as external. Evaluating such a diagram leads to a contribution to the effective int interaction S [φ] at scale Λ". Λ!

For general scales Λ" and Λ equation (4.20) is extremely diﬃcult to handle; the integral on the right is a full path integral in an interacting theory. To make progress we consider the case that we lower the scale Λ only inﬁnitesimally, setting Λ =Λ δΛ. To lowest " − order in δΛ, the χ propagator reduces to

d 1 (χ) 1 Λ − δΛ iΛpˆ (x y) DΛ (x, y)= d 2 2 dΩ e · − (4.22) (2π) Λ + m d 1 !S − as the range of momenta shrinks down, where dΩ denotes an integral over a unit (d 1)- − sphere in momentum space. This is a huge simplification! Since every χ propagator comes with a factor of δΛ, to lowest order in δΛ we need only consider diagrams with a single χ propagator. Since φ is treated as an external field, we have only two possible classes of diagram: either the χ propagator links together two separate vertices in Sint[φ, χ] or else it joins a single vertex to itself. This diagrammatic represention of the process of integrating out degrees of freedom is shown in figure 4. It has a very clear intuitive meaning. The mode χ appearing in the propagator is the highest energy mode left in the scale Λ theory. It thus probes the eff shortest distances we can reliably access using SΛ . When we integrate this mode out, we can no longer resolve distances 1/Λ and our view of the ‘local’ interaction vertices is correspondingly blurred. The graphs on the rhs of figure 4 represent new contributions to the n–point φ vertex in the lower scale theory coming respectively from two nearby vertices joined by a χ field, or a higher point vertex with a χ loop attached. Below scale Λ we image that we are unable to resolve the short distance χ propagator.

– 40 – 4.2.1 Polchinski’s equation We can write an equation for the change in the effective action that captures the information in the Feynman diagrams in figure 4. It was first obtained by Ken Wilson and is

∂Sint[φ] δSint δSint δ2Sint Λ Λ = ddx ddy D (x, y) D (x, y) , (4.23) − ∂Λ δφ(x) Λ δφ(y) − Λ δφ(x) δφ(y) ! * + where DΛ(x, y) is the propagator (4.22) for the mode at energy Λ that is being integrated out. The variations of the effective interactions tell us how this propagator joins up the various vertices. Notice in the second term that since Sint[φ] is local, both the δ/δφ variations must act at the same place if we are to get a non–zero result. On the other hand, the first term generates non–local contributions to the effective action since it links fields at x to fields at a different point y. In position space we expect a propagator at scale Λ2 + m2 to lead to a potential e √Λ2+m2 r/rd 3 so this non–locality is mild and we can ∼ − − expanding the fields in δSint/δφ(y) as a series in (x y). This leads to new contributions − to interactions involving derivatives of the fields, just as we saw in section 3.3 in one Sint[φ] dimension. Finally, the minus signs in (4.23) comes from expanding e− to obtain the Feynman diagrams. It’s convenient to rewrite equation (4.23) as

Sint[φ] 2 ∂ e− d d δ Sint[φ] Λ = d x d yD (x, y) e− , (4.24) ∂Λ − Λ δφ(x) δφ(y) ! in which form it is known as Polchinski’s exact renormalization group equation14. Polchinski’s equation shows that the process of integrating out degrees of freedom results in a form of heat equation, with ‘Laplacian’

δ2 ∆= ddx ddyD (x, y) (4.25) Λ δφ(x) δφ(y) ! on the space of fields. We thus anticipate that the process of renormalization will be somewhat akin to heat flow, with t ln Λ playing the role of renormalization group ‘time’15. ≡ 4.3 Renormalization group flow

The most important property of heat ﬂow on Rn is that it is a strongly smoothing operation. n Expanding a function f : R R>0 R as × →

f(x, t)= f˜k(t) uk(x) #k in terms of a basis of eigenfunctions uk(x) of the Laplacian, under heat ﬂow the coeﬃcients λ t evolve as f˜k(t)=f˜k(0) e− k . Consequently, all components f˜k(t) corresponding to positive

14Polchinski actually wrote a slightly more general version of the momentum space version of this equation, which he arrived at by a somewhat diﬀerent method to the one we have used here. 15In the AdS/CFT correspondence, this RG time really does turn into an honest direction: into the bulk of anti–de Sitter space!

– 41 – eigenvalues λk are quickly damped away, with only the constant piece surviving. This just corresponds to the well– known fact that a heat spreads out from areas of high concentration (say near a flame) until the whole room is at constant temperature. Something very similar happens under renormalization group flow. As we shall soon int see, almost every interaction (or operator) that can appear in SΛ corresponds to a positive eigenvalues of the RG Laplacian, and so is quickly suppressed as we lower the cut–off through RG time t. Via the metric rescaling argument given in section 4.1.1, we can equivalently say that the effect of these operators becomes rapidly less important as we keep the cut–off fixed, but probe the theory at long distances. Operators that are suppressed in as we head into the IR are known as irrelevant. On the other hand, finitely many (and typically very few) operators will correspond to eigenfunctions with negative eigenvalues. These operators become increasingly important as we lower the cut–off, or as we probe the theory in the IR. Such operators are called relevant. The remaining case is marginal operators, which have vanishing eigenvalues and so neither increase nor decrease under RG flow. Let me point out that certain operators may correspond to non–zero eigenvalues that are nonetheless extremely small. Thus, while these operators will ultimately be either relevant or irrelevant, for sufficiently small RG evolution they will behave like marginal operators. Such operators are called either marginally relevant or marginally irrelevant and play an important role phenomenonlogically. I also emphasize that, just as with the Laplacian on Rn, the eigenfunctions themselves may look completely different to any individual term you choose to include neatly in the effective interaction Seff . A simple–looking individual operator that appears in (4.1) or is explicitly inserted into a correlation Oi function could actually consist of many RG eigenfunctions. We say that operators mix, because a given operator transforms under RG flow into its dominant eigenfunction. The picture of the RG flow of couplings as a form of heat flow is very powerful. In the infinite dimensional space of theories, whose coordinates are all possible couplings g in { i} the effective action, we define the critical surface to be the surface where the couplings to all relevant operators vanish. I’ve stated that there are only finitely many relevant operators in any QFT, whereas there are infinitely many irrelevant ones, so the critical surface is infinite dimensional. If we pick any QFT on this critical surface, under RG flow all the irrelevant operators in Sint will be exponentially suppressed. Consequently, all these theories16 flow towards a critical point where all the β-functions vanish and only marginal operators remain. This is analogous to heat flow on a Riemannian manifold; heat spreads out from an initial spike until the whole room is at constant temperature. A region in the critical surface within the domain of attraction of a critical point is sketched in figure 5.

We usually denote the couplings at a critical point as gi∗ — they are the coordinates of the critical point in the space of theories.

The theory at a critical point gi∗ is very special. The β-functions βi(gj∗) vanish by deﬁnition, so (4.12) shows that correlation functions in this theory are independent of the

16There are a few exotic examples where the theories ﬂow to a limiting cycle rather than a ﬁxed point.

– 42 – Figure 5: Theories on the critical surface flow (dashed lines) to a critical point in the IR. Turning on relevant operators drives one away from the critical surface (solid lines), with flow lines focussing on the (red) trajectory emanating from the critical point. overall length scale s of the metric. The metric appears in the action, so rescaling the metric leads to a change δ δS ddx δgµν(x) ln = ddx δgµν(x) = ddxgµν(x) T (x) δgµν(x) Z − δgµν(x) − $ µν % ! ! 1 2 ! (4.26) in the partition function, where Tµν is the stress tensor. Scale invariance of a theory at µ g thus implies that T µ = 0. In fact, all known examples of Lorentz–invariant, unitary i∗ $ % QFTs that are scale invariant are actually invariant under the larger group of conformal transformations17. Near to a critical point we have non–zero β-functions ∂g Λ i =B δg + (δg2) (4.27) ∂Λ ij j O &gj∗+δgj & & where δg = g g , and where B &is a constant (infinite dimensional!) matrix. Diagonal- i i − i∗ ij izing B we obtain ∂σ Λ i = (∆ d)σ + (σ2) (4.28) ∂Λ i − i O where, at least at this linearized level, σi is the coupling to an eigenoperator of the RG flow with eigenvalue labelled by ∆ d.(d is the dimension of space–time.) If we can find i − 17It’s a theorem that this is always true in two dimensions. It is believed to be true also in higher dimensions, but the question is actually a current hot topic of research.

– 43 – ∆i, then to this order the RG ﬂow for σi is

∆ d Λ i− σ (Λ) = σ (Λ") . (4.29) i Λ i ( " ) Classically, expected a dimensionless coupling to scale with a power of Λ determined by the explicit powers of Λ included in the action in (4.1). Just as for the correlation function in (4.10), near a critical point the net eﬀect of integrating out degrees of freedom is to modify this scaling so that the coupling (to an eigenoperator) scales with a power of Λ determined by the eigenvalues of the linearized β-function matrix B . The quantity γ := ∆ d is ij i i − called the anomalous dimension of the operator, mimicking the anomalous dimension

γφ of the field itself, while the quantity ∆i itself is called the scaling dimension of the operator. If the quantum corrections vanished then the scaling dimension would coincide with the naive mass dimension of an operator obtained by counting the powers of fields and derivatives it contains. Notice that while the β-functions vanish at a critical point by definition, there is no reason for the anomalous dimensions of fields or operators to vanish. Now consider starting near a critical point and turning on the coupling to any operator with ∆i >d. This coupling becomes smaller as the cut–off is lowered, or as we probe the theory in the IR, and so the corresponding operator is irrelevant as turning them on just makes us flow back to gi∗. These operators thus parametrize the critical surface. Classically, we can obtain operators with arbitrarily high mass dimension by including more and more fields and derivatives. This is why the critical surface is infinite dimensional. On the other hand, couplings with ∆i

– 44 – Dimension Relevant operators Marginal operators d =2 φ2k for all k 0 (∂φ)2, φ2k(∂φ)2 for all k 0 ≥ ≥ d =3 φ2k for k =1, 2 (∂φ)2, φ6 d =4 φ2 for 3 (∂φ)2, φ4 ≤ d>4 φ2 for 0 k 3 (∂φ)2 ≤ ≤ Table 1: Relevant & marginal operators in a Lorentz invariant theory of a single scalar ﬁeld in various dimensions. Only the operators invariant under φ φ are shown. Note →− that the kinetic term (∂φ)2 is always marginal, and the mass term φ2 is always relevant. the same at large distances. This is the reason you can do physics! To study a problem at a given energy scale you don’t ﬁrst need to worry about what the degrees of freedom at much higher energies are doing. They are, quite literally, irrelevant.

4.4 Critical phenomena Let’s now consider the behaviour of the two–point correlation function Γ(2) = φ(x) φ(y) $ % at a critical point. From (4.9) it obeys

1 (2) 1 (2) Z− Γ (x, y; gi(Λ)) = Z− Γ (x, y; gi(Λ")) (4.30) Λ Λ Λ! Λ! and by Lorentz invariance it can only be a function of the separation x y . For a theory | − | at a critical point the coupling is independent of scale, so gi(Λ) = gi(Λ")=gi∗. The anomalous dimension γφ(gi∗) := γ∗ is likewise scale independent, so by (4.11) the wavefunction 2γ∗ renormalization factor obeys ZΛ/ZΛ! = (Λ"/Λ) . Therefore the scaling form (4.12) of (2) d 2(1 γ∗) (2) the renormalization group equation says that ΓΛ (sx, sy; gi∗)=s − − ΓΛ/s(x, y; gi∗). Hence for a theory (CFT) at a critical point

(2) 1 Γ (x, y; gi∗) , (4.31) Λ ∝ x y 2∆φ | − | where ∆ =(d 2)/2+γ and the proportionality constant is independent of the insertion φ − φ points. This power–law behaviour of correlation functions is characteristic of scale–invariant theories. In a theory where the interactions between the φ insertions was due to some massive state traveling from x to y, we’d expect the potential to decay as e m x y / x y − | − | | − | where m is the mass of the intermediate state. As in (classical) electromagnetism, the pure power–law we have found for this correlator is a sign that our states are massless, so that their eﬀects are long–range. A very important example of such behaviour can be found in the theory of second– order18 phase transitions, where the Wilsonian renormalization group was born. At high temperatures iron is unmagnetized, as the microscopic magnetic spins (due ultimately to the electrons in the iron) are being jostled too much to do anything coherent.

18 Recall that a phase transition at temperature Tc is second order if the free energy at Tc vanishes to at least third order order parameter (such as magnetizatoion).

– 45 – As we gradually lower the temperature to the Curie temperature (around 1043K for iron), larger and larger regions of microscopic spins align, say along the direction of a weak external magnetic ﬁeld. Below the Curie temperature, As the phase transition is approached, the correlations between regions of diﬀerent magnetization grow DISCUSSION TO BE COMPLETED

4.5 The local potential approximation The idea of renormalization group flow as a form of heat flow, encapsulated in Polchinski’s equation (4.24), has provided us with great insight into the general properties of quantum field theories under renormalization. However, we still haven’t actually computed any concrete β-functions! The time has come to put that right. Like the Wilson and Polchinski approach of directly integrating out high–energy degrees of freedom, the techniques we’ll use in this section are still based on performing the path integral. They form a stepping– stone between the intuitive and general ideas presented above and the more practical but conceptually murky perturbative ideas we’ll meet in the following chapter. We first observe that, apart from the kinetic term, operators involving derivatives φk(∂φ)2 are irrelevant whenever d>2. This suggests that we can restrict attention to the case that the action takes the form 1 Seff [ϕ]= ddx ∂µϕ∂ ϕ + V (ϕ) (4.32) Λ 2 µ ! * + so that the only derivative term is the kinetic term. We take this potential to have the form d k(d 2) g2k 2k V (ϕ)= Λ − − ϕ (4.33) (2k)! #k so that V ( ϕ)=V (ϕ) and the couplings g are dimensionless as before. Neglecting the − 2k derivative interactions is known as the local potential approximation; it is important because it will tell us the shape of the effective potential experienced by a slowly varying field. Splitting the field ϕ = φ + χ into its low– and high–energy modes as before, we expand the action as an infinite series

eff eff d 1 2 1 2 1 3 S [φ + χ]=S [φ]+ d x (∂χ) + χ V ""(φ)+ χ V """(φ)+ . (4.34) Λ Λ 2 2 3! ··· ! * + Notice that we have chosen a definition of φ so that it sits at a minimum of the potential, V "(φ) = 0. This can always be arranged by adding a constant to φ, which is certainly a low–energy mode. Now consider integrating out the high–energy modes χ. As before, we lower the cut–off infinitesimally, setting Λ =Λ δΛ and working just to first order in δΛ. In any given " − Feynman graph, each χ loop comes with an integral of the form

d d 1 d p Λ − d ( )=δΛ d dΩ ( ) Λ δΛ< p Λ (2π) ··· (2π) Sd 1 ··· ! − | |≤ ! −

– 46 – d 1 d where dΩ denotes an integral over a unit S − R and ( ) represents the propagators ⊂ ··· and vertex factors involved in this graph. The key point is that since each loop integral comes with a factor of δΛ, to lowest non–trivial order in δΛ we need consider at most 1-loop diagrams for χ. Suppose a particular graph involves an number vi vertices containing i powers of χ and arbitrary powers of φ. Euler’s identity tells us that a connected graph with e edges and - loops obeys e v = - 1 , (4.35) − i − #i In computing the rhs of (4.17) χ is the only propagating field and, furthermore, since we are integrating out χ completely, there are no external χ lines. Thus we also have the identity 2e = ivi (4.36) #i since every χ propagator is emitted and absorbed at some (not necessarily distinct) vertex. Eliminating e from (4.35) gives i 2 - = 1 + − v . (4.37) 2 i #i Since we only want to keep track of 1-loop diagrams, we see that only the vertices with i =2χ lines (and arbitrary numbers of φ lines) are important. We can thus truncate the difference Seff [φ + χ] Seff [φ] in (4.34) to Λ − Λ

(2) d 1 2 1 2 S [χ]= d x ∂χ + V ""(φ) χ (4.38) 2 2 ! * + so that χ appears only quadratically. The diagrams that can be constructed from this action are shown in ﬁgure 6. If we make the temporary assumption that the low–energy ﬁeld φ is actually constant, then in momentum space the quadratic action S(2) becomes

d (2) d p 1 2 1 S [χ]= d χ˜(p) p + V ""(φ) χ˜( p) Λ δΛ< p Λ (2π) 2 2 − ! − | |≤ * + (4.39) d 1 Λ − δΛ 2 = d (Λ + V ""(φ)) dΩ χ˜(Λpˆ)˜χ( Λpˆ) 2(2π) d 1 − !S − using the fact that these modes have energies in a narrow shell of width δΛ. Performing the path integral over χ is now straightforward. If the narrow shell contains N momentum modes, then from standard Gaussian integration

N/2 eﬀ δ S [φ] S2[χ,φ] π e− Λ = χ e− = C . (4.40) D Λ2 + V (φ) ! ( "" ) On a non–compact manifold, N is actually inﬁnite. To regularize it, we place our theory in a box of linear size L and impose periodic boundary conditions. The momentum is

– 47 – . . .

. . . .

. . .

+ . + + ...

. .

. . . . .

Figure 6: Diagrams contributing in the local potential approximation to RG flow. The dashed line represents a χ propagator at the cut–off scale Λ, while the solid lines represent external φ fields. All vertices are quadratic in χ.

d then quantized as pµ =2πnµ/L for nµ Z so that there is one mode per (2π) volume in ∈ Euclidean space–time. The volume of space–time itself is Ld. Thus d 1 Vol(S − ) d 1 d N = Λ − δΛ L (4.41) (2π)d which diverges as the volume Ld of space–time becomes inﬁnite. However, we can obtain a (correct) ﬁnite answer once we recognize that the cause of this divergence was ou simplifying assumption that φ was constant. For spatially varying φ, we would instead obtain

eff d 1 d 2 δΛS [φ]=a Λ − δΛ d x ln Λ + V ""(φ) (4.42) ! 3 4 where the factor of Ld ln[Λ2 + V (φ)] in (4.40) has been replaced by an integral over M. × "" The constant Vol(Sd 1) 1 a := − = (4.43) 2(2π)d (4π)d/2 Γ(d/2) is proportional to the surface area of a (d 1)-dimensional unit sphere. Expanding the rhs − of (4.42) in powers of φ leads to a further infinite series of φ vertices which combine with those present at the classical level in V (φ). Once again, integrating out the high–energy field χ has lead to a modification of the couplings in this potential. We’re now in position to write down the β-functions. Including the contribution from both the classical action and the quantum correction (4.42), the β-function for the φ2k coupling is 2k dg2k k(d 2) ∂ 2 Λ =[k(d 2) d]g aΛ − ln Λ + V ""(φ) . (4.44) dΛ − − 2k − ∂φ2k &φ=0 3 4& For instance, the first few terms in this expansion give & & dg2 ag4 Λ = 2g2 dΛ − − 1+g2 2 dg4 ag6 3ag4 Λ =(d 4)g4 + 2 (4.45) dΛ − − 1+g2 (1 + g2) 3 dg8 ag8 15ag4g6 30ag4 Λ = (2d 6)g6 + 2 3 dΛ − − 1+g2 (1 + g2) − (1 + g2)

– 48 – as β–functions for the mass term, φ4 and φ6 vertices. There are several things worth noticing about the expressions in (4.45). Firstly, each term on the right comes from a particular class of Feynman graph; the first term is the scaling behaviour of the classical φ2k vertex, the second term involves a single χ propagator with both ends joined to the same valence 2k + 2 vertex, the third (when present) involves a pair of χ propagators joining two vertices of total valence 2k + 4, etc.. Secondly, we note that these Feynman diagrams are different to the ones that appeared in (4.23). By taking the local potential approximation, we have neglected any possible derivative terms that may have contributed to the running of the couplings in V (φ). The effect of this is seen in the higher–order terms that appear on the rhs of (4.45). From the point of view of the Wilson–Polchinski renormalization group equation, the local potential approximation effectively amounts to solving the β-function equations that follow from (4.23), writing the derivative couplings in terms of the non–derivative ones, and then substituting these back into the remaining β-functions for non–derivative couplings to obtain (4.45). The message is that the price to be paid for ignoring possible couplings in the effective action is more complicated β-functions. We will see this again in chapter 6, where β-functions will no longer be determined purely at one loop. 2 2 Finally, recall that g2 = m /Λ is the mass of the φ field in units of the cut–off. If this mass is very large, so g 1, then the quantum corrections to the β-functions in (4.45) 2 / are strongly suppressed. As for correlation functions near to, but not at, a critical point, this is as we would expect. A particle of mass m leads to a potential V (r) e mr/rd 3 in ∼ − − position space, so should not affect physics on scales r m 1. / − 4.5.1 The Gaussian critical point From the discussion of heat flow above, we expect that the limiting values of the couplings in the deep IR will be a critical point of the RG evolution (4.44). The simplest type of critical point is the Gaussian fixed–point where g =0 k>1, corresponding 2k ∀ to a free theory. Every one of the Feynman diagrams shown on the right of the Wilson renormalization group equation in figure 4 involves a vertex containing at least three fields (either χ or φ), so if we start from a theory where the couplings to each of these vertices are precisely set to zero, then no interactions can ever be generated. Indeed, in the local potential approximation we see from (4.45) that the Gaussian point is indeed a fixed–point of the RG flow, with the mass term β-function β = 2g simply compensating for the 2 − 2 scaling of the explicit power of Λ introduced to make the coupling dimensionless. Last term you used perturbation theory to study φ4 theory in four dimensions. Using perturbation theory means that you considered this theory in the neighbourhood of the Gaussian critical point so that the couplings could be treated as ‘small’. Let’s examine this again using our improved understanding of Renormalization Group flow. Firstly, to find the behaviour of any coupling near to the free theory, as in equation 4.27 we should linearize the β-functions around the critical point. We’ll use our results (4.45) for a theory with an arbitrary polynomial potential V (φ). To linear order in the couplings, only the

– 49 – ﬁrst two terms on the rhs of (4.45) contribute, giving

β =(k(d 2) d) g ag 2k − − 2k − 2k+2 1 (4.46) d==4 (2k 4) g g − 2k − 16π2 2k+2 where δg = g g = g since g = 0 for the Gaussian critical point. The second 2k 2k − 2∗k 2k 2∗k line gives the result in four dimensions. For k>2 It shows that all the operators φ2k with k>3 are irrelevant in d = 4, at least perturbatively around the free theory. This is why last term you studied φ4 theory without bothering to write down any higher order terms in the action: they’re there, but their effects are negligible in the IR. The φ4 coupling itself is particularly interesting. We’ve seen that the φ6 interaction is irrelevant in d = 4 near the Gaussian fixed point, so at low energies we may neglect it. β4 then vanishes to linear order, so that the φ4 coupling is marginal at this order. To study its behaviour, we need to go to higher order. From (4.45) we have ∂g 1 g g2 3 β =Λ 4 = 6 +3 4 g2 (4.47) 4 ∂Λ −16π2 1+g (1 + g )2 ≈ 16π2 4 ( 2 2 ) to quadratic order, where we’ve again dropped the g6 term. Equation (4.47) is solved by 1 3 = C 2 ln Λ (4.48a) g4(Λ) − 16π where C is an integration constant, or equivalently 16π2 g (Λ) = (4.48b) 4 3 ln(µ/Λ) in terms of some arbitrary scale µ>Λ. There are several important things to learn from this result. Firstly, we see that g4(Λ) decreases as Λ 0, ultimately being driven to zero. However, the scale dependence of → g4 is rather mild; instead of power–law behaviour we have only logarithmic dependence on the cut–off. Thus the φ4 coupling, which was marginal at the classical level, because marginally irrelevant once quantum effects are taken into account. In the deep IR, we see only a free theory. Secondly, away from the IR we notice that the integration constant µ determines a scale at which the coupling diverges. If we try to follow the RG trajectories back into the UV, perturbation theory will certainly break down before we reach Λ µ. The fact that ≈ the couplings in the action can be traded for energy scales µ at which perturbation theory breaks down is a ubiquitous phenomenon in QFT known as dimensional transmutation. We’ll meet it many times in later chapters. The question of whether the φ4 coupling really diverges as we head into the UV or just appears to in perturbation theory is rather subtle. More sophisticated treatments back up the belief that it does indeed diverge: in the UV we lose all control of the theory and in fact we do not believe that φ4 theory really exists as a well–defined continuum QFT in four dimensions. This has important phenomenological implications for the Standard Model, through the quartic coupling of the scalar Higgs boson; take the Part III Standard Model course if you want to find out more.

– 50 – The fact that the φ4 coupling is not a free constant, but is determined by the scale and can even diverge at a finite scale Λ = µ should be worrying. How can we ever trust perturbation theory? The final lesson of (4.48b) is that if we want to use perturbation theory, we should always try to choose our cut–off scale so as to make the couplings as small as possible. In the case of φ4 theory this means we should choose Λ as low as possible. In particular, if we want to study physics at a particular length scale -, then our best chance for a weakly coupled description is to integrate out all degrees of freedom on length scales shorter than -, so that Λ - 1. ∼ − 4.5.2 The Wilson–Fisher critical point The conclusion at the end of the previous section was that φ4 theory does not have a continuum limit in d = 4. Since the only critical point is the Gaussian free theory we reach at low energies, four dimensional scalar theory is known as a trivial theory. It’s interesting to ask whether there are other, non–trivial critical points away from four dimensions. In general, finding non–trivial critical points is a difficult problem. Wilson and Fisher had the idea of introducing a parameter ε := 4 d which is treated as ‘small’ so that − one is ‘near’ four dimensions. One then hopes that results obtained via the ε–expansion may remain valid in the physically interesting cases of d = 3 or even d = 2. From the local WF potential approximation (4.44) Wilson & Fisher showed that there is a critical point gi where 1 1 gWF = ε + (ε2) ,gWF = ε + (ε2) (4.49) 2 −6 O 4 3a O and gWF εk for all k>2. We require ε> 0 to ensure that V (φ) 0 as φ so that 2k ∼ → | | → ∞ the theory can be stable. To find the behaviour of operators near to this critical point, once again we linearize WF the β-functions of (4.45) around g2k . Truncating to the subspace spanned by (g2,g4) we have ∂ δg ε/3 2 a(1 + ε/6) δg Λ 2 = − − 2 . (4.50) ∂Λ %δg4' % 0 ε '%δg4' The matrix has eigenvalues ε/3 2 and ε, with corresponding eigenvectors − 1 a(3 + ε/2) σ2 = ,σ4 = − (4.51) %0' % 2(3 + ε) ' respectively. In d =4 ε dimensions we have − 1 1 1 ε a = = + (1 γ + ln 4π)+ (ε2) (4.52) (4π)d/2 Γ(d/2) 16π2 32π2 − O &d=4 ε & − & where we have used the recurrence& relation Γ(z + 1) = z Γ(z) and asymptotic formula 2 Γ( ε/2) = γ + (ε) (4.53) − −ε − O for the Gamma function as ε 0, where γ is the Euler–Mascheroni constant γ 0.5772. → ≈ Since ε is small the first eigenvalue is negative, so the mass term φ2 is a relevant perturbation

– 51 – g2

G g4

WF II

Figure 7: The RG flow for a scalar theory in three dimensions, projected to the (g2,g4) subspace. The Wilson–Fisher and Gaussian fixed points are shown. The blue line is the projection of the critical surface. The arrows point in the direction of RG flow towards the IR. of the Wilson–Fisher fixed point. On the other hand, the operator a(3+ε/2)φ2+2(3+ε)φ4 − corresponding to σ4 corresponds to an irrelevant perturbation. The projection of RG flows to the (g2,g4) subspace is shown in figure 7. Although we have only seen the existence of the Wilson–Fisher fixed point when 0 < ε 1, more sophisticated techniques can be used to prove its existence in both d = 3 and ( d = 2 where it in fact corresponds to the Ising Model CFT. As shown in figure 7, both the Gaussian and Wilson–Fisher fixed–points lie on the critical surface, and a particular RG trajectory emanating from the Gaussian model corresponding to turning on the operator σ4 ends at the Wilson–Fisher fixed point in the IR. Theories on the line heading vertically out of the Gaussian fixed–point correspond to massive free theories, while theories in region I are massless and free in the deep UV, but become interacting and massive in the IR. These theories are parametrized by the scalar mass and by the strength of the interaction at any given energy scale. Theories in region II are likewise free and massless in the UV but interacting in the IR. However, these theories have g2 < 0 so that the mass term is negative. This implies that the minimum of the potential V (φ) lies away from φ = 0, so for theories in region II, φ will develop a vacuum expectation value, φ = 0. The RG $ %2 trajectory obtained by deforming the Wilson–Fisher fixed point by a mass term is shown in red. All couplings in any theory to the right of this line diverge as we try to follow the RG back to the UV; these theories do not have well–defined continuum limits.

– 52 – 4.5.3 Zamolodchikov’s C–theorem Polchinski’s equation showed that renormalization group flow could be understood as a form of heat flow. It’s natural to ask whether, as for usual heat flow, this can be thought of as a gradient flow so that there is some real positive function C(gi, Λ) that decreases monotonically along the flow. Notice that this implies C = const. at a fixed point gi∗, and that C(gi∗, Λ) >C(gi∗∗, Λ") whenever a fixed point gi∗∗ may be reached by perturbing the theory a fixed point gi∗ by a relevant operator and flowing to the IR. In 1986, Alexander Zamolodchikov found such a function C for any unitary, Lorentz invariant QFT in two dimensions. Consider a two dimensional QFT whose (improved) energy momemtum tensor is given by T (x). This is a symmetric 2 2 matrix, so has three independent components. Intro- µν × ducing complex coodinates z = x +ix andz ¯ = x ix , we can group these components 1 2 1 − 2 as ∂xµ ∂xν 1 T := T = (T T iT ) zz ∂z ∂z µν 2 11 − 22 − 12 ∂xµ ∂xν 1 T := T = (T T +iT ) (4.54) z¯z¯ ∂z¯ ∂z¯ µν 2 11 − 22 12 ∂xµ ∂xν 1 T := T = (T + T ) zz¯ ∂z ∂z¯ µν 2 11 22 where Tzz¯ = Tzz¯. This stress tensor is conserved, with the conservation equation being

µ 0=∂ Tµν = ∂z¯Tzz + ∂zTz¯z¯ (4.55) in terms of the complex coordinates. Note that the stress tensor is a smooth function of z andz ¯. The two–point correlation functions of these stress tensor components are given by 1 T (z,z¯) T (0, 0) = F ( z 2) $ zz zz % z4 | | 4 T (z,z¯) T (0, 0) = G( z 2) (4.56) $ zz zz¯ % z3z¯ | | 16 T (z,z¯) T (0, 0) = H( z 2) $ zz¯ zz¯ % z 4 | | | | where the explicit factors of z andz ¯ on the rhs follow from Lorentz invariance, which also requires that the remaining functions F , G and H depend on position only through z . | | Like any correlation function, these functions will also depend on the couplings and scale Λ used to define the path integral. The two–point function T (z,z¯) T (0) appearing here satisfies an important posi- $ zz¯ zz¯ % tivity condition. Using canonical quantization, we insert a complete set of QFT states to find Hτ T (z,z¯) T (0) = 0 Tˆ (z,z¯)e− n n Tˆ (0, 0) 0 $ zz¯ zz¯ % $ | zz¯ | %$ | zz¯ | % n # (4.57) Enτ 2 = e− n Tˆ (0, 0) 0 |$ | zz¯ | %| n # so that this two–point function is positive definite, and it follows that H( z 2) is also positive | | definite.

– 53 – Zamolodchikov now used a combination of this positivity condition and the current conservation equation to construct a certain quantity C(gi, Λ) that decreases monotonically along the RG ﬂow. In terms of the two–point functions, current conservation (4.55) for the energy momentum tensor becomes

4F " + G" 3G = 0 and 4G" 4G + H" 2H = 0 (4.58) − − − where F = dF ( z 2)/d z 2 etc. We define the C-function to be " | | | | 3 C( z 2) := 2F G H (4.59) | | − − 8 which obeys dC/d z 2 = 3H/4 by the current conservation equations. But by the posi- | | − tivity of the two–point function T (z,z¯) T (0) this means that $ zz¯ zz¯ % dC r2 < 0 (4.60) dr2 so C decreases monotonically under two dimensional scaling transformations, or equivalently under two dimensional RG flow. The value of C at an RG fixed point can be shown to be the central charge of the CFT. Ever since it was first proposed, physicists have searched for a generalization of Zamolod- chikov’s theorem to RG flows in higher dimensions. The two–dimensional quantity Tzz¯ is µ just the trace T µ of the energy momentum tensor and in 1988 John Cardy proposed that a µ certain term – known as “a” — in the expansion of the two–point correlator of T µ plays the role of Zamolodchikov’s C in any even number of dimensions. Cardy’s conjecture was veri- fied to all orders in perturbation theory the following year by DAMTP’s own Hugh Osborn, while a complete, non–perturbative proof was finally given in 2011 by Zohar Komargodski & Adam Schwimmer.

– 54 – 5 Symmetries of the Path Integral

5.1 Noether’s theorem One of the most important results in classical mechanics and classical ﬁeld theory is Noether’s theorem, stating that local symmetries of the action corresponds to a conserved charge. Let’s recall how to derive this. Consider the transformation

δ!φ(x)=#f(φ, ∂µφ) (5.1) for some function f(φ, ∂µφ) of the ﬁelds and their derivatives and a small parameter #. The transformation is local if the function f depends on the values of the ﬁeld and its derivatives only at the point x M. The transformation (5.1) is a symmetry if the action is invariant, ∈ δS[φ] = 0, whenever the parameter # is constant. Because it is invariant when # is constant, if # is now allowed to depend on position the change in the action must be proportional to the derivative of #. In other words,

δ S[φ]= jµ(x) ∂ #(x)ddx (5.2) ! − µ !M for some function jµ(x) known as the current19. However, when the equations of motion hold, the action is invariant under any small change in the fields. In particular, it will be invariant under the change (5.1) even if the parameter # depends on position. Integrating by parts and choosing our function #(x) to vanish on the boundary of M we find that µ ∂µj = 0 (5.3) whenever the equations of motion are satisfied. We define the charge Q corresponding to this transformation by µ Q[N] := jµ dS (5.4) !N where N is any codimension one hypersurface in the space–time M with (d 1)–dimensional µ − volume element dS . If N0,1 are two such hypersurfaces bounding a region M ! of space– time then µ µ µ Q[N1] Q[N0]= jµ dS jµ dS = jµ dS − N1 − N0 N1 N0 ! ! ! − (5.5) µ µ d = jµ dS = ∂ jµ d x =0 !∂M ! !M ! by the conservation equation (5.3), so that Q[N] depends on the choice of N only through its homology class. The most common application of this result is to take the surfaces N1,0 to be constant time slices of Minkowski space–time. The statement that Q[N1]=Q[N0] then becomes the statement that the charge Q is conserved.

19The minus sign is a convention, included for later convenience. More geometrically, on a Riemannian 1 manifold (M, g) I would deﬁne the current as a 1-form j Ω (M) and write this relation as δ!S[φ]= ∈ j d# where d is the exterior derivative and is the Hodge star. The current conservation law then − M ∗ ∗ becomes d j = 0. If you know a little diﬀerential geometry, feel free to rewrite the relations below in this ! ∗ language.

– 55 – Figure 8: Classically, the charge (5.4) associated to a local symmetry is independent of the hypersurface over which it is integrated.

As a simple example, consider the Lagrangian 1 = ∂ φ∂¯ µφ + V ( φ 2) (5.6) L 2 µ | | for a complex scalar field. This Lagrangian is invariant under the U(1) transformation φ eiαφ that rotates the phase of φ by a constant amount α. Taking α to be infinitesimally → small, we have δφ =iαφ , δφ¯ = iαφ¯ (5.7) − and the corresponding current is j =i ∂ φφ¯ φ∂¯ φ . µ µ − µ 5.2 Ward identities " # The derivation of Noether’s theorem used the classical equations of motion, so we must re– examine this in quantum theory. Suppose that some local field transformation φ φ (φ) → ! leaves the product of the action and path integral measure invariant, i.e.,

Seff [φ] Seff [φ ] φ e− Λ = φ! e− Λ ! . (5.8) D D In most cases, the symmetry transformation will actually leave both the action and measure invariant separately, but we only need this weaker condition. For example, under a Lorentz µ transformation with parameters M ν, a scalar field transforms as φ(x) φ (x)=φ(Mx). → ! The action is invariant if it is built from Lorentz invariant combinations of the field and its derivatives, while the path integral measure is invariant provided we integrate over modes in a Lorentz invariant manner: this is one reason it is convenient to impose the cut–off pµp Λ2 in Euclidean signature. µ ≤ Given any such symmetry, we can immediately deduce an important restriction on correlation functions of operators on a compact space. We first consider a class of operators whose only variation under the transformation φ φ comes from their dependence on φ. → ! Such operators transform as (φ) (φ ). At least on a compact space–time manifold O →O !

– 56 – M we have

Seff [φ] Seff [φ ] φ e− Λ (φ(x )) (φ(x )) = φ! e− Λ ! (φ!(x )) (φ!(x )) D O1 1 ···On n D O1 1 ···On n ! ! Seff [φ ] = φ! e− Λ ! (φ!(x )) (φ!(x )) D O1 1 ···On n ! (5.9) The first equality here is a triviality: we’ve simply relabeled φ by φ! as a dummy variable in the path integral. The second equality is non–trivial and uses the assumed symmetry (5.8) under the transformation φ φ . We see that the correlation function obeys → ! (φ(x )) (φ(x )) = (φ!(x )) (φ!(x )) (5.10) %O1 1 ···On n & %O1 1 ···On n & so that it is invariant under the transformation. For example, consider the transformation φ(x) φ (x)=φ(x a) of translation → ! − through a vector a. If the action & path integral measure are translation invariant and the operators depend on x only through their dependence on φ(x), then we have Oi (x ) (x ) = (x a) (x a) %O1 1 ···On n & %O1 − ···On n − & for any such vector a. Thus the correlator depends only on the relative positions (x x ). i − j Similarly, if the action & measure are invariant under rotations (or Lorentz transformations) x Lx then a correlation function of scalar operators will obey → (x ) (x ) = (Lx ) (Lx ) . %O1 1 ···On n & %O1 1 ···On n & Combining this with the previous result shows that the correlator can depend only on the rotational (or Lorentz) invariant distances (x x )2 between the insertion points. i − j Similarly, the phase transformation φ φ =eiαφ, φ¯ φ¯ =e iαφ¯ that is a symmetry → ! → ! − of (5.6) implies that correlation functions built from local operators of the form = φri φ¯si Oi obey n iα (ri si) 1(x1) n(xn) =e i=1 − 1(x1) n(xn) . %O ···O & ! %O ···O & Allowing α to take different (constant) values shows that this correlator vanishes unless

i ri = i si. The symmetry thus imposes a selection rule on the operators we can insert if we wish to obtain a non–zero correlator. $ $ As in the classical theory, any continuous symmetry comes with an associated current. Suppose that φ φ = φ+δ φ is a symmetry of the path integral when # is an inﬁnitesimal → ! ! constant parameter. Then the variation of the action and path integral measure must be proportional to ∂µ# when # depends on position, so that

Seff [φ ] Seff [φ] µ d = φ! e− Λ ! = φ e− Λ 1 j ∂ # d x (5.11) Z D D − µ ! ! % !M & to lowest order. Notice that the current here may include possible changes in the path integral measure as well as in the action. The zeroth order term agrees with the partition function on the left, so the first order term must vanish and we have

∂ # jµ(x) ddx = #(x) ∂ jµ(x) =0. (5.12) µ % & − µ% & !M !M

– 57 – For this to hold for arbitrary # we must have ∂ jµ = 0 so that the expectation value of µ% & the current obeys a conservation law, just as in classical physics. Now let’s see how the current insertions affect more general correlation functions. Consider a class of local operators that transform under φ φ + δ φ as → ! (φ) (φ + δ φ)= (φ)+δ (5.13) O →O ! O !O to lowest order, where we have defined δ := δ φ∂ /∂φ. Then !O ! O n n Seff [φ] Seff [φ ] φ e− Λ (φ(x )) = φ! e− Λ ! (φ!(x )) D Oi i D Oi i ! 'i=1 ! 'i=1 n n (5.14) Seff [φ] µ d = φ e− Λ 1 j ∂ # d x (x )+ δ (x ) (x ) . D − µ  Oi i !Oi i Oj j  ! % !M & i=1 i=1 j=i ' * '#   where again the first equality is a triviality and the second follows upon writing φ! = φ+δ!φ and expanding both φ e S[φ!] and the operators to first order in the variable parameter D ! − #(x). The #-independent term on the rhs exactly matches the lhs, so the remaining terms must cancel. To first order in # this gives

n n ddx#(x) ∂ jµ(x) (x ) = δ (x ) (x ) . (5.15) µ % Oi i & − % !Oi i Oj j & !M - i=1 . i=1 j=i ' * '# Note that the derivative hits the whole correlation function. We’d like to strip oﬀ the parameter #(x) and obtain a relation purely among the correlation functions themselves. In order to do this, we must handle the fact that the operator variations on the rhs are located only at the points x , . . . , x M. We thus write { 1 n}∈ δ (x )= ddxδd(x x )#(x) δ (x ) !Oi i − i Oi i !M as an integral, so that all terms in (5.15) are proportional to #(x). Since this may be chosen arbtrarily, we obtain ﬁnally

n n ∂ jµ(x) (x ) = δd(x x ) δ (x ) (x ) . (5.16) µ% Oi i & − − i % Oi i Oj j & i=1 i=1 j=i ' * '# This is known as the Ward identity for the symmetry represented by φ φ + δφ. It → states that the divergence of a correlation function involving a current jµ vanishes everywhere except at the locations of other operator insertions, and is the modiﬁcation of the conservation law ∂ jµ(x) for the expectation value of the current itself. µ% & Ward identities can be derived for any symmetry, but the name is often associated to the transformations

iα iα ψ ψ! =e ψ, ψ¯ ψ¯! =e− ψ¯ ,A A! = A → → µ → µ µ

– 58 – which for constant α are symmetries of the QED action

1 S [A, ψ]= d4x F µνF + ψ¯(iD/ m)ψ . (5.17) QED 4 µν − ! % & The transformation of the path integral measure is

1 1 δψ!(x) − δψ¯!(x) − ψ ψ¯ ψ! ψ¯! = ψ ψ¯ det det , (5.18) D D →D D D D δψ(y) δψ¯(y) / 0 / 0 where the inverse of the usual Jacobian arises because if θ is a Grassmann variable and θ! = aθ for some constant a, then Grassmann integration 1 = dθ! θ! = d(aθ) aθ implies d(aθ)=a 1dθ. The determinants in (5.18) cancel against one another provided we are − 1 1 integrating over an equal number of ψ and ψ¯ modes (and in any case are field independent). Thus these transformations are indeed symmetries of the path integral. We promote α to a local parameter — this is not a gauge transformation because the gauge field Aµ itself remains unaffected. The path integral measure depends only on the fields ψ and ψ¯, not their derivatives, so if we can regulate this in a way that preserves the local symmetry, the only contribution to the current will come from the action. One finds jµ = ψγ¯ µψ. Now consider the correlation function ψ(x )ψ¯(x ) . Since δψ ψ the Ward % 1 2 & ∝ identity becomes

∂ jµ(x)ψ(x )ψ(x ) = δd(x x ) ψ(x )ψ¯(x ) + δd(x x ) ψ(x )ψ¯(x ) . (5.19) µ% 1 2 & − − 1 % 1 2 & − 2 % 1 2 & This identity is traditionally viewed in momentum space. Introducing the Fourier transformed correlators

µ 4 4 4 ip x ik1 x1 ik2 x2 µ M (p, k ,k ) := d x d x d x e · e · e− · j (x)ψ(x )ψ¯(x ) 1 2 1 2 % 1 2 & ! (5.20) 4 4 ik1 x1 ik2 x2 M (k ,k ) := d x d x e · e− · ψ(x )ψ¯(x ) 0 1 2 1 2 % 1 2 & ! the Ward identity (5.19) becomes

ip M µ(p, k ,k )=M (k + p, k ) M (k ,k p) (5.21) µ 1 2 0 1 2 − 0 1 2 − in momentum space. Diagrammatically, this identity is where the cross represents the insertion of the Fourier transformed current ˜µ(p). Note that unlike a Feynman diagram for scattering amplitudes, there is no requirement that the momenta in this diagram are on–shell; they are just the Fourier transforms of the insertion points. The correlator ψ(x )ψ¯(x ) is the exact 2–point function of the electron in the quan- % 1 2 & tum theory, or in other words the position space electron propagator, including all loop corrections. On the other hand, the correlator jµ(x)ψ(x )ψ¯(x ) can be related to the % 1 2 & 3–point function A (x)ψ(x )ψ¯(x ) using the Dyson–Schwinger equations. This is the % µ 1 2 & exact electron–photon vertex, again including all quantum corrections. In the classical action, the electron kinetic terms are closely related to the electon– photon vertex by the requirement of gauge invariance. However, we have seen that quantum

– 59 – corrections can cause the coefficients of both the kinetic terms and the vertices to vary with energy scale. How then can we be sure that gauge invariance remains valid in the quantum theory? As you will explore further in the problems, the Ward identity (5.21) is the first signal that all is well. However, it is important to note that we have derived it under the assumption that the path integral measure can be defined in a way that is compatible with the local symmetry.

5.2.1 Charges as generators Let’s integrate the Ward identity over some region M M with boundary ∂M = N N , ! ⊆ ! 1 − 0 just as we studied classically. We’ll ﬁrst choose M to contain all the points x , . . . , x ! { 1 n} so that the integral receives contributions from all of the terms on the rhs of (5.16). Then

n Q[N ] (x ) Q[N ] (x ) = δ (x ) (x ) (5.22) % 1 Oi i &−% 0 Oi i & % Oi i Oj j & i i i=1 j=i ' ' * '# µ where the charge Q[N]= N jµdS just as in the classical case. In particular, if M ! = M and M is closed (i.e., compact without boundary) then we obtain 1 n 0= δ (x ) (x ) (5.23) % Oi i Oj j & i=1 j=i * '# telling us that if we perform the symmetry transform throughout space–time then the correlation function is simply invariant, δ = 0. This is just the inﬁnitesimal version % i Oi& of the result we had before in (5.10). 2 Next, we consider the case that only one of the x s lies in M , say x M but x / M i ! 1 ∈ ! j ∈ ! for j =2, . . . , n. In this case only one of the δ-functions on the rhs of the Ward identity is satisﬁed and we obtain n Q[N ] (x ) Q[N ] (x ) = δ (x ) (x ) . (5.24) % 1 Oi i &−% 0 Oi i & % O1 1 Oj j & 'i 'i j'=2

– 60 – It is interesting to view this equation in the canonical picture. The operator insertions become time–ordered products, so assuming t1 is the earliest time

n n Qˆ[N1] ˆi(xi) = ˆj(xj) Qˆ[N1] ˆ1(x1) T 3 O 4 T  O  O 'i=1 j'=2  n n   Qˆ[N0] ˆi(xi) = ˆj(xj) ˆ1(x1)Qˆ[N0] T 3 O 4 T  O  O 'i=1 j'=2 

Taking the limit that M ! shrinks to a time interval of zero width centred on t1, the integrated Ward identity (5.24) becomes

n n 0 ˆ (x ) Q,ˆ ˆ (x ) 0 = 0 ˆ (x ) δ ˆ (x ) 0 (5.25) % |T  Oj j  O1 1 | & % |T  Oj j  O1 1 | & j'=2  ; < j'=2  in the operator picture. This relation holds for arbitrary local operators, and therefore

Q,ˆ ˆ(x) = δ ˆ(x) (5.26) O O ; < holds as an operator relation. We say that the charges generate the transformation laws of the operators. The I should mention that the condition that M be closed cannot be relaxed lightly. On a manifold with boundary, to define the path integral we must specify some boundary conditions for the fields. The transformation φ φ may now affect the boundary condi- → ! tions, which lead to further contributions to the rhs of the Ward identity. For a relatively trivial example, the condition that the net charges of the operators we insert must be zero becomes modified to the condition that the difference between the charges of the incoming and outgoing states (boundary conditions on the fields) must be balanced by the charges of the operator insertions. A much more subtle example arises when the space–time is non–compact and has infinite volume. In this case, the required boundary conditions as x are that our | | → ∞ fields take some constant value φ0 which lies at the minimum of the effective potential. S[φ] Because of the suppression factor e− , such field configurations will dominate the path integral on an infinite volume space–time. However, it may be that the (global) minimum of the potential is not unique; if V (φ) is minimized for any φ and our symmetry ∈M transformations move φ around in the symmetry will be spontaneously broken. M You’ll learn much more about this story if you’re taking the Part III Standard Model course.

5.3 Eﬀective Field Theory In the previous section we’ve studied the behaviour of correlation functions under global symmetry transformations of the ﬁelds. These correlation functions are to be computed by carrying out the whole path integral; to actually evaluate we must integrate %O1 ···On&

– 61 – over all the quantum fields. In previous chapters, we learned that the structure of the Wilsonian effective action changes with energy scale, so it’s sensible to ask whether the symmetries of a scale Λ effective action are also symmetries of the effective action at scale Λ! < Λ obtained by integrating out high–energy modes. To understand this, suppose we split a field ϕ(x)=φ(x)+χ(x) into its low– and high–energy Fourier components as in (4.16). If a transform ϕ ϕ! is a symmetry of the Seff [ϕ ] Seff [ϕ] → scale Λ theory so that ϕ e Λ ! = ϕ e Λ , then we can write the transformation D ! − D − of the path integral measure as

δφ!(x) δφ!(x) δϕ!(x) δφ(y) δχ(y) ϕ ϕ! = ϕ det = φ χ det (5.27) D →D D δϕ(y) D D  δχ!(x) δχ!(x)  / 0 δφ(y) δχ(y)   and we can only view this as a product of transformations of the measures for low– and high–energy modes separately if δφ!(x)/δχ(y) = 0 or δχ!(x)/δφ(y) = 0 so that the transformation does not mix modes. If the low and high energy parts of the measure are separately invariant, then

eff eff eff eff S [φ] SΛ [φ+χ] SΛ [φ!+χ!] S [φ!] e− Λ! = χ e− = χ! e− =e− Λ! , (5.28) D D ! ! where the first equality is the definition of the low–energy effective action, the second uses the assumed symmetry of the scale Λ action and the measure χ on the space of high– D energy modes. The final equality provides the desired result that the low–energy effective action will be invariant under the same symmetries as the high–energy effective action. This is an important result as it means we can safely put aside the worry that integrating out fields in a Lorentz invariant way could ever generate any Lorentz violating terms at low energies, and reassures us that terms of the form φ2k+1 can never appear if the microscopic action obeys S [φ]=S [ φ]. Λ Λ − However, the combination of renormalization group flow and symmetry is much more powerful than this. In trying to construct low energy effective actions, we should simply identify the relevant degrees of freedom for the system we wish to study and then write down all possible interactions that are compatible with the expected symmetries. At low energies, the most important terms in this action will be those that are least suppressed by powers of the scale Λ. Thus, to describe some particular low–energy phenomenon, we simply write down the lowest dimension operators that are capable of causing this effect. Let’s illustrate this by looking at several examples.

5.3.1 Why is the sky blue? As a first example, we’ll use effective field theory to understand how light is scattered by the atmosphere. Visible light has a wavelength between around 400nm and 700nm, while atmospheric N has a typical size of 7 10 6nm, nearly a million times smaller. Thus, 2 ∼ × − when sunlight travels through the atmosphere we do not expect to have to understand all the details of the microscopic N2 molecules, so we neglect all its internal degrees of freedom and model the N2 by a complex scalar field φ so that excitations of φ correspond to creation

– 62 – of an N2 molecule (with excitations of φ¯ creating anti–Nitrogen). Importantly, because the N2 molecules are electrically neutral, φ is uncharged so Dµφ = ∂µφ and Nitrogen does not couple to light via a covariant derivative. The presence of the atmosphere explicitly breaks Lorentz invariance, defining a preferred rest frame with 4-velocity uµ = (1, 0, 0, 0), so the kinetic term of the φ field is 4 1 ¯ µ d x 2 φu ∂µφ showing that the field φ has mass dimension 3/2. The lowest dimension couplings between φ and A we can write down are 1 µ φ 2F µνF and φ 2uµuνF F κ , | | µν | | µκ ν = each of which have mass dimension seven in d = 4. Schematically therefore, the effective= ,j i, interactions responsible for this scattering are i, j

j j p int 4 g1 2 2 g2 2 2 p i S [A, φ]= d x φ F + φ (u F ) + (5.29)i 2 " p Λ 8Λ3 8Λ3 · ··· p " 2 ! p ! p! ; < | | | where the couplings g1,2(Λ) are dimensionless and Λ is the cut–off scale. In the case| − ij ij − δ at hand, the obvious cut–off scale is the inverse size of the N2 molecule whoseδ orbital electrons are ultimately responsible for the scattering. We expect our effective theory really " ! i i ! " i contains infinitely many further terms involving higher powers of φ, F and their derivatives.i + 2 2 + However,p on dimensional grounds these will all be suppressed by higher powersp of Λ and = 2 2 = y so will betr negligible at energies Λ. The φ F terms themselvestr must be retained if we y ij , ij D D otat ihthe with contracts i want to understand how light can be scattered by φ at all. i contracts with the x γ Now let’s consider computing a scattering amplitude φ + γ φ + γ usingγ the the- x 2 2 2 2 → ory photon (5.29). The transverse the vertices calculated φ . We F and φ (u F ) each. involve We calculated two copies the of transverseφ and two photon copies ! · !A of the photon, so can both contributeA to this scattering at tree–level. In particular, for g2 we find n contributes Aand Aand contributes h ne on index . The i k i2g. The index on γ | γ| 2 ie ∼ ieΛ3 − φ φ − niginstantaneous ending ending instantaneous (6.83) ff Because the interaction proceeds via the fieldstrength F rather than A directly, it involves ff (6.83) a derivative of each the photon, bringing down a power of the photon’s momentum. For contributes contributes 2 3 6.4.1 Naive Feynmanmassless) 0 particles Rules such as the photon, k = ω, so the amplitude g2ω0 )/Λ and the | | A∝ Rules Feynman Naive 6.4.1 tree–levely scattering cross–section takes the form y We want to determine the Feynman are rules rules the for this fermions, theory. For theory. For this fermions, for rules the Feynman rules the are determine to want We 0 3 index telling us the component of of component the us telling index 3 0 − We− x denote, the photon an with by a comes wavy line the of line. end Each Each line. end wavy a ω of by 4 the line photon the comes, x denote withWe an the same| y ! as those( 2 given in Section 5. The new pieces are: 2 are: pieces 2 new The 5. Section in 2 ( given !y those | as same the , δ 2 σRayleigh = g2 6 2 δ, (5.30) 1 |A| ∝ Λ 0 ) 1 • − x ! ) 0 !x − • characteristic of Rayleigh scattering. Loop corrections to this cross–section will involve (6.84) | π propagatorγ e in (6.33): it is is it (6.33): in eγ propagator π| (6.84) ) 4 ( 3 i( 4 ) y higher powers ofi the coupling g2/Λ and so will be suppressed for photon energies Λ. y , x − x ( Since the cross–section increases rapidly with frequency, blue light is scattered much more ( − opnn fteguefil,w ol r to try 2 could we · ip field, gauge the of component component of the gaugeip· field,2 we could try to =0 than red light, so theThe daylight vertex sky in a direction away vertex fromThe the sun appears blue. =0 ic hc ilcpueti em nfc,it fact, In ! p term. this e capture will which piece 0 0 piece which will capturee p! this term. In fact, it ν ν | A A | µ, | p 4 00 Our treatment• of the scattering using the simple effective action (5.29) is only justified 00 4 p | µ, 4 • 4 =0 =0 ) d D index on the line. photon photon the on line.index D d ) " j π if ω Λ, where Λ was the inverse size of the N2 molecules responsible for the scattering π j " = (2 , (2 = ne,with index, 3 ν ν , 3 index, with , 2 , by given is space, position in which, The interaction non-local non-local The interaction which, in position space, is given by , 2 = 0 ) = , , 1 # ) 0 # 1 , otherwise =0 y • • y =0 otherwise , " i contributes a factor of i " hc correctly which | y ! 0 of factor a contributes 0 !y| which correctly µ =0 = − – 63 – − = =0 µ ν µ x (x µ ν γ γ − x ! ( δ δ !x − ie µ, π µ, ie | π 4 4 | − − j i i j 2 in working for paid we’ve price the is This messy. rather are rules Feynman These These Feynman rules are rather messy. This is the price we’ve paid for working2 p in p i 2 " i p p! p " 2 | ! p + something into expressions these massage can we that show now We’ll gauge. Coulomb Coulomb gauge. We’ll now show that we can massage these expressions+ into| something ! p | | p! | | o the with start Let’s invariant. Lorentz and simple more much much more simple and Lorentz invariant. Let’s start with the o | | − − ij δ the from comes it Since interaction. interaction. Since it comes from the δ ij 0 a include to propagator the redefine redefine the propagator to include a 0 " ! i i ! " i have we space, momentum in look we if form: this in nicely quite fits fits quite nicely in this form: if we look in momentum space,i we have + + 2  2 above gets replaced by bv esrpae by replaced gets above i p  p i    γ γ   ie   )= )= ie p ( (p − ν ν − ow a obn h o-oa neato ihtetases htnpoaao by propagator photon transverse   the with µ interaction non-local the combine can we so so we can combine the non-localµ interaction with the transverse photon propagator by   D D   propagator photon new a defining defining a new photon propagator  4 – 141 – – 141 –

piece in the instantaneous interaction. ic nteisatnosinteraction. instantaneous the in piece 2 2 ) 0 0 ) opnn aigcr fteisatnositrcin enwneed now We interaction. instantaneous the of care γ e taking component 0 = = 0 componenteγ taking care of the instantaneous interaction. We now need µ µ

ihti rpgtr h aypoo ienwcrisa carries now line photon wavy the propagator, this With With this propagator, the wavy photon line now carries a h extra the the extra ocag u etxsihl:the slightly: vertex our change to to change our vertex slightly: the consfrte( the for accounts accounts for the ( at a microscopic level. It will thus fail for photons of too high energy, or if the visible light enters a region where the scattering is done by larger particles. In particular, as water droplets coalesce in the atmosphere they can easily reach sizes in excess of the wavelength of visible light. In this case the relevant scale Λ is the inverse size of the water droplet, and (5.29) will be unreliable for light in the visible spectrum; there are inﬁnitely many higher order terms φ 2rF 2s/M 2s+3r 4 (and also further derivative interactions) that will | | − be just as important. Thus there’s no reason to expect that higher frequencies will be scattered more. Clouds are white.

5.3.2 Why does light bend in glass? In vacuum, the lowest–order gauge, Lorentz and parity invariant action we can write for the photon is of course the Maxwell action

1 4 µν 1 3 1 SM[A]= d xF Fµν = dt d x #0E E B B (5.31) 4µ 3,1 4 3,1 · − µ · 0 !R !R / 0 0 where µ0 is the magentic permeability of free space. For later convenience, we’ve written 2 this term out in non–relativistic notation, using c =1/µ0#0 to introduce the electric permittivity #0 in the electric term. In the presence of other sources, we should add a new term to this action that describe interactions between the photons and the sources. Effective field theory provides a powerful way to think about these new interaction terms. We suppose the degrees of freedom we expect to be important at some scale Λ can be described by some field(s) Φ, which may have arbitrary spin, charge etc.. Then in general we’d expect the new action to be

int S = SM[A]+SΛ [A, Φ] . (5.32) The Euler–Lagrange equations become

µ ∂ Fµν = µ0Jν , (5.33)

int ν where the current Jµ(x) := δSΛ /δA (x) is defined to be the variation of the new interaction terms. Together with the Bianchi identity ∂[κFµν] = 0, these give Maxwell’s equations. For example, consider a piece of glass. Glass is an insulator, so the Fermi surface lies in a band gap. Thus, so long as the light with which we illuminate our glass has sufficiently low–frequency, the electrons will be unable to move and the insulator has no relevant degrees of freedom. In this case, we must have Sint[A, Φ] = Sint[A], so the interaction Lagrangian must be a sum of gauge invariant terms built from A. However, while the local structure of glass is invariant under rotations and reflections, a lump of glass is certainly not a Lorentz invariant as it has a defines a preferred rest frame. So in writing our effective interactions, there’s no reason to impose Lorentz invariance. Thus for glass we should add a term 1 Sint[A]= dt d3x (χ E E χ B B + ) (5.34) Λ 2 e · − m · ··· !glass where the dimensionless couplings χe(Λ) and χm(Λ) are respectively the electric and magnetic susceptibilities of the glass. (Invariance under reflections rules out any E B ·

– 64 – Figure 9: Different polarizations of light travel through birefringent materials such as calcite at different speeds, leading to multiple imaging. term.) Higher–order polynomials in E, B and their derivatives are certainly allowed, but by dimensional analysis must come suppressed by a power of the electron bang–gap energy int Λ. The field equations obtained from SM + SΛ show that light travels through the glass with reduced speed given by 1 1 c2 = + χ , glass (# + χ ) µ m 0 e / 0 0 This leads to Snell’s Law at an interface and the appearance of bending. Notice that our EFT argument doesn’t tell us anything about the values of χe or χm; for that we’d need to know more about the microphysics of the silicon in the glass. However, it does predict that integrating out the high energy degrees of freedom (the electrons in the glass) will lead to an effective Lagrangian that at low–energies must look like (5.34). Since E and B each have mass dimension +1, any higher powers of these fields would be suppressed by powers of the energy scale required to excite electrons. Similarly, a crystal such as calcium carbonate or quartz has a lattice structure that breaks rotational symmetry. For such materials there is no reason for the effective La- grangian to be rotationally symmetric, so we should expect different permeabilities and permittivities for the different components of E and B: 1 S = dt d3x ((χ ) E E (χ ) B B ) . (5.35) crystal 2 e ij i j − m ij i j ! This leads to different speeds of propagation for the different polarization states of light, resulting in the phenomenon of birefringence. (See figure 9.) Ancient Viking texts describe a “sólarstein” that could be used to determine the direction of the Sun even on a cloudy day, and was an important navigational aid. It’s believed that this sunstone was a form of calcium carbonate. Near the Arctic, sunlight is quite

– 65 – Figure 10: Higher–order terms in the effective action lead to nonlinear optical effects such as the generation of these harmonics from the incoming laserbeam on the left. Notice that the scattered light has a higher frequency than the incoming red light. (Figure taken from the Nonlinear Optics group at Universiteit Twente, Netherlands.) strongly polarized, with the polarization reduced in directions away from the sun due to random scattering from the atmosphere. CaCO3 is sensitive to this polarization, so by moving a crystal around one can detect the direction of the Sun. As we’ve emphasized above, the effective actions (5.34)-(5.35) are only the lowest–order terms in the infinite series we obtain from integrating out high–energy modes corresponding to the atoms in the glass or crystal. The next order terms take the schematic form20

α β γ δ ε dt d3x E4 + B4 + E2B2 + (∂E)2 + (∂B)2 ∼ Λ2 Λ2 Λ2 Λ2 Λ2 ! % & and on dimensional grounds will be suppressed by an energy scale Λ of order the excitation energy of the insulating material. Such higher order terms lead to Euler–Lagrange equations that are nonlinear in the electric and magnetic fields. We’d expect them to become important as the energy density of the electric or magnetic fields grows to become appreciable on the scale Λ. Their presence implies that, contrary to what the lowest–order action claims, a powerful laser will not pass directly through an insulating material in a straight line, but rather will be scattered in a very complicated non–linear way. Indeed this is observed (see figure 10) and is the starting–point for the whole field of Nonlinear Optics.

5.3.3 Quantum Gravity as an EFT The ﬁnal example of an Eﬀective Field Theory I’d like to discuss is General Relativity. Including (as we should) a cosmological constant λ, the Einstein–Hilbert action for General

20Alternatively, we can take # = #(E, B) and µ = µ(E, B) so that the permittivity and permeability are themselves functions of the ﬁelds.

– 66 – Relativity is 1 S [g]= d4x√ g λ + R(g) (5.36) EH − 16πG ! % N & µν where R = g Rµν is the Ricci scalar. The Riemann tensor involves the second derivative of the metric and so has mass dimension +2, showing that the Newton constant GN must have mass dimension 2 in four dimensions, as is well–known. The cosmological constant − has mass dimension 4. Thus the couplings shown in (5.36) are both relevant and may be expected to dominate the behaviour in the IR. In the spirit of effective field theory however, we should expect that the Einstein– Hilbert action is just the first term in an infinite series of possible higher–order couplings. Diffeomorphism invariance restricts these higher–order terms to be products of the metric and covariant derivatives of the Riemann tensor and the first few terms are

Seﬀ [g]= d4x√ g c Λ4 + c Λ2R + c R2 + c RµνR + c RµνρκR + (5.37) Λ − 0 1 2 3 µν 4 µνρκ ··· ! A B where the couplings ci are dimensionless. In fact, it can be shown that a linear combination of the couplings c2, c3 and c4 is proportional to the four–dimensional Gauss–Bonnet term which is topological and does not aﬀect perturbation theory. DISCUSSION TO BE COMPLETED

5.4 Emergent symmetries Physics of the 20th Century was largely driven by symmetry principles, with the recog- nition of global and local symmetries being key tools in unlocking the secrets of physics from hadronic interactions to electroweak physics and the Standard Model, and from su- perconductivity to Bose–Einstein condensates. It’s therefore important to ask whether the global21 symmetries we see, say in the Standard Model, are really fundamental. In fact, it’s often the case that the small number of relevant and marginal operators available in any given theory are invariant under a wider range of field transformations than the infinite class of irrelevant operators. At low energies the effects of these irrelevant operators are highly suppressed, so it may appear as though the theory has the larger symmetry group. Thus, symmetry can be emergent in the low–energy theory, irrespective of whether or not it is present in the microscopic theory. As an example, consider a theory of electromagnetism coupled to several generations of charged fermions, denoted ψ , each with the same charge e. We might imagine that ψ i − i describe the three generations of charged leptons in the Standard Model. The most general Lorentz– and gauge–invariant Lagrangian we can write down for these fields that contains only relevant and marginal operators is 1 [A, ψ ]= Z F µνF L i −4e2 3 µν (Z ) ψ¯ (iD/)ψ +(Z ) ψ¯ (iD/)ψ + M ψ¯ ψ + M¯ ψ¯ ψ , − L ij Lj Li R ij Rj Ri ij Li Rj ij Ri Lj *i,j A (5.38)B

21We’ll consider gauge ‘symmetry’ in detail in chapter 7. As we’ll see there, gauge transformations do not really correspond to a symmetry at all, but rather a redundancy in our description of Nature.

– 67 – where 1 1 ψ := (1 + γ )ψ ,ψ := (1 γ )ψ Li 2 5 i Ri 2 − 5 i are the left– and right–handed parts of the fermions, where Z3 and ZL,R are possible 22 wavefunction renormalization factors for the photon for and leptons, and where ML,R are lepton mass terms. For the Lagrangian (5.38) to be real, the matrices ZL,R must be Hermitian, while their eigenvalues must be positive if we are to have the correct sign kinetic terms. If the wavefunction renormalization matrices (ZL,R)ij are non–diagonal then the form of (5.38) suggests that processes such as ψ ψ + γ are allowed, so that the absence of 2 → 1 such a process in the Standard Model would seem to indicate an important new symmetry.

However, this is a mirage. We introduce renormalized fields ψL,R! defined by ψL = SLψL! and ψR = SRψR. The Lagrangian for the new fields takes the same form, but with new matrices

ZL! = SL† ZLSL ,ZR! = SR† ZRSR ,M! = SL† MSR .

Now take SL to have the form SL = ULDL, where UL is the unitary matrix that diagonalizes the positive–deﬁnite Hermitian matrix ZL, and DL is a diagonal matrix whose entries and the inverses of the eigenvalues of ZL. Such an SL ensures that ZL! = 1, and we can arrange ZR = 1 similarly. This condition does not completely ﬁx the unitary matrix UL, because if

ZL! = 1 then it is unchanged by conjugation by a further unitary matrix. We can use this remaining freedom to diagonalize the mass matrix M. The polar decomposition theorem implies that any complex square matrix M can be written as M = VH where V is unitary and H is a positive semi–definite Hermitian matrix. Thus, we perform a further field !! !! redefinition ψL! = SL! ψL and ψR! = SR! ψR with SL! =(SR! )†V † and choose SR! to be the unitary matrix that diagonalizes H. In terms of the new fields the Lagrangian (5.38) becomes finally (dropping all the primes) 1 [A, ψ]= Z F µνF ψ¯ (iD/)ψ + ψ¯ (iD/)ψ m ψ¯ ψ m ψ¯ ψ L −4e2 3 µν − Li Li Ri Ri − i Li Ri − i Ri Li *i 1 A B = Z F µνF ψ¯ (iD/ m )ψ . −4e2 3 µν − i − i i i * (5.39) This form of the Lagrangian manifestly shows that the ‘new’ fields ψi have conserved individual lepton numbers. It’s easy to write down an interaction that would violate these µ individual lepton numbers, such as Yijklψ¯iγ ψj ψ¯kγµψl. However, all such operators have mass dimension > 4 and so are suppressed in the low–energy effective action. Lepton number conservation is merely an accidental property of the Standard Model, valid23 only at low–energies.

22 It is conventional to denote the photon wavefunction renormalization factor by Z3. 23In fact, certain non–perturbative processes known as sphalerons lead to a very small violation of lepton number even in with dimension 4 operators. However the diﬀerence B L between baryon number and lepton − number is precisely conserved in the Standard Model, yet is believed to be just an accidental symmetry.

– 68 – Just as light could be scattered by a neutral particle using a φ2F 2 interaction as above, higher dimension operators can lead to processes such as proton decay that are impossible according to the dimension 4 operators that dominate the low–energy behaviour. ≤ Thus, although such processes are highly suppressed, they are very distinctive signatures of the presence of higher dimension operators. Experimental searches for proton decay put important limits on the scale at which the new physics responsible for generating these interactions comes into play. Sorting out the details in various diﬀerent possible extensions of the Standard Model is one of the main occupations of particle phenomenologists. In fact, there are arguments to suggest that there are no continuous global symmetries in a quantum theory of gravity. Certainly; no one has succeeded in ﬁnding such symmetries in string theory (global symmetries do exist, but they are always discrete). From this perspective, all the continuous symmetries that guided the development of so much of 20th Century physics may be low–energy accidents.

– 69 –