Phys 151: Introduction to Quantum Field Theory

Fall 2022

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Basic Info

Lecturer: Petr Hořava (email: horava@berkeley.edu)
Office: 401 Physics South.
Office hours: Thu, 2:30-4:00pm..
GSI: Emil Albrychiewicz
Reader: Pablo Castaño.

Over the past several decades, our community has recognized that quantum field theory (QFT) -- more generally, many-body theory -- represents the leading paradigm and universal language in modern theoretical physics, and an absolutely essential ingredient in our current understanding of the Universe on an astonishingly diverse range of scales. The basic ideas and techniques of QFT are at the core of our understanding of high-energy particle physics and cosmology, as well as phenomena in condensed matter, statistical mechanics, quantum information, nonequilibrium mesoscopic dynamics, AMO physics, and even finance. QFT also naturally leads to its logical extension -- string theory -- which in turn provides a unified framework for reconciling the quantum paradigm with the other leading paradigm of the 20th century physics: that of general relativity, in wich gravity is understood as the geometry of spacetime.

At the core of the modern understanding of QFT is the so-called Wilsonian framework: A way of understanding how interacting systems with many degrees of freedom reorganize themselves as we change the scale at which we observe the system. This makes concepts and techniques of QFT remarkably universal, and applicable to just about every area of physics. As a result, a solid understanding of the basic structure, ideas and techniques of QFT is indispensable not only to high-energy particle theorists and experimentalists, or condensed matter theorists, but also to string theorists, astrophysicists and cosmologists, as well as an increasing number of mathematicians.

This course will provide the introduction to the principles of QFT, mostly in -- but not limited to -- the special case of the relativistic regime. The focus will be two-fold: Primarily, on developing a "big-picture" understanding of the basic ideas and concepts of QFT, and secondly, on developing the techniques of QFT (albeit at a slightly less ambitious and detailed level than in our graduate-level QFT courses 232A and 232B).

More specifically, the main themes that we will focus on in this semester are the following:

• Path integral re-formulation of Quantum Mechanics.
• From quantum particles to quantum fields.
• Quantization of free fields: Canonical and path-integral formulations. Bosonic and fermionic fields.
• Interactions: Perturbation theory, the logistics of Feynman diagrams.
• Importance of topological invariants in QFT.
• Basics of the renormalization group ideas.
• Renormalization process in perturbation theory.
• Basics of quantization in theories with gauge invariance.

In the process, we will illustrate the abstract concepts using concrete examples, ranging from the relativistic fields of the Standard Model, to relativistic and non-relativistic field theories in low spacetime dimensions, to Fermi liquids, topological insulators, etc., without requiring any prior knowledge of those examples.

The main required textbook is going to be:

A. Zee, Quantum Field Theory in a Nutshell. 2nd edition (Princeton U.P., 2010).

The second book is a new and intriguing introductory text,

J. Donoghue and L. Sorbo, A Prelude to Quantum Field Theory (Princeton U.P., 2022),

which is aimed at precisely the right level at which our course will begin: To provide a bridge from the ideas and concepts of Quantum Mechanics (assumed to be known to the student) to the ideas and concepts of Quantum Field Theory, comparing and contrasting what is similar and what is different in these two ways of thinking about quantum systems. This book is very readable and short (less than 150 pages!), and therefore can serve as a possible life-boat if any student starts feeling in danger of drowning in the vast sea of Quantum Field Theory as covered in the incredibly large number of textbooks on this subject.

I also strongly recommend

M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory (Perseus, 1995),

a textbook that is often used as the go-to text in many graduate-level courses on relativistic QFT, and which can serve as a "second-opinion book" for the students in our Physics 151 when they encouter a new topic in Zee and want to see more.

Occasionally, I will introduce examples or concepts not covered in any of these three books, in which case I will provide a specific reference to a good source in the literature, ofter in the form of an arXiv paper.

In the discussion of the Klein paradox, I was following the presentation from the beginning of the following little introductory book:
L. Alvarez-Gaumé and M.A. Vázquez-Mozo, Invitation to Quantum Field Theory (Springer, 2012).

Week 2: We continued our detailed derivation of the path-integral reformulation of quantum mechanics. Specifically, we paid a lot of attention to the Gaussian integrals and various of their cousins, since they play such a central role in the path integral formalism, both in quantum mechanics and later in QFT. We treated the important case, where the Gaussian has a purely imaginary coefficient of the quadratic term in the exponential, by naive brute-force analytic continuation (as in [Zee]). (In the math literature, this continued version of the Gaussian integral is often referred to as the "Fresnel integral".) For a much more mathematically rigorous direct treatment of the Fresnel version of the Gaussian integral, see Howie Haber's notes at UCSC.

Week 3: We continued developing our understanding of the path-integral reformulation of quantum mechanics, looking at several simple but illuminating examples that can be solved exactly. This included a nonrelativistic particle on a circle, which illustrated the importance of topological invariants; the evaluation of the partition function of the harmonic oscillator at non-zero temperature, which illustrated the parallel between thermal systems and the evolution in imaginary time; and a few other examples. (Most of those examples that are not in [Zee] can be found discussed in detail in the very elegant and accurate book,
J. Zinn-Justin, Path Integrals in Quantum Mechanics(Oxford U.P., 2005).)
Although this book goes way beyond the scope of this course, it is an extremely valuable source of insights into the path integral method. For example, the Discussion paragraph on page 36 of this book parallels very closely our derivation and discussion of the scaling dimensions of the typical path that contributes to the path integral of the free particle, as we presented it in detail in class.)
We also discussed the semiclassical limit (in the form of the steepest-descent approximation), and started moving from the path integral for a few degrees of freedom, to many degrees of freedom, to the "continuum limit" that defines field theory.

Week 4: We moved from the path integral of very many degrees of freedom to the continuum limit in terms of a field theory. We familiarized ourselves with the treatment of Gaussian integrals for fields, and discussed the role of the various "propagators" that solve the equation for inverting the quadratic form in the action for field theory (at least we did that for the simplest example of a scalar field, for now). We saw an example of low-energy "emergent" or "accidental" symmetries. We saw that in the presence of fields, external sources that couple to them experience forces, which can be thought of as mediated by the exchange of quantum particles associated with the fields. The properties of those particles are encoded in the propagator of the field, with the on-the-mass-shell condition of the particle (i.e., in particular its mass) determined by where the propagator grows very large. Thus, quantum fields imply particles, which in turn imply forces caused by those particles.

Week 5: I introduced the notion of a "scaling dimension", both at the classical level and in principle for the full quantum theory, using as examples some very simple free fields. Yet, we were able to see an example of a non-trivial "renormalization-group flow", from a free massless theory with one field-theory degree of freedom at short distances, to the theory with zero degrees of freedom at long distances. We also evaluated the classical scaling dimensions of various coupling constants appearing in additional non-Gaussian terms that we added to the action, such as the lambda coupling that multiplies the term quartic in the scalar field, and found how these scaling dimensions depend intriguingly on the spacetime dimension d. This led us to classify all couplings into "relevant", "marginal" and "irrelevant", depending on whether the scaling dimension is positive, zero or negative. Lastly, we wanted to be able to evaluate quantum corrections to the classical values of the scaling dimensions, which is leading us to develop the perturbation theory for interacting theories, and rewrite it in the combinatorially nice language of Feynman diagrams.

Week 6: We focused on the systematic introduction of Feynman diagrams and Feynman rules, as a main tool for organizing perturbative expansions in interacting field theories in the powers of a "small" coupling constant. First, we discussed the systematics of the diagrams for a baby example of a "field theory" in zero spacetime dimensions, and then -- using the example of the relativistic λφ^4 theory -- developed the Feynman rules both in coordinate space and then in momentum space. We introduced the notion of the s-point Green's function, in the expansion of the partition function into the powers of the source J. We explained how the number of "loops" of a Feynman diagram is properly defined, and saw in our scalar example that the number of loops counts the power of Planck's constant in the semiclassical expansion. We also introduced other interesting examples of QFTs, such as relativistic electrodynamics, or non-relativistic scalar fields, and discussed how the assignment of scaling dimensions changes in those cases. Using the propagator of the free electromagnetic and the free gravitational field, we explained why the force caused between equal-sign charges is always repulsive, while gravity is always attractive. Corresponding Sections in [Zee] are: I.7, I.5.

Week 7: We spent more time familiarizing ourselves with Feynman diagrams in the relativistic scalar theory in four spacetime dimensions, and have seen first UV divergences in the loop diagrams. We performed one-loop renormalization in the λφ^4 theory, and understood that the parameters originally in the action are "bare", dependent on the UV cutoff; in order to relate them to measurable quantities, they must be replaced systematically with "renormalized" (or "physical", in the language of [Zee]) quantities, which are UV-cutoff independent, but now depend on the newly introduced "renormalization group scale" μ -- intuitively, the characteristic energy scale at which we observe the system. We introduced the superficial degree of divergence, and showed that in the relativistic scalar theory in four dimensions, only diagrams with up to four external legs are superficially divergent. We completed the one-loop renormalization by considering the two-point function. We looked at examples of "renormalizable" and "nonrenormalizable" theories, and interpreted the latter using the language of "effective field theory". Einstein gravity in the metric formulation (in dimensions greater than two) is an example of a nonrenormalizable quantum field theory, and we talked about its limits of validity and about the cosmological constant problem. Corresponding Sections in [Zee] are: III.1, III.2 and III.3.

Week 8: In the first lecture, we discussed "renormalized perturbation theory", in which the expansion is written directly in terms of the renormalized physical coupling, not the bare coupling. The Feynman rules were accordingly modified, for example with the propagators now containing the physical masses, and we clarified the origin of so-called "counterterms". The primary reference for this material is Section III.3 of [Zee]. In addition, as a preparation for the mid-term, we also reviewed a few additional nonrelativistic free field theories, and their scaling properties.
Then, in the second lecture, we focused on another important class of relativistic fields, described by "spinors". After motivating the construction historically as in Dirac's original work, by requiring that the field satisfy a first-order relativistic differential equation of motion, we switched to a more modern mathematical approach, and developed from first principles the theory of Clifford algebras, which automatically exist in any spacetime dimension -- in fact, in any signature (p,q) -- equipped with a nondegenerate metric. As a consequence of Chevalley's theorem from 1954, we argued that each such Clifford algebra has a unique irreducible matrix representation, up to isomorphism. Hence, we can freely think of the elements of the Clifford algebra as matrices, which then naturally act on a new vector space of the appropriate rank, the "vector space of spinors". We studied a few low-dimensional examples (0+1 dimensions, 1+0 dimensions, 1+1 dimensions, 2+0 dimensions, ... ), and we saw how the specific properties of spinors sensitively depend on the number of dimensions and the signature of spacetime: While a generic spinor is complex-valued (and referred to as the "Dirac" spinor), sometimes we can impose a reality condition (in which case the spinors will be called "Majorana"), and in even dimensions, we can always impose a chirality condition (and call the chiral spinors -- which come in two types: left-chiral and right-chiral -- "Weyl").

Week 9: We continued with the discussion of spinors, and argued using the path integral that they must be quantized as objects with Fermi statistics ("fermions"). We introduced classical anticommuting Grassmannian variables (="supermathematics"), including the appropriate notion of integration (the Berezin integral), and then focused on the Dirac field in four spacetime dimensions. On Friday, we had a guest lecturer speaking about internal symmetries, the Noether theorem in field theory, and about spontaneous symmetry breaking.

Week 10: We started a more systematic study of the interplay between symmetries and renormalization. In order to evaluate whether the potential has a minimum at zero field once one-loop quantum effects are taken into account, we developed the functional formalism of the "effective action" and "effective potential". We calculated the renormalized Coleman-Weinberg effective potential for a scalar field in four dimensions, by correctly including counterterms and imposing appropriate normalization conditions, and found that at zero mass, the quantum corrections tend to destabilize the vacuum at zero field and break the reflection symmetry spontaneously. This was another way how to see the techniques of renormalization in action, to appreciate how the dependence on a mass scale (essentially, the RG scale) is generated by the quantum effects, and also that to keep the calculation under systematic control, it would be better to have another expansion parameter (such as the number N of field components, so that we could expand in the powers of 1/N).

Week 11: We looked once again at the superficial degree of divergence in some simple four-dimensional theories, and observed what [Zee] calls the "Weisskopf phenomenon": The one-loop correction to the masses of fermions due to the Yukawa interaction does not in fact diverge linearly as the superficial degree of divergence suggests, but only logarithmically. We generalized this phenomenon substantially, and extended it to the notion of "Technical Naturalness", as formalized by G. 't Hooft: A parameter or a collection of parameters (such as masses, couplings, etc.) can be naturally small when as we set them to zero, the system exhibits an enhanced symmetry. The original reference, in which the principle of Technical Naturalness was first clearly formulated, is:
G. 't Hooft, Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking (in: NATO ASI Proceedings, 1979).
This principle plays a central role in many areas of fundamental physics, and we discussed some of the most pressing Naturalness puzzles: The Higgs mass hierarchy puzzle, and the cosmological constant puzzle.
Then, returning to the question of spontaneous symmetry breaking of continuous internal symmetries, we have stated a preliminary form of Goldstone's theorem (guaranteeing the existence of a massless, or in nonrelativistic systems gapless, excitations associated with the spontaneously broken symmetries). In order to be able to prove Goldstone's theorem, we had to fill in one gap in our understanding of QFT so far: Namely, we needed to develop a canonical quantization, leading to a much better understanding of the operator picture of QFT and the structure of the Hilbert space of states of quantum fields.

Week 12: We continued our systematic derivation of the canonical quantization of free relativistic fields, and found the precise multi-particle interpretation of the Hilbert space of states in terms of Fock spaces. Bosonic as well as fermionic creation and annihilation operators played a central role. We discussed the operator interpretation of the free Feynman propagator, in terms of the vacuum expectation value of the time-ordered product of two quantum fields, and suggested that the time ordering is precisely required in order to reproduce the well-known i-epsilon prescription that we derived early on from the path integral. Since Friday is a holiday, there was only one lecture this week.

Week 13: We have proven Goldstone's theorem, and discussed its implications for both relativistic and nonrelativistic systems. In the nonrelativistic case, we distinguished between Type A and Type B Nambu-Goldstone modes, and discussed how the possibility of Type B modes in nonrelativistic systems explains why the number of Nambu-Goldstone modes can sometimes be smaller than the number of independent broken symmetry generators. We also presented an example of spontaneously broken spacetime symmetries (in a relativistic system with membranes), and showed that in that case, the number of Nambu-Goldstone modes can also be smaller than the number of spontaneously broken independent symmetry generators. Turning the argument of Goldstone's theorem around, we proved the Coleman-Hohenberg-Mermin-Wagner theorem, which shows that spontaneous breaking on continuous internal bosonic symmetries is impossible in 1+1 relativistic dimensions, simply by showing that the would-be Nambu-Goldstone mode does not exist quantum mechanically, due to the infrared divergences; and we discussed how this theorem can be evaded by going nonrelativistic.
In our second lecture this week, we discussed in detail the calculus of the "Wilsonian approach" to renormalization (and the renormalization group), first by studying simple relativistic scalar theories in various dimensions, and then also touching on a few nonrelativistic examples. After Thanksgiving, we will continue this discussion by focusing on the Wilsonian approach to two nonrelativistic systems: The surface growth problem, and the Fermi liquids. In the meantime, I can recommend some truly excellent resources for those two systems. First, the surface growth problem (and how it translates into a QFT in the path integral formulation) is covered in a very approachable form in the beautiful book
A.-L. Barabasi and H.E. Stanley, Fractal Concepts in Surface Growth (Cambridge U.P., 1995),
which can be an excellent and entertaining Thanksgiving-break reading material, as well as a good resource for solving some of the Problems from HW set 11. Then, the renormalization-group approach to the Fermi liquids (as well as simpler, relativistic systems) is covered in great pedagogical detail in the beautiful review paper
R. Shankar, Renormalization-Group Approach to Interacting Fermions, Rev. Mod. Phys. 66 (1994) 129.
This latter review is a particularly instructive and lucid introduction to the Wilsonian method in general, not just for the Fermi liquids. Enjoy!

Week 14: Thanksgiving week, no lectures (per the University declaration of Wed, Thu and Fri as official "non-instruction days"). Happy Thanksgiving, everyone!

Week 15: After the Thansgiving break, we continued our discussion of the Wilsonian renormalization process using the example of Fermi liquids in D+1 dimensions, first focusing on identifying where the "slow" and "fast" modes are in the non-interacting system of nonrelativistic fermions. We found that the notorious steps of the Wilsonian approach (Step 1: Integrate out fast modes, Step 2: Rescale things appropriately so that you can compare the theory from after Step 1 to the theory before Step 1) must be preceded by what I usually call "Step 0": Identify the correct ground state, and find out where the slow and fast modes are. In this discussion, we were primarily following the Shankar review cited above. In our second lecture this week, we used our prior analysis of the free Fermi liquids to show that the system exhibits an infinite number of classically marginal couplings -- a rather rare and beautiful phenomenon, even more so because it works for the Fermi liquids generically in any number of spacetime dimensions. We also talked about another system with infinitely many classically marginal couplings: The relativistic nonlinear sigma model in 1+1 dimensions. The role of the marginal couplings is played by the target-space metric in this model, and the conditions of quantum-mechanical conformal invariance (i.e., the vanishing of the beta function) imply (at one loop) the full non-linear Einstein equations for this metric! This result was first shown by then a Berkeley graduate student Dan Friedan in 1980 :) In the second part of our second lecture, we returned to the idea of the large-N expansion in QFT, with N being the number of components of a field. This leads to results that in the leading order in 1/N are often solvable exactly, and are nonperturbative in the original coupling. We focused on simple 1+1 dimensional examples: the Gross-Neveu model, and the nonlinear sigma model whose target space is an N-dimensional sphere. Besides Chapter VII.4 of [Zee], I recommend an excellent introduction to the concepts of the large N expansion, J. Maldacena, TASI 2003 lectures on AdS/CFT, arXiv:hep-th/0309246. Especially the first few chapters of these lectures are an excellent overview of the basic logic of this approach, stressing how it connects to string theory and holography.

HW 1 (posted Fri, Aug 26; due on Fri, Sept 2): Problem I.2.1 from [Zee] (on page 16). This asks you to derive Eqn. (5) from Section I.2 of [Zee]; since we will derive the closely related Eqn. (4) from Section I.2 in our lecture on Wednesday, Aug 31, you may want to wait with completing this HW assignment until after that lecture.

HW 2 (posted Fri, Sept 2; due on Fri, Sept 9): This homework assignment contains two problems. First problem: Derive Eqn. (24) from Section I.2 in [Zee] (on page 15) -- this way, you will effectively prove Wick's theorem :) Second problem: Derive Eqn. (28) from Section I.2 in [Zee] (on page 16) -- deriving this result will make you ready to take the semiclassical expansion of path integrals, and develop a systematic perturbation theory.

HW 3 (posted Fri, Sept 9; due on Fri, Sept 16): At the end of today's lecture, we derived the equation for the "propagator" D(x-y) of the free field, Eqn. (17) of Section I.3 of [Zee]. As we will discuss in more detail in the Wed, Sept 14 lecture, this equation has several distinct solutions. The one given in Eqn. (22) on page 23 in [Zee] is called the "Feynman propagator," and plays the central role in relativistic quantum field theory. This week's homework contains two problems, first dealing with the Feynman propagator, while the second deals with different solutions of the same equation. Specifically, solve Problems I.3.1 and I.3.3 (from pages 24-5) in [Zee].

HW 4 (posted Fri, Sept 16; due on Fri, Sept 23): Two problems from [Zee]: Problem I.4.1 (on page 31), and Problem I.3.2 (on page 24).

HW 5 (posted Fri, Sept 23; due on Fri, Sept 30): First, read the un-numbered subchapter entitled "Renormalization group flow" on pages 359-60 of Chapter VI.8 of [Zee], and then solve Problem VI.8.1.
Next, returning to the concept of Feynman diagrams, in our baby example of a zero-dimensional field theory: Identify the term in Eqn. (4) of Chapter I.7 of [Zee] which is of linear order in the coupling constant lambda and of fourth order in the source J (as illustrated in Figure I.7.1); I am asking you to derive this in two different ways -- the first way, by following the strategy explained just below Figure I.7.1 at the top of page 45; and then by following the strategy leading to the un-numbered Equation on page 47 of [Zee] (between Eqns. (5) and (6) of that page). Verify that both derivations are indeed giving the same result.
Finally, solve problem I.7.2 on page 60 of [Zee].

HW 6 (posted Fri, Sept 30; due on Fri, Oct 7): Problems I.5.1 (on page 39), I.7.1, I.7.3 and I.7.4 (on page 60 of [Zee]).

No new HW was posted for the week of Fri, Oct 7. Instead, the Mid-Term Take-Home Exam was distributed to the students on Monday, Oct 10. The exam is due on Fri, Oct 14.

HW 7 (posted Fri, Oct 14; due on Fri, Oct 21): Problems III.1.1 and III.1.3 (on page 168 of [Zee]). Plus, two additional Problems: First, verify Eqn. (5) of Section III.3 (at the bottom of page 176 of [Zee]). Second, derive Eqns. (11), (12) and (13) from that same Section (on pages 178 and 179).

HW 8 (posted Fri, Oct 21; due on Fri, Oct 28): First, read pages 93 to 100 of Section II.1 in [Zee], where he goes over the properties of spinors in Minkowski spacetime of 3+1 dimensions. Then, solve Problem II.1.1 (on p. 105). Next, solve Problem II.5.1 (on p. 131). Finally, solve Problem I.10.1 (on p. 80).

HW 9 (posted Fri, Oct 28; due on Fri, Nov 4): Problems III.3.3 and III.3.4 (on p. 181 of [Zee]), and Problems IV.3.3 and IV.3.4 (on p. 244).

HW 10 (posted Fri, Nov 4; due on Monday, Nov 14): There are just two problems in this week's homework set. First, Problem IV.1.1 (on p. 230 in [Zee]). And then the second one: Using the expansion of the free scalar field into plane waves with creation and annihilator operators as coefficients (presented in Eqn. (11) of Chapter I.8 on p. 63 of [Zee]), and assuming the standard commutation relations of the creation and annihilation operators as shown in Eqn. (12) of Chapter I.8, prove by a direct calculation that (i) the field and it canonically conjugate momentum satisfy the equal-time commutation relation shown in Eqn. (8) on the previous page 62, and that (ii) the momentum-momentum as well as field-field equal-time commutation relations written down in parenthesis in Zee on the line just below Eqn. (8) are also satisfied.

HW 11 (posted on Fri, Nov 18; due in two weeks, on Fri, Dec 2): Problem I.8.3 (on p. 69 of [Zee]) is an exercise in using creation and annihilation operators to prove our claim about the Feynman propagator as made in the lectures one week earlier. Then, the following four problems refer to the QFT that describes surface growth (and which will be discussed in the lecture on Wednesday, Nov 30): Problems VI.6.2 and VI.6.3 (p. 349), and Problems VI.8.3 and VI.8.4 (on p. 368). Finally, using the Wilsonian method as described in lectures and on p. 362-3 of [Zee], verify the scaling relation given in Eqn. (15) of Chapter VI.8 on p. 363; how will the result change for terms in the action which have not only n fields but also some (even) number 2m of derivatives acting on them? (See Footnote 2 on p. 363 for an example of such a higher-order term, with n=4 and m=2.)

HW 12 (posted on Fri, Dec 2; due on Fri, Dec 9): This homework set is our last one in this semester! We will only grade it on the coarsest grading scale: "+" or "-". The focus is on the large-N expansion technique, in the example of the Gross-Neveu model, as discussed on pages 402-405 in Chapter VII.4 of [Zee]. All Eqn. numbers below will refer to the equations on those pages. Problem 1: Consider the theory as described by Eqn. (17). By integrating out explicitly the auxiliary field σ in the path integral, verify how this theory is equivalent to the one in Eqn. (16). Problem 2: Starting again with the same theory in Eqn. (17), now perform the Gaussian integral over the fermions instead, and verify that the result reproduces the effective action in Eqn. (18). Problem 3: Why is the infinite N limit equivalent to the classical limit of this representation of the Gross-Neveu model? Problem 4: Now consider the explicit form of the effective potential as given in Eqn. (19), and use it to derive the renormalization group equation for the coupling g, as quoted in Eqn. (21). Problem 5: Verify that the dynamically generated mass of the fermions, as given in Eqn. (22), is independent of the renormalization-group scale, and explain why this dynamically generated mass is an effect nonperturbative in g.

horava@berkeley.edu