QFTAndTheSM

Since the modes of an electromagnetic field have the same classical equations as a simple harmonic oscillator, we can quantize them in the same way. This is known as second quantization. At the risk of oversimplifying things a little, that is all there is to quantum field theory. The rest is just quantum mechanics. The two steps really are (1) interpret these modes as having energy E = hbar omega and (2) quantize each mode as a harmonic oscillator. In that sense we are only quantizing once.

There are two new features in second quantization:

1. We have many quantum mechanical systems – one for each p⃗ – all at the same time.
2. We interpret the nth excitation of the p⃗ harmonic oscillator as having n particles.

For the Lamb shift calculation, the sum was also over photon states, which was divergent. It diverged because the phase space for photons, d^3k, is larger than the suppression, 1/|k| , due to the energies of the intermediate excited states.

There was actually nothing wrong with Oppenheimer’s calculation. He did get the answer that OFPT predicts. What he missed was that there are other infinities that eventually cancel this infinity (for example, the electron mass is infinite too, so in fact his conclusion was on the right track).

The LSZ reduction formula: relates S-matrix elements ⟨f|S|i⟩ for n asymptotic momentum eigenstates to an expression involving the quantum fields φ(x): The factors of [] + m2 project onto the S-matrix: [] + m2 becomes −p2 + m2 in Fourier space, which vanishes for the asymptotic states. These factors will therefore remove all terms in the time-ordered product except those with poles of the form 1/(p2 - m2) , corresponding to propagators of on-shell particles. Only the terms with poles for each factor of p2 − m2 will survive, and the S-matrix is given by the residue of these poles. Thus, the physical content of the LSZ formula is that the S-matrix projects out one-particle asymptotic states from the time-ordered product of fields.

The LSZ reduction says that to calculate an S-matrix element, multiply the time-ordered product of fields by some [] + m2 factors and Fourier transform.

the [] + m2 terms will kill anything that does not have a divergence, that will be anything but the exact initial and final state we want.

It is easy to think that LSZ is totally trivial, but it is not.

LSZ does not distinguish elementary particles, which we define to mean particles that have corresponding fields appearing in the Lagrangian, from any other type of particle.

quantum field theory is very flexible: it works if you use fields that do not correspond to elementary particles (effective field theories) or if you scatter particles that do not have corresponding fields (such as bound states)

in a classical theory, correlation functions satisfy the same differential equations as the fields within the correlation functions. In a quantum theory, that is true only up to δ-functions, which in this context are also called contact interactions. These contact interactions allow virtual particles to be created and destroyed, which permits closed loops to form in the Feynman diagrammatic expansion, as we will now see.

In a classical theory, as noted above, the contact interactions are absent. It was these contact interactions that allowed us to contract two fields within a correlation function to produce a term in the expansion with fewer fields. For example, <φ1φ2φ3φ4> = iδ12⟨φ3φ4⟩ + · · · . In the classical theory, all that can happen is that the fields will proliferate.

A good way to start understanding the consistency requirements of the physical universe is with a discussion of spin. There is a deep connection between spin and Lorentz invariance that is obscure in non-relativistic quantum mechanics.

The polarizations of light are easy to understand at the classical level since light is a field, but how can an individual photon be polarized? For the electron, we can at least think of it as a spinning top, so there is a classical analogy, but photons are massless and structureless, so what exactly is spinning? The answers to these questions follow from an understanding of Lorentz invariance and the requirements of a consistent quantum field theory.

In addition, we want unitary representations. The unitary representations of the Poincare ́ group are only a small subset of all the representations of the Poincare ́ group. For example, as we will discuss, the 4-vector representation, Aμ, is not unitary. But the unitary ones are the only ones from which we will be able to compute Poincare ́-invariant matrix elements, so we have to understand how to find them. Thus, Particles transform under irreducible unitary representations of the Poincare ́ group. This statement can even be interpreted as the definition of what a particle is.

By the way, there is an even stronger requirement on physical theories: the S-matrix must be unitary. Requiring a unitary S-matrix constrains the dynamics of the theory, while demanding unitary representations of the Poincare ́ group is just a statement about free-particle states. Implications of unitarity of the S-matrix is the subject of Chapter 24.

We look at the irreducible representations because those are the building blocks with which we can construct the most general description of nature.

there are no finite-dimensional unitary representations of the Poincare ́ group.

Knowing what the representations of the Poincare ́ group are is a great start, but we still have to figure out how to construct a unitary interacting theory of particles in these representations. To do that, we would like to embed the irreducible representations into objects with space-time indices. That is, we want to squeeze states of spin 0 , 1/2 , 1, 3/2 , 2 etc. into scalar fields φ(x), vector fields Vμ(x), tensor fields Tμν(x), spinor fields ψ(x) etc. That way we can write down simple-looking Lagrangians and develop general methods for doing calculations. We see an immediate complication: tensors have 1,4,16,64,…,4n elements, but spin states have 1, 3, 5, 7, . . . , 2j + 1 physical degrees of freedom.

The transformation Aμ → −Aμ under C may seem strange, since a vector field is real, so it should not transform under an operation that switches particles with antiparticles. Since particles and antiparticles have opposite charge and Aμ couples proportionally to charge, this transformation is needed to compensate for the transformation of the charged fields.

There is an important lesson here: you could take C : Aμ → Aμ, but then the Lagrangian would not be invariant. Thus, rather than trying to figure out how C acts, the right question is: How can we enlarge the action of the transformation C, which we know for Dirac spinors, to a full interacting theory so that the symmetry is preserved? Whether we interpret C with the words “takes particles to antiparticles,” has no physical implications. In contrast, a symmetry of a theory does have physical implications: preservation of the symmetry gives a superselection rule – certain transitions cannot happen. An important example is that C invariance forces matrix elements involving an odd number of photons to vanish,

Although QED had great successes at tree-level, that is at leading-order in the fine-structure constant α, it appeared in the 1930s incapable of making quantitative predictions at higher orders. For example, the infinite contribution of the Coulomb potential to the electron mass in the classically theory was still infinite in QED; and QED could not be used to compute corrections to the energy levels of the hydrogen atom. By the late 1930s, the experts generally believed that QED was incomplete, if not wrong.

As we will see in Part III, the key to performing calculations in QED beyond leading order in α is to carefully relate observable quantities to other observable quantities.

The Ward–Takahashi identity not only implies the usual Ward identity and gauge invariance, but since it is non-perturbative it will also play an important role in the renormalization of QED.

It has important implications. In Section 19.5, we will show that it implies that charge conservation survives renormalization, which is highly non-trivial. The reason it is so powerful is that it applies not just to S-matrix elements, but to general correlation functions. It also implies the regular Ward identity, as we will show below.

The program of systematically making testable predictions about long-distance physics in spite of formally infinite short-distance fluctuations is known as renormalization. Because physics at short and long distance decouples, we can deform the theory at short distance any way we like to get finite answers – we are unconstrained by physically justifiable models. In fact, our most calculationally efficient deformation will be analytic continuation to d = 4 − ε dimensions with ε → 0. The beauty of renormalization is that the existence of a physical cutoff is totally irrelevant: quantitative predictions about long-distance physics do not care what the short-distance cutoff really is, or even whether or not it exists.

Observables are finite and in-principle calculable functions of other observables.

loops can produce behavior different from anything possible at tree-level. In particular, non-analytic behavior, such as ln(s) , is characteristic of loop effects.

In many cases, the non-analytic behavior will comprise the entire physical prediction associated with the loop.

Despite the difficulty of probing extremely high energies in QED experimentally, one can at least ask what would happen if we attempted scattering at Q ≫ m. From Eq. (16.66) we can see that at some extraordinarily high energies, Q ∼ 10²⁸⁶ eV, the loop correction, the logarithm, is as important as the tree-level value, the 1. Thus, perturbation theory is breaking down. At these scales, the 2-loop value will also be as large as the 1-loop and tree-level values, and so on. The scale where this happens is known as a Landau pole.

QED has a Landau pole: perturbation theory breaks down at short distances.

This means that QED is not a complete theory in the sense that it does not tell us how to compute scattering amplitudes at all energies.

it is helpful to navigate away from observables such as the Lamb shift or the Coulomb potential, which are particular to one type of correction, to thinking of general observables. Unfortunately, the question of what is observable and what is not is extremely subtle and has no precise definition in quantum field theory. For example, one might imagine that S-matrix elements are observable; in many cases they are actually infinite due to IR divergences,

In particular, we will find that in QED, while there are an infinite number of divergent graphs contributing to the S-matrix, the divergences can be efficiently categorized and renormalized through the one-particle irreducible subgraphs

We can choose different values for the counterterms which differ from these by finite numbers and GR(p/) will still be finite. Any prescription for choosing the finite parts of the counterterms is known as a subtraction scheme. Not only must observables in a renormalized theory be finite, but they also must be independent of the subtraction scheme, as we will see. Nevertheless, there are some smart choices for subtraction schemes and some not-so-smart choices.

The two subtraction schemes most often used in quantum field theory are the on-shell subtraction scheme and the minimal subtraction (MS) scheme. Minimal subtraction is by far the simplest scheme and the one used in almost all modern quantum field theory calculations.

We saw that UV divergences were artifacts of not computing something physical, since the UV-divergent answer was calculated using parameters in a Lagrangian that were not defined based on observables. More precisely, we saw that the normalizations of the electron and photon fields were not observable, and so these fields could be rescaled by wavefunction renormalization factors Z2 = 1 + δ2 and Z3 = 1 + δ3, with the counterterms δ3 and δ2 dependent on the UV regularization and subtraction scheme. We also saw that the bare electric charge parameter e0 appearing in the Lagrangian and the bare Lagrangian electron mass parameter m0 could be redefined keeping physical quantities (such as the charge measured by Coulomb’s law at large distances and the location of the pole in the electron propagator) finite. This introduced two new counterterms, δ1 and δm. We found that these same counterterms, and the four associated renormalization conditions that define them to all orders in perturbation theory, made all the 2-, 3- and 4-point functions we have so far considered finite.

The next question we will address is: Will this always be the case? Can these same four counterterms remove all of the infinities in QED? If so, QED is renormalizable. In a renormalizable theory, all UV divergences can be canceled with a finite number of counterterms.

because all propagators have some factor of loop momentum in them (by definition of 1PI), a 1-loop diagram can never be more divergent than its superficial degree of divergence.

the superficial degree of divergence depends only on the external particles in the process (this is essentially just dimensional analysis since the only scale available is Λ).

All divergences can be removed by counterterms corresponding to superficially divergent 1PI amplitudes.

there are an infinite number of superficially divergent Feynman diagrams for a theory with a massive vector boson, and hence such theories are not renormalizable. That is not to say that they cannot be renormalized (they can!), but only that all of the UV divergences cannot be canceled with a finite number of counterterms. non-renormalizable theories can be renormalized, but only by continually adding terms to the Lagrangian to provide counterterms to cancel divergences. While such a procedure seems like it would destroy the predictivity of a theory, in fact non-renormalizable theories are still extremely predictive.

A remarkable and important fact is that non-renormalizable theories are predictive not just at tree-level but also at loop-level, through calculable quantum corrections.

Renormalizable theories are simple and beautiful: with just a handful of measurements, an infinite number of predictions can be made. These theories are surpassed in their mathematical beauty only by finite theories (which are free of UV divergences) such as string theory or certain supersymmetric gauge theories.

In some sense, the merit of renormalizability also limits its usefulness: to predict an infinite number of observables in perturbation theory with a finite number of parameters one must actually evaluate the loops!

Luckily, however, not all of quantum field theory consists of computing loops in renormalizable theories. If one’s goal is to make predictions that can be compared to experiment, it is often better to use a non-renormalizable theory rather than a renormalizable one. By isolating the relevant degrees of freedom for a physical problem, one can construct an appropriate effective theory which has a limited range of applicability, but is much more predictive in that range than the corresponding renormalizable theory on which it is based. These effective theories are generally non-renormalizable, but they are still predictive at the quantum level.

the Schro ̈dinger equation is useful even if we do not know about Lorentz invariance. The reason is that in the non-relativistic limit |p⃗| ≪ m, the higher-order terms generally have a very small effect. Nevertheless, through specially designed experiments, these coefficients can in fact be measured. For example, a1 contributes to the fine structure of the hydrogen atom, and a2 contributes to the hyperfine structure. So even if a1 and a2 were not known from the Dirac equation, they could be measured from the hydrogen atom. Once measured, they could then be used to make predictions: the fine structure of helium, for example, or lots of other things.

the Schro ̈dinger equation made quantum predictions many years before the Dirac equation was discovered.

It is also important to note that the Schro ̈dinger equation is not predictive for momenta |p⃗| >~ m, since all of the higher-order terms are then important. Thus, the Schro ̈dinger equation is predictive at low energy, but also indicates the scale at which perturbation theory breaks down.

We saw that although the scalar mass gets quadratically divergent corrections, for example from a fermion loop, these divergences can be removed with counterterms. The resulting physical pole mass must be determined from experiment as a renormalization condition. we saw that there can be a large difference between the pole mass and the MS mass for a scalar.

to have a scalar whose mass m2P ≪ Λ2 requires that the bare mass parameter m2 in the Lagrangian must be m2 ≈ Λ2 + m2P .

Much of our intuition for fine-tuning and naturalness comes from condensed matter physics.

The renormalization group is one of those brilliant ideas that lets you get something for nothing through clever reorganization of things you already know. It is hard to under-estimate the importance of the renormalization group in shaping the way we think about quantum field theory. The phrase renormalization group refers to an invariance of observables under changes in the way things are calculated.

the fact that the theory is independent of something means one can set up differential equations such as dX/dL=0, where X is some observable. Solving these differential equations leads to a trajectory in the space of theories. The term renormalization group (RG) or renormalization group evolution refers to the flow along these trajectories.

The Narrow width approximation says that near a resonance we can treat the resonant particle as being on-shell. In the narrow-width approximation, the production and decay of the resonance can be treated separately – there can be no interference between production and decay.

Factorization when intermediate particles go on-shell is a gen- eral consequence of unitarity

A much more important mass definition is the MS mass, mR, discussed in Section 18.4. The MS mass is not defined by any pole prescription. Instead, it is a renormalized quantity which must be extracted from scattering processes that depend on it. Recall from Chapter 18 that a mass definition is equivalent to a subtraction scheme that is a prescription for determining the finite parts of counterterms. For the MS mass, one simply sets the finite parts of the counterterms to zero.

MS masses are particularly useful for particles that do not form asymptotic states and cannot be identified as resonances, such as quarks. For example, there is no way to extract the bottom-quark mass from a Breit– Wigner distribution. MS masses are also important for precision physics, as we will see in Chapter 31.

, it is impossible to come up with a convention-independent way to compare these fields at different points. Moreover, it is also impossible to compute ∂μφ(x), since the derivative is a difference, and the difference depends on the phase choices. This motivates defining a new field W (x, y) called a Wilson line. It is a kind of bi-local field that depends on two points.

The point of this is that now |W(x, y) φ(y) − φ(x)| is independent of our choice of a local phase convention.

In this way, the gauge field is introduced as a connection, allowing us to compare field values at different points, despite their arbitrary phases

A theory with a conserved and Lorentz-covariant energy-momentum tensor can never have a massless particle of spin 2.

The Weinberg–Witten theorem does not say anything useful about the Standard Model, But it does say that if you started with a theory without gravity, say only with scalars, spinors and gauge fields, which does have a conserved energy-momentum tensor, you would never have some kind of phase transition that gives you a massless graviton, String theory and the anti de Sitter/conformal field theory (AdS/CFT) correspondence get around this by having gravity emerge in a different space-time

Yang–Mills theory is the natural generalization of electrodynamics to systems with many fields. If we have N fields φi, then the Lagrangian L = −φ⋆i []φi is invariant under a global SU(N ) symmetry, under which φi → Uij φj for U ∈ SU(N).

It is perhaps worth emphasizing that the important feature of these theories is not gauge invariance (which is an unphysical feature of Lagrangians) but the existence of massless spin-1 particles that are charged, that is, they carry quantum numbers.

In Chapter 25, gauge invariance was motivated as allowing us to choose a different SU(N) convention at different points. We saw that we could compare field values at different points in a convention-independent way if we used Wilson lines W (x, y) defined so that W (x, y) φ (y) transforms as φ(x). Expanding such a Wilson line out for small deviationsled to W(x,x+δx)=1−igδxμAaμTa, where Ta are the generators of SU(N) in the fundamental representation. In this way, we found that a local theory needs one gauge field Aaμ for each generator, and thus the gauge fields Aaμ transform in the adjoint representation of SU(N).

Next, we found that, in computing the propagator for the gauge boson, we had to gauge-fix, as in QED. But in the non-Abelian case, the covariant gauge-fixing (Rξ gauges), when done properly through the path integral, generated new particles called Faddeev–Popov ghosts, which have spin-0 but fermionic statistics. These particles never appear as external states but must be included in internal lines for consistency. That we need these ghosts is a horrible consequence of the Lagrangian formulation of field theory. There is no observable consequence of ghosts, we just need them to describe an interacting theory of massless spin-1 particles using a local manifestly Lorentz-invariant Lagrangian.

to maintain manifest Lorentz invariance in a perturbative gauge theory, it seems ghosts are unavoidable.

Why are the matrix elements for gluon scattering such a mess and the final answer so simple? The root of the problem is our insistence on manifest locality. In fact, the entire formalism of quantum field theory that we have developed so far is based on describing interactions among particles in terms of local Lagrangians.

While the local Lagrangian description has its advantages, such as manifest Lorentz invariance, it also has disadvantages. In Chapter 8, we encountered subtleties in trying to write a Lagrangian for a massless spin-1 particle that would only propagate the two physical degrees of freedom. We needed to have a redundancy of description, called gauge invariance,

π0 → γγ, provides an important way to measure the number of colors of quarks. Historically, this was one of first constraints on QCD [Adler, 1969], and it remains one of the easiest ways to measure the number of colors.

the π0 → γγ rate calculation has a tumultuous history:

In the 1940s, when π0 → γγ was of particular interest, neither quantum field theory nor the profound importance of symmetries were well understood. Early attempts at this decay rate were producing non-gauge invariant amplitudes.

Why the η′ is so heavy is known as the U(1) problem. It is called that because the Goldstone boson corresponding to the axial diagonal U(1) is apparently missing.

Since the symmetry is anomalous, it is not a symmetry. If there is no symmetry, it cannot be spontaneously broken and there can be no Goldstone boson.

the heavy η′ tells us that non-perturbative configurations must be important.

Anomaly matching is one the few tools we have for making concrete statements about non-perturbative theories.

First we will discuss constraints on the gauge sector of the electroweak theory. At tree- level, many observables depend only on the three parameters g, g′ and v (or equivalently αe, sin2θw and the Fermi constant GF ). The dominant radiative corrections to many of these observables are from virtual top-quark- and Higgs-boson-loop contributions to the W-boson, Z-boson and photon propagators. Corrections to the gauge boson propagators are called oblique corrections. Oblique corrections provided important indirect information about the mass of the top quark and Higgs boson before these particles were seen directly, and they continue to provide important constraints on beyond-the-Standard-Model physics.

Thus, if you propose a model of physics beyond the Standard Model, you can calculate S and T as a shortcut to comparing with electroweak precision data.

if we can get the results of the full quantum theory with classical physics, why bother with loops at all? That is, why not just start from the effective action? The answer is that effective actions are in general highly non-local and hopelessly unconstrained.

If we start with a local classical action and then perform the loops, things such as locality, Lorentz invariance and causality are easier to check and confirm.

a construction of an effective action from first principles (unitarity, analyticity etc.) is essentially equivalent to the S-matrix program of the 1950s and 1960s. This approach may eventually prove predictive, but at this point only effective actions derived from local classical Lagrangians are known to be consistent.

An important application of the Coleman–Weinberg approach is to the effective potential for the Standard Model Higgs boson. It will let us answer the very important question: Is the Standard Model vacuum stable? If not, there must be physics beyond the standard model coming in to make it stable.

For our vacuum to be absolutely stable, this potential should be positive for all h, which means the coefficient of the logarithm should be positive.

we find absolute stability holds if mh > 247.7 GeV. This bound assumes the potential can be trusted for all h; however, for mh = 247.7 GeV, the potential only goes negative at a value well above the Planck scale where quantum gravity becomes strong. Asking that the potential be positive up to MPl ≈ 1019 GeV gives a weaker bound, mh > 221 GeV. For h<= MPl, we do not have to worry about quantum gravity; however, we do have to worry about large logarithms.

In other words, if mh < 125.8 GeV we are 95% confident that our patch of the universe will eventually tunnel into a more stable vacuum.

That the real Higgs boson mass is close to the bound of instability is an intriguing and unexplained fact.