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A comprehensive introduction to quantum field theory, covering classical fields, canonical quantization, Feynman diagrams, and perturbation theory. The notes include examples, exercises, and references for further reading.
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- P b μ|pi = pμ|pi,
- 3.1 Canonical quantization
- 4 Theories and Lagrangians
- ✪ = eψγμψ.
- 5 Towards computational rules: Feynman diagrams
- ✪ uα(p, s )
- 5.3 An example: Compton scattering
- Kμ.
- 6.2 Symmetries in the quantum theory
- × SU(2)
- 7 Anomalies
- 7.1 Axial anomaly
- U(1)A : u± −→ e±iαu±.
- , Jμ =
- 0) − (0 − N) = 2N. (7.24)
- Jμ
- 8.1 Removing infinities
- Ophysical = lim (8.1) Λ→∞ [O(Λ)bare + ∆O(Λ) ] ,
- H = H⋆ + δλ O, (8.41)
- S[φ′ a, μ] = Z d4x L0[φ′ a] + Xi gi(μ)Oi[φ′ a] . (8.45)
- ✪ ✪ α†(k)α(k), (2π)3 2ωk
- Appendix: A crash course in Group Theory
- GeneratedCaptionsTabForHeroSec
This is compatible with the Lorentz invariance of the normalization that we have checked above
So far we have contented ourselves with requiring a number of properties to the quantum scalar field: existence of asymptotic states, locality, microcausality and relativistic invariance. With these only ingredients we have managed to go quite far. The previous can also be obtained using canonical quantization. One starts with a classical free scal...
Up to this point we have used a scalar field to illustrate our discussion of the quantization procedure. However, nature is richer than that and it is necessary to consider other fields with more complicated behavior under Lorentz transformations. Before considering other fields we pause and study the properties of the Lorentz group.
The quantization of interacting field theories poses new problems that we did not meet in the case of the free theories. In particular in most cases it is not possible to solve the theory exactly. When this happens the physical observables have to be computed in perturbation theory in powers of the coupling constant. An added problem appears when c...
As the basic tool to describe the physics of elementary particles, the final aim of Quantum Field Theory is the calculation of observables. Most of the information we have about the physics of subatomic particles comes from scattering experiments. Typically, these experiments consist of arranging two or more particles to collide with a certain ener...
✪ 11The contribution of each diagram comes also multiplied by a degeneracy factor that takes into account in how many ways a given Wick contraction can be done. In QED, however, these factors are equal to 1 for many diagrams. Outgoing antifermion: α =⇒ vα(p, s) Incoming photon: μ =⇒ ǫμ(k, λ) Outgoing photon: μ =⇒ ǫμ(k, λ)∗ Here we have assumed that...
To illustrate the use of Feynman diagrams and Feynman rules we compute the cross section for the dispersion of photons by free electrons, the so-called Compton scattering: γ(k, λ) + e−(p, s) −→ γ(k′, λ′) + e−(p′, s′). In brackets we have indicated the momenta for the different particles, as well as the polarizations and spins of the incoming and ou...
∂(∂μφ) − Actually a conserved current implies the existence of a charge
We have seen that in canonical quantization the conserved charges Qa associated to symmetries by Noether’s theorem are operators implementing the symmetry at the quantum level. Since the charges are conserved they must commute with the Hamiltonian
symmetry. This symmetry acts independently in the left- and right-handed spinors as uL,R dL,R
So far we did not worry too much about how classical symmetries of a theory are carried over to the quantum theory. We have implicitly assumed that classical symmetries are preserved in the process of quantization, so they are also realized in the quantum theory. This, however, does not have to be necessarily the case. Quantizing an interacting fie...
Probably the best known examples of anomalies appear when we consider axial symmetries. If we consider a theory of two Weyl spinors u±
Using Noether’s theorem, there are two conserved currents, a vector current Jμ =
A u− − u† − +u+ −u† −u− u† − +u+ u† . The associated conserved charges are given, for the vector current by L
Therefore we conclude that the coupling to the electric field produces a violation in the conservation of the axial charge per unit time given by ∆QA ∼ eE. This implies that ∂μJμ
A the relevant quantity is the correlation function
From its very early stages, Quantum Field Theory was faced with infinities. They emerged in the calculation of most physical quantities, such as the correction to the charge of the electron due to the interactions with the radiation field. The way these divergences where handled in the 1940s, starting with Kramers, was physically very much in the s...
where ∆O(Λ) represents the regularized quantum corrections. To make this qualitative discussion more precise we compute the corrections to the elec-tric charge in Quantum Electrodynamics. We consider the process of annihilation of an electron-positron pair to create a muon-antimuon pair e−e+ → μ+μ−. To lowest order in the electric charge e the only...
where δλ is the perturbation of the coupling constant corresponding to O (we can also consider per-turbations in more than one coupling constant). At the same time thinking of the λa’s as coordinates in the space of all Hamiltonians this corresponds to moving slightly away from the position of the fixed point. The question to decide now is in which...
This Wilsonian interpretation of renormalization sheds light to what in section 8.1 might have looked just a smart way to get rid of the infinities. The running of the coupling constant with the energy scale can be understood now as a way of incorporating into an effective action at scale μ the effects of field excitations at higher energies E > μ....
[ = 0. (9.1) This means that any states with a well-defined number of particle excitations will preserve this number at all times. The situation, however, changes as soon as interactions are introduced, since in this case particles can be created and/or destroyed as a result of the dynamics. Another case in which the number of particles might chang...
In this Appendix we summarize some basic facts about Group Theory. Given a group G a represen-tation of G is a correspondence between the elements of G and the set of linear operators acting on a vector space V , such that for each element of the group g ∈ G there is a linear operator D(g) satisfying the group operations
A PDF file of notes from lectures on quantum field theory by Luis Alvarez-Gaume and Miguel A. Vazquez-Mozo. The lectures cover topics such as classical and quantum fields, gauge fields, symmetries, anomalies, renormalization, and special topics in particle physics and string theory.
- Luis Alvarez-Gaume, Miguel A. Vazquez-Mozo
- 2010
1 A Brief History of Quantum Field Theory Quantum fleld theory (QFT) is a subject which has evolved considerably over the years and continues to do so. From its beginnings in elementary particle physics it has found applications in many other branches of science, in particular condensed matter physics but also as far afleld as biology and ...
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In each case, this failure is telling us that once we enter the relativistic regime we need a new formalism in order to treat states with an unspecified number of particles. This formalism is quantum field theory (QFT). Answer 2: Because all particles of the same type are the same. This sound rather dumb.
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A PDF file of lecture notes on the conceptual basis of quantum field theory, covering scalar, spinor and gauge elds, renormalization, anomalies, and topological twists. The notes are based on the author's book \"Relativistic Quantum Field Theory\" and are intended for advanced students and researchers.
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2 Algebraic quantum mechanics 2.1 Postulates of quantum mechanics The standardformalism of quantum theory startswith a complexHilbert space H, whose elements φ∈ H are called state vectors. (For convenience some basic definitions concerning operators on Hilbert space are collected in Appendix A.) The key postulates of quantum mechanics say that:
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1.5 The quantum mechanical harmonic oscillator Problems 2 Classical Field Theory 2.1 From N-point mechanics to field theory 2.2 Relativistic field theory 2.3 Action for a scalar field 2.4 Plane wave solution to the Klein-Gordon equation 2.5. Symmetries and conservation laws Problems Quantum Field Theory 3.1 Canonical field quantisation
