"Renormalization is used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions."
Discusses the issues that arise when we try to reconcile quantum fields with the observed phenomenon of divergent integrals, and introduces the techniques of regularization and renormalization.
Quantum Field Theory (QFT): The study of particle interactions in the framework of quantum mechanics, including the concept of fields and their dynamics.
Regularization: The process of introducing a finite cut-off in QFT calculations to avoid divergent integrals.
Divergence: A mathematical term used to describe an infinite value that arises in QFT calculations due to the presence of an infinite number of degrees of freedom.
Renormalization: The process of removing divergences that arise in QFT calculations by arbitrarily redefining the parameters of the theory.
Wilsonian Renormalization: A modern approach to renormalization that involves integrating out the high-energy degrees of freedom and recasting the theory in terms of effective field theory.
Perturbative Renormalization: A method for calculating physical observables in QFT using a systematic expansion in powers of the coupling constants.
Non-perturbative Renormalization: A method that deals with strong-coupling regimes in QFT by using various techniques such as lattice simulations, Monte Carlo methods, and functional integrals.
Schwinger-Dyson equations: A set of non-linear differential equations that govern the behavior of correlation functions in a QFT.
Anomalous dimensions: The scaling dimensions of operators in QFT that determine their behavior under rescaling.
Effective field theory: A theory that describes the long-distance behavior of a QFT by integrating out the high-energy degrees of freedom.
Running of coupling constants: The dependence of the parameters of a QFT on the energy scale at which they are measured.
Beta function: The differential equation that governs the running of coupling constants in a QFT.
Asymptotic freedom: The phenomenon in QFT where the coupling constant becomes smaller at high energies.
Conformal field theory: A QFT that is invariant under conformal transformations.
Gauge theory: A QFT that describes the interaction between particles and fields in terms of gauge symmetries.
Lattice QCD: A non-perturbative approach to simulating QCD, the gauge theory describing the strong nuclear force, on a lattice.
Wilson loop: A lattice QCD observable that describes the behavior of particles moving through gauge fields.
Chiral symmetry breaking: A phenomenon in QCD where the symmetry between left- and right-handed particles is spontaneously broken.
Instantons: Solutions to the QCD equations of motion that describe tunneling between different topological sectors.
Effective action: The action that describes the dynamics of a QFT after integrating out the high-energy degrees of freedom.
Regularization: In order to compute the expected value of a quantum field theory, one has to use an integral. However, this integral is typically divergent, meaning the result is infinite. Regularization is a technique of introducing an artificial regulator into the integrals which prevents the divergence from occurring. This allows one to make calculations and take the limit as the regulator is removed.
Cut-off Renormalization: Cut-off Renormalization is a type of regularization, where the integral is limited to regions in momentum space where the momentum is less than a fixed, upper cut-off. This type of renormalization is useful because it doesn't produce large, unphysical masses, but it does show how divergences appear quantitatively.
Dimensional Renormalization: It is a type of regularization that utilizes the fact that the infinities produced in quantum field theory are usually due to short distance limits. Dimensional regularization extends space-time to more than 4 dimensions, in order to achieve a less singular behavior.
Mass Renormalization: Mass Renormalization is a type of correction to a particle's mass so that it cancels out the contribution due to the vacuum. It is one of the most common types of renormalization.
Wavefunction Renormalization: Wavefunction Renormalization involves investigating the effects of quantum fluctuations upon the probability density of particles. The purpose of this type of renormalization is to ensure that the probability of detecting a particle follows a Gaussian distribution around the expected value, rather than a distribution with infinite variance.
Running Coupling Renormalization: Running Coupling Renormalization allows for the evolution of a physical coupling constant, such as the strong force coupling, with energy. In this type of renormalization, the coupling constant is regarded as a function of the energy scale.
Operator Renormalization: Operator Renormalization accounts for the divergences that arise when expanding a field operator in powers of the bare coupling constant. Its purpose is to ensure that, as a result of the perturbative calculations, the correlation functions are finite and physically meaningful.
Lattice Renormalization: Lattice Renormalization replaces continuous space-time with a discretized lattice in calculation. It is useful in situations where simulating quantum field theories is computationally difficult.
"...it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian."
"Renormalization mathematically replaces the initially postulated mass and charge of an electron with the experimentally observed mass and charge."
"Renormalization specifies relationships between parameters in the theory when parameters describing large distance scales differ from parameters describing small distance scales."
"When describing spacetime as a continuum, certain statistical and quantum mechanical constructions are not well-defined. To define them, or make them unambiguous, a continuum limit must carefully remove 'construction scaffolding' of lattices at various scales."
"Renormalization procedures are based on the requirement that certain physical quantities (such as the mass and charge of an electron) equal observed (experimental) values."
"Initially viewed as a suspect provisional procedure even by some of its originators..."
"On the basis of the breakthrough renormalization group insights of Nikolay Bogolyubov and Kenneth Wilson..."
"The focus is on variation of physical quantities across contiguous scales while distant scales are related to each other through 'effective' descriptions."
"All scales are linked in a broadly systematic way..."
"Wilson clarified which variables of a system are crucial and which are redundant."
"Renormalization is distinct from regularization, another technique to control infinities by assuming the existence of new unknown physics at new scales."
"Renormalization was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in perturbation theory."
"...renormalization eventually was embraced as an important and self-consistent actual mechanism of scale physics in several fields of physics and mathematics."
"A continuum limit must carefully remove 'construction scaffolding' of lattices at various scales."
"...the observed measurement represents areas of quantum field theory that require deeper derivation from theoretical bases."
"...the actual physics pertinent to each is extracted with the suitable specific computational techniques appropriate for each. Wilson clarified which variables of a system are crucial and which are redundant."
"Renormalization is used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions."
"Accounting for the interactions of the surrounding particles shows that the electron-system behaves as if it had a different mass and charge than initially postulated."
"A cloud of virtual particles, such as photons, positrons, and others surrounds and interacts with the initial electron."