Regularization (physics)
Encyclopedia
Introduction
In physicsPhysics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, especially quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator (for example, the minimal distance
in space which is useful if the divergences arise from short-distance physical effects). The correct physical result is obtained in the limit in which the regulator goes away (in our example, 
), but the virtue of the regulator is that for its finite value, the result is finite.However, the result usually includes terms proportional to expressions like
which are not well-defined in the limit 
. Regularization is the first step towards obtaining a completely finite and meaningful result; in quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
it must be usually followed by a related, but independent technique called renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
. Renormalization is based on the requirement that some physical quantities — expressed by seemingly divergent expressions such as
— are equal to the observed values. Such a constraint allows one to calculate a finite value for many other quantities that looked divergent.The existence of a limit as ε goes to zero and the independence of the final result from the regulator are nontrivial facts. The underlying reason for them lies in universality
Universality (dynamical systems)
In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together...
as shown by Kenneth Wilson and Leo Kadanoff
Leo Kadanoff
Leo Philip Kadanoff is an American physicist. He is a professor of physics at the University of Chicago and a former President of the American Physical Society . He has contributed to the fields of statistical physics, chaos theory, and theoretical condensed matter physics.-Biography:Kadanoff...
and the existence of a second order phase transition. Sometimes, taking the limit as ε goes to zero is not possible. This is the case when we have a Landau pole
Landau pole
In physics, the Landau pole is the momentum scale at which the coupling constant of a quantum field theory becomes infinite...
and for nonrenormalizable couplings like the Fermi interaction. However, even for these two examples, if the regulator only gives reasonable results for
and we are working with scales of the order of 
, regulators with 
still give pretty accurate approximations. The physical reason why we can't take the limit of ε going to zero is the existence of new physics below Λ.It is not always possible to define a regularization such that the limit of ε going to zero is independent of the regularization. In this case, one says that the theory contains an anomaly
Anomaly (physics)
In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics an anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking...
. Anomalous theories have been studied in great detail and are often founded on the celebrated Atiyah–Singer index theorem
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
or variations thereof (see, for example, the chiral anomaly
Chiral anomaly
A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...
).
Specific types of regularization include
- Dimensional regularizationDimensional regularizationIn theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini for regularizing integrals in the evaluation of Feynman diagrams; in other words, assigning values to them that are meromorphic functions of an auxiliary complex parameter d, called the...
- Pauli–Villars regularization
- Lattice regularization
- Zeta function regularizationZeta function regularizationIn mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators...
- Causal regularizationCausal perturbation theoryCausal perturbation theory is a mathematically rigorous approach to renormalization theory, which makesit possible to put the theoretical setup of perturbative quantum field theory on a sound mathematical basis....
- Hadamard regularization
- Point-splitting regularization
Conceptual problem
Perturbative predictions by quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
about quantum scattering of elementary particles, implied by a corresponding Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
densities, are computed using the Feynman rules, a regularization method to circumvent ultraviolet divergences so as to obtain finite results for Feynman diagrams containing loops, and a renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
scheme. Regularization method results in regularized n-point Green’s functions (propagator
Propagator
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...
s), and a suitable limiting procedure (a renormalization scheme) then leads to perturbative S-matrix elements. These are independent of the particular regularization method used, and enable one to model perturbatively the measurable physical processes (cross sections, probability amplitudes, decay widths and lifetimes of excited states). However, so far no known regularized n-point Green’s functions can be regarded as being based on a physically realistic theory of quantum-scattering since the derivation of each disregards some of the basic tenets of conventional physics (e.g., by not being Lorentz-invariant, by introducing either unphysical particles with negative metric or wrong statistic, or discrete space-time, or lowering the dimensionality of space-time, or some combination thereof. . . ). So the available regularization methods are understood as formalistic technical devices, devoid of any direct physical meaning. In addition, there are qualms about renormalization. For a history and comments on this more than half-a-century old open conceptual problem, see e.g.
Pauli´s conjecture
As it seems that the vertices of non-regularized Feynman series adequately describe interactions in quantum scattering, it is taken that their ultraviolet divergences are due to the asymptotic, high-energy behavior of the Feynman propagators. So it is a prudent, conservative approach to retain the vertices in Feynman series, and modify only the Feynman propagators to create a regularized Feynman series. This is the reasoning behind the formal Pauli-Villars covariant regularization by modification of Feynman propagators through auxiliary unphysical particles, cf and representation of physical reality by Feynman diagrams.In 1949 Pauli
Wolfgang Pauli
Wolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...
conjectured there is a realistic regularization, which is implied by a theory that respects all the established principles of contemporary physics. So its propagators (i) do not need to be regularized, and (ii) can be regarded as such a regularization of the propagators used in quantum field theories that might reflect the underlying physics. The additional parameters of such a theory do not need to be removed (i.e. the theory needs no renormalization) and may provide some new information about the physics of quantum scattering, though they may turn out experimentally to be negligible. By contrast, any present regularization method introduces formal coefficients that must eventually be disposed of by renormalization.
Opinions
DiracPaul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
was persistently, extremely critical about procedures of renormalization. So in 1963 :“… in the renormalization theory we have a theory that has defied all the attempts of the mathematician to make it sound. I am inclined to suspect that the renormalization theory is something that will not survive in the future,…” So he was expecting a realistic regularization.
About the skepticism that there is a realistic regularization Salam
Abdus Salam
Mohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk (Urdu: محمد عبد السلام, pronounced , (January 29, 1926– November 21, 1996) was a Pakistani theoretical physicist and Nobel laureate in Physics for his work on the electroweak unification of the...
's remark in 1972 is still relevant: ”Field-theoretic infinities first encountered in Lorentz's computation of electron have persisted in classical electrodynamics for seventy and in quantum electrodynamics for some thirty-five years. These long years of frustration have left in the subject a curious affection for the infinities and a passionate belief that they are an inevitable part of nature; so much so that even the suggestion of a hope that they may after all be circumvented - and finite values for the renormalization constants computed - is considered irrational. Compare Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
's postscript to the third volume of his Autobiography The Final Tears, 1944-1967 (George Allen and Unwin, Ltd., London 1969) p.221: ‘In the modern world, if communities are unhappy, it is often because they have ignorances, habits, beliefs, and passions, which are dearer to them than happiness or even life. I find many men in our dangerous age who seems to be in love with misery and death, and who grow angry when hopes are suggested to them. They think hope is irrational and that, in sitting down to lazy despair, they are merely facing facts.’ ”
However, in ’t Hooft’s opinion : “History tells us that if we hit upon some obstacle, even if it looks like a pure formality or just a technical complication, it should be carefully scrutinized. Nature might be telling us something, and we should find out what it is.”
By Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
: “One can distinguish between two main procedures for a theoretical physicist. One of them is to work from the experimental basis . . . The other procedure is to work from the mathematical basis. One examines and criticizes the existing theory. One tries to pin-point the faults in it and then tries to remove them. The difficulty here is to remove the faults without destroying the very great successes of the existing theory.” The difficulty with a realistic regularization is that so far there is none, although nothing could be destroyed by its bottom-up approach; and there is no experimental basis for it.
Minimal realistic regularization
Considering distinct theoretical problems, Dirac in 1963 suggested: ”I believe separate ideas will be needed to solve these distinct problems and that they will be solved one at a time through successive stages in the future evolution of physics. At this point I find myself in disagreement with most physicists. They are inclined to think one master idea will be discovered that will solve all these problems together. I think it is asking too much to hope that anyone will be able to solve all these problems together. One should separate them one from another as much as possible and try to tackle them separately. And I believe the future development of physics will consist of solving them one at a time, and that after any one of them has been solved there will still be a great mystery about how to attack further ones.”According to Dirac: ”Quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
is the domain of physics that we know most about, and presumably it will have to be put in order before we can hope to make any fundamental progress with other field theories, although these will continue to develop on the experimental basis.”
Preceding two Dirac’s remarks suggest that we should start searching for a realistic regularization in the case of quantum electrodynamics (QED) in the four-dimensional Minkowski spacetime, starting with the original QED Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
density.
The path-integral formulation provides the most direct way from the Lagrangian density to the corresponding Feynman series in its Lorentz-invariant form. The free-field part of the Lagrangian density determines the Feynman propagators, whereas the rest determines the vertices. As the QED vertices are considered to adequately describe interactions in QED scattering, it make sense to modify only the free-field part of the Lagrangian density so as to obtain such regularized Feynman series that the Lehmann-Symanzik-Zimmermann reduction formula provides a perturbative S-matrix that: (i) is Lorentz invariant and unitary; (ii) involves only the QED particles; (iii) depends solely on QED parameters and those introduced by the modification of the Feynman propagators—for particular values of these parameters it is equal to the QED perturbative S-matrix; and (iv) exhibits the same symmetries as the QED perturbative S-matrix. Let us refer to such a regularization as the minimal realistic regularization, and start searching for the corresponding, modified free-field parts of the QED Lagrangian density.
Transport theoretic approach
In the Kinetic theoryKinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...
of gases, one can extend the relativistic differential equations of classical fluid dynamics to shorter distances through the relativistic Boltzmann
Boltzmann equation
The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of one particle in rarefied gas...
integro-differential transport equation with the ´Stosszahlansatz´ assumption. This is written as a momentum-space integral over the product of one-particle distribution functions of two independent four-vector variables: of space-time positions and of four-momenta of the constituent particles. For examples see, and the references cited therein.
According to Bjorken and Drell, it would make physical sense to sidestep ultraviolet divergence
Ultraviolet divergence
In physics, an ultraviolet divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very high energy , or, equivalently, because of physical phenomena at very short distances. An infinite answer to a question that should have a...
s by using more detailed description than can be provided by differential field equations. And Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
noted about a differential equations :“. . . for neutron diffusion it is only an approximation that is good when the distance over which we are looking is large compared with the mean free path. If we looked more closely, we would see individual neutrons running around.” And then he wondered, “Could it be that the real world consists of little X-ons which can be seen only at very tiny distances? And that in our measurements we are always observing on such a large scale that we can’t see these little X-ons, and that is why we get the differential equations? . . .Are they [therefore] also correct only as a smoothed-out imitation of a really much more complicated microscopic world?”
Already in 1938, Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
proposed that a quantum field theory can provide only an idealized, large-scale description of quantum dynamics, valid for distances larger than some fundamental length, expected also by Bjorken and Drell in 1965. The preceding Feynman's remark provides a possible physical reason for its existence.
All these remarks suggest presuming that X-ons exist and using certain results of the Kinetic theory in constructing the minimal realistic regularization, so as to obtain a particular finite theory of QED scattering . Which indicates that to avoid ultraviolet divergences and model faster-than-light effects it suffices to appropriately change only the free-field Lagrangians, introducing an additional four vector variable, while retaining their locality in space-time and Lorentz invariance.