The latter need not be razed immediately, and may ultimately glean supportive rigging from components not yet constructed. In short, the theoretician hopes that the axiomatization will effectively separate sense from nonsense, and that this will serve to make possible substantial progress towards the development of a mature theory. Grounding in a rigorous mathematical framework can be an important part of the exercise, and that was a key aspect of the axiomatization of QFT by Wightman.
It was further refined in the late s by Bogoliubov, who explicitly placed axiomatic QFT in the rigged Hilbert space framework Bogoliubov et al. Rigged Hilbert space entered the axiomatic framework by way of the domain axiom, so this axiom will be discussed in more detail below. In QFT, a field is characterized by means of an operator rather than a function.
A field operator may be obtained from a classical field function by quantizing the function in the canonical manner — see Mandl , pp. Field operators that are relevant for QFT are too singular to be regarded as realistic, so they are smoothed out over their respective domains using elements of a space of well-behaved functions known as test functions. There are many different test-functions spaces Gelfand and Shilov , Chapter 4.
It was later determined that some realistic models require the use of other test-function spaces. Streater and Wightman As noted earlier, the appropriateness of the rigged Hilbert space framework enters by way of the domain axiom. Concerning that axiom, Wightman says the following in the notation introduced above, which differs slightly from that used by Wightman. In Bogoliubov et al. This serves to justify a claim they make earlier in their treatise:.
In both approaches, a field is an abstract system having an infinite number of degrees of freedom. Sub-atomic quantum particles are field effects that appear in special circumstances. In algebraic QFT, there is a further abstraction: the most fundamental entities are the elements of the algebra of local and quasi-local observables, and the field is a derived notion. The term local means bounded within a finite spacetime region, and an observable is not regarded as a property belonging to an entity other than the spacetime region itself.
The term quasi-local is used to indicate that we take the union of all bounded spacetime regions. In short, the algebraic approach focuses on local or quasi-local observables and treats the notion of a field as a derivative notion; whereas the axiomatic approach as characterized just above regards the field concept as the fundamental notion.
The two approaches are mutually complementary — they have developed in parallel and have influenced each other by analogy Wightman For a discussion of the close connections between these two approaches, see Haag , p. Those criticisms motivated mathematically inclined physicists to search for a mathematically rigorous formulation of QFT.
Axiomatic versions of QFT have been favored by mathematical physicists and most philosophers. With greater mathematical rigor it is possible to prove results about the theoretical structure of QFT independent of any particular Lagrangian.
Axiomatic QFT provides clear conceptual frameworks within which precise questions and answers to interpretational issues can be formulated. In Wightman QFT, the axioms use functional analysis and operator algebras and is closer to LQFT since its axioms describe covariant field operators acting on a fixed Hilbert space. However, axiomatic QFT approaches are sorely lacking with regards to building empirically adequate models. Even though there is a canonical mathematical framework for quantum mechanics, there are many interpretations of that framework, e.
QFT has two levels that require interpretation: 1 which QFT framework should be the focus of these foundational efforts, if any, and 2 how that preferred framework should be interpreted. Since 1 involves issues about mathematical rigor and pragmatic virtues, it directly bears on the focus of this article.
The lack of a canonical formulation of QFT threatens to impede any metaphysical or epistemological lessons that might be learned from QFT. Fraser , argues that the interpretation of QFT should be based on the mathematically rigorous approach of axiomatic formulations of QFT. Swanson and Egg, Lam, and Oldofedi are good overviews of the debate between Fraser and Wallace for an extended analysis see James Fraser The debate covers many different philosophical topics in QFT, which makes it more challenging to pin down exactly what is essential to the arguments for both sides for one view of what is essential for the debate, see Egg, Lam, and Oldofedi One issue is the role of internal consistency established by mathematical rigor versus empirical adequacy.
LQFT has a collection of calculational techniques including perturbation theory, path integrals, and renormalization group methods. One criticism of LQFT is that the calculational techniques it uses are not mathematically rigorous. Since exactly solvable free QFT models are more mathematically tractable than interacting QFT models, perturbative QFT treats interactions as perturbations to the free Lagrangian assuming weak coupling.
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For strongly coupled theories like quantum chromodynamics that idealization fails. Using perturbation theory, approximate solutions for interacting QFT models can be calculated by expanding S-matrix elements in a power series in terms of a coupling parameter.
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However, the higher order terms will often contain divergent integrals. Typically, renormalization of the higher order terms is required to get finite predictions. Two sources of divergent integrals are infrared long distance, low energy and ultraviolet short distance, high energy divergences. Infrared divergences are often handled by imposing a long distance cutoff or putting a small non-zero lower limit for the integral over momentum. A sharp cutoff at low momentum is equivalent to putting the theory in a finite volume box.
Ultraviolet divergences are often handled by imposing a momentum cutoff to remove high momentum modes of a theory. That is equivalent to freezing out variations in the fields at arbitrarily short length scales. Putting the system on a lattice with some finite spacing can also help deal with the high momentum. Dimensional regularization, where the integral measure is redefined to range over a fractional number of dimensions, can help with both infrared and ultraviolet divergences.
The last step in renormalization is to remove the cutoffs by taking the continuum limit i. The hope is that the limit is well-defined and there are finite expressions of the series at each order. James Fraser identifies three problems for perturbative QFT. James Fraser argues that 1 and 2 do not pose severe problems for perturbative QFT because it is not attempting to build continuum QFT models. It is building approximate physical quantities — not mathematical structures that are to be interpreted as physical systems.
Baker and Swanson note that LQFT makes false or unproven assumptions such as the convergence of certain infinite sums in perturbation theory. Dyson gives a heuristic argument that quantum electrodynamic perturbation series do not converge. Baker and Swanson also argue that the use of long distance cutoffs is at odds with cosmological theory and astronomical observations which suggest that the universe is spatially infinite. Even in the weak coupling limit where perturbation theory can be formally applied, it is not clear when the perturbative QFT gives an accurate approximation of the underlying physics.
When there are 4 dimensions, the theory is also trivial if additional technical assumptions hold see Swanson p. Another area where questions of mathematical rigor arise within perturbative QFT is the use of path integrals. The following details come mainly from Hancox-Li More specifically, the action is a functional of quantum fields. The functional integral over the action ranges over all possible combinations of the quantum fields values over spacetime.
Informally, the sum is being taken over all possible field configurations. As Swanson notes, the path integral requires choosing a measure over an infinite dimensional path space, which is only mathematically well-defined in special cases. For example, if the system is formulated on a hypercubic lattice, then the measure can be defined see section 1. Another way of having a well-defined measure is to restrict attention to a finite dimensional subspace.
But if functions are allowed to vary arbitrarily on short length scales, then the integral ceases to be well-defined Wallace , p. All of the correlation functions i. To deal with 1 , physicists do the following procedures Hancox-Li , pp. But this construction is purely formal and not mathematically defined. The rules used to manipulate the Lagrangian, and hence the partition function, are not well-defined.
Wallace argues that renormalization group techniques have overcome the mathematical deficiencies of older renormalization calculational techniques for more details on the renormalization group see Butterfield and Bouatta , Fraser , Hancox-Li a, b, According to Wallace, the renormalization group methods put LQFT on the same level of mathematical rigor as other areas of theoretical physics. It provides a solid theoretical framework that is explanatorily rich in particle physics and condensed matter physics, so the impetus for axiomatic QFT has been resolved.
Renormalization group techniques presuppose that QFT will fail at some short length scale, but the empirical content of LQFT is largely insensitive to the details at such short length scales. James Fraser and Hancox-Li b argue that the renormalization group does more than provide empirical predictions in QFT.
The renormalization group gives us methods for studying the behavior of physical systems at different energy scales, namely how properties of QFT models depend or do not depend on small scale structure. The renormalization group provides a non-perturbative explanation of the success of perturbative QFT.
Hancox-Li b discusses how mathematicians working in constructive QFT use non-perturbative approximations with well controlled error bounds to prove the existence or non-existence of ultraviolet fixed points. Hancox-Li argues that the renormalization group explains perturbative renormalization non-perturbatively. The renormalization group can tell us whether certain Lagrangians have an ultraviolet limit that satisfies the axioms a QFT should satisfy. Thus, the use of the renormalization group in constructive QFT can provide additional dynamical information e.
Fraser takes QFT to be the union of quantum theory and special relativity. QFT is not a truly fundamental theory since gravity is absent. LQFT gives us an effective ontology. The renormalization group tell us that QFT cannot be trusted in the high energy regimes where quantum gravity can be expected to apply, i.
There are, however, other options to consider. Some philosophers have rejected the seemingly either-or nature of the debate between Wallace and Fraser to embrace more pluralistic views. On these pluralistic views, different formulations of QFT might be appropriate for different philosophical questions. LQFT supplies various powerful predictive tools and explanatory schemas. It can account for gauge theories, the Standard Model of particle physics, the weak and strong nuclear force, and the electromagnetic force.
However, the collection of calculational techniques are not all mathematically well-defined. LQFT provides QFT theories at only certain length scales and cannot make use of unitarily inequivalent representations since LQFT uses cutoffs which renders all representations finite dimensional and unitarily equivalent by the Stone-von Neumann theorem. Axiomatic QFT is supposed to provide a rigorous description of fundamental QFT at all length scales, but that conflicts with the effective field theory viewpoint where QFT is only defined for certain lengths.
But if axiomatic QFT capture what all QFTs have in common, then effective field theories should be captured by it as well. Within the axiomatic approach, Wightman QFT has many sophisticated tools for building concrete models of QFT in addition to rigorously proving structural results like the PCT theorem and the spin statistics theorem. But Wightman QFT relies on localized gauge-dependent field operators that do not directly represent physical properties.
It has topological tools to define global quantities like temperature, energy, charge, particle number which use unitarily inequivalent representations. But AQFT has difficulty constructing models. The nontrivial solutions it constructs are supposed to correspond to Lagrangians that particle physicists use. This ensures that various axiomatic systems have a physical connection to the world via the empirical success of LQFT. While constructive QFT has done this for some models with dimensions less than 4, it has not yet been accomplished for a 4 dimensional Lagrangian that particle physicists use.
Any model that satisfies the Osterwalder-Schrader axioms will automatically satisfy the Wightman axioms. Constructive QFT tries to construct the functional integral measures for path integrals by shifting from Minkowski spacetime to Euclidean spacetime via a Wick rotation what follows is based on section four of Hancox-Li The Osterwalder-Schrader axioms are related to the Wightman axioms by the Osterwalder-Schrader Reconstruction Theorem which states that any set of functions satisfying the Osterwalder-Schrader axioms determines a unique Wightman model whose Schwinger functions form that set.
It allows the constructive field theorists to use the advantages of Euclidean space for defining a measure while ensuring that they are constructing models that exist in Minkowski spacetime. It still has to be verified that the solution corresponds to a renormalized perturbation series that physicists derive for the corresponding Lagrangian in LQFT. This is crucial since, as Swanson points out, it is unclear whether perturbation theory is an accurate guide for the underlying physics described by LQFT.
Those models correspond to the Lagrangians of interest to particle physicists. Another tool of constructive QFT is the use of asymptotic series, which can tell us which function the perturbative series is asymptotic to, which perturbative QFT does not. Constructive QFT tries to determine some properties of non-perturbative solutions to the equations of motion which guarantee that certain methods of summing asymptotic expansions will lead to a unique solution see Hancox-Li pp. Roughly, a function is asymptotic to a series expansion when successive terms of the series provide an increasingly accurate description of how quickly the function grows.
Resonance expansions in quantum mechanics
The difference between the function and each order of the perturbation series is approximately small. But there are many different functions that have the same asymptotic expansion. Ideally, we want there to be a unique function because then there is a unique non-perturbative solution. The concept of strong asymptoticity requires that the difference between the function and each order of the series is smaller than what was required by asymptoticity. A strongly asymptotic series uniquely determines a function.
If there is a strong asymptotic series, then the function can be uniquely reconstructed from the series by Borel summation. The Borel transform of the series is given by dividing the coefficients each term in the series by a factorial of the order of that term and then integrating to recover the exact function.
In constructive QFT, the goal is to associate a unique function with a renormalized perturbation series and some kind of Borel summability is the main candidate so far, though the Borel transform cannot remove large-order divergences. The asymptotic behavior of the renormalized perturbation series can be extremely sensitive to the choice of regularization and render it asymptotic to a free field theory even if it appears to describe nontrivial perturbations see Swanson p. Table of contents 19 chapters Table of contents 19 chapters Introduction Pages The algebraic structure of the space of states Pages The topological structure of the space of states Pages Generalized eigenvectors and the nuclear spectral theorem Pages A remark concerning generalization Pages References on chapter I Pages Introduction Pages The Moller wave operators Pages The Hardy class functions on a half plane Pages References for chapter II Pages Rigged Hilbert spaces of Hardy class functions Pages Functional for Ho and Hl Pages The Hardy class functions on a half plane.
Rigged Hilbert spaces of Hardy class functions. The spaces? Functional for Ho and Hl. The RHS model for decaying states. Dynamical semigroups. Virtual states. The Grand Design. In Stock. The Physics Behind The Behind Physics for the Life Sciences Second Edition. Effective Computation in Physics. Classical Mechanics, Volume 1 Tools and Vectors. Group Theory For Physicists.