Quantum theory of fields. Foundations
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.
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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.
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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. 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. 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.
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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.
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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.
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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.