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A number of further aspects intensified the unease about the standard formulation of QFT. The first one is that quantities like total charge, total energy or total momentum of a field are unobservable since their measurement would have to take place in the whole universe. Accordingly, quantities which refer to infinitely extended regions of space-time should not appear among the observables of the theory as they do in the standard formulation of QFT. The physical counterpart of the problem is that it would require an infinite amount of energy to measure a field at a point of space-time.

One way to handle this situation—and one of the starting points for axiomatic reformulations of QFT—is not to consider fields at a point but instead fields which are smeared out in the vicinity of that point using certain functions, so-called test functions. The third important problem for standard QFT which prompted reformulations is the existence of inequivalent representations.

In other words, we are merely dealing with two different ways for representing the same physical reality, and it is possible to switch between these different representations by means of a unitary transformation, i.

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Representations of some given algebra or group are sets of mathematical objects, like numbers, rotations or more abstract transformations e. This means that the combination of any two elements in the representation space, say a and b , leads to a third element which corresponds to the element that results when you combine the elements corresponding to a and b in the algebra or group that is represented. In von Neumann gave a detailed proof of a conjecture by Stone that the canonical commutation relations CCRs for position coordinates and their conjugate momentum coordinates in configuration space fix the representation of these two sets of operators in Hilbert space up to unitary equivalence von Neumann's uniqueness theorem.

This means that the specification of the purely algebraic CCRs suffices to describe a particular physical system. In quantum field theory, however, von Neumann's uniqueness theorem looses its validity since here one is dealing with an infinite number of degrees of freedom. Now one is confronted with a multitude of inequivalent irreducible representations of the CCRs and it is not obvious what this means physically and how one should cope with it.

Since the troublesome inequivalent representations of the CCRs that arise in QFT are all irreducible their inequivalence is not due to the fact that some are reducible while others are not a representation is reducible if there is an invariant subrepresentation, i. Since inequivalent irreducible representations short: IIRs seem to describe different physical states of affairs it is no longer legitimate to simply choose the most convenient representation, just like choosing the most convenient frame of reference. The acuteness of this problem is not immediately clear, since prima facie it is possibly that all but one of the IIRs are physically irrelevant, i.

However, although apparently this applies to most of the available IIRs, it seems that a number of irreducible representations of the CCRs remain that are inequivalent and physically relevant. According to the algebraic point of view algebras of observables rather than observables themselves in a particular representation should be taken as the basic entities in the mathematical description of quantum physics; thereby avoiding the above-mentioned problems from the outset. Thus sticking to the usual Hilbert space formulation tacitly implies choosing one particular representation.

Another point where algebraic formulations are advantageous derives from the fact that two quantum fields are physically equivalent when they generate the same algebras of local observables. Such equivalent quantum field theories belong to the same so-called Borchers class which entails that they lead to the same S -matrix. The choice of a particular field system is to a certain degree conventional, namely as long as it belongs to the same Borchers class. Thus it is more appropriate to consider these algebras, rather than quantum fields, as the fundamental entities in QFT.

A prominent attempt to axiomatise QFT is Wightman's field axiomatics from the early s. Wightman imposed axioms on polynomial algebras P O of smeared fields, i.

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While the usage of unbounded field operators makes Wightman's approach mathematically cumbersome, Algebraic Quantum Field Theory AQFT —arguably the most successful attempt to reformulate QFT axiomatically—employs only bounded operators. One of the crucial ideas of AQFT is taking so-called nets of algebras as basic for the mathematical description of a quantum physical system. Against this approach Haag argued that inequivalent representations can be understood physically by realizing that the important physical information in a quantum field theory is not contained in individual algebras but in the net of algebras, i.

The crucial point is that it is not necessary to specify observables explicitly in order to fix physically meaningful quantities. The very way how algebras of local observables are linked to space-time regions is sufficient to supply observables with physical significance.

It is the partition of the algebra A loc of all local observables into subalgebras which contains physical information about the observables, i. Physically the most important notion of AQFT is the principle of locality which has an external as well as an internal aspect.

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The external aspect is the fact that AQFT considers only observables connected with finite regions of space-time and not global observables like the total charge or the total energy momentum vector which refer to infinite space-time regions. This approach was motivated by the operationalistic view that QFT is a statistical theory about local measurement outcomes with all the experimental information coming from measurements in finite space-time regions. Accordingly everything is expressed in terms of local algebras of observables.

The basic structure upon which the assumptions or conditions of AQFT are imposed are local observables, i. States can thus be understood as assignments of expectation values to observables. One can group the assumptions of AQFT into relativistic axioms, such as locality and covariance, general physical assumptions, like isotony and spectrum condition, and finally technical assumptions which are closely related to the mathematical formulation.

As a reformulation of QFT, AQFT is expected to reproduce the main phenomena of QFT, in particular properties which are characteristic of it being a field theory, like the existence of antiparticles, internal quantum numbers, the relation of spin and statistics, etc.

Pdf Mathematical Aspects Of Quantum Field Theory

That this aim could not be achieved on a purely axiomatic basis is partly due to the fact that the connection between the respective key concepts of AQFT and QFT, i. It turned out that the main link between observable algebras and quantum fields are superselection rules , which put restrictions on the set of all observables and allow for classification schemes in terms of permanent or essential properties.

In recent years, QFT has received a lot of attention in the philosophy of physics. While most philosophers of physics who are skeptical about this approach remained largely silent, Wallace , launched an eloquent attack on the predominance of AQFT for foundational studies about QFT.

To be sure, Wallace emphasizes, his critique is not directed against the use of algebraic methods, e. So what may justify this drastic conclusion? On the one hand, Wallace points out that, the problem of ultraviolet divergences, which initiated the search for alternative approaches in the s, was eventually solved in CQFT via the renormalization group techniques. On the other hand, AQFT never succeeded in finding realistic interacting quantum field theories in four dimensions such as QED that fit into their framework. She argues that consistency plays a central role in choosing between different formulations of QFT since they do not differ in their respective empirical success and AQFT fares better in this respect.

Moreover, Fraser questions Wallace's crucial point in defense of CQFT, namely that the empirically successful application of renormalization group techniques in QFT removes all doubts about CQFT: The fact that renormalization in condensed matter physics and QFT are formally similar does not license Wallace's claim that there are also physical similarities concerning the freezing out of degrees of freedom at very small length scales.

And if that physical analogy cannot be sustained, then the empirical success of renormalization in CQFT leaves the physical reasons for this success in the dark, in contrast to the case of condensed matter physics, where the physical basis for the empirical success of renormalization is intelligible, namely the fact that matter is discrete at atomic length scales. As a consequence, despite of the formal analogy with renormalization in condensed matter physics the empirical success of renormalization in CQFT does not, as Wallace claims, discredit the idea to work with arbitrarily small regions of spacetime, as it is done in AQFT.

Kuhlmann b also advocates AQFT as the prime object for foundational studies, focusing on ontological considerations. He argues that for matters of ontology AQFT is to be preferred over CQFT because, like ontology itself, AQFT strives for a clear separation of fundamental and derived entities and a parsimonious selection of basic assumptions.

CQFT, on the other hand is a grown formalism that is very good for calculations but obscures foundational issues. AQFT is suited and designed to illuminate the basic structure of QFT, but it is not and never will be the appropriate framework for the working physicist. Ontology is concerned with the most general features, entities and structures of being. One can pursue ontology in a very general sense or with respect to a particular theory or a particular part or aspect of the world.

With respect to the ontology of QFT one is tempted to more or less dismiss ontological inquiries and to adopt the following straightforward view. There are two groups of fundamental fermionic matter constituents, two groups of bosonic force carriers and four including gravitation kinds of interactions.

Mathematical Aspects of Quantum Field Theory

As satisfying as this answer might first appear, the ontological questions are, in a sense, not even touched. Saying that, for instance the down quark is a fundamental constituent of our material world is the starting point rather than the end of the philosophical search for an ontology of QFT. The main question is what kind of entity, e. The answer does not depend on whether we think of down quarks or muon neutrinos since the sought features are much more general than those ones which constitute the difference between down quarks or muon neutrinos.

The relevant questions are of a different type. What are particles at all? Can quantum particles be legitimately understood as particles any more, even in the broadest sense, when we take, e. Could it be more appropriate not to think of, e. Many of the creators of QFT can be found in one of the two camps regarding the question whether particles or fields should be given priority in understanding QFT. While Dirac, the later Heisenberg, Feynman, and Wheeler opted in favor of particles, Pauli, the early Heisenberg, Tomonaga and Schwinger put fields first see Landsman Today, there are a number of arguments which prepare the ground for a proper discussion beyond mere preferences.

It seems almost impossible to talk about elementary particle physics, or QFT more generally, without thinking of particles which are accelerated and scattered in colliders. Nevertheless, it is this very interpretation which is confronted with the most fully developed counter-arguments. There still is the option to say that our classical concept of a particle is too narrow and that we have to loosen some of its constraints. After all, even in classical corpuscular theories of matter the concept of an elementary particle is not as unproblematic as one might expect.

For instance, if the whole charge of a particle was contracted to a point, an infinite amount of energy would be stored in this particle since the repulsive forces become infinitely large when two charges with the same sign are brought together. The so-called self energy of a point particle is infinite.

The Unreasonable Effectiveness of Quantum Physics in Modern Mathematics -- Robbert Dijkgraaf

Probably the most immediate trait of particles is their discreteness. Obviously this characteristic alone cannot constitute a sufficient condition for being a particle since there are other things which are countable as well without being particles, e. It seems that one also needs individuality , i. Teller discusses a specific conception of individuality, primitive thisness , as well as other possible features of the particle concept in comparison to classical concepts of fields and waves, as well as in comparison to the concept of field quanta, which is the basis for the interpretation that Teller advocates.

A critical discussion of Teller's reasoning can be found in Seibt Since this discussion concerns QM in the first place, and not QFT, any further details shall be omitted here. French and Krause offer a detailed analysis of the historical, philosophical and mathematical aspects of the connection between quantum statistics, identity and individuality. See Dieks and Lubberdink for a critical assessment of the debate. Also consult the entry on quantum theory: identity and individuality.

There is still another feature which is commonly taken to be pivotal for the particle concept, namely that particles are localizable in space. While it is clear from classical physics already that the requirement of localizability need not refer to point-like localization, we will see that even localizability in an arbitrarily large but still finite region can be a strong condition for quantum particles.

Bain argues that the classical notions of localizability and countability are inappropriate requirements for particles if one is considering a relativistic theory such as QFT. Eventually, there are some potential ingredients of the particle concept which are explicitly opposed to the corresponding and therefore opposite features of the field concept. Whereas it is a core characteristic of a field that it is a system with an infinite number of degrees of freedom , the very opposite holds for particles.

A particle can for instance be referred to by the specification of the coordinates x t that pertain, e. A further feature of the particle concept is connected to the last point and again explicitly in opposition to the field concept. In a pure particle ontology the interaction between remote particles can only be understood as an action at a distance.

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In contrast to that, in a field ontology, or a combined ontology of particles and fields, local action is implemented by mediating fields. Finally, classical particles are massive and impenetrable, again in contrast to classical fields. The easiest way to quantize the electromagnetic or: radiation field consists of two steps.