Definition of quantum field theory. Examples of quantum field theory in a Sentence Recent Examples on the Web On the theoretical side, quantum mechanics was soon superseded by quantum field theory , where the fields are also quantized.
First Known Use of quantum field theory , in the meaning defined above. Learn More About quantum field theory. Share quantum field theory Post the Definition of quantum field theory to Facebook Share the Definition of quantum field theory on Twitter. Time Traveler for quantum field theory The first known use of quantum field theory was in See more words from the same year. Dictionary Entries Near quantum field theory quantum evolution quantum field theory quantum gravity See More Nearby Entries.
Style: MLA. More from Merriam-Webster on quantum field theory Britannica. Get Word of the Day daily email! Test Your Vocabulary. EFTs describe relevant phenomena only in a certain domain since the Lagrangian contains only those terms that describe particles which are relevant for the respective range of energy.
EFTs are inherently approximative and change with the range of energy considered. EFTs are only applicable on a certain energy scale, i. Influences from higher energy processes contribute to average values but they cannot be described in detail. This procedure has no severe consequences since the details of low-energy theories are largely decoupled from higher energy processes.
Both domains are only connected by altered coupling constants and the renormalization group describes how the coupling constants depend on the energy. The main idea of EFTs is that theories, i. The physics changes by switching to a different energy scale, e. The dependence of theories on the energy scale distinguishes QFT from, e. Nevertheless, laws from different energy scales are not completely independent of each other. A central aspect of considerations about this dependence are the consequences of higher energy processes on the low-energy scale.
On this background a new attitude towards renormalization developed in the s, which revitalizes earlier ideas that divergences result from neglecting unknown processes of higher energies. Low-energy behavior is thus affected by higher energy processes.
Since higher energies correspond to smaller distances this dependence is to be expected from an atomistic point of view. According to the reductionist program the dynamics of constituents on the microlevel should determine processes on the macrolevel, i. However, as, for instance hydrodynamics shows, in practice theories from different levels are not quite as closely connected because a law which is applicable on the macrolevel can be largely independent of microlevel details.
For this reason analogies with statistical mechanics play an important role in the discussion about EFTs. The basic idea of this new story about renormalization is that the influences of higher energy processes are localizable in a few structural properties which can be captured by an adjustment of parameters. This new attitude supports the view that renormalization is the appropriate answer to the change of fundamental interactions when the QFT is applied to processes on different energy scales.
The price one has to pay is that EFTs are only valid in a limited domain and should be considered as approximations to better theories on higher energy scales. This prompts the important question whether there is a last fundamental theory in this tower of EFTs which supersede each other with rising energies. Some people conjecture that this deeper theory could be a string theory, i. Or should one ultimately expect from physics theories that they are only valid as approximations and in a limited domain?
Williams argues that EFTs by no means undermine a realist interpretation of QFT, provided one adopts a more refined notion of scientific realism. Wallace and Fraser discuss what the successful application of renormalization methods in quantum statistical mechanics means for their role in QFT, reaching very different conclusions.
Egg et al. However, there is a crucial difference. While the standard model is a theory with a fixed ontology understood in a prephilosophical sense , i. This section deals with only some particularly important proposals that go beyond the standard model, but which do not necessarily break up the basic framework of QFT. The standard model of particle physics covers the electromagnetic, the weak and the strong interaction.
However, the fourth fundamental force in nature, gravitation, has defied quantization so far. Although numerous attempts have been made in the last 80 years, and in particular very recently, there is no commonly accepted solution up to the present day. One basic problem is that the mass, length and time scales quantum gravity theories are dealing with are so extremely small that it is almost impossible to test the different proposals.
The most important extant versions of quantum gravity theories are canonical quantum gravity, loop theory and string theory. Canonical quantum gravity approaches leave the basic structure of QFT untouched and just extend the realm of QFT by quantizing gravity.
Other approaches try to reconcile quantum theory and general relativity theory not by supplementing the reach of QFT but rather by changing QFT itself.
String theory, for instance, proposes a completely new view concerning the most fundamental building blocks: It does not merely incorporate gravitation but it formulates a new theory that describes all four interactions in a unified way, namely in terms of strings see next subsection. While quantum gravity theories are very complicated and even more remote from classical thinking than QM, SRT and GRT, it is not so difficult to see why gravitation is far more difficult to deal with than the other three forces.
Electromagnetic, weak and strong force all act in a given space-time. In contrast, gravitation is, according to GRT, not an interaction that takes place in time, but gravitational forces are identified with the curvature of space-time itself. Thus quantizing gravitation could amount to quantizing space-time, and it is not at all clear what that could mean. Also, see the entry on quantum gravity. String theory is one of the most promising candidates for bridging the gap between QFT and general relativity theory by supplying a unified theory of all natural forces, including gravitation.
The basic idea of string theory is not to take particles as fundamental objects but strings that are very small but extended in one dimension.
This assumption has the pivotal consequence that strings interact on an extended distance and not at a point. This difference between string theory and standard QFT is essential because it is the reason why string theory also encompasses the gravitational force which is very difficult to deal with in the framework of QFT.
It is so hard to reconcile gravitation with QFT because the typical length scale of the gravitational force is very small, namely at Planck scale, so that the quantum field theoretical assumption of point-like interaction leads to untreatable infinities.
To put it another way, gravitation becomes significant in particular in comparison to strong interaction exactly where QFT is most severely endangered by infinite quantities. The extended interaction of strings brings it about that such infinities can be avoided. In contrast to the entities in standard quantum physics strings are not characterized by quantum numbers but only by their geometrical and dynamical properties. A basic geometrical distinction is the one between open strings, i.
The central dynamical property of strings is their mode of excitation, i. Reservations about string theory are mostly due to the lack of testability since it seems that there are no empirical consequences which could be tested by the methods which are, at least up to now, available to us.
But there are also other peculiar features of string theory which might be hard to swallow. One of them is the fact that preferred models of string theory need space-time with 10, 11 or even 26 dimensions. An intuitive idea can be gained by thinking of a macaroni which is a tube, i. Due to the problems of string theory, many physicists have abandoned it, but not all.
Correspondingly, string theory has also received some attention within the philosophy of physics community in recent years. Dawid see Other Internet Resources below argues that string theory has significant consequences for the philosophical debate about realism, namely that it speaks against the plausibility of anti-realistic positions.
Also see Dawid Johansson and Matsubara assess string theory from various different methodological perspectives, reaching conclusions in disagreement with Dawid Standard introductory monographs on string theory are Polchinski and Kaku Greene is a very successful popular introduction. There are three main motives for reformulating conventional QFT. The first motive is operationalism, the second one mathematical rigour and the third one finding a way to deal with the availability of inequivalent Hilbert space representations for systems with an infinite number of degrees of freedom, such as fields.
While in principle the three motives are independent of one another there are multiple interconnections in their actual implementation. One way in which the three motives are connected is the following: In QFT the quests for operationalism and mathematical rigour seem to go hand in hand, i. Moreover, it leads to an algebraic formulation that avoids privileging one amoung various available inequivalent representations, which tacitly happens in conventional QFT.
The first motive—operationalism—is not so higly valued any more today, and for good reasons see entry on Operationalism. Nevertheless, it was, not only in physics, very strong in and around the s, when axiomatic reformulations of QFT entered the scene. Accordingly, the impact of operationalism must not by overlooked. 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. From an operationalist perspective equally troublesome as point-like quantities are global quantities, like total charge, total energy or total momentum of a field.
They 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. In any case, however, it has been important in the formation of axiomatic reformulations of QFT.
Another operationalist reason for favouring algebraic formulations derives from the fact that two quantum fields are physically equivalent when they generate the same algebras of local observables.
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. The resulting operationalistic view of QFT is that it is a statistical theory about local measurement outcomes, expressed in terms of local algebras of observables. So far, we focussed on the operationalist motives for reformulating QFT and some of its consequences.
Now we will distinguish different, partly competing ways of implementing these general ideas. The second motive—mathematical rigour—consists foremost in the quest towards a concise axiomatic formulation, instead of the grab bag of conventional QFT, with its numerous mathematically dubious, even though successful, approximation techniques.
This quest comprises three parts, namely, first, the choice of those entities upon which the axioms are to be imposed, second, the choice of appropriate axioms, and, third, the proof that one has actually achieved an axiomatic reformulation of conventional QFT, which can reproduce all the established empirical and theoretical successes.
While axiomatic approaches are clear and sharp on the first two counts, their success is more limited with respect to the third. In general, one can say there are valuable successes with respect to very general theoretical insights, such as the connection of spin and statistics as well as non-localizability, while the weak point is the lack of realistic models for interacting quantum field theories. Since the fundamental entities in axiomatic reformulations of QFT are algebras of smeared field operators or of observables instead of quantum fields , reformulating QFT in algebraic terms and in axiomatic terms are enterprises with a large factual overlap.
Both originated in the s and influenced each other in their formation. The crucial axioms are covariance , microcausality spacelike separated field operators required to either commute or anticommute , and spectrum condition positive energy in all Lorentz frames, so that the vacuum is a stable ground state.
One shortcoming of this approach is that field operators are gauge-dependent and thereby arguably not qualified as directly representing physical quantities. AQFT takes so-called nets of algebras as basic for the mathematical description of quantum systems, i. The insight behind this apporoach is that the net structure of algebras, i. In this rather abstract setting, physical states are identified as positive, linear, normalized functionals which map elements of local algebras to real numbers.
States can thus be understood as assignments of expectation values to observables. Via the so-called Gelfand-Neumark-Segal construction, one can recover the concrete Hilbert space representations in the conventional formalism.
AQFT then imposes a whole list of axioms on the abstract algebraic structure, namely relativistic axioms in particular locality and covariance , general physical assumptions e. As a reformulation of QFT, AQFT is expected to reproduce the main features of QFT, like the existence of antiparticles, internal quantum numbers, the relation of spin and statistics, etc. 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.
One main link are superselection rules , which put restrictions on the set of all observables and allow for classification schemes in terms of permanent or essential properties. The empirical success of renormalization in CQFT leaves the physical reasons for this success in the dark, argues Fraser, unlike in condensed matter physics, where its success is due to the fact that matter is discrete at atomic length scales.
The third important problem for standard QFT which prompted reformulations is the existence of inequivalent representations. 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. Representations of some given algebra or group are sets of mathematical objects, like numbers, rotations or more abstract transformations e.
This means that the specification of the purely algebraic CCRs suffices to describe a particular physical system. Now one is confronted with a multitude of unitarily inequivalent representations UIRs 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 unitarily inequivalent representations seem to describe different physical states of affairs it would no longer be legitimate to simply choose the most convenient representation, just like choosing the most convenient frame of reference.
In principle all but one of the UIRs could be physically irrelevant, i. However, it seems that at least some irreducible representations of the CCRs are inequivalent and physically relevant. These considerations motivate the algebraic point of view that algebras of observables rather than observables themselves in a particular representation should be taken as the basic entities in the mathematical description of QFT, so that the above-mentioned problems are to some degree avoided from the outset.
However, obviously this cannot just be the end of the story. Even if UIRs are not basic, it is still necessary to say what the availability of different UIRs means, physically and thereby ontologically. One of the most fundamental interpretative obstacles concerning QFT is the question which formalism to consider and to then identify which parts of the respective formalism carry the physical content, and which parts are surplus structure, from an ontological point of view.
While Hilbert space conservativism seems to be the default position, often adopted without further justification, algebraic imperialism usually comes with an explicit justification. Hilbert space conservatism dismisses the availability of a plethora of UIRs as a mathematical artifact with no physical relevance. In contrast, algebraic imperialism argues that instead of choosing a particular Hilbert space representation, one should stay on the abstract algebraic level.
The selection of a particular faithful representation is a matter of convenience without physical implications. It may provide a more or less handy analytical apparatus. The coexistence of UIRs can be readily understood by looking at ferromagnetism for infinite spin chains see Ruetsche At high temperatures the atomic dipoles in ferromagnetic substances fluctuate randomly.
Below a certain temperature the atomic dipoles tend to align to each other in some direction. Since the basic laws governing this phenomenon are rotationally symmetrical, no direction is preferred. Since there is a different ground state for each direction of magnetization, one needs different Hilbert space representations—each containing a unique ground state—in order to describe symmetry breaking systems.
Correspondingly, one has to employ inequivalent representations. To conclude, it is difficult to say how the availability of UIRs should be interpreted in general. Clifton and Halvorson b propose seeing this as a form of complementarity. Accordingly, she advocates taking UIRs more seriously than in these extremist approaches. The Unruh effect constitutes a severe challenge to a particle interpretation of QFT, because it seems that the very existence of the basic entities of an ontology should not depend on the state of motion of the detectors.
Teller — tries to dispel this problem by pointing out that while the Minkowski vacuum has the definite value zero for the Minkowski number operator, the particle number is indefinite for the Rindler number operator, since one has a superposition of Rindler quanta states. This means that there are only propensities for detecting different numbers of Rindler quanta but no actual quanta.
Clifton and Halvorson b argue, contra Teller, that it is inapproriate to give priority to either the Minkowski or the Rindler perspective. Both are needed for a complete picture. The Minkowski as well as the Rindler representation are true descriptions of the world, namely in terms of objective propensities. Arageorgis, Earman and Ruetsche argue that Minkowski and Rindler or Fulling quantization do not constitute a satisfactory case of physically relevant UIRs.
First, there are good reasons to doubt that the Rindler vacuum is a physically realizable state. Second, the authors argue, the unitary inequivalence in question merely stems from the fact that one representation is reducible and the other one irreducible: The restriction of the Minkowski vacuum to a Rindler wedge, i.
Therefore, the Unruh effect does not cause distress for the particle interpretation—which the authors see to be fighting a losing battle anyhow—because Rindler quanta are not real and the unitary inequivalence of the representations in question has nothing specific to do with conflicting particle ascriptions. The occurrence of UIRs is also at the core of an analysis by Fraser She restricts her analysis to inertial observers but compares the particle notion for free and interacting systems.
Fraser argues, first, that the representations for free and interacting systems are unavoidably unitarily inequivalent, and second, that the representation for an interacting system does not have the minimal properties that are needed for any particle interpretation—e. Bain has a diverging assessment of the fact that only asymptotically free states, i. For Bain, the occurrence of UIRs without a particle or quanta interpretation for intervening times, i.
Bain concludes that although the inclusion of interactions does in fact lead to the break-down of the alleged duality of particles and fields it does not undermine the notion of particles or fields as such. Baker points out that the main arguments against the particle interpretation—concerning non-localizability e. Malament and failure for interacting systems Fraser —may also be directed against the wave functional version of the field interpretation see field interpretation iii above.
First, a Minkowski and a Rindler observer may also detect different field configurations. Second, if the Fock space representation is not apt to describe interacting systems, then the unitarily equivalent wave functional representation is in no better situation: Interacting fields are unitarily inequivalent to free fields, too.
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. 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. 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.
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.
The branch of quantum physics that is concerned with the theory of fields; it was motivated by the question of how an atom radiates light as its electrons jump from excited states. Fluctuations of vacuum fields are irregular , but their averaged effects can be calculated using quantum field theory QFT.
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