What is field theory




















However, when we are looking at the problem with particle hat on, we usually don't say it's a field. For instance, when describing real objects as consisting of atoms, we are usually talking about statistical mechanics, or condensed matter physics. Only when we move to the realm of continuum mechanics, we say that there are fields. There is much more to be said on the topic but this post got already too long so I'll stop here.

If you have any questions, ask away! A field theory is a physical description of reality in which the fundamental entities are fields, i.

Examples of fields are the temperature in a room, for each location in the room, a temperature can be specified, although in most cases temperature will be pretty uniform, unless for instance if you just turned on a heater, then there will be a temperature gradient. The gravitational field in Newtonian mechanics is a description of what the force of attraction on a test particle is as generated by a large mass. This field is vector-valued. Another example of a vector-valued field is the velocity field in a fluid.

It gives the velocity of each infinitesimal piece of fluid at some instant t. The electromagnetic field is specified by giving the value of an antisymmetric rank-2 tensor at each space-time location.

The novelty of quantum mechanics with respect to classical mechanics is that it has to incorporate the discreteness of the action. That's what we call quantizing. In particular, this means that energy will be quantized under certain circumstances namely due to some boundary conditions or potentials limiting the amount of possible states.

In classical mechanics, systems typically have a finite amount of degrees of freedom. For instance, the 1D harmonic oscillator has two degrees of freedom, the position and the momentum of the oscillator.

In quantum mechanics, the energy of the oscillator which is a combination of position and momentum becomes quantized. Quantum field theory takes this one step further, instead of quantizing systems with a finite amount of degrees of freedom, it tackles systems with an infinite amount, in other words fields.

The way non-interacting fields are quantized is reminiscent of the way the harmonic oscillator is quantized except you now have an infinite amount of oscillators. This brings with it a lot of technical complications. A non field theory is a theory where effectively there are fixed number of point particles or rigid bodies. A field theory is a theory where there are so many particles or body that they form a density or distribution.

Let me illustrate with a simple example. Take a given electric field in a one dimensional space. Imagine two point charges. To determine the motion of these two charges, we calculate the force from the electric field and the force between the two charges.

The energy for example is the sum of the energies of the particles plus the sum of the interaction energies between the particles. Imagine the same thing, but now with a billion charges.

Instead of treating the charges as points, we treat them as one entity, the charge density. So for non field theories, the object of interest are discrete whereas in field theories the objects of interest are densities or as we call them fields. One particular point of interest in this difference. For a discrete field, the particles are fixed. For the particle number to change dynamically you have to introduce some mechanism for it to change.

Looking at the line charge example, with one electric field, the charge density in most regions is zero i. Change the electric field and you might get overall charge mostly zero but two regions where the charge is positive and one where the charge is negative. So the total charge is the same but there is a larger total positive charge and a larger total negative charge. This is effectively particle creation, which you get for free in the field theory. Sign up to join this community.

The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. While individuals may share a goal, the fields that they must pass through to achieve this goal are all different. Field changes can affect the person both in the present and in how that individual develops in future. That rendition of history is shaped to some degree by the current field conditions.

Skip to content. 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. 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 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. 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. First, one Fourier analyses the vector potential of the classical field into normal modes using periodic boundary conditions corresponding to an infinite but denumerable number of degrees of freedom.

Second, since each mode is described independently by a harmonic oscillator equation, one can apply the harmonic oscillator treatment from non-relativistic quantum mechanics to each single mode. The result for the Hamiltonian of the radiation field is. These commutation relations imply that one is dealing with a bosonic field.

In order to see this, one has to examine the eigenvalues of the operators. Due to the commutation relations 5. The interpretation of these results is parallel to the one of the harmonic oscillator. That is, equation 5. This is a rash judgement, however.

For instance, the question of localizability is not even touched while it is certain that this is a pivotal criterion for something to be a particle. All that is established so far is that certain mathematical quantities in the formalism are discrete.

However, countability is merely one feature of particles and not at all conclusive evidence for a particle interpretation of QFT yet.

It is not clear at this stage whether we are in fact dealing with particles or with fundamentally different objects which only have this one feature of discreteness in common with particles. The degree of excitation of a certain mode of the underlying field determines the number of objects, i.

However, despite of this deviation, says Teller, quanta should be regarded as particles: Besides their countability another fact that supports seeing quanta as particles is that they have the same energies as classical particles.

Teller has been criticized for drawing unduly far-reaching ontological conclusions from one particular representation, in particular since the Fock space representation cannot be appropriate in general because it is only valid for free particles see, e. In order to avoid this problem Bain proposes an alternative quanta interpretation that rests on the notion of asymptotically free states in scattering theory. For a further discussion of the quanta interpretation see the subsection on inequivalent representations below.

It is a remarkable result in ordinary non-relativistic QM that the ground state energy of e. In addition to this, the relativistic vacuum of QFT has the even more striking feature that the expectation values for various quantities do not vanish, which prompts the question what it is that has these values or gives rise to them if the vacuum is taken to be the state with no particles present.

If particles were the basic objects of QFT how can it be that there are physical phenomena even if nothing is there according to this very ontology? Before exploring whether other potentially necessary requirements for the applicability of the particle concept are fulfilled let us see what the alternatives are.

Proceeding this way makes it easier to evaluate the force of the following arguments in a more balanced manner. Since various arguments seem to speak against a particle interpretation, the allegedly only alternative, namely a field interpretation, is often taken to be the appropriate ontology of QFT.

So let us see what a physical field is and why QFT may be interpreted in this sense. Thus a field is a system with an infinite number of degrees of freedom, which may be restrained by some field equations.

Whereas the intuitive notion of a field is that it is something transient and fundamentally different from matter, it can be shown that it is possible to ascribe energy and momentum to a pure field even in the absence of matter. This somewhat surprising fact shows how gradual the distinction between fields and matter can be. Thus there is an obvious formal analogy between classical and quantum fields: in both cases field values are attached to space-time points, where these values are specified by real numbers in the case of classical fields and operators in the case of quantum fields.

Due to this formal analogy it appears to be beyond any doubt that QFT is a field theory. But is a systematic association of certain mathematical terms with all points in space-time really enough to establish a field theory in a proper physical sense?

Is it not essential for a physical field theory that some kind of real physical properties are allocated to space-time points? This requirement seems not fulfilled in QFT, however. Teller ch. Only a specific configuration , i.

There are at least four proposals for a field interpretation of QFT, all of which respect the fact that the operator-valuedness of quantum fields impedes their direct reading as physical fields.

The main problem with proposal i , and possibly with ii , too, is that an expectation value is the average value of a whole sequence of measurements, so that it does not qualify as the physical property of any actual single field system, no matter whether this property is a pre-existing or categorical value or a propensity or disposition.

But this is also a problem for the VEV interpretation: While it shows nicely that much more information is encoded in the quantum field operators than just unspecifically what could be measured, it still does not yield anything like an actual field configuration. While this last requirement is likely to be too strong in a quantum theoretical context anyway, the next proposal may come at least somewhat closer to it.

Correspondingly, it is the most widely discussed extant proposal; see, e. In effect, it is not very different from proposal i , and with further assumptions for i even identical. However, proposal ii phrases things differently and in a very appealing way. The basic idea is that quantized fields should be interpreted completely analogously to quantized one-particle states, just as both result analogously from imposing canonical commutation relations on the non-operator-valued classical quantities.

Thus just as a quantum state in ordinary single-particle QM can be interpreted as a superposition of classical localized particle states, the state of a quantum field system, so says the wave functional approach, can be interpreted as a superposition of classical field configurations.

In practice, however, QFT is hardly ever represented in wave functional space because usually there is little interest in measuring field configurations.

The multitude of problems for particle as well as field interpretations prompted a number of alternative ontological approaches to QFT. Auyang and Dieks propose different versions of event ontologies. In recent years, however, ontic structural realism OSR has become the most fashionable ontological framework for modern physics.

While so far the vast majority of studies concentrates on ordinary QM and General Relativity Theory, it seems to be commonly believed among advocates of OSR that their case is even stronger regarding QFT, in light of the paramount significance of symmetry groups also see below —hence the name group structural realism Roberts Explicit arguments are few and far between, however.

Cao b points out that the best ontological access to QFT is gained by concentrating on structural properties rather than on any particular category of entities. The central significance of gauge theories in modern physics may support structural realism. Lyre claims that only ExtOSR is in a position to account for gauge theories. Moreover, it can make sense of zero-value properties, such as the zero mass of photons.

Category theory could be a promising framework for OSR in general and QFT in particular, because the main reservation against the radical but also seemingly incoherent idea of relations without relata may depend on the common set theoretic framework. See SEP entries on structural realism 4.

Superselection sectors are inequivalent irreducible representations of the algebra of all quasi-local observables. Since we are dealing with quantum physical systems many properties are dispositions or propensities ; hence the name dispositional trope ontology. A trope bundle is not individuated via spatio-temporal co-localization but because of the particularity of its constitutive tropes.

Morganti also advocates a trope-ontological reading of QFT, which refers directly to the classification scheme of the Standard Model. In other words the state space of an elementary system shall have no internal structure with respect to relativistic transformations. Put more technically, the state space of an elementary system must not contain any relativistically invariant subspaces, i. If the state space of an elementary system had relativistically invariant subspaces then it would be appropriate to associate these subspaces with elementary systems.

The requirement that a state space has to be relativistically invariant means that starting from any of its states it must be possible to get to all the other states by superposition of those states which result from relativistic transformations of the state one started with. Doing that involves finding relativistically invariant quantities that serve to classify the irreducible representations. Regarding the question whether Wigner has supplied a definition of particles, one must say that although Wigner has in fact found a highly valuable and fruitful classification of particles, his analysis does not contribute very much to the question what a particle is and whether a given theory can be interpreted in terms of particles.

What Wigner has given is rather a conditional answer. For instance, the pivotal question of the localizability of particle states, to be discussed below, is still open. Kuhlmann a: sec. It thus appears to be impossible that our world is composed of particles when we assume that localizability is a necessary ingredient of the particle concept. So far there is no single unquestioned argument against the possibility of a particle interpretation of QFT but the problems are piling up.

The Reeh-Schlieder theorem is thus exploiting long distance correlations of the vacuum. Or one can express the result by saying that local measurements do not allow for a distinction between an N-particle state and the vacuum state.

Malament formulates a no-go theorem to the effect that a relativistic quantum theory of a fixed number of particles predicts a zero probability for finding a particle in any spatial set, provided four conditions are satisfied, namely concerning translation covariance, energy, localizability and locality. The localizability condition is the essential ingredient of the particle concept: A particle—in contrast to a field—cannot be found in two disjoint spatial sets at the same time.

It requires that the statistics for measurements in one space-time region must not depend on whether or not a measurement has been performed in a space-like related second space-time region. A relativistic quantum theory of a fixed number of particles, satisfying in particular the localizability and the locality condition, has to assume a world devoid of particles or at least a world in which particles can never be detected in order not to contradict itself.

One is forced towards QFT which, as Malament is convinced, can only be understood as a field theory. This is even the case arbitrarily close after a sharp position measurement due to the instantaneous spreading of wave packets over all space. Note, however, that ordinary QM is non-relativistic.

A conflict with SRT would thus not be very surprising although it is not yet clear whether the above-mentioned quantum mechanical phenomena can actually be exploited to allow for superluminal signalling.

The local behavior of phenomena is one of the leading principles upon which the theory was built. This makes non-localizability within the formalism of QFT a much severer problem for a particle interpretation. According to Saunders it is the localizability condition which might not be a natural and necessary requirement on second thought. One can only require for the same kind of event not to occur at both places.

The question is rather whether QFT speaks about things at all. One thing seems to be clear. Does the field interpretation also suffer from problems concerning non-localizability? This procedure leads to operator-valued distributions instead of operator-valued fields.

The lack of field operators at points appears to be analogous to the lack of position operators in QFT, which troubles the particle interpretation. However, for position operators there is no remedy analogous to that for field operators: while even unsharply localized particle positions do not exist in QFT see Halvorson and Clifton , theorem 2 , the existence of smeared field operators demonstrates that there are at least point-like field operators.

Symmetries play a central role in QFT. In order to characterize a special symmetry one has to specify transformations T and features that remain unchanged during these transformations: invariants I. The basic idea is that the transformations change elements of the mathematical description the Lagrangians for instance whereas the empirical content of the theory is unchanged.

There are space-time transformations and so-called internal transformations. Whereas space-time symmetries are universal, i. The invariance of a system defines a conservation law, e. Inner transformations, such as gauge transformations, are connected with more abstract properties. Symmetries are not only defined for Lagrangians but they can also be found in empirical data and phenomenological descriptions. If a conservation law is found one has some knowledge about the system even if details of the dynamics are unknown.

The analysis of many high energy collision experiments led to the assumption of special conservation laws for abstract properties like baryon number or strangeness. Evaluating experiments in this way allowed for a classification of particles. This phenomenological classification was good enough to predict new particles which could be found in the experiments. Free places in the classification could be filled even if the dynamics of the theory for example the Lagrangian of strong interaction was yet unknown.

As the history of QFT for strong interaction shows, symmetries found in the phenomenological description often lead to valuable constraints for the construction of the dynamical equations.

Arguments from group theory played a decisive role in the unification of fundamental interactions. In addition, symmetries bring about substantial technical advantages. For example, by using gauge transformations one can bring the Lagrangian into a form which makes it easy to prove the renormalizability of the theory.



0コメント

  • 1000 / 1000