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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.

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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. 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.

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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. 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 2. The interpretation of these results is parallel to the one of the harmonic oscillator.

Advanced quantum theory, Lecture 1

That is, equation 2. Note that Pauli's exclusion principle is not violated since it only applies to fermions and not to bosons like photons.

The corresponding interpretation for the annihilation operator a r k is parallel: When it operates on a state with a given number of photons this number is lowered by one. 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.

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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.

Quantum Field 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. A classical point particle can be described by its position x t and its momentum p t , which change as the time t progresses. So there are six degrees of freedom for the motion of a point particle corresponding to the three coordinates of the particle's position and three more coordinates for its momentum. In the case of a classical field one has an independent value for each single point x in space, where this specification changes as time progresses.

Thus a field is a system with an infinite number of degrees of freedom, which may be restrained by some field equations.


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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.

Relativistic Quantum Mechanics and Field Theory

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. According to this reconstruction theorem all the information that is encoded in quantum field operators can be equivalently described by an infinite hierarchy of n -point vacuum expectation values, namely the expectation values of all products of quantum field operators at n in general different space-time points, calculated for the vacuum state.

Since this collection of vacuum expectation values comprises only definite physical values it qualifies as a proper field configuration, and, Wayne argues, due to Wightman's theorem, so does the equivalent set of quantum field operators. Thus, and this is the upshot of Wayne's argument, an ascription of quantum field operators to all space-time points does by itself constitute a field configuration, namely for the vacuum state, even if this is not the actual state.

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.