Abstract
Quantum field theory (QFT) shares many of its philosophical problems with quantum mechanics. This applies in particular to the quantum measurement process and the connected interpretive problems, to which QFT contributes hardly any new aspects, let alone solutions. The question as to how the objects described by the theory are spatially embedded was already also discussed for quantum mechanics. However, the new mathematical structure of QFT promises new answers, which renders the spatiotemporal interpretation of QFT the pivotal question. In this chapter, we sketch the mathematical characteristics of QFT and show that a particle as well as a field interpretation breaks down.
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Notes
- 1.
See, e.g. the 1925 Como lecture of Bohr (1961), in particular p. 54.
- 2.
A note for more advanced readers: A manifestly relativistically invariant notation is also possible. Furthermore, we should also mention that the commutation relations (6.3) are valid for bosonic fields only, such as the electromagnetic field in particular. In the framework of QFT, however, one can in addition to these interaction fields also describe material “particles”, with half-integer spin, in terms of fields. For such fermionic fields, e.g. the Dirac field for electrons, one requires anti-commutation relations instead of Eq. 6.3. In the following, we also make use of the so-called Heisenberg picture, i.e. we will be working with time-dependent operators.
- 3.
The Schrödinger equation violates the requirement of special relativity which states that the laws of nature must maintain their form when one goes via Lorentz transformations from the inertial frame of reference of one observer to that of another. The Maxwell equations, for example, fulfil this requirement: light, in particular, has the same velocity c in vacuum for all these observers.
- 4.
As usual in QFT, we will call it the “free theory” in the following.
- 5.
In particular, there are solutions with negative energies, which would lead to endless cascades towards energetically more favourable states at lower energies. In retrospect, one can argue that it cannot be expected that relativistic processes could be described by a single-particle theory, since the energy-mass equivalence \(E=mc^2\) in special relativity permits the formation of particle-antiparticle pairs (Peskin and Schroeder 1995, Chap. 2). An additional problem consists in the fact that the normalization of the states determined by the Klein–Gordon equation is no longer time-independent, which undermines their interpretation as probability densities. The Klein–Gordon equation indeed fulfils the requirements of special relativity, but not those of quantum mechanics (Srednicki 2007, Chap. 1).
- 6.
- 7.
In the Lagrange formalism, the relevant equation of motion (for fields the field equation) can be derived by subjecting the Lagrangian density to a variation procedure which is expressed by the Euler–Lagrange equation (or more precisely: which leads to that equation). Thus, for example, inserting the Lagrangian density of electrodynamics into the Euler–Lagrange equation permits one to derive the Maxwell equations.
- 8.
The frequency \(\omega \) = 2\(\pi \)\(\nu \) is connected with the momentum via \(\hbar \omega _\mathbf {p} = \sqrt{|\mathbf {{p}}|^2c^2 + m^2c^4}\). In many textbooks on QFT, the solutions of the Klein–Gordon equation are not formulated in terms of the momenta \(\mathbf {{p}}\), but rather using the wavevectors \(\mathbf {{k}}\), which are related via \(\mathbf {{p}} = \hbar \mathbf {{k}}\) (and, if we use the above-mentioned unit convention \(\hbar =1\), are in fact identical).
- 9.
The “0” on the right-hand side denotes the null vector, which should not be confused with the vacuum state \( | 0 \rangle \)!
- 10.
In fact, at this point we are already working with the so-called Fock space representation of the commutation relations, which we will introduce systematically in the next Sect. 6.3.3.
- 11.
For the relationship between commutation relations and the space of states, cf. Mandl and Shaw (2010), Sects. 1.2.2 and 3.1.
- 12.
The possibility of describing variable numbers of particles does not entail that fundamental interactions themselves are described. We will continue to work with the so-called free theory from the previous section. Creation and annihilation operators do not describe dynamic processes in that theory. In fact, Haag’s theorem even says that the description of interactions is generally excluded within the framework of this theory. This limitation to a free theory has important consequences for its interpretation (see Sect. 6.4.2). For the actual treatment of scattering processes, this is not as important as it might seem, since there we are mainly dealing with asymptotically free states, i.e. far from the scattering process, and this “far from the scattering process” is attained almost immediately following the interaction.
- 13.
This alternative means that the ontological significance of the states created is here not (yet) determined.
- 14.
Since, according to Eq. 6.9, \(a^{\dagger } ({\mathbf {p}})\) and \(a^{\dagger } ({\mathbf {p'}})\) commute, the two-particle state \(a^{\dagger } ({\mathbf {p}}) a^{\dagger } (\mathbf{p'}) | 0 \rangle \) is identical to the state \(a^{\dagger } ({\mathbf {p'}}) a^{\dagger } ({\mathbf {p}}) | 0 \rangle \) with exchanged creation operators. Many-particle states are thus symmetric under permutations of the creation operators. Furthermore, arbitrarily many “Klein–Gordon particles” can be created with the same momentum or in the same field mode \({\mathbf {p}}\). As we have seen in Chap. 3, this means that “Klein–Gordon particles” must be bosons.
- 15.
Of course there are also systems in quantum physics with distinguishable particles, namely many-particle systems with different types of particles. These are also described by the Hilbert-space formalism and therefore also permit the typical superpositions.
- 16.
We use here \( {h}_1 \otimes _{s} {h}_2 \equiv {h}_1 \otimes {h}_2 + {h}_2 \otimes {h}_1\).
- 17.
- 18.
Since most operators are differential operators in Schrödinger’s version of quantum mechanics, we called this approach the “calculus approach”.
- 19.
In the following, some still-unexplained concepts will occasionally appear in the text, and they will be placed in quotation marks or parentheses, either to ensure that the statements made are not incorrect, and/or to guarantee to readers who wish to pursue their interests further in the literature that they will be able to follow the concepts employed there. In a first reading of this chapter, these concepts can be ignored, however.
- 20.
See Sect. 3.1.4.
- 21.
One notorious example for the significance of inequivalent representations is the so-called Unruh effect, according to which what appears to be a vacuum for one observer takes the form of a thermal bath of particles for another, accelerated observer. The deeper reason for this apparent paradox is that different inequivalent representations are associated with the two observers, which means among other things that they experience different vacuum states. In fact, different inequivalent representations are even systematically related to different vacuum states. This is the basis of the so-called GNS construction, which plays an important role in the relation between AQFT and conventional QFT—because, beginning with observable algebras, different operator representations lead to different vacuum states.
- 22.
Ruetsche (2003) discusses the philosophical consequences of this in detail.
- 23.
- 24.
This requirement does not contradict the possibility of non-local EPR correlations (see Chap. 4).
- 25.
- 26.
A compact treatment of the various deficits of conventional QFT can be found in Kuhlmann (2012), Sect. 4.1.
- 27.
An overview of the various interpretations of QFT can be found in Kuhlmann 2012, Sect. 5.1.2.
- 28.
Wigner’s group-theoretical classification of the elementary particles (Wigner 1939) also gives no definition of the particle concept, in contrast to what is often assumed. What Wigner defines instead is “elementarity” (see Kuhlmann 2010, Sect. 8.1.2). This can be seen already by the fact that spatial localizability plays no role in Wigner’s definition.
- 29.
Relativistic classical particles must obey the energy condition expressed by Eq. (6.4), owing to the equivalence of mass and energy.
- 30.
Instead of “ordinally countable” sometimes one simply says “countable” and contrasts it with “aggregable” (here: “cardinally countable”). Teller (1995) in particular uses alternative terminology.
- 31.
Fraser (2008) expresses this as follows: The countability and energy conditions are fulfilled. However, since countability is often understood in the ordinal sense—-for example by Teller —but in the present connection we are concerned with the cardinal sense, we speak instead of “discreteness”.
- 32.
Baker (2013, p. 267) argues conversely that the situation could be similar to that of atoms in superposition states, which nevertheless does not cause us to doubt the existence of atoms.
- 33.
Teller (1995) argues in opposition to this view that probabilistic statements are a generic feature of quantum physics, and that both problems can be resolved with a propensity interpretation of quantum-mechanical probabilities (see Sect.2.2.2). Teller (1995) discusses the first problem on pp. 31–33 and the second one on pp. 110–112. An accessible treatment and critical discussion of Teller’s arguments is given by Huggett and Weingard (1996) .
- 34.
Note that Teller presumes that it is not reasonable to assume quantum objects to be primitively individuated. However, as we have seen from the discussion on the possibility of weak distinguishability in Sect. 3.2.3, this presumption has been frequently criticized, especially in recent years.
- 35.
- 36.
This holds in spite of the fact that the Fock space is used in perturbation-theoretical calculations within conventional quantum mechanics.
- 37.
Bain (2011) has formulated an alternative quanta interpretation for the asymptotically free theory.
- 38.
According to Baker (2013) , the permutations refer to the order in which the charges are added (to “algebraic states”).
- 39.
This was formulated somewhat more cautiously than in many other textbooks by Peskin and Schroeder (1995, p. 22): “It is quite natural to call these excitations particles, since they are discrete entities that have the proper relativistic energy-momentum relation. (By a particle, we do not mean something that must be localized in space; \(a_\mathbf{p}^{\dagger }\) creates particles in momentum eigenstates)”.
- 40.
- 41.
In the mathematically correct algebraic formulation of QFT (see Sect. 6.3.5), the quantities are not associated with points, but rather with finite regions of spacetime; furthermore, they are not associated with individual operators, but with algebras of operators. For the following arguments, this makes no essential difference, however.
- 42.
Baker (2009) has analysed some additional difficulties of a field interpretation.
- 43.
- 44.
Maurin (2013) gives an up-to-date overview.
- 45.
One concern is the boundary problem (Campbell 1990): Consider a blue sheet of paper that you tear apart. Are there suddenly two blue tropes now? Or were they “in” the original sheet already. If yes, then we have acquired a general problem: where does one colour trope end and another one begin on a macroscopic level?.
- 46.
The term “particular” is less misleading here than “individual thing”, since single tropes are precisely not things.
- 47.
Kuhlmann (2010), Chaps. 11–15.
- 48.
- 49.
One proposed solution is the “Swiss army approach” by Ruetsche (2011) . However, it is not clear whether the problem is actually solved by such a multi-perspective approach, rather than merely explicitly formulated.
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Exercises
Exercises
-
1.
Gather information about theories of light from the history of physics. Why was Newton’s theory of light not considered satisfactory? Compare the mathematical descriptions of particles and of waves. What is, in your opinion, the principal difference?
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2.
Is there anything in classical physics which is neither a particle nor a field?
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3.
State two arguments which in your opinion provide the strongest support for considering quantum field theory to be a theory of particles. What objections can be raised against these arguments?
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4.
Would it be helpful in your opinion to extend the concepts of “particles” and “fields”, i.e. by referring not only to those entities as “particles” or “fields” which exhibit all the characteristics of classical particles or classical fields?
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Kuhlmann, M., Stöckler, M. (2018). Quantum Field Theory. In: The Philosophy of Quantum Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-78356-7_6
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