Abstract
This chapter considers the role of the fundamental principles of quantum theory and principle thinking in quantum field theory, beginning with quantum electrodynamics. The fundamental principles of relativity will be addressed as well, in view of their role in quantum electrodynamics and quantum field theory. Dirac’s work, in particular his derivation of his relativistic equation of the electron by combining the principles of relativity and quantum theory, is the main focus of this chapter, in parallel with Heisenberg’s work as the main focus in the discussion of quantum mechanics in Chap. 2. Heisenberg’s work, especially his paper introducing quantum mechanics, which Dirac studied very carefully, was a major influence on Dirac’s thinking throughout, I would argue, all of his work. This influence, however, does not diminish the originality and creativity of Dirac’s thinking, which, ultimately, led him to the discovery of his equation for the relativistic electron and antimatter, one of the greatest discoveries of fundamental physics. After a general introduction given in Sect. 6.1, Sect. 6.2 addresses Dirac’s discovery of his equation. It argues, along the lines of the argument developed in Chap. 2 in considering Heisenberg’s work, that Dirac’s discovery was that of a mathematical machinery, technology, responding to and, in some key respects, specifically as concerns the role of antimatter, anticipating the architecture of high-energy quantum phenomena, as manifested in the experimental technology that defines them. Although not a quantum field theory or even quite quantum electrodynamics, Dirac’s theory of the electron, based in his equation, provided some of the key physical, mathematical, and epistemological ingredients of quantum electrodynamics or quantum field theory as a viable nonrealist theory, to be considered here in terms of a particular concept of quantum field. Dirac’s equation, which expressly considered electrons as particles, was not a field equation, but given its essentially quantum-field-theoretical nature, it would also be difficult to see it merely in terms of relativistic quantum mechanics of particles, as some suggest. Sec. 6.3 addresses the architecture of quantum field theory, as grounded, in addition to the QD and QP/QS principles, which are, just as in quantum mechanics, primary, in the combination of the RWR principle and the particle-transformation, PT, principle. The PT principle emerged as a result of Dirac’s discovery of antimatter, an unintended consequence of his equation. New concepts of both “elementary particle” and “quantum field” are, I argue, required by the combination of both principles, and I shall suggest such concepts here. These concepts allow one to pose and relate the questions “What is a quantum field?” and “What is an elementary particle?” in a new way, even if not answer them. Sec. 6.4 discusses the role of renormalization, and comments on the future of quantum theory and fundamental physics given the current state of quantum field theory.
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Notes
- 1.
The subject of multiple parallel or (both terms are used) alternative Universes, which entered fundamental physics with the many-worlds interpretation of quantum mechanics, has acquired new prominence (on different grounds) in the wake of string and brane theories, leading, unsurprisingly, to a controversy and, equally unsurprisingly (given the appeal of the idea and its controversial nature) to numerous popular expositions. See, for example, (Susskind 2006) and (Vilenkin 2007), which consider two among current proposals, and (Greene 2011), which offers a more comprehensive survey of the subject. These books also contain further references, including to technical literature. See also “Multiverse” (Wikipedia). For an alternative, “sequential,” view of Universes, each emerging from the preceding one, see (Penrose 2012). I shall, however, not be concerned with these subjects here.
- 2.
These relationships were given a new dimension by J. Maldacena’s work on the so-called AdS/CFT (Anti-de-Sitter/Conformal-Field-Theory) correspondence. See “AdS/CFT Correspondence” (Wikipedia). For a technical introduction, see (Năstase 2015).
- 3.
I shall only be concerned with standard versions of QED and QFT, and not with alternative formalisms, for example, those along the lines of Bohmian mechanics. I will also not discuss algebraic quantum field theories (AQFT), although much of my argument would apply to them. For an introduction to the current state of QFT, including AQFT, see (Kuhlman 2015) and references there, which include most standard physical and philosophical treatments of the subject, such as, to give a representative physical and a representative philosophical example (Weinberg 1996; Teller 1995). An exceptionally lucid nontechnical account of QED is given by Feynman (Feynman 1985). For a clear and elegant nontechnical account of more advanced developments, such as quantum chromodynamics (QCD), and some future prospects, see (Wilczek 2009).
- 4.
The key steps leading to this understanding, and some of them, such as Dirac’s hole theory, were quite ingenious in their own right, are well known and have been discussed in literature (e.g., Schweber 1994, 56–69). W. Furry and R. Oppenheimer’s 1934 paper (Furry and Oppenheimer 1934) was, arguably, the first to present the current view, stated here and discussed in Sect. 6.3, although, as will also be discussed in Sect. 6.3, earlier work of Dirac, Jordan, and others on second quantization was important for developing this view as well.
- 5.
Later on, Dirac lost his confidence in QED. This was not because he lost his faith in the fundamental principles of quantum theory, but rather because of the theory’s inability to give these principles a proper (which for Dirac also meant mathematically elegant) mathematical expression. However, inelegant and even messy, and, to some, mathematically questionable, if legitimate at all, as it might be, the theory initiated by Dirac proved to be a remarkable success, from antimatter to the Higgs boson. I shall return to this Janus-like nature of QED and QFT below.
- 6.
At this point (the statement dates from 1949), by “new fundamental features of atomicity” Bohr clearly refers to his concept of atomicity, perhaps influenced as much by the developments of QED, as by the development of quantum mechanics. QED and QFT played a major role in the development of Bohr’s thinking concerning quantum theory and complementarity. Bohr made a major contribution to the field by a classic treatment of quantum-field measurement in his collaborations with Rosenfeld (Bohr and Rosenfeld 1933; Bohr and Rosenfeld 1950). The latter, an updated version of their 1933 article, takes into account the intervening developments, including the renormalization of QED, accomplished just then. I shall return to this work below. For a detailed treatment of Bohr’s engagement with QED and QFT, see (Plotnitsky 2012a, pp. 89–106).
- 7.
- 8.
As discussed earlier, at the time, Heisenberg and others, including Bohr, only assumed a form of proto-RWR-principle, or even allowed for a certain residual realism, even while seeing the role of measurement as irreducible.
- 9.
The situation, again, acquires a new dimension in view of the AdS/CFT correspondence (see note 2).
- 10.
Einstein, as I noted, developed a major interest in Dirac’s equation, as a spinor equation, and he used it, in his collaborations with W. Mayer, as part of his program for the unified field theory, conceived as a classical-like field theory, modeled on general relativity, and in opposition to quantum mechanics and, by then, quantum field theory. Accordingly, he only considered a classical-like spinor form of Dirac’s equation, thus depriving it of (Einstein might have thought “freeing” it from) its quantum features, most fundamentally, discreteness (h did not figure in Einstein’s form of Dirac’s equation), and probability. Einstein hoped but failed to derive discreteness from the underlying field-continuity. As noted above, by this point Einstein abandoned the principle approach in favor of the constructive approach or rather, given that he still used fundamental principles, a mixed approach anchored in the constructive one. His use of Dirac’s equation was part of this new way of thinking. He was primarily interested in the mathematics of spinors, which he generalized in what he called “semivectors” (van Dongen 2010, pp. 96–129). It is worth noting that, unlike Einstein, O. Klein (for example, in his version of the Kaluza-Klein theory) always took quantum principles, especially discreteness, as primary, rather than aiming, as did Einstein, or earlier Schrödinger, to derive quantum discreteness from an underlying continuity of a conceptually classical-like field theory. This is hardly surprising coming from a long-time assistant of Bohr. Klein’s thinking, which led to several major contributions, was always quantum-oriented. It is just that the Klein-Gordon equation did not manage to bring quantum theory and relativity together successfully. The equation itself was later used in meson theory. Of course, Dirac’s equation, too, was a unification of quantum mechanics and special relativity, but not of the kind Einstein wanted.
- 11.
It is worth keeping in mind that von Neumann’s book, which appeared in 1932 (after Dirac’s 1930 Principles), was not yet published at the time. Related articles by von Neumann were published, but they would have been unlikely to be familiar to Dirac.
- 12.
As also discussed earlier, Dirac might have assumed that the mathematical model defined by his equation provides such a representation, but even if so, his article does not claim it does.
- 13.
Although one might see the sets of effects in question here as a kind of bundle of properties, the present, nonrealist, view is essentially different from Kuhlman’s realist Dispositional Trope Ontology (DTO) mentioned above, because the latter assumes such bundles or tropes as quantum level properties (Kuhlman 2010, 2015).
- 14.
See, again, (Kuhlman 2015), which clearly confirms this point.
- 15.
The wave function of QM formalism in low-energy regimes, say, for an electron in an atom, can be recast, quite elegantly, in terms of annihilation and creation operators as well, by the procedure known as “second quantization,” which was one the first steps toward QED. One sometimes speaks, appealingly but loosely, of the first quantization as making particles into waves, and the second quantization as making waves particles again, in a new sense. While the procedure was developed to deal with quantum many-body systems, and reflects the indistinguishability of elementary particles of the same type, it is applicable even to a single electron in an atom, normally described by a wave function, thus replaced by annihilation and creation operators. In high-energy regimes, governed by QED or QFT, it is meaningless to ever speak of a single electron even in the hydrogen atom. In Pais’s words: “the hydrogen atom can no longer be considered to consist of just one proton and one electron. Rather it contains infinitely many particles” (Pais 1986, p. 325). Besides, we still have quarks plus gluons inside this proton, in the same transformational existence, although we cannot (because of the “confinement” of quarks) register their effects apart from those of this proton.
- 16.
The latter were the main subject of Bohr and Rosenfeld’s paper, written in response to L. Landau and R. Peierls’s argument, contested by Bohr and Rosenfeld, concerning a possible inapplicability of the uncertainty relations in quantum field theory (Bohr and Rosenfeld 1933).
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Plotnitsky, A. (2016). The Principles of Quantum Theory, Dirac’s Equation, and the Architecture of Quantum Field Theory. In: The Principles of Quantum Theory, From Planck's Quanta to the Higgs Boson. Springer, Cham. https://doi.org/10.1007/978-3-319-32068-7_6
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