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A Matter of Principle: The Principles of Quantum Theory, Dirac’s Equation, and Quantum Information

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I am working out a quantum theory about it for it is really most tantalizing state of affairs.—James Joyce, Finnegans Wake.

Abstract

This article is concerned with the role of fundamental principles in theoretical physics, especially quantum theory. The fundamental principles of relativity will be addressed as well, in view of their role in quantum electrodynamics and quantum field theory, specifically Dirac’s work, which, in particular Dirac’s derivation of his relativistic equation of the electron from the principles of relativity and quantum theory, is the main focus of this article. I shall also consider Heisenberg’s earlier work leading him to the discovery of quantum mechanics, which inspired Dirac’s work. I argue that Heisenberg’s and Dirac’s work was guided by their adherence to and their confidence in the fundamental principles of quantum theory. The final section of the article discusses the recent work by D’Ariano and coworkers on the principles of quantum information theory, which extend quantum theory and its principles in a new direction. This extension enabled them to offer a new derivation of Dirac’s equations from these principles alone, without using the principles of relativity.

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Notes

  1. Throughout this article, unless specified otherwise, by quantum theory I refer to the standard versions of quantum mechanics, quantum electrodynamics, and quantum field theory, especially their mathematical structures, rather than alternative theories of quantum phenomena, such as Bohmian theories, for example. By quantum phenomena, I refer to those observed physical phenomena in considering which Planck’s constant, h, must be taken into account; and by quantum objects, I refer to those entities in nature that, through their interactions with measuring instruments, are responsible for the appearance of quantum phenomena.

  2. Among exceptions are A. Zeilinger’s article [5], J. Bub’s article on quantum mechanics as a principle theory on Einstein’s definition [6], an earlier approach to Heisenberg’s discovery of quantum mechanics by the present author [7, pp. 9–16], and most recently, Ungar and Smolin’s book, which builds on Smolin’s earlier work [8]. The principles grounding Smolin’s argument are, however, not the principles of quantum theory considered in this article. Indeed, most of his key principles, beginning with Leibniz’s principle of sufficient reason, which grounds Smolin’s argument, are in conflict with the key principles of quantum theory advocated here. (Smolin’s argumentation is in conflict with some of the principles of relativity, both special and general, as well.) There is some overlap. The gauge-invariance principle, extensively used by Smolin, is consistent with the principles advocated here and is especially important in quantum field theory. Also, Smolin’s view of mathematics and its role in physics is in accord with the present argument and the overall (non-Platonist) philosophical stance adopted in this article.

  3. I distinguish “the spirit of Copenhagen” from “the Copenhagen interpretation,” a rubric that I shall avoid, because there is no single such interpretation. Indeed, some interpretations designated “Copenhagen interpretations” only partially conform to the spirit of Copenhagen as understood here, and some do not conform to it at all.

  4. For a further discussion of the concept of reality without realism, see [17].

  5. I distinguish causality, which is an ontological category, describing reality, from determinism, which is an epistemological category, describing part of our knowledge of reality, specifically our ability to predict the state of a system, at least as defined by an idealized model, exactly at any moment of time once we know its state at a given moment of time. Determinism is sometimes used in the same sense as causality, and in the case of classical mechanics (which deals with single objects or a sufficiently small number of objects), causality and determinism, as defined here, coincide. Once a system is large enough, one needs a superhuman power to predict its behavior exactly, as was famously noted by P. S. Laplace. However, while it follows automatically that noncausal behavior, considered at the level of a given model, cannot be handled deterministically, the reverse is not true. The underlined qualification is necessary because we can have causal models of processes in nature that may not be causal.

  6. As will be seen, D’Ariano et al adopt a principle of causality of that type [10, pp. 3, 11]. See also [22] and [23] for further discussions of causality in quantum theory.

  7. In doing so, quantum theory also suggested that it may true more generally, for example, as noted earlier, in general relativity [19], but the subject would require a separate treatment.

  8. Although Bohr does not appear to have made or possibly subscribed to the stronger claim, assumed in this article, of the impossibility of even any conception of quantum objects and their behavior, this claim may be seen as made in the spirit of Copenhagen. Not all interpretations in the spirit of Copenhagen adopt this more radical view and some stop short of that of Bohr.

  9. Bell’s and the Kochen-Specker theorems, dealing with the EPR-type phenomena (for discrete variables) may be seen as lending support to this principle, in part in view of the question of locality, insofar as the latter could be maintained if one adopt the RWR-principle. The meaning and implications of these theorems and related findings, and of the concepts involved, such as locality, are under debate. I shall return to the question of locality in Sect. 5.

  10. In the late 1930s, following his exchanges with Einstein concerning the EPR-type experiments, Bohr rethought this principle in terms of his concept of phenomenon [24, v. 2, p. 64]. For the discussion of Bohr’s views by the present author, see [28] and [29].

  11. I have considered Heisenberg’s discovery of quantum mechanics, in terms of principles, in detail in [7, pp. 77–137]. See also [6] for a related but a somewhat different view of Bohr’s key principles.

  12. The principle of complementarity, as formulated here, reflects more Bohr’s later works, from 1929 on (the concept itself was introduced in 1927), impacted by his debate with Einstein. In these works the principle is exemplified by the complementary nature of the position and the momentum measurements, always mutually exclusive and as such correlative to the uncertainty relations. See [28, 29], for the development of Bohr’s views.

  13. For details, see [7, pp. 77–137].

  14. Einstein 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, and his use on 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.” While relevant, including in the context of the quantum-informational (principle) derivation of Dirac’s equation in [11, Sect. 5], the subject is beyond my scope here. It is extensively discussed in [16]. It is worth noting that, unlike Einstein, Klein (for example, in his version of the Kaluza-Klein theory) always took quantum principles, especially discreteness, as primary, rather than aiming, as Einstein, to derive quantum discreteness from an underlying continuity of a classical-like field theory. That 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, albeit not of the kind Einstein wanted.

  15. The statement is cited in [41], which considers the question of identity and indistinguishability of elementary particles (i.e., their indistinguishability from each other within the same particle type, e.g., electrons vs. photons) from a realist perspective. See also [42], for a comprehensive realist treatment of the subject.

  16. Although some of these principles are mathematical, they reflect and often express profound physical principles, as does, for example, the gauge symmetry principle, found already in Maxwell’s electrodynamics, but especially important in general relativity and quantum field theory, as well as in most proposals for quantum gravity. Thus, quantum electrodynamics is an abelian gauge theory with the symmetry group U(1) (this group is commutative), and it has one gauge field, with the photon being the gauge boson. The Standard Model is a non-abelian gauge theory with the symmetry group U(1)\(\times \)SU(2)\(\times \)SU(3) and broken symmetries, and it has a total of twelve gauge bosons: the photon, three weak bosons, and eight gluons.

  17. The title was reprised by S. Weinberg’s 1996 article, reflecting on a more advance stage of quantum field theory, without, however, answering the question either [44].

  18. “Theory” here refers primarily to the mathematical structure of quantum theory, rather than to its mechanical or dynamical aspects, such as, in the case of the finite-dimensional quantum theory, found the quantum mechanics of discrete variables (spin). See Note 1 above.

  19. Among the key predecessors here are C. Fuchs’s work, which, however, more recently “mutated” to a somewhat different program, that of quantum Bayesianism or QBism (e.g., [49]), and Hardy [50], equally motivated by the aim of deriving quantum mechanics from a more natural set of principles, postulates, or axioms. Hardy’s paper was, arguably, the first rigorous derivation of that type. Neither of these two approaches is constructive or realist, nor, again, is that of D’Ariano et al. See [9] for further references. The different terms just mentioned (all of them are use by D’Ariano et al as well) do not affect the essential aspects of the programs in question at the moment. All these attempts refer to finite-dimensional quantum theories. Let me add that, while emphasizing the role of fundamental principles in quantum theory, the present article does not claim that a sufficient understanding of quantum theory itself, say, quantum mechanics, from such principles has been achieved. This remains an open question, even more so when dealing with continuous variables (to which my discussion has been restricted thus far), where the application of the principles of quantum information is more complex as well.

  20. References inside this and other passages cited from [9, 10] and [11] are adjusted to follow the numbering of references in the present article.

  21. Bub’s article, cited earlier, also considers quantum mechanics as a principle theory in order to account for the EPR-type experiments and quantum entanglement [6].

  22. I think, that, in accordance with the definition given at the outset, “postulate” may be a better term, because one can hardly have self-evidence of “axioms,” but this is a secondary matter, which, as I said, does not really affect the essence of the situation.

  23. As explained earlier (Note 6), this principle is different from that of classical causality (indeed already by virtue the principle’s appeal to probability), while being consistent with relativity.

  24. On the other hand, the article provides “an analytical description of the QCA for the narrow-band states of quantum field theory in terms of a dispersive Schrödinger equation holding at all scales” [11, pp. 1, 4].

  25. This concept has further implications for our understanding of the EPR-type experiments and related problematics, as well as for the question of causality, because relativity (both special and general) is a classically causal theory, while quantum theory, including as thus extended beyond the Fermi scale, is not. It is only causal relativistically or, again, locally in the sense here defined. These subjects would, however, require separate treatments, which cannot be undertaken here.

  26. For yet another type of alternative approach, see [67].

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Acknowledgments

I am grateful to G. Mauro D’Ariano for sharing, in many invaluable discussions, his thinking and his knowledge of quantum theory. I would also like to thank Lucien Hardy, Gregg Jaeger, Andrei Khrennikov, and Paolo Perinotti for productive exchanges that helped my work on this article. I would like to add that the authors mentioned here, as well as the present author, have each published a series of papers on quantum foundations in the Proceedings of Växjö conferences on quantum foundations during the last decade. I gratefully acknowledge the role of these conferences in my work.

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    Arkady Plotnitsky

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Plotnitsky, A. A Matter of Principle: The Principles of Quantum Theory, Dirac’s Equation, and Quantum Information. Found Phys 45, 1222–1268 (2015). https://doi.org/10.1007/s10701-015-9928-z

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