Indeterminacy and the Limits of Classical Concepts: The Transformation of Heisenberg’s Thought Kristian Camilleri University of Melbourne This paper examines the transformation which occurs in Heisenberg’s under- standing of indeterminacy in quantum mechanics between 1926 and 1928. After his initial but unsuccessful attempt to construct new quantum concepts of space and time, in 1927 Heisenberg presented an operational deªnition of concepts such as ‘position’ and ‘velocity’. Yet, after discussions with Bohr, he came to the realisation that classical concepts such as position and momentum are indispensable in quantum mechanics in spite of their limited applicabil- ity. This transformation in Heisenberg’s thought, which centres on his theory of meaning, marks the critical turning point in his interpretation of quan- tum mechanics. 1. Introduction The publication of Heisenberg’s paper ‘The Physical Content of Quantum Kinematics and Mechanics’ in March 1927 is widely regarded as having made a major contribution to the development of the orthodox interpreta- tion of quantum mechanics (Heisenberg, [1927] 1983). In the paper Heisenberg argued that the accuracy with which we can know both the position and momentum of a particle, such as an electron, is subject to an in principle limitation. The more precisely we can determine the particle’s position, the less precisely we know its momentum, and vice versa. The uncertainty or indeterminacy principle, as it is commonly known, has been the subject of detailed historical and philosophical investigation (Beller 1985; Beller 1999:65–101; Jammer 1974:56–84). However, it has gone largely unnoticed that at the time when Heisenberg was formulating his thoughts on indeterminacy, his own philosophical view was in a state of transformation, particularly with regard to the use of concepts like ‘po- sition’ and ‘momentum’ in quantum mechanics. In this paper I show that Perspectives on Science 2007, vol. 15, no. 2 ©2007 by The Massachusetts Institute of Technology 178 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc.2007.15.2.178 by guest on 28 September 2021 Perspectives on Science 179 in the critical period between 1926 and 1928 Heisenberg’s thought passes through three phases, each of which constitutes a different approach to the development of a new conceptual framework for quantum mechanics, where it was recognized that the concepts ‘position’ and ‘momentum’ have only limited applicability. After the development of matrix mechanics in 1925, Heisenberg be- came convinced that quantum mechanics demanded a fundamentally new conception of space and time. This was made explicit in his correspon- dence with Pauli in the second half of 1926, during which time Heisen- berg expressed the view that space and time may be statistical concepts, or alternatively that space-time might have a discontinuous structure. Nev- ertheless, by February 1927 he had abandoned this line of thought and in- stead turned his attention to a redeªnition of the concepts of position and velocity through an operational analysis.1 This approach, which is the one we ªnd in his uncertainty principle paper, represents a conscious attempt on Heisenberg’s part to employ the same method of analysis to quantum mechanics as Einstein had in redeªning the concept of simultaneity in the special theory of relativity. However, Heisenberg’s commitment to opera- tional viewpoint was short lived. After fractious discussions with Bohr in 1927, Heisenberg gradually came to realize that this method of deªning concepts was untenable, and he subsequently abandoned the operational theory of meaning which had characterized his 1927 paper. This realiza- tion signiªes an important turning point in Heisenberg’s philosophy, in which he abandoned a veriªcation theory of meaning, and ªnally accepted Bohr’s doctrine concerning the indispensability of classical concepts. In bringing to light the shift that occurs in Heisenberg’s thinking dur- ing this period, this paper serves to draw attention to two important points which have often been ignored or misunderstood in the secondary literature. First, Heisenberg’s introduction of the imaginary gamma-ray microscope was not intended primarily to demonstrate the limits of preci- sion in measurement. Though it certainly did this, its real purpose was to deªne the concept of position through an operational analysis. This be- comes evident once we situate Heisenberg’s use of imaginary gamma-ray microscope within the context of his concerns over the meaning of con- cepts in quantum theory. Secondly, this paper offers a different perspective on the Bohr- Heisenberg dialogue in Copenhagen in 1927 to the one usually presented. 1. It should be noted that Heisenberg did return to the idea of discontinuous space- time in 1930, but these later speculations must be seen in the context of his efforts to build a new quantum ªeld theory, and did not form part of his understanding of quantum me- chanics, with which we are concerned here (Carazza & Kragh, 1995). Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc.2007.15.2.178 by guest on 28 September 2021 180 Indeterminacy and the Limits of Classical Concepts Much of the literature takes the view that the discussions between Bohr and Heisenberg over the gamma-ray microscope centred on the sig- niªcance of wave-particle duality. This is primarily because the wave- particle dilemma of light and matter appears to have been Bohr’s central concern at this time, in contrast to Heisenberg emphasis on discontinuity (Stepansky 1997; Jammer 1974:66; Tanona 2004). Recent scholarship has shown, however, that Bohr and Heisenberg never reached agreement on this issue (Beller, 1999:226–7; Camilleri, 2005:69–94; Camilleri, 2006). Yet the discussions with Bohr did leave a lasting impression on Heisen- berg for another reason—they convinced him that classical concepts such as ‘position’ and ‘momentum’ were indispensable in quantum mechanics despite the limitations on their applicability. This marks a crucial turning point in Heisenberg’s thought. In order to see more clearly the overall shift in Heisenberg’s thinking, it is necessary to turn ªrst to his attempts to ªnd an interpretation of quantum concepts through new concepts of space and time. 2. New Concepts of Space and Time in Quantum Theory In the period between 1925 and 1927, the concept of a particle in quan- tum mechanics underwent a radical transformation. In the three-man pa- per on matrix mechanics, Born, Heisenberg and Jordan agued that “the motion of electrons cannot be described in terms of the familiar concepts of space and time” (Born, Heisenberg & Jordan [1926] 1967: 322). Schrö- dinger rejected this view, preferring to think of the electron as a matter wave spread throughout space (Schrödinger 1928). Though physicists dis- agreed on whether the electron was in reality a particle or a wave, a num- ber of them shared the view that quantum mechanics would require an entirely new conceptual framework, in which concepts such as ‘position’ and ‘velocity’ would no longer occupy a central place.2 Writing a decade later, the philosopher Ernst Cassirer gave clear expression to this line of thought: We must squarely face the problems thus created. There seems to be no return to the lost paradise of classical concepts; physics has to undertake the construction of a new methodological path...Ifit appears that certain concepts, such as those of position, velocity, or of the mass of an individual electron can no longer be ªlled with deªnite empirical content, we have to exclude them from the theo- retical system of physics, important and fruitful though their func- tion may have been (Cassirer [1936] 1956:194–5). 2. His view continued to be expressed by physicists such as Max von Laue and Erwin Schrödinger well into the 1930s (Beller 1988:159–161). Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc.2007.15.2.178 by guest on 28 September 2021 Perspectives on Science 181 The view here expressed by Cassirer stands in sharp contrast to the inter- pretation of quantum mechanics developed by Bohr, Heisenberg and Pauli in the late 1920s, according to which classical concepts such as position and momentum remain indispensable—even in quantum mechanics. Yet, if we look closely at Heisenberg’s efforts to develop new conceptual foun- dations for quantum mechanics prior to 1927, we ªnd that his point of view shares much in common with the one expressed in the passage quoted above. Indeed, the attempt to ªnd new quantum concepts was in- tegral to Heisenberg’s project of reconstructing the foundations of quan- tum mechanics between 1925 and 1927. As his correspondence with Pauli during this period shows, Heisenberg’s early efforts to resolve the difªculties associated with quantum theory took the form of an attempt to construct a new conception of space and time in quantum theory. Throughout 1926, Heisenberg had repeatedly emphasized the restric- tions imposed by quantum theory for a space-time description. In his pa- per on the many-body problem published in June that year he argued that while electrons behave as thought they were particles in the sense that they possess mass and electric charge, it is not possible to give “a descrip- tion of the motion of the corpuscles in our familiar space-time concepts” (Heisenberg 1926:412). Here Heisenberg argued that the new quantum mechanics forces to abandon all hope of describing the motion of an elec- tron in space and time “unless one were to consider a space, whose struc- ture [Maßbestimmung] differs substantially from the Euclidean one, as rep- resenting ‘ordinary’ space” (Heisenberg 1926:412–3). In a letter to Pauli written on 28 October, Heisenberg set out his thoughts on the problem: [In quantum theory] it makes no sense, for example to speak about the position of a particle with deªnite velocity.