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American Journal of Physics

Lagrangian Formulation and Mechanical Interpretation

Mario Bunge

Published Apr 1, 1957• 15 citations
Publisher

Abstract

The question is asked whether, as is often claimed, the Lagrangian formulation of a physical theory provides its mechanical interpretation. For purposes of this enquiry, an analysis of physical theories is performed, with special emphasis on classical mechanics, electromagnetism, and quantum mechanics. Three aspects are distinguished and defined: the formulation (statement), the representation (transformation properties), and the interpretation (meaning). Confusion of formalism with meaning is criticized, and is suggested as a source of some misunderstandings of recent reinterpretations of quantum mechanics. Properties and advantages of Lagrangian formulations are listed, and it is recalled that their chief usefulness lies in that they are the most helpful precisely when mechanisms are either ignored or absent. It is concluded that Lagrangian formulations contribute to the demechanization of physics and that the program of reduction to mechanics has not succeeded in physics.

DOI
10.1119/1.1934403

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The question is asked whether, as is often claimed, the Lagrangian formulation of a physical theory provides its mechanical interpretation. For purposes of this enquiry, an analysis of physical theories is performed, with special emphasis on classical mechanics, electromagnetism, and quantum mechanics. Three aspects are distinguished and defined: the formulation (statement), the representation (transformation properties), and the interpretation (meaning). Confusion of formalism with meaning is criticized, and is suggested as a source of some misunderstandings of recent reinterpretations of quantum mechanics. Properties and advantages of Lagrangian formulations are listed, and it is recalled that their chief usefulness lies in that they are the most helpful precisely when mechanisms are either ignored or absent. It is concluded that Lagrangian formulations contribute to the demechanization of physics and that the program of reduction to mechanics has not succeeded in physics.
All of the mathematical formulations of quantum mechanics describe the interaction between radiation and atoms, but we show here that only Einstein has been able to establish a solid logical connection with classical theory. His methods are used to interpret the other mathematical models by requiring that observables be described in a physically-defined coordinate system and that there be strict adherence to the conservation of momentum. The properties of momentum exchange are included in a quantum oscillator, or clock, by replacing the non-relativistic Hamiltonian model with a relativistically correct Lagrangian model. Equal contributions are assigned to photon, electron, and nucleus; and field boundaries are defined that coincide with the electron shells. The fields of the particles superpose linearly, but otherwise their physical integrity is maintained throughout. The Lagrangian model accounts for the 720 degree rotation of a wave function as an initial 360 degree electromagnetic wave cycle (photon absorption) followed by a second wave cycle (photon emission), yielding two wave cycles to correspond with one complete electron cycle. The abstract space of quantum mechanics is replaced by a real classical space.
Some aspects of the interpretation of quantum theory are discussed.It is emphasized that quantum theory is formulated in a Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutator relations between 'canonically conjugated' coordinate and momentum operators leads to a wrong version of quantum mechanics.In this connection the Feynman integral formalism is also discussed.In this formalism the measure is not well-defined and there is no idea how to distinguish between the true version of quantum mechanics and an incorrect one; it is rather a mnemonic rule to generate perturbation series from an undefined zero order term.The origin of time is analyzed in detail by the example of atomic collisions.It is shown that the time-dependent Schrödinger equation for the closed three-body (two nuclei + electron) system has no physical meaning since in the high impact energy limit it transforms into an equation with two independent time-like variables; the time appears in the stationary Schrödinger equation as a result of extraction of a classical subsystem (two nuclei) from a closed three-body system.Following the Einstein-Rosen-Podolsky experiment and Bell's inequality the wave function is interpreted as an actual field of information in the elementary form.The relation between physics and mathematics is also discussed.
This chapter, based on the background information of the previous chapter, will review the formalism of quantum mechanics – its fundamental mathematical structure (the basic contents of Hilbert space representation and operator algebra representation) and physical assumptions (kinematics, dynamics, and measurement theory). These are the premise and foundation for discussing the quantum histories interpretation. Then, in combination with the viewpoints of some literature in recent years, we will explore what is the interpretation of quantum mechanics, analyze the relationship between the theory of quantum mechanics and its interpretation, and declare the author’s basic stance on these two issues. These two are the primary issues that all interpretive work in the philosophy of physics will face (not limited to the interpretation of quantum mechanics), and the necessary clarification here is also conducive to the subsequent discussion. Three common terms – theory, formalism, and interpretation – are explicated and differentiated by offering the necessary definitions of their meanings, based on the works of Acuña and Ruetsche. The clarification will help to make the subsequent related discussions more precise.
A Lagrangian description is presented here that can be used in conjunction with a particle interpretation of quantum mechanics. The advantages of this description are that the field equations and the particle equations of motion can both be deduced from a single Lagrangian density expression and that conservation of energy and momentum are assured. This Lagrangian formulation is then applied to the special case of the well-known Bohm model as an example. It is thereby demonstrated that such a formulation is compatible with the predictions of quantum mechanics.
The foundations and meaning of quantum theory became a central issue to Albert Einstein and Niels Bohr since the onset of their impassioned debate in the 1920s, enriched by the contributions of many other distinguished scientists and philosophers.The questions are not settled down at all, despite the great achievements of the theory, its impressive accordance with experiment and predictive power.The fundamental and technological applications range from cosmology to biology, with the development of invaluable instruments and the design of new materials.Is quantum mechanics a complete or an incomplete theory?Is there an objective reality independent of the observer or is the reality created by the measurements?Are hidden-variable theories justifiable?Is there a quantum theory founded in a local-causal and non-linear approach that formally contains the orthodox linear theory as a special case?Can such a formulation unify classical and quantum physics?Are Heisenberg's uncertainty relations valid in all cases?Here, the subject is addressed as an adaptation of our contribution to the Colloquium "Quantal aspects in Chemistry and Physics.
It has often been remarked that Bohr's writings on the interpretation of quantum mechanics make scant reference to the mathematical formalism of quantum theory; and it has not infrequently been suggested that this is another symptom of the general vagueness, obscurity and perhaps even incoherence of Bohr's ideas. Recent years have seen a reappreciation of Bohr, however. In this article we broadly follow this "rehabilitation program". We offer what we think is a simple and coherent reading of Bohr's statements about the interpretation of quantum mechanics, basing ourselves on primary sources and making use of and filling lacunas in|recent secondary literature. We argue that Bohr's views on quantum mechanics are more firmly connected to the structure of the quantum formalism than usually acknowledged, even though Bohr's explicit use of this formalism remains on a rather global and qualitative level. In our reading, Bohr's pronouncements on the meaning of quantum mechanics should first of all be seen as responses to concrete physical problems, rather than as expressions of a preconceived philosophical doctrine. In our final section we attempt a more detailed comparison with the formalism and conclude that Bohr's interpretation is not far removed from present-day non-collapse interpretations of quantum mechanics.