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The mathematical style of modern physics

Narasimhaiengar Mukunda

Published Feb 1, 1987• 0 citations
OpenAlex

Abstract

In the author's opinion, two main features are characteristic for modern applications of mathematics to physics: the use of symmetries and the use of unobservable quantities (antioperationalistic approach). He gives a brief history of symmetry in physics: first, symmetry as a tool for describing useful features of given theories, then symmetry as a tool for restricting possible equations (e.g. the Dirac equation was thus obtained) and, finally, symmetry as a creative tool allowing one to obtain equations almost uniquely (Yang-Mills, etc.). Whenever equations are invariant with respect to some group, in the case that our observation data are invariant with respect to some subgroup $G$, one cannot tell the state $x$ from the state $gx$, where $g\\in G$, so the parameters telling $x$ from $gx$ become unobservable---of this kind are potentials in electrodynamics, coordinate effects in general relativity, etc. This unobservability can be ultimate (e.g. phase of wave function in quantum theory) or only approximate---in all other theories where symmetry is only approximate. \n \n \n

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In the author's opinion, two main features are characteristic for modern applications of mathematics to physics: the use of symmetries and the use of unobservable quantities (antioperationalistic approach). He gives a brief history of symmetry in physics: first, symmetry as a tool for describing useful features of given theories, then symmetry as a tool for restricting possible equations (e.g. the Dirac equation was thus obtained) and, finally, symmetry as a creative tool allowing one to obtain equations almost uniquely (Yang-Mills, etc.). Whenever equations are invariant with respect to some group, in the case that our observation data are invariant with respect to some subgroup GGG, one cannot tell the state xxx from the state gxgxgx, where ginGg\\in GginG, so the parameters telling xxx from gxgxgx become unobservable---of this kind are potentials in electrodynamics, coordinate effects in general relativity, etc. This unobservability can be ultimate (e.g. phase of wave function in quantum theory) or only approximate---in all other theories where symmetry is only approximate. \n \n \n
General remarks As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics – the Schrödinger equation – lies beyond the potentialities of modern mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties. Symmetry concepts come about in physics in two ways. First, since any physical process occurs in a real space, we have to make use of one or another coordinate system. The isotropy and homogeneity of space make physically meaningful only those mathematical relationships that remain unchanged under rotations of the axes of the coordinate system, and that impose fairly rigorous constraints on the possible physical laws. Second, every physical object and process features a symmetry which should be taken into account by the physical theory. A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum. The group of rotations of a three-dimensional space stands apart in atomic spectroscopy. This is mostly due to the high accuracy of the central field approximation, on which the entire modern theory of complex atoms and ions is based.
Abstract Group theory is a branch of mathematics appropriate for studying symmetries of physical systems. A symmetry is a natural attribute of the physical world and therefore is the starting point of any physical law. The foundation of dynamics is in Lagrangian theory or equivalently in Hamiltonian theory. A symmetry of a physical system is expressed by the invariance of its Lagrangian or Hamiltonian, or ultimately of its equations of motion, under certain transformations. When these transformations form groups it is convenient and useful to study symmetries through group theory. Group theory can explain observed regularities and helps to simplify and unify the physics of apparently different systems. It is a valuable tool that helps in predicting the behaviour of systems for which a symmetry has been found.
This brief and rudimentary chapter on QED has been included to close the circle in the story telling. The aim has been to emphasize the significance of the notion of the planewave in the theory development. With no more than the idea of the planewave and considering the Maxwell equations in free space inside in a cubic box, the Hamiltonian operator for describing the electromagnetic field was obtained. Thereby, the reader was guided to utilize what they already know, while fully respecting the order of operators. A thorough discussion of the QED following the same style of writing which has been pursued in this book would require an independent treatise. Accomplishing such a project would certainly be a noble undertaking. A volume of this kind would also require the inclusion of the theory of special relativity and of a fully-fletched treatment of spin phenomena. The available books on QED, despite being voluminous and comprehensive, are not easily accessible to engineers and technologists. The discussion in this chapter might convince the reader that with reasonable effort the necessary mathematical tools can be developed and satisfactorily understood. The ultimate vision is that the engineers and technologists, once they have acquired a working knowledge of the methods, they will start enriching the quantum field theory by their own unique creative style of thinking. The first five references in the following list provide further detailed information on the structures of the scaffolding matrices in the mathematical physics. The remaining references may assist the reader in digging deeper in QED. The reader can expect to read comprehensive, and at the same time, accessible treatments of the QED, the spin phenomena, and the special theory of relativity in later MQPET Volumes.
Introduction The application of group theory to study physical problems and their solutions provides a formal method for exploiting the simplifications made possible by the presence of symmetry. Often the symmetry that is readily apparent is the symmetry of the system/object of interest, such as the three-fold axial symmetry of an NH 3 molecule. The symmetry exploited in actual analysis is the symmetry of the Hamiltonian. When alluding to symmetry we usually include geometrical, time-reversal symmetry, and symmetry associated with the exchange of identical particles. Conservation laws of physics are rooted in the symmetries of the underlying space and time. The most common physical laws we are familiar with are actually manifestations of some universal symmetries. For example, the homogeneity and isotropy of space lead to the conservation of linear and angular momentum, respectively, while the homogeneity of time leads to the conservation of energy. Such laws have come to be known as universal conservation laws. As we will delineate in a later chapter, the relation between these classical symmetries and corresponding conserved quantities is beautifully cast in a theorem due to Emmy Noether. At the day-to-day working level of the physicist dealing with quantum mechanics, the application of symmetry restrictions leads to familiar results, such as selection rules and characteristic transformations of eigenfunctions when acted upon by symmetry operations that leave the Hamiltonian of the system invariant.
Abstract Symmetry principles play an important part in theoretical physics, both as a means of understanding conservation laws and as computational devices providing quick and elegant solutions to problems that would otherwise be tedious to solve. In quantum theory, their role is especially significant since they not only clarify basic concepts and simplify the formal development of the subject, but they also provide detailed information about physical systems for which no explicit dynamical models exist. However, the methods used to investigate symmetry in quantum systems are often based on abstract group-theoretical techniques, which bypass the canonical formalism. They include no general rules for going over to the classical limit and for directly relating the symmetries of the quantum system to those of a corresponding classical system.
Hamiltonians whose symbols are not simply real valued, but matrix or, more generally, endomorphism valued functions appear in many places in physics, examples being the Dirac equation, multicomponent wave equations like electrodynamics in media, and Yang-Mills theories, and the Born-Oppenheimer approximation in molecular physics. The aim of this paper is to give a completely geometric approach to the WKB approximation od such systems, and to reduce the problem ``as far as possible'' to the scalar case. A star-product formulation of quantum mechanics proves to be particularly useful in this context. As opposed to other approaches in the literature which restrict themselves to the use of the Moyal product and thus to the study of trivial bundles (or local trivializations) over ℜ2n\real^{2n}ℜ2n, we will consider general bundles over arbitrary symplectic manifolds. Here, Fedosov's construction \cite{fedosov} will be the adequate tool, since it gives an explicit construction for star products in this general setting.