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Lecture notes in physics

Symmetry and Conservation Laws

Tomáš Brauner

Published Jan 1, 2024• 0 citations
Publisher

Abstract

This chapter introduces the general concept of symmetry in both classical and quantum physics. The part devoted to classical physics stresses the importance of conservation laws and the role of Noether’s theorem for their understanding. A fairly general discussion of Noether’s theorem is offered, including ambiguities in the definition of the Noether current, possible existence of generalized, tensor conservation laws, and the inverse of Noether’s theorem. The Hamiltonian formalism is then used to motivate the transition from classical to quantum physics, and the definition of symmetry in the latter. In anticipation of the discussion of spontaneous symmetry breaking in the following chapter, the text here emphasizes the difference between implementing symmetry in terms of transformations on the algebra of physical observables, and in terms of unitary mappings on the Hilbert space of states.

DOI
10.1007/978-3-031-48378-3_4

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This chapter introduces the general concept of symmetry in both classical and quantum physics. The part devoted to classical physics stresses the importance of conservation laws and the role of Noether’s theorem for their understanding. A fairly general discussion of Noether’s theorem is offered, including ambiguities in the definition of the Noether current, possible existence of generalized, tensor conservation laws, and the inverse of Noether’s theorem. The Hamiltonian formalism is then used to motivate the transition from classical to quantum physics, and the definition of symmetry in the latter. In anticipation of the discussion of spontaneous symmetry breaking in the following chapter, the text here emphasizes the difference between implementing symmetry in terms of transformations on the algebra of physical observables, and in terms of unitary mappings on the Hilbert space of states.
The previously developed 'continuous temporal development and Noether's theorem' approach to the association of symmetries and conservations of the laws of nature, part of the foundation of any (classical or quantum) Lagrangian theory, is generalised significantly. Within this framework a strong result is obtained for the inverse Noether's theorem, associating symmetries with given conservations. A general symmetry-conservation formalism is proposed, into which both the previously developed 'linear temporal development and conserved eigenheit' approach, part of the foundation of any quantum theory and also applicable to certain classical ones, and the 'continuous temporal...' approach, as different as they are, fit nicely.
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.
Symmetries are defined in histories-based generalized quantum mechanics paying special attention to the class of history theories admitting quasitemporal structure (a generalization of the concept of `temporal sequences' of `events' using partial semigroups) and logic structure for `single time histories'. Symmetries are classified into orthochronous (those preserving the `temporal order' of `events') and non-orthochronous. A straightforward criterion for physical equivalence of histories is formulated in terms of orthochronous symmetries; this criterion covers various notions of physical equivalence considered by Gell-Mann and Hartle as special cases. In familiar situations, a reciprocal relationship between traditional symmetries (Wigner symmetries in quantum mechanics and Borel-measurable transformations of phase space in classical mechanics) and symmetries defined in this work is established. In a restricted class of theories, a definition of conservation law is given in the history language which agrees with the standard ones in familiar situations; in a smaller subclass of theories, a Noether type theorem (implying a connection between continuous symmetries of dynamics and conservation laws) is proved.
This chapter examines the assumptions behind Noether’s theorem connecting symmetries and conservation laws, taking an algebraic approach to compare classical and quantum versions of this theorem. In both classical and quantum mechanics, observables are naturally elements of a Jordan algebra, while generators of one-parameter groups of transformations are naturally elements of a Lie algebra. Noether’s theorem holds whenever we can map observables to generators in such a way that each observable generates a one-parameter group that preserves itself. In ordinary complex quantum mechanics, this mapping is multiplication by the square root of ?1. In the more general framework of unital JB-algebras, Alfsen and Shultz call such a mapping a ‘dynamical correspondence’ and show its presence allows us to identify the unital JB-algebra with the self-adjoint part of a complex C*-algebra. However, to prove their result, they impose a second, more obscure, condition on the dynamical correspondence. This expresses a relation between quantum and statistical mechanics, closely connected to the principle that ‘inverse temperature is imaginary time’.
Symmetries are defined in histories-based theories paying special attention to the class of history theories admitting quasitemporal structure (a generalization of the concept of `temporal sequences' of `events' using partial semigroups) and logic structure for `single-time histories'. Symmetries are classified into orthochronous (those preserving the `temporal order' of `events') and nonorthochronous. A straightforward criterion for physical equivalence of histories is formulated in terms of orthochronous symmetries; this criterion covers various notions of physical equivalence of histories considered by Gell-Mann and Hartle as special cases. In familiar situations, a reciprocal relationship between traditional symmetries (Wigner symmetries in quantum mechanics and Borel-measurable transformations of phase space in classical mechanics) and symmetries defined in this work is established. In a restricted class of theories, a definition of conservation law is given in the history language which agrees with the standard ones in familiar situations; in a smaller subclass of theories, a Noether type theorem (implying a connection between continuous symmetries of dynamics and conservation laws) is proved. The formalism evolved is applied to histories (of particles, fields or more general objects) in general curved spacetimes. Sharpening the definition of symmetry so as to include a continuity requirement, it is shown that a symmetry in our formalism implies a conformal isometry of the spacetime metric.
Chapter 8 is devoted to the study of the dynamics of quantum systems, based on the formalism presented in Chapter 7 and previous chapters. The operator notation and its matrix representation are used without distinction, moving freely from one to the other as convenient. This allows us to study the evolution of the system from different perspectives, depending on the description adopted. The chapter includes a discussion of the statistical information contained in the quantum formalism, a formal derivation of the Heisenberg inequalities and a clarification of their physical meaning. A section is dedicated to quantum canonical and related transformations. More advanced topics, which provide a valuable complement, include Noether’s theorem on the relationship between the symmetry properties and the conserved dynamical variables of the system.