This chapter provides a fairly systematic analysis of when quantities that are variant under a dynamical symmetry transformation should be regarded as unobservable, or redundant, or unreal; of when models related by a dynamical symmetry transformation represent the same state of affairs; and of when mathematical structure that is variant under a dynamical symmetry transformation should be regarded as surplus. In most of these cases, the answer is ‘it depends’: that is, it depends on the details of the symmetry in question. A central feature of the analysis is that in order to draw any of these conclusions for a dynamical symmetry, it needs to be understood in terms of its possible extensions to other physical systems, in particular to measurement devices.
This chapter provides a fairly systematic analysis of when quantities that are variant under a dynamical symmetry transformation should be regarded as unobservable, or redundant, or unreal; of when models related by a dynamical symmetry transformation represent the same state of affairs; and of when mathematical structure that is variant under a dynamical symmetry transformation should be regarded as surplus. In most of these cases, the answer is ‘it depends’: that is, it depends on the details of the symmetry in question. A central feature of the analysis is that in order to draw any of these conclusions for a dynamical symmetry, it needs to be understood in terms of its possible extensions to other physical systems, in particular to measurement devices.
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 ginG, 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
The roots of the modern understanding of symmetries in physics can be traced to the work of Sophus Lie on transformations of differential equations in the nineteenth century. In this chapter, the history of the subject is used to emphasize the distinction between the symmetry of the dynamics of a physical system and the symmetry of its specific state. The central concept of spontaneous symmetry breaking is then introduced as the discrepancy between the two notions of symmetry. It is shown how the presence of spontaneous symmetry breaking can be detected through the sensitivity of measurements on the physical state to external perturbations. This naturally leads to the concepts of the order parameter and the vacuum manifold. The rest of the chapter is devoted to highlighting some subtle features of spontaneous symmetry breaking. These include the properties of the ground state and the tower of low-lying excited states in a finite volume, and the absence of well-defined unitary operators representing broken symmetry in the thermodynamic limit.
In this final part I analyse the modal interpretations from a more philosophical point of view. In Chapter 12 I start by arguing that modal interpretations describe noumenal states of affairs and that therefore metaphysically tenable interpretations need only to meet the criteria of Consistency and Internal Completeness. Then I analyse the relations between properties, states and outcomes of measurements in the modal description of reality. I end by discussing how modal interpretations, when restricted to the description of measurement outcomes, recover the standard formulation of quantum mechanics. Chapter 13 concerns the relations between the properties ascribed to composite systems and subsystems. I show that the property ascriptions of the bi and spectral modal interpretations can be characterised as holistic and non-reductionistic, whereas the property ascription of the atomic modal interpretation is non-holistic and to a large extent reductionistic. I argue that notwithstanding the lack of reductionism, the bi and spectral modal interpretations are empirically adequate and metaphysically tenable. I also discuss the possibility of saving the metaphysical tenability of the atomic modal interpretation by taking holistic properties as dispositional properties. In Chapter 14 I give a survey of the possibilities and impossibilities of modal interpretations and in Chapter 15 I end with general conclusions.
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.
A symmetry is an equivalence of different physical situations, and is therefore strictly associated with the concept of invariance . One of the main aims of physics is the search in nature for symmetries and, therefore, for observables that are invariant under certain classes of transformation. In this context, it becomes particularly interesting to look for the conditions under which a certain symmetry is eventually broken. We have already discussed some symmetries of quantum mechanics: time translations (Ch. 3), space translations (Ch. 2) and rotations (Ch. 6). Another example of invariance is represented by the indistinguishability of identical particles, which we discussed in the previous chapter. Symmetries are induced by physical transformations that generate some invariant properties (see Sec. 8.1). In Sec. 8.2, the continuous transformations induced by rotations and space translations are analyzed, whereas in Sec. 8.3, space reflection and time reversal ( discrete symmetries) are discussed. As we shall see in the next chapter, time reversibility is lost in the measurement process. The rigorous mathematical framework of a theory of transformations and of their invariance is called group theory (the subject of Sec. 8.4). Quantum transformations and symmetries Active and passive transformations We have seen that time evolution may be represented either in the Schrödinger picture (state vectors evolve and observables are kept constant) or in the Heisenberg picture (states vectors remain constant and observables evolve) (see Subsec. 3.6.1 and Sec. 5.3).
Gauge theories are conventionally interpreted by identifying gauge-related configurations as one and the same physical state. We argue that this identification is untenable. The key distinction is between dynamics and physical identity: a dynamics determines how configurations evolve, whereas physical identity, an ontological relation between configurations, is independent of that dynamics. We apply this principle to the local QED transformation (Aμ,ψ)→(Aμ−∂μΛ,eieΛ/ℏψ). Although this transformation is a symmetry of the massless electromagnetic dynamics, a dynamics on the same configuration variables containing a gauge-field mass term distinguishes the two configurations. If they were one and the same physical state, their identity could not depend on this change of dynamics; hence the gauge-related configurations cannot be identified as one physical state. The Higgs mechanism does not change the conclusion: enlarging the configuration space restores gauge covariance without establishing gauge redundancy. The result extends to non-Abelian gauge theories and, with diffeomorphisms replacing gauge transformations, to general relativity. Gauge covariance is thus a symmetry of the equations, not a principle of physical identity. Gauge-related configurations are physically distinct, and the electromagnetic potential Aμ should consequently be regarded as a real physical field rather than a mathematical redundancy.