Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Philosophy of Science

Spontaneous Symmetry Breaking and Chance in a Classical World

Chuang Liu

Published Jul 1, 2003• 8 citations
Publisher

Abstract

This essay explores the nature of spontaneous symmetry breaking (SSB) in connection with a cluster of interrelated concepts such as Curie's symmetry principle, ergodicity, and chance and stability in classical systems. First, a clarification of the two existing senses of SSB is provided and an argument developed for a proposal for SSB, in which not only the possibilities but also the actual breakings are referred to. Second, a detailed analysis is given of classical SSB that answers the questions: (i) how we are justified in regarding it as a matter of chance, and (ii) why the breakings in it are equally probable. The answer provides some support to the applicability of ergodicity in special systems (such as ours).

DOI
10.1086/376786

Related papers

Philosophical Aspects of Spontaneous Symmetry Breaking

Giacomo Schwarz

This essay expounds the algebraic framework describing general physical the-ories, within which the phenomenon of spontaneous symmetry breaking (SSB) makes its appearance in infinite quantum systems. This is in contrast with the fact that a large class of theories- both classical and quantum, finite and infinite- are termed, in the conventional account of classical and quantum mechanics, as exhibiting SSB. This discrepancy will be understood in the light of an interpretation that finds the symmetry breaking to be in some respects stronger in the algebraic account than is generally the case in the conventional picture. The case of SSB in the standard account of quantum field theory (QFT) will then be discussed, and it will be argued that, although one would expect a connection with the algebraic account to be possible, this turns out to be problematic. Finally the role of the idealisation of infinite systems, crucial to algebraic SSB, will be dis-cussed.

•May 4, 2012•0 citations

Symmetry Breaking through Superselection by Boundary Conditions

Silvester G. A. Borsboom, José P. Dupont

Spontaneous symmetry breaking (SSB) is central to modern physics but is conventionally defined only for infinite systems, raising challenges for its interpretation in finite, real-world setups. This paper argues that the key to resolving this issue lies in the underappreciated role of boundary conditions in quantum systems. Inspired by both the relational approach to symmetries and the physical mechanism behind symmetry breaking, we formulate a relational interpretation of SSB: a finite system exhibits SSB relative to a reference environment which can induce perturbations across the boundary. This eliminates the need for the thermodynamic limit, offering a more physical picture of SSB that emphasizes the observable consequences of the interactions that real-life systems inevitably have with their environment. We show how, in this relational interpretation, SSB for both lattice systems and (gauge) field theories should be understood as subtle, rather than spontaneous, symmetry breaking, still in contrast to explicit symmetry breaking. We also explain how algebraic definitions of SSB for infinite systems relate to the intuitive picture of SSB in finite systems and illustrate how asymptotic boundary conditions push the environment "to infinity". In this way, our relational interpretation of SSB provides a unified conceptual framework applicable to symmetry-breaking in systems of any size.

physics.hist-phCross-listedarXiv:2606.15272•Jun 16, 2026•0 citations

Spontaneous symmetry breaking in quantum systems: Emergence or reduction?

Nicolaas P. Landsman

Beginning with Anderson (1972) , spontaneous symmetry breaking ( ssb ) in infinite quantum systems is often put forward as an example of (asymptotic) emergence in physics , since in theory no finite system should display it. Even the correspondence between theory and reality is at stake here, since numerous real materials show ssb in their ground states (or equilibrium states at low temperature), although they are finite. Thus against what is sometimes called ‘Earman's Principle’, a genuine physical effect (viz. ssb ) seems theoretically recovered only in some idealisation (namely the thermodynamic limit), disappearing as soon as the idealisation is removed. We review the well-known arguments that (at first sight) no finite system can exhibit ssb , using the formalism of algebraic quantum theory in order to control the thermodynamic limit and unify the description of finite- and infinite-volume systems. Using the striking mathematical analogy between the thermodynamic limit and the classical limit, we show that a similar situation obtains in quantum mechanics (which typically forbids ssb ) versus classical mechanics (which allows it). This discrepancy between formalism and reality is quite similar to the measurement problem (now regarded as an instance of asymptotic emergence), and hence we address it in the same way, adapting an argument of the Landsman and Reuvers (2013) that was originally intended to explain the collapse of the wave-function within conventional quantum mechanics. Namely, exponential sensitivity to (asymmetric) perturbations of the (symmetric) dynamics as the system size increases causes symmetry breaking already in finite but very large quantum systems. This provides continuity between finite- and infinite-volume descriptions of quantum systems featuring ssb and hence restores Earman's Principle (at least in this particularly threatening case).

arXiv:1305.4473•Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics•Sep 13, 2013•54 citations

Spontaneous Symmetry Breaking: Quantum Statistical Mechanics versus Quantum Field Theory

Doreen Fraser

Philosophical analysis of spontaneous symmetry breaking (SSB) in particle physics has been hindered by the unavailability of rigorous formulations of models in quantum field theory (QFT). A strategy for addressing this problem is to use the rigorous models that have been constructed for SSB in quantum statistical mechanics (QSM) systems as a basis for drawing analogous conclusions about SSB in QFT. On the basis of an analysis of this strategy as an instance of the application of the same mathematical formalism to different domains and as an instance of drawing analogies between domains, I conclude that certain structural explanations can be exported from QSM to QFT but that causal explanations cannot.

Philosophy of Science•Nov 19, 2012•2 citations

Spontaneous symmetry breaking under Bose--Einstein condensation

Vyacheslav I. Yukalov

Due to high current interest to Bose-Einstein condensation, the related topics, such as spontaneous gauge symmetry breaking, ergodic decomposition, and particle fluctuations, are intensively discussed in literature. These discussions, unfortunately, involve quite a number of controversies and confusions. The goal of the present brief survey is to clarify some of these confusions, concentrating on the principal points, such as the relationship between the conditions of Bose-Einstein condensation, the Bogolubov method of quasiaverages, spontaneous gauge symmetry breaking, ergodic decomposition, the presumed existence of nonthermodynamic particle fluctuations leading to the so-called ``grand canonical catastrophe", and the requirements for system stability.

arXiv:2606.12606•Jun 10, 2026•0 citations

Spontaneous Symmetry Breaking Induced by Quantum Monitoring

Luis Pedro García-Pintos, Diego Tielas, Adolfo del Campo

Spontaneous symmetry breaking (SSB) is responsible for structure formation in scenarios ranging from condensed matter to cosmology. SSB is broadly understood in terms of perturbations to the Hamiltonian governing the dynamics or to the state of the system. We study SSB due to quantum monitoring of a system via continuous quantum measurements. The acquisition of information during the measurement process induces a measurement backaction that seeds SSB. In this setting, by monitoring different observables, an observer can tailor the topology of the vacuum manifold, the pattern of symmetry breaking, and the nature of the resulting domains and topological defects.

arXiv:1808.08343•Physical Review Letters•Aug 29, 2019•18 citations
This essay explores the nature of spontaneous symmetry breaking (SSB) in connection with a cluster of interrelated concepts such as Curie's symmetry principle, ergodicity, and chance and stability in classical systems. First, a clarification of the two existing senses of SSB is provided and an argument developed for a proposal for SSB, in which not only the possibilities but also the actual breakings are referred to. Second, a detailed analysis is given of classical SSB that answers the questions: (i) how we are justified in regarding it as a matter of chance, and (ii) why the breakings in it are equally probable. The answer provides some support to the applicability of ergodicity in special systems (such as ours).
This essay expounds the algebraic framework describing general physical the-ories, within which the phenomenon of spontaneous symmetry breaking (SSB) makes its appearance in infinite quantum systems. This is in contrast with the fact that a large class of theories- both classical and quantum, finite and infinite- are termed, in the conventional account of classical and quantum mechanics, as exhibiting SSB. This discrepancy will be understood in the light of an interpretation that finds the symmetry breaking to be in some respects stronger in the algebraic account than is generally the case in the conventional picture. The case of SSB in the standard account of quantum field theory (QFT) will then be discussed, and it will be argued that, although one would expect a connection with the algebraic account to be possible, this turns out to be problematic. Finally the role of the idealisation of infinite systems, crucial to algebraic SSB, will be dis-cussed.
Spontaneous symmetry breaking (SSB) is central to modern physics but is conventionally defined only for infinite systems, raising challenges for its interpretation in finite, real-world setups. This paper argues that the key to resolving this issue lies in the underappreciated role of boundary conditions in quantum systems. Inspired by both the relational approach to symmetries and the physical mechanism behind symmetry breaking, we formulate a relational interpretation of SSB: a finite system exhibits SSB relative to a reference environment which can induce perturbations across the boundary. This eliminates the need for the thermodynamic limit, offering a more physical picture of SSB that emphasizes the observable consequences of the interactions that real-life systems inevitably have with their environment. We show how, in this relational interpretation, SSB for both lattice systems and (gauge) field theories should be understood as subtle, rather than spontaneous, symmetry breaking, still in contrast to explicit symmetry breaking. We also explain how algebraic definitions of SSB for infinite systems relate to the intuitive picture of SSB in finite systems and illustrate how asymptotic boundary conditions push the environment "to infinity". In this way, our relational interpretation of SSB provides a unified conceptual framework applicable to symmetry-breaking in systems of any size.
Beginning with Anderson (1972) , spontaneous symmetry breaking ( ssb ) in infinite quantum systems is often put forward as an example of (asymptotic) emergence in physics , since in theory no finite system should display it. Even the correspondence between theory and reality is at stake here, since numerous real materials show ssb in their ground states (or equilibrium states at low temperature), although they are finite. Thus against what is sometimes called ‘Earman's Principle’, a genuine physical effect (viz. ssb ) seems theoretically recovered only in some idealisation (namely the thermodynamic limit), disappearing as soon as the idealisation is removed. We review the well-known arguments that (at first sight) no finite system can exhibit ssb , using the formalism of algebraic quantum theory in order to control the thermodynamic limit and unify the description of finite- and infinite-volume systems. Using the striking mathematical analogy between the thermodynamic limit and the classical limit, we show that a similar situation obtains in quantum mechanics (which typically forbids ssb ) versus classical mechanics (which allows it). This discrepancy between formalism and reality is quite similar to the measurement problem (now regarded as an instance of asymptotic emergence), and hence we address it in the same way, adapting an argument of the Landsman and Reuvers (2013) that was originally intended to explain the collapse of the wave-function within conventional quantum mechanics. Namely, exponential sensitivity to (asymmetric) perturbations of the (symmetric) dynamics as the system size increases causes symmetry breaking already in finite but very large quantum systems. This provides continuity between finite- and infinite-volume descriptions of quantum systems featuring ssb and hence restores Earman's Principle (at least in this particularly threatening case).
Philosophical analysis of spontaneous symmetry breaking (SSB) in particle physics has been hindered by the unavailability of rigorous formulations of models in quantum field theory (QFT). A strategy for addressing this problem is to use the rigorous models that have been constructed for SSB in quantum statistical mechanics (QSM) systems as a basis for drawing analogous conclusions about SSB in QFT. On the basis of an analysis of this strategy as an instance of the application of the same mathematical formalism to different domains and as an instance of drawing analogies between domains, I conclude that certain structural explanations can be exported from QSM to QFT but that causal explanations cannot.
Due to high current interest to Bose-Einstein condensation, the related topics, such as spontaneous gauge symmetry breaking, ergodic decomposition, and particle fluctuations, are intensively discussed in literature. These discussions, unfortunately, involve quite a number of controversies and confusions. The goal of the present brief survey is to clarify some of these confusions, concentrating on the principal points, such as the relationship between the conditions of Bose-Einstein condensation, the Bogolubov method of quasiaverages, spontaneous gauge symmetry breaking, ergodic decomposition, the presumed existence of nonthermodynamic particle fluctuations leading to the so-called ``grand canonical catastrophe", and the requirements for system stability.
Spontaneous symmetry breaking (SSB) is responsible for structure formation in scenarios ranging from condensed matter to cosmology. SSB is broadly understood in terms of perturbations to the Hamiltonian governing the dynamics or to the state of the system. We study SSB due to quantum monitoring of a system via continuous quantum measurements. The acquisition of information during the measurement process induces a measurement backaction that seeds SSB. In this setting, by monitoring different observables, an observer can tailor the topology of the vacuum manifold, the pattern of symmetry breaking, and the nature of the resulting domains and topological defects.