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. Based on 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.
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. Based on 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.
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.
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.
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).
QM∞ includes quantum field theory and the thermodynamic limit of quantum statistical mechanics — theories which, unlike the 'ordinary' quantum theories typically discussed by philosophers, concern infinite systems. The mathematical framework appropriate for the presentation of a theory of ordinary quantum mechanics is essentially unique. For theories of QM∞, this is not so. This prompts interpretive questions—for instance, about what makes a quantum theory the quantum theory it is — this work aims to chronicle. Having introduced a formal apparatus (operator algebra theory) suited to pursue these questions, the book articulates a variety of accounts of the content of quantum theories, accounts responsive to QM∞'s characteristic non-uniqueness. To evaluate these accounts, the book examines QM∞ settings (e.g. spontaneous symmetry breaking, cosmological particle creation, superconductivity) in which that characteristic non-uniqueness seems to matter, with a view toward determining which accounts sustain the uses to which the non-uniqueness is put. This approach not only brings work on the foundations of quantum theories in contact with the foundational investigation of other sorts of physical theories (e.g., thermodynamics, statistical mechanics, solid state physics, general relativity, and cosmology), it also brings the philosophy of physics in contact with other sorts of philosophy of science (e.g., accounts of explanation, reduction, and explanationist defenses of scientific realism). The book concludes that received notions of physical content and physical modality must be revised if they are to apply usefully to particular physical theories.
The basic problem how spontaneous symmetry breaks is studied rigorously for local fields in a finite domain. Another basic problem is to investigate whether or not there exist lower dimensions (of the domain to which a field H is applied) below which no spontaneous symmetry breaking (SSB) occurs. This is studied using scaling, renormalization group and mean-field theory for partial fields in an infinite domain. Our main results are the following: (i) The total magnetization is proportional to the square of the spontaneous magnetization when a field is applied only to a finite region (an extension of Yang's result [Phys. Rev. 85 (1952) 808], M = N m 2 s for a fixed single spin). A general formulation is given to calculate the proportionality coefficient which is shown to vanish as the field goes to zero. This indicates no SSB in this situation. (ii) On the other hand, if we apply partial fields to a d '-dimensional region and make them vanish, we find that SSB occurs, at least, for d '> d ' c =β/ν. In the region 0< d '≤ d ' c , the situation is subtle, as is exemplified by an exactly soluble model with such a field on one-side boundary as depends on the system-size.
Spontaneous symmetry breaking (SSB) in quantum systems, such as ferromagnets, is normally described as (or as arising from) degeneracy of the ground state; however, it is well established that this degeneracy only occurs in spatially infinite systems, and even better established that ferromagnets are not spatially infinite. I review this well-known paradox, and consider a popular solution where the symmetry is explicitly broken by some external field which goes to zero in the infinite-volume limit; although this is formally satisfactory, I argue that it must be rejected as a physical explanation of SSB since it fails to reproduce some important features of the phenomenology. Motivated by considerations from the analogous classical system, I argue that SSB in finite systems should be understood in terms of the approximate decoupling of the system's state space into dynamically-isolated sectors, related by a symmetry transformation; I use the formalism of decoherent histories to make this more precise and to quantify the effect, showing that it is more than sufficient to explain SSB in realistic systems and that it goes over in a smooth and natural way to the infinite limit.