Spontaneous symmetry breaking in quantum systems: Emergence or reduction?

Abstract

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 idealization (namely the thermodynamic limit), disappearing as soon as the idealization 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 author 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). Motto"The characteristic behaviour of the whole could not, even in theory, be deduced from the most complete knowledge of the behaviour of its components, taken separately or in other combinations, and of their proportions and arrangements in this whole. This is what I understand by the 'Theory of Emergence'. I cannot give a conclusive example of it, since it is a matter of controversy whether it actually applies to anything." (Broad, 1925, p. 59) * Final version, to be published in Studies in History and Philosophy of Modern Physics.

Deduction and definability in infinite statistical systems

“…If one understands the quantities of each finite subsystem of the infinite spin chain to be represented by a C*-algebra, then one can construct a continuous field of C*-algebras (See Dixmier (1977) and Landsman (1998Landsman ( , 2013) to represent the limiting procedure for the entire algebra of quantities. That is, one starts with the quantum theory of a finite number of subsystems and lets the number of subsystems grow larger and larger, analyzing how the entire algebra of quantities changes.…”

Obituary for a Flea

“…Even though the post-measurement state is not equal to (11), it is close enough to this state to conclude that tracing out the system will yield a non-pure state. By the previous reasoning, the wave function of the flea model cannot represent the state of the pointer variable in a straightforward way.…”

“…In the second approach, called asymptotic Bohrification the commutative and non-commutative C*-algebras, no longer related by an inclusion relation, are glued together in a bundle called a countinuous field of C*-algebras. Rather than discuss this approach in full generality, we first consider the motivating example of the flea approach to the problem of outcomes, as introduced by Landsman and his student Reuvers in [14], and later embedded by Landsman in the asymptotic Bohrification programme in [11]. The terminology which we adopt here was introduced by Landsman in [13].…”

“…σ z i and the limit N → ∞, where N is the number of sites on the lattice. This example was treated in the setting of asymptotic Bohrification in [11]. The only thing that we need to add is a flea, for example in the shape of a matrix W = diag (0, 0, .…”

Quantum spin systems versus Schroedinger operators: A case study in spontaneous symmetry breaking

“…"While idealizations are useful and, perhaps, even essential to progress in physics, a sound principle of interpretation would seem to be that no effect can be counted as a genuine physical effect if it disappears when the idealizations are removed." (Earman, 2004) As argued in detail in Landsman (2013Landsman ( , 2017, the solution to his paradox lies in Earman's very principle itself, which (contrapositively) implies what we call Butterfield's Principle:…”