Quantum dynamics in the thermodynamic limit

Abstract

The description of spontaneous symmetry breaking that underlies the connection between classically ordered objects in the thermodynamic limit and their individual quantum mechanical building blocks is one of the cornerstones of modern condensed matter theory and has found applications in many different areas of physics. The theory of spontaneous symmetry breaking however, is inherently an equilibrium theory, which does not address the dynamics of quantum systems in the thermodynamic limit. Here, we will use the example of a particular antiferromagnetic model system to show that the presence of a so-called thin spectrum of collective excitations with vanishing energy -one of the well-known characteristic properties shared by all symmetry-breaking objects-can allow these objects to also spontaneously break time-translation symmetry in the thermodynamic limit. As a result, that limit is found to be able, not only to reduce quantum mechanical equilibrium averages to their classical counterparts, but also to turn individual-state quantum dynamics into classical physics. In the process, we find that the dynamical description of spontaneous symmetry breaking can also be used to shed some light on the possible origins of Born's rule. We conclude by describing an experiment on a condensate of exciton polaritons which could potentially be used to experimentally test the proposed mechanism. 1: IntroductionCombining many elementary particles into a single interacting system may result in collective behaviour that qualitatively differs from the properties allowed by the physical theory governing the individual building blocks. This realisation -immortalised by P.W. Anderson in his famous phrase 'More is Different' [1]-not only forms the basis of much of the research being done in condensed matter physics today, but has also found applications in areas ranging from string theory to cosmology. The theory of Spontaneous Symmetry Breaking which formalises these ideas first took shape over fifty years ago [2,3,4,5,6,7], and was completed in the context of quantum magnetism only two decades ago by the detailed description of the classical state as a combination of thin spectrum states, emerging as N → ∞ because of the singular nature of the thermodynamic limit [8,9,10]. The same description of the classical state emerging from the thin spectrum has since been shown to also directly apply to the cases of quantum crystals, antiferromagnets, Bose-Einstein condensates and superconductors [11,12,13,14,15].The connection between the quantum mechanical properties of microscopic particles and the classical behaviour of symmetry broken macroscopic objects has now again come to the forefront of modern science because of our technological capability to create ever larger and heavier quantum superpositions in the laboratory. Superconducting flux qubits harbour counterrotating streams of supercurrent consisting of up to 10 11 Cooper pairs [16,17,18], while Bose Einstein condensates of the order of 10 5 Rubidium atoms can be routinely brought i...

Broken Time Translation Symmetry as a Model for Quantum State Reduction

“…The mechanism of spontaneous symmetry breaking is traditionally formulated in an equilibrium description, thereby pre-empting any possibility of finding a state that does not obey the unitary symmetry. We have recently shown that this is not a necessary constraint, and that it is possible to give a dynamical description of spontaneous symmetry breaking in quantum mechanics which does indeed allow even the time translation symmetry to break down [5,6]. The time evolution resulting from the spontaneous breakdown of unitarity turns out to be surprisingly familiar: it reproduces precisely the quantum state reduction process observed whenever we try to measure a quantum state with an effectively classical measuring apparatus.…”

“…If the non-unitary field is a given, static field, it will always favour the component of a superposed wavefunction that is closest to its centre. Even if the location of that centre changes from one run of the experiment to another, this could never lead to a distribution of experimental outcomes that is correlated to the weight of individual components in the initial wavefunction [5,6,28]. However, we saw in the previous section that the conflict between general covariance and quantum mechanics can give rise to a non-unitary perturbation of Schrödinger's equation that is best modelled by the introduction of a time dependent stochastic variable, as in equation (16).…”

“…The time it takes for a delocalised state to spontaneously localise due the presence of an order parameter field is proportional to 1/(N mω 2 ). If ω 2 is of the correct order of magnitude, this timescale could be immeasurably long for microscopic objects while being immeasurably short for macroscopic objects [5,6]. If we assume that the order parameter field results from the incompatibility of general covariance and unitarity, as we argued before, then the combination of ω 2 = Gρ and M is indeed of the correct order of magnitude to ensure that elementary particles, molecules and even supercurrents should be considered microscopic, while tables, chairs and pointers should be treated as macroscopic objects.…”

Spontaneous symmetry breaking in quantum systems: Emergence or reduction?

“…The work of van Wezel (2007van Wezel ( , 2008van Wezel ( , 2010, which was brought to our attention only after the first version of this paper appeared on the arXiv, analyzes the spontaneous breakdown of continuous symmetries in a way compatible with (and indeed predating) our treatment the discrete case, including the fundamental observation about the instability of the symmetric ground state of a finite system under 'infinitesimal' asymmetric perturbations. Van Wezel (2010) also discusses the connection between ssb and the measurement problem, though his proposed solution to this problem is quite different from ours.…”

“…This works because in these models (in which the energy spectrum is discrete for any N < ∞), as N → ∞, on the one hand all energy levels merge into a continuum, but on the other, they split into pairs whose energy difference is exponentially small. In addition, the instability of the ground state has been derived in a different way by Narnhofer & Thirring (1996, 1999 for Z2, as well as by van Wezel (2007van Wezel ( , 2008van Wezel ( , 2010 for SU (2).…”