A mathematical analysis of a pendulum system reveals the relevance to quantum systems of the classical concept of ‘monodromy’ — why a falling cat always lands the right way up.
A central problem in modern physics is to find effective methods for quantizing classical dynamical systems — modifying the classical equations to incorporate the effects of quantum mechanics. One of the main obstacles is the disparity between the linearity of quantum theory and the nonlinearity of classical dynamics. Taking a big step forward, R. H. Cushman et al. have analysed a quantum version of the spring pendulum, whose resonant state was first discussed by Enrico Fermi and which is a standard model for the carbon dioxide molecule (Phys. Rev. Lett. 93, 024302; 2004).
Cushman et al. show that when this system is quantized, the allowed states, or eigenstates, fail to form a perfect lattice, contrary to simpler examples. Instead, the lattice has a defect, a point at which the regular lattice structure is destroyed. They show that this defect can be understood in terms of an important classical phenomenon known as monodromy. A quantum-mechanical cliché is Schrödinger's cat, whose role is to dramatize the superposition of quantum states by being both alive and dead. Classical mechanics now introduces a second cat, which dramatizes monodromy through its ability always to land on its feet (Fig. 1). The work affords important new insights into the general problem of quantization, as well as being a beautiful example of the relation between nonlinear dynamics and quantum theory.
The underlying classical model here is the swing–spring, a mass suspended from a fixed point by a spring (Fig. 2a). The spring is free to swing like a pendulum in any vertical plane through the fixed point, and it can also oscillate along its length by expanding and contracting. The Fermi resonance occurs when the spring frequency is twice the swing frequency. The same resonance occurs in a simplified model of the two main classical vibrational modes of the carbon dioxide molecule (Fig. 2b), and the first mathematical analysis of the swing–spring was inspired by this model.
Using a modern technique of analysis known as reduction, which exploits the rotational symmetry of a system, Cushman et al. show that this particular resonance has a curious implication, which manifests itself physically as a switching phenomenon. Start with the spring oscillating vertically but in a slightly unstable state. The vertical ‘spring mode’ motion quickly becomes a ‘swing mode’ oscillation, just like a clock pendulum swinging in some vertical plane. However, this swing state is transient and the system returns once more to its spring mode, then back to a swing mode, and so on indefinitely. The surprise is that the successive planes in which it swings are different at each stage. Moreover, the angle through which the swing plane turns, from one occurrence to the next, depends sensitively on the amplitude of the original spring mode.
The apparent paradox here is that the initial state has zero angular momentum — the net spin about the vertical axis is zero. Yet the swing state rotates from one instance to the next. Analogously, a falling cat that starts upside down has no angular momentum about its own longitudinal axis, yet it can invert itself, apparently spinning about that axis. The resolution of the paradox, for a cat, is that the animal changes its shape by moving its paws and tail in a particular way. At each stage of the motion, angular momentum remains zero and is thus conserved, but the overall effect of the shape changes is to invert the cat. The final upright state also has zero angular momentum, so there is no contradiction of conservation. This effect is known as the ‘geometric phase’, or monodromy, and is important in many areas of physics and mathematics.
The central topic of the paper is this: how does monodromy show up when the system is quantized? The answer, obtained in the specific context of the carbon dioxide molecule, is both elegant and remarkable.
A molecule of carbon dioxide can be modelled classically as a central carbon atom, attached symmetrically by identical springs to two oxygen atoms, with the springs inclined at an obtuse angle (Fig. 2b). The molecule has three main vibrational modes. The two most important modes are symmetric stretching, where both springs change their lengths in synchrony, and bending, where the angle between the two springs oscillates. These modes are analogous to the spring and swing modes of a swing–spring. The third main mode, asymmetric stretching, occurs when the two springs oscillate out of phase with each other, and it can be removed from consideration by averaging over a vibrational cycle. The result is a ‘reduced Hamiltonian’, or energy function, which is simpler than the exact Hamiltonian but is still a good model.
The quantum energy–momentum lattice of the molecule consists of the eigenstates of this Hamiltonian, that is, the pure vibrational modes. For a fixed energy, these modes correspond to two classical ‘constants of motion’ — angular momentum and a quantity related to the rotational symmetry. The eigenstates can be characterized by two quantum numbers, which are integers, so these eigenstates form a regular planar lattice like a chessboard.
However, there is an extra quantum number, related to another classical variable, called the ‘action’. The new phenomenon here is that, because of monodromy, the action is defined only locally and cannot be consistently extended across the entire lattice. For fixed quantum numbers in the lattice, this additional quantum number can take on infinitely many values, at equally spaced points at right angles to the chessboard. The simplest structure of this kind is a three-dimensional cubic lattice — an infinite stack of chessboards, vertically above each other. Monodromy implies that the totality of all sets of quantum numbers does not form a cubic lattice. Instead, it has a single topological defect where the regularity of the lattice structure breaks down.
This analysis is important because it suggests, and supports, a general principle. The most significant features of the quantum-mechanical description of a classical system occur at its singularities. The singularities introduce defects into the ensemble of quantum eigenstates, but they also organize the structure of those defects. Everywhere else, quantization works just as in previous, simpler examples. The authors suggest several directions for future progress, mostly to develop the growing use of nonlinear dynamics in the understanding of quantization. But the most tantalizing is the possibility of detecting quantum monodromy experimentally. Maybe we will soon be able to see how Schrödinger's cat turns itself upside down.
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Journal of Biosciences (2007)