NEWS AND VIEWS

Classical and quantum physics mix

A neat way of talking of quantum and classical physics in the same language will be of interest in itself even if it does not avoid the need for a quantum theory of gravity; but that would be a huge extra.

MIXING together classical mechanics and photons whose emission is stimulated) are The Poisson bracket is defined as quantum mechanics is a long-standing source dealt with as such, while the structural ele­ (ajd J8-at/d xg), where ax and Ok denote ofembarrassment, usually arising quite early ments ofthe laser (which define the shape of partial differentiation with respect to x and k in the textbooks of quantum mechanics. the resonating cavity) appear simply as pa­ respectively. (Plainly the test is trivially There is, for example, the simplest of all the rameters in the equations. This fuss does satisfied for the original variables x and k.) problems in quantum mechanics, that of a not imply that the standard treatments of What Anderson does is simply to put the particle in a box. The heuristic value of this mixed problems are in some sense wrong, two conditions together, arriving at a test for example is that it stands proxy for an elec­ but that they are at least inelegant and may pairs of conjugate variables (A and B) with tron or some other charged particle in a fixed leave people with an awkward sense of the form [A,B] + i{A,B} = i. At the most electrostatic potential, as in the interior of a insecurity. trivial level, this equation is plainly valid; if metal for example. That seems to be part of the reason why A and B are a conjugate pair of quantum Beginning students are not, however, Arlen Anderson, from Imperial College variables, only the first term applies (and invited to wonder how the electrostatic box London (but also affiliated to the Newton equals i by definition), but if they are a itself is constructed. Ifthey were (and ifthe Institute at Cambridge) has embarked on a classical pair, only the second is relevant dimensions of the box were of the same scheme for including both quantum and and is identically true. order of magnitude as the wavelength of the classical variables in a common theory (phys. More generally, he defines a new bracket electron), they would quickly conclude that Rev. Lett. 74,621-625; 1995). As he puts it, expression for any two variables as [A,Br, the electrostatic field that constitutes the he seeks to "assuage mathematical doubts say, by the expression above, [A,B] + i{A,B}; box can have been made only from micro­ about how classical and quantum variables he departs from others who have followed scopic objects, say the ions of a crystal may coexist in a single theory". But reading this route in the rules for evaluating [A,B)' lattice, which are themselves, in principle, between the lines, the long-term objective is for mixtures of classical and quantum vari­ also subject to the laws of quantum mechan­ to find a way of handling the field theories of ables. Following the standard procedures of ics. So why treat the electron in a box as a particle physics self-consistently against the quantum mechanics, there then tumbles out quantum object, yet consider the electro­ background of a classical theory of gravity, an expression for the time rate of change of static potential that confines it as a classical that of Einstein's general relativity. an arbitrary function A of quantum and given? Anderson acknowledges earlier work in classical variables (and in which the time is Similar problems arise in more taxing the same direction by I. V. Aleksandrov and not explicit) in the familar form of -i[A,H]', fashion in textbook discussions ofthe prob­ by W. Boucher and J. Traschen, saying that where H is the Hamiltonian of the system. lems of quantum measurement. The the distinctive contribution is to demon­ The physics of this formalism is not as Heisenberg gedanken microscope used to strate the 'back-reaction' of the quantum obscure as it may seem (but Anderson's demonstrate the inescapable uncertainty in system on a classical system with which it paper illustrates the point with two work­ a measurement of the position of an electron is coupled. This effectively takes the form able - and neatly worked - examples). is a case in point. The idea is to estimate the of an extra source of noise in the evolution The deficiency of representing the variables position of the electron from the scattering of the classical system whose effect is to of the classical parts of a mixed system as of photons. The precision of the measure­ induce a correlation between the values parameters in quantum equations of motion ments is inevitably a function of the fre­ of classical observables (the quantum word is that they are thereby given the status of quency of the photons, but the greater the for 'measurable quantity' or 'variable') givens, and cannot be changed. But even if frequency, the greater the momentum they and those attained by the coupled quantum the classical parts are so far away from the will transfer to the electron, which offsets system. conditions required of quantum systems that the potential benefit of working with higher The algebraic technology of putting to­ they will not, for example, be quantized in frequencies. gether quantum and classical mechanics the sense that only some states are accessi­ The argument demonstrates what the turns out to be simple enough, perhaps de­ ble to them, there is every reason why their authors of the textbooks want to show, that ceptively so. The starting point is the defini­ continual interaction with a quantum part Heisenberg's Uncertainty Principle is un­ tion of a conjugate pair of variables (say q will affect the probability distribution of the avoidable, but hardly anybody stops to ask and p, perhaps the position of a particle and states in which they may be found. whether the microscope itself is a quantum its momentum in that direction), to which Whether these developments will solve object. If, by chance, they do, the answer the Uncertainty Principle applies in quan­ the problem of the gravitational field is will be that it is a macroscopic object, the tum mechanics, which is that [q,p] = i (where another matter. After two decades of serious uncertainty in the position of its optical [q,p] == qp - pq, i is V( -1) and the units attempts to produce a quantum theory of elements (calculable from the same uncer­ are such that h127r = 1). gravitation, it is forgivable that people should tainty principle) will be shown to be much The classical analogue of this relation­ now be wondering whether it is really nec­ less than the uncertainty in the position of ship is familiar from Hamilton's nineteenth­ essary to try. The clamant need for quantum the electron, and that will be that. century formulation of classical mechanics, gravity arises only in exceptional circum­ Thus the world has become used to the which makes it possible to test whether, stances, at the edges of black holes and at idea that quantum physics and classical phys­ given one pair of conjugate variables, say x similar exceptional locations. But to follow ics are a little like oil and water. They do not and k, functional combinations ofthem, say even Anderson's new approach, people will mix. Where some interaction is of the es­ j(x,k) and g(x,k), are also conjugate vari­ have to learn to do quantum mechanics in sence, as in the behaviour of a laser, the ables. The test is that the so-called Poisson curved space-time. That will not be child's quantum elements (excited atoms and the bracket, called {f,g}, should be equal to 1. play. John Maddox NATURE . VOL 373 . 9 FEBRUARY 1995 469