Institute for History and Foundations of Science, Utrecht University, P.O. Box 80.000, 3508 TA Utrecht, The Netherlands
Received 11 July 2001; accepted 1 November 2001 Time is often said to play an essentially different role from position in quantum mechanics: whereas position is represented by a Hermitian operator, time is represented by a c-number.This difference is puzzling and has given rise to a vast literature and many efforts at a solution. It is argued that the problem is only apparent and that there is nothing in the formalism of quantum mechanics that forces us to treat position and time differently. The apparent problem is caused by the dominant role point particles play in physics and can be traced back to classical mechanics. ©2002 American
Association of Physics Teachers.
I. INTRODUCTION From the early days, the role time plays in quantum mechanics has caused concerns. For example, von Neumann, in his famous book complains: ‘‘First of all we must admit that this objection points at an essential weakness which is, in fact, the chief weakness of quantum mechanics: its nonrelativisticcharacter, which distinguishes the time t from the three space coordinates x, y, z, and presupposes an objective simultaneity concept. In fact, while all other quantities especially those x, y, z closely connected with t by the Lorentz transformation are represented by operators, there corresponds to the time an ordinary number-parameter t, just as in classical mechanics.’’1 Of course, it is true thatelementary quantum mechanics is not relativistic, but it is not true that the three space coordinates are operators in quantum mechanics. Seventy years later little seems to have changed when we read: ‘‘Moreover, space and time are treated very differently in quantum mechanics. The spatial coordinates are operators, whereas time is a parameter.’’2 Most textbooks written during the intervening periodtell us that time is exceptional in quantum mechanics and many efforts to deal with this problem have appeared in the literature.3 In the following, I will show that time does not pose a special problem for quantum mechanics.4 II. TIME IN CLASSICAL MECHANICS Quantum mechanics is modeled on classical Hamiltonian mechanics. In Hamiltonian mechanics a physical system is described by N pairs ofcanonical conjugate dynamical variables, Q k and k , which satisfy the following Poissonbracket relations:5 Qk ,
l kl ,
Q k ,Q l
The canonical variables deﬁne a point in the 2N-dimensional phase space of the system. The time evolution of the system is generated by the Hamiltonian, a function of the canonical variables, H H(Q k , k ), dQ k /dt Q k ,H , d
k /dt k ,H
2We assume that H does not explicitly depend on time. The Q k and k are generalized variables; they need not be positions and momenta, but may be angles, angular momenta, and the like. However, if the system is a collection of point particles, the canonical variables are usually taken to
301 Am. J. Phys. 70 3 , March 2002 http://ojps.aip.org/ajp/
be the positions qn and momenta pn of theparticles threevectors are in bold type and the subscript denotes the nth particle . Let us consider the relation of the Hamiltonian formalism with space and time. In all of physics, with the exception of Einstein’s theory of gravity general relativity , physical systems are assumed to be situated in a three-dimensional Euclidean space. The points of this space are given by Cartesian coordinates x(x,y,z). Together with the time parameter t, they form the coordinates of a continuous, independently given, space-time background. How the existence of this space and time background is to be justiﬁed is an important and difﬁcult question into which I will not enter. I will just take this assumption as belonging to the standard formulation of classical and quantum mechanics and of special...