We have found that the usual measuring procedure for preselected and postselected ensembles of quantum systems gives unusual results. Under some natural conditions of weakness of the measurement, its result consistently defines a new kind of value for a quantum variable, which we call the weak value. A description of the measurement of the weak value of a component of a spin for an ensemble of preselected and postselected spin-2 particles is presented.PACS numbers: 03.65.Bz This paper will describe an experiment which measures a spin component of a spin--, ' particle and yields a result which is far from the range of "allowed" values. We shall start with a brief description of the standard measuring procedure. Considering measurements on an ensemble of preselected and postselected systems, we shall define a new concept: a weak value of a quantum variable. And, finally, we shall describe the measurement of the weak value on the example of a spin--, ' particle.In quantum theory, the result of a measurement of a variable A which has discrete eigenvalues a; must necessarily be one of those values. The Hamiltonian of the standard measurement procedure ' is H = -g(t)qA, where g(t) is a normalized function with a compact support near the time of measurement, and q is a canonical variable of the measuring device with a conjugate momentum tr. The initial state of the measuring device in the ideal case has to be such that tr is well defined.After the interaction (I) we can ascertain the value of A from the final value of tr: A Btr.As a reasonable approximation for a real situation, we may take the initial state of the measuring device as a Gaussian in the q (and consequently also in the tr) representation. For this case, the Harniltonian (1) leads to the transformation t I H dte a2I4(-tea) ' g-(2) where g; a; i A =a;) is the initial state of our system. If the spread of the tt distribution hatt is much smaller than the differences between the a;, then, after the interaction, we shall be left with the mixture of Gaussians located around a; correlated with different eigenstates of A. A measurement of tt will then indicate the value of A.In the opposite limit, where htr is much bigger than all a;, the final probability distribution will be again close to a Gaussian with the spread hatt. The center of the Gaussian will be at the mean value of A: (A) =g; i a; i a;.One measurement like this will give no information because htr»(A); but we can make this same measurement on each member of an ensemble of W particles prepared in the same state, and that will reduce the relevant uncertainty by the factor I/JN, while the mean value of the average will remain (A). By enlarging the number X of particles in the ensemble, we can make the measurement of (A) with any desired precision. The outcome of the measurement is the average of the obtained values tr of the measuring devices. As we explained earlier, it will yield, for a sufficiently large ensemble, the value (A). We now raise the question: Can we change the above outcome by taking in...

We study the instantaneity of the state-reduction process in relativistic quantum mechanics. The conclusion of various authors that this instantaneity will restrict the set of relativistic observables to purely local ones [i.e., that the measurement of any nonlocal property of a system at a well-defined time would give rise to violations of relativistic causality) is found to be erroneous, and experiments (of a kind not encountered before in measurement theory) are described whereby certain nonlocal properties of some simple physical systems can be measured at a well-defined time without violating causality. The attempts of certain authors to reconcile the reduction process with the covariance of the relativistic quantum state are considered and found wanting, and it is argued that the covariance of relativistic quantum theories resides exclusively in the experimental probabilities, and not in the underlying quantum states. The problem of nonlocal measurement is considered in general: distinctions (which are not to be met with in the nonrelativistic case) arise in relativistic quantum mechanics between what can be measured for fermions and what can be measured for bosons, between what can be measured for individual systems and what can be measured for ensembles, and between what kinds of states can be verified by measurement and what kinds of states can be prepared by measurement; and these pose difficult questions about the nature of measurement itself. L INTRODUCTIONBy now it i s well known that the change of state associated with the measurement process, particularly the instantaneity of this change, will produce novelties and difficulties in relativistic quantum mechanics which a r e not to be met with in the nonrelativistic theory. Two of these a r e in the form of paradoxes which we have begun to reexamine in a recent paper.' They a r i s e roughly a s follows:(1) Suppose that a particle i s initially localized2 to within some finite region of space-time A, and that at some well-defined time t, a measurement of the momentum of the particle i s carried out. Whatever value i s obtained for the momentum, the measurement will instantaneously redistribute the probability uniformly throughout all space, since it will certainly cause the wave function to collapse onto sorne eigenfunction of the momentum (Fig. 1). Thus, if the position of the particle i s measured at time t , +~, a nonzero probability exists that the particle may be found in a region entirely spacelike separated f r o m A. Apparently the momentum measurement process i s capable of moving the particle around at superluminal velocities. Thereby a paradox arises: on the one hand we may formally attribute a given(2) Suppose that at time t = -a a f r e e particle has been prepared in a momentum eigenstate and that at time t = 0 the same particle is found at the origin by means of a detector which has been positioned there, and which interacts locally with the particle. The wave function associated with the particle will change instantaneously at t = 0, ...

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