ABSTRACT: A generalization of Weinberg's nonlinear quantum theory is used to model a reported violation of the predictions of orthodox quantum theory.

This work concerns the possibility of causal anomalies. By a causal anomaly I mean a theoretical or empirical situation in which the occurrence or nonoccurrence of an observable event at one time must apparently depend upon a

Considerations of the Einstein-Podolsky-Rosen [1] and Bell's-Theorem [2] type entail [3] -- if many-world's interpretations are excluded -- the occurrence of causal anomalies on the theoretical level, provided certain predictions of quantum theory are at least approximately valid. However, those anomalies cannot manifest on the empirical level if the quantum predictions hold exactly [4]. On the other hand, slight departures from the exact validity of the quantum predictions [5] could lead to small but observable causal anomalies [6].

Empirical causal anomalies have been reported in the past in experiments that appear, at least superficially, to have been conducted in accordance with scientific procedures [7], and the protocols are becoming ever more stringent [8]. I do not enter into the difficult question of assessing the reliability of these reports. The scientific community generally looks upon them with skepticism. But at least part of this skepticism originates not from specific challenges to the protocols and procedures of the works of, for example, Jahn, Dobyns and Dunne [7], but from the belief that such results are not compatible with well-established principles of physics, and hence to be excluded on theoretical grounds. However, it turns out that small modifications of the standard quantum principles would allow some of the most impossible sounding of the reported phenomena to be accommodated. According to the report in Ref. [8], it would appear that in certain experimental situations willfull human acts, selected by pseudorandom numbers generated at one time, can shift, relative to the randomness predicted by normal quantum theory, the timings of radioactive decays that were detected and recorded months earlier on floppy discs, but that were not observed at that time by any human observer. Such an influence of an observer backward in time on atomic events seems completely at odds with physical theory. However, a slight modification of normal quantum theory can accommodate the reported data. In the scientific study of any reported phenomena it is hard to make progress without a theoretical description that ties them in a coherent way into the rest physics.

The purpose of the present work is to construct, on the basis of an extension of Weinberg's nonlinear generalization of quantum theory [5], a theoretical model that would accommodate causal anomalies of the kind described above. Specifically, the present work shows that the reported phenomena, although incompatible with the main currents of contemporary scientific thought, can be theoretically modeled in a coherent and relatively simple way by combining certain ideas of von Neumann and Pauli abut the interpretation of quantum theory with Weinberg's nonlinear generalization of the quantum formalism.

To retain the mathematical structure of quantum theory

Following, then, the mathematics, instead of intuition, I shall adopt the assumption that the Schrodinger equation holds uniformly in the

In fact, this point of view is not incompatible with the Copenhagen interpretation, which, although epistemological rather than ontological in character [11], rests on the central fact that in science we deal, perforce, with connections between human observations: the rest of science is a theoretical imagery whose connection to reality must remain forever uncertain.

According to this point of view, expressed however in ontological terms, the various possibilities in regard to the detection of a radioactive decay remain in a state of "possibility" or "potentiality," even after the results are recorded on magnetic tape: no reduction of the wave packet occurs until some pertinent mental event occurs.

By adopting this non-common-sense point of view, we shift the problem raised by the reported results from that of accounting for an influence of willful thoughts occurring at one time upon radioactive decays occurring months earlier to the simpler problem of accounting for the biasing of the probabilities for the occurrence of the thoughts themselves, i.e., a biasing relative to the probabilities predicted by orthodox quantum theory. This latter problem is manageable: Weinberg [5] has devised a nonlinear quantum mechanics that is very similar to quantum theory, but that can produce probabilities that are biased, relative to the probabilities predicted by linear quantum mechanics. Gisin [6] has already pointed out that Weinberg's theory can lead to causal anomalies.

According to the interpretation of quantum theory adopted here, the mechanical recording of the detection of the products of a radioactive decay generates a separation of the physical world into a collection of superposed "channels" or "branches": the physical world, as represented by the wave function of the universe, divides into a superposition of channels, one for each of the different possible recorded (but unobserved) results. Contrary to common sense the recorded but unobserved numbers remain in a state of superposed "potentia," to use the word of Heisenberg. Later, when the human observer looks at the device, the state of his brain will separate into a superposition of channels corresponding to the various alternative macroscopic possibilities, in the way described by von Neumann [9]. FInally, when the

If the probabilities associated with the various alternative possibilities for the brain state are those given by orthodox quantum theory, then there can be no systematic positive bias of the kind reported: the probabilities associated with the alternative possible brain events will necessarily, according to the orthodox theory, as explained by von Neumann, agree with those that were determined earlier from the probabilities of the alternative possible detections of radioactive decays: there could be no biasing of those probabilities due to a subsequent willful intent of an observer. However, a generalization of Weinberg's nonlinear quantum mechanics allows the probabilities for the possible reductions of the state of the brain of the observer to be biased, relative to those predicted by orthodox quantum theory, by features of the state of the brain of the conscious observer. If such a feature were the activity of the brain that is associated with "intent," then the effect of the anomalous term in the Hamiltonian would be to shift the quantum probabilities corresponding to the various alternative possible conscious events toward the possibilities linked to his positive intent.

We turn, therefore, to a description of Weinberg's theory, in the context of the problem of the shifting of the probabilities away from those predicted by orthodox quantum theory, and toward those defined by an "intent" represented by particular features of the state of the brain of the observer.

Weinberg's nonlinear quantum theory is rooted in the fact that the quantum-mchanical equations of motion for a general quantum system are just the classical equations of motion for a very simple kind of classical system, namely a collection of classical simple harmonic oscillators. Thus a natural way to generalize quantum theory is to generalize this simple classical system.

This example shows that the reported phenomena, although contrary to orthodox ideas about causality, can be model within a Weinberg-type of nonlinear quantum theory if the Hamiltonian function

If there are in nature nonlinear contributions of the kind indicated...then it seems likely that biological systems would develop in such a way as to exploit the biasing action. The biasing states, illustrated in the model by the state

The Weinberg nonlinearities were intially introduced in the present context because of Gisin's result, which showed that these nonlinearities could lead to causal anomalies of the Einstein-Podolsky-Rosen (EPR) kind. However, the considerations given above indicate that those nonlinearities alone cannot produce anomalies of the kind reported in Ref. [8]: a nonreal

Because the nonlinear aspect is not obviously needed, one could try to revert to a linear theory. Yet it is important to recognize that in the modeling of acausal effects one has available the more general nonlinear framework.

If the purported acausal phenomena is a real physical eitect and is explainable in terms of a nonreal

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