While denying that the transition from the possible to the actual in the measurement operation is connected with the act of registration of the measurement result by the mind of the observer, Heisenberg states that the discontinuous change in the probability function due to the second measurement takes place with the act of registration, because it is a discontinuous change of our knowledge in the instant of registration, a change that has its image in the discontinuous change in the probability function.  Bohm quotes this passage in his article, and he concludes that until an observer actually perceives the result of observation, so that he can write a new wave function representing the actual state to which the previous possibilities have collapsed as a result of his perception, there is no actuality at all as far as anything that can appear in the theory is concerned, but only the set of possibilities.  Bohm illustrates his view of Heisenberg’s subjectivism with a hypothetical experiment involving a set of Geiger counters arranged in a grid and toward which a free electron is directed.  He supposes a point in time at which the electron has already entered the grid system and has triggered off one of the counters, and furthermore supposes that no observer has yet looked to see which counter has been triggered.  Bohm says that on Heisenberg’s view, at the supposed point in time when no observer has seen which counter has been triggered, one knows the objective possibilities, namely that the counter in question must be one of those located where the amplitude of the electron wave function is appreciable; but if one tries to describe the physical actuality of which counter has been triggered, there is no way in the theory to do so, because the probability function describes only psychic actualities.  In other words until an observer actually perceives which counter has operated, so that he can write a new wave function representing the actual state to which the previous possibilities have collapsed as a result of his perception, there is no actuality described by the theory but only the set of possibilities. Thus the physical actualities play no part at all in the theory, because no predicted result would be changed if the theory were developed without mentioning the physical actualities.  It is noteworthy that Bohm seems not merely to be saying that the Copenhagen interpretation is subjectivist because it is probabilistic; he is not merely criticizing Heisenberg’s Copenhagen interpretation by assuming the subjectivist interpretation of probability as the only probability interpretation.  Bohm’s criticism is the claim that the subjectivism is in Heisenberg’s Copenhagen interpretation, and that Heisenberg’s argument unintentionally but logically implies that to be is to be perceived by the registering mind of the observer.  In fact this is not Heisenberg’s view.  Heisenberg’s thesis is that to be is to be produced by the disturbing physical apparatus used by the observer.  Thus Bohm’s thought experiment involving the grid of Geiger counters demonstrates only that there may be a time interval between production and perception of the new actuality.
          The confusion seems to arise from Heisenberg’s decision to give the quantum probability function both a subjective and an objective interpretation.  Normally these two interpretations are distinguished as alternatives, and for good reason: On the objective interpretation the probability function is a statement in the object language like any other theory in physics with a semantics describing the real physical world.  Thus interpreted the probability function is an object language statement with a semantics describing the potentia ontology and the ontology of indeterminism.  On the subjective interpretation the probability function is a statement in a metalanguage for physics with a semantics describing the physicist’s state of knowledge or ignorance expressed by the object language, and it consists of statements making statistical estimates of measurement error.  Heisenberg chose to combine these two interpretations, presumably because in quantum theory it is not possible to write a separate probability function for the measurement error.  In classical physics, which traditionally assumes an ontology of determinism, variations in repeated measurements under the same experimental conditions are assumed to be due entirely to randomly distributed measurement errors.  These could be represented statistically by the standard deviation about the calculated mean of the measurement values, also known as the standard error of the estimate of the true measurement value, or by a probability function based on a normalization inversely related to the deviations from the mean.  In quantum theory, on the other hand, the indeterminist ontology introduces a random variation not originating in measurement errors even though the indeterminacy operates in the measurement process.  Therefore, the two sources of variation are inseparable, and the effects of both are taken up in the probability function of quantum theory. What is noteworthy is that Heisenberg does not state in his exposition that the discontinuity in the probability function is due to measurement errors occurring in the second measurement.  He says it is a discontinuous change in our knowledge in the instant of registration that has its image in the discontinuous change of the probability function, and that is occasioned by the disturbance produced by measurement process.  Thus the objective interpretation would seem to be the operative one in this passage, because the probability function is viewed as constituting the experimenter’s knowledge rather than describing that knowledge.  The knowledge that Heisenberg says is the image of the new probability function is the semantics that describes the new physical actuality realized by the action of the measurement apparatus.
          However, Bohm prefers to construe this passage to mean that the probability function should be taken with a subjective interpretation, and that it describes knowledge, which Heisenberg calls psychical instead of physical.  This places the quantum theory entirely in the metalanguage for physics.  Thus Bohm says that the physical actualities play no part whatsoever in the theory, since no predicted result would be changed in any way at all, if the theory were developed without mentioning them.  Then Bohm says that to avoid subjectivism, Heisenberg adopts the completely metaphysical assumption of physical entities, which play no part in the theory, but which are introduced to avoid what would otherwise be an untenable philosophical position.
          Having exposed to his satisfaction the inconsistency in the Copenhagen interpretation, namely the subjectivism he believes implied in Heisenberg’s exposition and contradicted by Heisenberg’s ad hoc attempt to introduce physical actuality, Bohm then goes on to say that it was due to this problem that he himself was led to criticize the Copenhagen interpretation years earlier, and that while trying to find a way to remedy the absence of the actuality function, he developed his own alternative interpretation.  In his alternative interpretation, namely the hidden-variable interpretation, he proposes in addition to the Schrödinger wave function the existence of a particle having a well defined position and momentum, which interacts with the wave in a certain prescribed manner.  The position of this particle plays the part of an actuality function in the sense that, when the wave function spreads out over many possibilities, this particle function determines which of these possibilities is actually present.
          Consider next Bohm’s semantical critique of Heisenberg’s version of the Copenhagen interpretation.  His semantical critique is distinctive, because it exploits Heisenberg’s distinction between on the one hand everyday concepts and on the other hand the Newtonian concepts of classical physics that are said to be refinements of the everyday concepts.  Bohm rejects the Copenhagen thesis that the classical concepts are necessary for describing macrophysical objects such as the equipment used in microphysical experiments, and thus maintains that alternatives to the Copenhagen interpretation are conceptually possible.  He maintains contrary to Heisenberg that the everyday concepts actually used in ordinary experience including the physicist’s description of his laboratory equipment may be refined to topological concepts, and need not be the Cartesian-coordinate concepts of Newtonian physics.  He therefore believes that the topological concepts are more fundamental in the mathematical sense for the description of space and time than the Cartesian concepts, and that the latter must be translated from the former for Newtonian physics.
          Bohm exemplifies this idea with the problem of location an ordinary pencil.  The location is not ordinarily stated in terms of a coordinate system such as latitude and longitude.  What is actually done in ordinary experience is to locate the pencil as laying upon a desk within a certain room in a certain house, which is located on a certain street, etc.  Thus the pencil is located with the aid of a series of topological relations, in which one entity is within or upon another entity.  He then says that the laboratory physicist also uses topological relations in his work.  In no experiment does he ever locate anything by giving an exact coordinate, which is to say an infinite number of decimals.  Rather what he does in practice for making a measurement is to place a pointer between certain marks on a scale, thus locating it by the topological relation of between.  In every experiment the notion of precisely defined coordinates is just an abstraction, which is approximated when a topologically described experimental result is translated into the Cartesian language of continuous coordinates.  Bohm adds that everyday concepts could be refined in other ways than to topological concepts, but that for description of space and time, the topological concepts are most appropriate for physical theory.
          Bohm then suggests a topological formulation of the quantum theory, and says that these nonclassical concepts make possible new kinds of experimental predictions, which cannot be considered in the framework of the Copenhagen interpretation, and which according to Heisenberg’s conclusions are not possible.  Bohm says that there is a remarkable analogy between the mathematics of topology and that of the modern quantum mechanical field theory, and that utilization of this analogy can make possible the development of a topological formulation, which while leading to the results of the usual quantum theory in suitable limiting cases, nevertheless possesses certain genuinely novel features with regard both to its mathematical formalism and to its experimental predictions.  He also says that he cannot go into the details in this paper, and he is not known to have done so in any other paper he has published.  This novation with or without the inspiring analogy is a promissory note backed by an as-yet-unearned income, because Bohm does not set forth an explicit topological formulation of the quantum theory.  If he actually had set forth such a new formulation, and if its novel experimental predictions were found to be superior to those made by the current quantum theory, such as resolving the renormalization problem, then his new topological formulation would be a revolutionary development in microphysical theory, and much more than merely a new interpretation of the quantum theory.

Bohm and Bell on the EPR Experiment and Nonlocality

          In 1935 Einstein, Podolsky and Rosen (conventionally abbreviated as “EPR”) published an article in the Physical Review titled “Can Quantum- Mechanical Description of Physical Reality Be Considered Complete?”  Their negative answer implies that the current statistical quantum theory is inadequate, and that further development is needed that would presumably involve identifying presently unknown factors conventionally referred to as hidden variables.  The authors firstly set forth a necessary condition for completeness, according to which every element of the physical reality must have a counterpart in the physical theory.  And they secondly set forth a sufficient ontological condition for affirming the reality of a physical quantity, which consists in the possibility of predicting with certainty the physical quantity under investigation without disturbing the physical system. The three authors propose a hypothetical or gedanken experiment, now conventionally known as the “EPR experiment”, which assumes among other things the experimental correctness of the quantum theory and Heisenberg’s indeterminacy relations, but which concludes to a demonstration of the present quantum theory’s incompleteness. 
            There are several equivalent versions of this now famous experiment including some that have since actually been performed.  The authors postulate two particles initially interacting, such that their properties are correlated, and then subsequently separated spatially by being sent off in opposite directions, so that they can no longer interact but still retain their correlated properties.  One of the implicit assumptions of the argument is that there is no instantaneous action at a distance, so that the spatial separation of the two particles precludes the measurement of one particle from disturbing the other particle in any way.  This assumption has been called either separability or locality.  In this thought experiment the noteworthy properties are the noncommuting observables, position and momentum.  If the momentum of one of the particles is measured, then since its momentum is correlated to the momentum of the second particle, the momentum of the second is also known by the measurement of the first.  Or if the position of the first particle is measured, then since its position is correlated to the position of the second particle, the position of the second is also known by the measurement of the first.  But according to Heisenberg’s indeterminacy relations no quantum wave/particle can simultaneously have both position and momentum as determinate properties The selection of which quantity is determinate is made by the measurement action, a selection which is the free and arbitrary choice of the experimenter.  The second particle has no interaction with the first at the time that the first particle is measured, so the second particle cannot know, as it were, which of the noncommuting properties the experimenter selected as the determinate property of the first particle.  Yet paradoxically the second particle’s determinate property is always correlated to that of the first.  The authors, Einstein, Podolsky and Rosen, conclude that the paradox can only be resolved by recognizing that in fact both particles always had both determinate position and determinate momentum from the time of their separation, and that the current quantum theory fails to represent the physical reality of the situation completely.  The current quantum theory, in other words, is incomplete.
          Bohr responded to this argument in an article with the same title appearing in a later issue of the same journal in the same year.  He takes issue with EPR’s criterion for physical reality, reaffirms his principle of complementarity, and maintains contrary to EPR that quantum theory is not incomplete.  He says that because it is impossible to control the reaction of the object on the measuring instruments, the interaction between object and measuring devices conditioned by the very existence of the quantum of action entails the necessity of a final renunciation of the classical ideal of causality and a radical revision of our attitude towards the problem of physical reality.  Bohr discusses this aspect of measurement in the context of the two-slit experiment of electron diffraction, which is not a hypothetical experiment but an actual one.  He references Heisenberg’s uncertainty principle, and says that the uncertainty of momentum of the incident particle is inseparably connected with an exchange of momentum between the particle and the diaphragm.  This impossibility of a closer analysis of the reactions between the particle and the measuring instrument is an essential property of any arrangement where there is a feature of individuality completely foreign to classical physics.  Any attempt to take into account the momentum exchanged between the particle and the separate parts of the apparatus, would imply conclusions about the course of such phenomena, such as what particular slit the particle passes on its way to the photographic plate.  This would be quite incompatible with the fact that the probability of the particle reaching a given place on the photographic plate is determined not by the presence of any particular slit, but by the position of all the slits.  Bohr explains that complementarity is due to this impossibility in the field of quantum theory of accurately controlling the reaction of the object on the measuring instrument, i.e. the transfer of momentum in the case of position measurements and the displacement in the case of momentum measurements.  And he concludes that in such cases the physicist is not dealing with an incomplete description characterized by the arbitrary picking out of different elements of physical reality at the cost of sacrificing other such elements, but with a rational discrimination between essentially different experimental arrangements which are suited either for an unambiguous use of the idea of space location or for the legitimate application of the conservation laws of momentum.  There is nothing in this rebuttal by Bohr that was not previously known to physicists and to EPR at the time of their famous paper, and Bohr’s arguments cannot be said to have been responsive to the particulars of EPR’s thought experiment.
          In a section titled “The Paradox of Einstein, Podolsky and Rosen” in his Quantum Theory Bohm says that the EPR criticism of quantum theory has been shown to be unjustified, and in a footnote to this statement he references Bohr’s critique of EPR published in Physical Review. At this time Bohm was sympathetic to the Copenhagen interpretation, and critical of Einstein’s views.  In addition to EPR’s necessary condition for a complete physical theory and their sufficient condition for recognizing an element of reality, Bohm says that there are two additional assumptions implicit in the EPR argument.  These assumptions are firstly that the world can be correctly analyzed in terms of distinct and separately existing elements of reality, and secondly that every one of these elements must be a counterpart of a precisely defined mathematical quantity appearing in a complete theory.  Bohm attacks these two implicit assumptions.  He says that the one-to-one correspondence between mathematical theory and well defined elements of reality exist only at the classical level.  At the quantum level, on the other hand, the properties described by the wave function are not well defined properties, but are only potentialities which are more definitely realized in interaction with an appropriate classical system such as a measuring apparatus.
          For his own critique of EPR, Bohm offers a modified but equivalent version of the EPR experiment for his analysis.  His version considers the spin of the two separated and correlated particles.  The second particle’s spin is always correlated to the measurement axis, i.e. the spin component, chosen for measurement of the first particle, regardless of the component selected by the experimenter for measurement.  On the EPR interpretation precisely defined elements of reality must therefore exist in the second particle corresponding to the simultaneous definition of all three dimensional components of spin.  And since the Schrödinger wave function can specify at most only one of these components at a time with precision, it cannot provide a complete description of all elements of reality existing in the second particle.  But Bohm maintains that the wave function provides the most complete description of physical reality consistent with the actual structure of matter, because on his view no component of spin of a given variable exists with a precisely defined value until interaction with the measuring apparatus has taken place.  As soon as the first particle interacts with the measuring apparatus a given spin component is determined.  As a result the definite phase relations between the wave functions of the two particles are destroyed, and the wave function of the other particle will take a form that guarantees the development of the opposite value of spin, if the second particle interacts with an apparatus measuring the same component of spin.  Bohm therefore says that wave function describes the propagation of correlated potentialities.

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