SCHRÖDINGER AND THE LOSS OF IDENTITY

SCHRÖDINGER AND THE LOSS OF IDENTITY

The most well-known articulation of the Received View can be-and typically has been-attributed to Schrödinger. We have already noted his concern with particle identifiability and individuality in his consideration of the N! problem. Indeed, this concern fed directly into the construction of his wave mechanics. Following the development of quantum statistics within the framework of the new mechanics, Schrödinger-one of the more philosophically inclined of the quantum physicists ¹⁵⁷ -returned to his earlier concern and on numerous occasions expressed the claim that particles could no longer be regarded as individuals. However, and significantly, he also took very seriously-more so than most physicists-the implications of this claim for the interpretation of quantum mechanics. We have already noted the tension that arises within the orthodox interpretation if particles are regarded as non-individuals and Schrödinger argued that this tension was so great, it actually undermined the orthodox view itself. What was required, he insisted, was a new interpretation capable of accommodating this non-individuality.

Perhaps the most well-known statements of the Received View appear in a series of public lectures ¹⁵⁸ given by Schrödinger at the Institute for Advanced Studies in Dublin in 1950, subsequently published as Science and Humanism. ¹⁵⁹ He writes that, in the face of quantum physics,

. we have . been compelled to dismiss the idea that . a particle is an individual entity which retains its 'sameness' forever. Quite the contrary, we are now obliged to assert that the ultimate constituents of matter have no 'sameness' at all. ¹⁶⁰

And he continues,

I beg to emphasize this and I beg you to believe it: It is not a question of our being able to ascertain the identity in some instances and not being able to do so in others. It is beyond doubt that the question of 'sameness', of identity, really and truly has no meaning. ¹⁶¹

We shall return to the issue of how one might represent and understand this idea of identity no longer having any meaning in subsequent chapters, ¹⁶² but for the moment we simply want to consider, first of all, the grounds for such claims 20320c211u , and secondly, what Schrödinger took their implications to be.

These claims were based on two sets of considerations. ¹⁶³ The first has to do with the re-identifiability or trans-temporal identity of particles. We recall from Chapter 1 the requirement of spatio-temporal continuity as a necessary condition of trans-temporal identity. Schrödinger's concern here is precisely that this condition fails in the quantum context and thus, significantly, that the very idea of individual particles must be abandoned. This concern can be traced back to his unhappiness with the claim that particles possess values of every observable and his own formulation of no-hidden-variables arguments. ¹⁶⁴ When the observables are position and momentum, this sense of metaphysical disquiet led him to conclude first, that particles could not be attributed definite spatio-temporal trajectories and further, that the idea of particles which do not have such trajectories was simply incoherent. As he put it in a letter to Margenau, "To me, giving up the path seems giving up the particle". ¹⁶⁵

According to Bitbol, this step, from giving up trajectories, to giving up particles, is grounded in Schrödinger's view of spatio-temporal continuity as the Principle of Individuality itself. ¹⁶⁶ As Schrödinger explicitly acknowledged, this idea of individual particles travelling along well-defined trajectories can be traced back to the pre-Socratic atomists. ¹⁶⁷ In his letter to Margenau,

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Schrödinger also insisted that the role of spatio-temporal continuity had been widely recognized within classical physics and could be explicitly found in Boltzmann's work. Boltzmann was, of course, one of Schrödinger's great scientific heroes, ¹⁶⁸ and in this context he noted Boltzmann's emphasis on the importance of clear and detailed models ¹⁶⁹ which functioned as consistency checks. ¹⁷⁰ The construction of such models must meet the demand expressed in the continuity postulate but according to quantum mechanics the latter is unfillable, and this ". is intimately connected with what I have earlier called the lack of individuality of a particle". ¹⁷¹ In other words, under the impact of the uncertainty relations together with the no-hidden-variables arguments, Space-Time Individuality cannot be maintained in the quantum context. ¹⁷²

According to Schrödinger, such considerations alone suffice to strongly question the adequacy of the particle concept, ¹⁷³ but as he also remarks, other reasons can be given. Thus, in a 1949 seminar he writes that the implications of the 'new' quantum statistics will ". play a great role in our future attitude towards atomism". ¹⁷⁴ He continues,

If you think about those new statistics rationally and soundly, you see that they are illogical, when applied to the individuals or individual events. ¹⁷⁵

Schrödinger tries to illustrate what he means using coin tosses or 'casts': ¹⁷⁶ if one casts a coin six times, say, classical, Maxwell-Boltzmann counting would

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yield the combination of three heads and three tails as having the highest probability. The suggestion that the possibilities of zero, one, two, ., six heads are all equally probable-which is what Bose-Einstein statistics yields-seems 'illogical'. Likewise, it is illogical to assume that heads can only appear once in all six casts or not at all, as Fermi-Dirac statistics dictates. How could we account for such bizarre behaviour of the coins? As Schrödinger suggests, one possibility would be to introduce some form of dependence between the probabilities of each cast, but this would have to be 'most intricate'. ¹⁷⁷ Alternatively, he argues, we must reconceive our classical notions of individuality. Thus in the above analogy of coin tosses, the tosses themselves are analogous to the particles and 'heads' and 'tails' correspond to the states (the boxes in our balls and boxes illustration). Now, if we consider just a single toss or cast, Fermi-Dirac statistics seems entirely appropriate, since 'heads' and 'tails' are mutually exclusive. But now what is analogous to the electron is not the cast but the notion 'heads' and "Heads is not an individual but a property of an individual, namely of a cast". ¹⁷⁸ The corresponding analogy for Bose-Einstein statistics-and this has been reproduced many times in discussions of these matters ¹⁷⁹ -would be that of money in a bank account; "But again, the shillings and pennies in your bank account are not individuals". ¹⁸⁰

He subsequently stressed the importance of quantum statistics even more explicitly:

There is a general reason for not regarding the elementary particles-electrons, protons, light-quanta, mesons-as individuals, a reason very well known to everybody. When you are dealing with a system that contains equal particles you must wipe out their individuality, lest you get quite the wrong results. ¹⁸¹

The appropriate statistics must be invoked, or equivalently, the wave function for the system must be symmetrized or anti-symmetrized appropriately, and, Schrödinger emphasizes, this is the case whether you have a large number of particles-as in a gas containing 10²⁰ molecules-or only a few-as in the case of the two electrons in the helium atom. ¹⁸²

He then makes two further points which will be significant for our discussion of the philosophical implications of the Received View. First of all,

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he states that this lack of individuality implies that the particles cannot be regarded as labelled in any way:

This means much more than that the particles or corpuscles are all alike. It means that you must not even imagine any one of them to be marked 'by a red spot' so that you could recognize it later as the same. ¹⁸³

Secondly, however, he insists that ". waves can easily be marked, by their shape or modulation". ¹⁸⁴ Just as the light-waves that hit your eye are the same that were emitted by the sun, or the sound waves that reach your ear are the same as those produced by leather hitting willow on the cricket pitch, so the waves of quantum mechanics have to be treated as individuals:

You know that if you deal with such many-body problems as the He-molecule or a gas of 10²⁰ molecules, by the method of wave-mechanics, the proper modes have to be regarded as distinguished from one another, they have to be treated as true individuals. You must not apply Fermi-Dirac statistics or Bose-Einstein statistics to them, but ordinary Boltzmann statistics: then you obtain the correct results, the same as you get by applying the new-fangled statistics to the non-individual corpuscles. ¹⁸⁵

Here we see another expression of that form of duality that runs through the history of quantum statistics, where the apparent loss of individuality of the particles is metaphysically compensated for, as it were, by the individuality of the underlying states, or waves. What is the principle of individuality in the latter case? Schrödinger makes perhaps the most obvious suggestion, that it has to do with the 'shape or modulation' of the waves, that is, with their form. ¹⁸⁶

Now, these considerations have a two-fold implication for Schrödinger's interpretation of quantum mechanics: the first-very well known-has to do with his focus on a wave-based ontology; the second-less well known-relates to certain structuralist aspects of this ontology which bear a resemblance to Born's speculations. With regard to the first, Schrödinger's critique of the metaphysics of individual particles can be seen as a preparatory basis for his 'ontology of ψ-waves'. ¹⁸⁷ The second implication emerges from Schrödinger's

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discussion of the problem of how we obtain individual objects from assemblies of apparently non-individual ones. It is here that the structuralist facet of his view is revealed. He begins by reflecting on our observational knowledge of states of affairs, which he calls 'true observables'. ¹⁸⁸ It is these states, he insists, which really constitute matter, as it is meant by the philosopher. He continues,

The way in which particles constitute matter is thus a very strange and novel one. We must investigate it more closely. Particles, having no individuality, constitute pieces of matter that have. They do it by giving rise to observables. What we usually call the building material is of a fundamentally different nature from what is built up of it. In current quantum mechanics, this different nature is expressed by the twofold set of mathematical entities we use: vectors and tensors, or wave functions and operators. ¹⁸⁹

How, then, is this transition from non-individual particles to individual, "truly observable, pieces of matter" achieved? ¹⁹⁰ Schrödinger's answer seems rather weak at first glance: a number of non-individual particles ". coalesce to form a more extended gathering of a build or constitution not copied in the immediate neighbourhood by a number of similar gatherings of the same build". ¹⁹¹ It is this 'coalescence' or 'gathering' which results in observables, or matter 'in the meaning of the philosopher'.

Schrödinger notes two objections which this approach must meet. The first has to do with individuating the pieces of matter and he asks how the individuality of his pocket knife, understood as a ". system of observables referring in the quantum mechanical way to a gathering of particles",would be maintained when faced with other, apparently indistinguishable, pocket knives. As he says, this is trivial. The knife can be marked or scratched or distinguished and hence individuated spatio-temporally.The second objection is more interesting and bears on the recurrent issue of the experimental foundation of the metaphysics of individuality. Schrödinger notes that we sometimes speak of observables referring to a single particle; but if the latter is a non-individual how can the correspondence between observables and individuality be maintained?In experimental practice, however, we never encounter single particles, for reasons already outlined above. Typically such talk arises in the context of certain thought experiments in which similar particles are effectively removed from the immediate 'neighbourhood', a procedure Schrödinger refers to as "smoothing out of the rest of the world". ¹⁹⁵ Then, or so the thought experiment continues, one only has the single particle to form a 'gathering' and effectively produce matter. But, insists Schrödinger, such thought experiments are extremely misleading and encourage us to transfer the metaphysics of individuality from its legitimate domain to circumstances where it cannot be applied: ¹⁹⁶

These thought experiments are precisely what has given rise to the wrong idea that even a single particle is an identifiable piece of matter, and thus that matter is composed of particles. ¹⁹⁷

His answer to the question, "How does individuality arise at all in objects composed of non-individuals?" ¹⁹⁸ is then used to cast light on the deeper metaphysical question, "What are these particles then, if they are not individuals?". ¹⁹⁹ The answer to this question is given by describing what is 'permanent' to a particle, despite its 'lack of individuality'. And both the answer and Schrödinger's way of approaching it are remarkably similar to Born's above.

He begins with the example of an iron paperweight in the shape of a Great Dane which accompanied Schrödinger on his various moves, was left behind when he fled Austria and was subsequently recovered and returned into his possession by a friend. He then asks, on what grounds can he be sure it is the same iron Great Dane? The answer is that "[I]t is clearly the peculiar form or shape (German: Gestalt) that raises the identity beyond doubt, not the material content". ²⁰⁰

Schrödinger takes this example to be analogous to the case of particles and the above 'coalescence' or 'gathering' is now understood in structural terms,

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as the individuality of bodies composed of particles, ". arises out of the structure of their composition, out of shape or form, or organization .". ²⁰¹ In particular, the 'old idea'-outlined in Chapter 1-that individuality is grounded in the identity of substantial matter is now seen as a ". gratuitous and almost mystical addition". ²⁰² It has been replaced by the 'new idea' that ". what is permanent in these ultimate particles or small aggregates is their shape and organization". ²⁰³ Since Aristotle, Schrödinger claims, we have been led to believe that 'shape' or 'form' must be the shape or form of something, some underlying substantial substratum which carries individuality, but when it comes to the ultimate particles we must give up this way of thinking. Such particles are, ". pure shape, nothing but shape; what turns up again and again in successive observations is this shape, not an individual speck of material". ²⁰⁴

Of course, by 'shape' or Gestalt here Schrödinger does not mean geometrical shape; the latter features only as part of our crude and childish models. ²⁰⁵ In the more sophisticated models constructed by science, the Gestalt is given by sets of invariant properties and in his 'William James' lectures of 1954, Schrödinger emphasizes the fundamental importance of invariance. ²⁰⁶ Indeed, in a passage which bears a close resemblance to Born's remarks, noted above, Schrödinger states that our basic idea of a 'thing', as something permanent in the world, is the result of this fundamental process of forming invariants and he writes that "[T]he most fundamental invariants that we form at an early stage are the things in our surroundings, including our own body". ²⁰⁷

This process gives us our view of 'what the world is really like', but, he insists, such a view (of permanent things) is not inevitable; indeed, he compares it to 'scientific constructs' and as such, he emphasizes, it is liable to revision and improvement just as scientific theories are. And, of course, it must be revised when we attempt to carry it over from the observable world around us to the quantum domain. In the latter, the invariants give us only the kind (all electrons are fermions, have the same charge, etc.); ²⁰⁸ to get from the kind to the particular thing in the observable, everyday world we must individuate, via space-time trajectories, or substance or whatever, but such individuation must

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be abandoned in the quantum realm. ²⁰⁹ Nevertheless, there is permanence, as Schrödinger emphasizes.

That Schrödinger's reflections on the philosophical implications of the 'Received View' should display elements of a structuralist tendency is not so surprising, given his familiarity with the writings of two of the great structuralists of the 1920s and 30s, namely the neo-Kantian philosopher Cassirer and the great scientist and science popularizer, Eddington. Both explicitly accepted the non-individuality of quantum particles and incorporated it within their structuralist metaphysics; however, as interesting as they are, we shall not consider their views here. ²¹⁰

We have explored Schrödinger's metaphysics in some detail because of his central importance in articulating and disseminating the 'Received View'. However, it also raises issues of a formal, logico-mathematical nature, as was also perceived in the period following the establishment of quantum mechanics.