> YZXܥhc eFNƋ"]vvv"ȭȭȭȭȭح(ȭ8Q 6666666888888Xm86LR66668666666666Rȭȭ6666Weyl and von Neumann: Symmetry,
Group Theory, and Quantum Mechanics
Otvio Bueno
Department of Philosophy
California State University, Fresno
Fresno, CA 937408024, USA
email: otavio_bueno@csufresno.edu
1. introduction
In this paper, I shall discuss the heuristic role of symmetry in the mathematical formulation of quantum mechanics. I shall first set out the scene in terms of Bas van Fraassens elegant presentation of how symmetry principles can be used as problemsolving devices (see van Fraassen [1989] and [1991]). I will then examine in what ways Hermann Weyl and John von Neumann have used symmetry principles in their work as a crucial problemsolving tool.
In order to examine the issue, I will initially compare the grouptheoretic approach to quantum mechanics provided by Weyl [1931] with von Neumanns Hilbert space formalism (von Neumann [1932]). In Weyls approach, group theory is, of course, the crucial feature, and Weyl explores symmetry to establish crucial representation theorems about quantum theory. Although group theory doesnt play a major role in von Neumanns Hilbert space formalism, by the late 1930s von Neumann came to be dissatisfied with the Hilbert spaces approach (Rdei [1997]). After all, this approach didnt provide an adequate notion of probability for quantum systems with an infinite number of degrees of freedom. The alternative framework advanced by von Neumann was articulated in terms of his theory of operators (Murray and von Neumann [1936]) and what we now call von Neumann algebras (Rdei [1998]). According to von Neumann, this was the appropriate setting in terms of which we should formulate quantum mechanics (Birkhoff and von Neumann [1936]). However, in order to guarantee (via a convenient representation theorem) that the notion of probability is appropriately formulated in this new setting, grouptheoretic notions have to be introduced (Bub [1981]). So Weyls emphasis on grouptheoretic techniques was eventually vindicated, and the centrality of group theory in von Neumanns later approach to QM becomes manifest.
What is the significance of this? I shall explore one consequence of this situation to recent debates about structural realism (SR) and empiricism in physics (Worrall [1989], Ladyman [1998], and French [1999]). It is tempting for the structural realist to employ the centrality of group theory as an argument for SR. After all, by using group theory and symmetry arguments, representation theorems are established, and given the structural invariance guaranteed by these theorems, they indicate the (mathematical) structures that the structural realist should be realist about.
I will argue that, although persuasive, this line of argument should be resisted. Firstly, it is hard to reconcile the realist reading of Weyl and von Neumann that is presupposed here with their antirealist attitudes towards mathematics and physics. Secondly, from the fact that certain mathematical tools (grouptheoretic techniques) are useful for solving a given problem (to establish convenient representation theorems in QM), it doesnt follow that we have to believe that the structures generated by these techniques truly describe the world. Usefulness and problemsolving abilities are pragmatic factors (van Fraassen [1989] and [1991]), and they are best understood in the context of a structural empiricist view (Bueno [1997], [1999] and [2000]).
2. symmetry, representation and problemsolving
The notion of symmetry has widespread use in contemporary physics. A number of important results in quantum mechanics and relativity theory have been established on the basis of symmetry considerations. This is not by chance. Symmetry principles have important heuristic consequences, providing insightful problemsolving techniques (see van Fraassen [1989] and [1991]). The notion of symmetry also has different meanings, depending on the context in which it is used. The meaning of symmetry ranges from a broad, general modeltheoretic sense (that can be used to systematize the different uses of the term) to more specific senses, restricted to particular domains of physics. In this section, I shall briefly review the general sense of symmetry. In this general sense, symmetry is a transformation that leaves the relevant structure invariant. Relevance is, of course, a contextual matter, and this generates more specific forms of symmetries, which depend on what is taken to be relevant in a given context. Later we will have opportunity to contrast the general sense of symmetry with its more specific meanings.
If we characterize the notion of symmetry in this way (as a transformation that leaves the relevant structure invariant), it becomes clear that symmetry bears a close connection with representation. In order for one even to formulate the notion of symmetry, it is presupposed that some form of representation is adopted, since we need some notion of representation to talk about relevant structure. In other words, it is only in a representational context that symmetry has its bite. Since the general notion of symmetry seems to presuppose a modeltheoretic context, and given that the this context also provides a fruitful framework to articulate the notion of representation, I shall start the discussion by recalling a few features of the semantic (or modeltheoretic) approach to science, in particular in the form articulated by van Fraassen.
According to van Fraassen, to present a scientific theory is to specify a class of structures, its models (see van Fraassen [1980], [1989] and [1991]). These models are used to represent, in particular, the states of a certain physical system. The states are characterized by physical magnitudes (observables) that pertain to the system and can take certain values (van Fraassen [1991], p. 26). In classical mechanics, for instance, the history of a system (its evolution in time) can be represented as a trajectory in the space of possible states of the system (its statespace). Such a trajectory is a map from time to the statespace. As an example, consider a classical mechanical system (see Varadarajan [1968]). Its states can be completely specified by a 2ntuple (x1,..., xn, p1,..., pn), where (x1,..., xn) represents the configuration and (p1,..., pn) the momentum vector of the system in a given time instant. Thus, the possible states of the system are represented by points of the open set (n((n (where (n is the ndimensional space of ntuples of real numbers). The systems evolution is then determined by its Hamiltonian. In other words, if the state of the system at the time t is represented by (x1(t),...,xn(t), p1(t),..., pn(t)), the functions xi(t), pi(t) (with 1 ( i ( n) satisfy the equations:
(1) dxi/dt = (H/(pi, and dpi/dt = (H/(xi (with 1 ( i ( n).
The set of all possible states of the system satisfying (1) gives us its statespace.
What is important in this representation is that it provides an immediate way of studying logical relations between statements about the systems states. Van Fraassen calls these statements, whose truth conditions are represented in the models of the system, elementary statements ([1991], pp. 2930). As an example, let A be the statement the system X has state s at time t. Now, A is true exactly if the state of X at t is (represented by a point) in the region SA of its statespace. In other words, the statespace provides information about the physical system X. The crucial point for van Fraassen is that the family of statements about X receives some structure from the geometric character of the statespace. For instance, in quantum mechanics, if A attributes a pure state to X, the region SA it determines in the statespace is a subspace. In this way, by investigating the structure of the statespace, one investigates the logic of elementary statements (van Fraassen [1991], p. 30). Of particular interest are of course the following logical relationships: (i) the elementary statement A implies the elementary statement B if SA is a subset of SB; (ii) C is the conjunction of A and B if SC is the g.l.b. (greatest lower bound), with respect to the implication relation, of SA and SB.
Van Fraassens statespaces represent information about structure in science; focusing, in particular, on empirical information and the study of the underlying logic of statements about physical systems.
But what is the point of these representations? In a nutshell, to solve problems. This is quite clear in van Fraassens case, and he stresses, in particular, the heuristic role of symmetry in theory construction (see van Fraassen [1989], and [1991], pp. 2126). The idea is that in order to solve a problem we have first to devise an adequate representation of it ( a model of the situation described by the problem. This model introduces certain variables, and what we are looking for is a rule, a function, from certain inputs (the data of the problem) to one, and usually only one, output (the problems solution). The representation of the problem brings certain symmetries, which are crucial for its solution. Now, it often happens that we may not know how to solve a problem P1 (stated in terms of the model we initially devised), but we may well know how to solve a related problem P2. One way of solving P1 is then to show that it is essentially P2. This requires that we provide a transformation (lets call it T) from P1 into P2 that leaves P1 essentially the same. What this means is that the structure that characterizes P1 (the relationships between its variables) is preserved by T. The underlying heuristic move is then clear: the same problems have the same solutions (van Fraassen [1991], p. 25). The symmetries of the problem, being preserved by T, allow this heuristic move to get off the ground.
The important point here is that symmetries are, in general, transformations that leave the relevant structure invariant. In this way, the two components of the problem of structure meet: by specifying certain interrelationships between structures (appropriate symmetries), we have a powerful technique to solve problems, which in turn increases our ways of representing information in mathematics and in science. Thus, structure can play a decisive role in solving problems in both science and mathematics. As a result, it comes as a surprise that this point is not stressed more often in the context of structuralism (be it platonist or nominalist), since I think here lies one of the strengths of the program. One of the advantages of moving to a structuralist stance derives from the rich heuristic setting it provides ( not only for the scientist and the mathematician (since structures have a crucial role in problemsolving), but also for the philosopher (given that, moving one level up, structures also allow us to represent how problems are solved). This is a point we shall have opportunity to consider in more detail below.
3. group theory, hilbert spaces and quantum mechanics
As is well known, in 1925 and 1926, two entirely distinct formulations of quantum mechanics were devised. On the one hand, Heisenberg, Born, Jordan and Dirac formulated, in a series of papers in 1925, the socalled matrix mechanics; on the other hand, in 1926, also in a series of works, Schrdinger articulated wave mechanics. The two formulations couldnt be more different. Matrix mechanics is expressed in terms of a system of matrices defined by algebraic equations, and the underlying space is discrete. Wave mechanics is articulated in a continuous space, which is used to describe a fieldlike process in a configuration space governed by a single differential equation. However, despite these differences, the two theories seemed to have the same empirical consequences. For example, they gave coinciding energy values for the hydrogen atom.
It was crucial to establish that the two theories were equivalent. Partially successful attempts in this direction were made by Schrdinger and Dirac. The attempts didnt succeed since, in Schrdingers case, what was established was only a mapping assigning one matrix to each waveoperator, but not the converse (Schrdinger [1926]; for a discussion, see Muller [1997], pp. 4958). Dirac, in turn, did establish the equivalence between the two theories, but his method required the introduction of the socalled (functions, which are inconsistent (Dirac [1930]).
It is in this context that von Neumann formulated his approach in terms of Hilbert spaces (von Neumann [1932]). Von Neumann noticed that the mathematical spaces used in the formulation of wave and matrix mechanics were very different (one space was discrete, the other continuous). However, if we consider the functions defined over these spaces, we obtain particular cases of Hilbert spaces. This suggested that the latter provide the appropriate framework to develop quantum mechanics. And von Neumanns celebrated equivalence proof established an isomorphism between such Hilbert spaces.
But there was a further reason for the introduction of Hilbert spaces. They provide a straightforward setting for the introduction of probability in quantum mechanics. This is a crucial issue, given the irreducibly probabilistic character of the theory. And in fact, in a paper written in 1927 with Hilbert and Nordheim, the problem of introducing probability into quantum mechanics had been explicitly addressed (see Hilbert, Nordheim and von Neumann [1927]). The approach was articulated in terms of the notion of the amplitude of the density for relative probability (for a discussion, see Rdei [1997]). But it faced a serious technical difficulty (which was acknowledged by the authors): the assumption was made that every operator is an integral operator, and therefore Diracs problematic function had to be assumed. As a result, an entirely distinct account was required to adequately introduce probability in quantum mechanics. And this led to von Neumanns use of Hilbert spaces.
In other words, by 1927 quantum mechanics could be seen as a semicoherent assemblage of principles and rules for applications. And von Neumann provided a systematic approach to overcome this. But around the same time Weyl provided a different approach. His 1931 book was an attempt to impose a degree of coherence via the introduction of grouptheoretic techniques. Weyls approach, similarly to von Neumanns, was concerned with foundational questions, although not exactly the same sort of questions. As Mackey points out ([1993], p. 249), Weyl distinguished two questions in the foundations of quantum mechanics (Weyl [1927]): (a) How does one arrive at the selfadjoint operators which correspond to various concrete physical observables? (b) What is the physical significance of these operators, i.e. how are physical statements deduced from such operators? According to Weyl, (a) has not been adequately treated, and is a deeper question; whereas (b) has been settled by von Neumanns formulation of quantum mechanics in terms of Hilbert spaces. But to address (a) Weyl needed a different framework altogether; he needed group theory.
According to Weyl, group theory reveals the essential features which are not contingent on a special form of the dynamical laws nor on special assumptions concerning the forces involved ([1931], p. xxi). And he continues:
Two groups, the group of rotations in 3dimensional space and the permutation group, play here the principal role, for the laws governing the possible electronic configurations grouped about the stationary nucleus of an atom or an ion are spherically symmetric with respect to the nucleus, and since the various electrons of which the atom or ion is composed are identical, these possible configurations are invariant under a permutation of the individual electrons. (ibid.; italics omitted.)
In particular, the theory of group representation by linear transformations, the mathematically most important part of group theory, is exactly what is necessary for an adequate description of the quantum mechanical relations (ibid.). As Weyl establishes, all quantum numbers, with the exception of the socalled principal quantum number, are indices characterizing representations of groups (ibid.; italics omitted). Moreover, as Weyl shows, Heisenbergs uncertainty relations and Paulis exclusion principle can be obtained via group theory (for a discussion, see Mackey [1993], and French [1999] and [2000]). Given these considerations, Weyls conclusion does not seem surprising: We may well expect that it is just this part of quantum physics [the one formulated grouptheoretically] which is most certain of a lasting place (ibid.).
But it is not only in the foundations of quantum mechanics that group theory has a decisive role; it is also crucial for the application of quantum theory. This role was explored, in particular, by Wigner (see Wigner [1931]). Here we find an important difference between Weyls and Wigners use of grouptheoretic techniques in quantum mechanics (see Mackey [1993], and French [2000]). Weyl explored group theory at the foundational level (indicating how to obtain grouptheoretically quantum mechanical principles). Wigner, on the other hand, was particularly concerned with the application of quantum mechanics (this is the main theme of his 1931 book). And as he argues, we cannot apply Schrdingers equation directly, but we need to introduce group theoretic results to obtain the appropriate idealizations (see French [2000]). In Wigners own words:
The actual solution of quantum mechanical equations is, in general, so difficult that one obtains by direct calculation only crude approximations to the real solutions. It is gratifying, therefore, that a large part of the relevant results can be deduced by considering the fundamental symmetry operations. (Wigner [1931], p. v)
In particular, group theory allows the physicist to overcome the mathematical intractability of the many body problem (involved in a system with more than two electrons), and in this way, it allows to relate quantum mechanics to the data (French [2000]). Thus, group theory enters both at the foundational level and at the level of application.
However, in order for the use of group theory to get off the ground, one has to adopt the prior reformulation of quantum mechanics in terms of Hilbert spaces. It is from the representation of the state of a quantum system in terms of Hilbert spaces that a grouptheoretic account of symmetric and antisymmetric states can be provided (see Weyl [1931], pp. 185191). The grouptheoretic approach also depends on the Hilbert space representation to introduce probability into quantum mechanics. Moreover, at the application level, despite the need for idealizations to apply quantum theory, Schrdingers equation is still crucial (putting constraints on the accepted phenomenological models), and the representation of states of a quantum system in terms of Hilbert spaces has to be used. In other words, group theory is not an independent mathematical framework to articulate quantum mechanics ( the Hilbert spaces representation is crucial. Roughly speaking, we can say that von Neumanns Hilbert spaces representation is sandwiched between Weyls foundational use of group theory and Wigners application program. Hence, there is a close interdependence between group theory and Hilbert spaces theory in the proper formulation of quantum mechanics.
4. trying to make sense of this predicament:
the move toward structural realism
Structural realism revisited. According to structural realism, scientific theories capture the structure of the world, and that is why they are so successful in predicting novel phenomena. However, capturing the structure of the world is compatible with introducing radically different ontologies (the same structure may be instantiated in different ways). So our theories may well be false, as far as their ontological commitments are concerned.
This is certainly an attractive idea. Recent developments of this proposal can be found in the works of a number of people; in particular, Worrall [1989], Chiappin [1989], Zahar [1996], Ladyman [1998], French [1999], and French and Ladyman [2000].
As Ladyman [1998] indicates, there are two formulations of structural realism. An epistemic formulation (that, to some extent, can be found in Worrall [1989]), according to which (1) structural realism is an epistemological claim about our knowledge of structural preservation in science, following Poincar and Russell (all that we can know about the world is structure), and (2) this claim is formulated in terms of the syntactic approach. According to the ontological reformulation (the one favored by Ladyman), (1() structural realism is a metaphysical claim (all there is to the world is structure), and (2() this claim is better formulated via the semantic approach.
In both cases, there is a clear realist component, since scientific theories uncover the truth about the world, by capturing the structure of the universe.
Structural realism and quantum mechanics. Given the need for both group theory and Hilbert spaces in the formulation of quantum mechanics, which of the resulting structures should the structural realist be realist about: (i) to both grouptheoretic structures and those derived from Hilbert spaces, (ii) neither of them, (iii) only one of them, or (iv) the other? I shall argue that the commitment to either (iv) or (iii) is arbitrary, (ii) is incoherent with realism about structure, and (i) is ontologically problematic. Therefore, independently of the option taken, the structural realist runs into trouble. Now, since quantum mechanics is one of our best tested theories, if a philosophical proposal is unable to accommodate this theory, it faces a serious problem.
Why would it be arbitrary for the structural realist to believe, say, only on the grouptheoretic structures used in quantum mechanics? Because, as noticed above, the grouptheoretic representation depends on the formulation provided by Hilbert spaces. The representation theorem relating irreducible representations of a group and subspaces of a Hilbert space depends on the adequacy of the latter as a procedure to represent states of a quantum system. Moreover, the introduction of probability in quantum mechanics is typically done in terms of Hilbert spaces; thus, group theory depends on the latter. Now, given the crucial role of the Hilbert spaces representation, which is acknowledged and explored by Weyl, it would be ad hoc for the structural realist only to believe on the structures arising from group theorys side. Thus option (iv) is no option.
On the other hand, it would be similarly ad hoc for the structural realist only to believe on the Hilbert space formalism. As is stands, the latter doesnt provide crucial ontological questions, such as the nature of quantum particles, whereas, as we saw, it is straightforward to claim (following the grouptheoretic approach) that such particles are certain sets of invariants. In other words, given the close interplay between group theory and Hilbert spaces in quantum mechanics, it is arbitrary to believe in one and not in the other. This excludes option (iii).
What if the structural realist withholds belief in both structures altogether? In other words, what if he or she is neither a realist about Hilbert spaces nor about grouptheoretic structures? In that case, the structural realist would relinquish realism. The point of the position is to defend realism about structures, and that is exactly what this option denies (after all, Hilbert spaces and grouptheoretic structures are the crucial structures involved in quantum mechanics). Thus, since option (ii) is incompatible with realism about structures, it is unacceptable to the structural realist.
The final option is then to be realist about both Hilbert spaces and grouptheoretic structures employed in quantum mechanics. But there are a number of difficulties with this move. Firstly, there is no single picture of the quantum world associated with the two kinds of structure. We need an interpretation of quantum mechanics to explore this sort of question. But the fact that there is a plurality of interpretations is enough to indicate that the quantum mechanical formalism is unable to settle the question about the picture provided by the theory. In other words, it is not enough simply to cling to the formalism and claim that the structures that satisfy the latter, whatever they are, are the ones the structural realist believes. For without an interpretation, no clear picture emerges from the formalism.
However, if the structural realist moves beyond the formalism and actually articulates an interpretation, nonstructural factors will have to be introduced. No interpretation is purely structural. We have to assert things like: That part of the theory refers to that. In other words, we need a pragmatic move to run any interpretation. As a result, we have moved beyond structuralism. Moreover, if we consider particular interpretations of quantum mechanics, they incorporate nonstructural components. For example, the distinction made in the modal interpretation between value states and dynamic states is not structural. In van Fraassens characterization, the value state is fully specified by stating which observables have values, and which they are; the dynamic state is completely determined by stating how the system will develop if acted upon in a particular way, and how it will develop if isolated (van Fraassen [1991], p. 275). Value states and dynamics states are not representations of quantum states (they are not an interpretation of the system); they are states of the system. In this sense, although they can be represented by structures, such states are not structures themselves.
Secondly, quantum mechanics is certainly more unified with the introduction of group theory, and some ontological questions (e.g. about quantum particles) can be better addressed grouptheoretically. However, Hilbert spaces are also needed (for instance, as noticed above, to introduce probability in quantum theory). But the ontological status of these spaces is far less clear. Such spaces certainly provide an important way of representing the states of a quantum system; but why should this be an argument for the existence of anything like a multidimensional Hilbert space in reality? Why is the usefulness of a representation an argument for its truth? It goes without saying that this conflates pragmatic and epistemic reasons. And even the realist should be careful in not conflating these reasons (it requires a great deal of anthropocentrism to warrant this conflation!).
Therefore, since the last option, (i), is not available to the structural realist, I conclude that there are no grounds to claim that the approach accommodates quantum mechanics.
Scientific change and structural losses. If we still consider quantum mechanics, I think additional problems emerge. Shortly after the publication of his 1932 book, von Neumann became increasingly dissatisfied with the Hilbert space formalism (for an illuminating discussion of this episode, see Rdei [1997]). As we noticed above, one of his major motivations for the use of Hilbert spaces derived from the need for introducing probability in quantum theory. Now, as part of the development of his ideas on quantum logic (see Birkhoff and von Neumann [1936]), von Neumann noticed that the geometry associated with a quantum system whose degrees of freedom is finite is a projective geometry. However, if we consider a system whose degrees are infinite, the associated geometry changes dramatically. It actually leads to a new kind of mathematical structure, constructed by von Neumann in his formulation of continuous geometry (see von Neumann [1981]). Now, what happens is that, for such infinite quantum systems, the Hilbert space formalism doesnt generate the appropriate notion of probability (see Rdei [1997]). What is required is an entirely now conceptual setting: in terms of von Neumanns type II1 algebras.
Thus, according to von Neumann, one should move from Hilbert spaces to von Neumann algebras to properly accommodate the introduction of probability in quantum mechanics. However, the resulting structures ( on the one hand, what is called a type II1 factor algebra, on the other, the algebra of bounded operators in a Hilbert space ( are not equivalent: the former is modular and nonatomic, the latter is modular, atomic and nondistributive if the dimension of the Hilbert space is greater or equal to 2 (see Rdei [1997], p. 505). But the two algebras lead to the same empirical results. In this sense, both frameworks are adequate for the development of quantum mechanics. Conceptually, however, the type II1 algebra is better, since it leads to the appropriate probabilities for quantum systems (for a discussion, see Rdei [1997]). Now, by moving to this algebra, a substantial structural change happens, since the atomicity of the algebra of bounded operators cannot be recovered. And it is unclear how the structural realist can accommodate this structural loss.
Laudan [1996] indicates a number of other cases of structural losses in science. For example, in the move from Cartesian to Newtons mechanics, all the vortices of Descartess theory have been lost. Although they explained the direction of the planets orbits, the structural realist may say that such vortices were essentially a metaphysical theory, which wasnt well confirmed enough. However, given the crucial role played, in quantum mechanics, by the algebra of bounded operators in a Hilbert space, the same reply cant be used here. I think this poses a further problem for structural realism.
5. finding an alternative: structural empiricism?
The difficulties considered above indicate the possibility of an alternative account. It shares with structural realism the emphasis on structure. But it completely abandons realism. Structures are employed not to capture the structure of the world; they only help us to represent the observable level. Thus the account sides with an empiricist interpretation, and can be called structural empiricism (see Bueno [1999], and van Fraassen [2000]).
By dropping the realist component, all the problems above can be overcome. After all, without realism, structures dont have to be cashed out metaphysically. Moreover, with empiricism, a clear approach to the issue of identity and individuality of quantum particles is taken (see van Fraassen [1991]). Given that metaphysics is resisted in this approach, there is no incompatibility with it and the (nonmetaphysical) nature of structure. Furthermore, without the realist component, the structural empiricist has no difficulties in accommodating the role of group theory and Hilbert spaces in quantum mechanics. The empiricist withholds belief in anything that goes beyond the observable. Moreover, the existence of structural losses in science, being an argument against realism, is very much well taken by the structural empiricist.
In this way, structural empiricism doesnt face the difficulties that challenge structural realism. It may face others; but that remains to be seen.
references
Birkhoff, G., and von Neumann, J. [1936]: The Logic of Quantum Mechanics, Annals of Mathematics 37, pp. 823843. (Reprinted in von Neumann [1962], pp. 105125.)
Bub, J. [1981]: Hidden Variables and Quantum Logic ( A Sceptical Review, Erkenntnis 16, pp. 275293.
Bueno, O. [1997]: Empirical Adequacy: A Partial Structures Approach, Studies in History and Philosophy of Science 28, pp. 585610.
Bueno, O. [1999]: What is Structural Empiricism? Scientific Change in an Empiricist Setting, Erkenntnis 50, pp. 5985.
Bueno, O. [2000]: Empiricism, Mathematical Change and Scientific Change, Studies in History and Philosophy of Science 31, pp. 269296.
Butterfield, J., and Pagonis, C. (eds.) [1999]: From Physics to Philosophy. Cambridge: Cambridge University Press.
Chiappin, J.R.N. [1989]: Duhems Theory of Science: An Interplay Between Philosophy and History of Science. Ph.D. thesis, University of Pittsburgh.
Dirac, P.A.M. [1930]: The Principles of Quantum Mechanics. Oxford: Clarendon Press.
French, S. [1999]: Models and Mathematics in Physics: The Role of Group Theory, in Butterfield and Pagonis (eds.) [1999], pp. 187207.
French, S. [2000]: The Reasonable Effectiveness of Mathematics: Partial Structures and the Application of Group Theory to Physics, forthcoming in Synthese.
French, S., and Ladyman, J. [2000]: Remodelling Structural Realism, forthcoming in Synthese.
Greffe, J.L., Heinzmann, G., Lorenz, K. (eds.) [1996]: Henri Poincar: Science and Philosophy. Berlin: Akademie Verlag.
Hilbert, D., Nordheim, L., and von Neumann, J. [1927]: ber die Grundlagen der Quantenmechanik, Mathematische Annalen 98, pp. 130. (Reprinted in von Neumann [1961].)
Ladyman, J. [1998]: What is Structural Realism?, Studies in History and Philosophy of Science 29, pp. 409424.
Laudan, L. [1996]: Beyond Positivism and Relativism. Boulder: Westview Press.
Mackey, G.W. [1993]: The Mathematical Papers, in Wigner [1993], pp. 241290.
Muller, F.A. [1997]: The Equivalence Myth of Quantum Mechanics, Studies in History and Philosophy of Modern Physics 28, pp. 3561; 219247.
Murray, F.J., and von Neumann, J. [1936]: On Rings of Operators, Annals of Mathematics 37, pp. 116229.
Rdei, M. [1997]: Why John von Neumann did not Like the Hilbert Space Formalism of Quantum Mechanics (and What He Liked Instead), Studies in History and Philosophy of Modern Physics 28, pp. 493510.
Rdei, M. [1998]: Quantum Logic in Algebraic Approach. Dordrecht: Kluwer Academic Publishers.
Schrdinger, E. [1926]: ber das Verhltnis der HeisenbergBornJordanschen Quantenmechanik zu der meinen, in Schrdinger [1927], pp. 4561.
Schrdinger, E. [1927]: Collected Papers on Wave Mechanics. (Translated by J.F. Shearer.) New York: Chelsea.
van Fraassen, B.C. [1980]: The Scientific Image. Oxford: Clarendon Press.
van Fraassen, B.C. [1989]: Laws and Symmetry. Oxford: Clarendon Press.
van Fraassen, B.C. [1991]: Quantum Mechanics: An Empiricist View. Oxford: Clarendon Press.
Varadarajan, V.S. [1968]: Geometry of Quantum Theory. (Volume 1.) Princeton: van Nostrand.
von Neumann, J. [1932]: Mathematical Foundations of Quantum Mechanics. (The English translation, by Robert T. Beyer, of the original German edition was first published in 1955.) Princeton: Princeton University Press.
von Neumann, J. [1961]: Collected Works, vol. I. Logic, Theory of Sets and Quantum Mechanics. (Edited by A.H. Taub.) Oxford: Pergamon Press.
von Neumann, J. [1962]: Collected Works, vol. IV. Continuous Geometry and Other Topics. (Edited by A.H. Taub.) Oxford: Pergamon Press.
von Neumann, J. [1981]: Continuous Geometries with a Transition Probability, Memoirs of the American Mathematical Society 34, No. 252, pp. 1210.
Weyl, H. [1927]: Quantenmechanik und Gruppentheorie, Zeit. fr Phys. 46, pp. 146.
Weyl, H. [1931]: The Theory of Groups and Quantum Mechanics. (Translated from the second, revised German edition by H.P. Robertson. The first edition was published in 1928.) New York: Dover.
Wigner, E.P. [1993]: The Collected Works of Eugene Paul Wigner. The Scientific Papers, volume 1. (Edited by Arthur S. Wightman.) Berlin: SpringerVerlag.
Worrall, J. [1989]: Structural Realism: The Best of Both Worlds?, Dialectica 43, pp. 99124.
Zahar, E. [1996]: Poincars Structural Realism and his Logic of Discovery, in Greffe et al. (eds.) [1996], pp. 4568.
My thanks go to Steven French, James Ladyman, and Bas van Fraassen for really helpful discussions.
For a detailed critical discussion, and references, see Muller [1997].
Diracs 1930 work represents a further attempt to lay out a coherent basis for the theory. However, as von Neumann perceived, neither Weyls nor Diracs approaches offered a mathematical framework congenial for the introduction of probability at the most fundamental level, and (initially at least) this was one of the major motivations for the introduction of Hilbert spaces.
As French points out: the fundamental relationship underpinning [some applications of group theory to quantum mechanics] is that which holds between the irreducible representations of the group and the subspaces of the Hilbert space representing the states of the system. In particular, if the irreducible representations are multidimensional then the appropriate Hamiltonian will have multiple eigenvalues which will split under the effect of the perturbation (French [2000]). In this way, under the action of the permutation group the Hilbert space of the system decomposes into mutually orthogonal subspaces corresponding to the irreducible representations of this group (ibid.; see also Mackey [1993], pp. 242247). As French notes, of these representations the most well known are the symmetric and antisymmetric, corresponding to BoseEinstein and FermiDirac statistics respectively, but others, corresponding to socalled parastatistics are also possible (ibid.).
How could the world possibly be the way this representation (in terms of Hilbert spaces and group theory) says it is? This is, of course, the typical foundational question (see van Fraassen [1991]). The way to answer this question is by providing an interpretation of quantum mechanics.
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BI'm still thinking about what to present in the symmetry workshop
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