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arXiv:quant-ph/0411174 · AIP conference proceedings

Quantum theory as a statistical theory under symmetry

Inge Svein Helland

Published Jan 1, 2005• 3 citations
arXivHTMLPDF

Abstract

Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic quantum theory.A total parameter space Φ, equipped with a group G of transformations, gives the mental image of some quantum system, in such a way that only certain components, functions of the total parameter φ can be estimated. Choose an experiment/question a, and get from this a parameter space Λa, perhaps after some model reduction compatible with the group structure. As in statistics, it is important always to distinguish between observations and parameters, in particular between (minimal) observations ta and state variables (parameters) λa. Let Ka be the L2 ‐space of functions of ta, and let Ha be the image of Ka under the expectation operator of the model. The measure determining the L2‐spaces is the invariant measure under the maximal subgroup which induces a transformation on Λa.There is a unitary connection between Ka and Ha, and then under natural conditions between Ha and Hb for a and b are different. Thus there exists a common Hilbert space H such that Ha equals UaH for a unitary Ua. In agreement with the common formulation of quantum mechanics, the vectors of H are taken to represent the states of the system. The state interpretation is then: An ideal experiment Ea has been performed, and the result of this experiment, disregarding measurement errors, is that the parameter λa is equal to some fixed number λka. This essentially statistical construction leads under natural assumptions to the basic axioms of quantum mechanics, and thus implies a new statistical interpretation of this traditionally very formal theory. The probabilities are introduced via Born’s formula, and this formula is proved from general, reasonable assumptions, essentially symmetry assumptions.The theory is illustrated by a simple macroscopic example, and by the example of a spin 12 particle. As a last example we show a connection to inference between related macroscropic experiments under symmetry. The result of this last example seems to have potential for being generalized considerably.

DOI
10.1063/1.1874567

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Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic quantum theory.A total parameter space Φ, equipped with a group G of transformations, gives the mental image of some quantum system, in such a way that only certain components, functions of the total parameter φ can be estimated. Choose an experiment/question a, and get from this a parameter space Λa, perhaps after some model reduction compatible with the group structure. As in statistics, it is important always to distinguish between observations and parameters, in particular between (minimal) observations ta and state variables (parameters) λa. Let Ka be the L2 ‐space of functions of ta, and let Ha be the image of Ka under the expectation operator of the model. The measure determining the L2‐spaces is the invariant measure under the maximal subgroup which induces a transformation on Λa.There is a unitary connection between Ka and Ha, and then under natural conditions between Ha and Hb for a and b are different. Thus there exists a common Hilbert space H such that Ha equals UaH for a unitary Ua. In agreement with the common formulation of quantum mechanics, the vectors of H are taken to represent the states of the system. The state interpretation is then: An ideal experiment Ea has been performed, and the result of this experiment, disregarding measurement errors, is that the parameter λa is equal to some fixed number λka. This essentially statistical construction leads under natural assumptions to the basic axioms of quantum mechanics, and thus implies a new statistical interpretation of this traditionally very formal theory. The probabilities are introduced via Born’s formula, and this formula is proved from general, reasonable assumptions, essentially symmetry assumptions.The theory is illustrated by a simple macroscopic example, and by the example of a spin 12 particle. As a last example we show a connection to inference between related macroscropic experiments under symmetry. The result of this last example seems to have potential for being generalized considerably.
The aim of this paper is to show a connection between an extended theory of statistical experiments on the one hand and the foundation of quantum theory on the other hand. The main aspects of this extension are: One assumes a hyperparameter space ΦΦΦ common to several potential experiments, and a basic symmetry group G associated with this space. The parameter θ_{a} of a single experiment, looked upon as a parametric function θa(⋅)θ_{a}(\cdot)θa​(⋅) on ΦΦΦ, is said to be permissible if G induces in a natural way a new group on the image space of the function. If this is not the case, it is arranged for by changing from G to a subgroup GaG_{a}Ga​. The Haar measure of this subgroup (confined to the spectrum; see below) is the prefered prior when the parameter is unknown. It is assumed that the hyperparameter itself can never be estimated, only a set of parametric functions. Model reduction is made by restricting the space of complex `wave' functions, also regarded as functions on ΦΦΦ, to an irreducible invariant subspace M\mathcal{M}M under G. The spectrum of a parametric function is defined from natural group-theoretical and statistical considerations. We establish that a unique operator can be associated with every parametric functions θa(⋅)θ_{a}(\cdot)θa​(⋅), and essentially all of the ordinary quantum theory formalism can be retrieved from these and a few related assumptions. In particular the concept of spectrum is consistent. The scope of the theory is illustrated on the one hand by the example of a spin 1/2 particle and a related EPR discussion, on the other hand by a simple macroscopic example.
We derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The basic setting is a set A\mathcal{A}A of incompatible experiments, and a transformation group G on the cartesian product Π of the parameter spaces of these experiments. The set of possible parameters is constrained to lie in a subspace of Π, an orbit or a set of orbits of G. Each possible model is then connected to a parametric Hilbert space. The spaces of different experiments are linked unitarily, thus defining a common Hilbert space H. A state is equivalent to a question together with an answer: the choice of an experiment a∈Aa\in\mathcal{A}a∈A plus a value for the corresponding parameter. Finally, probabilities are introduced through Born’s formula, which is derived from a recent version of Gleason’s theorem. This then leads to the usual formalism of elementary quantum mechanics in important special cases. The theory is illustrated by the example of a quantum particle with spin.
The aim of the paper is to derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The main extensions, which also can be motivated from an applied statistics point of view, relate to symmetry, the choice between complementary experiments and hence complementary parametric models, and use of the fact that there for simple systems always is a limited experimental basis that is common to all potential experiments. Concepts related to transformation groups together with the statistical concept of sufficiency are used in the construction of the quantummechanical Hilbert space. The Born formula is motivated through recent analysis by Deutsch and Gill, and is shown to imply the formulae of elementary quantum probability/ quantum inference theory in the simple case. Planck's constant, and the Schroedinger equation are also derived from this conceptual framework. The theory is illustrated by one and by two spin 1/2 particles; in particular, a statistical discussion of Bell's inequality is given. Several paradoxes and related themes of conventional quantum mechanics are briefly discussed in the setting introduced here.
Taking several statistical examples, in particular one involving a choice of experiment, as points of departure, and making symmetry assumptions, the link towards quantum theory developed in Helland (2005a,b) is surveyed and clarified. The quantum Hilbert space is constructed from the parameters of the various experiments using group representation theory. It is shown under natural assumptions that a subset of the set of unit vectors of this space, the generalized coherent state vectors, can be put in correspondence with questions of the kind: What is the value of the (complete) parameter? - together with a crisp answer to that question. Links are made to statistical models in general, to model reduction of overparametrized models and to the design of experiments. It turns out to be essential that the range of the statistical parameter is an invariant set under the relevant symmetry group.
A version of quantum theory is derived from a set of plausible assumptions related to the following general setting: For a given system there is a set of experiments that can be performed, and for each such experiment an ordinary statistical model is defined. The parameters of the single experiments are functions of a hyperparameter, which defines the state of the system. There is a symmetry group acting on the hyperparameters, and for the induced action on the parameters of the single experiment a simple consistency property is assumed, called permissibility of the parametric function. The other assumptions needed are rather weak. The derivation relies partly on quantum logic, partly on a group representation of the hyperparameter group, where the invariant spaces are shown to be in 1-1 correspondence with the equivalence classes of permissible parametric functions. Planck's constant only plays a role connected to generators of unitary group representations.
A different approach towards quantum theory is proposed in this paper. The basis is taken to be conceptual variables, physical variables that may be accessible or inaccessible, i.e., it may be possible or impossible to assign numerical values to them. In an epistemic process, the accessible variables are just ideal observations as observed by an actor or by some communicating actors. Group actions are defined on these variables, and using group representation theory this is the basis for developing the Hilbert space formalism here. Operators corresponding to accessible conceptual variables are derived as a result of the formalism, and in the discrete case it is argued that the possible physical values are the eigenvalues of these operators. The Born formula is derived under specific assumptions. The whole discussion here is a supplement to the author's book [1]. The interpretation of quantum states (or eigenvector spaces) implied by this approach is as focused questions to nature together with sharp answers to those questions. Resolutions if the identity are then connected to the questions themselves; these may be complementary in the sense defined by Bohr. This interpretation may be called a general epistemic interpretation of quantum theory. It is similar to Zwirn's recent Convival Solipsism, and also to QBism, and more generally, can be seen as a concrete implementation of Rovelli's Relational Quantum Mechanics. The focus in the present paper is, however, as much on foundation as on interpretation. But the simple consequences of an epistemic interpretation for some so called quantum paradoxes are discussed. Connections to statistical inference theory are discussed in a preliminary way, both through an example and through a brief discussion of quantum measurement theory.