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.
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.
We derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The basic setting is a set 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∈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.
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 Φ 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(⋅) on Φ, 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 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 Φ, to an irreducible invariant subspace 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(⋅), 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.
Essential elements of quantum theory are derived from an epistemic point of view, i.e., the viewpoint that thetheory has to do with what can be said about nature. This gives a relationship to statistical reasoning and to other areas of modelling and decision making. In particular, a quantum state can be defined from an epistemic point of view to consist of two elements: A (maximal) question about the value of some parameter together with the answer to that question. Quantization itself can be approached from the point of view of model reduction under symmetry.
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 conceptual variable is any variable defined by a person or by a group of persons. Such variables may be inaccessible, meaning that they cannot be measured with arbitrary accuracy on the physical system under consideration at any given time. An example may be the spin vector of a particle; another example may be the vector (position and momentum). In this paper, a space of inaccessible conceptual variables is defined, and group actions are defined on this space. Accessible functions are then defined on the same space. Assuming this structure, the basic Hilbert space structure of quantum theory is derived: Operators on a Hilbert space corresponding to the accessible variables are introduced; when these operators have a discrete spectrum, a natural model reduction implies a new model in which the values of the accessible variables are the eigenvalues of the operator. The principle behind this model reduction demands that a group action may also be defined also on the accessible variables; this is possible if the corresponding functions are permissible, a term that is precisely defined. The following recent principle from statistics is assumed: every model reduction should be to an orbit or to a set of orbits of the group. From this derivation, a new interpretation of quantum theory is briefly discussed: I argue that a state vector may be interpreted as connected to a focused question posed to nature together with a definite answer to this question. Further discussion of these topics is provided in a recent book published by the author of this paper.