A Linear Representation of Form Invariance

Abstract

According to the principles of group theory, every group $\mathcal G$ of transformations $G_{\rho }$ can be represented by an isomorphic group $\mathcal{G}_L$ of linear transformations $\mathbf{G}_{\rho }$ of a vector space. The transformations of event and parameter, introduced in Chap. 6, are nonlinear. The probability distributions p and w are not elements of a vector space. Is it possible to define form invariance in terms of linear transformations of a vector space? This would give the possibility of distinguishing classes of transformations, such as orthogonal or unitary ones, that can occur in form invariance, from other ones that cannot occur. Section 6.4 has given a hint of the way of constructing linear representations: one must express probabilities by probability amplitudes. The present chapter pursues this route and defines linear representations of models with the symmetries of translation and dilation.