We analyze in detail the quantization of a simple noncommutative model of spontaneous symmetry breaking in zero dimensions taking into account the noncommutative setting seriously. The connection to the counting argument of Feynman diagrams of the corresponding theory in four dimensions is worked out explicitly. Special emphasis is put on the motivation as well as the presentation of some well-known basic notions of quantum field theory which in the zero-dimensional theory can be studied without being spoiled by technical complications due to the absence of divergencies. C 2002 Elsevier Science (USA) CONTENTS 1. Introduction. 2. Zero-dimensional ϕ 4 -theory. 2.1. n-point functions of zero-dimensional ϕ 4 -theory. 2.2. The various generating functionals. 2.3. Renormalization. 3. A model with spontaneous symmetry breaking. 3.1. Definition of the model and the problems with the Goldstone particle. 3.2. Proper definition of the theory and recursive determination of (n) . 3.3. Feynman propagators and "counting" of diagrams. 3.4. Renormalization. 4. Noncommutative formulation of a zero-dimensional Higgs model and its quantization. 4.1. Noncommutative gauge theories. 4.2. Zero-dimensional noncommutative model of SSB and motivation for the matrix calculus. 4.3. The matrix calculus. 4.4. Differential equations for Z , W , and . 4.5. Recursive determination of . 5. Ghost contributions within the noncommutative formulation. 5.1. Problems with the noncommutative integration of ghost terms. 5.2. Higher orders of including ghost contributions. 6. Conclusion and outlook. Appendixes. A. Kummer's equation and the Function 1 F 1 . B. Integrals with Gaussian weights. C. Determination of for ϕ 4 -theory in zero dimensions. D. Determination of ren for zero-dimensional ϕ 4 -theory. E. Naive treatment of the model with SSB. F. Correct treatment of the model with SSB. G. Proofs of the rules for the matrix derivative. H. The full differential equation for within the matrix formulation. I. Higher orders of within the matrix formulation. J. The full set of differential equations for .