By Badis Ydri (auth.)

These lecture notes supply a scientific advent to matrix types of quantum box theories with non-commutative and fuzzy geometries.

The e-book at the start makes a speciality of the matrix formula of non-commutative and fuzzy areas, by way of an outline of the non-perturbative remedy of the corresponding box theories. for example, the section constitution of non-commutative phi-four thought is taken care of in nice aspect, with a separate bankruptcy at the multitrace strategy. The final bankruptcy deals a common advent to non-commutative gauge theories, whereas appendices around out the text.

Primarily written as a self-study consultant for postgraduate scholars – with the purpose of pedagogically introducing them to key analytical and numerical instruments, in addition to invaluable actual types in functions – those lecture notes also will profit skilled researchers by means of supplying a reference advisor to the basics of non-commutative box thought with an emphasis on matrix types and fuzzy geometries.

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2 n;m 2 2 Q ; ˇ/ D exp. ˛ ; ˛/ D 0. In order for this to be true an entire function of the variables ˛ and ˇ defined precisely by the above double series must vanish on the domain given by ˇ D ˛ in C2 . A known result from complex analysis states that if an entire twovariable function vanishes on the domain ˇ D ˛ for all ˛ it must vanish identically Q everywhere in C2 . ˛ ; ˇ/ D 0 for all ˛ and ˇ and as a consequence the operator itself vanishes identically, viz AO D 0. It is a general result that the diagonal matrix elements of an operator AO (bounded1 or a polynomial in a and aC ) in the coherent states basis determine completely the operator.

D< ˛ ˇj exp. 33) The wave function < ˛j > determines the state j > completely. It is called the symbol of the state j >. This provides a functional realization of the Hilbert space H as we will now show. 35) is absolutely convergent for all z 2 C. The proof goes as follows. 36) p The ratio of the n C 1th term to the nth term is j˛j= n C 1 which goes to 0 as n ! 0. ˛/ is an entire or integral function. We note that only entire functions subjected to some growth restriction are in fact allowed.

Is an entire or integral function. We note that only entire functions subjected to some growth restriction are in fact allowed. ˛/ endowed with the above scalar product form a Hilbert space F which provides the so-called Fock-Bargmann representation of the Hilbert space H. ˛/. ˛/m r p 2 nCm e . 1/nCm n C m C 2 . 39) Obviously < njm >D 0 if n ¤ m and < njn >D 1 as it should be. 1 Heisenberg Algebra and Coherent States 27 Operator Symbols A large class of operators acting on the Hilbert space H can also be represented by functions called their symbols.