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Lectures on Super Analysis Lectures on Super Analysis —– Why necessary and What’s that? Towards a new approach to a system of PDEs arXiv:1504.03049v4 [math-ph] 15 Dec 2015 e-Version1.5 December 16, 2015 By Atsushi INOUE NOTICE: COMMENCEMENT OF A CLASS i Notice: Commencement of a class Syllabus Analysis on superspace —– a construction of non-commutative analysis 3 October 2008 – 30 January 2009, 10.40-12.10, H114B at TITECH, Tokyo, A. Inoue Roughly speaking, RA(=real analysis) means to study properties of (smooth) functions defined on real space, and CA(=complex analysis) stands for studying properties of (holomorphic) functions defined on spaces with complex structure. On the other hand, we may extend the differentiable calculus to functions having definition domain in Banach space, for example, S. Lang “Differentiable Manifolds” or J.A. Dieudonn´e“Trea- tise on Analysis”. But it is impossible in general to extend differentiable calculus to those defined on infinite dimensional Fr´echet space, because the implicit function theorem doesn’t hold on such generally given Fr´echet space. Then, if the ground ring (like R or C) is replaced by non-commutative one, what type of analysis we may develop under the condition that newly developed analysis should be applied to systems of PDE or RMT(=Random Matrix Theory). In this lectures, we prepare as a “ground ring”, Fr´echet-Grassmann algebra having count- ably many Grassmann generators and we define so-called superspace over such algebra. On such superspace, we take a space of super-smooth functions as the main objects to study. This procedure is necessary not only to associate a Hamilton flow for a given 2d 2d system × of PDE which supports to resolve Feynman’s murmur, but also to make rigorous Efetov’s result in RMT. (1) Feynman’s path-integral representation of the solution for Schr¨odinger equation (2) Dirac and Weyl equations, Feynman’s murmur (3) Why is such new algebra necessary?Differential operator representations of 2d 2d-matrices × (4) Fr´echet-Grassmann algebra R and superspace Rm|n (5) Elementary linear algebra on superspace, super-determinant and super-trace, etc (6) Differential calculus on superspace; super smooth functions and implicit function theorem, etc (7) Integral calculus on superspace; integration by parts, change of variables under integral sign, etc (8) Fourier transformations on Rm|n and its application (9) Path-integral representation of a fundamental solution of Weyl equation Home Page:http://www.math.titech.ac.jp/‘inoue/SLDE2-08.html (Closed after my retirement from TITech at 31th March 2009) To audience: Please keep not only intellectual curioisity but also have patience to follow at least 3 lectures. ii For what and why, this lecture note is written I delivered 14 lectures, each 90 minutes, for graduate students at Tokyo Institute of Technology, started in Autumn 2008. Since there is a special tendency based not only on Japanese culture but rather asian aesthetic feeling, students hesitate to make a question during class. Reasons about this tendency seems based on modesty and timidity and feeling in his or her noviciate, or more frankly speaking, they are afraid of making a stupid question before friends which may probably exhibit their ignorance, which stems from their sense of guilty that they haven’t been studied sufficiently enough, or bother those appreciation for lectures by stopping by questions. In general, some one’s any questions on a lecture is very constructive that makes clear those which are not easy to understand for audience but also corrects miss understandings of speaker himself. Though “Instantaneous response towards your uncomfortable feeling” is embodied by very young peoples saying “Why, Mamy?”, but it seems rather difficult to do so in student society. Even though, please make a question without hesitation in any time. Not only primitive stays near radical but also your slight doubt makes slow down the speed of speaker’s explanation which gives some time to other students to help consider and to follow up. To make easy to pose such questions, I prepare pre-lecture note for a week before my lecture and corrected version of it after lecture with answers to questions if possible and necessary, those are posted on my home-page. This lecture note is translated and revised from them. Especially, I make big revision in Chapter 8 and Chapter 9. At the delivering time of my lectures, I haven’t yet clarified sufficiently the characterization of super-smooth functions but also the definition of integral on superspace which admits naturally the change of variables under integral sign. Therefore, I change significantly these representations in this notes from those original lectures. I owe much to my colleague Kazuo Masuda whose responses to my algebra related questions help me very much not only to my understanding but also to correct some defects in other’s publications with counter-examples. Concerning the characterization of super- smoothness, we need to prepare Cauchy-Riemann equation for them which is deeply connected with the countably infinite Grassmann generators. In his naive definition of integral, Berezin’s formula of change of variables under integral sign holds only for integrand having a compact support. This point is ameliorated when we modify the method of V.S. Vladimirov and I.V. Volovich [139] or A. Rogers [115] which relates to the different recognition of body part of “super space” from Berezin. Leaving from syllabus, I give some application of super analysis to Random Matrix Theory (=RMT) and of SUSYQM=SUper SYmmetric Quantum Mechanics with Witten’s index. I gather some facts which are not explained fully during these lectures in the last chapter named “Miscellaneous”. (1) I give a precise proof of Berezin’s formula of change of variables under integral sign. I confess Rothstein’s paper is not understandable to me, even a question letter to him without response and FORWHATANDWHY,THISLECTURENOTEISWRITTEN iii an explanation in Rogers’ book is also outside my scope. (2) Function spaces on superspace and Fourier transformation for supersmooth functions are breifly introduced. (3) As a typical simplest example of application of superanalysis, I give another proof of Qi’s result concerning an example of weakly hyperbolic equations, and (4) I give also an example for a system version of Egorov’s theorem which begins with Bernardi’s question. (5) I mentioned in the lecture that the problem posed by I.M. Gelfand at ICM congress address in 1954 concerning functional derivative equations related to QED or turbulence, which is more precisely explained there, and finally, (6) in his famous paper, E. Witten introduced Supersymmetric Lagrangian, derived from his de- ∗ formed dφ dφ +dφdφ. Here, we derive “the classical symbol of this operator” as the supersymmetric ∗ i j extension of Riemannian metric gij(q)dq dq , which is merely my poor interpretation from physi- cists calculation. (7) As an example where we need the countably infinite number of Grassmann generators, we consider Weyl equation with electro-magnetic external field. Besides whether it is phyisically meanigfull, we solve the Hamilton equation corresponding to this equation by degree of Grass- mann generators. Though, references are delivered at each time in lectures, but I gathered them at the last part. Finally, this note may be unique since I confess several times “their arguments are outside of my comprehension” which are not so proud of but I do so. Because not only I feel ashamed of my unmatureness of differential geometry and algebra’s intuition, but also I hope these confessions encourage to those young mathematicians who try to do some thing new! Contents Notice: Commencement of a class i For what and why, this lecture note is written ii Chapter 1. A motivation of this lecture note 1 1.1. Necessity of the non-commutative analysis and its benefit 1 1.2. The first step towards Dirac and Weyl equations 13 Chapter 2. Super number and Superspace 21 2.1. Super number 21 2.2. Superspace 27 2.3. Rogers’ construction of a countably infinite Grassmann generators 28 Chapter 3. Linear algebra on the superspace 31 3.1. Matrix algebras on the superspace 31 3.2. Supertrace, superdeterminant 33 3.3. An example of diagonalization 39 Chapter 4. Elementary differential calculus on superspace 47 4.1. Gˆateaux or Fr´echet differentiability on Banach spaces 47 4.2. Gˆateaux or Fr´echet differentiable functions on Fr´echet spaces 48 4.3. Functions on superspace 51 4.4. Super differentiable functions on Rm|n 58 4.5. Inverse and implicit function theorems 69 Chapter 5. Elementary integral calculus on superspace 75 5.1. Integration w.r.t. odd variables – Berezin integral 75 5.2. Berezin integral w.r.t. even and odd variables 80 5.3. Contour integral w.r.t. an even variable 83 5.4. Modification of Rogers, Vladimirov and Volovich’s approach 86 5.5. Supersymmetric transformation in superspace – as an example of change of variables 92 Chapter 6. Efetov’s method in Random Matrix Theory and beyond 95 6.1. Results – outline 96 6.2. The derivation of (6.1.7) with λ iǫ(ǫ> 0 fixed) and its consequences 98 − 6.3. The proof of semi-circle law and beyond that 107 6.4. Edge mobility 109 6.5. Relation between RMT and Painlev´etranscendents 114 Chapter 7. Fundamental solution of Free Weyl equation `ala Feynman 117 v vi CONTENTS 7.1. A strategy of constructing parametrices 118 7.2. Sketchy proofs of the procedure mentioned above 119 7.3. Another construction of the solution for H-J equation 123 7.4. Quantization 128 7.5. Fundamental solution 132 Chapter 8. Supersymmetric Quantum Mechanics and Its Applications 143 8.1.
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