2. Non-relativistic invariant differential operators and equations
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Vladimir K. Dobrev
Abstract
We give a review of some group-theoretical results related to non-relativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We recall the fact that there is a hierarchy of equations on the boundary, invariant w. r. t. the Schrödinger algebra. The derivation of this hierarchy uses a mechanism introduced first for semisimple Lie groups and adapted to the non-semisimple Schrödinger algebra. This requires development of the representation theory of the Schrödinger algebra, which is reviewed in some detail. In Section 2.1 the Schrödinger equation is reviewed as an invariant differential equation in the (1 + 1)-dimensional case. On the boundary this was done in [175] (extending the approach in the semisimple group setting [161]), constructing actually an infinite hierarchy of invariant differential equations, the first member being the free heat/Schrödinger equation). In Section 2.1.4 the extension of this construction is reviewed to the bulk combining techniques from [6] and [175]. In Section 2.2 the Schrödinger equation is reviewed as an invariant differential equation in the general (n + 1)-dimensional case following [8, 190]. The general situation is very complicated and requires separate study of the cases n = 2N and n = 2N + 1. In Section 2.3 the (3 + 1)-dimensional case is reviewed separately and in more detail, since it is most important for physical applications. In Section 2.4 the q-deformation is reviewed of the Schrödinger algebra in the (1 + 1)-dimensional case; cf. [176]. In Section 2.5 the difference analogues of the Schrödinger algebra in the (n + 1)-dimensional case are reviewed; cf. [177].
Abstract
We give a review of some group-theoretical results related to non-relativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We recall the fact that there is a hierarchy of equations on the boundary, invariant w. r. t. the Schrödinger algebra. The derivation of this hierarchy uses a mechanism introduced first for semisimple Lie groups and adapted to the non-semisimple Schrödinger algebra. This requires development of the representation theory of the Schrödinger algebra, which is reviewed in some detail. In Section 2.1 the Schrödinger equation is reviewed as an invariant differential equation in the (1 + 1)-dimensional case. On the boundary this was done in [175] (extending the approach in the semisimple group setting [161]), constructing actually an infinite hierarchy of invariant differential equations, the first member being the free heat/Schrödinger equation). In Section 2.1.4 the extension of this construction is reviewed to the bulk combining techniques from [6] and [175]. In Section 2.2 the Schrödinger equation is reviewed as an invariant differential equation in the general (n + 1)-dimensional case following [8, 190]. The general situation is very complicated and requires separate study of the cases n = 2N and n = 2N + 1. In Section 2.3 the (3 + 1)-dimensional case is reviewed separately and in more detail, since it is most important for physical applications. In Section 2.4 the q-deformation is reviewed of the Schrödinger algebra in the (1 + 1)-dimensional case; cf. [176]. In Section 2.5 the difference analogues of the Schrödinger algebra in the (n + 1)-dimensional case are reviewed; cf. [177].
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- 1. Relativistic and nonrelativistic holography 1
- 2. Non-relativistic invariant differential operators and equations 51
- 3. Virasoro algebra and super-Virasoro algebras 91
- 4. Affine Lie (super-)algebras 143
- Epilogue 197
- Bibliography 199
- Author Index 231
- Subject Index 233
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- 1. Relativistic and nonrelativistic holography 1
- 2. Non-relativistic invariant differential operators and equations 51
- 3. Virasoro algebra and super-Virasoro algebras 91
- 4. Affine Lie (super-)algebras 143
- Epilogue 197
- Bibliography 199
- Author Index 231
- Subject Index 233
