http://redberry.cc/ 2026-04-19T11:54:59+00:00 http://redberry.cc/ http://redberry.cc/lib/tpl/redberry/images/favicon.ico text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:applying_and_manipulating_transformations?rev=1448109188&do=diff Applying and manipulating transformations Next topic: Substitutions ---------- All transformations in Redberry are first-class objects which means that they can be assigned to variables and share a common way for their applying and manipulation. Consider the syntax used for applying text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:calculating_one-loop_counterterms?rev=1448109188&do=diff Calculating one-loop counterterms Background Redberry provides a set of tools for calculation of one-loop counterterms in curved space-time. The theoretical formalism based on the extended t' Hooft and Veltman method of the background calculations was developed by Pronin and Stepanyantz ( text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:einstein_notation?rev=1448109188&do=diff Einstein notation Next topic: Standard form of mathematical expressions ---------- Redberry distinguishes covariant (lower) and contravariant (upper) indices. The corresponding property of single index called state. Two indices are considered to be contracted if and only if they have similar names and types but different states. As a consequence of this convention there are some natural restrictions on general structure of the expressions. text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:expression_tree_traversal_and_modification?rev=1448109188&do=diff Expression tree traversal and modification text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:getting_started?rev=1448109188&do=diff Getting started Next topic: Inputting and printing mathematical expressions ---------- Installing and running Redberry In order to install Redberry follow the instructions on the Installation page. The majority of code snippets presented on this site can be executed by wrapping them with the following Groovy code: text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:inputting_and_printing_mathematical_expressions?rev=1448109188&do=diff Inputting and printing mathematical expressions Next topic: Tensors and Indices ---------- Inputting expressions The notation used for inputting mathematical expressions in Redberry is almost the same as in many other CASs. For inputting tensors, Redberry uses text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:list_of_transformations?rev=1448109188&do=diff List of common transformations Next topic: Overview of HEP features ---------- Here is a list of basic transformations available in Redberry: Apply index mapping applies mapping of indices to tensors: See Mappings of indices. Collect collects terms by patterns: See Collect . CollectScalars text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:mappings_of_indices?rev=1448109188&do=diff Mappings of indices Next topic: Tree traversal ---------- Basics Perhaps the most significant difference between tensor- and symbol-oriented computer algebra systems lies in the comparison of mathematical expressions. In the symbol-oriented CASs result of atomic comparison problem (determination of whether two expressions are equal, i.e. operation that is the main building block of such complicated routines as pattern matching) is just a logical true or false, while in the case of tensori… text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:notes_on_internal_architecture?rev=1448109188&do=diff Notes on internal architecture Next topic: Setting up matrix objects ---------- One of the applications of finding mappings of indices is testing whether two expressions are equal (to within dummy indices relabelling and interchanging indices of simple tensors according to their symmetries). Probably, this operation is the most frequent operation that arises in any calculation (like reduction of similar terms). This problem can be solved by finding mapping of indices which maps free indices… text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:overview_of_hep_features?rev=1448109188&do=diff Overview of HEP features Next topic: Mappings of indices ---------- Besides general features for manipulation with tensors and a wide range of general-purpose transformations, Redberry provides a set of transformations and functions required for high energy physics (HEP). Calculation of one-loop counterterms text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:permutations_and_permutation_groups?rev=1448109188&do=diff Permutations and permutation groups Next topic: Programming with Redberry ---------- Permutations Permutations play a very important role in tensor algebra. For example, symmetries of tensors are specified in terms of permutations of indices. Besides, permutations and permutation groups are used in many routines and algorithms with tensors (e.g. text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:programming_with_redberry?rev=1448109188&do=diff Programming with Redberry Next topic: Notes on internal architecture ---------- Basics Redberry interface is written in Groovy and is intended to be used within the Groovy environment. Groovy is a general-purpose programming language and one can use all features and programming language constructs that are available in Groovy: looping, branching, functions, lambda-expressions, lists, classes etc. Besides, Redberry provides a specialized domain-specific programming constructs which are usef… text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:representation_of_derivatives?rev=1448109188&do=diff Representation of derivatives Next topic: Applying and manipulating transformations ---------- Representation of derivatives in Redberry is very similar to other systems and is very close to standard mathematical sense of this concept. However, presence of indices brings new features of derivatives with respect to indexed arguments. text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:setting_up_matrix_objects?rev=1448109188&do=diff Setting up matrix objects Next topic: Calculating one-loop counterterms ---------- Simplest case Matrices are tensors with a subset of indices marked as matrix indices. These indices have nonmetric types, which means that raising and lowering are forbidden. The typical examples of matrix objects are Dirac text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:standard_form_of_mathematical_expressions?rev=1448109188&do=diff Standard form of mathematical expressions Next topic: Symmetries of tensors ---------- A core feature of any CAS is its ability to reduce arbitrary expression to some standard form (SF) in which it is then used everywhere in manipulations. This approach facilitates comparison and matching of expressions and gives a way for more robust and fast algorithms for almost all CAS operations. Redberry uses the same paradigm: any intermediate and resulting expression is guaranteed to be in the text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:substitutions?rev=1448109188&do=diff Substitutions Next topic: List of common transformations ---------- Basics The most frequent transformation in all computations is a substitution. Here we shall discuss the usage aspects of substitutions, while the idea of the underlying algorithms can be found in Mappings of indices. The very important feature of any tensorial CAS is automatic relabelling of dummy indices in the case of dummy indices clash. Redberry takes care about it in all types of substitutions. Consider, for example… text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:symmetries_of_tensors?rev=1448109188&do=diff Symmetries of tensors Next topic: Representation of derivatives ---------- Specifying symmetries of tensors One of the distinctive features of tensors is the presence of symmetries. Consider symmetries under permutations of indices. Permutational symmetries in Redberry can be defined for indices of text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:tensors_and_indices?rev=1448109188&do=diff Tensors and Indices Next topic: Einstein notation ---------- There are three central object types in Redberry: Tensor, Indices, and Transformation. Objects of the same type share common properties and can be manipulated in a common way. Tensors and their indices are considered in this section, while transformations are discussed in text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:tree_traversal?rev=1448109188&do=diff Tree traversal Next topic: Permutations and permutation groups ---------- Each Tensor in Redberry is a container of its child tensors, so any complicated expression becomes a hierarchical tree of such containers. Iteration over direct child elements of a tensor described in Tensors and Indices. Besides, there are special tools for iteration and modification over a whole tree. text/html 2015-11-21T12:33:08+00:00 http://redberry.cc/documentation:guide:types_of_indices_and_metric?rev=1448109188&do=diff Types of indices and metric