Added Note: consider this a short antiquated introductory babbling, and then go to the main Tensor page at Wikipedia. What does "tensor" mean? It is unfortunate that the word has several meanings in the various contexts of mathematics and physics; it is even more unfortunate that these meanings have become a matter of specialized jargon, and that they are rarely, if ever, disambiguated in such a way that a student of mathematics, or mathematical or theoretical physics becomes enlightened on such a fundamental concept, or the distinctions in the various usages of the word soon enough. Fortunately or unfortunately, the various meanings of "tensor" do actually have a commonality, so the problem becomes to recognize that commonality and also to disambiguate the different contextual meanings within that commonality. That is the problem that the following addresses, addressed in not so much detailed mathematics as in the concept that creates the details. There are three essential meanings to "tensor", in three different contexts; while these meanings are related, they are not equivalent. The three contexts are: 1. Abstract algebra which uses the context of the "tensor product" derived from the concept of "Cartesian product"; 2. Group theory, the area of continuous groups and (analytic) Lie groups in particular (tensor with respect to a given group); 3. Differential geometry, where special kinds of "geometric objects" are singled out as tensors, and tensor densities. Properly, this the context of Tensor Analysis, analysis because it includes the concepts of differential and integral calculus. These are more properly called "tensor fields" defined over manifolds which include the objects of study in differential geometry. I will discuss each of these separately, and try to show how a general concept of "tensor" applies to each. Before beginning, I should say that in the end, it is the second context that will dominate the concept, but that it is often the important group behind the concept that is most often slighted in the literature. For the purposes of building that concept, I will take them in the above order. Prelude On Nomenclature In abstract algebra, complicated algebraic structures are either constructed by, or decomposed into, "tensor products" of simpler structures. In addition to these tensor products there are also "direct products"; another combining form is the "direct sum". These are more general than the tensor product which is very often reserved on certain occasions to be direct products of linear (vector) spaces. In group theory, there is an additional important combining form called a "semidirect product", a typical example of which is the combining, in an affine Euclidean space, of a group of rotations with a group of translations. Prelude On Metatheory The way of abstract algebra is to define a category of algebraic objects axiomatically, e.g., the category of groups. This happens to be perhaps the most powerful (useful) in the body of mathematics as well as in physical theory, because groups always seem to appear in the sense of "groups of transformations". The concept of "transformation" is an immediate one to all of existence since existence presupposes a mechanism of transformation whereby a thing at time t1 is to be recognized as "the same thing" at a later time t2. Without a concept of time there can be no concept of displacement and no concept of change. Are there philosophical and physical problems with that statement? Of course there are. For the time being, bear with me - though one could simply ask what "time" is. A Few Fundamental Problems Of Philosophy The temporal aspect of such existential transformations can be avoided in order simply to speak of structural transformations and a bit of philosophy contained in the age old question of, "when are two things 'the same'?". Of course, two distinct things are *never* the same, else they would not be distinct. The question must be better posed. Doing so, means that we have to be a little more careful about the concept of 'the same', and understand that if the absolute sense leads nowhere, that a contextual or structural sense must be adopted to realize the intuitive sense in back of the question. The necessity of specifying exactly what structure it is that should be the same is obvious. The intuitive idea in back of the question of "sameness" is not just a cute philosophical question to let you know that philosophers, just for the fun of it, ask impossible questions that seem to make sense, but then don't or seem silly. When such a question is made specific enough, it becomes a highly practical question to which humans in their endeavors and activities need to have answers - maybe not *an* answer, but answers - at least one. A few examples: A group of people get together to form a "government". Governments change. What is the structural point of that process of changing where the current government is no longer 'the same' government that was formed? This might be the quandary of a typical thoughtful historian, or citizen. A composer writes music, (or any artist creates a work of art), and the central problem of any composer or artist is to create a human illusion that everything written or created has a coherence - that it has a universe of discourse that belongs to itself. What are the criteria through which such a coherence is achieved? This construction will necessarily introduce a concept of "sameness" between the parts of the work, for it is that sameness that makes the coherence. A writer writes words that somehow also have a coherence or "subject" that is exposed in an understandable way to the reader. What are the criteria for the writing to be understandable, and that there exists a subject? The subject here is a prototype of the concept of "tensor", largely because I've said so - but how do I know that my intent "is the same" as the result of my efforts? My effort does not ensure my intent. These things are not automatically the same, yet I would want some measure of how close or how far apart intent and result are. Are they the same? I must rely on the reader to transform from word to thought as I have transformed thought to word. Tensors have very much to do with transformations and how things transform with a change of viewpoint. When are two distinct 'things' ('Ding an sich' - Heidegger) considered to be "the same 'kinds' of things"? When we ask *how* two things are the same, and *when* the conditions for sameness are met, this is the actual and more accurate statement of the question. To a very large extent this is completely arbitrary, and so a systematics and metasystematics is important in order to clarify both thought and communication. The one thing that must be gotten used to is that humans *never* perceive reality, but rather organize stimuli according to their human natures. Therefore, we always build insufficient models that must never be confused with reality. This is the scientific condition; it is also a human condition and an epistemological constraint that is a fact of life, and most probably a condition of fundamental existence based in deep roots of information theory. Mathematicians are really rather simple minded, as are physicists; they tend to pose problems that they have a sense are solvable without inordinate difficulty - sometimes they guess wrongly. It is guessing, and one way or another, both enterprises are about the art of constructing selfconsistent models. That is an antiplatonic viewpoint, which is to say that the body of mathematics is basically a creative aspect of the commonalities of the human nervous system; this body does not exist as the discovery of an 'a priori' existence: Platonic ideals are merely emotionally and perceptually motivated fictions, useful though they may be. An ontology of Platonic ideals is by definition unverifiable. The Fundamental Formal Problems Of Algebra I'll repeat now that the way of abstract algebra is to define a category of algebraic objects axiomatically. There are many different things in the category that is defined, so the first problem that arises is the "structure problem". The structure problem is determining the structure of the category by cataloging the various objects as to their individual properties that are intrinsic and not determined by the category axioms, the relationships of equivalence that exist between the objects by isomorphisms, and the relationships that exist in similarity by homomorphisms. These very thoughts are the origins of Category Theory. The structure problem for the category of groups is nowhere near completely solved, not even for finite groups. [PUT someplace else?] The structure theory for finite dimensional complex Lie groups was partially solved years ago. Roughly, according to a classical theorem (Levi's theorem) on the subject, any Lie group is a semidirect product of a solvable algebra by a semisimple algebra, meaning that the structure problem is broken into determining the structure problems for solvable and semisimple algebras. The Representation Problem The representation problem applies to a particular algebraic structure in a given category, which is to say, to each and every member of the appropriate category. This has to do with how abstract structures ultimately appear in numerical forms, those numerical forms being either numbers or matrices whose elements are numbers. A representation of an abstract algebraic structure is a structure preserving homomorphism from the abstract structure into an algebra of matrices. The "into" is rarely an onto, and generally any given algebraic structure has many different representations. This happens to be very important to the mathematics of quantum theory. Emergence of "combination principles" As elements of the the category both combine and decompose, so also do the representations of elements of the category. In both cases the notion of tensor or direct product appears as a combining principle. So does another combining principle of direct sum. These notions allow a new concept to appear that is called irreducibility, which means pretty much what it says: if a structure or any of its representations cannot be expressed either as a direct sum or a direct product of smaller structures, it is called "irreducible"; the important thing to specify in any meaning of "irreducibility" is the set of means by which reduction is possible. Since one can generate all structures or representations from the irreducible ones, isolating the subcategory of irreducible elements is enough to generate the category by the specified means through which irreducibility is defined, and so studying the irreducibles makes either the structure problem or the representation problem easier. Direct Sum The direct sum and direct product are easily understood conceptually with a simple example using 2x2 matrices of abstract numerics. Let | a b | | e f | X := | | and Y := | | | c d | | g h | The direct sum of X and Y (X + Y) = Z can be represented as a 4x4 matrix | X 0 | Z := | | | 0 Y | where it can be understood that (X + Y) is equivalent to (Y + X), so direct sum is commutative, and behaves like ordinary addition. Notice that (X + Y) is not equal to (Y + X); and equivalence would be in terms of canonical forms. Direct Product The direct product of X x Y is not necessarily commutative | aY bY | X x Y := | | = | cY dY | | ae af be bf | | ag ah bg bh | | ce cf de df | | cg ch dg dh | In many cases a direct product can be expressed as a direct sum, or a weighted direct sum. When this happens, the structures or representations have all the conditions necessary to be a ring. This does happen with groups and one can speak of the ring of representations of a group. This is an important mathematical fact for quantum theory. Tensor Product The tensor product grows from the idea of the "Cartesian Product" of the one dimensional x-axis as a vector space, and the one dimensional y-axis in elementary algebra to form the two dimensional Cartesian plane. If there are ------------------------------------- In differential geometry the understanding of a tensor is actually what a physicist or mathematician would properly call a "tensor field"; this might best be understood in terms of fibre bundles with the manifold M of dimension n in question being the base space, and GL(n, R) or GL(n, C) being the group G. The bundle is not necessarily principle. A Fibre Bundle To say that a bundle is "principle" is almost a sneaky way of saying that it is trivial since then it is "merely" the topological product of it base space and its fibre space, which is to say the easiest generalization of the Cartesian plane (X x Y) as a principle bundle over X as base space. The fibre is a tensor algebra over a point of M whose elements transform according to some tensor product of fundamental representations of G that are induced by general coordinate substitutions on M. This is the idea in general relativity of "general covariance", that the laws of physics should be stated and stateble as equations of tensorial fields, in which case the the equations are form invariant under arbitrary coordinate substitutions, or more precisely be diffeomorphically form invariant. A diffeomorphism is a mapping that is a differentiable homeomorphism (topological equivalence). This starts to get "interesting" when the global topology of M is considered since to do this sort of thing properly, the coordinates should either have no singularities, or have singularities that are removable. This is not always possible, and it is generally not possible unless the manifold is conformally Euclidean. Removable and Nonremovable singularities. Planes & Spheres Integrable and Nonintegrable structures - "Holonomy" Homology v. Cohomology "A space and the space of functions over the space are intertwined; they mirror each other's topological structure." Homotopy? A Clifford Bundle fibre = algebra of completely antisymmetric covariant or contravariant tensors associated with a metric or pseudometric on the underlying manifold. Differential forms and DeRahm cohomology.TOC

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Created: August 14, 2003

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