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
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
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
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
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
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
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
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.
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.
The direct sum and direct product are easily understood conceptually
with a simple example using 2x2 matrices of abstract numerics.
| 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
| 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.
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.
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
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
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."
A Clifford Bundle
fibre = algebra of completely antisymmetric covariant or contravariant
tensors associated with a metric or pseudometric on the underlying
Differential forms and DeRahm cohomology.
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Created: August 14, 2003
Last Updated: July 7, 2007