Talk:Tensor/Archive 4

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Characterization of spinors

Latest comment: 10 years ago34 comments3 people in discussion

It probably makes sense to open a discussion on this here. I interpret the edit summary

It isn't "nonsense". A tensor is an O(n)-equivariant function on a principal homogeneous space for the action of O(n) and a spinor is a Spin(n)-equivariant function. These both require a notion of orthonormality.

as saying that a notion of a metric tensor is necessary, with which I agree. The concept of a spinor inherently must be independent of the choice of basis, though, so while the concepts of orthogonality and of normalization are necessary for rotations and spinors to be defined, this places no constraints on the choice of basis. The properties of the entities mentioned is meaningless to me, but should not be relevant to my point. Perhaps we could add the concept of a metric tensor or bilinear form back after my removal of reference to a basis? (Although, this is already implied by rotations being defined, so I hardly see the point.)

The separate matter of the "transformation law for tensors" is partially my confusing it with the use of the term "transformation law" elsewhere in the article to refer to the transformation of components under a change of basis, which does not apply here. I would like to see a different choice of words here to make this distinction clearer. —Quondum 18:12, 16 March 2014 (UTC)

A tensor is an O(n)-equivariant function from a principal homogeneous space P for O(n) (that is, orthonormal frames in a Euclidean space) with values in a representation

(V,\rho )

of O(n),

f:P\to V

. The "transformation law" is the equivariance of this function under the orthogonal group, which acts both on the space of frames and on V by virtue of the representation,

f(u.g)=\rho (g^{-1})f(u).

This is precisely the same way the term "transformation law" is used throughout the article, although possibly with different groups in mind (the general linear rather than rotation group). The space of orthonormal frames appears explicitly in this definition of a tensor, and it is actually necessary in order to make sense out of spinors as a generalization of tensors. A spinor is an equivariant function on a principal homogeneous space for Spin(n) with values in a spin representation. (At least, this is the bridge between the mathematician's way of thinking of a tensor or spinor and the physicist's or geometer's.) Sławomir Biały (talk) 18:35, 16 March 2014 (UTC)

I guess I'm bumping into the usual different-levels-of-abstraction problem, or perhaps a lack of clarity about the definition being used at each point in the article (tensors are defined multiple ways in the article), and the language appropriate to each varies accordingly. You have at least now worded it so that it is clear that the transformation being referred to is a change of basis, rather than some rotation of the tensor itself; I had assumed otherwise. —Quondum 21:30, 16 March 2014 (UTC)

In physics, the difference in perspective is one of active and passive transformations. In the "indices with transformation law" perspective, the transformations act "passively": a change of basis induces a compensating change on the components of the tensor. This is the approach that the section on spinors takes (or attempts to), and is a more typical way of thinking of spinors in physics. Mathematicians tend to think of tensors as elements of the representation spaces of groups themselves (that is, the relevant groups act "actively"). I personally think that as generalizations of tensors, it makes more sense to think of spinors in the former rather than latter way, but that's probably a matter of taste. Sławomir Biały (talk) 22:25, 16 March 2014 (UTC)

I suppose the idea of an active rotation on a manifold presents problems that a passive transformation does not, making the latter more attractive in the physics context. I hadn't thought that this would work (in a sense, a passive transformation does nothing), which was why I incorrectly assumed that an active transformation was meant. In a pure tensor algebra (i.e. not as a field on a manifold), either could work. It is also perhaps not so obvious (at least to me) why fermions and bosons as spinors should act like groups to fit the mathematicians' concept; they seem to be wavefunctions to be acted upon, not to have a pointwise group action. As idle speculation on the generalization of tensors to spinors, I wonder whether spinors can be accommodated naturally within a tensor algebra, considering that they fit into a Clifford algebra, which in turn can be expressed as a quotient of a tensor algebra. —Quondum 00:29, 17 March 2014 (UTC)

From a mathematical perspective, the spinor bundle is a covering space for the tensor bundle.

I'm confused by fermions and bosons as spinors; surely only fermions are spinorial. Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:31, 18 March 2014 (UTC)

A particular mathematical characterization need not be unique. I was speculating on a link that suggests a possible representation of spinors within the tensor algebra, albeit summing tensors of differing orders. Not really an appropriate topic for a talk page, though.

The question isn't uniqueness but relevance; spinors are only relevant for Fermions. Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:30, 19 March 2014 (UTC)

Bosons: Certainly not important, possibly a misconception on my side: bosons are spin-1 fields, so I guessed that they may fall within the same framework. The observation about active vs. passive rotations applies to any field (bundle). —Quondum 03:42, 19 March 2014 (UTC)

No, Bosons are fields with integral spins, e.g., 0, 1, 2. They are represented by scalar, vector and tensor fields. Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:30, 19 March 2014 (UTC)

Yes, naturally. I was being sloppy, only mentioning one case. The only real point is that one needs to generalize the treatment in some way to accommodate the (fermion) spinors. —Quondum 19:33, 19 March 2014 (UTC)

From a mathematical perspective, tensors are not tied into O(n), SO(n), U(n), SU(n) or a metric. Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:31, 18 March 2014 (UTC)

Certainly there is nothing specific to the notion of a tensor that requires that the structure group be any particular classical group. I'm not sure why that's relevant here though. As generalizations of tensors, spinors certainly require the structure group be the orthogonal group. Sławomir Biały (talk) 21:22, 18 March 2014 (UTC)

Or, presumably, an indefinite orthogonal group? —Quondum 03:42, 19 March 2014 (UTC)

Actually any orthogonal group associated to a quadratic form. Sławomir Biały (talk) 10:53, 19 March 2014 (UTC)

There's nothing in the notion of a tenser that requires that there be a structure group, althou you could always say that the structure group is GL(n). It's relevant to the claim A tensor is an O(n)-equivariant function on a principal homogeneous space for the action of O(n). Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:30, 19 March 2014 (UTC)

Well I disagree. Certainly in geometry, there is always a structure group underlying the notion of a tensor. In geometry, a tensor is defined to be equivariant map from a principal homogeneous space to a representation of a group. What can this mean if there is no group?? Sławomir Biały (talk) 20:03, 19 March 2014 (UTC)

Linking some terms for easy access: equivariant map, principal homogeneous space, group representation.

The terms might compactly tell you what you are referring to, Sławomir, but they are making my head spin. I have recently become a little familiar with a geometry defined in terms of a group action on a set (Klein's Erlangen program), but am not sure that this even relates to what you are talking about. I am most familiar with tensors defined in terms of a vector space and its dual space, and this can be done in the absence of a metric tensor. Without further structure, such a vector space (and hence the associated tensor algebra) exhibits GL(n) symmetry, which contains all the O(p,q) groups as subgroups, but there is no natural way to identify any one of these without an associated quadratic form. It seem to me that there is no way to tie any specific orthogonal group to the general (nonmetric) tensor concept, though it is possible that I've missed what you are really saying. —Quondum 00:47, 20 March 2014 (UTC)

It's all fairly concrete in the "Definition" box of this article. If f is just a generic basis of a vector space (and R is an element of GL(n)), then this defines the notion of a "GL(n) tensor". If f is an orthonormal basis and R is a rotation, then you get an "O(n) tensor". This actually makes sense for any class of frames of a vector space that is acted on freely and transitively by a group. But of course the group is part of the data needed to specify the notion of tensor. Sławomir Biały (talk) 01:31, 20 March 2014 (UTC)

Okay, I'm getting a glimmer (I think). It seems you are generalizing that definition of a tensor to incorporate objects such as spinors by allowing restriction of the group of transformations. The restriction on the bases is not as clear to me. Since we can presumably find a representation for any group, for Spin(n) we simply find a matrix representation, and the set of suitable bases is the principal homogeneous space. The only purpose of each basis appears to be to act as an element of this space, and is thus pretty abstract: not obviously related to a spacetime basis. You might even mean frame.

If this is even halfway close, with spinors being tensors (T) rather than transformations (R) in this picture, it seems that §Spinors could do with some rewording. —Quondum 03:40, 20 March 2014 (UTC)

The space of orthonormal bases is not simply connected. This is thought of in physics as a path-dependence under rotations: two paths from a frame A to a frame B through rotations are not always deformable one into the other. There is an additional discrete invariant, the "spin", that is attached to the frame. Spinors are then tensorial objects whose components transform under the "spin group", that is the group of rotations with spin, and are sensitive to the difference in spin of the frames. Sławomir Biały (talk) 17:37, 20 March 2014 (UTC)r

I am surprised at the description in the latest edits). I had expected that the tensor representation of a spinor would naturally be the double cover of an orthogonal-type tensor. I know that such linearized representations of the spin groups exist without the need for additional discrete variables. The transformation law for the added discrete variable on the frame (and hence for the corresponding tensor) would be somewhat contrived. —Quondum 02:03, 21 March 2014 (UTC)

I'm not sure what isn't being effectively communicated here. The spinors I am describing are linear representations of the spin group. The spin group has an additional discrete symmetry in addition to the continuous symmetry of the orthogonal group, which is the fundamental group acting by covering transformations. In the language of groups and representations, spinors are representations of the spin group that can actually detect these covering transformations (by a change of sign), unlike the tensor representations of the spin group which cannot detect which sheet of the cover you're on. Sławomir Biały (talk) 03:13, 21 March 2014 (UTC)

We're probably just not using quite the same language; the change of sign is exactly what I expect. The sentence It is possible to attach an additional discrete invariant to each frame called the "spin" that incorporates this path dependence, and which turns out to have values $\pm 1$ is no doubt what is confusing me. I think I should become more familiar with the subject area before abusing more of your time. —Quondum 05:18, 21 March 2014 (UTC)

Offtopic trolling

The following discussion has been closed. Please do not modify it.

The Spinor bundle is a vector bundle, but not part of the Tensor bundle. Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:10, 20 March 2014 (UTC)

ITYM the "Definition" box of Tensor#As multidimensional arrays in this article.

I've never seen a definition of the tensor bundle that included a structure group, although such a definition is customary for the more general Fiber bundle. Note that Spinor bundle refers to the Orthonormal frame bundle rather than to the Tangent bundle. Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:10, 20 March 2014 (UTC)

Ok, but who is talking about "the tensor bundle"? We are here discussing the general notion of a tensor, which of course includes a structure group in its definition. Sławomir Biały (talk) 17:04, 20 March 2014 (UTC)

There is no of course; I have never seen a definition of a tensor in any paper or book that included a structure group. Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:36, 21 March 2014 (UTC)

Well it's been pretty standard in the literature at least since the work in the 1930s of Cartan (and possibly Weyl). I believe also Chern's 1951 mimeographed notes on differential geometry contain a discussion of tensors with a structure group, although it was probably much later popularized by Kobayashi and Nomizu's influential textbook "Foundations of differential geometry". Modern accounts can be found in Sharpe "Differential geometry", Kollar, Michor, and Slovak "Natural operations in differential geometry", or probably any other textbook that discusses principal bundles. But in any case, it is unclear how this discussion actually pertains to improving this article. Sławomir Biały (talk) 23:11, 21 March 2014 (UTC)

It's the frame bundle that is a principal bundle. Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:12, 23 March 2014 (UTC)

Do you have a point? Sławomir Biały (talk) 20:20, 23 March 2014 (UTC)

Yes; my point is that your claim is bogus. As best I can determine you are confusing tensors with frames. If you have a source that actual defines tensors using a structure group, please quote the relevant text or provide a page number; when the index has only one reference to structure group and that is not for a discussion of tensors, one has to wonder why you cited the book as supporting evidence. Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:51, 26 March 2014 (UTC)

If you think I have conflated tensors with frames, then you have trouble with reading comprehension, and further discussion seems rather fruitless. Cartan defines tensor in the way I have done above in this thread, although in slightly more archaic terms. Sharpe defines tensors almost exactly as I have done. Chern, Kobayashi-Nomizu and Kollar et al (if I recall) define tensorial forms on a principle bundle. If you are interested in learning about such things, then I suggest that you go and read the books. I'm not interested in being browbeaten by a WP:RANDY on a point that increasingly departs from anything even remotely connected with the text of the article. Sławomir Biały (talk) 16:09, 26 March 2014 (UTC)

Actually, that was simply the most charitable explanation; the alternative was that you were just blowing smoke, which would explain your refusal to provide page numbers, much less quotes. You gave three references, each of which demolishes your claim.

If, as you claim, Cartan defined tensor as you do, then why have you been unable to cite that definition?

To set the record straight, Kobayashi, Soshichi; Nomizu, Katsumi (1963). "1. Differential manifolds". Foundations of Differential Geometry. Interscience Publishers. p. I-5. {{cite book}}: Cite has empty unknown parameters: |editor[n]-last=, |authorlink[n]=, |laydate=, |author-name-separator=, |laysummary=, |nopp=, |chapterurl=, |trans_title=, |editor[n]-first=, |trans_chapter=, |month=, |author-separator=, |lastauthoramp=, and |editor[n]-link= (help); Unknown parameter |separator= ignored (help), Koláǩ, Ivan; Michor, Peter W.; Slovák, Jan (1993). "1.7. The Tangent Bundle". Natural Operations in Differential Geometry. Springer-Verlag. p. 7. ISBN 3-540-56235-4. {{cite book}}: Cite has empty unknown parameters: |editor[n]-last=, |authorlink[n]=, |editor[n]-first=, |laydate=, |author-name-separator=, |laysummary=, |chapterurl=, |nopp=, |editor[n]-link=, |trans_title=, |month=, |trans_chapter=, |first[n]=, |last[n]=, |author-separator=, and |lastauthoramp= (help); Unknown parameter |separator= ignored (help)CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) and Sharpe, R. W. (1997). "§ 4. Tangent Vectors, Bundles anf Fields". Differentail Geometry Cartan's Generalization of Klein's Erlanger Program. pp. 46–47. ISBN 0-387-94732-9. {{cite book}}: Cite has empty unknown parameters: |editor[n]-last=, |authorlink[n]=, |editor[n]-first=, |editor[n]-link=, |laydate=, |author-name-separator=, |laysummary=, |nopp=, |chapterurl=, |trans_title=, |trans_chapter=, |first[n]=, |month=, |last[n]=, |author-separator=, and |lastauthoramp= (help); Unknown parameter |separator= ignored (help) all define the tangent bundle without parametrizing with a structure group,

as does Steenrod, Norman (1951). The Topology of Fiber Bundles. Princeton University Press. {{cite book}}: Cite has empty unknown parameters: |editor[n]-last=, |authorlink[n]=, |editor[n]-first=, |laydate=, |author-name-separator=, |laysummary=, |nopp=, |chapterurl=, |trans_title=, |trans_chapter=, |first[n]=, |month=, |last[n]=, |author-separator=, and |editor[n]-link= (help); Unknown parameter |separator= ignored (help); their definitions of the tensor bundle are similar.

Now, they define other things in other places, but those have noting to do with the point in dispute.

BTW, resorting to argumentum ad hominem does not help your case, and is no repacement for specific citations. Shmuel (Seymour J.) Metz Username:Chatul (talk) 23:04, 3 April 2014 (UTC)

Ok, here is a specific citation: Sharpe, "Differential geometry", p. 194. Cartan, "Theory of spinors", section II.23. Kobayashi and Nomizu p. 75-76. Although given the overall tenor of your comments here, I do not expect you to understand these citations. I believe that you are trolling this thread. So unless you have something that specifically relates to part of the article, I consider this discussion closed. Sławomir Biały (talk) 23:29, 3 April 2014 (UTC)

Tensor densities

Latest comment: 10 years ago19 comments6 people in discussion

Hi

I had a quick question and I was hoping I could get some clarifiaction.

I was taught that tensor densities transform as:

${\mathfrak {T}}_{\nu '_{1}\cdots \nu '_{m}}^{\mu '_{1}\cdots \mu '_{n}}=\left\vert {\frac {\partial x'}{\partial x}}\right\vert ^{w}{\frac {\partial x^{\mu '_{1}}}{\partial x^{\mu _{1}}}}\cdots {\frac {\partial x^{\mu '_{n}}}{\partial x^{\mu _{n}}}}{\frac {\partial x^{\nu _{1}}}{\partial x^{\nu '_{1}}}}\cdots {\frac {\partial x^{\nu _{m}}}{\partial x^{\nu '_{m}}}}{\mathfrak {T}}_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}$

That is the new (prime) co-odinates in the numerator of the Jacobian/determinant. However I have come accross that tensor densities transform as:

${\mathfrak {T}}_{\nu '_{1}\cdots \nu '_{m}}^{\mu '_{1}\cdots \mu '_{n}}=\left\vert {\frac {\partial x}{\partial x'}}\right\vert ^{w}{\frac {\partial x^{\mu '_{1}}}{\partial x^{\mu _{1}}}}\cdots {\frac {\partial x^{\mu '_{n}}}{\partial x^{\mu _{n}}}}{\frac {\partial x^{\nu _{1}}}{\partial x^{\nu '_{1}}}}\cdots {\frac {\partial x^{\nu _{m}}}{\partial x^{\nu '_{m}}}}{\mathfrak {T}}_{\nu _{1}\cdots \nu _{m}}^{\mu _{1}\cdots \mu _{n}}$

That is the new (prime) co-odinates in the denominator of the Jacobian/determinant.

The first one tells us that g is a scalar density of -2, the second one tells us that g is a scalar density of +2.

Which one is correct? I heard that both are and it is just a convention (like (-,+,+,+) vs. (+,-,-,-)) but I thought I'd ask.

P.S. I am inclined (I prefer) the definition with the prime in the numerator. This is because I was taught that the volume element transforms as

$dV'=\left\vert {\frac {\partial (x'_{1},\cdots ,x'_{n})}{\partial (x_{1},\cdots ,x_{n})}}\right\vert dV$

(So the volume element has weight +1 and g has wieght -2 (Using the prime-in-numerator definition))

Thank you! — Preceding unsigned comment added by 206.45.28.59 (talk) 19:43, 3 July 2014 (UTC)

It's obviously a convention, and I've seen both. But the "right" choice is to give the determinant of the metric tensor a weight of +2. The reason is that intuitively a density is something that should have a well-defined integral (because a density represents a density of stuff on a manifold). The Jacobian change of variables formula dictates what the weight must be in order for this to hold. Sławomir Biały (talk) 23:04, 3 July 2014 (UTC)

Hi Slawomir

Thank you for your reply. If it is a convention then how can one be 'right' or more 'right'? — Preceding unsigned comment added by 206.45.181.185 (talk) 03:11, 4 July 2014 (UTC)

Well. I did put "right" in quotation marks. It is of course, only a convention. But the opposite convention seems rather hard to defend, in my opinion. I don't see that you've presented any argument for it, other than appeal to authority. Sławomir Biały (talk) 11:28, 4 July 2014 (UTC)

Should this article even mention tensor densities? Tensor densities are generalizations of tensor fields not tensors. (I guess you could define something analogous for tensors on a single vector space, but do people ever do that?) T R 12:14, 7 July 2014 (UTC)

This is a complicated subject, to be sure. I would like to see the mention confined to a link in the See also section. IMO, we should not make the claim that they are generalizations of tensors. My reasoning for this is thus: (at least to me) it seems that they don't even make sense in the abstract picture of tensors (in the sense that they cannot be defined in this picture). As such, they are not generalizations of tensors, but rather of their representation as bundles over the manifold of basis vectors at each point on the space (the "other" manifold having the differential structure that we're really interested in). This point is distinct from and does not negate yours. —Quondum 13:03, 7 July 2014 (UTC)

They are generalizations of tensors in that they are tensor products of the usual tensor bundles with certain natural line bundles (which are trivial, but not naturally trivial). Sławomir Biały (talk) 14:33, 7 July 2014 (UTC)

You are mentioning terms that are a challenge to me. Tensor bundles seem to fit into the usual picture, so my issue would only be with "certain natural line bundles" (which I do not think of as natural because they are not restricted to being scalar tensors and introduce an unnecessary parameterization into the abstract picture). But yes, I must concede that in this sense they can be regarded as a generalization even in the abstract sense; unfortunately it is a sense that weakens a very powerful symmetry/invariant of the tensor bundle: that of basis independence, without any apparent benefit (i.e. it does not seem to encompass a broader class of problems, at least not in the context of physics). As such, it seems a bit pretentious to make the point that it is a generalization. What's the point? —Quondum 18:09, 7 July 2014 (UTC)

I'm (as usual) confused. I don't really understand your point Q. It's not the same thing as saying I disagree. I wouldn't know what to disagree with to begin with. Tensor are unusually well-behaved objects. For every well-behaved object you'll have ℵ_{ω +42} objects that aren't well-behaved. But some of them aren't that bad and resemble tensors enough to call them generalizations whatever you formally mean by that. Transformation properties is a prime example if you want a more mundane characterization than Sławomir's bundles

. The oldest (still alive) characterization of tensors is probably in therms of transformation properties. These generalize to tensor densities. They do appear in the physics literature, and, I'm sure, in the mathematics literature (hence too, and correctly so, in this article). That should ensure that they have some sort of use. The Levi-Civita symbol (viewed as a field) is a tensor density. How would you characterize it if not as a tensor density? Would you lump it together with all other non-tensors? YohanN7 (talk) 19:52, 7 July 2014 (UTC)

And oh, if your point is reducing the section to a mostly a link because there's a main article, then I understand and support because of easier maintenance. YohanN7 (talk) 20:13, 7 July 2014 (UTC)

I'll be happy to discuss this separately, but I can see that I'm risking overtaxing the patience of others who don't seem to think that I necessarily have a point in my brash challenge of the value of a well-established topic as a motivation for, in an article on a well-behaved object, removing a section to a less well-behaved object. I was not trying to motivate the removal of a notable topic from Wikipedia. —Quondum 22:45, 7 July 2014 (UTC)

I still don't understand. But this time it is linguistic problems on my part. I'm not a native English speaker and the first sentence is about three times longer than I can handle. (I did try to break it down, but failed.) YohanN7 (talk) 23:50, 7 July 2014 (UTC)

Sorry, I'm just saying that I think of tensor densities as having little valueand hence should not be emphasized in this article, but clearly others here (and many eminent authors) do not share my view. —Quondum 05:09, 8 July 2014 (UTC)

No, it's actually an interesting question I think to determine in what sense tensor densities are "natural". The way in which tensor bundles are "natural" is that they are characterized as functors on the category of smooth manifolds. (Actually another class of bundles has this properties:jet bundles, which also deserve mentioning is this article.) But densities seem to require additional structure on the category: presumably they're functors on the category of smooth manifolds with a free Z_2 action. I've not had a chance to look at how this works in detail though. Sławomir Biały (talk) 00:35, 8 July 2014 (UTC)

I have to wrap my mind around too many new words and concepts in one sentence to make proper sense of this with any confidence. Since I think in terms of a non-holonomic basis allowing us to decouple the tangent basis from the coordinates and the differential structure of the manifold. Also, one can drop the metric while maintaining a volume form. I'm just throwing this in, since the added degrees of arbitrariness facilitate reasoning about naturalness. —Quondum 05:09, 8 July 2014 (UTC)

It is at any rate awkward that tensor densities are described more fully here than in tensor density. I suggest suggest most of the material here is merged into the main article. YohanN7 (talk) 06:50, 8 July 2014 (UTC)

I think we got derailed from my original question. Tensor densities are a generalization of tensor fields, which are (mostly) discussed in a different article. Is it logical to discuss them as generalizations of tensors in this article? For readers unfamiliar with tensors the uses of the the term tensor both for tensor fields and individual tensors is mightily confusing. I fear that tensor densities appearing in this article will only exacerbate that. (It is somewhat weird that this article spends more words on tensor densities than on tensor fields.)T R 09:21, 8 July 2014 (UTC)

There is already stuff about tensor densities over at tensor field: Twisting by a line bundle. YohanN7 (talk) 11:55, 8 July 2014 (UTC)

TR, I apologize for the derailment. I agree with and support what you say. —Quondum 14:57, 8 July 2014 (UTC)

Tensor products of infinite dimensional spaces

Latest comment: 9 years ago30 comments8 people in discussion

(split from preceding thread)

I would argue that the lead should take into account the infinite dimensional cases, where a bit more care is needed in the definitions and where things are less cannonical. Shmuel (Seymour J.) Metz Username:Chatul (talk) 21:02, 1 December 2014 (UTC)

Who exactly would read such a high-brow introduction? Sławomir Biały (talk) 23:02, 1 December 2014 (UTC)

I have never heard of anyone who actually uses tensors on an infinite dimensional space. JRSpriggs (talk) 05:00, 2 December 2014 (UTC)

Fock space and constructions around it? YohanN7 (talk) 06:48, 2 December 2014 (UTC)

Although Fock space is an example of (a) tensor product(s) of an infinite dimensional vector space, I've never seen anybody refer to the elements of Fock space as "tensors". (The closest you'll get is the occasional use of the term "pure tensor" to refer to a state that is the tensor product of two other states.)T R 08:59, 2 December 2014 (UTC)

Nor have I heard it. But it struck me that the elements (state vectors) are tensors by one definition of tensor. YohanN7 (talk) 09:28, 2 December 2014 (UTC)

Off topic, but already in ordinary QM, multi-particle states are introduced as tensor products in some books. The notation then quickly hides it away. I think Shankar does it that way. YohanN7 (talk) 10:03, 2 December 2014 (UTC)

An element of Fock space is a tuple of symmetrized or antisymmetrized tensors, each member of the tuple representing a particular number of particles. In mathematical developments of Fock space that I have seen, square integrability and all but a finite number of tuple elements being nonzero are usually assumed. Although coherent states can be thought as a large N limit of a Fock state for bosons. While tensor products are ubiquitous in QM, they are often products of different spaces, so transformations need to handled on a case by case basis. For instance, the state of an electron in a hydrogen atom is the tensor product of the orbital state and the spin state. These transform differently and operations like tensor contraction across spaces don't make sense. So there typically isn't a lot to be gained by considering tensor transformations vs transformations of the individual spaces. --Mark viking (talk) 22:07, 2 December 2014 (UTC)

This discussion raises the question: Is everything which is a linear combination of tensor products of two (or more) vector spaces actually a 'tensor'? Or does the word "tensor" imply more than that? JRSpriggs (talk) 01:36, 3 December 2014 (UTC)

The section "Using tensor products" of this article defines a tensor as an element of a tensor product of vector spaces. Is that what you mean? Mgnbar (talk) 02:22, 3 December 2014 (UTC)

The section "Using tensor products" defines a tensor as an element of the tensor product of a vector space with (multiple copies of) itself or its dual. The thing described by JRSpriggs is more general than that, as it is an element a tensor product of (not necessarily the same) vector space. If the answer to his first question is yes, it raises the further question of elements of tensor products of groups or modules are also referred to as "tensors".

In physics literature, I believe the answers is no though. "Tensor" exclusively refers the more narrow definition currently in the article. (or tensor fields).T R 12:58, 3 December 2014 (UTC)

Surely not? The definition in § Using tensor products is overly simplified and hence not a rigorous definition at all. I am not referring to the choice of vector space (to a vector space and its dual), but to the ordering within the tensor product (braiding), which is in this case evidently omitted for simplicity. Just as a (2,0)-tensor is not in general equal to its transpose, a (1,1)-tensor is not even be comparable to its transpose (it inhabits a space that is not identified with the original space: V⊗V^∗ ≠ V^∗⊗V). For correctness, it should be defined as the ordered product of arbitrary number of vector spaces (which may or may not be restricted to a vector space and its dual). A lot of physics literature is sloppy by a mathematician's standards (it is normally is), but at least we should assume that it is being sloppy about a mathematically rigorous underlying concept. So I'd suggest that the physics literature does implicitly define the tensor product more rigorously than in the article, even if most authors do not go beyond using a vector space and its dual. —Quondum 18:15, 3 December 2014 (UTC)

I agree on the braiding part. For the record, the physics literature is generally on point on that matter (since they use (abstract) index notation). The sloppiness here is completely inherited from the mathematics sources used. (Mostly Lee, which for some reason is no longer cited as a source.) This does not mean that anybody used the much more general definition that you propose. I would even go as far as saying that that definition makes no sense, because many concepts related to tensors (such as type and rank) only make sense if the tensor are constrained to elements of tensor products of a vector space with itself and its dual.T R 20:12, 3 December 2014 (UTC)

Though I disagree with your examples (type as defined here is a special case of a more general concept, rank seems to me to be a nonexample, and contraction is in any event defined only for a vector space and its dual), I am *not* suggesting that we elaborate on the general case in this article. (Despite being a stunning notation, abstract index notation is ambiguous about braiding unless a convention such as lexicographical order is used, but I take your point about the sloppiness.) All I'm suggesting is that a suitable definition would be along the lines of an ordered product of vector spaces over the same field, which can then immediately be followed by a restriction for this article to a vector space and its dual. If we want to treat the general case, this should go into a more abstractly oriented article. —Quondum 20:37, 3 December 2014 (UTC)

Do you have any sources that use the term "tensor" in the general sense that you propose. Are there sources that refer to an element of a "heterogeneous" tensor product as a "tensor". There may be, but I have never seen them. (Off-topic: In my understanding lexicographical ordering is a standard part of (abstract) index notation, it is probably the single most common use of the \phantom command in LaTeX.)T R 22:26, 3 December 2014 (UTC)

The answer is, it depends. From a geometric POV , tensors are geometric objects that have particular transformation properties induced by the geometric space they are in; those transformation properties may be said to define the type of tensor he object is. If the set of transformations are all linear, then linear superpositions of tensors of the same type are again a tensor of that type. In physics, all the transformations I have seen, either global or local in the tangent space, are linear, so it would seem that closure under superposition holds. Maybe there are spaces with inherently nonlinear transformation properties--I don't know--but if there were, it would make defining even the underlying vector space difficult.

Mathematically, things are a bit more subtle. If the vector spaces are actually infinite dimensional Hilbert spaces, then metric completion under the inner product norm needs to be considered. As Tensor product of Hilbert spaces discusses, there is a natural way to do this for finite tensor products of Hilbert spaces. But for infinite products of Hilbert spaces, only a restricted set of superpositions (see Tensor product of Hilbert spaces#Infinite tensor products) may make sense. For more general topological spaces, like Banach spaces, even forming tensor products becomes hairy, see Topological tensor product. This may be what Chatul is referring to above. --Mark viking (talk) 20:43, 3 December 2014 (UTC)

By the way, in an old discussion on this talk page, one editor asserted that the term "tensor" does not include elements of tensor products over general modules, and another editor (me) was unable to find reliable sources saying otherwise. Mgnbar (talk) 21:04, 3 December 2014 (UTC)

Hmm, well its true that the current mathematical fashion is to talk categorically about mappings of tensor spaces rather than transformations of tensor elements, but the tensors are still there. A quick search shows discussion of simple tensors in the module context in a textbook on abstract algebra and a set of course notes (p. 3) and monomial tensors (a synonym) in another set of course notes (p. 394). --Mark viking (talk) 22:16, 3 December 2014 (UTC)

I think I have seen "pure tensors" used in a similar sense, but in any of the cases where these terms are used, does man ever refer to a linear combination of "simple/monomial/pure tensors" as a "tensor"?T R 22:37, 3 December 2014 (UTC)

Of course, what else would they be called? Here is another ref that explicitly states on page 2 that an element of a module tensor space is called a tensor and can be expressed as an R-linear combination of simple tensors. --Mark viking (talk) 23:10, 3 December 2014 (UTC)

Given that a tensor product of modules is again a module, it would not be unthinkable to call its elements, "elements". In fact, that is what mathematicians do most of the time. In mathematics, there is no real need for a special word for "element of a tensor product of spaces", hence there not being one was a distinct possibility. (Even when an obvious term existed).

Given the existence of sources that use the term such, I am not against adding something that reflects that usage in the article. A printed source would obviously be better than self-published notes to support whatever we add.T R 12:50, 4 December 2014 (UTC)

I don't think it is appropriate for an encyclopedia article to emphasize edge cases like this. The fact that there exists some person who calls elements of a tensor product of modules a "tensor" does not mean that we should do the same, particularly when most mathematicians would not. For the vast majority of sources, "tensor" means the usual familiar notion (finite dimensional, associated to tangent vectors). WP:WEIGHT is relevant. Sławomir Biały (talk) 14:21, 4 December 2014 (UTC)

While I agree with not emphasizing terminology in cases where it is not commonly used, we should also avoid making the implication that it does not apply. For example, to restrict its use to contexts where it is defined in terms of tangent vectors threatens to blur the concept with a tensor field, a blurring that we should carefully avoid. We should avoid unnecessarily confining the definition of a concept to a narrower use than ~~its correct~~ some reasonable definition. For example, we should avoid defining a linear transform as a matrix, and we should avoid defining vector spaces as being only over the real and complex fields. That other terms may be preferentially used instead in some contexts does not imply that ~~the original~~ a nominal general term does not apply. —Quondum 15:26, 4 December 2014 (UTC)

I object to your use of the words "correct" and "original" they suggest things that have no basis.T R 19:34, 4 December 2014 (UTC)

Better?

—Quondum 21:05, 4 December 2014 (UTC)

Wait, are you asserting that although mathematicians carefully call certain universal products of modules "tensor products", and they carefully call the elements of a spanning set for the tensor product space "simple/pure/monomial tensors", when it comes to linear superpositions of those simple tensors, they apply the forgetful functor and regress to calling those linear combinations "elements"? Hard to believe. I have not seen simple/pure/monomial tensors called anything but tensors in a module context. Do you have sources for this convention? --Mark viking (talk) 20:14, 4 December 2014 (UTC)

I don't think I asserted anything resembling that. But the fact is, really not that many people are ultimately concerned with the tensor products of modules, let alone what to call its elements. This is a non-issue. Sławomir Biały (talk) 22:22, 4 December 2014 (UTC)

Here is a Bourbaki reference, using the tensor nomenclature in a module context: Algebra I chapter III, section 5: Tensor algebras, Tensors, page 492. --Mark viking (talk) 20:34, 4 December 2014 (UTC)

Following on from what Mark says here, it would be reasonable to refer to an element of the space spanned by a set of simple tensors, or to a tensor, but to refer to only an element without the following of phrase (unless this is linguistically implied) would be ... kinda silly? —Quondum 21:17, 4 December 2014 (UTC)

Can we please refocus this discussion? I think it is not unreasonable to include in the "generalizations" section a discussion of tensor products of modules. That section needs other work too, such as moving the mention of monoidal category there (from the "History" section) with expansion. Any takers? Sławomir Biały (talk) 22:02, 4 December 2014 (UTC)

Re: Are there sources that refer to an element of a "heterogeneous" tensor product as a "tensor".

Latest comment: 9 years ago11 comments3 people in discussion

I've added a couple such sources to the article... One is more physics oriented, the other pure math. 86.121.137.79 (talk) 13:56, 19 January 2015 (UTC)

As a third source see https://books.google.com/books?id=NSXCaGSVaX4C&pg=PA262 although he (Knapp) say that some call them "pure tensors". I'm not sure what's pure about them... and I've not encountered that term before. 86.121.137.79 (talk) 14:23, 19 January 2015 (UTC)

See my reply below. These should not be referred to as "tensors" in the context of the present article. This article is about the usual, standard, meaning of the term, not some niche meaning that one or two authors might use. Sławomir Biały (talk) 14:26, 19 January 2015 (UTC)

Who defines "standard" tensors? 86.121.137.79 (talk) 14:32, 19 January 2015 (UTC)

The majority of sources in multilinear algebra, differential geometry, continuum mechanics, mathematical physics, etc. For example, Bourbaki, Algebre, Tome III; Kobayashi and Nomizu, "Foundations of differential geometry"; Hawking and Ellis "The large scale structure of space-time"; Arfken and Weber "Mathematical methods for physicists"; Feynman, Leighton, Sands, Volume 2; Schaum's Outline of Vector Analysis; etc. Sławomir Biały (talk) 14:50, 19 January 2015 (UTC)

Examples of tensor products on infinite dimensional vector spaces

https://books.google.com/books?id=QJg1agw0_-8C&pg=PA38 mentions the result of the tensor product of two Hilbert spaces (albeit poorly typeset). There are probably others. 86.121.137.79 (talk) 14:09, 19 January 2015 (UTC)

Having said that, a more useful construction in the context of Hilbert spaces, called Hilbert tensor product is given in https://books.google.com/books?id=wjzZCLzx6hUC&pg=PA186. 86.121.137.79 (talk) 14:14, 19 January 2015 (UTC)

This is the subject of the article tensor product of Hilbert spaces. More general topological tensor products exist. However, it is not very common to refer to such things as "tensors". This is especially problematic in the context of the current article, where the concept of tensor is the geometrical/algebraic notion. This also precludes discussion of "heterogeneous tensors" that you refer to above. This article is about the usual, standard, notion of tensor. Not the various variants that some comparatively small number of authors employ. Sławomir Biały (talk) 14:25, 19 January 2015 (UTC)

Here is one more example:

Dirac, P. A. M. (1945), Unitary representations of the Lorentz group, vol. 183, Proc. Roy. Soc. A, pp. 284–295, Bibcode:1945RSPSA.183..284D, doi:10.1098/rspa.1945.0003

Dirac doesn't explicitly mention tensor products, but it is clearly what he intends. This theory of expansors (elements of this tensor product) of his, generalizes in a certain sense the notion of a tensor as ordinarily perceived. This was elaborated by Harish-Chandra leading to the theory of expinors,

Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. Roy. Soc. A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047

and, in the by-bass, he found (being first in his PHD thesis for Dirac, I believe) the classification of the infinite-dimensional unitary representations of the Lorentz group. YohanN7 (talk) 14:27, 19 January 2015 (UTC)

Such things certainly exist, and are even the topic of an earlier thread. But it is a mistake in the context of the present article to conflate these with the notion of "tensor" as covered here. It would be confusing in the extreme, for example, if we were to go around saying "spinors are tensors". Sławomir Biały (talk) 14:34, 19 January 2015 (UTC)

You are right, but the topic is (here in the talk section) not off topic since "Tensor products of infinite dimensional spaces" is the title of the present thread and since "tensors are elements of a tensor product of vector spaces" is in the article. This deserves an "nb" (nota bene, note box, or whatever it is called, a footnote distinguished from a usual citation) imo.

B t w, feel free to have a look in my sandbox if you want to, but don't hold me responsible for the content you might find there, it's by far not ready for anything (but please point out errors if you do look, it is not the same thing as holding me responsible

). The "expansors" are close to the top. YohanN7 (talk) 14:49, 19 January 2015 (UTC)

Lead

Latest comment: 9 years ago24 comments8 people in discussion

An IP user is going about to define tensors in the lead using the transformation properties definition. This will undoubtedly be reverted (by someone with rollback authority). The added material could perhaps (maybe in modified form) fit into a new section Tensor#Definition. YohanN7 (talk) 10:35, 3 November 2014 (UTC)

Hi could someone please explain what's wrong with that definition? I found the previous wikipedi definition extremely confusing and took the transform properties one directly from Schaum's Tensor Calculus book. I'm trying to understand the topic but as I understand it from Schaum's, a vector or matrix by itself is /not/ a tensor, and only becomes one if it also fulfills those properties? — Preceding unsigned comment added by 143.167.9.247 (talk) 12:31, 3 November 2014 (UTC)

There are many ways to understand a basic topic such as this. Read the article to get an idea for some of these other ways. Reading the article will also help you understand the consensus that has emerged about how Wikipedia should explain tensors.

In particular, some editors find the treatment that you espouse misguided/misleading/offensive/incorrect/whatever. There has been a lot of argument over the years. So propose major changes on the talk page. Cheers. Mgnbar (talk) 12:40, 3 November 2014 (UTC)

Nothing is inherently wrong with the definition of a tensor as a multi-dimensional array with a transformation law. See the section Tensor#As multidimensional arrays where this definition is discussed at length. It is one among several equivalent ways of defining a tensor. A linear transformation is always a tensor, and as a matrix can be regarded as a linear transformation, it also gives a tensor. (In elementary linear algebra, one of the topics covered is the change in a matrix relative to a change in basis.) It is true that a tensor is more than just an array of numbers. Tensors can be represented as arrays of numbers in a basis, but they undergo a transformation when we change that basis. Thinking of tensors as tensor fields at the beginning is not very helpful I think, unless one is already very familiar with the idea of what the Jacobian matrix represents: it is just the change of basis matrix for the coordinate vectors at a point induced by a change in a curvilinear coordinate system. If one already knows this about the Jacobian, then going from a pointwise definition of a tensor to that of a tensor field is rather obvious. If one doesn't already know about Jacobians, then presenting a definition of tensor that involves the Jacobian is unhelpful and confusing. I suggest that you are confused because this is what Schaum's outlines has done, not because of our article on the subject. Sławomir Biały (talk) 12:51, 3 November 2014 (UTC)

Many thanks for the fast replies -- I can see that the article is the way it is due to a long and careful discussion then, so Wikipedia 1 Schaums Nil :-) — Preceding unsigned comment added by 143.167.9.247 (talk) 13:01, 3 November 2014 (UTC)

Unfortunately, there really are no good treatments out there (no Royal road). Most people learn tensors just by using them until eventually one either understands them or is so used to them that understanding is no longer required. Sławomir Biały (talk) 13:07, 3 November 2014 (UTC)

(ec)Slightly more specifically. The definition you gave is actually for a tensor field, i.e. something that assigns a tensor to each point in space. Somewhat confusingly the term "tensor" is used for both concepts (even within the same text). This point generally leads to a lot of confusion. This article is strictly about tensors on a single vector space, i.e. arrays that behave in a certain way when the basis of that vector space is changed. Tensor field are treated in a separate article.

Note that by definition a (geometric) vector (e.g. an element of a tangent space) has the right transformation properties to be a (first order, contravariant) tensor. Similarly, linear maps have the right transformation properties to be type (1,1) tensors. In fact, tensor can be defined entirely in terms of mathematical objects that intrinisically have the right transformation properties (i.e. multilinear maps).

Even more abstractly, tensors can simply be defined as the elements of the tensor product of a vector spaces with copies of itself and its dual. (This was another point where the text you wrote was (at best) misleading).T R 13:04, 3 November 2014 (UTC)

The IP editor has motivated me to examine the citations in this article. A lot of the citations in the Definitions section are to Kline, Morris (1972). Mathematical thought from ancient to modern times. We should probably be using more standard tensor-specific texts. Also there should probably be more citations in the introduction. Mgnbar (talk) 13:09, 3 November 2014 (UTC)

Agree. The Morris Kline references, are mostly due to it being the only text I could find (at the time) that simultaneously treats various approaches to defining tensors. It should definitely be supplemented with more standard references using the various approaches individually.T R 13:15, 3 November 2014 (UTC)

John M Lee's Introduction to smooth manifolds defines the tensor product concretely, proceeds to show that it satisfies the universal property. As a problem, it is outlined that any two definitions that satisfy the universal property are canonically isomorphic. Could we use this? YohanN7 (talk) 22:39, 30 November 2014 (UTC)

Lee does have a fairly good treatment linking some of the definitions we provide. Please do some refs where applicable.T R 13:02, 3 December 2014 (UTC)

A linear transformation is always a tensor?

Re: "a linear transformation is always a tensor" said above. Is this really true? If so please give a proof (here at least). I've added a note with what I found on that after some confusion/question emerged at http://math.stackexchange.com/questions/1108842/why-is-a-linear-transformation-a-1-1-tensor/ There are more refs & discussion over there. 86.121.137.79 (talk) 13:34, 19 January 2015 (UTC)

The tensor-hom adjunction is a natural isomorphism of

End(V)

with

V^{*}\otimes V

. See, for example, Bourbaki Algebre, tome III. Sławomir Biały (talk) 14:36, 19 January 2015 (UTC)

Isomorphism iff V is finite dimensional. But I see you always assume that (and perhaps Bourbaki does too). Unlike the source I cited in the article, Bamberg & Sternberg (1991). A Course in Mathematics for Students of Physics: Volume 2. Cambridge University Press. p. 669. 86.121.137.79 (talk) 14:45, 19 January 2015 (UTC)

Yes, we do here follow the vast majority of sources in assuming finite dimensionality of the spaces involved. In fact, even the source you just referred to actually does assume finite dimensions, but they may not say so explicitly. (On page 666, they use the double duality isomorphism, which only exists in finite dimensions, at least for the algebraic dual space.) The "Generalizaitons" section has a short discussion of infinite dimensions. In infinite dimensions, the naive notions of "tensor product" and "linear transformation" do not generalize in a useful way. Sławomir Biały (talk) 15:26, 19 January 2015 (UTC)

As far as I can tell, Bourbaki only talks of a homomorphism

E^{*}\otimes _{A}F\to \mathrm {Hom} _{A}(E,F)

, not of an isomorphism, see http://books.google.com/books?id=STS9aZ6F204C&pg=PA268. (bottom of page) 86.121.137.79 (talk) 15:38, 19 January 2015 (UTC)

Yes, but over (finite-dimensional) vector spaces, this is an isomorphism. Sławomir Biały (talk) 15:41, 19 January 2015 (UTC)

Sure, but my point is that Bourbaki does not appear to share your wiki agenda. 86.121.137.79 (talk) 15:44, 19 January 2015 (UTC)

If you are here to claim that any significant number of sources using the word "tensor" mean it to refer to something in infinite dimensions, then you are just plain wrong. A confrontational attitude certainly does not make you any less wrong. Even Bourbaki, about as general as one can get, ultimately restricts attention to finitely generated projective modules (so, effectively, "tensor fields"). If you are here to learn some linear algebra, that's great for you. But it might be better if you came back after assimilating more sources. Since I don't really appreciate being accused of having an "agenda", I consider further discussion on this matter closed. Sławomir Biały (talk) 15:50, 19 January 2015 (UTC)

Even in the context of vectors, Bourbaki does not state it as an isomorphism, but as a homomorphism with isomorphism in the finite-dimensional case: http://books.google.com/books?id=STS9aZ6F204C&pg=PA308 This is called being hoisted on your own petard. TLDR posturing on civility doesn't change the facts. 86.121.137.79 (talk) 16:09, 19 January 2015 (UTC)

I never claimed that Bourbaki restricted attention to finite dimensional vector spaces. If you look at an early section, I was actually for referencing Bourbaki for the more general case. In this section, you asked (presumably in good faith), for a proof. There is a complete proof in III.5.6 of Bourbaki that, as a bonus, also holds for finitely-generated projective modules (i.e., for tensor fields). But for whatever reason, this thread has become a vicious attack on my own personal motives as an editor. That's just absurd. Nobody said you had to like my answer to your original question, but there is simply no excuse for this abuse. Sławomir Biały (talk) 16:17, 19 January 2015 (UTC)

And, for some reason, the abuse continues offwiki. Unbelievable! Sławomir Biały (talk) 17:31, 19 January 2015 (UTC)

I removed the recently-added blurb about linear transformations and tensors in infinite dimensions. The definitions of "tensor" given in the article are inequivalent in infinite dimensions, because various identifications are made under double-duality, so making precise statements about what

V\otimes V^{*}

means as a submodule of Hom(V,V) is an error. Anyway, it's offtopic for the article, and I have added a note here that the vector spaces involved are finite-dimensional. Sławomir Biały (talk) 11:59, 20 January 2015 (UTC)

The note belongs in the article itself, either up front or wherever it is relevant, e.g., in Tensor#Examples. Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:15, 29 January 2015 (UTC)

Definition/intro proposal

Latest comment: 9 years ago18 comments4 people in discussion

If tensors "belong" to mathematics, then this is about the worst article I've seen on wikipedia. There needs to be a basic definition early on. All of the physics specific things should be in sections in later parts of the article. My proposal for a basic definition: Tensors are elements of a tensor product. Alternatively, tensors are vectors specified with 0 or more indices. — Preceding unsigned comment added by 137.141.220.130 (talk) 16:47, 31 March 2015 (UTC)

It is not clear what you mean by this. Tensors may be described/defined from an abstract mathematical perspective, or from the perspective of linear algebra, or from a geometric perspective, or from the perspective of a physicist, or ... you get my drift. This article seems to be aimed more at the physicist than a mathematician with its emphasis on geometry, and it doesn't even start "In mathematics [...]". Your first proposal is given adequately near the start of the body as one of the alternative definitions in Tensor § Using tensor products, and your second proposal does not qualify as a definition. If you specifically feel that a compact definition is needed in the lead rather than merely a description, this is very difficult to address, considering the many ways to define a tensor. —Quondum 17:45, 31 March 2015 (UTC)

I find the entire lead-in to be extremely confusing from a mathematical perspective. The first sentence essentially boils down to "tensors are a way of specifying linear functions", which I doubt anyone would say is 100% correct. Furthermore, tensors appear in many different areas in mathematics including geometry, however at the most fundamental level they are an algebraic object. I think the entire article (or at least the lead-in) should be rewritten, and I would like to start a conversation on how to go about it. For one, if the community thinks that tensors most fundamentally a mathematical object, then the article should be written from a mathematical perspective. My proposed definition was the best I could come up with which might be accessible to a 2nd year math or physics undergraduate.137.141.220.130 (talk) 18:09, 31 March 2015 (UTC)

There are at least three mathematical ways of defining a tensor. These are described in the relevant section of the article, that is devoted to the definitions. One is as a multilinear map from some Cartesian product of copies of a vector space and its dual. Another is an equivariant function on a torsor for the general linear group. Another is an element of a tensor product of copies of a vector space and its dual space. Also, please note that none of these is the definition you have suggested: "Tensors are elements of a tensor product." This is incorrect: a tensor (in the sense of the word in overwhelmingly prevailing use) is not just an element of some random tensor product. It specifically concerns tensor products of a vector space and its dual. One can probably find a few counterexamples to this in the literature, but the article should not be written from this minoritarian perspective. Sławomir Biały (talk) 18:23, 31 March 2015 (UTC)

I don't have access to the literature at this moment, however among algebraists and presumably among other mathematicians it is quite common to refer to arbitrary elements of tensor products as tensors. This usage can be seen on multiple wikipedia articles involving tensors. Thus the question is what definition should be used. I guess that is why there seem to be so many flame wars on these talk pages.137.141.220.130 (talk) 19:40, 31 March 2015 (UTC)

I'm not questioning that some mathematicians use the word "tensor" to refer to something more general that is not the subject of this article. The issue is one of WP:PRIMARYTOPIC. We already have an article tensor product, that already covers the use "among algebraists and presumably among other mathematicians". (Really? All of them?) Sławomir Biały (talk) 20:34, 31 March 2015 (UTC)

We all suffer from observational bias. In my experience as a mathematician, most mathematicians use the tensor product definition of tensor. Evidently in your experience this is not the case. However it is clear from the talk pages that many people with a math background are coming to this page and finding the treatment rather confusing. Perhaps the lead-in could state which perspective this article is written for?137.141.220.130 (talk) 21:06, 31 March 2015 (UTC)

"In my experience as a mathematician, most mathematicians use the tensor product definition of tensor." I'm willing to grant this even if I don't agree with it (e.g., in non-linear analysis tensors are much more commonly defined using the multilinear functions approach). But I find it rather unbelievable that "most mathematicians" refer to an element of an arbitrary tensor product as a "tensor". Usually such things are understood in terms of a fixed tensor algebra. There may be occasions on which it is expedient to call something of the form

V\otimes W

a "tensor", where there is no relation between the vector spaces V and W, but these are the exception rather than the rule.

"We all suffer from observational bias." Precisely my point. Show an article on tensors to a student in a graduate algebra course, they think it should be written from the point of view of tensor products through-and-through, because this is the only thing they were exposed to by their teachers (and thus naturally think that their way is the "true way"). Show it to a graduate student specializing in category theory, and they are shocked that the article is not written from the point of view of monoidal categories. Show it to someone whose only real mathematical background is functional analysis, and they start talking about tensor products in infinite dimensions. Students familiar with relativity theory will think of tensors as tensor fields with coordinate transitions. And so on...

We need to be more post-modern in our thinking. There is a common notion of tensor that underlies these various disciplines that have generalized the notion in as many ways. The article should be written from a perspective that incorporates these various points of view. It should not give special privilege to any treatment as the "correct" one. So you'll not get much sympathy from me on the idea that the article should be written this-way-or-that. Sławomir Biały (talk) 22:03, 31 March 2015 (UTC)

Perhaps it would be helpful to acknowledge that finding what the "best" way to define a tensor is is impossible for our purposes. The OP has made another point, though: that of understandability of the presentation. I'm afraid that this too is problematic to improve; even the question of how to improve it has proven highly contentious in the past. So I wouldn't start with high hopes here. A specific improvement might be suggested, but others are likely to object and little progress might be made. —Quondum 03:43, 1 April 2015 (UTC)

When I started this post, I was not aware that the algebra definition of tensor (an element of a tensor product) and the physics definition (the subject of this webpage) were different. For that I apologize. It also makes me unqualified to edit this page. Nonetheless, much confusion could be avoided if in the lead-in it was made clear that the physics definition is a specific case of the algebra definition. I've been asking around some of my mathematical aquaintances, and many of them in various disciplines are unaware of the difference in definitions. Most (including the non-algebraists) think of a tensor as an element of a tensor product. One (an analyst) only knew the physics definition. Several (analysts and geometers) knew of both definitions, but preferred the algebra definition. Perhaps a good deal of confusion comes from the fact that when tensor products are defined in most algebra courses, people don't usually formally define a tensor. But due to terms like "pure tensor" everyone uses the natural language to refer to the elements of a tensor product as tensors. At any rate, by making it clear in the introduction the different definitions, much confusion could be avoided. To add to the confusion, most of the examples in the history section of this article are based on the algebra definition and not the physics definition. That should probably be cleaned up as well. I still think a better lead-in would start with something like "A tensor is an element of a tensor product. However in physics and some area of mathematics, a tensor only refers to the elements of specific types of tensor products. This is the subject of this article...."137.141.220.130 (talk) 15:05, 2 April 2015 (UTC)

It's a mistake to characterize this as a math versus physics issue. As I said previously, there are at least three definitions of tensor that appear in the mathematics literature. These are related by natural isomorphisms. Also, I am unconvinced that even most algebraists regularly call an element of an arbitrary tensor product a "tensor". Bourbaki, for example, only refers to elements of the tensor algebra of

V+V^{*}

as a tensor, and I've not seen any similarly authoritative source that refers to an element of an arbitrary tensor product as a "tensor". This article is not about some notion in physics. It concerns the primary notion of tensor, as it appears in both mathematics and physics. Mathematicians are presumably clever enough to figure out how to follow the little blue links to the article tensor product, which concerns this distinct but related notion. Sławomir Biały (talk) 15:27, 2 April 2015 (UTC)

I am convinced that you are mistaken that this article's definition of tensor is the primary one in mathematics. Bourbaki is a bit dated, so it may not reflect modern usage. The algebra and the physics definitions are similar enough that I originally thought that they are the same, but some physicist has done an awful job of explaining it. I'm sure many others have made the same mistake. It is this confusion the article ought to clear up.137.141.220.130 (talk) 15:37, 2 April 2015 (UTC)

Again, there is no "algebra" and "physicist" definition. This is a false dichotomy, and further discussion predicated on this mistaken dichotomy is not likely to be constructive. If you believe the Bourbaki reference is dated, the onus is on you to find sources if similar pedigree that reflect a more contemporary usage. You cannot just dismiss the source because it is old. I've looked in a number of sources that I consider "standard references", e.g., Bourbaki, "Algebra", Kobayashi and Nomizu, "Foundations of differential geometry", Gelfand "Lectures on linear algebra", and the Springer encyclopedia of mathematics. I'm not seeing any support for your supposedly standard "algebra" point of view in these fairly authoritative sources. It's not for want of trying. (Also note, none of these are written by physicists.) Sławomir Biały (talk) 16:12, 2 April 2015 (UTC)

While this doesn't count as a source, the wikipedia article on tensor products (in its current form) uses the term "tensors" in my algebraic sense many times. To me this shows that this sense of the word "tensor" is very common, and the difference in meanings ought to be addressed in this article early on to help avoid confusion.137.141.220.130 (talk) 16:36, 2 April 2015 (UTC)

I'm not disagreeing that sometimes people call elements of a tensor product "tensors", in a setting where there is no risk of confusion. But the primary usage, as indicated by a preponderance of sources in both mathematics and physics, appears to be the notion addressed in this article, refers to an element of the tensor algebra of

V\oplus V^{*}

. Overwhelmingly, when someone says "Let T be a tensor", this is what is intended. Yes, in algebra, as well as in physics. Sławomir Biały (talk) 16:50, 2 April 2015 (UTC)

I agree that if someone says "Let T be a tensor", then they are definitely referring to the definition in this article. The problem is that anyone who does not know that this definition exists, and who knows about tensor products, will think they know what that person means when they do not. I do not see the harm in adding some clarifying language early on. Furthermore, Bourbaki's tensors are far more general than the tensors in this article in that they are elements of the tensor products of modules, not vector spaces. So to me this article is written from a physics perspective. Which I don't mind, again I just think some clarifying language ought to be added to help avoid confusion.137.141.220.130 (talk) 18:34, 2 April 2015 (UTC)

I think the statement "far more general" overstates things rather substantially. Bourbaki's tensors are precisely the kind of "tensors" referred to in this article, just with "modules over a commutative ring" replacing "vector spaces". The article even states that these definitions of tensor are equivalent for finitely generated projective modules (which is ultimately the case of most interest anyway in Bourbaki). So, I don't really see the confinement of the present article chiefly to the case of vector spaces to be a legitimate complaint. Sławomir Biały (talk) 19:23, 2 April 2015 (UTC)

I have added a remark to the Using tensor products section, because I think that the anonymous poster's request for clarification in the article is reasonable. It could probably use a citation. Mgnbar (talk) 20:27, 2 April 2015 (UTC)

"All of this can be generalized, essentially without modification, to vector bundles or coherent sheaves."

Latest comment: 9 years ago9 comments3 people in discussion

I'm afraid this reeks the smell of original research. It cites Bourbaki (in a way) but I don't think Bourbaki's textbook supports this claim. It is true that a finitely projective module behaves like vector spaces regarding tensor products (the important topic in tensor product of modules) but it seems a bit of stretch to claim that basically a tensor analysis works over a non-field. For instance, that not all element in the tensor product of modules are pure tensors seems like already a big difference. -- Taku (talk) 12:56, 9 April 2015 (UTC)

Not all elements of the tensor product of vector spaces are pure tensors, in exactly the same way. —Quondum 13:34, 9 April 2015 (UTC)

Of course (my excuse is that I didn't have my morning coffee). I just meant to say that a funny thing could happen if the base is not a field; for example, you can't choose a basis for one thing and so components don't make sense; that's what I wanted to mean by pure tensor.

All that's being said is that tensor products can be defined either by their universal property or as multilinear maps over vector bundles and coherent sheaves. This is a standard fact, and has nothing to do with tensor analysis. Sławomir Biały (talk) 14:06, 9 April 2015 (UTC)

Thank you for clarifying edit, which addresses the above concern. I do agree that the "constructions" can be done more generally (in fact, there is no need to mention bundles or sheaves; the proper setup is, as I linked before, to use sheaf of modules, but that's besides the point.) -- Taku (talk) 14:26, 9 April 2015 (UTC)

The context is in the recent dispute that somehow "tensors in infinite dimensions" are a fruitful enough generalization of the tensor concept to warrant rewriting the article from that perspective. No, it is not. When one leaves the category of finite dimensional vector spaces, there are many different notions of tensor. Some of these behave more or less like we expect, others do not. Sławomir Biały (talk) 15:06, 9 April 2015 (UTC)

Also, not all definitions are equivalent over sheaves of modules. One needs finitely generated projective modules for the isomorphism

Hom(E,F)

and

E^{*}\otimes F

(and other similar identifications). Sławomir Biały (talk) 21:35, 9 April 2015 (UTC)

I understand the context, but certainly you can't expect the readers to be aware of internal disputes. We have to be conservative here, to keep our reputation intact. As for more mathematical matter, I'm not sure if I follow your point. Some parts generalize; some other don't. For me, a tensor is really "components"; so elements in tensor product of modules doesn't feel like tensors to me (and of course it's purely algebraoc construction devoid of geometric meaning, unless you define geometry as algebra like I might do:). -- Taku (talk) 16:32, 10 April 2015 (UTC)

Yes, I think we are in full agreement on that point. But there has been a consistent push to move this article in the other, non-geometrical, direction. Such content has been added largely to appease such (in my opinion somewhat misguided) ideas. Sławomir Biały (talk) 18:49, 10 April 2015 (UTC)

I would not agree with the "a tensor is components" perspective, since there are equally valid component-free and indeed (to me, at least) more intuitive approaches that avoid components in their definition. This does not change the perspective that this article is about an intrinsically geometric object, or an algebra for manipulating a vector space. So I also agree with the perspective that the scope of the article should not be pushed in the direction of generalization. OTOH, the algebraic approach to geometry has moved towards the abstract zone with discrete geometries etc., so one does need to choose what rings to permit. It seems to me that this article is firmly restricted to the field of real numbers, which I think is sensible. The only points of dispute might arise from physicists wanting to apply the term in an extended sense, e.g. to Hilbert spaces or over complex numbers. —Quondum 20:11, 10 April 2015 (UTC)

Tensor fields section in wrong place?

Latest comment: 9 years ago9 comments4 people in discussion

Currently, the section "Tensor fields" is a subsection of "As multidimensional arrays", suggesting that tensor fields are particular to this view of tensors. Should we also add a subsection to "Using tensor products", explaining that tensor fields are sections of a tensor product of the tangent and cotangent bundles? Actually I propose that we put both of these treatments into a new top-level section after Definition and before Examples. Mgnbar (talk) 20:14, 11 April 2015 (UTC)

It does not seem to to fit in the § Definition section at all, so perhaps it should be removed from this section. I would not be averse to deleting it.

Your characterization is also not all that accessible, and not fully general (it does not technically have to be related to the tangent space of a manifold, but I know that this is nitpicking). I would suggest limiting the mention of tensor fields to little more than a mention somewhere else in the article. —Quondum 20:39, 11 April 2015 (UTC)

I wasn't sure how generally the term tensor field is applied to sections of other vector bundles. I wasn't trying to take a stance.

If we don't mention tensor fields at all, then we have to remove material throughout the article, including Riemann curvature, covariant derivatives, many of the applications, tensor densities, etc. Mgnbar (talk) 21:02, 11 April 2015 (UTC)

Perhaps I'm overdoing it. There is nothing wrong with mentioning tensor fields under examples and applications, which is where it predominantly occurs in the article. But that does not mean that we need to define them under § Definition. Indeed, tensor fields is an application of tensors (and they could even be defined under that section). —Quondum 22:02, 11 April 2015 (UTC)

This discussion runs the risk of overlooking a very important point of view. In physics, engineering, and applied mathematics, it is very typical to think of tensors as multiindexed fields exhibiting compatible changes by the Jacobian under coordinate transformations (that is, tensor fields, in the multidimensional array perspective). One can find this as a definition of tensors in countless textbooks on these subjects. I think we should be wary of pandering too much to the perspective of "pure" mathematics, where such definitions often only appear as corollaries to some more general construction. (This has been a big theme on this discussion page, with many mathematicians complaining that the article should be rewritten to accommodate their favorite dogmatic point of view.) Yet still, for a majority of uses, tensors are still thought of in this old-fashioned, early 20th-century way. I don't see any reason why we can't just let this definition peacefully coexist with the others. Sławomir Biały (talk) 13:36, 12 April 2015 (UTC)

You seem to be responding to the "indices vs. tensor products" issue. But my post is not about that issue. I'm happy that the article treats tensors from multiple viewpoints. Why should tensor fields be presented as if they are compatible with only one of those viewpoints? Mgnbar (talk) 14:37, 12 April 2015 (UTC)

Slawekb, I don't follow entirely what you're trying to say. You seem to be saying that we should keep §§ As multidimensional arrays, with which I agree. It is only §§§ Tensor fields that should go. If you are saying that the term tensor is a name that is used to mean a field that transforms locally under the requisite transformations, I'd reply that it violates WP:NAD to include a homonym. If you are saying that we should specifically mention transformations that are constrained to a holonomic basis, thus relating the transformations to the differential structure, IMO this constraint does not belong under § Definitions, especially as is not a defining property of tensor fields at all, merely a convenient choice. —Quondum 16:09, 12 April 2015 (UTC)

There are many high quality sources that define a tensor field as a multidimensional array in a coordinate system that transforms under the Jacobian of the coordinate change. The issue of the basis being holonomic is not just a convenience. It is essential to the definition: there is no other basis, except those admitted by the patching condition. Once one has a notion of tensor, then one can introduce other bases, but otherwise the definition becomes circular. Sławomir Biały (talk) 17:16, 12 April 2015 (UTC)

And, yes, the article should absolutely mention that the word tensor is often used for tensor fields. It should also be mentioned that tensor is frequently used for thingies that transform under certain representations of the Lorentz group. (Some elements of the spacetime algebra are of this sort.) ~~This should go into the lead already to frame what this article is and is not about.~~ (Part of this is at the very top already.) This article is in the scope of the physics project as well. On several occasions there have been edits to the article where the editor has confused the issues (believing this article is about tensor fields/and or Lorentz transformation related stuff.) I support the idea of having a top level section or a subsection of applications on tensor fields where the most common definition is presented, not much more. YohanN7 (talk) 17:34, 12 April 2015 (UTC)