A Differential Forms or Tensors for Theoretical Physics Today

[giulio_hep]

By the way, since @PrecPoint has mentioned the covariant exterior derivative, now it is maybe interesting to notice that, via the Chern-Weil theory, the coefficients of the characteristic polynomial of the curvature form (= covariant exterior derivative) do not depend on the choice of connection, but they can be seen as an obstruction to find global sections and are topological invariants, appearing also in many applications in physics, for example the geometric quantization. A similar treatment can be done in Higgs bundles... As an aside, since @lavinia added a comment about the relationship between the two disciplines and a metaphor, you can hear Nigel Hitchin quoting J W von Goethe in the beginning of his Higgs bundles talk:

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and right away it is something entirely different.

 

[kay bei]

means that one can form the "totally-symmetric part" of a tensor and the "totally-antisymmetic part" of a tensor (not unlike finding the real-part of a complex number and the imaginary-part of a complex number).

For example,
M_{ab} =\displaystyle \frac{(M_{ab} +M_{ba})}{2} + \frac{(M_{ab} -M_{ba})}{2} = M_{(ab)} + M_{[ab]},
where M_{(ab)} is the symmetric-part and M_{[ab]} is the antisymmetric-part.

If M_{ab}=M_{(ab)} (or equivalently, M_{[ab]}=0), then M_{ab} is said to be symmetric.

If M_{ab}=M_{[ab]} (or equivalently, M_{(ab)}=0), then M_{ab} is said to be antisymmetric.

But in general, since M_{ab}=M_{(ab)} + M_{[ab]},
the general tensor M_{ab} is neither symmetric nor antisymmetric.

Robphy, that is a great analogy with complex numbers. So then from what I understand, a Tensor with a non-zero symmetric part cannot be written in differential forms (because forms are antisymmetric). So from my understanding, you can rewrite the antisymmetric type (0,n) tensor in a differential form notation. And that is all it is, a convenient notation with properties which can often help intuition when working in higher dimensions and often times saving space in calculations.

I am also aware that the generalization of antisymmetric tensor to higher dimensions is called alternating tensor. So my question here is, does the definition of differential form as antisymmetric (0,n) tensor generalize to , a differential form is a alternating type (0,n) tensor?

 

[weirdoguy]

I am also aware that the generalization of antisymmetric tensor to higher dimensions

There is no generalization, tensors (including antisymmetric ones) are defined on vector spaces of any arbitrary dimension. So (in this context) they are as general as they can be. You can use different names, but that does not change the object you are talking about. In most cases, alternating tensor is a different name for an (totally) antisymmetric tensor.

 

[giulio_hep]

So then from what I understand, a Tensor with a non-zero symmetric part cannot be written in differential forms (because forms are antisymmetric).

Notice that multilinear forms are covariant tensors.
Differential (forms) are totally antisymmetric (covariant tensors).

And that is all it is, a convenient notation...

It is much more than notation convenience: it is the natural language to describe the notions of volume and orientation.

 

[kay bei]

Notice that multilinear forms are covariant tensors.
Differential (forms) are totally antisymmetric (covariant tensors).
It is much more than notation convenience: it is the natural language to describe the notions of volume and orientation.

Oh okay, so differential forms are a subset of multilinear forms. And more specifically differential forms are totally antisymmetric multilinear forms. So can we write any Tensor as multilinear form? Why are differential forms much more important to differential geometry than multilinear forms? Is it differential forms or is it multilinear forms that provide the natural language for volume and orientation? Can you suggest a good textbook or set of lecture notes for physicists or engineers on differential geometry which provides a good explanation of all of this. It would be good to have a problem based book and informal style of mathematics as used by many physicists.

 

[giulio_hep]

Is it differential forms or is it multilinear forms that provide the natural language for volume and orientation?

Absolutely differential forms. If it is permitted here, I'd simply suggest you this succinct but effective answer.

And Terence Tao wrote a short introduction to explain why:

The integration on differential forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples of cohomology.

 

[afi1188]

To answer the initial question, forms are a subset of tensors...So obviously not all formulas using tensors can be converted to differential forms equivalent.

If i may ask a question as i am not a specialist: Is there any known computational/logic process (by this i mean kind of an algorithm...that can be implemented using a computer for example) that allows to convert the syntax of an equation using diff forms to equivalent equation using tensors (and the reverse, of course with the requirement that the equation only uses tensors within the forms subset) ? ...of course i know that for a given equation a mathematician would be able to do it...

thanks

see below extract form Fortney A visual introduction to diff forms...
https://i.stack.imgur.com/aJU2u.jpg

 

[WWGD]

To answer the initial question, forms are a subset of tensors...So obviously not all formulas using tensors can be converted to differential forms equivalent.

If i may ask a question as i am not a specialist: Is there any known computational/logic process (by this i mean kind of an algorithm...that can be implemented using a computer for example) that allows to convert the syntax of an equation using diff forms to equivalent equation using tensors (and the reverse, of course with the requirement that the equation only uses tensors within the forms subset) ? ...of course i know that for a given equation a mathematician would be able to do it...

thanks

see below extract form Fortney A visual introduction to diff forms...
https://i.stack.imgur.com/aJU2u.jpg

Not clear on your question: Every Differential Form is a tensor ( but obviously not vice versa). In particular, they are alternating antisymmetric tensors.

 

[afi1188]

i know that…my question was about the existence of an algorithm/method to convert a formula using forms onto a formula using tensors….mainly is this doable using a computer for example?
the reverse should also be possible in the specific cases where the tensors involved are all aternating and antisymétric
i understand that forms are a tool used for easy computational results in specific cases/conditions,
tensors being the reference in more general cases : required for example when a metric tensor is involved like einstein field equations.

 

[weirdoguy]

method to convert a formula using forms onto a formula using tensors

Forms are tensors, so every formula involving forms is, by definition of a form, also a formula involving tensors.

 

[afi1188]

yes i totally agree, but i think syntactically using forms or tensors with indices makes the formula very different.

 

[weirdoguy]

Both forms and general tensors can be written with, or without indices. One has to remember what is the origin of index notation. All tensors are multilinear mappings and can be written out as a linear combinations of tensor products of bases of tangent and cotangent spaces (wedge product is an antisymmetrized tensor product) and that is where index notation comes in. You have forms, e.g.:
$$f=f_{ij}\textsf{d}x^i\wedge\textsf{d}x^j$$
and you have general tensors, e.g.:
$$a=a_{ijk}\textsf{d}x^i\otimes\textsf{d}x^j\otimes\textsf{d}x^k$$
or:
$$b=b_{ij}^{\phantom{ij}k}\textsf{d}x^i\otimes\textsf{d}x^j\otimes\frac{\partial}{\partial x^k}$$
Of course I assume Einstein's summation convention.

 

[lavinia]

By the way, since @PrecPoint has mentioned the covariant exterior derivative, now it is maybe interesting to notice that, via the Chern-Weil theory, the coefficients of the characteristic polynomial of the curvature form (= covariant exterior derivative) do not depend on the choice of connection, but they can be seen as an obstruction to find global sections and are topological invariants, appearing also in many applications in physics, for example the geometric quantization. A similar treatment can be done in Higgs bundles... As an aside, since @lavinia added a comment about the relationship between the two disciplines and a metaphor, you can hear Nigel Hitchin quoting J W von Goethe in the beginning of his Higgs bundles talk:

@giulio_hep

A couple of points:

Invariant polynomials evaluated on the matrix of curvature 2 forms are not independent of the connection. Two connections determine different forms in each dimension. But forms of the same dimension do represent the same cohomology class. That is: they are closed forms that differ by an exact form. For complex vector bundles, each cohomology class is a multiple of one of the Chern classes . By this is meant that each is a multiple of the image of a Chern class under the coefficient homomorphism $H^{*}(M;Z)→ H^{*}(M;C)$.

I do not believe that Chern classes of complex manifolds are in general topological invariants of the manifold. Rational Pontryagin classes of a real manifold ,though, are. These are Chern classes of the complexified tangent bundle (not the tangent bundle) modulo torsion. Also I believe that the top Chern class is the Euler class of the tangent sphere bundle and so is a topological invariant as well.

If one does all of this by the Weil homomorphism then one gets forms on the tangent bundle of a principal Lie group bundle so the forms are not cohomology classes of the manifold. However they are pullbacks of forms on the manifold.

For the OP: This area of mathematics originated in geometry and only later came to Physics. The subject began with the first studies of geometric invariance by Gauss in the early 19'th century. The Gauss Bonnet Theorem started the subject off. In Chern-Weil language it says that the Gauss curvature multiplied by the volume element of a closed oriented surface is 2π times its first Chern class and is 2π times the Euler class of the tangent bundle. Chern and Weil generalized Gauss's original insight. Nowadays these geometric ideas are daily bread for Physicists.

 

[Infrared]

I do not believe that Chern classes of complex manifolds are in general topological invariants of the manifold.

I think $\mathbb{C}P^n$ and $\overline{\mathbb{C}P^n}$ work as an example. If you want "topological invariant" to include orientation, I guess you should also fix $n$ to be even.

 

[lavinia]

I think $\mathbb{C}P^n$ and $\overline{\mathbb{C}P^n}$ work as an example. If you want "topological invariant" to include orientation, I guess you should also fix $n$ to be even.

So one has two complex structures on the tangent bundle that have different Chern classes.

 

[giulio_hep]

This area of mathematics originated in geometry and only later came to Physics. ... Nowadays these geometric ideas are daily bread for Physicists.

It sounds like Michael Atiyah says the opposite:

The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics.

 

[WWGD]

It sounds like Michael Atiyah says the opposite:

The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics.

I don't doubt they both feed of each other. Main difference is Mathematics is not bound by Physical reality. This includes Theoretical Physics in a general sense.

 

[lavinia]

I don't doubt they both feed of each other. Main difference is Mathematics is not bound by Physical reality. This includes Theoretical Physics in a general sense.

@WWGD

Interestingly, Chomsky in one of his non-political talks wonders why Mathematics is so effective in describing physical phenomena and suggests that the Universe is a mathematical object.

From this point of view Physical reality is one of many possible mathematical objects as is the 3 sphere or the dihedral group of order 8.

 

[WWGD]

@WWGD

Interestingly, Chomsky in one of his non-political talks wonders why Mathematics is so effective in describing physical phenomena and suggests that the Universe is a mathematical object.

From this point of view Physical reality is one of many possible mathematical objects as is the 3 sphere or the dihedral group of order 8.

I suspect it all comes from the subconscious, though I have only vague ideas of how, none of them verifiable at this point.

 

[lavinia]

I suspect it all comes from the subconscious, though I have only vague ideas of how, none of them verifiable at this point.

Perhaps that subconscious picture of the world is the "a priori synthetic" of Kant's philosophy. I suspect these early thinkers in the time of the Enlightenment were immersed in philosophy and metaphysics.

BTW: The algebra of forms is defined for any smooth vector bundle over a manifold. Pedantically, it is just the algebra of skew symmetric tensors on the bundle. But if one looks at the definition of exterior derivative it is clear that it cannot be extended to other bundles than the tangent bundle. I wonder if there are any generalizations though.

 

[giulio_hep]

it was believed that the flat Euclidean geometry is the geometry of physical space (regarded by Immanuel Kant as being necessarily true as an « a priori synthetic » proposition) until Einstein’s great discovery that space-time, though locally flat, is in fact curve

From "Non Local Aspects of Quantum Phases" by J. ANANDAN, also noticing:

the electromagnetic field strength of a magnetic monopole belongs to a Chern class that is an element of the second de Rham cohomology group.

More generally I'd say that a differential structure with a tangent bundle is almost always assumed in physics (both classical and quantum, and e.g. including here even Penrose spinor bundles) and I can hardly imagine a generalization to other bundles than those ones.