A Differential Forms or Tensors for Theoretical Physics Today

[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.