Dimensionality of the space of p-forms

  • Thread starter center o bass
  • Start date
  • Tags
    Space
In summary, a p-form is a completely anti-symmetric tensor, which means that the form is completely determined by a set of indicies where each i is less than or equal to the previous i. So in n-dimensions we have then $$\frac{n!}{p!(n-p)!}$$independent components of the tensor. Then it's stated that this is the dimensionality of the space of p-forms in n-dimensions. But why is this called the dimensionality? Even though the components ##A_{\mu_1 \cdots \mu_k \cdots \mu_l \cdots \mu_p}##
  • #1
center o bass
560
2
A p-form ##\alpha## is a completely anti-symmetric tensor. This means that the form is completely determined by a set of indicies ##\left\{i_1, i_2, \ldots, i_p\right\}## where ##i_1 < i_2 <...<i_p## since the other components can be found by permutation of these. So in ##n##- dimensions we have then

$$\frac{n!}{p!(n-p)!}$$

independent components of the tensor. Then it's stated that this is the dimensionality of the space of p-forms in n-dimensions. But why is this called the dimensionality? Even though the components ##A_{\mu_1 \cdots \mu_k \cdots \mu_l \cdots \mu_p}## and ##A_{\mu_1 \cdots \mu_l \cdots \mu_k \cdots \mu_p}## are dependent, the basis tensors ##e^{\mu_1}\otimes \cdots\otimes e^{ \mu_k } \otimes \cdots \otimes e^{\mu_l }\otimes \cdots \otimes e^{\mu_p}## and ##e^{\mu_1}\otimes \cdots\otimes e^{ \mu_l } \otimes \cdots \otimes e^{\mu_k }\otimes \cdots \otimes e^{\mu_p}## are not, and thus the corresponding terms in

$$\alpha = \alpha_{\mu_1 \mu_2 \cdots \mu_p} e^{\mu_1} \otimes e^{\mu_2} \cdots \otimes e^{\mu_p} $$

can not be written as linear combinations of each other. I would think that the dimensionality would be equal to that of the number of independent basis vectors. Why is this not the case for forms?
 
Physics news on Phys.org
  • #2
Because we can take [itex]\dfrac{n!}{p!(n-p)!}[/itex] distinct matrices, representing these tensors in some coordinate system, having "1" for each of the independent components and "0" for all others. And every such tensor can then be written as a linear combination of those tensors. The number of independent basis vector is the number of independent components. The "basis vectors" you give are NOT all "completely anti-symmetric".
 
  • #3
So what you're saying is that the combinatorical symbol is the dimension -in the space of anti-symmertric covariant tensors- while it's not in the space of ordinary covariant p-tensors. That males sense! Thanks! :)
 

1. What is the dimensionality of the space of p-forms?

The dimensionality of the space of p-forms is determined by the number of independent p-forms that can be constructed in a given space. It is denoted by the symbol mp and can be calculated using the formula mp = n!/p!(n-p)!, where n is the dimension of the space.

2. How does the dimensionality of the space of p-forms relate to the dimension of the space?

The dimensionality of the space of p-forms is closely related to the dimension of the space. In fact, the dimension of the space is equal to the sum of the dimensions of all the p-form spaces, from p=0 to p=n, where n is the dimension of the space.

3. Can the dimensionality of the space of p-forms change?

Yes, the dimensionality of the space of p-forms can change depending on the dimension of the space. For example, if the dimension of the space is increased, the dimensionality of the space of p-forms will also increase. Additionally, different spaces may have different dimensionality of the space of p-forms.

4. What is the significance of the dimensionality of the space of p-forms in mathematics?

The dimensionality of the space of p-forms plays an important role in various mathematical fields such as differential geometry, topology, and algebraic geometry. It helps to understand the geometric and topological properties of a space and is used in the development of mathematical models and theories.

5. How is the dimensionality of the space of p-forms applied in physics?

In physics, the dimensionality of the space of p-forms is used in various theories and models, such as Maxwell's equations in electromagnetism and Einstein's theory of general relativity. It also plays a crucial role in understanding the properties and behavior of physical systems in different dimensions.

Similar threads

  • Differential Geometry
Replies
29
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
829
  • Differential Geometry
Replies
2
Views
3K
  • Differential Geometry
Replies
1
Views
2K
Replies
1
Views
1K
  • Special and General Relativity
Replies
33
Views
4K
  • Differential Geometry
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Special and General Relativity
Replies
1
Views
3K
  • Differential Geometry
Replies
4
Views
6K
Back
Top