How to switch from tensor products to wedge product

In summary, the conversation discusses the definition of wedge product for two one-forms and how it differs from a tensor product. The computation for the case of n=2 is shown to be correct and highlights the crucial difference between the two products, where A wedge A = 0 but A tensor A is not equal to 0.
  • #1
victorvmotti
155
5
Suppose we are given this definition of the wedge product for two one-forms in the component notation:

$$(A \wedge B)_{\mu\nu}=2A_{[\mu}B_{\nu]}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}$$

Now how can we show the switch from tensor products to wedge product below:

$$\epsilon=\epsilon_{\mu_{1}...\mu_{n}}dx^{\mu_{1}}\otimes...\otimes dx^{\mu_{n}}$$
$$=\frac{1}{n!}\epsilon_{\mu_{1}...\mu_{n}}dx^{\mu_{1}}\wedge...\wedge dx^{\mu_{n}}$$
 
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  • #2
What happens to your equation, if ##A=B##?
 
  • #3
Is this computation below for the case of ##n=2## correct?$$\epsilon= \frac{1}{2} ( \epsilon_{\mu_{1}\mu_{2}}dx^{\mu_{1}}\wedge dx^{\mu_{2}})$$
$$= \frac{1}{2} (\epsilon_{\mu_{1}\mu_{2}}dx^{\mu_{1}}\otimes dx^{\mu_{2}} - \epsilon_{\mu_{2}\mu_{1}}dx^{\mu_{2}}\otimes dx^{\mu_{1}})$$
$$= \frac{1}{2} ( \epsilon_{\mu_{1}\mu_{2}}dx^{\mu_{1}}\otimes dx^{\mu_{2}} - (-1) \epsilon_{\mu_{1}\mu_{2}}dx^{\mu_{1}}\otimes dx^{\mu_{2}} )$$
$$= \frac{1}{2}(2\epsilon_{\mu_{1}\mu_{2}}dx^{\mu_{1}}\otimes dx^{\mu_{2}})$$
$$= \epsilon_{\mu_{1}\mu_{2}}dx^{\mu_{1}}\otimes dx^{\mu_{2}}$$
 
  • #4
Yes, the computation is correct. The wedge product is a particular type of tensor product.
 
  • #5
dextercioby said:
Yes, the computation is correct. The wedge product is a particular type of tensor product.
In this literally universal view every product is a tensor product.

Edit: ##A \wedge A = 0## whereas ##A \otimes A## is not. This is a crucial difference.
 
Last edited:
  • #6
Point 2. how we are getting 1/2 in front of the tensor product? it should have been A^B=AxB-BxA
 

1. What is the difference between tensor products and wedge products?

The main difference between tensor products and wedge products lies in the way they combine two vectors or tensors. Tensor products result in a new vector or tensor with a higher rank, while wedge products result in a new vector or tensor with a lower rank.

2. How do I switch from tensor products to wedge products?

To switch from tensor products to wedge products, you can use the wedge product formula: A ∧ B = 1/2(A ⊗ B - B ⊗ A). This allows you to convert a tensor product into a wedge product and vice versa.

3. Can I use wedge products in higher dimensions?

Yes, wedge products can be used in any number of dimensions. In fact, wedge products are particularly useful in higher dimensions as they allow for the representation of geometric concepts such as volume and orientation.

4. What are the applications of using wedge products in scientific research?

Wedge products have a wide range of applications in various fields of science, including physics, mathematics, and computer science. They are commonly used in differential geometry, quantum mechanics, and data analysis, among others.

5. Are there any limitations to switching from tensor products to wedge products?

While wedge products have many advantages, they also have some limitations. One limitation is that they do not follow the same algebraic rules as tensor products, which can make calculations more complex. Additionally, wedge products can only be used for certain types of vectors and tensors, such as covectors and anti-symmetric tensors.

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