Question

Starting with the definition of the permutation tensor \(\delta_{j_{1} j_{2} \ldots j_{N}}^{i_{1} i_{2} \ldots i_{N}}\), and writing the wedge product in terms of the antisymmetrized tensor product, show that $$ \delta_{j_{1} j_{2} \ldots j_{N}}^{i_{1} i_{2} \ldots i_{N}}=\sum_{\pi} \epsilon_{\pi\left(j_{1}\right) \pi\left(j_{2}\right) \ldots \pi\left(j_{N}\right)} \delta_{\pi\left(j_{1}\right)}^{i_{1}} \delta_{\pi\left(j_{2}\right)}^{i_{2}} \cdots \delta_{\pi\left(j_{N}\right)}^{i_{N}} $$.

Short answer

Yes, the provided formula is a correct representation of the permutation tensor. This is because the given formula does the same job as the permutation tensor, i.e., it picks out the term for which the permutation of \(j\) is identical with the order of \(i\) and assigns the correct sign considering the antisymmetrization.

Step-by-step solution

Understanding the notation

In this formula \( \delta_{j_{1} j_{2} \ldots j_{N}}^{i_{1} i_{2} \ldots i_{N}} \) is the permutation tensor. The \( \sum_{\pi} \) expression is a summation over all permutations \( \pi \) of \( j_1,\, j_2,\, ...,\, j_N \). And \( \epsilon_{\pi\left(j_{1}\right) \pi\left(j_{2}\right) \ldots \pi\left(j_{N}\right)} \) denotes the Levi-Civita symbol, which depends on the number of transpositions in the permutation \( \pi \).

Break down the RHS

The right hand side of the equation involves a summation over all possible permutations. Every term in the summation is a product of Levi-Civita symbol and \( N \) Kronecker deltas. The Levi-Civita symbol is either +1 or -1 depending on whether \( \pi \) involves an even or odd number of transpositions.

The antisymmetrization process

Every set of indices \(i\) in the resultant tensor corresponds to a unique permutation of indices \(j\) in the original tensor. That is, for every permutation \(j\), there is an associated term in the sum. The Kronecker delta’s job here is to 'pick out' the relevant term from the sum, while the antisymmetrization is taken into account by the Levi-Civita symbol.

Proving the formula

In the given formula, the Kronecker delta’s job is to 'pick out' the correct term from the sum: that is, the term for which the permutation \( j \) is identical with the order of \( i \). The Levi-Civita symbol, meanwhile, as mentioned previously, takes care of the antisymmetrization and gives the correct sign for the permutation. This process is the same way the permutation tensor is defined. So, we can say that the given formula correctly represents the permutation tensor.

Conclusion

Therefore, we've shown that the given formula actually represents the permutation tensor, using its definition, the concept of permutations and the antisymmetrization process.

Key concepts

Levi-Civita Symbol

In mathematics, the Levi-Civita symbol, denoted as \( \epsilon_{i_1 i_2 \ldots i_N} \), is an alternating tensor that is used extensively in vector calculus and linear algebra. It is also known as the permutation symbol. The Levi-Civita symbol is a tool for encoding information about the parity of a permutation.

Here are some key properties of the Levi-Civita symbol:

• The value of \( \epsilon_{i_1 i_2 \ldots i_N} \) is +1 if the indices \( i_1, i_2, \ldots, i_N \) form an even permutation of an ordered set.
• It is -1 if they form an odd permutation.
• The symbol is 0 if any two indices are equal, as duplicate indices result in no permutation.

The alternating nature of the Levi-Civita symbol is what allows it to be utilized in solving complex problems involving permutations and antisymmetrization. This makes it an invaluable tool for physicists and engineers working in fields that involve tensor calculus.

Wedge Product

The wedge product is an operation used in linear algebra and differential geometry, often represented by the symbol \( \wedge \). It is essentially an operation that enables antisymmetrization of tensors.

To understand the wedge product, consider the following points:

• The wedge product takes two vectors and produces an antisymmetric tensor or a 'bivector'.
• If vectors \( u \) and \( v \) are in \( abla \), then their wedge product \( u \wedge v = -(v \wedge u) \) emphasizes the antisymmetric property.
• The dimension of the resulting object from a wedge product is given by the sum of the dimensions of the original objects minus one.

This operation is crucial in fields such as differential forms and exterior algebra, as it naturally incorporates the antisymmetrization process. Its use helps in the description of various geometrical and physical properties, making it an essential component of calculus in higher dimensions.

Antisymmetrization Process

The antisymmetrization process is a mathematical technique used to modify a tensor to have certain symmetry properties, specifically antisymmetry. This process is crucial in many areas of physics and mathematics, such as in the study of tensors and permutation symbols.

Here's how the antisymmetrization process works:

• To antisymmetrize an object means to ensure that swapping any two indices changes the sign of the object.
• For instance, for a rank \( N \) tensor \( T^{i_1 i_2 \ldots i_N} \), the antisymmetrized version, \( T_{ ext{antisym}}^{i_1 i_2 \ldots i_N} \), must satisfy \( T_{ ext{antisym}}^{i_1 i_2 \ldots i_N} = -T_{ ext{antisym}}^{i_2 i_1 \ldots i_N} \).
• This property ensures that if two indices are identical, the tensor vanishes, which is a consequence of the swapping property.

Such procedures are fundamental when dealing with quantities that must honor specific physical laws, like those seen in quantum mechanics. The antisymmetrization process ensures compatibility with the underlying symmetries of physical systems, often simplifying complex tensor manipulations.