Question

Suppose that \(\mathcal{V}\) is a symplectic vector space and \(\mathbf{v}, \mathbf{v}^{\prime} \in \mathcal{V}\) are expressed in a canonical basis of \(\mathcal{V}\) with coefficients \(\left\\{x_{i}, y_{i}, z_{i}\right\\}\) and \(\left\\{x_{i}^{\prime}, y_{i}^{\prime}, z_{i}^{\prime}\right\\}\). Show that $$ \omega\left(\mathbf{v}, \mathbf{v}^{\prime}\right)=\sum_{i=1}^{n}\left(x_{i} y_{i}^{\prime}-x_{i}^{\prime} y_{i}\right) . $$

Short answer

The symplectic product of vectors \(\mathbf{v}\) and \(\mathbf{v}'\) is \(\omega(\mathbf{v}, \mathbf{v}') = \sum_{i=1}^{n} (x_{i}y_{i}' - x_{i}'y_{i})\). This result was obtained by expressing the vectors in the canonical basis, calculating the symplectic product of the vectors and then simplifying the resulting expression using the properties of the symplectic form on the canonical basis.

Step-by-step solution

Express the vectors in terms of canonical basis

Expressing \(\mathbf{v}\) and \(\mathbf{v}'\) in a canonical basis, we have \(\mathbf{v} = \sum_{i=1}^{n} x_{i}e_{i} + y_{i}f_{i}\), and \(\mathbf{v}' = \sum_{i=1}^{n} x_{i}'e_{i} + y_{i}'f_{i}\), where \((e_{i}, f_{i})\) are base vectors in the symplectic space \(\mathcal{V}\).

Calculate the symplectic product of the vectors

The symplectic product \(\omega\) is given by \(\omega(\mathbf{v}, \mathbf{v}') = \omega(\sum_{i=1}^{n} x_{i} e_i + y_{i} f_i , \sum_{j=1}^{n} x_{j}' e_j + y_{j}' f_j)\). By the linearity of a symplectic form we can rewrite the symplectic product as a double sum over the products of the coefficients with the symplectic forms on the canonical basis: \(\omega(\mathbf{v}, \mathbf{v}') = \sum_{i=1}^{n} \sum_{j=1}^{n} (x_i x_j' \omega(e_i, e_j) + x_i y_j' \omega(e_i, f_j) + y_i x_j' \omega(f_i, e_j) + y_i y_j' \omega(f_i, f_j))\).

Simplify the Symplectic Form on Canonical Basis

In any canonical basis for a symplectic space \(\mathcal{V}\), we have two important properties \(\omega(e_i, e_j) = \omega(f_i, f_j) = 0\) and \(\omega(e_i, f_j) = -\omega(f_i, e_j) = \delta_{ij}\), where \(\delta_{ij}\) is the Kronecker delta, which equals 1 when \(i = j\), and 0 otherwise. Substituting these back into the expression for the symplectic product gives \(\omega(\mathbf{v}, \mathbf{v}') = \sum_{i=1}^{n} \sum_{j=1}^{n} (0 + x_i y_j' \delta_{ij} - y_i x_j' \delta_{ij} + 0)\). The Kronecker delta \(\delta_{ij}\) ensures that the only terms remaining in the double sum are those where \(i = j\), hence our final result is \(\omega(\mathbf{v}, \mathbf{v}') = \sum_{i=1}^{n} (x_{i}y_{i}' - x_{i}'y_{i})\).

Key concepts

Canonical Basis

A canonical basis in a symplectic vector space is a special set of basis vectors arranged in pairs, typically denoted as \( (e_i, f_i) \). These pairs are designed to simplify the symplectic structure of the space. Each vector \( \mathbf{v} \) in the symplectic vector space \( \mathcal{V} \) can be expressed using these basis vectors.

For example, if \( \mathbf{v} \, \) is written in terms of the canonical basis, it becomes \( \mathbf{v} = \sum_{i=1}^{n} x_i e_i + y_i f_i \), where \( x_i \) and \( y_i \) are coefficients that specify the vector in the space.

This canonical basis helps to standardize the form of vectors and assists in calculations involving symplectic products (which we'll cover next). The use of canonical basis eases the extraction of information related to the symplectic structure in a more manageable form.

Symplectic Product

The symplectic product, often denoted by \( \omega \), is a fundamental operation in symplectic vector spaces. It is a bilinear form that takes two vectors from a symplectic space and returns a scalar. This product is antisymmetric, meaning \( \omega(\mathbf{v}, \mathbf{v}') = -\omega(\mathbf{v}', \mathbf{v}) \).

In the given problem, the symplectic product is expressed as:

• \[ \omega(\mathbf{v}, \mathbf{v}') = \sum_{i=1}^{n} \left( x_i y_i' - x_i' y_i \right) \]

This equation shows how the coefficients of the vectors in a canonical basis interact through the symplectic product. It highlights how swapping coefficients (in this case, \( x_i \) with \( y_i' \) and vice versa) leads to alternating signs. This is a direct result of the antisymmetry property.

The calculation of the symplectic product involves the symmetry properties of the basis vectors, significantly simplified by using the canonical basis.

Kronecker Delta

The Kronecker delta function, denoted by \( \delta_{ij} \), is a vital component in many areas of mathematics, including symplectic geometry. It is a simple function that's defined as:

• \( \delta_{ij} = 1 \) when \( i = j \)
• \( \delta_{ij} = 0 \) when \( i eq j \)

In the context of the symplectic product, the Kronecker delta is crucial for simplifying the double sum obtained when expanding the symplectic product in a canonical basis.

When performing calculations, the Kronecker delta acts like a filter, letting only terms with \( i = j \) persist in the sum. This leads to the cancellation of irrelevant terms, resulting in a cleaner and more straightforward expression like the final symplectic product formula. Essentially, it means only mutual interactions between corresponding terms are considered, aligning well with properties of the canonical basis.

Thus, the Kronecker delta significantly simplifies the computation and underscores its importance in symplectic geometry.

Linearity of Symplectic Form

Linearity is a fundamental property of the symplectic form \( \omega \), meaning it respects the operations of scalar multiplication and vector addition. When assessing the symplectic product of two vectors, this linearity implies that \( \omega \) distributes over the components of each vector when they are expressed in a canonical basis.

This is evident in how we expanded the symplectic product during the solution:

• \( \omega(\mathbf{v}, \mathbf{v}') = \sum_{i=1}^{n} \sum_{j=1}^{n} (x_i x_j' \omega(e_i, e_j) + x_i y_j' \omega(e_i, f_j) + y_i x_j' \omega(f_i, e_j) + y_i y_j' \omega(f_i, f_j)) \)

Because of linearity, each component of the vectors contributes individually to the result, and each term is processed separately. The symplectic form maintains its linear structure even when expanded over complex expressions.

This characteristic is integral in decomposition and simplification, serving as the groundwork that makes the canonical basis and the Kronecker delta effective tools in simplifying symplectic product calculations.