Non-degenerate Poisson bracket and even-dimensional manifold

[lalbatros]

From this reference:

titled From Classical to Quantum Mechanics,

I quote the following: ( $$\xi^i$$ are coordinate functions)

Let M be a manifold of dimension n. If we consider a non-degenerate Poisson bracket, i.e. such that

$$\{\xi^i,\xi^j\} \equiv \omega^i^j$$

is an inversible matrix, we may define the inverse $$\omega_i_j$$ by requiring

$$\omega_i_j \omega^j^k = \delta_i^k$$

We define a tensorial quantity

$$\omega \equiv \frac{1}{2}\;\omega_i_j \; d\xi^i \wedge d\xi^j$$

which turns out to be a non-degenerate 2-form.
This implies that the dimension of the manifold M is necessarily even.

My questions are the following:

I don't understand the two statement that I have put in red above.
What is a non-degenerate 2-form?
Why does this one above 'turns out' to be non-degenerate?
Why does that imply that M is even?
Additional comments would be welcome. Like concerning the meaning of $$\omega$$ above.

In addition, I guess the point here by a shorter way: I think that all odd-dimensional antisymmetric matrices are singular. Is there a link with the language used above?

Warm thanks in advance,

Michel

 

[matt grime]

A non-degenerate two form is essentially a non-degenerate symplectic form on the tangent bundle at all points. This implies that the tangent bundle has even dimension. Which is what the things in red are saying.

given any two-form, it is not necessarily non-degenerate, just as any symplectic form is not necessarily non-degenerate.

'it turns out' means 'in this case with these hypotheses we can prove it is'

 

[M98Ranger]

Thank you for your questions. Let me try to explain the statements in red and address your questions.

A non-degenerate Poisson bracket is one where the matrix \omega^i^j is invertible, meaning that it has a well-defined inverse matrix. In this context, the statement \omega_i_j \omega^j^k = \delta_i^k means that when we multiply the two matrices, we get the identity matrix \delta_i^k, which has 1s on the diagonal and 0s elsewhere. This is a way to define the inverse of \omega^i^j.

A non-degenerate 2-form is a tensorial quantity that is defined by the matrix \omega_i_j, as shown in the equation \omega \equiv \frac{1}{2}\;\omega_i_j \; d\xi^i \wedge d\xi^j. This 2-form is non-degenerate because it is defined using the inverse of \omega^i^j, which means that it has a well-defined inverse as well.

The fact that the manifold M must be even-dimensional is a consequence of the non-degenerate Poisson bracket. This is because the 2-form \omega is non-degenerate, and in order for a 2-form to be non-degenerate, the manifold it is defined on must have an even dimension. This is a mathematical result and is not specific to this context.

As for the meaning of \omega, it is a tensorial quantity that is used to define the Poisson bracket in classical mechanics. It is related to the symplectic structure of the manifold M and plays a crucial role in the transition from classical mechanics to quantum mechanics.

Your observation about odd-dimensional antisymmetric matrices being singular is correct. In fact, in this context, the non-degeneracy of the Poisson bracket is equivalent to the antisymmetry of the matrix \omega^i^j. This means that if the matrix is not antisymmetric, it is not non-degenerate, and vice versa.

I hope this clarifies the statements and answers your questions. If you have any further questions, please don't hesitate to ask. Thank you.