Lorentz Invariant Phase Space: Symplectic Geometry

[gasar8]

I have an assignment to show that specific intensity over frequency cubed $$\frac{I}{\nu^3},$$ is Lorentz invariant and one of the main topics there is to show that the phase space is Lorentz invariant. I did it by following J. Goodman paper, but my professor wants me to show this in another way, using symplectic geometry (wedge products), which I am not really familiar with and Liouville's theorem. I found this on Wiki stating that "that the Lie derivative of volume form is zero along every Hamiltonian vector field." Does this prove the also Lorentz invariance?

Also I found http://www.damtp.cam.ac.uk/research/gr/members/gibbons/dgnotes3.pdf notes on symplectic geometry (chapter 13 on p76). Does the statement that $\omega$ is closed (or $d\omega = 0$), mean that it is invariant? Is this proof enough for Lorentz invariance?

 

[Ambitwistor]

Hello,

Thank you for sharing your question and providing helpful background information. I would like to clarify a few points and provide some insights that may help you in your assignment.

Firstly, the specific intensity over frequency cubed, \frac{I}{\nu^3}, is a quantity that is commonly used in physics and astrophysics to describe the energy flux per unit frequency. It is a measure of the brightness or luminosity of an object and is often used in the study of electromagnetic radiation. The Lorentz invariance of this quantity means that it remains unchanged under Lorentz transformations, which are mathematical transformations that describe how physical quantities change when observed from different reference frames in special relativity.

Now, coming to your question about showing Lorentz invariance using symplectic geometry and Liouville's theorem, let me first explain a few key concepts. Symplectic geometry is a branch of mathematics that deals with the geometry of phase space, which is the space of all possible states of a physical system. It is a powerful tool in theoretical physics, particularly in the study of Hamiltonian systems. Liouville's theorem, on the other hand, is a fundamental result in symplectic geometry that states that the volume of a region in phase space is conserved under Hamiltonian flow, which is the flow of a physical system described by a Hamiltonian.

Now, in order to show Lorentz invariance using symplectic geometry, we need to consider the symplectic structure of phase space. This structure is described by a two-form, \omega, which is a mathematical object that assigns a number to pairs of vectors in phase space. This two-form is also known as the symplectic form and it plays a crucial role in symplectic geometry. The statement that \omega is closed (or d\omega = 0) means that it is invariant under small deformations of phase space, which is a key property in symplectic geometry. However, this does not necessarily imply Lorentz invariance.

To show Lorentz invariance, we need to consider the Hamiltonian flow of a physical system. This flow is described by a set of equations known as Hamilton's equations, which are derived from the Hamiltonian of the system. Now, according to Liouville's theorem, the volume of a region in phase space remains constant under this flow. This means that if we