Symplectic group $Sp(2N)$ in Srednicki's book

Question

There is a question in Mark Srednicki's Book (Problem 24.4, p.160) about $Sp(2N)$, but I am not sure I understand the significance (application?) of this group. In that chapter, he talks about $SO(N)$ and $SU(N)$, which are part of the Standard Model and from accepted answer of this I gather that the structure of the SM does not contain $Sp(2N)$. Is there a reason for Srednicki to give that group as an example, or is it given only as an exercise for a different group from those mentioned in the chapter?

Answer 1

Well, Srednicki also briefly mentions it on page 410 but doesn't explicitly use it anywhere else in the book.

This group though is of great importance in physics. At the classical level, $Sp(2N,\mathbb{R})$ is the group of linear transformations preserving the antisymmetric bilinear form $\Omega$ which determines the symplectic geometry of the phase space ${\mathcal{M}=(p^N,q^N)}$. So, you encounter this group each time you have a system with $N$ degrees of freedom which can be described by Hamiltonian equations. (As you quantize the system, you end up with the double cover of $Sp(2N,\mathbb{R})$, the group $Mp(2N,\mathbb{R})$.)

Besides this, there are some low-dimensional correspondences: ${Spin(3)=Sp(1)=SU(2)}$, ${Spin(4)=Sp(1)\times Sp(1)}$, ${Spin(5)=Sp(2)}$, which are also important in phyiscs,

As a reference, I would recommend Peter Woit's book "Quantum Theory, Groups and Representations".

Answer 2

It is true that symplectic geometry plays a pivotal role in the Hamiltonian formulation, but this aspect is not really explored in Srednicki's book in any depth. Non-abelian Lie groups are mostly discussed in the context of Yang-Mills gauge theory. Unitarity imposes conditions on the gauge group, cf. e.g. this & this Phys.SE posts.

• The non-compact symplectic group $Sp(2N,\mathbb{F})$ is mentioned in Problem 24.4. It is not appropriate as a gauge group.

• Srednicki mentions in the text below eq. (69.23) a compact symplectic group. Presumably he means $Sp(N)\equiv USp(2N)\equiv U(2N)\cap Sp(2N,\mathbb{C})$, which is the topic of this Phys.SE post.

Confusingly, he uses the same notation $Sp(2N)$ in both cases.