Srednicki — Quantum Field Theory · Problem 24.4

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · symplectic-group-generators

Problem

The elements of the group SO

(N)

can be defined as

N \times N

matrices

R

that satisfy

R_{ii'}R_{jj'}\delta_{i'j'} = \delta_{ij} \,. \tag{24.14}

The elements of the symplectic group Sp

(2N)

can be defined as

2N \times 2N

matrices

S

that satisfy

S_{ii'}S_{jj'}\eta_{i'j'} = \eta_{ij} \,, \tag{24.15}

where the symplectic metric

\eta_{ij}

is antisymmetric,

\eta_{ij} = -\eta_{ji}

, and squares to minus the identity:

\eta^2 = -I

. One way to write

\eta

\eta = \begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix} \,, \tag{24.16}

where

I

is the

N \times N

identity matrix. Find the number of generators of Sp

(2N)