We know that the system (Hamiltonian) of $N$ harmonic oscillators possesses $SU(N)$ symmetry, where \begin{equation} H=\hbar \omega \sum_{i=1}^{N}\left( a_{i}^{\dagger } a_{i} +\frac{1}{2}\right) . \end{equation} We can see $H$ is invariant under $U(N)$ transformation: \begin{equation} Ua_{i}^{\dagger } U^{\dagger } =\sum_{j=1}^{N} a_{j}^{\dagger } U_{ji} \ , \end{equation} where $U_{ij} \in U( N)$. Actually, the dynamical group of $H$ is larger $Sp( N,\mathbb{R})$, see How to show that an $N$-dimensional SHO's dynamics symmetry is $SU(N)$?.
However, we know that the free part of the Hamiltonian of QFT can be viewed as an infinite harmonic oscillator in momentum space: \begin{equation} H_{0} =\int \mathrm{d}^{3} kk^{0}\left( a^{\dagger }(\boldsymbol{k}) a(\boldsymbol{k}) +\frac{1}{2} \delta ^{3} (\boldsymbol{k} -\boldsymbol{k} )\right) \ , \end{equation}
where the discrete index $i$ is replaced with continuous momentum $\boldsymbol{k}$. It seems that $H_{0}$ have $U( \infty )$ symmetry (or larger $Sp( \infty ,\mathbb{R})$ symmetry), which seems weird.
So my question is:
- What's the formal mathematical description of this kind of "infinite" symmetry?
- Is there any implication of this symmetry in physics? (If exists/non-trivial?)
