Symmetries and Non-relativistic Quantum Fields

Abstract

In our study (chapters 25 and 26) of quantization using complex structures on phase space, we found that using the Poisson bracket, quadratic polynomials of the (complexified) phase space coordinates provided a symplectic Lie algebra $\mathfrak {sp}(2d,\mathbf C)$, with a distinguished $\mathfrak {gl}(d,\mathbf C)$ sub-Lie algebra determined by the complex structure (see section 25.2). In section 25.3, we saw that these quadratic polynomials could be quantized as quadratic combinations of the annihilation and creation operators, giving a representation on the harmonic oscillator state space, one that was unitary on the unitary sub-Lie algebra $\mathfrak {u}(d)\subset \mathfrak {gl}(d,\mathbf C)$.