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Symmetries and Non-relativistic Quantum Fields

In: Quantum Theory, Groups and Representations

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  • Peter Woit

    (Columbia University, Department of Mathematics)

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 $$\mathfrak {u}(d)\subset \mathfrak {gl}(d,\mathbf C)$$ .

Suggested Citation

  • Peter Woit, 2017. "Symmetries and Non-relativistic Quantum Fields," Springer Books, in: Quantum Theory, Groups and Representations, chapter 0, pages 477-491, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-64612-1_38
    DOI: 10.1007/978-3-319-64612-1_38
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