16 Quadratic Polynomials and the Symplectic Group

In chapters 14 and 15 we studied in detail the Heisenberg Lie algebra as the Lie algebra of linear functions on phase space. After quantization, such functions will give operators $Q_j$ and $P_j$ on the state space $H$. In this chapter we’ll begin to investigate what happens for quadratic functions with the symplectic Lie algebra now the one of interest.

The existence of non-trivial Poisson brackets between homogeneous order two and order one polynomials reflects the fact that the symplectic group acts by automorphisms on the Heisenberg group. The significance of this phenomenon will only become clear in later chapters, where examples will appear of interesting observables coming from the symplectic Lie algebra that are quadratic in the $Q_j$ and $P_j$ and act not just on states, but non-trivially on the $Q_j$ and $P_j$ observables.

The identification of elements $L$ of the Lie algebra $\mathfrak{sp}(2d,\mathbf{R})$ with ordertwo polynomials ${\mu }_L$ on phase space $M$ is just the moment map for the action of the symplectic group $Sp(2d,\mathbf{R})$ on $M$. Quantization of these quadratic functions will provide quantum observables corresponding to any Lie subgroup $G\subset Sp(2d,\mathbf{R})$ (any Lie group $G$ that acts linearly on $M$ preserving the symplectic form). Such quantum observables may or may not be “symmetries”, with the term “symmetry” usually meaning that they arise by quantization of a ${\mu }_L$ such that $\{{\mu }_L,h\}=0$ for h the Hamiltonian function.

The reader should be warned that the discussion here is not at this stage physically very well-motivated, with much of the motivation only appearing in later chapters, especially in the case of the observables of quantum field theory, which will be quadratic in the fields, and act by automorphisms on the fields themselves.

中文目录

Chapter contents

• 16.1 The symplectic group
  • 16.1.1 The symplectic group for $d=1$
  • 16.1.2 The symplectic group for arbitrary d
• 16.2 The symplectic group and automorphisms of the Heisenberg group
  • 16.2.1 The adjoint representation and inner automorphisms
  • 16.2.2 The symplectic group as automorphism group
• 16.3 The case of arbitrary d
• 16.4 For further reading

Previous chapter: 15 Hamiltonian Vector Fields and the Moment Map

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