38 Symmetries and Non-relativistic Quantum Fields

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 sp(2d, $C$), 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 $\mathfrak{u}(d)\subset \mathfrak{gl}(d,\mathbf{C})$

The non-relativistic quantum field theory of chapters 36 and 37 is an infinite dimensional version of this, with the dual phase space now the space ${\mathcal{H}}_1$ of solutions of the Schr¨odinger equation. When a group $G$ acts on ${\mathcal{H}}_1$ preserving the Hermitian inner product (and thus the symplectic and complex structures), generalizing the formulas of sections 25.2 and 25.3 should give a unitary representation of such a group $G$ on the multi-particle state space $S^{\ast }({\mathcal{H}}_1)$ .

In sections 36.5 and 37.2 we saw how this works for $G=\mathbf{R}$ , the group of time translations, which determines the dynamics of the theory. For the case of a free particle, the field theory Hamiltonian is a quadratic polynomial of the fields, providing a basic example of how such polynomials provide a unitary representation on the states of the quantum theory by use of a quadratic combination of the quantum field operators. In this chapter we will see some other examples of how group actions on the single-particle space ${\mathcal{H}}_1$ lead to quadratic operators and unitary transformations on the full quantum field theory. The moment map for these group actions gives the quadratic polynomials on ${\mathcal{H}}_1$ , which after quantization become the quadratic operators of the Lie algebra representation.

中文目录

Chapter contents

• 38.1 Unitary transformations on H₁
• 38.2 Internal symmetries
  • 38.2.1 U(1) symmetry
  • 38.2.2 $U(n)$ symmetry
• 38.3 Spatial symmetries
  • 38.3.1 Spatial translations
  • 38.3.2 Spatial rotations
  • 38.3.3 Spin $\frac{1}{2}$ fields
• 38.4 Fermionic fields
• 38.5 For further reading

Previous chapter: 37 Multi-particle Systems and Field Quantization

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