26 Complex Structures and Quantization
The Schr¨odinger representation of uses a specific choice of extra structure on classical phase space: a decomposition of its coordinates into positions and momenta . For the unitarily equivalent Bargmann-Fock representation a diferent sort of extra structure is needed, a decomposition of coordinates on phase space into complex coordinates and their complex conjugates . Such a decomposition is called a “complex structure” and will correspond after quantization to a choice that distinguishes annihilation and creation operators. In previous chapters we used one particular standard choice , but in this chapter will describe other possible choices. For each such choice we’ll get a diferent version of the Bargmann-Fock construction of a Heisenberg group representation. In later chapters on relativistic quantum field theory, we will see that the phenomenon of antiparticles is best understood in terms of a new possibility for the choice of J that appears in that case.
Chapter contents
- 26.1 Complex structures and phase space
- 26.2 Compatible complex structures and positivity
- 26.3 Complex structures and quantization
- 26.4 Complex vector spaces with Hermitian inner product as phase spaces
- 26.5 Complex structures for d=1 and squeezed states
- 26.6 Complex structures and Bargmann-Fock quantization for arbitrary d
- 26.7 For further reading
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正文：英文 · OCR 机器稿 · 待校对
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来源版本：2025-10-20
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