26.6 Complex structures and Bargmann-Fock quantization for arbitrary d

Generalizing from $d = 1$ to arbitrary $d,$ the additional piece of structure introduced by the method of annihilation and creation operators appears in the following ways:

• As a choice of positive compatible complex structure J, or equivalently a decomposition

M \otimes C = M_{J}^{+} \oplus M_{J}^{-}

• As a division of the quantizations of elements of $M \otimes C$ into creation (coming from $M_{J}^{+})$ and annihilation (coming from $M_{J}^{-})$ operators. There will be a corresponding J-dependent definition of normal ordering of an operator $O$ that is a product of such operators, symbolized by $: O : $

• As a distinguished vector $∣0 ⟩_{J},$ , the vector in $H$ annihilated by all annihilation operators. Writing such a vector in the Schr¨odinger representation as a position-space wavefunction in $L^{2} (R^{d})$ , it will be a Gaussian function generalizing the $d = 1$ case of equation 26.20, with $τ$ now a symmetric matrix with positive-definite imaginary part. Such matrices parametrize the space of positive compatible complex structures, a space called the “Siegel upper half-space”.

• As a distinguished subgroup of $S p (2 n, R)$ , the subgroup that commutes with $J$ . This subgroup will be isomorphic to the group $U (n)$ . The Siegel upper half-space of positive compatible complex structures can also be described as the coset space $S p (2 n, R) / U (n)$

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26.6 Complex structures and Bargmann-Fock quantization for arbitrary d

26.6 Complex structures and Bargmann-Fock quantization for arbitrary d

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