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An Approach to Anti-Wick Ordering of Bosonic Fields

John E. Gough, Hideyasu Yamasita

TL;DR

This work develops a systematic, operator-algebraic approach to anti-Wick (Berezin/Toeplitz) ordering for bosonic fields by embedding commuting classical-like variables into a doubled boson system with a normal operator C = A + B^*, followed by a partial vacuum expectation that implements anti-Wick quantization as a completely positive map. It leverages the complex-wave representation and Bargmann-Segal machinery to realize a Bargmann-type isomorphism between the underlying Hilbert space and a reproducing kernel Hilbert space of anti-holomorphic functions, and extends the framework to functions over general separable Hilbert spaces. The paper also provides a concrete field-theoretic formulation of anti-Wick ordering using two copies of Fock space, defines a commutative Z-algebra, and shows how anti-Wick ordering emerges from a partial trace that preserves complete positivity. Together, these results unify anti-Wick quantization with Cohen-class quantization, yield explicit integral representations, and clarify the factorization and positivity properties that make anti-Wick ordering particularly tractable in both finite and infinite-dimensional settings.

Abstract

We present a new technique for putting general boson fields into ant-Wick ordered form. The anti-Wick map associates an operator with a given function of complex variables, and we show that it may be realized as composition of a mapping to a commutative sub-algebra of a doubled-up boson algebra followed by a partial conditional expectation onto one of the factors.

An Approach to Anti-Wick Ordering of Bosonic Fields

TL;DR

This work develops a systematic, operator-algebraic approach to anti-Wick (Berezin/Toeplitz) ordering for bosonic fields by embedding commuting classical-like variables into a doubled boson system with a normal operator C = A + B^*, followed by a partial vacuum expectation that implements anti-Wick quantization as a completely positive map. It leverages the complex-wave representation and Bargmann-Segal machinery to realize a Bargmann-type isomorphism between the underlying Hilbert space and a reproducing kernel Hilbert space of anti-holomorphic functions, and extends the framework to functions over general separable Hilbert spaces. The paper also provides a concrete field-theoretic formulation of anti-Wick ordering using two copies of Fock space, defines a commutative Z-algebra, and shows how anti-Wick ordering emerges from a partial trace that preserves complete positivity. Together, these results unify anti-Wick quantization with Cohen-class quantization, yield explicit integral representations, and clarify the factorization and positivity properties that make anti-Wick ordering particularly tractable in both finite and infinite-dimensional settings.

Abstract

We present a new technique for putting general boson fields into ant-Wick ordered form. The anti-Wick map associates an operator with a given function of complex variables, and we show that it may be realized as composition of a mapping to a commutative sub-algebra of a doubled-up boson algebra followed by a partial conditional expectation onto one of the factors.

Paper Structure

This paper contains 8 sections, 6 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Complex Wave Representation
  3. Isomorphism
  4. Reproducing Kernel Hilbert Spaces
  5. Generalization to Functions over a Hilbert Space
  6. Anti-Wick Quantization
  7. Anti-Wick Ordering of Fields
  8. Discussion and Conclusion

Key Result

Proposition 3

The pairs $(\hat{A}, \hat{A}^\ast)$ and $(\hat{B} , \hat{B}^\ast )$ are mutually adjoint and satisfy

Figures (1)

  • Figure 1: The anti-Wick mapping taking a function $f(\alpha^\ast , \alpha)$ to an operator $\hat{f} = \mathscr{A} (f)$ based on $\alpha \to \hat{A}, \alpha^\ast \to \hat{A}^\ast$. We obtain $\mathscr{A}$ as a composition of an embedding $\varphi$ of commutative functions in a dilation followed by a partial vacuum expectation $\mathscr{E}$.

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Remark 7
  • Proposition 8
  • Remark 9
  • Corollary 10