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Reconstruction of Quantum Fields

Nicolás Medina Sánchez, Borivoje Dakić

TL;DR

This work reframes the passage from single-particle to many-particle quantum physics as a quotient of the tensor algebra by an indistinguishability-encoding ideal, producing a Fock space with preserved particle-number grading. It shows that, under an external/internal factorization and three operational constraints, the resulting relations are necessarily quadratic and PBW-compatible, connecting to Yang–Baxter equations and Koszul algebras. The authors construct a full creation–annihilation framework (transfields) via a cross-map and derive a GL(d) representation, with a concrete finite-order example illustrating a nontrivial transtatistical species. A key conceptual outcome is the Koszul duality link between transbosonic and transfermionic statistics, suggesting deep algebraic structure underlying generalized quantum statistics and potential extensions to quantum field theory with causal constraints.

Abstract

One of the traditional ways of introducing bosons and fermions is through creation and annihilation algebras. Historically, these have been associated with emission and absorption processes at the quantum level and are characteristic of the language of second quantization. In this work, we formulate the transition from first to second quantization by taking quotients of the state spaces of distinguishable particles, so that the resulting equivalence classes identify states that contain no information capable of distinguishing between particles, thereby generalising the usual symmetrisation procedure. Assuming that the resulting indistinguishable-particle space (i) admits an ordered basis compatible with how an observer may label the accessible modes, (ii) is invariant under unitary transformations of those modes, and (iii) supports particle counting as a mode-wise local operation, we derive a new class of creation-annihilation algebras. These algebras reproduce the partition functions of transtatistics-maximal generalisations of bosons and fermions consistent with these operational principles.

Reconstruction of Quantum Fields

TL;DR

This work reframes the passage from single-particle to many-particle quantum physics as a quotient of the tensor algebra by an indistinguishability-encoding ideal, producing a Fock space with preserved particle-number grading. It shows that, under an external/internal factorization and three operational constraints, the resulting relations are necessarily quadratic and PBW-compatible, connecting to Yang–Baxter equations and Koszul algebras. The authors construct a full creation–annihilation framework (transfields) via a cross-map and derive a GL(d) representation, with a concrete finite-order example illustrating a nontrivial transtatistical species. A key conceptual outcome is the Koszul duality link between transbosonic and transfermionic statistics, suggesting deep algebraic structure underlying generalized quantum statistics and potential extensions to quantum field theory with causal constraints.

Abstract

One of the traditional ways of introducing bosons and fermions is through creation and annihilation algebras. Historically, these have been associated with emission and absorption processes at the quantum level and are characteristic of the language of second quantization. In this work, we formulate the transition from first to second quantization by taking quotients of the state spaces of distinguishable particles, so that the resulting equivalence classes identify states that contain no information capable of distinguishing between particles, thereby generalising the usual symmetrisation procedure. Assuming that the resulting indistinguishable-particle space (i) admits an ordered basis compatible with how an observer may label the accessible modes, (ii) is invariant under unitary transformations of those modes, and (iii) supports particle counting as a mode-wise local operation, we derive a new class of creation-annihilation algebras. These algebras reproduce the partition functions of transtatistics-maximal generalisations of bosons and fermions consistent with these operational principles.

Paper Structure

This paper contains 30 sections, 10 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. From first to second quantization as a quotient
  3. Operational indistinguishability and information loss
  4. Fock space as a quotient algebra:
  5. Segal's theorem
  6. Quadratic spaces and generalised statistics
  7. Internal degeneracies and quantum grammar
  8. Order implies quadratic generation
  9. U(1) grading
  10. U(d) invariance
  11. Quadratic relations and the generating ideal
  12. Yang--Baxter constraints and PBW property
  13. Single--mode partition function
  14. Creation--annihilation algebras and transfields
  15. Annihilation algebra
  16. ...and 15 more sections

Key Result

Corollary 1

Let $\mathcal{V}$ be a subspace of the algebra of covariant tensors over a complex Hilbert space $H$ that is and assume that $\mathcal{V}$ is maximal among subspaces with these properties that do not contain all tensors from some rank onward. Then $\mathcal{V}$ is either the symmetric ideal or the skew--symmetric ideal.

Theorems & Definitions (23)

  • Definition 1: Characteristic and fully characteristic ideals
  • Corollary 1: Segal
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Theorem 2: Quadratic realisation of transtatistics
  • ...and 13 more