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arXiv:2512.16775

Reconstruction of Quantum Fields: CCR, CAR and Transfields

Nicolás Medina Sánchez, Borivoje Dakić

quant-phmath-phSubmitted Dec 18, 2025• Announced Dec 19, 2025• 0 citations• Revised Apr 14, 2026 (v3)
v1 · Dec 18, 2025v2 · Dec 23, 2025v3 · Apr 14, 2026
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Abstract

One of the traditional ways of introducing bosons and fermions is through creation-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, the maximal generalisations of bosons and fermions consistent with these operational principles.

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One of the traditional ways of introducing bosons and fermions is through creation-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, the maximal generalisations of bosons and fermions consistent with these operational principles.
Identical quantum particles exhibit only two types of statistics: bosonic and fermionic. Theoretically, this restriction is commonly established through the symmetrization postulate or (anti)commutation constraints imposed on the algebra of creation and annihilation operators. The physical motivation for these axioms remains poorly understood, leading to various generalizations by modifying the mathematical formalism in somewhat arbitrary ways. In this work, we take an opposing route and classify quantum particle statistics based on operationally well-motivated assumptions. Specifically, we consider that a) the standard (complex) unitary dynamics defines the set of single-particle transformations, and b) phase transformations act locally in the space of multi-particle systems. We develop a complete characterization, which includes bosons and fermions as basic statistics with minimal symmetry. Interestingly, we have discovered whole families of novel statistics (dubbed transtatistics) accompanied by hidden symmetries, generic degeneracy of ground states, and spontaneous symmetry breaking – effects that are (typically) absent in ordinary statistics.
In this chapter we describe a technique to deal with identical particles that is called second quantization . Despite being a technique, second quantization helps a lot in understanding physics. One can learn and endlessly repeat newspaper-style statements particles are fields , fields are particles without grasping their meaning. Second quantization brings these concepts to an operational level. Since the procedure of second quantization is slightly different for fermions and bosons, we have to treat the two cases separately. We start with considering a system of identical bosons, and introduce creation and annihilation operators (CAPs) that change the number of particles in a given many-particle state, thereby connecting different states in Fock space. We show how to express occupation numbers in terms of CAPs, and then get to the heart of the second quantization: the commutation relations for the bosonic CAPs. As a next step, we construct field operators out of the CAPs and spend quite some time (and energy) presenting the physics of the particles in terms of these field operators. Only afterward, do we explain why all this activity has been called second quantization. In Section 3.4 we then present the same procedure for fermions, and we mainly focus on what is different as compared to bosons. The formalism of second quantization is summarized in Table 3.1.
We present an algebraic method to derive the structure at the basis of the mapping of bosonic algebras of creation and annihilation operators into fermionic algebras, and vice versa, introducing a suitable identification between bosonic and fermionic generators. The algebraic structure thus obtained corresponds to a deformed Grassmann-type algebra, involving anticommuting Grassmann-type variables. The role played by the latter in implementing gauge invariance in second quantization within our procedure is then discussed. This discussion includes the application of the mapping to the case of the bosonic and fermionic harmonic oscillator Hamiltonians.
Summary of the lectures: 1. Creation and annihilation operators for the system of indistinguishable particles 1.1 The permutation group and the states of a system of indistinguishable particles 1.2 Dimension of the Hilbert space of a system of indistinguishable particles 1.3 Definition and properties of the creation and annihilation operators 1.4 TheFockspace 1.5 The representations of state vectors and operators 1.5.1 N-particlewave-functions 1.5.2 The second-quantization representation of operators 1.5.3 Examples 1.6 Evolution of operators in the Heisenberg picture 1.7 Statistical operators of indistinguishable particles 1.7.1 The averages of the s-particle operators 1.7.2 The general structure of the one-particle statistical operator 2 Quadratic Hamiltonian and the diagonalization 2.1 Diagionalization of the Hamiltonian quadratic in the fermion operators 2.2 Diagionalization of the Hamiltonian quadratic in the boson operators 2.2.1 Diagonalization of the quadratic bosonic Hamiltonian possessing a zero mode 2.3 Long range order, condensation and the Bogoliubov spectrum of weakly interacting Bose gas 2.3.1 The excitation spectrum of weakly interacting Bose gas inabox 2.3.2 The excitation spectrum of an interacting Bose gas in an externalpotential 2.4 The Jordan-Wigner transformation: fermionization of interacting 1D spin chains
Abstract Chapter 2 provides a review of pertinent aspects of the quantum mechanics of systems composed of many particles. It focuses on the foundations of quantum many-particle physics, the many-particle Hilbert spaces to describe large assemblies of interacting systems composed of Bosons or Fermions, which lead to the versatile formalism of second quantization as a convenient and eminently practical language ubiquitous in the mathematical formulation of the theory of many-particle systems of quantum matter. The main objects in which the formalism of second quantization is expressed are the Bosonic or Fermionic creation and annihilation operators that become, in the position basis, the quantum field operators.