Second Quantized Fermion and Boson Fields

• Second quantized fermion and boson fields are quantum fields constructed on Fock space via Clifford algebra, unifying internal quantum numbers like spin, charges, and families.
• The approach derives canonical (anti)commutation relations naturally from the algebraic parity of Clifford-odd and even basis vectors, eliminating the need for imposed postulates.
• This unified framework provides insights into gauge field constructions, supersymmetry-like pairings, and extensions to ghost sectors and gravitational interactions in high-dimensional theories.

A second quantized field theory of fermions and bosons is most fundamentally characterized by the operator structure of Fock space, wherein quantum fields are elevated to operator-valued distributions built from creation and annihilation operators that satisfy canonical (anti)commutation relations. In contemporary research, Clifford algebra methods provide a powerful and unifying formalism to construct both fermionic and bosonic second quantized fields from a single algebraic origin, tightly connecting spin, internal charges, gauging, and family structure. Within this framework, the anticommutativity of fermion operators and commutativity of boson operators emerge naturally from the algebraic parity (odd vs. even products) of Clifford algebra elements, rather than being imposed as postulates (Borstnik, 2021, Borštnik, 2023, Borštnik, 2022, Borstnik, 2023, Borstnik, 2022, Borštnik, 9 Dec 2025).

1. Algebraic Construction of Fermion and Boson Basis Vectors

Second quantized fields are constructed by tensoring “basis vectors” of internal space—realized as products of Clifford algebra generators $\gamma^a$—with plane-wave solutions or mode functions in spacetime. The core structure consists of:

• Clifford-odd basis vectors: Superpositions of odd products of $\gamma^a$, forming creation operators for fermionic quanta. Explicitly,

$$B_\text{f}^{(i)} = \sum_{a_1<\dots<a_{2k+1}} C^{(i)}_{a_1\cdots a_{2k+1}} \gamma^{a_1} \cdots \gamma^{a_{2k+1}}$$

These vectors inherit mutual anticommutation due to the anticommuting nature of the Clifford generators, reflecting Fermi statistics.

• Clifford-even basis vectors: Superpositions of even products of $\gamma^a$, giving rise to bosonic quanta:

$$B_\text{b}^{(j)} = \sum_{b_1<\dots<b_{2\ell}} D^{(j)}_{b_1\cdots b_{2\ell}} \gamma^{b_1} \cdots \gamma^{b_{2\ell}}$$

These commute among themselves, as required for Bose statistics.

Nilpotent and projector building blocks constructed from Clifford generators provide eigenstates of a chosen Cartan subalgebra, guaranteeing simultaneous diagonalization of internal charges and spin (Borstnik, 2021, Borštnik, 2023, Borštnik, 2022). Specifically, a nilpotent is $(k) = \frac{1}{2}(\gamma^a + ik\gamma^b)$ with $(k)^2 = 0$, and a projector is $[k] = \frac{1}{2}(1 + \frac{i}{k}\gamma^a\gamma^b)$ with $[k]^2 = [k]$.

2. Fock Space and Second Quantization Postulates

Constructed operators generated from these basis vectors and plane-wave factors act on the Fock vacuum. For momentum $p$, the operators are:

$$c_{\mathbf{p}}^{(i)\dagger} := B_\text{f}^{(i)} \otimes e^{ip\cdot x} \qquad a_{\mathbf{q}}^{(j)\dagger} := B_\text{b}^{(j)} \otimes e^{iq\cdot x}$$

Operating on the total vacuum $|\text{vac} \otimes 0_p\rangle$, these realize the canonical (anti)commutation relations:

$$\{c_{\mathbf{p}}^{(i)}, c_{\mathbf{p}'}^{(i')\dagger}\} = (2\pi)^3 \delta^{(3)}(\mathbf{p}-\mathbf{p}')\delta_{ii'} \quad \{c,c\}=\{c^\dagger,c^\dagger\}=0$$

$$[a_{\mathbf{q}}^{(j)}, a_{\mathbf{q}'}^{(j')\dagger}] = (2\pi)^3 \delta^{(3)}(\mathbf{q}-\mathbf{q}')\delta_{jj'} \quad [a,a]=[a^\dagger,a^\dagger]=0$$

Hence, the operator algebra is not assumed but derived from the Clifford algebraic structure (Borstnik, 2021, Borstnik, 2022, Borštnik, 2022).

3. Internal Degrees of Freedom: Unified Spins, Charges, and Families

The approach uses a single Clifford algebra, typically in high-dimensional even-$d$ spacetime ($d\geq 14$), to encode all internal quantum numbers. The odd Clifford basis corresponds to spinor irreducible representations of $SO(13,1)$ combined with “family” indices, resulting from the action of a second set of Clifford generators $\tilde{\gamma}^a$ generating additional family structure (Borstnik, 2021, Borstnik, 2023, Borštnik, 2023). Explicitly, in $d=14$,

• Each family consists of $2^6 = 64$ Clifford-odd basis states.
• Multiple families arise from the structure of the Clifford algebra and its representation.

Clifford-even basis vectors form adjoint representations of the same Lorentz algebra and, when projected into 4D, correspond to the gauge bosons and scalar Higgs fields of the Standard Model. Under dimensional reduction, internal directions distinguish vector ($\mu=0,1,2,3$) from scalar ($\sigma\geq 5$) bosons (Borštnik, 9 Dec 2025).

4. Paired Structure and Emergent Supersymmetry

In any even dimension $d=2(2n+1)$, the internal (Clifford) space splits into equal numbers of odd and even basis vectors:

$$\#\{\text{fermion~basis}\} = \#\{\text{boson~basis}\} = 2^{d-1}$$

This parity-based matching realizes an algebraic “supersymmetry,” distinct from conventional supersymmetry (which exchanges spacetime spins $s\leftrightarrow s+1/2$). Here, the correspondence is between Clifford odd and even structures, with each bosonic creation operator mapped to a bilinear of fermion operators (and vice versa) (Borstnik et al., 19 Feb 2025, Borštnik, 2023). This mapping is tightly connected to the internal structure of the Clifford algebra and can be generalized to include string-like degrees of freedom or extensions to odd dimensions, where ghost sectors (fields with reversed statistics) naturally emerge and play the role of Faddeev–Popov ghosts in gauge fixing (Borštnik, 2023, Borstnik et al., 19 Feb 2025).

5. Applications: Gauge Fields, Yukawa Couplings, and Phenomenology

The Clifford-even basis vectors directly encode the adjoint multiplets of gauge fields, with suitable projection yielding standard $SU(3)_\text{c}\times SU(2)_\text{w}\times U(1)_Y$ vector bosons and scalars (Higgs, as well as predicted new scalars). Gauge transformations correspond to Clifford-algebraic commutators, and the gauge field action in the unified higher-dimensional context is:

$$S = \int d^d x E \left[\bar{\psi} \gamma^a p_{0a} \psi + \tfrac{1}{2g^2} R(\omega^{(S)},\omega^{(\tilde{S})})\right]$$

with spin connections $\omega_{ab\mu}$ expanded as $A_\mu^A f^{AB}_C B_\text{b}^{(C)}$ in terms of Clifford-even basis vectors (Borstnik, 2021, Borštnik, 2022, Borštnik, 9 Dec 2025).

Phenomenological consequences include the automatic emergence of family replication, anomaly cancellation, quantization of charges, and predictions for additional chiral families and scalar states (Borštnik, 2023, Borštnik, 9 Dec 2025).

6. Relation to Operator Algebra, Fock Spaces, and Jordan–Wigner Transform

The algebraic construction of fermion and boson Fock spaces via Clifford algebra generalizes the customary operator approach (Shchesnovich, 2013, Gudder, 2018, Gudder, 27 Jul 2025):

• Antisymmetrization of the Clifford-odd subspace leads to the canonical fermion Fock space, while symmetrization of the Clifford-even subspace yields the boson Fock space.
• The algebraic structure renders both the field mode expansions and second-quantized Hamiltonians as direct consequences of the internal algebraic parity, with explicit construction of operator bases via nilpotents and projectors.
• The formalism generalizes to finite, infinite, and mixed-dimensional systems, and allows connections to algebraic bosonization/fermionization and extensions such as deformed Grassmann algebras (Lingua et al., 25 Sep 2024, Gudder, 27 Jul 2025).

7. Extensions: Odd-Dimensional Spaces, Ghosts, and Gravity

In odd-dimensional $d=2n+1$ spaces, the standard parity-based division of Clifford odd/even breaks down: certain Clifford-odd basis vectors commute (ghost-bosons) and some Clifford-even anticommmute (ghost-fermions). This reorganization is critical for gauge-fixing procedures and the emergence of Faddeev–Popov ghost fields essential for the quantum consistency of gauge theories (Borštnik, 2023, Borstnik et al., 19 Feb 2025). Furthermore, gravitational degrees of freedom—modeled as higher-rank Clifford-even operators—are included on equal footing, and the algebraic construction can incorporate string-theoretical generalizations and potential renormalizability improvements in high energy completions (Borštnik, 9 Dec 2025, Borštnik, 2023, Borstnik et al., 19 Feb 2025, Borštnik, 10 May 2024).

---

Key references:

• (Borstnik, 2021): How do Clifford algebras show the way to the second quantized fermions with unified spins, charges and families, and to the corresponding second quantized vector and scalar gauge fields.
• (Borštnik, 2022): Clifford odd and even objects offer the description of the internal space of fermions and bosons, respectively, opening new insight into the second quantization of fields.
• (Borštnik, 2023): Clifford odd and even objects in even and odd dimensional spaces describing internal spaces of fermion and boson fields.
• (Borstnik, 2023): How Clifford algebra helps understand second quantized quarks and leptons and corresponding vector and scalar boson fields, opening a new step beyond the standard model.
• (Borstnik et al., 19 Feb 2025): A trial to understand the supersymmetry relations through extension of the second quantized fermion and boson fields, either to strings or to odd dimensional spaces.
• (Borštnik, 9 Dec 2025): Internal spaces of fermion and boson fields, described with the superposition of odd and even products of $γ^{a}$, enable understanding of all the second-quantised fields in an equivalent way.
• (Lingua et al., 25 Sep 2024): Boson-fermion algebraic mapping in second quantization.
• (Gudder, 27 Jul 2025): Geometric Algebras and Fermion Quantum Field Theory.
• (Gudder, 2018): Toy Models for Quantum Field Theory.
• (Shchesnovich, 2013): The second quantization method for indistinguishable particles (Lecture Notes in Physics, UFABC 2010).