Clifford/Fock Construction Framework

• Clifford/Fock construction is a unifying algebraic framework that models quantum states and spinors using Clifford algebras and Fock spaces.
• It constructs a complete Extended Fock Basis for Cl(m, m), decomposing multivectors into simple spinor substructures and clarifying symmetry via eigenvalue equations.
• The approach offers significant computational advantages, using matrix isomorphism and parallelizable algorithms to enhance both theoretical and applied physics research.

The Clifford/Fock construction provides a rigorous and unifying framework for describing quantum states, spinors, and their algebraic and geometric structures using the machinery of Clifford algebras and Fock spaces. Central themes include the explicit construction of bases for Clifford algebras, the decomposition of the algebra into spinor subspaces, the correspondence between simple spinors and totally null planes, and algorithmic and physical implications of these algebraic structures. This formulation encompasses both the classical use of Fock space in quantum theory and modern approaches to Clifford algebra in geometry and representation theory.

1. Extended Fock Basis: Structure and Construction

The Extended Fock Basis (EFB) is introduced as a complete basis for the Clifford algebra $Cl(m, m)$, generalizing the traditional Fock basis from spinorial subspaces to the entire algebra (Budinich, 2010). EFB elements are built using the null (Witt) basis

$$p_i = \frac{1}{2}(\gamma_{2i-1} + \gamma_{2i}), \quad q_i = \frac{1}{2}(\gamma_{2i-1} - \gamma_{2i}), \quad i=1, \dots, m$$

The construction involves choosing, at each level $i$, one out of four possibilities (depending on q$_i$p$_i$, p$_i$, q$_i$, or $\Pi_i$), resulting in $4^m = 2^{2m}$ basis elements—a full spanning set for $Cl(m, m)$.

Unlike the traditional Fock basis, which is confined to minimal left ideals defined by a vacuum spinor, the EFB extends to all multivectors as a linear combination of simple spinors, thus reducing much of the analysis of $Cl(m, m)$ to the simplest case of $Cl(1, 1)$.

2. Spinor Subspaces as Left Eigenvectors of the Volume Element

Each EFB element is a left eigenvector of the Clifford algebra's volume element

$\Gamma = \gamma_1\gamma_2\cdots\gamma_{2m}$

The eigenvalue equation is

$\Gamma\Psi = \eta\Psi$

with $\eta = \prod_{i=1}^m h_i$ governed by the $h$-signature (each $h_i = \pm 1$ identifies which null vector appears first at level $i$). The Clifford algebra $Cl(m, m)$ thus decomposes as a direct sum of $2^m$ spinor subspaces, each labeled by a fixed $h$-signature, and every EFB element is a simple spinor left eigenvector of $\Gamma$. This decomposition clarifies the internal symmetry structure and the direct sum of minimal left ideals.

3. Simple Spinors, Totally Null Planes, and Nonzero Coordinates

Every EFB element is a simple (or pure) spinor. For any simple spinor $\Psi$, there exists a maximal totally null plane (TNP) $M(\Psi) \subset V$ (with $V$ the underlying vector space), defined by

$v\Psi = 0,\quad \forall v \in M(\Psi)$

Maximality demands $\dim M(\Psi) = m$. In matrix representations, EFB elements with constant $h$- and $g$-signatures have only one nonzero entry per column, directly mirroring the TNP structure; all other coordinates vanish.

The composition properties of simple spinors are also captured: if two simple spinors from the same subspace have TNPs intersecting in $m-2$ dimensions, any linear combination remains simple. In the case $m=3$ (6-dimensional vector spaces), four linearly dependent spinors can be chosen such that any linear combination is simple—a special low-dimensional phenomenon highlighted in the paper.

4. Algorithmic and Computational Implications

The EFB framework offers a dramatic computational improvement by mapping $Cl(m, m)$ to a matrix algebra $F(2^m)$ such that Clifford multiplication can be performed much faster—specifically a factor $2^m$ reduction versus standard gamma-matrix algorithms. EFB-based algorithms allow direct multiplication in the basis, with spinor subspace decomposition facilitating parallelization and block-based computations.

The basis also enables a natural matrix isomorphism, unifying spinors and multivectors and exposing a dual symmetry: maximal TNPs in the vector space correspond one-to-one to totally simple planes (spinor spaces) of dimension $m$. This duality streamlines both symbolic and numeric Clifford algebra computations.

5. Physical and Geometric Applications

The EFB and Clifford/Fock construction have direct applications to quantum field theory and geometry. The full expansion of the Clifford algebra in simple spinors is closely tied to the modeling of mirror particles and quantum fields, where multiple distinct spinor spaces are critical for encapsulating physical degrees of freedom.

Moreover, the generalized Fock space picture produced by the EFB provides alternative geometric interpretations, for instance, allowing for constraint equations for simple spinors with $m>3$, and facilitating the paper of possible configurations of quantum states in field-theoretic models.

6. Exceptional Low-Dimensional Behavior

The case $m=3$ (dimension six) has exceptional properties concerning the composition of simple spinors and the number of nonzero coordinates. For $m=3$, the usual bounds (e.g., $k\leq m$ or $k\leq m+1$ for linearly independent simple spinors whose combinations remain simple) are surpassed: four simple spinors may be chosen such that any linear combination is simple, even though they are linearly dependent in $h$-signature. This underlines subtle algebraic and geometric features present only in low-dimensional Clifford algebras.

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In conclusion, the Clifford/Fock construction based on the Extended Fock Basis provides a canonical decomposition and efficient computational approach to Clifford algebras and their spinor representations. It accomplishes a unification of algebraic, geometric, and computational perspectives, and supports both quantum-mechanical and field-theoretic applications. The explicit matrix isomorphism, direct sum decomposition, correspondence to totally null planes, and recognition of exceptional dimensional cases together constitute a robust and flexible foundation for ongoing theoretical and applied research.