[Paper Review] How Clifford algebra can help understand second quantization of fermion and boson fields

N. S. Mankoč Borštnik|arXiv (Cornell University)|Oct 7, 2022

International Science and Diplomacy4 citations

TL;DR

This paper demonstrates that Clifford algebra provides a unified framework for understanding second quantization of both fermion and boson fields by associating fermionic internal states with odd-grade Clifford objects and bosonic states with even-grade objects. The anticommutation relations of fermion creation/annihilation operators and the commutation relations of boson fields naturally emerge from the algebraic structure, offering a geometric origin for second quantization postulates in a higher-dimensional spacetime (d ≥ 13+1).

ABSTRACT

In the review article in Progress in Particle and Nuclear Physics (vol.121(2021) 103890)) the authors present the achievements so far of the spin-charge-family theory, which offers the explanation for all the so far observed properties of elementary fermion and boson fields, if the space-time is higher than d=(3+1), it must be $d\ge (13+1)$. Fermions interact with gravity only. Ref. PPNP (vol.121(2021) 103890)) presents, in addition to a rather detailed review of all the achievements of this theory so far, also an explanation for the postulates of the second quantization for fermionic fields: The internal space of fermions described with "basis vectors" represented by the Clifford odd objects manifests all the properties of fermion fields, including the anticommutativity of their creation and annihilation operators. This paper shows that even Clifford algebra objects provide a description of the internal space of boson fields manifesting all known properties of boson fields, explaining as well the reasons for the second quantization postulates for boson fields. Properties of fermion and boson fields with the internal spaces described by the Clifford odd and Clifford even objects are demonstrated on the toy model with $d=(5+1)$.

Motivation & Objective

• To provide a geometric and algebraic foundation for second quantization of fermion and boson fields using Clifford algebra.
• To explain why fermion fields obey anticommutation relations and boson fields obey commutation relations through the grading of Clifford algebra objects.
• To demonstrate that the internal space of fermions is naturally described by odd products of gamma matrices, while bosons arise from even products.
• To show that the second quantization postulates for both fields emerge from the algebraic properties of the Clifford algebra without ad hoc assumptions.
• To extend the spin-charge-family theory to include a consistent description of gauge bosons as second quantized fields via even-grade Clifford structures.

Proposed method

• Utilizes the Clifford algebra structure in d = (13+1) spacetime dimensions, where gamma matrices γ^a and their duals ˜γ^a generate two orthogonal subalgebras.
• Models fermion internal states as superpositions of odd-grade products of γ^a matrices, which inherently anticommute.
• Models boson internal states as superpositions of even-grade products of γ^a matrices, which commute under tensor product operations.
• Constructs creation and annihilation operators as tensor products of these basis vectors with momentum/coordinate bases, preserving algebraic statistics.
• Applies the Lorentz generator decomposition S^ab = (i/4)(γ^aγ^b − γ^bγ^a) and ˜S^ab = (1/4)(˜γ^a˜γ^b − ˜γ^b˜γ^a) to relate symmetries to gauge fields and charges.
• Uses the toy model in d = (5+1) to illustrate the algebraic structure before extending to higher dimensions.

Experimental results

Research questions

• RQ1 How can Clifford algebra provide a geometric origin for the second quantization postulates of fermion fields?
• RQ2 Why do creation and annihilation operators for fermions anticommute, and can this be derived from algebraic structure rather than postulated?
• RQ3 Can the second quantization of boson fields similarly be derived from the algebraic properties of even-grade Clifford objects?
• RQ4 How do the internal degrees of freedom of fermions and bosons in higher dimensions (d ≥ 13+1) reproduce the observed quantum numbers of the Standard Model?
• RQ5 What is the role of the two distinct Clifford subalgebras (odd and even grades) in generating fermionic and bosonic statistics respectively?

Key findings

• Fermion creation and annihilation operators inherit anticommutativity from the odd-grade Clifford basis vectors, reproducing the second quantization postulates without postulation.
• Boson fields, described by even-grade Clifford basis vectors, naturally give rise to commutation relations in the second quantized framework.
• In the d = (5+1) toy model, the algebraic structure of Clifford even and odd objects correctly reproduces the statistics of fermions and bosons.
• The internal space of fermions is described by superpositions of odd products of γ^a matrices, which are anticommuting and form a Clifford algebra of odd grade.
• The internal space of bosons is described by even products of γ^a matrices, which commute and form a Clifford algebra of even grade.
• The theory unifies the second quantization of both fermions and bosons under a single algebraic framework based on Clifford algebra grading in d ≥ (13+1) spacetime dimensions.