Quantization of massive fermions in vacuum and external fields

This paper presents the quantization of massive Majorana neutrinos in background matter using Weyl spinors and propagator construction, alongside a Hamiltonian formalism analysis of both these neutrinos and classical Dirac particles in vacuum.

The Gist

Imagine the universe as a giant, complex orchestra. Usually, when physicists talk about the "music" of particles like neutrinos, they assume the sheet music is written in a very specific, strange language where the notes don't just add up normally—they "anti-add" (if you play a note and then play it again, it cancels out). This is the standard way of describing Majorana neutrinos, which are particles that are their own antiparticles.

This paper, written by Maxim Dvornikov, is like a music theorist saying, "Wait a minute. What if we tried to write the sheet music for these neutrinos using a different, more traditional language first, and then translate it?"

Here is a breakdown of what the paper does, using everyday analogies:

1. The Main Goal: Two Ways to Describe a Ghost

Neutrinos are like ghosts: they have mass, but they barely interact with anything. The big mystery is whether they are "Dirac" particles (like a distinct person with a twin) or "Majorana" particles (like a person who is their own twin).

The author focuses on Majorana neutrinos.

• The Standard Approach: Usually, physicists treat these particles as "Grassmann variables." Think of this as a magical, anti-commuting language where the order of words matters in a weird way (A times B is the negative of B times A). This is the standard "quantum" language.
• The Author's Approach: This paper asks, "Can we describe these same particles using normal, everyday numbers (c-numbers) first, like a classical wave, and then turn them into quantum particles?"

2. Part I: The Neutrino in a Crowd (Background Matter)

In the first part of the paper, the author studies a massive Majorana neutrino moving through "background matter" (like a neutrino traveling through the dense core of a star or the Earth).

• The Analogy: Imagine a dancer (the neutrino) moving through a crowded room (the background matter). The crowd pushes and pulls on the dancer, changing how they move.
• What the paper does: The author writes down the "rules of motion" (the Lagrangian) for this dancer. Then, they solve the math to see exactly how the dancer moves.
• The Result: They successfully write down the "sheet music" (the wave function) for this dancer and figure out how to count the dancers (quantization). They also calculate the "propagator," which is essentially a map showing how a ripple from one dancer reaches another dancer through the crowd. This map is crucial for understanding how neutrinos change their "flavor" (like changing from a "muon" type to an "electron" type) while traveling.

3. Part II: The Classical "Ghost" Before the Magic

Here is where the paper gets a bit philosophical. The author points out that in standard physics, we usually assume you can't have a "classical" version of a Majorana neutrino because the math for a normal number cancels itself out.

• The Analogy: It's like saying you can't have a "classical" version of a shadow because shadows only exist when light hits an object.
• The Fix: The author uses a specific mathematical toolkit called Hamiltonian dynamics (a way of tracking energy and momentum) to prove that you can describe these particles using normal numbers first.
• The Metaphor: Think of it as building a wooden model of a car before you build the real, metal car. The author builds a "classical wooden model" of the neutrino using normal numbers, showing that it behaves consistently, even though it's usually thought to be impossible.

4. Part III: The Twin Brother (The Dirac Fermion)

After mastering the "self-twin" (Majorana) neutrino, the author applies the same wooden-model technique to a Dirac fermion.

• The Analogy: If the Majorana neutrino is a person who is their own twin, a Dirac fermion is a person with a distinct twin brother.
• What happens: The author takes this "classical wooden model" of the Dirac particle, runs it through their special math machine, and successfully turns it into a real quantum particle.
• The Payoff: They show that this new method produces the exact same energy and momentum formulas that physicists have been using for decades. It's like proving that a new, unusual way of baking a cake results in the exact same delicious taste as the traditional recipe.

Summary: Why Does This Matter?

The paper doesn't claim to discover a new particle or a new force. Instead, it's a mathematical proof of concept.

• The Claim: The author shows that you can describe these tricky, "ghost-like" particles using normal, everyday numbers first, and then convert them into quantum particles without breaking the laws of physics.
• The Takeaway: It offers an alternative perspective. While most physicists use the "magic anti-commuting language" (Grassmann variables) from the start, this paper says, "Look, you can also start with normal numbers and get the same result." It fills in some missing steps in previous research and corrects a few small errors in how this "classical-to-quantum" bridge was built before.

In short, the paper is a rigorous check of the "blueprints" for how we describe these elusive particles, ensuring that whether you build the house with magical bricks or normal bricks, the final structure stands up correctly.

Technical Summary

Technical Summary: Quantization of Massive Fermions in Vacuum and External Fields

Problem Statement
The paper addresses the theoretical challenges associated with the quantization of massive fermions, specifically Majorana neutrinos in background matter and Dirac fermions in vacuum. A central issue is the representation of these fields: while the standard approach treats classical fermionic fields using anticommuting Grassmann variables, an alternative Hamiltonian dynamics formalism utilizes commuting c-number variables. The author notes that previous attempts to describe massive Majorana neutrinos in external fields using Weyl spinors (Ref. [1], [2]) or Hamilton dynamics (Ref. [3], [4]) have left gaps, particularly regarding the rigorous derivation of propagators and the clarification of classical field theory descriptions where mass terms might otherwise vanish for commuting variables. Furthermore, the quantization of Dirac fields constructed via this specific Hamiltonian formalism requires explicit verification to ensure consistency with standard results.

Methodology
The study employs a combination of canonical quantization and Hamiltonian dynamics formalism:

1. Majorana Neutrinos in Matter: The author begins by representing a massive Majorana neutrino as a two-component Weyl spinor $\eta$ interacting with background matter via an effective potential $g$. The Lagrangian is formulated, and the wave equation is derived. The field is expanded into plane waves with creation and annihilation operators ($a_\pm, a^\dagger_\pm$) corresponding to helicities. Canonical quantization is performed by establishing equal-time anticommutators for the field and its conjugate momentum.
2. Propagator Derivation: Using the quantized field expansion, the author constructs two types of propagators: $S(x-y)$ for the time-ordered product of $\eta$ and $\eta^\dagger$, and $\tilde{S}(x-y)$ for $\eta$ and $\eta^T$. These are derived in momentum space, explicitly accounting for the energy shifts caused by the background matter potential.
3. Hamiltonian Dynamics for Classical Fields: To address the description of classical fields using commuting c-numbers, the author adopts the Hamilton dynamics approach. A Hamiltonian $H_M$ is constructed for commuting variables $(\eta, \pi)$. The canonical equations of motion are derived, and it is demonstrated that these equations reproduce the known wave equations for Majorana neutrinos and antineutrinos. To resolve the issue of constructing a Lagrangian from these first-order equations, extended Lagrangians ($L_R$ and $L_L$) are formulated, analogous to the Routh formalism in classical mechanics.
4. Dirac Fermion Quantization: The formalism is extended to a massive Dirac fermion in vacuum. A Hamiltonian $H_D$ is defined for commuting c-number variables. The system is quantized by expanding the fields into plane waves with independent creation/annihilation operators ($a_s, b_s, c_s, d_s$). Constraints are imposed on these operators to recover the standard physical interpretation, ensuring the energy and momentum operators take the expected forms.

Key Contributions and Results

• Quantization of Majorana Neutrinos in Matter: The paper provides a detailed canonical quantization of massive Majorana neutrinos represented by Weyl spinors in background matter. Explicit expressions for the total energy and momentum of the field are derived in terms of number operators.
• Propagators in Matter: The author derives the exact propagators for massive Weyl neutrinos in background matter. For ultrarelativistic neutrinos, the propagator $S(p)$ is shown to reduce to a form consistent with previous studies on flavor oscillations (Ref. [6]), but with a rigorous derivation that fills gaps in earlier sketchy treatments. The vacuum limit of these propagators is shown to coincide with known results.
• Clarification of Classical Field Theory: The work clarifies the Hamiltonian description of Majorana neutrinos using commuting c-numbers. It demonstrates that despite the mass term vanishing for commuting variables in a standard Lagrangian approach, a consistent classical field theory exists within the Hamilton formalism. The author corrects inexactitudes found in Ref. [3] regarding the factors in the extended Lagrangians ($L_R$ and $L_L$).
• Dirac Field Quantization: The paper successfully applies the Hamiltonian formalism to quantize a Dirac field. By imposing specific constraints on the auxiliary operators ($c_s = -ia_s$, $d_s = ib_s$), the author derives the standard expressions for the total energy and momentum of the Dirac field, confirming the validity of the c-number approach for this system.

Significance and Claims
The paper claims to provide a rigorous and detailed treatment of Majorana neutrino quantization in external fields, specifically addressing the derivation of propagators which was previously incomplete. It validates the Hamilton dynamics formalism as a viable alternative to the standard Lagrangian approach for describing classical fermionic fields using commuting variables. The author explicitly states that while the treatment of classical fermionic fields using anticommuting Grassmann variables is the commonly recognized standard, this work offers an alternative point of view. The paper concludes modestly, noting that while the formalism is consistent for free fields, the construction of a consistent theory for fermionic fields interacting with other particles within this framework requires further, separate investigation. The results are presented as foundational for understanding neutrino flavor oscillations in matter and for exploring alternative classical descriptions of fermions.