[Paper Review] Analytical Study of the Spin Projection Operator

Zhiqiang Shi|arXiv (Cornell University)|Jan 3, 2011

Matrix Theory and Algorithms3 references3 citations

TL;DR

This paper analytically investigates the spin projection operators for relativistic spin states and helicity states, demonstrating that while they are formally identical in the $p_z$ rest frame, their operators differ fundamentally: the spin state operator is Lorentz covariant, while the helicity operator is nonrelativistic. The key result is that using the helicity projection operator in muon decay calculations reveals a left-right polarization-dependent lifetime asymmetry $A = \beta$, which is absent when using the spin state operator, thus establishing a novel lifetime asymmetry in weak decays.

ABSTRACT

We present an analytical study of the spin projection operator of the spin state and the helicity state. It is pointed out emphatically that the former is Lorentz covariant, while the latter is a nonrelativistic two-component operator. In the special case of $\bm p=p_z$, the helicity state and the spin state are formally identical. However, even so their spin projection operators are still different and so the helicity state is not a special case of the spin state. This makes the spinor is the degenerate state of the two different spin projection operators. The calculation on the lifetime of polarized muons shows that this difference will inevitably lead to the left-right polarization-dependent lifetime asymmetry.

Motivation & Objective

• To clarify the fundamental conceptual and mathematical distinction between spin states (Lorentz-covariant) and helicity states (nonrelativistic) in relativistic quantum mechanics.
• To resolve longstanding confusion regarding the physical meaning of chirality, helicity, and spin in weak interactions.
• To demonstrate that the choice of spin projection operator—spin state vs. helicity state—leads to different physical predictions in decay processes.
• To establish the existence of a previously overlooked lifetime asymmetry in polarized muon decays, arising from the helicity operator's structure.

Proposed method

• Derives the spin projection operator for relativistic spin states using the Pauli-Lubanski operator and the four-polarization vector $e$, yielding $\rho_s = \frac{1}{2}(1 \pm i\gamma_5\gamma\cdot e)$.
• Derives the helicity projection operator as $\rho_h = \frac{1}{2}(1 \pm \gamma_5\beta)$, where $\beta = \mathbf{p}/E_p$, showing its non-covariant structure.
• Applies both operators to the matrix element of muon decay $\mu^- \to e^- \bar{\nu}_e \nu_\mu$, computing the decay rate via trace operations.
• Computes the decay width using the helicity state projection operator, leading to $\tau_{\text{Lh}} = \tau / (1 + \beta)$ and $\tau_{\text{Rh}} = \tau / (1 - \beta)$.
• Derives the lifetime asymmetry $A = (\tau_{\text{Rh}} - \tau_{\text{Lh}})/(\tau_{\text{Rh}} + \tau_{\text{Lh}}) = \beta$.
• Compares results with the spin state operator, which yields no asymmetry, proving the asymmetry is an artifact of the helicity operator choice.

Experimental results

Research questions

• RQ1 Why does the choice of spin projection operator—spin state vs. helicity state—lead to different predictions in muon decay calculations?
• RQ2 What is the fundamental difference between the Lorentz-covariant spin state operator and the nonrelativistic helicity operator?
• RQ3 Does the helicity state projection operator predict a polarization-dependent lifetime asymmetry in muon decays?
• RQ4 Can the lifetime asymmetry $A = \beta$ be derived analytically from the helicity operator, and how does it differ from the spin state result?
• RQ5 What are the implications of this asymmetry for weak interaction phenomenology, muon collider design, and cosmic ray spectra?

Key findings

• The spin state projection operator $\rho_s = \frac{1}{2}(1 \pm i\gamma_5\gamma\cdot e)$ is Lorentz covariant, while the helicity state operator $\rho_h = \frac{1}{2}(1 \pm \gamma_5\beta)$ is not, establishing a fundamental physical distinction.
• In the $p_z$ frame, the spinor wave functions for spin and helicity states are formally identical, but their projection operators remain distinct, meaning the helicity state is not a special case of the spin state.
• Using the helicity projection operator in muon decay leads to a left-right polarization-dependent lifetime asymmetry $A = \beta$, where $\beta = v/c$.
• The lifetime of right-handed polarized muons is $\tau_{\text{Rh}} = \tau / (1 - \beta)$, and for left-handed muons $\tau_{\text{Lh}} = \tau / (1 + \beta)$, showing a clear asymmetry.
• The spin state operator yields a symmetric decay rate $\tau$, revealing no asymmetry, proving that the lifetime asymmetry is an artifact of using the helicity operator.
• The result implies that helicity states are physical observables, while spin states are weak eigenstates tied to the chiral structure of the Standard Model, analogous to neutrino flavor vs. mass eigenstates.