[Paper Review] Significance of Negative Energy States in Quantum Field Theory $(1) $

Shihao Chen|ArXiv.org|Mar 26, 2002

Noncommutative and Quantum Gravity Theories4 references3 citations

TL;DR

This paper proposes a novel quantization scheme for quantum electrodynamics (QED) by introducing symmetric, independent Lagrangians for positive- and negative-energy states ($\mathcal{L}_F$ and $γ_W$), leading to a vacuum energy of zero without normal-ordering. This resolves the cosmological constant problem and enables direct determination of the cosmological constant from astronomical data, while also correcting nonperturbative methods reliant on ground-state energy subtraction.

ABSTRACT

We suppose that there are both particles with negative energies described by L_{W} and particles with positive energies described by L_{F}, L_{W} and L_{F} are independent of each other before quantization, dependent on each other after quantization and symmetric, and L=L_{W} + L_{F}. From this we present a new quantization method for QED. That the energy of the vacuum state is equal to zero is naturally obtained. Thus we can easily determine the cosmological constant according to data of astronomical observation, and it is possible to correct nonperturbational methods which depend on the energy of the ground state in quantum field theory.

Motivation & Objective

• To resolve the inconsistency in conventional quantum field theory (QFT) where vacuum energy is divergent and unphysical.
• To address the cosmological constant problem by ensuring the vacuum energy is exactly zero.
• To correct nonperturbative methods in QFT that depend on subtraction of divergent ground-state energy.
• To provide a consistent foundation for QFT by reinterpreting negative energy states as physically meaningful rather than problematic.
• To unify solutions to five major problems in QFT: cosmological constant, divergences, electroweak asymmetry, triviality of $\varphi^4$-theory, and dark matter/cosmic voids.

Proposed method

• Introduce two independent Lagrangians: $\mathcal{L}_F$ for positive-energy particles and $\mathcal{L}_W$ for negative-energy particles, both symmetric and independent before quantization.
• Construct a total Lagrangian $\mathcal{L} = \mathcal{L}_F + \mathcal{L}_W$ without requiring normal-ordering of the Hamiltonian.
• Define field operators for both sectors with modified commutation relations, including negative-energy creation/annihilation operators.
• Derive equations of motion for fields $\psi_0$, $\underline{\psi}_0$, $A_{0\mu}$, and $\underline{A}_{0\mu}$ using Heisenberg picture evolution.
• Ensure $H_{F0}$ and $H_{W0}$ commute with the total Hamiltonian, making them constants of motion.
• Demonstrate that the vacuum state energy $E_0 = 0$ naturally, without redefinition or normal-ordering, via the Hamiltonian's structure.

Experimental results

Research questions

• RQ1 Can a consistent QFT be formulated without normal-ordering the Hamiltonian, while ensuring finite and zero vacuum energy?
• RQ2 How can negative energy states be physically interpreted without invoking antiparticles as the sole explanation?
• RQ3 Can the cosmological constant be determined directly from observational data if vacuum energy is zero?
• RQ4 How does a zero vacuum energy affect nonperturbative methods like Hartree-type approximations?
• RQ5 Can this framework unify solutions to multiple outstanding problems in quantum field theory?

Key findings

• The vacuum energy $E_0$ is exactly zero without any redefinition or normal-ordering, resolving the cosmological constant problem at the fundamental level.
• The cosmological constant $\lambda$ can be directly determined from astronomical observations, since $\rho_{\text{vac}} = 0$.
• The vacuum state has zero energy and zero charge, as shown in equation (3.31), independent of operator ordering.
• Nonperturbative methods such as Hartree-type approximations no longer require subtraction of divergent zero-point energy, as $E_0 = 0$.
• The theory avoids divergences in Feynman integrals by construction, as the vacuum energy is finite and zero from the start.
• The framework naturally incorporates both positive- and negative-energy states as symmetric, fundamental components of the theory, not artifacts of second quantization.