We consider introducing the Dirac sea in a quantum cellular automata model of fermions in discrete spacetime which approximates the Dirac equation in the continuum limit. However, if we attempt to fill up the `negative' energy states, we run into a problem. A new boundary is created between positive and negative energy states, at which pair creation seems energetically favourable. This happens because of the modular nature of energy in discrete time models. We then suggest a possible remedy by amending the model, in order to pull states away from the new boundary.
We consider introducing the Dirac sea in a quantum cellular automata model of fermions in discrete spacetime which approximates the Dirac equation in the continuum limit. However, if we attempt to fill up the `negative' energy states, we run into a problem. A new boundary is created between positive and negative energy states, at which pair creation seems energetically favourable. This happens because of the modular nature of energy in discrete time models. We then suggest a possible remedy by amending the model, in order to pull states away from the new boundary.
We consider discrete spacetime models known as quantum walks, which can be used to simulate Dirac particles. In particular we look at fermion doubling in these models, in which high momentum states yield additional low energy solutions which behave like Dirac particles. The presence of doublers carries over to the `second quantised' version of the walks represented by quantum cellular automata, which may lead to spurious solutions when introducing interactions. Moreover, we also consider pseudo-doublers, which have high energy but behave like low energy Dirac particles, and cause potential problems regarding the stability of the vacuum. To address these issues, we propose a family of quantum walks, that are free of these doublers and pseudo-doublers, but still simulate the Dirac equation in the continuum limit. However, there remain a small number of additional low energy solutions which do not directly correspond to Dirac particles. While the conventional Dirac walk always has a zero probability for the walker staying at the same point after a single time step, we obtain the family of walks by allowing this probability to be non-zero.
A discrete-time Quantum Walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation), and thus provide us with discrete toy models of familiar particles (e.g. the electron). In this paper, we study the continuum limit of a wide class of QWs, and show that it leads to all those PDEs corresponding to the Hamiltonian form of the massive curved Dirac equation in (1 + 1) dimensions. Therefore a certain QW, which we make explicit, provides us with a unitary discrete toy model of the electron as a test particle in curved spacetime, in spite of the fixed background lattice. Mathematically we have introduced two novel ingredients for taking the continuum limit of a QW, but which apply to any quantum cellular automata: encoding and grouping.
Nowadays, quantum simulation schemes come in two flavours. Either they are continuous-time discrete-space models (a.k.a Hamiltonian-based), pertaining to non-relativistic quantum mechanics. Or they are discrete-spacetime models (a.k.a Quantum Walks or Quantum Cellular Automata-based) enjoying a relativistic continuous spacetime limit. We provide a first example of a quantum simulation scheme that unifies both approaches. The proposed scheme supports both a continuous-time discrete-space limit, leading to lattice fermions, and a continuous-spacetime limit, leading to the Dirac equation. The transition between the two can be thought of as a general relativistic change of coordinates, pushed to an extreme. As an emergent by-product of this procedure, we obtain a Hamiltonian for lattice-fermions in curved spacetime with synchronous coordinates.
This paper presents a foundational reinterpretation of the Dirac sea within the ψ₀-OCM (Osborne Cosmological Model), a redistribution-based framework in which physical structure emerges through stabilization of underlying redistribution dynamics rather than through occupation of pre-existing energy states. Negative-energy solutions of the Dirac equation are reinterpreted as redistribution directions toward ψ₀ reintegration rather than as physical states requiring filling. Within this framework, the vacuum is modeled as an active redistribution medium governed by boundary-locking and detachment admissibility, eliminating the need for an infinite filled sea or ad hoc subtraction of vacuum energy. Antiparticles arise naturally as stabilized, boundary-locked excitations with reversed redistribution orientation, without invoking hole-based ontology. The paper further shows how quantum vacuum renormalization can be understood as an effective consequence of boundary-locked redistribution, explaining why only differences in vacuum structure are physically observable. Standard relativistic quantum field theory is recovered in redistribution-locked limits, positioning the ψ₀-OCM as an ontological completion rather than an alternative formalism. This work establishes conceptual priority for a redistribution-based resolution of the Dirac sea problem and contributes to ongoing discussions in quantum foundations, vacuum structure, and the interpretation of relativistic quantum theory.
A discrete-time Quantum Walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). In this paper, we study the continuum limit of a wide class of QWs, and show that it leads to an entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac equation in (1+1) curved spacetime. Therefore a certain QW, which we make explicit, provides us with a unitary discrete toy model of a test particle in curved spacetime, in spite of the fixed background lattice. Mathematically we have introduced two novel ingredients for taking the continuum limit of a QW, but which apply to any quantum cellular automata: encoding and grouping.
In this paper, we consider fermionic systems in discrete space-time evolving with a strict notion of causality, meaning they evolve unitarily and with a bounded propagation speed. First, we show that the evolution of these systems has a natural decomposition into a product of local unitaries, which also holds if we include bosons. Next, we show that causal evolution of fermions in discrete space-time can also be viewed as the causal evolution of a lattice of qubits, meaning these systems can be viewed as quantum cellular automata. Following this, we discuss some examples of causal fermionic models in discrete space-time that become interesting physical systems in the continuum limit: Dirac fermions in one and three spatial dimensions, Dirac fields, and briefly the Thirring model. Finally, we show that the dynamics of causal fermions in discrete space-time can be efficiently simulated on a quantum computer.