Lagrangian Pilot-Wave Model in QFT

• The Lagrangian pilot-wave model is a deterministic quantum framework that extends configuration space with a compact dimension to clearly define particle visibility and evolution.
• It deterministically implements particle creation and annihilation by smoothly transitioning particles between visible and invisible states via a sharp step function.
• The model achieves a robust effective collapse by employing a nonlocal guidance equation and an auxiliary decoherence mechanism, simplifying previous fractional visibility approaches.

The Lagrangian pilot-wave model encompasses a family of quantum field-theoretic and quantum-mechanical pilot-wave theories in which both particle trajectories and wave functionals evolve deterministically under a guidance equation derived from a variational principle. The key motivation is to incorporate processes such as particle creation and annihilation within a deterministic framework, to provide a mathematically coherent foundation for quantum field theory (QFT) in a pilot-wave language, while addressing the shortcomings of both stochastic jump processes and ad hoc "visibility" mechanisms. The model extends standard configuration space by introducing a compactified coordinate, enabling deterministic and continuous evolution even when particles become "hidden" or "revealed" in the physical three-dimensional space. Nonlocal, yet deterministic, guidance equations and an explicit decoherence mechanism form core ingredients. The formalism eliminates prior "smooth visibility" constructs, achieving a simpler and more robust foundation for deterministic quantum field theories with beable trajectories.

1. Deterministic Creation and Annihilation via Extra Compactified Dimension

The Lagrangian pilot-wave model realizes particle creation and annihilation deterministically by extending the configuration space to $\mathbb{R}^3 \times T$, with $T$ a compact dimension parameterized by $x_4$ and periodicity $x_4 + 2T = x_4$. In this construction:

• A particle is visible in three-dimensional space when $0 \le x_4 < T$.
• It is invisible when $T \le x_4 < 2T$.

The field operator takes the form: $$\varphi(\vec{x}, x_4) = \begin{cases} \varphi(\vec{x}) & \text{if } 0 \le x_4 < T \ 0 & \text{if } T \le x_4 < 2T \end{cases}$$

Acting on the vacuum,

$$\varphi(\vec{x}, x_4)|0\rangle = \begin{cases} |\vec{x}\rangle & 0 \le x_4 < T\ |0\rangle & T \le x_4 < 2T \end{cases}$$

The creation or annihilation events correspond to the deterministic, continuous crossing of the $x_4 = 0$ and $x_4 = T$ boundaries on the compactified dimension. Thus, rather than introducing stochastic jumps (as in Bell-type or stochastic Fock space models), the particle trajectory remains smooth and deterministic in the full extended space (Sverdlov, 2010).

2. Mathematical Structure: Guidance Equation and Nonlocal Determinism

The deterministic dynamics of the system are formulated by regarding the probability density $p(\vec{x}, x_4)$ as a "charge distribution" and mapping the configuration space problem to an electrostatic-like problem in higher dimensions. Specifically, the evolution is governed by a continuity equation: $\partial_t \rho = \nabla \cdot (p \vec{v})$

The guidance equation is constructed analogously to Coulomb's law in $4N$-dimensional compactified space: $$v(\vec{x}, x_4) \propto p(\vec{x}, x_4)^{-1} \sum_{\text{images}} \frac{\vec{x} - \vec{x}^{\prime} + R}{|\vec{x} - \vec{x}^{\prime} + R|^{4N}}$$ where the sum runs over image charges corresponding to the compactification, with $R$ the displacement in the compact dimension. The area of a $4N$-sphere is expressed as: $A = \frac{(2N-1)!}{(N-1)!} r^{2N-1}$ ensuring normalization within the geometry.

This nonlocal guidance field generates deterministic trajectories for particles, enforcing smooth behavior even across the visible/invisible boundaries, and ensures that transitions such as creation or annihilation are embedded in a fully continuous dynamical evolution.

3. Decoherence and Robust Effective Collapse: The $H_B$ Term

A shortcoming of earlier models involved the possibility of nonlocal guidance fields temporarily steering the beable into regions where the quantum state supports multiple branches ("overlapping universes"), with an insufficient mechanism for collapse (or effective collapse via decoherence). This model introduces an auxiliary Hamiltonian term, $H_B$, specifically engineered to suppress the amplitude of any branch not occupied by the beable (i.e., unlabeled by the actual particle trajectory):

$H = H_Q + H_B$

Where $H_Q$ is the usual QFT Hamiltonian, and

$$H_B(S) = -\frac{i}{\delta} \int_{S(|s\rangle)} d\mu(\vec{x})$$

for $S(|s\rangle)$ the configuration space area of the branch. The amplitude on a branch unsupported by the beable decays rapidly.

An intuitive mechanism involves continuous creation of auxiliary $C$-particles at a rate proportional to $\exp(-k|\vec{x} - \vec{x}_B|^2)$, which in turn decay the wave function amplitude in those regions by a prescribed dynamic: $$\partial_t x(\vec{x}) = a \nabla^2 x(\vec{x}) + b \exp(-k|\vec{x}-\vec{x}_B|^2) - C x(\vec{x}) p(\vec{x})$$ Subsequent substitution of these objects with a continuous field yields a deterministic, effective collapse mechanism that rapidly eliminates unwanted branches as soon as they are unoccupied by the beable, thereby enforcing locality at the level of observed outcomes and stabilizing the beable's branch occupancy.

4. Elimination of "Visibility" Functions and Model Simplification

The prior "visibility" concept used a smooth function $f(x_4)$, which introduced a fractional "degree of visibility" for particles in the boundary regions $0 \lesssim x_4 \lesssim T$, leading to ambiguous, "half-visible" states and complicating the theory's interpretation and mathematical rigor.

The current model eliminates this intermediary regime by employing a sharp cutoff (step function) in the definition of visibility. The notion of a particle being either fully present or absent in 3D space is implemented exactly: transition across $x_4$ boundaries instantly switches the particle from visible to invisible.

Importantly, the nonlocal guidance equation—derived from the analogy to electrostatics—ensures that, despite this sharp discontinuity in the field operator, particle velocities and overall dynamics remain continuous and differentiable, since nonlocal fields are smooth even in the presence of step-function sources. This results in a mathematically natural and physically transparent model without ambiguous transitional states (Sverdlov, 2010).

5. Comparison with Other Approaches and Model Consequences

Traditional pilot wave QFT approaches (such as Bell's jump process models or stochastic Fock-space beable theories) invoke truly discontinuous or stochastic processes for particle creation and annihilation. The Lagrangian model developed in (Sverdlov, 2010) sidesteps these features entirely:

• Advantages:
  • Fully deterministic, continuous evolution in the extended configuration space.
  • No need for probability-based jump processes.
  • Robust decoherence and effective collapse via the $H_B$ mechanism, unlike earlier nonlocal field-line blockage arguments.
  • Transparent continuity in particle dynamics through creation/annihilation events, even at the level of trajectories.
• Limitations and Considerations:
  • Relies on nonlocal (but deterministic) guidance fields, with computational complexity scaling rapidly in high-dimensional configuration space.
  • The mathematical machinery (method of images, high-dimensional Coulomb problems) demands careful analytical and numerical treatment for practical implementation.
  • The model's essential nonlocality, required for effective guidance and collapse, persists even as other artificial constructs are removed.

6. Summary Table: Key Model Ingredients

Feature	Mathematical Realization	Role in the Model
Creation/Annihilation	$x_4$ on $\mathbb{R}^3 \times T$	Deterministic, continuous events
Guidance Equation	Nonlocal Coulomb field (Eq. (11,14))	Generates smooth trajectories
Decoherence/Collapse	$H_B$ branch-destroying Hamiltonian	Suppresses unoccupied branches
Visibility	Exact step function in $x_4$	Removes fractional states
Configuration Space	Compactified, periodic $x_4$	Enables creation/annihilation

This table synthesizes the essential structures and dynamical mechanisms.

7. Implications and Extensions

This Lagrangian pilot-wave model introduces a robust, deterministic solution to the problem of particle creation and annihilation in quantum field theory by extending configuration space, invoking nonlocal guidance fields, and explicitly incorporating decoherence dynamics. The approach significantly improves mathematical and interpretive clarity over previous formulations that relied on stochastic Fock-space jumps or arbitrary visibility parametrizations.

A plausible implication is that this framework can serve as a technical foundation for deterministic pilot-wave formulations of QFT phenomena where the classical trajectory language is desired, provided one accepts intrinsic nonlocality at the level of the guiding field. Additionally, the explicit decoherence mechanism via $H_B$ opens practical avenues for implementing robust single-branch beable dynamics in toy model quantum field theories or quantum simulations derived from a pilot-wave perspective.