# On Bohmian Mechanics, Particle Creation, and Relativistic Space-Time: Happy 100th Birthday, David Bohm!

## Abstract

**:**

## 1. Introduction

#### 1.1. Significance of Bohmian Mechanics

#### 1.2. Laws of Bohmian Mechanics

#### 1.3. Properties of Bohmian Mechanics

“This idea seems to me so natural and simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored.”

## 2. Extension of Bohmian Mechanics to Particle Creation

#### 2.1. Bell’s Jump Process (In Its Continuum Version)

#### 2.2. An Ultraviolet Divergence Problem

#### 2.3. UV Problem Solved!

**Theorem**

**1**

#### 2.4. Particle Trajectories

## 3. Extension of Bohmian Mechanics to Relativistic Space-Time

#### 3.1. The Time Foliation

#### 3.2. The Single-Particle Case

#### 3.3. Law of Motion for Many Particles

**Theorem**

**2**

**.**If detectors are placed along any spacelike surface Σ (and if some reasonable assumptions about the evolution of ${\psi}_{\Sigma}$ are satisfied), then the joint distribution of detection events is $|{\psi}_{\Sigma}{|}^{2}$.

#### 3.4. Multi-Time Wave Functions

## 4. Outlook and Concluding Remarks

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Possible patterns of particle world lines in theories with particle creation and annihilation: (

**a**) a boson (dashed world line) is emitted by a fermion and absorbed by another; and (

**b**) a boson (dashed world line) decays into two fermions. (Reprinted from [13]).

**Figure 3.**The configuration space in Equation (5) of a variable number of particles; drawn are, for space dimension $d=1$, the first four sectors: (

**a**) the zero-particle sector has a single element, the empty configuration; (

**b**) the one-particle sector is a copy of physical space; (

**c**) the two-particle sector; and (

**d**) the three-particle sector. In addition, the configuration curve corresponding to Figure 2a is drawn; it jumps at time ${t}_{1}$ from the two-particle sector to the three-particle sector and at time ${t}_{2}$ back. (Reprinted from [13]).

**Figure 4.**An example of a natural candidate for the cut-off function $\phi (\xb7)$: a bump-shaped function that is a smooth and square-integrable approximation to a Dirac $\delta $ function and vanishes outside a small ball around the origin.

**Figure 5.**When using ${H}_{\mathrm{cutoff}}$, the emission and absorption of a y-particle happens, according to Equation (6), not exactly at the location of an x-particle, but at a separation that can be as large as the radius of the support of $\phi $. This does not happen with the alternative Hamiltonian defined by means of interior-boundary conditions.

**Figure 6.**An interior-boundary condition is a relation between the values of $\psi $ at two points: a point q on the boundary (that is, where two particles collide, such as $(x,x)$ in the two-particle sector) and a point ${q}^{\prime}$ in the interior of a lower sector (such as x).

**Figure 7.**Example of a spacelike foliation (i.e., slicing into spacelike hypersurfaces) of Minkowski space-time in $1+1$ dimensions.

**Figure 8.**The equation of motion of BM${}_{\mathcal{F}}$ specifies the tangent direction of a world line by means of the wave function evaluated at the configuration where all world lines intersect the same time leaf $\Sigma $.

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Tumulka, R. On Bohmian Mechanics, Particle Creation, and Relativistic Space-Time: Happy 100th Birthday, David Bohm! *Entropy* **2018**, *20*, 462.
https://doi.org/10.3390/e20060462

**AMA Style**

Tumulka R. On Bohmian Mechanics, Particle Creation, and Relativistic Space-Time: Happy 100th Birthday, David Bohm! *Entropy*. 2018; 20(6):462.
https://doi.org/10.3390/e20060462

**Chicago/Turabian Style**

Tumulka, Roderich. 2018. "On Bohmian Mechanics, Particle Creation, and Relativistic Space-Time: Happy 100th Birthday, David Bohm!" *Entropy* 20, no. 6: 462.
https://doi.org/10.3390/e20060462