## 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

- Bohm, D. A suggested interpretation of the quantum teory in terms of “hidden” variables, I and II. Phys. Rev.
**1952**, 85, 166–193. [Google Scholar] [CrossRef] - Einstein, A. Reply to criticisms. In Albert Einstein, Philosopher-Scientist; Schilpp, P.A., Ed.; Open Court: La Salle, IL, USA, 1949; p. 664. [Google Scholar]
- Bell, J.S. Against “measurement”. In Sixty-Two Years of Uncertainty; Miller, A.I., Ed.; Plenum Press: New York, NY, USA, 1990; Reprinted as chapter 23 of Speakable and Unspeakable in Quantum Mechanics, 2nd ed.; Bell, J.S.; Cambridge University Press: Cambridge, UK, 2004. Also reprinted in Phys. World
**1990**, 3, 33–40. [Google Scholar] - Bricmont, J. Making Sense of Quantum Mechanics; Springer: Heidelberg, Germany, 2016. [Google Scholar]
- Norsen, T. Foundations of Quantum Mechanics; Springer: Heidelberg, Germany, 2018. [Google Scholar]
- Dürr, D.; Teufel, S. Bohmian Mechanics; Springer: Heidelberg, Germany, 2009. [Google Scholar]
- Bohm, D.; Hiley, B.J. The Undivided Universe: An Ontological Interpretation of Quantum Theory; Routledge: London, UK, 1993. [Google Scholar]
- Tumulka, R. Bohmian mechanics. In The Routledge Companion to the Philosophy of Physics; Wilson, A., Ed.; Routledge: London, UK, 2018. [Google Scholar]
- Dürr, D.; Goldstein, S.; Zanghì, N. Quantum equilibrium and the origin of absolute uncertainty. J. Stat. Phys.
**1992**, 67, 843–907, Reprinted in Quantum Physics without Quantum Philosophy; Dürr, D.; Goldstein, S.; Zanghì, N.; Springer: Heidelberg, Germany, 2013. [Google Scholar] [CrossRef][Green Version] - Bell, J.S. Six possible worlds of quantum mechanics. In Proceedings of the Nobel Symposium 65: Possible Worlds in Arts and Sciences, Stockholm, Sweden, 11–15 August 1986. Reprinted as chapter 20 of Speakable and Unspeakable in Quantum Mechanics, 2nd ed.; Bell, J.S.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Philippidis, C.; Dewdney, C.; Hiley, B.J. Quantum interference and the quantum potential. Il Nuovo Cimento
**1979**, 52B, 15–28. [Google Scholar] [CrossRef] - Bell, J.S. Speakable and Unspeakable in Quantum Mechanics, 2nd ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Dürr, D.; Goldstein, S.; Tumulka, R.; Zanghì, N. Bohmian mechanics and quantum field theory. Phys. Rev. Lett.
**2004**, 93, 090402, Reprinted in Quantum Physics without Quantum Philosophy; Dürr, D.; Goldstein, S.; Zanghì, N.; Springer: Heidelberg, Germany, 2013. [Google Scholar] [CrossRef] [PubMed] - Landau, L.; Peierls, R. Quantenelektrodynamik im Konfigurationsraum. Z. Phys.
**1930**, 62, 188–200, English translation: Quantum electrodynamics in configuration space. In Selected Scientific Papers of Sir Rudolf Peierls with Commentary; Dalitz, R.H., Peierls, R., Eds.; World Scientific: Singapore, 1997; pp. 71–82. [Google Scholar] [CrossRef] - Schweber, S. An Introduction to Relativistic Quantum Field Theory; Harper: New York, NY, USA, 1961. [Google Scholar]
- Nelson, E. Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys.
**1964**, 5, 1190–1197. [Google Scholar] [CrossRef] - Bell, J.S. Beables for quantum field theory. Phys. Rep.
**1986**, 137, 49–54, Reprinted in Quantum Implications: Essays in Honour of David Bohm; Peat, F.D., Hiley, B.J., Eds.; Routledge: London, UK, 1987; p. 227. Also reprinted as chapter 19 of Speakable and Unspeakable in Quantum Mechanics, 2nd ed.; Bell, J.S.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar] [CrossRef] - Dürr, D.; Goldstein, S.; Tumulka, R.; Zanghì, N. Trajectories and particle creation and annihilation in quantum field theory. J. Phys. A Math. Gen.
**2003**, 36, 4143–4149. [Google Scholar] [CrossRef][Green Version] - Dürr, D.; Goldstein, S.; Tumulka, R.; Zanghì, N. Bell-type quantum field theories. J. Phys. A Math. Gen.
**2005**, 38, R1–R43. [Google Scholar] [CrossRef] - Vink, J.C. Quantum mechanics in terms of discrete beables. Phys. Rev. A.
**1993**, 48, 1808–1818. [Google Scholar] [CrossRef] [PubMed] - Vink, J.C. Particle trajectories for quantum field theory. Found. Phys.
**2018**, 48, 209–236. [Google Scholar] [CrossRef] - Teufel, S.; Tumulka, R. New type of Hamiltonians without ultraviolet divergence for quantum field theories. arXiv, 2015; arXiv:1505.04847. [Google Scholar]
- Lampart, J.; Schmidt, J.; Teufel, S.; Tumulka, R. Particle creation at a point source by means of interior-boundary conditions. Math. Phys. Anal. Geom.
**2018**, 21, 12. [Google Scholar] [CrossRef] - Lampart, J.; Schmidt, J. On the domain of Nelson-type Hamiltonians and abstract boundary conditions. arXiv, 2018; arXiv:1803.00872. [Google Scholar]
- Lampart, J. A nonrelativistic quantum field theory with point interactions in three dimensions. arXiv, 2018; arXiv:1804.08295. [Google Scholar]
- Dereziński, J. Van Hove Hamiltonians—Exactly solvable models of the infrared and ultraviolet problem. Ann. Henri Poincaré
**2003**, 4, 713–738. [Google Scholar] [CrossRef] - Teufel, S.; Tumulka, R. Avoiding ultraviolet divergence by means of interior-boundary conditions. In Quantum Mathematical Physics—A Bridge between Mathematics and Physics; Finster, F., Kleiner, J., Röken, C., Tolksdorf, J., Eds.; Birkhäuser: Basel, Switzerland, 2016; pp. 293–311. [Google Scholar]
- Georgii, H.-O.; Tumulka, R. Some jump processes in quantum field theory. In Interacting Stochastic Systems; Deuschel, J.-D., Greven, A., Eds.; Springer: Berlin, Germany, 2004; pp. 55–73. [Google Scholar]
- Keppeler, S.; Sieber, M. Particle creation and annihilation at interior boundaries: One-dimensional models. J. Phys. A Math. Theor.
**2016**, 49, 125204. [Google Scholar] [CrossRef] - Moshinsky, M. Boundary conditions for the description of nuclear reactions. Phys. Rev.
**1951**, 81, 347–352. [Google Scholar] [CrossRef] - Moshinsky, M. Boundary conditions and time-dependent states. Phys. Rev.
**1951**, 84, 525–532. [Google Scholar] [CrossRef] - Thomas, L.E. Multiparticle Schrödinger Hamiltonians with point interactions. Phys. Rev. D
**1984**, 30, 1233–1237. [Google Scholar] [CrossRef] - Yafaev, D.R. On a zero-range interaction of a quantum particle with the vacuum. J. Phys. A Math. Gen.
**1992**, 25, 963–978. [Google Scholar] [CrossRef] - Dürr, D.; Goldstein, S.; Teufel, S.; Tumulka, R.; Zanghì, N. Bohmian trajectories for Hamiltonians with interior–boundary conditions. In preparation. 2018. [Google Scholar]
- Bohm, D. Comments on an article of Takabayasi concerning the formulation of quantum mechanics with classical pictures. Prog. Theor. Phys.
**1953**, 9, 273–287. [Google Scholar] [CrossRef] - Dürr, D.; Goldstein, S.; Münch-Berndl, K.; Zanghì, N. Hypersurface Bohm–Dirac models. Phys. Rev. A
**1999**, 60, 2729–2736, Reprinted in Quantum Physics without Quantum Philosophy; Dürr, D.; Goldstein, S.; Zanghì, N.; Springer: Heidelberg, Germany, 2013. [Google Scholar] [CrossRef] - Tumulka, R. Closed 3-Forms and Random World Lines. Ph.D. Thesis, Mathematics Institute, Ludwig-Maximilians-Universität, München, Germany, 2001. Available online: http://edoc.ub.uni-muenchen.de/7/ (accessed on 11 June 2018).
- Sutherland, R. Causally symmetric Bohm model. Stud. Hist. Philos. Mod. Phys.
**2008**, 39, 782–805. [Google Scholar] [CrossRef][Green Version] - Sutherland, R. Lagrangian description for particle interpretations of quantum mechanics-entangled many-particle case. Found. Phys.
**2017**, 47, 174–207. [Google Scholar] [CrossRef] - Dürr, D.; Goldstein, S.; Norsen, T.; Struyve, W.; Zanghì, N. Can Bohmian mechanics be made relativistic? Proc. R. Soc. A
**2014**, 470, 20130699. [Google Scholar] [CrossRef] [PubMed] - Teufel, S.; Tumulka, R. Simple proof for global existence of Bohmian trajectories. Commun. Math. Phys.
**2005**, 258, 349–365. [Google Scholar] [CrossRef] - Lienert, M.; Tumulka, R. Born’s rule for arbitrary Cauchy surfaces. arXiv, 2017; arXiv:1706.07074. [Google Scholar]
- Struyve, W.; Tumulka, R. Bohmian trajectories for a time foliation with kinks. J. Geom. Phys.
**2014**, 82, 75–83. [Google Scholar] [CrossRef][Green Version] - Struyve, W.; Tumulka, R. Bohmian mechanics for a degenerate time foliation. Quantum Stud. Math. Found.
**2015**, 2, 349–358. [Google Scholar] [CrossRef][Green Version] - Tumulka, R. Bohmian mechanics at space-time singularities. II. Spacelike singularities. Gen. Relat. Gravit.
**2010**, 42, 303–346. [Google Scholar] [CrossRef] - Dirac, P.A.M. Relativistic quantum mechanics. Proc. R. Soc. Lond. A
**1932**, 136, 453–464. [Google Scholar] [CrossRef] - Dirac, P.A.M.; Fock, V.A.; Podolsky, B. On quantum electrodynamics. Phys. Z. Sowjetunion
**1932**, 2, 468–479, Reprinted in Selected Papers on Quantum Electrodynamics; Schwinger, J., Ed.; Dover: New York, USA, 1958. [Google Scholar] - Bloch, F. Die physikalische Bedeutung mehrerer Zeiten in der Quantenelektrodynamik. Phys. Z. Sowjetunion
**1934**, 5, 301–305. [Google Scholar] - Lienert, M.; Petrat, S.; Tumulka, R. Multi-time wave functions. J. Phys. Conf. Ser.
**2017**, 880, 012006. [Google Scholar] [CrossRef][Green Version] - Tomonaga, S. On a relativistically invariant formulation of the quantum theory of wave fields. Prog. Theor. Phys.
**1946**, 1, 27–42. [Google Scholar] [CrossRef] - Schwinger, J. Quantum electrodynamics. I. A covariant formulation. Phys. Rev.
**1948**, 74, 1439–1461. [Google Scholar] [CrossRef] - Lienert, M. Direct interaction along light cones at the quantum level. arXiv, 2017; arXiv:1801.00060. [Google Scholar]
- Petrat, S.; Tumulka, R. Multi-time Schrödinger equations cannot contain interaction potentials. J. Math. Phys.
**2014**, 55, 032302. [Google Scholar] [CrossRef][Green Version] - Nickel, L.; Deckert, D.-A. Consistency of multi-time Dirac equations with general interaction potentials. J. Math. Phys.
**2016**, 57, 072301. [Google Scholar][Green Version] - Droz-Vincent, P. Relativistic quantum mechanics with non conserved number of particles. J. Geom. Phys.
**1985**, 2, 101–119. [Google Scholar] [CrossRef] - Lienert, M. A relativistically interacting exactly solvable multi-time model for two mass-less Dirac particles in 1+1 dimensions. J. Math. Phys.
**2015**, 56, 042301. [Google Scholar] [CrossRef] - Lienert, M.; Nickel, L. A simple explicitly solvable interacting relativistic N-particle model. J. Phys. A Math. Theor.
**2015**, 48, 325301. [Google Scholar] [CrossRef] - Petrat, S.; Tumulka, R. Multi-time wave functions for quantum field theory. Ann. Phys.
**2014**, 345, 17–54. [Google Scholar] [CrossRef][Green Version] - Petrat, S.; Tumulka, R. Multi-time formulation of pair creation. J. Phys. A Math. Theor.
**2014**, 47, 112001. [Google Scholar] [CrossRef][Green Version] - Tumulka, R. A relativistic version of the Ghirardi–Rimini–Weber model. J. Stat. Phys.
**2006**, 125, 821–840. [Google Scholar] [CrossRef] - Bedingham, D.; Dürr, D.; Ghirardi, G.C.; Goldstein, S.; Tumulka, R.; Zanghì, N. Matter density and relativistic models of wave function collapse. J. Stat. Phys.
**2014**, 154, 623–631. [Google Scholar] [CrossRef] - Oppenheimer, J.R. Note on light quanta and the electromagnetic field. Phys. Rev.
**1931**, 38, 725–748. [Google Scholar] [CrossRef] - Colin, S.; Struyve, W. A Dirac sea pilot-wave model for quantum field theory. J. Phys. A Math. Theor.
**2007**, 40, 7309–7341. [Google Scholar] [CrossRef][Green Version] - Deckert, D.-A.; Esfeld, M.; Oldofredi, A. A persistent particle ontology for QFT in terms of the Dirac sea. arXiv, 2016; arXiv:1608.06141. [Google Scholar]

**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 $.

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).