Editing De Broglie–Bohm theory (section)

=== Curved space ===
To extend de Broglie–Bohm theory to curved space ([[Riemannian manifolds]] in mathematical parlance), one simply notes that all of the elements of these equations make sense, such as [[gradient]]s and [[Laplacian]]s. Thus, we use equations that have the same form as above. Topological and [[boundary conditions]] may apply in supplementing the evolution of Schrödinger's equation.

For a de Broglie–Bohm theory on curved space with spin, the spin space becomes a [[vector bundle]] over configuration space, and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space.<ref>{{cite journal | last1 = Dürr | first1 = D. | last2 = Goldstein | first2 = S. | last3 = Taylor | first3 = J. | last4 = Tumulka | first4 = R. | last5 = Zanghì | first5 = N. | year = 2007 | title = Quantum Mechanics in Multiply-Connected Spaces | journal = J. Phys. A | volume = 40 | issue = 12| pages = 2997–3031 | doi=10.1088/1751-8113/40/12/s08|arxiv = quant-ph/0506173 |bibcode = 2007JPhA...40.2997D | s2cid = 119410880 }}</ref>
The field equations for the de Broglie–Bohm theory in the relativistic case with spin can also be given for curved space-times with torsion.<ref>{{cite journal | last1 = Fabbri | first1 = Luca | date= 2022 | title = de Broglie-Bohm formulation of Dirac fields | journal = Foundations of Physics| volume =  52| issue = 6 | pages =  116| doi = 10.1007/s10701-022-00641-2| arxiv = 2207.05755 | bibcode = 2022FoPh...52..116F | s2cid = 250491612 }}</ref><ref>{{cite journal | last1 = Fabbri | first1 = Luca | date= 2023 | title = Dirac Theory in Hydrodynamic Form | journal = Foundations of Physics| volume =  53| issue = 3 | pages =  54| doi = 10.1007/s10701-023-00695-w | arxiv = 2303.17461 | bibcode = 2023FoPh...53...54F | s2cid = 257833858 }}</ref>

In a general spacetime with curvature and torsion, the guiding equation for the [[four-velocity]] <math>u^i</math> of an elementary [[fermion]] particle is<math display="block">u^i=\frac{e^i_\mu \bar{\psi}\gamma^\mu \psi}{\bar{\psi}\psi}, </math>where the wave function <math>\psi</math> is a [[Dirac spinor|spinor]], <math>\bar{\psi}</math> is the corresponding [[Dirac adjoint|adjoint]], <math>\gamma^\mu</math> are the [[gamma matrices|Dirac matrices]], and <math>e^i_\mu</math> is a [[tetrad formalism|tetrad]].<ref name="FG">{{cite journal | author=F. R. Benard Guedes, N. J. Popławski | date=2024 | title=General-relativistic wave-particle duality with torsion | journal=Classical and Quantum Gravity | volume=41 | issue=6 | pages=065011 | doi=10.1088/1361-6382/ad1fcb | arxiv=2211.03234 }}</ref> If the wave function propagates according to the [[Dirac equation in curved spacetime|curved]] Dirac equation, then the particle moves according to the [[Mathisson-Papapetrou-Dixon equations|Mathisson-Papapetrou equations]] of motion, which are an extension of the [[geodesics in general relativity|geodesic equation]]. This relativistic wave-particle duality follows from the [[conservation law|conservation laws]] for the [[spin tensor]] and [[stress-energy tensor|energy-momentum tensor]],<ref name="FG" /> and also from the covariant [[Heisenberg picture]] equation of motion.<ref>{{cite journal | author=S. K. Wong | date=1972 | title=Heisenberg equations of motion for spin-1/2 wave equation in general relativity | journal=International Journal of Theoretical Physics | volume=5 | issue=4 | pages=221–230 | doi=10.1007/BF00670477 }}</ref>