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Quantum potential

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The quantum potential or quantum potentiality is a central concept of the <a href="De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm formulation</a> of <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, introduced by <a href="David_Bohm" title="David Bohm">David Bohm</a> in 1952.

Initially presented under the name quantum-mechanical potential, subsequently quantum potential, it was later elaborated upon by Bohm and <a href="Basil_Hiley" title="Basil Hiley">Basil Hiley</a> in its interpretation as an information potential which acts on a quantum particle. It is also referred to as quantum <a href="Potential_energy" title="Potential energy">potential energy</a>, Bohm potential, quantum Bohm potential or Bohm quantum potential.

<td>Quantum potential

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<annotation encoding="application/x-tex">{\displaystyle \quad Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}R}{R}}}</annotation>

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</math><img src="./e1e51e0b287976b65c38843208a195e3dbb46561.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.697ex; height:5.843ex;" alt="{\displaystyle \quad Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}R}{R}}}" loading="lazy">

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In the framework of the de Broglie–Bohm theory, the quantum potential is a term within the <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> which acts to guide the movement of quantum particles. The quantum potential approach introduced by Bohm<a href="#cite_note-bohm-1952-I-1">[1]</a><a href="#cite_note-bohm-1952-II-2">[2]</a> provides a physically less fundamental exposition of the idea presented by <a href="Louis_de_Broglie" title="Louis de Broglie">Louis de Broglie</a>: de Broglie had postulated in 1925 that the relativistic <a href="Wave_function" title="Wave function">wave function</a> defined on spacetime represents a <a href="Pilot_wave" class="mw-redirect" title="Pilot wave">pilot wave</a> which guides a quantum particle, represented as an oscillating peak in the wave field, but he had subsequently abandoned his approach because he was unable to derive the guidance equation for the particle from a non-linear wave equation. The seminal articles of Bohm in 1952 introduced the quantum potential and included answers to the objections which had been raised against the pilot wave theory.

The Bohm quantum potential is closely linked with the results of other approaches, in particular relating to works of <a href="Erwin_Madelung" title="Erwin Madelung">Erwin Madelung</a> in 1927<a href="#cite_note-3">[3]</a> and <a href="Carl_Friedrich_von_Weizs%C3%A4cker" title="Carl Friedrich von Weizsäcker">Carl Friedrich von Weizsäcker</a> in 1935.<a href="#cite_note-4">[4]</a>

Building on the interpretation of the quantum theory introduced by Bohm in 1952, David Bohm and <a href="Basil_Hiley" title="Basil Hiley">Basil Hiley</a> in 1975 presented how the concept of a quantum potential leads to the notion of an "unbroken wholeness of the entire universe", proposing that the fundamental new quality introduced by quantum physics is <a href="Quantum_nonlocality" title="Quantum nonlocality">nonlocality</a>.<a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Relation_to_the_Schrödinger_equation">Relation to the Schrödinger equation</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Madelung_equations" title="Madelung equations">Madelung equations</a></div>

The <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>

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<annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\right)\psi \quad }</annotation>

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</math><img src="./84ff3c939a125d0ef69c3e681220941a2c41352e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.984ex; height:6.343ex;" alt="{\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\right)\psi \quad }" loading="lazy"></dd></dl>

is re-written using the polar form for the wave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =R\exp(iS/\hbar )}">

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<annotation encoding="application/x-tex">{\displaystyle \psi =R\exp(iS/\hbar )}</annotation>

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</math><img src="./1ccef4c4e216b453a0a46e6c41e452f0bc8f73a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.895ex; height:2.843ex;" alt="{\displaystyle \psi =R\exp(iS/\hbar )}" loading="lazy"> with real-valued functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is the amplitude (<a href="Absolute_value" title="Absolute value">absolute value</a>) of the wave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

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</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S/\hbar }">

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<annotation encoding="application/x-tex">{\displaystyle S/\hbar }</annotation>

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</math><img src="./8022b3260f2d9015f68fd62189e5ffd0113a7dc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.968ex; height:2.843ex;" alt="{\displaystyle S/\hbar }" loading="lazy"> its phase. This yields two equations: from the imaginary and real part of the Schrödinger equation follow the <a href="Continuity_equation" title="Continuity equation">continuity equation</a> and the quantum <a href="Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a> respectively.<a href="#cite_note-bohm-1952-I-1">[1]</a><a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Continuity_equation">Continuity equation</h3></div>

The imaginary part of the Schrödinger equation in polar form yields

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial R}{\partial t}}=-{\frac {1}{2m}}\left[R\nabla ^{2}S+2\nabla R\cdot \nabla S\right],}</annotation>

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</math><img src="./67d517d8a8558c9299b186725f07672c3e2f6dae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.294ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial R}{\partial t}}=-{\frac {1}{2m}}\left[R\nabla ^{2}S+2\nabla R\cdot \nabla S\right],}" loading="lazy"></dd></dl>

which, provided <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =R^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \rho =R^{2}}</annotation>

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</math><img src="./7d510dfb50093f9115982f0b7b9c08916c936dba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.119ex; height:3.176ex;" alt="{\displaystyle \rho =R^{2}}" loading="lazy">, can be interpreted as the <a href="Continuity_equation#Quantum_mechanics" title="Continuity equation">continuity equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \rho /\partial t+\nabla \cdot (\rho v)=0}">

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<annotation encoding="application/x-tex">{\displaystyle \partial \rho /\partial t+\nabla \cdot (\rho v)=0}</annotation>

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</math><img src="./f1d52051f6507d20ac329a602978a80c6741beb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.695ex; height:2.843ex;" alt="{\displaystyle \partial \rho /\partial t+\nabla \cdot (\rho v)=0}" loading="lazy"> for the probability density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> and the velocity field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\frac {1}{m}}\nabla S}">

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<annotation encoding="application/x-tex">{\displaystyle v={\frac {1}{m}}\nabla S}</annotation>

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</math><img src="./f21a97b1d99df9be2af27d95f69a353d0f0fa474.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.538ex; height:5.176ex;" alt="{\displaystyle v={\frac {1}{m}}\nabla S}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Quantum_Hamilton–Jacobi_equation">Quantum Hamilton–Jacobi equation</h3></div>

The real part of the Schrödinger equation in polar form yields a modified Hamilton–Jacobi equation<div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">

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<annotation encoding="application/x-tex">{\displaystyle -{\frac {\partial S}{\partial t}}={\frac {\|\nabla S\|^{2}}{2m}}+V+Q}</annotation>

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</math><img src="./92dc5c283362f7094966a99e50e4b1c3053411fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.517ex; height:6.009ex;" alt="{\displaystyle -{\frac {\partial S}{\partial t}}={\frac {\|\nabla S\|^{2}}{2m}}+V+Q}" loading="lazy">

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also referred to as quantum Hamilton–Jacobi equation.<a href="#cite_note-7">[7]</a> It differs from the classical <a href="Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a> only by the term<div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">

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<annotation encoding="application/x-tex">{\displaystyle Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}R}{R}}.}</annotation>

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</math><img src="./99edb5d0f7c99fc5fcfb447683ed5fb69b644ab9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.021ex; height:5.843ex;" alt="{\displaystyle Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}R}{R}}.}" loading="lazy">

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This term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

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</math><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy">, called quantum potential, thus depends on the <a href="Curvature#Graph_of_a_function" title="Curvature">curvature</a> of the amplitude of the wave function.<a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a>

In the limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar \to 0}">

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<annotation encoding="application/x-tex">{\displaystyle \hbar \to 0}</annotation>

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</math><img src="./b2a3125ca5aa17bb14bf452daa5c3224c7470ab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.083ex; height:2.176ex;" alt="{\displaystyle \hbar \to 0}" loading="lazy">, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is a solution of the (classical) Hamilton–Jacobi equation;<a href="#cite_note-bohm-1952-I-1">[1]</a> therefore, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is also called the Hamilton–Jacobi function, or <a href="Action_(physics)" title="Action (physics)">action</a>, extended to quantum physics.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

Hiley emphasised several aspects<a href="#cite_note-teleportation-P7-10">[10]</a> that regard the quantum potential of a quantum particle:

<ul><li>it is derived mathematically from the real part of the Schrödinger equation under <a href="Polar_coordinate_system" title="Polar coordinate system">polar decomposition</a> of the wave function,<a href="#cite_note-11">[11]</a> is not derived from a Hamiltonian<a href="#cite_note-information-quantum-theory-and-the-brain-P207-12">[12]</a> or other external source, and could be said to be involved in a <a href="Self-organization" title="Self-organization">self-organising process</a> involving a basic underlying field;</li>

<li>it does not change if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is multiplied by a constant, as this term is also present in the denominator, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

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</math><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"> is independent of the magnitude of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

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</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> and thus of field intensity; therefore, the quantum potential fulfils a precondition for nonlocality: it need not fall off as distance increases;</li>

<li>it carries information about the whole experimental arrangement in which the particle finds itself.</li></ul>

In 1979, Hiley and his co-workers Philippidis and Dewdney presented a full calculation on the explanation of the <a href="Two-slit_experiment" class="mw-redirect" title="Two-slit experiment">two-slit experiment</a> in terms of Bohmian trajectories that arise for each particle moving under the influence of the quantum potential, <a href="De_Broglie%E2%80%93Bohm_theory#Double-slit_experiment" title="De Broglie–Bohm theory">resulting in the well-known interference patterns</a>.<a href="#cite_note-13">[13]</a>

Also the shift of the interference pattern which occurs in presence of a magnetic field in the <a href="Aharonov%E2%80%93Bohm_effect" title="Aharonov–Bohm effect">Aharonov–Bohm effect</a> could be explained as arising from the quantum potential.<a href="#cite_note-14">[14]</a>

<div class="mw-heading mw-heading3"><h3 id="Relation_to_the_measurement_process">Relation to the measurement process</h3></div>

The <a href="Wave_function_collapse" title="Wave function collapse">collapse of the wave function</a> of the <a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen interpretation</a> of quantum theory is explained in the quantum potential approach by the demonstration that, after a measurement, "all the packets of the multi-dimensional wave function that do not correspond to the actual result of measurement have no effect on the particle" from then on.<a href="#cite_note-15">[15]</a> Bohm and Hiley pointed out that

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</style><blockquote class="templatequote">...the quantum potential can develop unstable bifurcation points, which separate classes of particle trajectories according to the "channels" into which they eventually enter and within which they stay. This explains how measurement is possible without "collapse" of the wave function, and how all sorts of quantum processes, such as transitions between states, fusion of two states into one and fission of one system into two, are able to take place without the need for a human observer.<a href="#cite_note-16">[16]</a></blockquote>

Measurement then "involves a participatory transformation in which both the system under observation and the observing apparatus undergo a mutual participation so that the trajectories behave in a correlated manner, becoming correlated and separated into different, non-overlapping sets (which we call 'channels')".<a href="#cite_note-17">[17]</a>

<div class="mw-heading mw-heading3"><h3 id="Quantum_potential_of_an_n-particle_system">Quantum potential of an n-particle system</h3></div>

The Schrödinger wave function of a <a href="Quantum_field_theory#Single-_and_many-particle_quantum_mechanics" title="Quantum field theory">many-particle quantum system</a> cannot be represented in ordinary <a href="Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a>. Rather, it is represented in <a href="Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a>, with three dimensions per particle. A single point in configuration space thus represents the configuration of the entire n-particle system as a whole.

A two-particle wave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)}</annotation>

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</math><img src="./cbee8c23bbeb7076a1a1152bb8cab8be580b4174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.176ex; height:2.843ex;" alt="{\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)}" loading="lazy"> of <a href="Schr%C3%B6dinger_field#Identical_particles" title="Schrödinger field">identical particles</a> of mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> has the quantum potential<a href="#cite_note-teleportation-P10-18">[18]</a>

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<annotation encoding="application/x-tex">{\displaystyle Q(\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {(\nabla _{1}^{2}+\nabla _{2}^{2})R(\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)}{R(\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)}}}</annotation>

</semantics>

</math><img src="./dfce830f1f5429c94b8ee2a9ec8e79cb2a3a8d17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.34ex; height:6.676ex;" alt="{\displaystyle Q(\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {(\nabla _{1}^{2}+\nabla _{2}^{2})R(\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)}{R(\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{1}^{2}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla _{1}^{2}}</annotation>

</semantics>

</math><img src="./d34854e0f3f44d0b4672654afc057b5af4cebc74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.99ex; height:3.176ex;" alt="{\displaystyle \nabla _{1}^{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{2}^{2}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla _{2}^{2}}</annotation>

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</math><img src="./38ec803e999bf76049d5b21e9158d5bd0e9900f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.99ex; height:3.176ex;" alt="{\displaystyle \nabla _{2}^{2}}" loading="lazy"> refer to particle 1 and particle 2 respectively. This expression generalizes in straightforward manner to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> particles:

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</munderover>

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<annotation encoding="application/x-tex">{\displaystyle Q(\mathbf {r_{1}} ,...,\mathbf {r_{n}} ,\,t)=-{\frac {\hbar ^{2}}{2R(\mathbf {r_{1}} ,...,\mathbf {r_{n}} ,\,t)}}\sum _{i=1}^{n}{\frac {\nabla _{i}^{2}}{m_{i}}}R(\mathbf {r_{1}} ,...,\mathbf {r_{n}} ,\,t)}</annotation>

</semantics>

</math><img src="./d319ff60203bf49ebf9ea8d24f225cad7e6eb005.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.938ex; height:7.009ex;" alt="{\displaystyle Q(\mathbf {r_{1}} ,...,\mathbf {r_{n}} ,\,t)=-{\frac {\hbar ^{2}}{2R(\mathbf {r_{1}} ,...,\mathbf {r_{n}} ,\,t)}}\sum _{i=1}^{n}{\frac {\nabla _{i}^{2}}{m_{i}}}R(\mathbf {r_{1}} ,...,\mathbf {r_{n}} ,\,t)}" loading="lazy"></dd></dl>

In case the wave function of two or more particles is separable, then the system's total quantum potential becomes the sum of the quantum potentials of the two particles. Exact separability is extremely unphysical given that interactions between the system and its environment destroy the factorization; however, a wave function that is a <a href="Superposition_principle" title="Superposition principle">superposition</a> of several wave functions of approximately disjoint <a href="Support_(mathematics)" title="Support (mathematics)">support</a> will factorize approximately.<a href="#cite_note-19">[19]</a>

<div class="mw-heading mw-heading3"><h3 id="Derivation_for_a_separable_quantum_system">Derivation for a separable quantum system</h3></div>

That the wave function is separable means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

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</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> factorizes in the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=\psi _{A}(\mathbf {r_{1}} ,\,t)\psi _{B}(\mathbf {r_{2}} ,\,t)}">

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=\psi _{A}(\mathbf {r_{1}} ,\,t)\psi _{B}(\mathbf {r_{2}} ,\,t)}</annotation>

</semantics>

</math><img src="./8bb9c9d7c72c57460ddcb198602d69ec8249cf96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.944ex; height:2.843ex;" alt="{\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=\psi _{A}(\mathbf {r_{1}} ,\,t)\psi _{B}(\mathbf {r_{2}} ,\,t)}" loading="lazy">. Then it follows that also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> factorizes, and the system's total quantum potential becomes the sum of the quantum potentials of the two particles.<a href="#cite_note-20">[20]</a>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=-{\frac {\hbar ^{2}}{2m}}({\frac {\nabla _{1}^{2}R_{A}(\mathbf {r_{1}} ,\,t)}{R_{A}(\mathbf {r_{1}} ,\,t)}}+{\frac {\nabla _{2}^{2}R_{B}(\mathbf {r_{2}} ,\,t)}{R_{B}(\mathbf {r_{2}} ,\,t)}})=Q_{A}(\mathbf {r_{1}} ,\,t)+Q_{B}(\mathbf {r_{2}} ,\,t)}</annotation>

</semantics>

</math><img src="./c76e42983afd6cf60c82ec33588699b6727833c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:77.179ex; height:6.676ex;" alt="{\displaystyle Q(\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=-{\frac {\hbar ^{2}}{2m}}({\frac {\nabla _{1}^{2}R_{A}(\mathbf {r_{1}} ,\,t)}{R_{A}(\mathbf {r_{1}} ,\,t)}}+{\frac {\nabla _{2}^{2}R_{B}(\mathbf {r_{2}} ,\,t)}{R_{B}(\mathbf {r_{2}} ,\,t)}})=Q_{A}(\mathbf {r_{1}} ,\,t)+Q_{B}(\mathbf {r_{2}} ,\,t)}" loading="lazy"></dd></dl>

In case the wave function is separable, that is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> factorizes in the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=\psi _{A}(\mathbf {r_{1}} ,\,t)\psi _{B}(\mathbf {r_{2}} ,\,t)}">

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=\psi _{A}(\mathbf {r_{1}} ,\,t)\psi _{B}(\mathbf {r_{2}} ,\,t)}</annotation>

</semantics>

</math><img src="./8bb9c9d7c72c57460ddcb198602d69ec8249cf96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.944ex; height:2.843ex;" alt="{\displaystyle \psi (\mathbf {r_{1}} ,\mathbf {r_{2}} ,\,t)=\psi _{A}(\mathbf {r_{1}} ,\,t)\psi _{B}(\mathbf {r_{2}} ,\,t)}" loading="lazy">, the two one-particle systems behave independently. More generally, the quantum potential of an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-particle system with separable wave function is the sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> quantum potentials, separating the system into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> independent one-particle systems.<a href="#cite_note-21">[21]</a>

<div class="mw-heading mw-heading2"><h2 id="Formulation_in_terms_of_probability_density">Formulation in terms of probability density</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Quantum_potential_in_terms_of_the_probability_density_function">Quantum potential in terms of the probability density function</h3></div>

Bohm, as well as other physicists after him, have sought to provide evidence that the <a href="Born_rule" title="Born rule">Born rule</a> linking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> to the <a href="Probability_density_function" title="Probability density function">probability density function</a>

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<annotation encoding="application/x-tex">{\displaystyle \rho =R^{2}\quad }</annotation>

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</math><img src="./a31d460df98bc29960f00837bda98fe8f43f282b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.441ex; height:3.176ex;" alt="{\displaystyle \rho =R^{2}\quad }" loading="lazy"></dd></dl>

can be understood, in a pilot wave formulation, as not representing a basic law, but rather a theorem (called <a href="Quantum_equilibrium_hypothesis" class="mw-redirect" title="Quantum equilibrium hypothesis">quantum equilibrium hypothesis</a>) which applies when a quantum equilibrium is reached during the course of the time development under the Schrödinger equation. With Born's rule, and straightforward application of the <a href="Chain_rule" title="Chain rule">chain</a> and <a href="Product_rule" title="Product rule">product rules</a>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}{\sqrt {\rho }}=\nabla \nabla \rho ^{1/2}=\nabla \left({\frac {1}{2}}\rho ^{-1/2}\nabla \rho \right)={\frac {1}{2}}\left[\left(\nabla \rho ^{-1/2}\right)\nabla \rho +\rho ^{-1/2}\nabla ^{2}\rho \right]}</annotation>

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</math><img src="./1e32521ad225db6a9e029fdc7cc8bbd89724bad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:69.461ex; height:6.176ex;" alt="{\displaystyle \nabla ^{2}{\sqrt {\rho }}=\nabla \nabla \rho ^{1/2}=\nabla \left({\frac {1}{2}}\rho ^{-1/2}\nabla \rho \right)={\frac {1}{2}}\left[\left(\nabla \rho ^{-1/2}\right)\nabla \rho +\rho ^{-1/2}\nabla ^{2}\rho \right]}" loading="lazy"></dd></dl>

the quantum potential, expressed in terms of the probability density function, becomes:<a href="#cite_note-22">[22]</a>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}{\sqrt {\rho }}}{\sqrt {\rho }}}=-{\frac {\hbar ^{2}}{4m}}\left[{\frac {\nabla ^{2}\rho }{\rho }}-{\frac {1}{2}}{\frac {(\nabla \rho )^{2}}{\rho ^{2}}}\right]}</annotation>

</semantics>

</math><img src="./5a6cf0d7e0d004371aaf68829a3ef802c454a557.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.496ex; height:7.509ex;" alt="{\displaystyle Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}{\sqrt {\rho }}}{\sqrt {\rho }}}=-{\frac {\hbar ^{2}}{4m}}\left[{\frac {\nabla ^{2}\rho }{\rho }}-{\frac {1}{2}}{\frac {(\nabla \rho )^{2}}{\rho ^{2}}}\right]}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Quantum_force">Quantum force</h3></div>

The quantum force <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{Q}=-\nabla Q}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle F_{Q}=-\nabla Q}</annotation>

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</math><img src="./6323466603796bb87417803b5990eed12707d4f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.708ex; height:2.843ex;" alt="{\displaystyle F_{Q}=-\nabla Q}" loading="lazy">, expressed in terms of the probability distribution, amounts to:<a href="#cite_note-maddox-bittner-2003-23">[23]</a>

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<annotation encoding="application/x-tex">{\displaystyle F_{Q}={\frac {\hbar ^{2}}{4m}}\left[{\frac {\nabla (\nabla ^{2}\rho )}{\rho }}-{\frac {\nabla (\nabla \rho \cdot \nabla \rho )}{2\rho ^{2}}}-\left({\frac {\nabla ^{2}\rho }{\rho }}-{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}\right){\frac {\nabla \rho }{\rho }}\right]}</annotation>

</semantics>

</math><img src="./19ce59c70d1a877ad89ec3c6e2f1df63ba79f567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.694ex; height:7.509ex;" alt="{\displaystyle F_{Q}={\frac {\hbar ^{2}}{4m}}\left[{\frac {\nabla (\nabla ^{2}\rho )}{\rho }}-{\frac {\nabla (\nabla \rho \cdot \nabla \rho )}{2\rho ^{2}}}-\left({\frac {\nabla ^{2}\rho }{\rho }}-{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}\right){\frac {\nabla \rho }{\rho }}\right]}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Formulation_in_configuration_space_and_in_momentum_space,_as_the_result_of_projections">Formulation in configuration space and in momentum space, as the result of projections</h3></div>

M. R. Brown and B. Hiley showed that, as alternative to its formulation terms of <a href="Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">-space), the quantum potential can also be formulated in terms of <a href="Momentum_space" class="mw-redirect" title="Momentum space">momentum space</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-space).<a href="#cite_note-24">[24]</a><a href="#cite_note-brown-hiley-25">[25]</a>

In line with David Bohm's approach, Basil Hiley and mathematician <a href="Maurice_de_Gosson" class="mw-redirect" title="Maurice de Gosson">Maurice de Gosson</a> showed that the quantum potential can be seen as a consequence of a <a href="Projection_(mathematics)" title="Projection (mathematics)">projection</a> of an underlying structure, more specifically of a <a href="Non-commutative_algebra" class="mw-redirect" title="Non-commutative algebra">non-commutative algebraic</a> structure, onto a subspace such as ordinary space (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">-space). In algebraic terms, the quantum potential can be seen as arising from the relation between <a href="Implicate_and_explicate_order_according_to_David_Bohm" class="mw-redirect" title="Implicate and explicate order according to David Bohm">implicate and explicate orders</a>: if a <a href="Non-commutative_algebra" class="mw-redirect" title="Non-commutative algebra">non-commutative algebra</a> is employed to describe the non-commutative structure of the quantum formalism, it turns out that it is impossible to define an underlying space, but that rather "<a href="Basil_Hiley#Projections_into_shadow_manifolds" title="Basil Hiley">shadow spaces</a>" (homomorphic spaces) can be constructed and that in so doing the quantum potential appears.<a href="#cite_note-brown-hiley-25">[25]</a><a href="#cite_note-26">[26]</a><a href="#cite_note-hiley-reappraisal-bohm-2005-27">[27]</a><a href="#cite_note-non-commutative-2005-28">[28]</a><a href="#cite_note-29">[29]</a> The quantum potential approach can be seen as a way to construct the shadow spaces.<a href="#cite_note-hiley-reappraisal-bohm-2005-27">[27]</a> The quantum potential thus results as a distortion due to the projection of the underlying space into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">-space, in similar manner as a <a href="Mercator_projection" title="Mercator projection">Mercator projection</a> inevitably results in a distortion in a geographical map.<a href="#cite_note-hiley-anpa-23-2001-30">[30]</a><a href="#cite_note-hiley-callaghan-2010-A-31">[31]</a> There exists complete symmetry between the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">-representation, and the quantum potential as it appears in configuration space can be seen as arising from the dispersion of the momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-representation.<a href="#cite_note-hiley-phase-32">[32]</a>

The approach has been applied to extended <a href="Phase_space" title="Phase space">phase space</a>,<a href="#cite_note-hiley-phase-32">[32]</a><a href="#cite_note-33">[33]</a> also in terms of a <a href="Duffin%E2%80%93Kemmer%E2%80%93Petiau_algebra" title="Duffin–Kemmer–Petiau algebra">Duffin–Kemmer–Petiau algebra</a> approach.<a href="#cite_note-34">[34]</a><a href="#cite_note-35">[35]</a>

<div class="mw-heading mw-heading2"><h2 id="Relation_to_other_quantities_and_theories">Relation to other quantities and theories</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Relation_to_the_Fisher_information">Relation to the Fisher information</h3></div>

It can be shown<a href="#cite_note-36">[36]</a> that the mean value of the quantum potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=-\hbar ^{2}\nabla ^{2}{\sqrt {\rho }}/(2m{\sqrt {\rho }})}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle Q=-\hbar ^{2}\nabla ^{2}{\sqrt {\rho }}/(2m{\sqrt {\rho }})}</annotation>

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</math><img src="./1cc2da21943ade0970fdd51aaa9f0d63bcf04ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.561ex; height:3.509ex;" alt="{\displaystyle Q=-\hbar ^{2}\nabla ^{2}{\sqrt {\rho }}/(2m{\sqrt {\rho }})}" loading="lazy"> is proportional to the probability density's <a href="Fisher_information" title="Fisher information">Fisher information</a> about the observable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}}">

<mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {x}}}</annotation>

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</math><img src="./18d95a7845e4e16ffb7e18ab37a208d0ab18e0e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\hat {x}}}" loading="lazy">

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<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}=\int \rho \cdot (\nabla \ln \rho )^{2}\,d^{3}x=-\int \rho \nabla ^{2}(\ln \rho )\,d^{3}x.}</annotation>

</semantics>

</math><img src="./c7def2fd07ed3cd52681015e39f504d6fae6f442.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.069ex; width:44.865ex; height:5.676ex;" alt="{\displaystyle {\mathcal {I}}=\int \rho \cdot (\nabla \ln \rho )^{2}\,d^{3}x=-\int \rho \nabla ^{2}(\ln \rho )\,d^{3}x.}" loading="lazy"></dd></dl>

Using this definition for the Fisher information, we can write:<a href="#cite_note-37">[37]</a>

<msup>

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</msup>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \langle Q\rangle =\int \psi ^{*}Q\psi \,d^{3}x=\int \rho Q\,d^{3}x={\frac {\hbar ^{2}}{8m}}{\mathcal {I}}.}</annotation>

</semantics>

</math><img src="./2b0c87dfde0c9bf9ccbf3032be501372e7a21e0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.22ex; height:6.176ex;" alt="{\displaystyle \langle Q\rangle =\int \psi ^{*}Q\psi \,d^{3}x=\int \rho Q\,d^{3}x={\frac {\hbar ^{2}}{8m}}{\mathcal {I}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Quantum_potential_as_energy_of_internal_motion_associated_with_spin">Quantum potential as energy of internal motion associated with spin</h3></div>

Giovanni Salesi, Erasmo Recami and co-workers showed in 1998 that, in agreement with the <a href="K%C3%B6nig's_theorem_(kinetics)" title="König's theorem (kinetics)">König's theorem</a>, the quantum potential can be identified with the <a href="Kinetic_energy" title="Kinetic energy">kinetic energy</a> of the internal motion ("<a href="Zitterbewegung" title="Zitterbewegung">zitterbewegung</a>") associated with the <a href="Spin_(physics)" title="Spin (physics)">spin</a> of a <a href="Spin-1/2" title="Spin-1/2">spin-1/2</a> particle observed in a center-of-mass frame. More specifically, they showed that the internal zitterbewegung velocity for a spinning, non-relativistic particle of constant spin with no precession, and in absence of an external field, has the squared value:<a href="#cite_note-38">[38]</a>

<msup>

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</mrow>

<mfrac>

<mrow>

<msup>

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</msup>

<msup>

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</msup>

</mrow>

<msup>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {V} ^{2}={\frac {(\nabla \rho \land \mathbf {s} )^{2}}{(m\rho )^{2}}}={\frac {(\nabla \rho )^{2}\mathbf {s} ^{2}-(\nabla \rho \cdot \mathbf {s} )^{2}}{(m\rho )^{2}}}}</annotation>

</semantics>

</math><img src="./f545a789b1685b2a7c99ffa34bf276b1f87b952f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.27ex; height:6.676ex;" alt="{\displaystyle \mathbf {V} ^{2}={\frac {(\nabla \rho \land \mathbf {s} )^{2}}{(m\rho )^{2}}}={\frac {(\nabla \rho )^{2}\mathbf {s} ^{2}-(\nabla \rho \cdot \mathbf {s} )^{2}}{(m\rho )^{2}}}}" loading="lazy"></dd></dl>

from which the second term is shown to be of negligible size; then with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {s} |=\hbar /2}">

</mrow>

</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\mathbf {s} |=\hbar /2}</annotation>

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</math><img src="./c2b15c3e08e46e1419d935c2f7554e6c684aa1b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.079ex; height:2.843ex;" alt="{\displaystyle |\mathbf {s} |=\hbar /2}" loading="lazy"> it follows that

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<mfrac>

</mfrac>

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<mfrac>

<mrow>

</mrow>

<mrow>

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</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\mathbf {V} |={\frac {\hbar }{2}}{\frac {\nabla \rho }{m\rho }}}</annotation>

</semantics>

</math><img src="./593a64760a11663525199d64137471088772f355.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.633ex; height:5.843ex;" alt="{\displaystyle |\mathbf {V} |={\frac {\hbar }{2}}{\frac {\nabla \rho }{m\rho }}}" loading="lazy"></dd></dl>

Salesi gave further details on this work in 2009.<a href="#cite_note-39">[39]</a>

In 1999, Salvatore Esposito generalized their result from spin-1/2 particles to particles of arbitrary spin, confirming the interpretation of the quantum potential as a kinetic energy for an internal motion. Esposito showed that (using the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation>

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</math><img src="./de68de3a92517953436c93b5a76461d49160cc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" loading="lazy">=1) the quantum potential can be written as:<a href="#cite_note-esposito-1999-40">[40]</a>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle Q=-{\frac {1}{2}}m\mathbf {v} _{S}^{2}-{\frac {1}{2}}\nabla \cdot \mathbf {v} _{S}}</annotation>

</semantics>

</math><img src="./9b45d07439382926dc90cd184f3c8d2db1d0d854.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.645ex; height:5.176ex;" alt="{\displaystyle Q=-{\frac {1}{2}}m\mathbf {v} _{S}^{2}-{\frac {1}{2}}\nabla \cdot \mathbf {v} _{S}}" loading="lazy"></dd></dl>

and that the <a href="De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">causal interpretation of quantum mechanics</a> can be reformulated in terms of a particle velocity

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =\mathbf {v} _{B}+\mathbf {v} _{S}\times \mathbf {s} }</annotation>

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</math><img src="./94a56b34ce5ad090770303bddf84b5d33806bdd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.84ex; height:2.343ex;" alt="{\displaystyle \mathbf {v} =\mathbf {v} _{B}+\mathbf {v} _{S}\times \mathbf {s} }" loading="lazy"></dd></dl>

where the "drift velocity" is

<msub>

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</msub>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{B}={\frac {\nabla S}{m}}}</annotation>

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</math><img src="./70ab1c0cd6d4d741c67c5641781a0aa9cfe9282a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.26ex; height:5.343ex;" alt="{\displaystyle \mathbf {v} _{B}={\frac {\nabla S}{m}}}" loading="lazy"></dd></dl>

and the "relative velocity" is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{S}\times \mathbf {s} }">

<msub>

</mrow>

</mrow>

</msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{S}\times \mathbf {s} }</annotation>

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</math><img src="./31c975238cedd6397bd3a21964c710a3101da9b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.599ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{S}\times \mathbf {s} }" loading="lazy">, with

<msub>

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</mrow>

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<mfrac>

<mrow>

<msup>

</mrow>

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</mrow>

<mrow>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{S}={\frac {\nabla R^{2}}{2mR^{2}}}}</annotation>

</semantics>

</math><img src="./ddbdc63eb094233ab3794213e540da6ee132cf4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.659ex; height:6.009ex;" alt="{\displaystyle \mathbf {v} _{S}={\frac {\nabla R^{2}}{2mR^{2}}}}" loading="lazy"></dd></dl>

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {s} }</annotation>

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</math><img src="./644ae690160e658898a141e568a7fb0ee6040004.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.056ex; height:1.676ex;" alt="{\displaystyle \mathbf {s} }" loading="lazy"> representing the spin direction of the particle. In this formulation, according to Esposito, quantum mechanics must necessarily be interpreted in probabilistic terms, for the reason that a system's initial motion condition cannot be exactly determined.<a href="#cite_note-esposito-1999-40">[40]</a> Esposito explained that "the quantum effects present in the Schrödinger equation are due to the presence of a peculiar spatial direction associated with the particle that, assuming the isotropy of space, can be identified with the spin of the particle itself".<a href="#cite_note-41">[41]</a> Esposito generalized it from matter particles to <a href="Gauge_boson" title="Gauge boson">gauge particles</a>, in particular <a href="Photon" title="Photon">photons</a>, for which he showed that, if modelled as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =(\mathbf {E} -i\mathbf {B} )/{\sqrt {2}}}">

</mrow>

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<msqrt>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi =(\mathbf {E} -i\mathbf {B} )/{\sqrt {2}}}</annotation>

</semantics>

</math><img src="./d1260058c005c81f2e2188c802b916c746d9df65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.983ex; height:3.176ex;" alt="{\displaystyle \psi =(\mathbf {E} -i\mathbf {B} )/{\sqrt {2}}}" loading="lazy">, with probability function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{*}\cdot \psi =(\mathbf {E} ^{2}+\mathbf {B} ^{2})/2}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \psi ^{*}\cdot \psi =(\mathbf {E} ^{2}+\mathbf {B} ^{2})/2}</annotation>

</semantics>

</math><img src="./690d14825eea83b0f3ffe588fc966020791bd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.599ex; height:3.176ex;" alt="{\displaystyle \psi ^{*}\cdot \psi =(\mathbf {E} ^{2}+\mathbf {B} ^{2})/2}" loading="lazy">, they can be understood in a quantum potential approach.<a href="#cite_note-42">[42]</a>

James R. Bogan, in 2002, published the derivation of a reciprocal transformation from the Hamilton-Jacobi equation of classical mechanics to the time-dependent Schrödinger equation of quantum mechanics which arises from a <a href="Gauge_transformation" class="mw-redirect" title="Gauge transformation">gauge transformation</a> representing spin, under the simple requirement of <a href="Conservation_of_probability" class="mw-redirect" title="Conservation of probability">conservation of probability</a>. This spin-dependent transformation is a function of the quantum potential.<a href="#cite_note-43">[43]</a>

<div class="mw-heading mw-heading2"><h2 id="Re-interpretation_in_terms_of_Clifford_algebras">Re-interpretation in terms of Clifford algebras</h2></div>

B. Hiley and R. E. Callaghan re-interpret the role of the Bohm model and its notion of quantum potential in the framework of <a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a>, taking account of recent advances that include the work of <a href="David_Hestenes" title="David Hestenes">David Hestenes</a> on <a href="Spacetime_algebra" title="Spacetime algebra">spacetime algebra</a>. They show how, within a nested hierarchy of Clifford algebras <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\ell _{i,j}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle C\ell _{i,j}}</annotation>

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</math><img src="./ec3bfe8aa68236a950f8d63f14dbff82158c63de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.671ex; height:2.843ex;" alt="{\displaystyle C\ell _{i,j}}" loading="lazy">, for each <a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a> an element of a <a href="Minimal_ideal" title="Minimal ideal">minimal left ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{L}(\mathbf {r} ,t)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{L}(\mathbf {r} ,t)}</annotation>

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</math><img src="./46a4a7744151ee317ae19a79b90353e048317a2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.814ex; height:2.843ex;" alt="{\displaystyle \Phi _{L}(\mathbf {r} ,t)}" loading="lazy"> and an element of a <a href="Right_ideal" class="mw-redirect" title="Right ideal">right ideal</a> representing its <a href="Paravector#Clifford_conjugation" title="Paravector">Clifford conjugation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{R}(\mathbf {r} ,t)={\tilde {\Phi }}_{L}(\mathbf {r} ,t)}">

<msub>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{R}(\mathbf {r} ,t)={\tilde {\Phi }}_{L}(\mathbf {r} ,t)}</annotation>

</semantics>

</math><img src="./92e0861257da6e215e4cea337551e0bad8c09701.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.856ex; height:3.176ex;" alt="{\displaystyle \Phi _{R}(\mathbf {r} ,t)={\tilde {\Phi }}_{L}(\mathbf {r} ,t)}" loading="lazy"> can be constructed, and from it the Clifford density element (CDE) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{c}(\mathbf {r} ,t)=\Phi _{L}(\mathbf {r} ,t){\tilde {\Phi }}_{L}(\mathbf {r} ,t)}">

<msub>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle \rho _{c}(\mathbf {r} ,t)=\Phi _{L}(\mathbf {r} ,t){\tilde {\Phi }}_{L}(\mathbf {r} ,t)}</annotation>

</semantics>

</math><img src="./bdaeab6e13a7abfa9d7cfb3641e28731d4ef6835.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.659ex; height:3.176ex;" alt="{\displaystyle \rho _{c}(\mathbf {r} ,t)=\Phi _{L}(\mathbf {r} ,t){\tilde {\Phi }}_{L}(\mathbf {r} ,t)}" loading="lazy">, an element of the Clifford algebra which is isomorphic to the standard <a href="Density_matrix" title="Density matrix">density matrix</a> but independent of any specific representation.<a href="#cite_note-44">[44]</a> On this basis, bilinear invariants can be formed which represent properties of the system. Hiley and Callaghan distinguish bilinear invariants of a first kind, of which each stands for the expectation value of an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> of the algebra which can be formed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {Tr}}B\rho _{c}}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\rm {Tr}}B\rho _{c}}</annotation>

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</math><img src="./46adf6678faeffc47b92d68355d897c0b957a661.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.5ex; height:2.676ex;" alt="{\displaystyle {\rm {Tr}}B\rho _{c}}" loading="lazy">, and bilinear invariants of a second kind which are constructed with derivatives and represent momentum and energy. Using these terms, they reconstruct the results of quantum mechanics without depending on a particular representation in terms of a wave function nor requiring reference to an external Hilbert space. Consistent with earlier results, the quantum potential of a non-relativistic particle with spin (<a href="Pauli_equation" title="Pauli equation">Pauli particle</a>) is shown to have an additional spin-dependent term, and the momentum of a relativistic particle with spin (<a href="Dirac_equation" title="Dirac equation">Dirac particle</a>) is shown to consist in a linear motion and a rotational part.<a href="#cite_note-45">[45]</a> The two dynamical equations governing the time evolution are re-interpreted as conservation equations. One of them stands for the <a href="Conservation_of_energy" title="Conservation of energy">conservation of energy</a>; the other stands for the <a href="Conservation_of_probability" class="mw-redirect" title="Conservation of probability">conservation of probability</a> and <a href="Angular_momentum_operator#Conservation_of_angular_momentum" title="Angular momentum operator">of spin</a>.<a href="#cite_note-clifford-direc-bohm-hj-46">[46]</a> The quantum potential plays the role of an internal energy<a href="#cite_note-47">[47]</a> which ensures the conservation of total energy.<a href="#cite_note-clifford-direc-bohm-hj-46">[46]</a>

<div class="mw-heading mw-heading2"><h2 id="Relativistic_and_field-theoretic_extensions">Relativistic and field-theoretic extensions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Quantum_potential_and_relativity">Quantum potential and relativity</h3></div>

Bohm and Hiley demonstrated that the non-locality of quantum theory can be understood as limit case of a purely local theory, provided the transmission of active information is allowed to be greater than the speed of light, and that this limit case yields approximations to both quantum theory and relativity.<a href="#cite_note-48">[48]</a>

The quantum potential approach was extended by Hiley and co-workers to quantum field theory in <a href="Minkowski_space" title="Minkowski space">Minkowski spacetime</a><a href="#cite_note-49">[49]</a><a href="#cite_note-50">[50]</a><a href="#cite_note-51">[51]</a><a href="#cite_note-52">[52]</a> and to curved spacetime.<a href="#cite_note-53">[53]</a>

Carlo Castro and Jorge Mahecha derived the Schrödinger equation from the Hamilton-Jacobi equation in conjunction with the continuity equation, and showed that the properties of the relativistic Bohm quantum potential in terms of the ensemble density can be described by the Weyl properties of space. In Riemann flat space, the Bohm potential is shown to equal the <a href="Weyl_curvature" class="mw-redirect" title="Weyl curvature">Weyl curvature</a>. According to Castro and Mahecha, in the <a href="Relativistic_wave_equations" title="Relativistic wave equations">relativistic case</a>, the quantum potential (using the <a href="D'Alembert_operator" title="D'Alembert operator">d'Alembert operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \Box }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Box }</annotation>

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</math><img src="./0b41c86866080910eec1f6f1749e3cedd78b62e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \Box }" loading="lazy"> and in the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \hbar =1}</annotation>

</semantics>

</math><img src="./6e5f56ce258c75510831a8a14f3e2970ef0a1467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.567ex; height:2.176ex;" alt="{\displaystyle \hbar =1}" loading="lazy">) takes the form

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle Q=-{\frac {1}{2m}}{\frac {\quad \Box {\sqrt {\rho }}}{\sqrt {\rho }}}}</annotation>

</semantics>

</math><img src="./0cd1daa7137b2ec05c6352eb08f0e0539d955b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.889ex; height:6.843ex;" alt="{\displaystyle Q=-{\frac {1}{2m}}{\frac {\quad \Box {\sqrt {\rho }}}{\sqrt {\rho }}}}" loading="lazy"></dd></dl>

and the quantum force exerted by the relativistic quantum potential is shown to depend on the Weyl gauge potential and its derivatives. Furthermore, the relationship among Bohm's potential and the Weyl curvature in flat spacetime corresponds to a similar relationship among Fisher Information and Weyl geometry after introduction of a <a href="Complex_number" title="Complex number">complex</a> momentum.<a href="#cite_note-54">[54]</a>

Diego L. Rapoport, on the other hand, associates the relativistic quantum potential with the metric scalar curvature (Riemann curvature).<a href="#cite_note-55">[55]</a>

In relation to the Klein–Gordon equation for a particle with mass and charge, Peter R. Holland spoke in his book of 1993 of a "quantum potential-like term" that is proportional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box R/R}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Box R/R}</annotation>

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</math><img src="./4fe3af4c6c6b3e63f6d8c580241ca72e01a6aa7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.499ex; height:2.843ex;" alt="{\displaystyle \Box R/R}" loading="lazy">. He emphasized however that to give the Klein–Gordon theory a single-particle interpretation in terms of trajectories, as can be done for nonrelativistic Schrödinger quantum mechanics, would lead to unacceptable inconsistencies. For instance, wave functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {x} ,t)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {x} ,t)}</annotation>

</semantics>

</math><img src="./e0d619b826c3569f03532d5d67084d378a43b5bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.843ex;" alt="{\displaystyle \psi (\mathbf {x} ,t)}" loading="lazy"> that are solutions to the <a href="Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a> or the <a href="Dirac_equation" title="Dirac equation">Dirac equation</a> cannot be interpreted as the probability amplitude for a particle to be found in a given volume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{3}x}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d^{3}x}</annotation>

</semantics>

</math><img src="./dc9c17db5d994411d2f06ba6818f2fc930e7c4ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.602ex; height:2.676ex;" alt="{\displaystyle d^{3}x}" loading="lazy"> at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> in accordance with the usual axioms of quantum mechanics, and similarly in the causal interpretation it cannot be interpreted as the probability for the particle to be in that volume at that time. Holland pointed out that, while efforts have been made to determine a Hermitian position operator that would allow an interpretation of configuration space quantum field theory, in particular using the <a href="Newton%E2%80%93Wigner_localization" title="Newton–Wigner localization">Newton–Wigner localization</a> approach, but that no connection with possibilities for an empirical determination of position in terms of a relativistic measurement theory or for a trajectory interpretation has so far been established. Yet according to Holland this does not mean that the trajectory concept is to be discarded from considerations of relativistic quantum mechanics.<a href="#cite_note-56">[56]</a>

Hrvoje Nikolić derived <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=-(1/2m)\,\Box R/R}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q=-(1/2m)\,\Box R/R}</annotation>

</semantics>

</math><img src="./b4343680f7f9c646f9b3f44b59751f634ab6bd9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.968ex; height:2.843ex;" alt="{\displaystyle Q=-(1/2m)\,\Box R/R}" loading="lazy"> as expression for the quantum potential, and he proposed a Lorentz-covariant formulation of the Bohmian interpretation of many-particle wave functions.<a href="#cite_note-57">[57]</a> He also developed a generalized relativistic-invariant probabilistic interpretation of quantum theory,<a href="#cite_note-58">[58]</a><a href="#cite_note-nikolicqft-59">[59]</a><a href="#cite_note-60">[60]</a> in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi |^{2}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\psi |^{2}}</annotation>

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</math><img src="./1f1766f8c8e0a96326d9379b65a63900b3be22ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.861ex; height:3.343ex;" alt="{\displaystyle |\psi |^{2}}" loading="lazy"> is no longer a probability density in space but a probability density in space-time.<a href="#cite_note-61">[61]</a><a href="#cite_note-62">[62]</a>

<div class="mw-heading mw-heading3"><h3 id="Quantum_potential_in_quantum_field_theory">Quantum potential in quantum field theory</h3></div>

Starting from the space representation of the field coordinate, a causal interpretation of the Schrödinger picture of relativistic quantum theory has been constructed. The Schrödinger picture for a neutral, spin 0, massless field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi \left[\psi (\mathbf {x} ,t)\right]=R\left[\psi (\mathbf {x} ,t)\right]e^{S\left[\psi (\mathbf {x} ,t)\right]}}">

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \Psi \left[\psi (\mathbf {x} ,t)\right]=R\left[\psi (\mathbf {x} ,t)\right]e^{S\left[\psi (\mathbf {x} ,t)\right]}}</annotation>

</semantics>

</math><img src="./b73792526dcf5cef5e95e35923cc3228809719d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.322ex; height:3.343ex;" alt="{\displaystyle \Psi \left[\psi (\mathbf {x} ,t)\right]=R\left[\psi (\mathbf {x} ,t)\right]e^{S\left[\psi (\mathbf {x} ,t)\right]}}" loading="lazy">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[\psi (\mathbf {x} ,t)\right],S\left[\psi (\mathbf {x} ,t)\right]}">

<mrow>

<mrow>

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</mrow>

<mrow>

<mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\left[\psi (\mathbf {x} ,t)\right],S\left[\psi (\mathbf {x} ,t)\right]}</annotation>

</semantics>

</math><img src="./9bf69dae4b8417c7300a76550c14e91b35217a2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.26ex; height:2.843ex;" alt="{\displaystyle R\left[\psi (\mathbf {x} ,t)\right],S\left[\psi (\mathbf {x} ,t)\right]}" loading="lazy"> real-valued <a href="Functional_(mathematics)" title="Functional (mathematics)">functionals</a>, can be shown<a href="#cite_note-63">[63]</a> to lead to

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

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<msup>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q\left[\psi (\mathbf {x} ,t)\right]=-(1/2R)\int d^{3}x\,\delta ^{2}R/\delta \psi ^{2}}</annotation>

</semantics>

</math><img src="./7cdc00104683fed4561dd9b4e2fa91db0447301a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.701ex; height:5.676ex;" alt="{\displaystyle Q\left[\psi (\mathbf {x} ,t)\right]=-(1/2R)\int d^{3}x\,\delta ^{2}R/\delta \psi ^{2}}" loading="lazy"></dd></dl>

This has been called the superquantum potential by Bohm and his co-workers.<a href="#cite_note-64">[64]</a>

Basil Hiley showed that the energy–momentum-relations in the Bohm model can be obtained directly from the <a href="Energy%E2%80%93momentum_tensor" class="mw-redirect" title="Energy–momentum tensor">energy–momentum tensor</a> of <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a> and that the quantum potential is an energy term that is required for local energy–momentum conservation.<a href="#cite_note-65">[65]</a> He has also hinted that for particle with energies equal to or higher than the <a href="Pair_creation" class="mw-redirect" title="Pair creation">pair creation</a> threshold, Bohm's model constitutes a <a href="Many-body_theory" class="mw-redirect" title="Many-body theory">many-particle theory</a> that describes also pair creation and annihilation processes.<a href="#cite_note-66">[66]</a>

<div class="mw-heading mw-heading2"><h2 id="Interpretation_and_naming_of_the_quantum_potential">Interpretation and naming of the quantum potential</h2></div>

In his article of 1952, providing an alternative <a href="Interpretation_of_quantum_mechanics" class="mw-redirect" title="Interpretation of quantum mechanics">interpretation of quantum mechanics</a>, Bohm already spoke of a "quantum-mechanical" potential.<a href="#cite_note-67">[67]</a>

Bohm and Basil Hiley also called the quantum potential an information potential, given that it influences the form of processes and is itself shaped by the environment.<a href="#cite_note-information-quantum-theory-and-the-brain-P207-12">[12]</a> Bohm indicated "The ship or aeroplane (with its automatic Pilot) is a self-active system, i.e. it has its own energy. But the form of its activity is determined by the information content concerning its environment that is carried by the radar waves. This is independent of the intensity of the waves. We can similarly regard the quantum potential as containing active information. It is potentially active everywhere, but actually active only where and when there is a particle." (italics in original).<a href="#cite_note-68">[68]</a>

Hiley refers to the quantum potential as internal energy<a href="#cite_note-hiley-reappraisal-bohm-2005-27">[27]</a> and as "a new quality of energy only playing a role in quantum processes".<a href="#cite_note-69">[69]</a> He explains that the quantum potential is a further energy term aside the well-known <a href="Kinetic_energy" title="Kinetic energy">kinetic energy</a> and the (classical) <a href="Potential_energy" title="Potential energy">potential energy</a> and that it is a nonlocal energy term that arises necessarily in view of the requirement of energy conservation; he added that much of the physics community's resistance against the notion of the quantum potential may have been due to scientists' expectations that energy should be local.<a href="#cite_note-70">[70]</a>

Hiley has emphasized that the quantum potential, for Bohm, was "a key element in gaining insights into what could underlie the quantum formalism. Bohm was convinced by his deeper analysis of this aspect of the approach that the theory could not be mechanical. Rather, it is organic in the sense of <a href="Alfred_North_Whitehead" title="Alfred North Whitehead">Whitehead</a>. Namely, that it was the whole that determined the properties of the individual particles and their relationship, not the other way round."<a href="#cite_note-71">[71]</a><a href="#cite_note-72">[72]</a>

<a href="Peter_R._Holland" title="Peter R. Holland">Peter R. Holland</a>, in his comprehensive textbook, also refers to it as quantum potential energy.<a href="#cite_note-73">[73]</a> The quantum potential is also referred to in association with Bohm's name as Bohm potential, quantum Bohm potential or Bohm quantum potential.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

The quantum potential approach can be used to model quantum effects without requiring the Schrödinger equation to be explicitly solved, and it can be integrated in simulations, such as <a href="Monte_Carlo_methods_for_electron_transport#Hydrodynamic_and_drift_diffusion_method" title="Monte Carlo methods for electron transport">Monte Carlo simulations using the hydrodynamic and drift diffusion equations</a>.<a href="#cite_note-74">[74]</a> This is done in form of a "hydrodynamic" calculation of trajectories: starting from the density at each "fluid element", the acceleration of each "fluid element" is computed from the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

</semantics>

</math><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy">, and the resulting divergence of the velocity field determines the change to the density.<a href="#cite_note-75">[75]</a>

The approach using Bohmian trajectories and the quantum potential is used for calculating properties of quantum systems which cannot be solved exactly, which are often approximated using semi-classical approaches. Whereas in <a href="Mean_field_theory" class="mw-redirect" title="Mean field theory">mean field approaches</a> the potential for the classical motion results from an average over wave functions, this approach does not require the computation of an integral over wave functions.<a href="#cite_note-76">[76]</a>

The expression for the <a href="#Quantum_force">quantum force</a> has been used, together with <a href="Bayesian_statistics" title="Bayesian statistics">Bayesian statistical analysis</a> and <a href="Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">expectation-maximisation</a> methods, for <a href="De_Broglie%E2%80%93Bohm_theory#Quantum_trajectory_method" title="De Broglie–Bohm theory">computing ensembles of trajectories</a> that arise under the influence of classical and quantum forces.<a href="#cite_note-maddox-bittner-2003-23">[23]</a>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Fundamental_articles">Fundamental articles</h3></div>

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</style><cite id="CITEREFBohm1952" class="citation journal cs1">Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I". Physical Review. 85 (2): 166–179. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1952PhRv...85..166B">1952PhRv...85..166B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.85.166">10.1103/PhysRev.85.166</a>.</cite> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20121018190801/https://www.nd.edu/~dhoward1/Bohm%20HV-I%20Phys%20Rev%201952.pdf">full text</a>)</li>

<li><cite id="CITEREFBohm1952" class="citation journal cs1">Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables", II". Physical Review. 85 (2): 180–193. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1952PhRv...85..180B">1952PhRv...85..180B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.85.180">10.1103/PhysRev.85.180</a>.</cite> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20121018190859/https://www.nd.edu/~dhoward1/Bohm%20HV-II%20Phys%20Rev%201952.pdf">full text</a>)</li>

<li>D. Bohm, B. J. Hiley, P. N. Kaloyerou: An ontological basis for the quantum theory, Physics Reports (Review section of Physics Letters), volume 144, number 6, pp. 321–375, 1987 (<a rel="nofollow" class="external text" href="http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/bohm_hiley_kaloyerou_1986.pdf">full text</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120319181802/http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/bohm_hiley_kaloyerou_1986.pdf">Archived</a> 2012-03-19 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>), therein: D. Bohm, B. J. Hiley: I. Non-relativistic particle systems, pp. 321–348, and D. Bohm, B. J. Hiley, P. N. Kaloyerou: II. A causal interpretation of quantum fields, pp. 349–375</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Recent_articles">Recent articles</h3></div>

<ul><li><cite id="CITEREFHeGaoCai2014" class="citation journal cs1">He, Dongshan; Gao, Dongfeng; Cai, Qing-yu (2014). "Spontaneous creation of the universe from nothing". Physical Review D. 89 (8) 083510. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1404.1207">1404.1207</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014PhRvD..89h3510H">2014PhRvD..89h3510H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.89.083510">10.1103/PhysRevD.89.083510</a>.</cite></li>

<li><cite id="CITEREFDe_GossonHiley2013" class="citation journal cs1">De Gosson, Maurice A.; Hiley, Basil (2013). "The symplectic egg in classical and quantum mechanics". American Journal of Physics. 81 (5): 328–337. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1304.4771">1304.4771</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013AmJPh..81..328D">2013AmJPh..81..328D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.4791775">10.1119/1.4791775</a>.</cite></li>

<li><cite id="CITEREFCarroll2005" class="citation arxiv cs1">Carroll, Robert (2005). "Fluctuations, gravity, and the quantum potential". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0501045">gr-qc/0501045</a>.</cite></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Overview">Overview</h3></div>

<ul><li>Davide Fiscaletti: About the Different Approaches to Bohm's Quantum Potential in Non-Relativistic Quantum Mechanics, Quantum Matter, Volume 3, Number 3, June 2014, pp. 177–199(23), <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1166%2Fqm.2014.1113">10.1166/qm.2014.1113</a>.</li>

<li><a href="Ignazio_Licata" title="Ignazio Licata">Ignazio Licata</a>, Davide Fiscaletti (with a foreword by <a href="Basil_Hiley" title="Basil Hiley">B.J. Hiley</a>): Quantum potential: Physics, Geometry and Algebra, AMC, Springer, 2013, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-00332-0</bdi> (print) / <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-00333-7</bdi> (online)</li>

<li><a href="Peter_R._Holland" title="Peter R. Holland">Peter R. Holland</a>: The Quantum Theory of Motion: An Account of the De Broglie-Bohm Causal Interpretation of Quantum Mechanics, Cambridge University Press, Cambridge (first published June 25, 1993), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-35404-8</bdi> hardback, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-48543-6</bdi> paperback, transferred to digital printing 2004</li>

<li><a href="David_Bohm" title="David Bohm">David Bohm</a>, <a href="Basil_Hiley" title="Basil Hiley">Basil Hiley</a>: The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge, 1993, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-415-06588-7</bdi></li>

<li>David Bohm, <a href="F._David_Peat" title="F. David Peat">F. David Peat</a>: <a href="Science%2C_Order_and_Creativity" class="mw-redirect" title="Science, Order and Creativity">Science, Order and Creativity</a>, 1987, Routledge, 2nd ed. 2000 (transferred to digital printing 2008, Routledge), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-415-17182-2</bdi></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><div class="reflist">

<li id="cite_note-bohm-1952-I-1">^ <a href="#cite_ref-bohm-1952-I_1-0">a</a> <a href="#cite_ref-bohm-1952-I_1-1">b</a> <a href="#cite_ref-bohm-1952-I_1-2">c</a> <cite id="CITEREFBohm1952" class="citation journal cs1">Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I". Physical Review. 85 (2): 166–179. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1952PhRv...85..166B">1952PhRv...85..166B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.85.166">10.1103/PhysRev.85.166</a>.</cite> (<a rel="nofollow" class="external text" href="https://www.nd.edu/~dhoward1/Bohm%20HV-I%20Phys%20Rev%201952.pdf">full text</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121018190801/https://www.nd.edu/~dhoward1/Bohm%20HV-I%20Phys%20Rev%201952.pdf">Archived</a> 2012-10-18 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>)

</li>

<li id="cite_note-bohm-1952-II-2"><a href="#cite_ref-bohm-1952-II_2-0">^</a> <cite id="CITEREFBohm1952" class="citation journal cs1">Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables", II". Physical Review. 85 (2): 180–193. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1952PhRv...85..180B">1952PhRv...85..180B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.85.180">10.1103/PhysRev.85.180</a>.</cite> (<a rel="nofollow" class="external text" href="https://www.nd.edu/~dhoward1/Bohm%20HV-II%20Phys%20Rev%201952.pdf">full text</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121018190859/https://www.nd.edu/~dhoward1/Bohm%20HV-II%20Phys%20Rev%201952.pdf">Archived</a> 2012-10-18 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>)

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFMadelung1927" class="citation journal cs1 cs1-prop-foreign-lang-source">Madelung, E. (1927). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/BF01400372">"Quantentheorie in hydrodynamischer Form"</a>. Zeitschrift für Physik (in German). 40 (3–4): 322–326. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1927ZPhy...40..322M">1927ZPhy...40..322M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01400372">10.1007/BF01400372</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1434-6001">1434-6001</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFWeizsäcker1935" class="citation journal cs1 cs1-prop-foreign-lang-source">Weizsäcker, C. F. v. (1935). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/BF01337700">"Zur Theorie der Kernmassen"</a>. Zeitschrift für Physik (in German). 96 (7–8): 431–458. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1935ZPhy...96..431W">1935ZPhy...96..431W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01337700">10.1007/BF01337700</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1434-6001">1434-6001</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> D. Bohm, B. J. Hiley: On the intuitive understanding of nonlocality as implied by quantum theory, Foundations of Physics, Volume 5, Number 1, pp. 93-109, 1975, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01100319">10.1007/BF01100319</a> (<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01100319">abstract</a>)

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> David Bohm, Basil Hiley: The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge, 1993, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-415-06588-7</bdi>, therein Chapter 3.1. The main points of the causal interpretation, p. 22–<a rel="nofollow" class="external text" href="https://books.google.com/books?id=vt9XKjc4WAQC&pg=PA23">23</a>.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> David Bohm, Basil Hiley: The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge, 1993, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-415-06588-7</bdi>, also as cited in: B. J. Hiley and R. E. Callaghan: Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation, Foundations of Physics, January 2012, Volume 42, Issue 1, pp 192-208 (published online 20 May 2011), <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10701-011-9558-z">10.1007/s10701-011-9558-z</a> (<a rel="nofollow" class="external text" href="https://archive.today/20130203033020/http://www.springerlink.com/content/w063vp65r3240550/">abstract</a>, <a rel="nofollow" class="external text" href="http://www.bbk.ac.uk/tpru/BasilHiley/Bohm-Vienna.pdf">2010 preprint by B. Hiley</a>)

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> See for ex. <a href="Robert_E._Wyatt" title="Robert E. Wyatt">Robert E. Wyatt</a>, <a href="Eric_R._Bittner" title="Eric R. Bittner">Eric R. Bittner</a>: Quantum wave packet dynamics with trajectories: Implementation with adaptive Lagrangian grids of the amplitude of the wave function, Journal of Chemical Physics, vol. 113, no. 20, 22 November 2000, <a rel="nofollow" class="external text" href="http://k2.chem.uh.edu/group/OldStuff/Papers/JCP-WB.pdf">p. 8898</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111002194145/http://k2.chem.uh.edu/group/OldStuff/Papers/JCP-WB.pdf">Archived</a> 2011-10-02 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> See also: <a href="Pilot_wave" class="mw-redirect" title="Pilot wave">Pilot wave#Mathematical formulation for a single particle</a>

</li>

<li id="cite_note-teleportation-P7-10"><a href="#cite_ref-teleportation-P7_10-0">^</a> B. J. Hiley: Active Information and Teleportation, <a rel="nofollow" class="external text" href="http://www.bbk.ac.uk/tpru/BasilHiley/ActInfoTeleWein.pdf#page=7">p. 7</a>; appeared in: Epistemological and Experimental Perspectives on Quantum Physics, D. Greenberger et al. (eds.), pages 113-126, Kluwer, Netherlands, 1999

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> B.J. Hiley: <a rel="nofollow" class="external text" href="http://www.birkbeck.ac.uk/tpru/BasilHiley/Vexjo2001W.pdf">From the Heisenberg picture to Bohm: A New Perspective on Active Information and it Relation to Shannon Information</a>, pp. 2 and 5. Published in: A. Khrennikov (ed.): Proc. Conf. Quantum Theory: reconsideration of foundations, pp. 141–162, Vaxjö University Press, Sweden, 2002

</li>

<li id="cite_note-information-quantum-theory-and-the-brain-P207-12">^ <a href="#cite_ref-information-quantum-theory-and-the-brain-P207_12-0">a</a> <a href="#cite_ref-information-quantum-theory-and-the-brain-P207_12-1">b</a> B. J. Hiley: Information, quantum theory and the brain. In: Gordon G. Globus (ed.), Karl H. Pribram (ed.), Giuseppe Vitiello (ed.): Brain and being: at the boundary between science, philosophy, language and arts, Advances in Consciousness Research, John Benjamins B.V., 2004, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>90-272-5194-0</bdi>, pp. 197-214, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Fvs3sw9ZVfsC&pg=PA207">p. 207</a>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> C. Philippidis, C. Dewdney, B. J. Hiley: <a rel="nofollow" class="external text" href="https://archive.today/20130203072211/http://www.springerlink.com/content/m48mn37340524112/">Quantum interference and the quantum potential</a>, Il nuovo cimento B, vol. 52, no. 1, 1979, pp.15-28, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02743566">10.1007/BF02743566</a>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> C. Philippidis, D. Bohm, R. D. Kaye: <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02721695">The Aharonov-Bohm effect and the quantum potential</a>, Il nuovo cimento B, vol. 71, no. 1, pp. 75-88, 1982, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02721695">10.1007/BF02721695</a>

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> Basil J. Hiley: The role of the quantum potential. In: G. Tarozzi, Alwyn Van der Merwe: Open questions in quantum physics: invited papers on the foundations of microphysics, Springer, 1985, pages 237 ff., therein <a rel="nofollow" class="external text" href="https://books.google.com/books?hl=en&id=QN9HW1Oi7d4C&oi=fnd&pg=PA239">page 239</a>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> D. Bohm, B. J. Hiley, P. N. Kaloyerou: An ontological basis for the quantum theory, Physics Reports (Review section of Physics Letters), volume 144, number 6, pp. 323–348, 1987 (<a rel="nofollow" class="external text" href="http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/bohm_hiley_kaloyerou_1986.pdf">abstract)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120319181802/http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/bohm_hiley_kaloyerou_1986.pdf">Archived</a> 2012-03-19 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> B. J. Hiley: The conceptual structure of the Bohm interpretation of quantum mechanics, In: K. V. Laurikainen, C. Montonen, K. Sunnarborg (eds.): Symposium on the Foundations of Modern Physics 1994 – 70 years of Matter Waves, Editions Frontières, pp. 99–118, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>2-86332-169-2</bdi>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GeGf9-q3AJkC&pg=PA106">p. 106</a>

</li>

<li id="cite_note-teleportation-P10-18"><a href="#cite_ref-teleportation-P10_18-0">^</a> B. J. Hiley: Active Information and Teleportation, <a rel="nofollow" class="external text" href="http://www.bbk.ac.uk/tpru/BasilHiley/ActInfoTeleWein.pdf#page=10">p. 10</a>; appeared in: Epistemological and Experimental Perspectives on Quantum Physics, D. Greenberger et al. (eds.), pages 113-126, Kluwer, Netherlands, 1999

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite id="CITEREFDürrGoldsteinZanghí1992" class="citation journal cs1">Dürr, Detlef; Goldstein, Sheldon; Zanghí, Nino (1992). "Quantum equilibrium and the origin of absolute uncertainty". Journal of Statistical Physics. 67 (5–6): 843–907. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0308039">quant-ph/0308039</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992JSP....67..843D">1992JSP....67..843D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01049004">10.1007/BF01049004</a>.</cite>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> David Bohm, Basil Hiley: The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge, 1993, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-415-06588-7</bdi>, transferred to digital printing 2005, therein Chapter 4.1. The ontological interpretation of the many-body system, p. 59

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> D. Bohm, B. J. Hiley, P. N. Kaloyerou: An ontological basis for the quantum theory, Physics Reports (Review section of Physics Letters), volume 144, number 6, pp. 323–348, 1987 (<a rel="nofollow" class="external text" href="http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/bohm_hiley_kaloyerou_1986.pdf#page=31">p. 351, eq. (12)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120319181802/http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/bohm_hiley_kaloyerou_1986.pdf#page=31">Archived</a> 2012-03-19 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a><--page=31 p. 351 is not(!) a typo-->

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> See for example the Introduction section of: Fernando Ogiba: <a rel="nofollow" class="external text" href="http://www.ptep-online.com/index_files/2011/PP-27-06.PDF">Phenomenological derivation of the Schrödinger equation</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111011141734/http://ptep-online.com/index_files/2011/PP-27-06.PDF">Archived</a> 2011-10-11 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Progress in Physics (indicated date: October 2011, but retrieved online earlier: July 31, 2011)

</li>

<li id="cite_note-maddox-bittner-2003-23">^ <a href="#cite_ref-maddox-bittner-2003_23-0">a</a> <a href="#cite_ref-maddox-bittner-2003_23-1">b</a> Jeremy B. Maddox, Eric R. Bittner: <a rel="nofollow" class="external text" href="http://ib.cnea.gov.ar/~garriza/qtmnotas/jcp-03-119-6465-Maddox-Bayesian_statistics.pdf">Estimating Bohm's quantum force using Bayesian statistics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111120094035/http://ib.cnea.gov.ar/~garriza/qtmnotas/jcp-03-119-6465-Maddox-Bayesian_statistics.pdf">Archived</a> 2011-11-20 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Journal of Chemical Physics, October 2003, vol. 119, no. 13, p. 6465–6474, therein p. 6472, eq.(38)

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFBrown1997" class="citation arxiv cs1">Brown, M. R. (1997). "The quantum potential: The breakdown of classical symplectic symmetry and the energy of localisation and dispersion". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/9703007">quant-ph/9703007</a>.</cite>

</li>

<li id="cite_note-brown-hiley-25">^ <a href="#cite_ref-brown-hiley_25-0">a</a> <a href="#cite_ref-brown-hiley_25-1">b</a> <cite id="CITEREFBrownHiley2000" class="citation arxiv cs1">Brown, M. R.; Hiley, B. J. (2000). "Schrodinger revisited: An algebraic approach". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0005026">quant-ph/0005026</a>.</cite>

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<li id="cite_note-26"><a href="#cite_ref-26">^</a> Maurice A. de Gosson: "The Principles of Newtonian and Quantum Mechanics – The Need for Planck's Constant, h", Imperial College Press, World Scientific Publishing, 2001, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-86094-274-1</bdi>

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<li id="cite_note-hiley-reappraisal-bohm-2005-27">^ <a href="#cite_ref-hiley-reappraisal-bohm-2005_27-0">a</a> <a href="#cite_ref-hiley-reappraisal-bohm-2005_27-1">b</a> <a href="#cite_ref-hiley-reappraisal-bohm-2005_27-2">c</a> B. J. Hiley: Non-commutative quantum geometry: A reappraisal of the Bohm approach to quantum theory, in: A. Elitzur et al. (eds.): Quo vadis quantum mechanics, Springer, 2005, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-22188-3</bdi>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6HEaNL40C0MC&pg=PA299">p. 299–324</a>

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<li id="cite_note-non-commutative-2005-28"><a href="#cite_ref-non-commutative-2005_28-0">^</a> B.J. Hiley: Non-Commutative Quantum Geometry: A Reappraisal of the Bohm Approach to Quantum Theory. In: Avshalom C. Elitzur, Shahar Dolev, Nancy Kolenda (eds.): Quo Vadis Quantum Mechanics? The Frontiers Collection, 2005, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6HEaNL40C0MC&pg=PA299">pp. 299-324</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-26669-0_16">10.1007/3-540-26669-0_16</a> (<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-26669-0_16">abstract</a>, <a rel="nofollow" class="external text" href="http://www.bbk.ac.uk/tpru/BasilHiley/TemplePaper%28sys9%2902.pdf">preprint</a>)

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> B.J. Hiley: Phase space description of quantum mechanics and non-commutative geometry: Wigner–Moyal and Bohm in a wider context, In: Theo M. Nieuwenhuizen et al (eds.): Beyond the quantum, World Scientific Publishing, 2007, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-277-117-9</bdi>, pp. 203–211, therein p. 204

</li>

<li id="cite_note-hiley-anpa-23-2001-30"><a href="#cite_ref-hiley-anpa-23-2001_30-0">^</a> Basil J. Hiley: Towards a Dynamics of Moments: The Role of Algebraic Deformation and Inequivalent Vacuum States, published in: Correlations ed. K. G. Bowden, Proc. ANPA 23, 104-134, 2001 (<a rel="nofollow" class="external text" href="http://www.birkbeck.ac.uk/tpru/BasilHiley/14MomentsANPA2001W.pdf">PDF</a>)

</li>

<li id="cite_note-hiley-callaghan-2010-A-31"><a href="#cite_ref-hiley-callaghan-2010-A_31-0">^</a> <cite id="CITEREFHileyCallaghan2010" class="citation arxiv cs1">Hiley, B. J.; Callaghan, R. E. (2010). "The Clifford Algebra approach to Quantum Mechanics A: The Schroedinger and Pauli Particles". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1011.4031">1011.4031</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math-ph">math-ph</a>].</cite>

</li>

<li id="cite_note-hiley-phase-32">^ <a href="#cite_ref-hiley-phase_32-0">a</a> <a href="#cite_ref-hiley-phase_32-1">b</a> B. Hiley: Phase space description of quantum mechanics and non-commutative geometry: Wigner-Moyal and Bohm in a wider context, in: Th. M. Nieuwenhuizen et al. (eds.): Beyond the Quantum, World Scientific, 2007, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-277-117-9</bdi>, p. 203–211, therein: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sCrtjHPcY3gC&pg=PA207">p. 207 ff.</a>

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<li id="cite_note-33"><a href="#cite_ref-33">^</a> <cite id="CITEREFNasiri2006" class="citation journal cs1">Nasiri, Sadollah (2006). "Quantum Potential and Symmetries in Extended Phase Space". Symmetry, Integrability and Geometry: Methods and Applications. 2: 062. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0511125">quant-ph/0511125</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006SIGMA...2..062N">2006SIGMA...2..062N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3842%2FSIGMA.2006.062">10.3842/SIGMA.2006.062</a>.</cite>

</li>

<li id="cite_note-34"><a href="#cite_ref-34">^</a> <cite id="CITEREFFernandesVianna1998" class="citation journal cs1">Fernandes, Marco Cezar B.; Vianna, J. David M. (1998). "On the Duffin-Kemmer-Petiau algebra and the generalized phase space". Brazilian Journal of Physics. 28 (4): 00. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1590%2FS0103-97331998000400024">10.1590/S0103-97331998000400024</a>.</cite>

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<li id="cite_note-35"><a href="#cite_ref-35">^</a> <cite id="CITEREFFernandesVianna1999" class="citation journal cs1">Fernandes, M. C. B.; Vianna, J. D. M. (1999). "On the Generalized Phase Space Approach to Duffin-Kemmer-Petiau Particles". Foundations of Physics. 29 (2): 201–219. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999FoPh...29..201F">1999FoPh...29..201F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1018869505031">10.1023/A:1018869505031</a>.</cite>

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<li id="cite_note-36"><a href="#cite_ref-36">^</a> <cite id="CITEREFTsekov2009" class="citation journal cs1">Tsekov, Roumen (2009). "Towards Nonlinear Quantum Fokker-Planck Equations". International Journal of Theoretical Physics. 48 (5): 1431–1435. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0808.0326">0808.0326</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009IJTP...48.1431T">2009IJTP...48.1431T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10773-008-9913-9">10.1007/s10773-008-9913-9</a>.</cite>

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<li id="cite_note-37"><a href="#cite_ref-37">^</a> Robert Carroll: On the Emergence Theme of Physics, World Scientific, 2010, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>981-4291-79-X</bdi>, Chapter 1 Some quantum background, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_1S6qKztt_UC&pg=PA1">p. 1</a>.

</li>

<li id="cite_note-38"><a href="#cite_ref-38">^</a> <cite id="CITEREFSalesiRecamiHernandez_F.Kretly1998" class="citation arxiv cs1">Salesi, G.; Recami, E.; Hernandez F., H.; Kretly, L. C. (1998). "Hydrodynamics of Spinning Particles". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9802106">hep-th/9802106</a>.</cite>

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<li id="cite_note-39"><a href="#cite_ref-39">^</a> <cite id="CITEREFSalesi1996" class="citation journal cs1">Salesi, Giovanni (1996). "Spin and Madelung Fluid". Modern Physics Letters A. 11 (22): 1815–1823. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0906.4147">0906.4147</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996MPLA...11.1815S">1996MPLA...11.1815S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0217732396001806">10.1142/S0217732396001806</a>.</cite>

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<li id="cite_note-esposito-1999-40">^ <a href="#cite_ref-esposito-1999_40-0">a</a> <a href="#cite_ref-esposito-1999_40-1">b</a> <cite id="CITEREFEsposito1999" class="citation arxiv cs1">Esposito, S. (1999). "On the role of Spin in Quantum Mechanics". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/9902019">quant-ph/9902019</a>.</cite>

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</li>

<li id="cite_note-42"><a href="#cite_ref-42">^</a> S. Esposito: Photon wave mechanics: A de Broglie–Bohm approach, <a rel="nofollow" class="external text" href="http://people.na.infn.it/~sesposit/doc/debroglie.pdf#page=8">p. 8 ff.</a>

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<li id="cite_note-43"><a href="#cite_ref-43">^</a> <cite id="CITEREFBogan2002" class="citation arxiv cs1">Bogan, James R. (2002). "Spin: The Classical to Quantum Connection". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0212110">quant-ph/0212110</a>.</cite>

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<li id="cite_note-44"><a href="#cite_ref-44">^</a> B. Hiley, R. E. Callaghan: The Clifford algebra approach to quantum mechanics A: The Schrödinger and Pauli particles, 14 March 2010, <a rel="nofollow" class="external text" href="http://www.bbk.ac.uk/tpru/BasilHiley/CliffordPauliBohm.pdf#page=6">p. 6</a>

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<li id="cite_note-45"><a href="#cite_ref-45">^</a> B. Hiley, R. E. Callaghan: The Clifford algebra approach to quantum mechanics A: The Schrödinger and Pauli particles, 14 March 2010, <a rel="nofollow" class="external text" href="http://www.bbk.ac.uk/tpru/BasilHiley/CliffordPauliBohm.pdf">p. 1-29</a>

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<li id="cite_note-clifford-direc-bohm-hj-46">^ <a href="#cite_ref-clifford-direc-bohm-hj_46-0">a</a> <a href="#cite_ref-clifford-direc-bohm-hj_46-1">b</a> B. Hiley: Clifford algebras and the Dirac–Bohm Hamilton–Jacobi equation, 2 March 2010, <a rel="nofollow" class="external text" href="http://www.bbk.ac.uk/tpru/BasilHiley/Bohm-Vienna.pdf#page=22">p. 22</a>

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<li id="cite_note-47"><a href="#cite_ref-47">^</a> B. J. Hiley: Non-commutative geometry, the Bohm interpretation and the mind–matter relationship, <a rel="nofollow" class="external text" href="http://www.bbk.ac.uk/tpru/BasilHiley/noncommgeobohm.pdf#page=14">p. 14</a>

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<li id="cite_note-48"><a href="#cite_ref-48">^</a> <cite id="CITEREFBohmHiley1989" class="citation journal cs1">Bohm, D.; Hiley, B.J. (1989). "Non-locality and locality in the stochastic interpretation of quantum mechanics". Physics Reports. 172 (3): 93–122. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1989PhR...172...93B">1989PhR...172...93B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-1573%2889%2990160-9">10.1016/0370-1573(89)90160-9</a>.</cite>

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<li id="cite_note-49"><a href="#cite_ref-49">^</a> P.N. Kaloyerou, Investigation of the Quantum Potential in the Relativistic Domain, PhD. Thesis, Birkbeck College, London (1985)

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<li id="cite_note-50"><a href="#cite_ref-50">^</a> P.N. Kaloyerou, Phys. Rep. 244, 288 (1994).

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<li id="cite_note-51"><a href="#cite_ref-51">^</a> P.N. Kaloyerou, in "Bohmian Mechanics and Quantum Theory: An Appraisal", eds. J.T. Cushing, A. Fine and S. Goldstein, Kluwer, Dordrecht,

155 (1996).

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<li id="cite_note-52"><a href="#cite_ref-52">^</a> D. Bohm, B. J. Hiley, P. N. Kaloyerou: An ontological basis for the quantum theory, Physics Reports (Review section of Physics Letters), volume 144, number 6, pp. 323–348, 1987 (<a rel="nofollow" class="external text" href="http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/bohm_hiley_kaloyerou_1986.pdf">PDF)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120319181802/http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/bohm_hiley_kaloyerou_1986.pdf">Archived</a> 2012-03-19 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>

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<li id="cite_note-53"><a href="#cite_ref-53">^</a> B. J. Hiley, A. H. Aziz Muft: The ontological interpretation of quantum field theory applied in a cosmological context. In: Miguel Ferrero, Alwyn Van der Merwe (eds.): Fundamental problems in quantum physics, Fundamental theories of physics, Kluwer Academic Publishers, 1995, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-3670-4</bdi>, <a rel="nofollow" class="external text" href="https://books.google.com/books?hl=en&id=1wUFoP7HC38C&oi=fnd&pg=PA141">pages 141-156</a>

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<li id="cite_note-54"><a href="#cite_ref-54">^</a> Carlo Castro, Jorge Mahecha: <a rel="nofollow" class="external text" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.75.6580&rep=rep1&type=pdf#page=93">On nonlinear quantum mechanics, Brownian motion, Weyl geometry and Fisher information</a>, submitted February 2005, In: F. Smarandache and V. Christianto (Eds.): Quantization in Astrophysics, Brownian Motion, and Supersymmetry, pp.73–87, MathTiger, 2007, Chennai, Tamil Nadu, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>81-902190-9-X</bdi>, page 82, eq.(37) ff.

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<li id="cite_note-55"><a href="#cite_ref-55">^</a> <cite id="CITEREFRapoport2007" class="citation book cs1">Rapoport, Diego L. (2007). "Torsion fields, Cartan-Weyl space-time, and state-space quantum geometries, Brownian motion, and their topological dimension". In Smarandache, F.; Christianto, V. (eds.). <a rel="nofollow" class="external text" href="https://archive.org/details/quantizationastr00smar_182">Quantization in Astrophysics, Brownian Motion, and Supersymmetry</a>. Chennai, Tamil Nadu: MathTiger. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/quantizationastr00smar_182/page/n286">276</a>–328. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.75.6580">10.1.1.75.6580</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-81-902190-9-9</bdi>.</cite>

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<li id="cite_note-56"><a href="#cite_ref-56">^</a> Peter R. Holland: The quantum theory of motion, Cambridge University Press, 1993 (re-printed 2000, transferred to digital printing 2004), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-48543-6</bdi>, p. 498 ff.

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<li id="cite_note-57"><a href="#cite_ref-57">^</a> Hrvoje Nikolić: <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10702-005-1128-1">Relativistic Quantum Mechanics and the Bohmian Interpretation</a>, Foundations of Physics Letters, vol. 18, no. 6, November 2005, pp. 549-561, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10702-005-1128-1">10.1007/s10702-005-1128-1</a>

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<li id="cite_note-58"><a href="#cite_ref-58">^</a> <cite id="CITEREFNikolic2008" class="citation arxiv cs1">Nikolic, H. (2008). "Time in relativistic and nonrelativistic quantum mechanics". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0811.1905">0811.1905</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/quant-ph">quant-ph</a>].</cite>

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<li id="cite_note-nikolicqft-59"><a href="#cite_ref-nikolicqft_59-0">^</a> <cite id="CITEREFNikolić2010" class="citation journal cs1">Nikolić, Hrvoje (2010). "QFT as Pilot-Wave Theory of Particle Creation and Destruction". International Journal of Modern Physics A. 25 (7): 1477–1505. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0904.2287">0904.2287</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010IJMPA..25.1477N">2010IJMPA..25.1477N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0217751X10047889">10.1142/S0217751X10047889</a>.</cite>

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<li id="cite_note-60"><a href="#cite_ref-60">^</a> <cite id="CITEREFNikolic2010" class="citation arxiv cs1">Nikolic, H. (2010). "Making nonlocal reality compatible with relativity". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1002.3226">1002.3226</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/quant-ph">quant-ph</a>].</cite>

</li>

<li id="cite_note-61"><a href="#cite_ref-61">^</a> Hrvoje Nikolić: <a rel="nofollow" class="external text" href="http://iopscience.iop.org/1742-6596/67/1/012035/pdf/jpconf7_67_012035.pdf">Bohmian mechanics in relativistic quantum mechanics, quantum field theory and string theory</a>, 2007 J. Phys.: Conf. Ser. 67 012035

</li>

<li id="cite_note-62"><a href="#cite_ref-62">^</a> See also: <a href="De_Broglie%E2%80%93Bohm_theory#Relativity" title="De Broglie–Bohm theory">De Broglie–Bohm theory#Relativity</a>

</li>

<li id="cite_note-63"><a href="#cite_ref-63">^</a> Peter R. Holland: The quantum theory of motion, Cambridge University Press, 1993 (re-printed 2000, transferred to digital printing 2004), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-48543-6</bdi>, p. 520 ff.

</li>

<li id="cite_note-64"><a href="#cite_ref-64">^</a> Basil Hiley: The conceptual structure of the Bohm interpretation of quantum mechanics, Kalervo Vihtori Laurikainen et al (ed.): Symposium on the Foundations of Modern Physics 1994: 70 years of matter waves, Editions Frontières, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>2-86332-169-2</bdi>, p. 99–117, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GeGf9-q3AJkC&pg=PA114">p. 144</a>

</li>

<li id="cite_note-65"><a href="#cite_ref-65">^</a> B. J. Hiley: The Bohm approach re-assessed (<a rel="nofollow" class="external text" href="http://www.bbk.ac.uk/tpru/RecentPublications.html">2010 preprint</a>), <a rel="nofollow" class="external text" href="http://www.bbk.ac.uk/tpru/BasilHiley/BohmReassed1.pdf#page=6">p. 6</a>

</li>

<li id="cite_note-66"><a href="#cite_ref-66">^</a> <cite id="CITEREFHiley2013" class="citation arxiv cs1">Hiley, B. J. (2013). "Bohmian Non-commutative Dynamics: History and New Developments". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1303.6057">1303.6057</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/quant-ph">quant-ph</a>].</cite>

</li>

<li id="cite_note-67"><a href="#cite_ref-67">^</a> <cite id="CITEREFBohm1952" class="citation journal cs1">Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I". Physical Review. 85 (2): 166–179. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1952PhRv...85..166B">1952PhRv...85..166B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.85.166">10.1103/PhysRev.85.166</a>.</cite> <a rel="nofollow" class="external text" href="https://www.nd.edu/~dhoward1/Bohm%20HV-I%20Phys%20Rev%201952.pdf#page=5">p. 170</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121018190801/https://www.nd.edu/~dhoward1/Bohm%20HV-I%20Phys%20Rev%201952.pdf">Archived</a> 2012-10-18 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>

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<li id="cite_note-68"><a href="#cite_ref-68">^</a> David Bohm: <a rel="nofollow" class="external text" href="https://archive.today/20111009151609/http://www.tkpi.org/content/meaning-and-information-david-bohm">Meaning And Information</a>, In: P. Pylkkänen (ed.): The Search for Meaning: The New Spirit in Science and Philosophy, Crucible, The Aquarian Press, 1989, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-85274-061-0</bdi>

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<li id="cite_note-69"><a href="#cite_ref-69">^</a> B.J. Hiley: Non-commutative quantum geometry: A reappraisal of the Bohm approach to quantum theory. In: Avshalom C. Elitzur, Shahar Dolev, Nancy Kolenda (es.): Quo vadis quantum mechanics? Springer, 2005, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-22188-3</bdi>, pp. 299 ff., therein <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6HEaNL40C0MC&pg=PA310">p. 310</a>

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<li id="cite_note-70"><a href="#cite_ref-70">^</a> <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=W_Ro2unZdFY">Basil Hiley & Taher Gozel, episode 5</a>, YouTube (downloaded 8 September 2013)

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<li id="cite_note-71"><a href="#cite_ref-71">^</a> B. J. Hiley: <a rel="nofollow" class="external text" href="http://www.bbk.ac.uk/tpru/BasilHiley/History_of_Bohm_s_QT.pdf">Some remarks on the evolution of Bohm's proposals for an alternative to quantum mechanics</a>, 30 January 2010

</li>

<li id="cite_note-72"><a href="#cite_ref-72">^</a> See also: <a href="Basil_Hiley#Quantum_potential_and_active_information" title="Basil Hiley">Basil Hiley#Quantum potential and active information</a>

</li>

<li id="cite_note-73"><a href="#cite_ref-73">^</a> <a href="Peter_R._Holland" title="Peter R. Holland">Peter R. Holland</a>: The quantum theory of motion, Cambridge University Press, 1993 (re-printed 2000, transferred to digital printing 2004), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-48543-6</bdi>, p. 72

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<li id="cite_note-74"><a href="#cite_ref-74">^</a> G. Iannaccone, G. Curatola, G. Fiori: <a rel="nofollow" class="external text" href="http://www.iannaccone.org/data/P47.pdf">Effective Bohm Quantum Potential for device simulators based on drift-diffusion and energy transport</a>, Simulation of Semiconductor Processes and Devices, 2004, vol. 2004, pp. 275–278

</li>

<li id="cite_note-75"><a href="#cite_ref-75">^</a> <cite id="CITEREFBittner2000" class="citation journal cs1">Bittner, Eric R. (2000). "Quantum tunneling dynamics using hydrodynamic trajectories". The Journal of Chemical Physics. 112 (22): 9703–9710. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0001119">quant-ph/0001119</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000JChPh.112.9703B">2000JChPh.112.9703B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.481607">10.1063/1.481607</a>.</cite>

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<li id="cite_note-76"><a href="#cite_ref-76">^</a> E. Gindensberger, C. Meier, J.A. Beswick: <a rel="nofollow" class="external text" href="http://www.car8.ups-tlse.fr/beswick/publis/JCP009369.pdf">Mixing quantum and classical dynamics using Bohmian trajectories</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120328094316/http://www.car8.ups-tlse.fr/beswick/publis/JCP009369.pdf">Archived</a> 2012-03-28 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Journal of Chemical Physics, vol. 113, no. 21, 1 December 2000, pp. 9369–9372

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