# Bohmian Mechanics

Consider the equa­tion $PV = nRT$. This equa­tion relates the pres­sure, volume, and tem­per­a­ture of an ideal gas. All of these con­cepts are macro­scopic — meaning that on the level of the mol­e­cules that make up the gas the meaning of ‘pres­sure,’ ‘volume,’ and ‘tem­per­a­ture’ dis­solves. One mol­e­cule cannot have a pres­sure, it cannot be said to rep­re­sent a volume of gas, and it does not pos­sesses tem­per­a­ture. All three of these con­cepts begin to take on meaning as we zoom out and con­sider a col­lec­tion of the mol­e­cules and account for their motions — as we tran­si­tion from a micro­scopic scale to a macro­scopic scale.

What does it mean to say that this equa­tion relates prop­er­ties of an ideal gas? What is an ideal gas? It means that energy con­ser­va­tion and closed system con­sid­er­a­tions apply. In the case of our gas it means that the interactions/collisions between the mol­e­cules are all com­pletely elastic. Gasses that exhibit mea­sur­able inelas­ticity in their inter­ac­tions cannot be accu­rately rep­re­sented by this equa­tion on all macro­scopic scales.

Why are we talking about all of this? Well the math­e­matics that best mimics the geo­metric struc­ture of qst to date is cap­tured by a set of equa­tions known as Bohmian mechanics. The Bohmian for­malism has been shown to make all the pre­dic­tions that the stan­dard model of quantum mechanics makes — iden­ti­cally — while remaining a deter­min­istic theory. However, Bohmian mechanics (and the stan­dard equa­tions of quantum mechanics) are inca­pable of incor­po­rating the geo­metric effects of gravity into their models.

Let’s explore a can­di­date reason for why this is the case. In order to make the Bohmian for­malism com­pletely rep­re­sen­ta­tive of the geom­etry of qst let’s treat the equa­tions in this for­malism as macro­scopic expres­sions of ide­al­ized inter­ac­tions of the quanta of space­time. Just like the equa­tion $PV = nRT$, the Bohmian for­malism assumes per­fect elas­ticity of the under­lying con­stituents in its macro­scopic expres­sions. It is pos­sible that all we have to do to bring gravity into the for­malism is to get to the under­lying struc­ture that relates the inter­ac­tions of the space­time quanta and include a small second-order inelas­ticity in those inter­ac­tions. This would be like mod­eling mol­e­c­ular inter­ac­tions and allowing them to have a slight inelas­ticity. Doing this might allow us to pro­duce an gen­eral equa­tion that cap­tures the behavior of ideal gasses and non-ideal gasses simultaneously.

For those inter­ested, here is the deriva­tion of the Bohmian set of equa­tions:

Let’s begin by addressing the objec­tive state of the wave func­tion on the micro­scopic level. (Microscopic level in this case means on the quantum or Planck scale.) If our system (a chosen domain of space­time) is com­posed of N par­ti­cles, then a com­plete descrip­tion of that system will nec­es­sarily include a spec­i­fi­ca­tion of the posi­tions Qi of each of those par­ti­cles. On its own, the wave­func­tion $\Psi$ does not pro­vide a com­plete descrip­tion of the state of that system. Instead, the com­plete descrip­tion of this quantum system must be given by $(Q, \Psi)$ where

$Q = (Q_1, Q_2, Q_3 \ldots Q_N) \in \mathbb{R}^{3N}$

is the con­fig­u­ra­tion of the system and

$\Psi = \Psi(q) = \Psi(q_1, q_2, \ldots q_N)$

a (nor­mal­ized) func­tion on the con­fig­u­ra­tion space — the super­spa­tial dimen­sions — is its wave function.

At this point, all we have to do in order to obtain our theory is specify the law of motion for the state $(Q, \Psi)$. Of course, the sim­plest choice we can make here would be one that is causally con­nected. In other words, one whose future is deter­mined by its present spec­i­fi­ca­tion, and more specif­i­cally whose average total state remains fixed — at least in the macro­scopic sense of the familiar four dimen­sions of space­time. To obtain this we simply need to chore­o­graph the par­ticle motions by first-order equa­tions that assume elastic inter­ac­tions. The evo­lu­tion equa­tion for $\Psi$ is Schrödinger’s equation:

$i\hbar\frac{\partial \Psi_t}{\partial t} = H\Psi_t = -\sum_{k = 1}^{N} \frac{\hbar^2}{2m_k} \nabla^2q_k \Psi_t + V\Psi_t$

Where $\Psi$ is the wave func­tion and V is the poten­tial energy of the system.

Therefore, in keeping with our pre­vious con­sid­er­a­tions, the evo­lu­tion equa­tion for Q should be:

$\frac{d Q_t}{dt} = \upsilon^{\Psi_t}(Q_t)$.

with $\upsilon^\Psi = (\upsilon^\Psi_1, \upsilon^\Psi_2,\upsilon^\Psi_3, \ldots \upsilon^\Psi_N)$

where $\upsilon^\Psi$ takes the form of a (velocity) vector field on our chosen con­fig­u­ra­tion space $\mathbb{R}^{3N}$. Thus the wave func­tion $\Psi$ reflects the motion of the par­ti­cles in our system in a macro­scopic averaged-over sense based on the under­lying assump­tion of elastic inter­ac­tion. These motions are coor­di­nated through a vector field that is defined on our spec­i­fied con­fig­u­ra­tion space.

$\Psi \mapsto \upsilon^\Psi$

If we simply require time-reverse sym­metry and sim­plicity to hold in our system (auto­matic neces­si­ties for a deter­min­istic theory) then,

$\upsilon^\Psi_k = \frac{\hbar}{m_k} Im \frac{\nabla q_k \Psi}{\Psi}$

Notice that there are no ambi­gu­i­ties here. The gra­dient $\nabla$ on the right-hand side is sug­gested by rota­tion invari­ance, the $\Psi$ in the denom­i­nator is a con­se­quence of homo­geneity (a direct result of the fact that the wave func­tion is to be under­stood pro­jec­tively, which is in turn an under­standing required for the Galilean invari­ance of Schrödinger’s equa­tion alone), the Im by time-reverse sym­metry which is imple­mented on $\Psi$ by com­plex con­ju­ga­tion in keeping with Schrödinger’s equa­tion, and the con­stant in front falls directly out of the require­ments for covari­ance under Galilean boosts.1

Therefore, the evo­lu­tion equa­tion for Q is

$\frac{dQ_k}{dt} = \upsilon^\Psi_k (Q_1, Q_2, \ldots Q_N) \equiv \frac{\hbar}{m_k} Im \frac{\nabla q_k \Psi}{\Psi} (Q_1, Q_2, \ldots Q_N)$

This com­pletes the for­malism of Bohmian mechanics that David Bohm con­structed in 1952.2 The math may appear daunting but the con­cepts are amaz­ingly simple. In our con­struc­tion we have con­sid­ered applying the analogy of a gas being made up of elas­ti­cally inter­acting con­stituents to the quanta of our spactime system. As an exten­sion of de Broglie’s pilot wave model3 this for­malism exhaus­tively depicts a non­rel­a­tivistic uni­verse of N par­ti­cles without spin.4 Spin must be included in order to account for Fermi and Bose-Einstein sta­tis­tics. The full form of the guiding equa­tion, which is found by retaining the com­plex con­ju­gate of the wave func­tion, accounts for all the appar­ently para­dox­ical quantum phe­nomena asso­ci­ated with spin. For con­sid­er­a­tions without spin the com­plex con­ju­gate of the wave func­tion can­cels because it appears in the numer­ator and the denom­i­nator of the equa­tion. The full form of the evo­lu­tion equa­tion is:

$\frac{dQ_k}{dt} = \frac{\hbar}{m_k}Im\left[\frac{\Psi^*\partial_k \Psi}{\Psi^*\Psi}\right](Q_1, Q_2, \ldots Q_N)$

Notice that the right-hand side of the guiding equa­tion is J/Q, the ratio for the quantum prob­a­bility cur­rent to the quantum prob­a­bility den­sity.5

Note that the ide­al­ized assump­tion in play here is that $\rho = \left|\Psi\right|^2$. In other words, the trans­for­ma­tion $\rho^\Psi \mapsto \rho^{\Psi_t}$ arises directly from Schrödinger’s equa­tion. If these evo­lu­tions are indeed com­pactable, then

$(\rho^\Psi)_t = \rho^{\Psi_t}$

is equi­variant. Therefore, under the time evo­lu­tion $\rho^\Psi$ retains its form as a func­tion of $\Psi$.

If you are inter­ested in taking part in red­eriving the Bohmian set from under­lying inter­ac­tions that are first-order elastic and second-order inelastic please send an email to qst@​einsteinsintuition.​com.

Notes:

1. Detlef Dürr, Sheldon Goldstein, and Nino Zanghí, ‘Quantum Physics Without Quantum Philosophy,’ pp. 5-6.

2. D. Bohm, ‘A sug­gested inter­pre­ta­tion of the quantum theory in terms of “hidden” vari­ables,’ Physical Rev. 85 (1952), pp. 166-193.

3. L. de Broglie, ‘La nou­velle dynamique des quanta,’ Electrons et Photons: Rapports et Discussions du Cinquieme Conseil de Physique tenu a Bruxelles du 24 au 29 Octobre 1927 sous les Auspices de l’Institut International de Physique Solvay, Gautheir – Villars, Paris, 1928, pp. 105-132.

4. Of course in the limit ħ/m = 0, the Bohm motion Qt approaches the clas­sical motion. See: D. Bohm and B. Hiley, ‘The Undivided Universe: an Ontological Interpretation of Quantum Theory,’ Routledge & Kegan Paul, London, 1993; Detlef Durr, Sheldon Goldstein, and Nino Zanghi, ‘Quantum Physics Without Quantum Philosophy,’ p. 7.

5. Sheldon Goldstein, ‘Bohmian Mechanics.’ For fur­ther exam­ples of how easily spin can be dealt with in the Bohmian for­malism see: J. S. Bell, 1966, pp. 447-452; D. Bohm, 1952, pp. 166-193; D. Dürr et al ‘A survey of Bohmian mechanics, Il Nuovo Vimento’ and ‘Bohmian mechanics, iden­tical par­ti­cles, paras­ta­tis­tics, and anyons’, In preparation.

1. Ben says:

Sent.

2. Jeff says: