6

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.

Comments (6)

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  1. Ben says:

    Please send your book.

  2. Jeff says:

    Please send your book. Very inter­ested to learn more.

  3. Branton says:

    Well if you’re emailing them – I’d like a copy too!

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