# 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: