Bohmian Mechanics (AKA Pilot Wave Theory)

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Bohmian Mechanics (AKA Pilot Wave Theory)

Unread post by paladin17 » Sun Dec 10, 2023 8:36 pm

I've recently read a really good book on Bohmian mechanics (named after David Bohm). The following are the notes that I took while reading. But first, let me give a brief overview of the topic.

The theory basically states: the wave function (governed by Schroedinger's equation) draws a certain "landscape" in space, and then the particles move on top of this landscape (but they are still actual "classical" particles, not waves). Hence another name for this theory - "pilot wave theory", where the pilot wave's (wave function's) role is to simply guide the motion of particles (produce a landscape of possible trajectories for them).

Bohmian mechanics is basically a much better version of regular quantum mechanics, since:
1) it gives the same results as regular quantum mechanics wherever actual experiments are possible;
2) it does not contain any paradoxes and does not need any additional "interpretations".
It is thus a completely self-contained theory with correct predictions and actual tangible ontology.

The latter property also leads to some surprizing results which are inaccessible to regular quantum mechanics: for example, in Bohmian mechanics the electron in a ground state of a hydrogen atom is stationary, i.e. it does not move at all. But it has a defined position (in contrast to regular quantum mechanics where the position cannot be determined in principle). Of course, we may not know the position, but it doesn't mean that it doesn't exist (which is the case in regular quantum mechanics; incidentally, if you need a brief introduction to the latter, you may watch my lecture).

I recommend reading the book for yourself. It is written in an engaging and lively tone, where the authors constantly polemize with regular misconceptions about Bohmian mechanics and quantum theory in general. There are a few chapters (specifically, from 12 to 15) which are incredibly heavy on high level mathematics (much higher than was actually needed, if you ask me), but you may just skip these, it won't matter in the grand scheme of things.
If you don't want to read it at all, here are my notes. Also, Stanford has a surprisingly well written article on philosophical/ontological aspects of the Bohmian theory, from which you can clearly see how weak (or even demonstrably wrong) any arguments against it are.

Just like previously, I'd make my own comments in italics, while literal quotes from the text would be given in "quotation marks". I always give the number of the page where the following statements are found (just in case).

____________________________________________

Chapter 1.
Introduction


P. 3
Schroedinger: there is a difference between a blurry photo and a photo of a blurry thing.

P. 4
Linearity of Schroedinger's equation means it doesn't describe reality (because it allows superposition states that are not observed). One needs additional tools on top of the wave function.
Measurement problem and circularity of QM.
We claim that we only study what can be measured. And these are particles. But the measurement apparatus is also made of particles. So what are we studying and how?..

P. 5
GRW theories and dynamic reduction models which erase the superpositions through some dynamic law.

P. 6
Bohmian mechanics is deterministic.
Born's statistical law is not an axiom but a theorem in Bohmian mechanics.

P. 7
Wave function collapse is not needed in Bohmian mechanics.
It also gives some (which?) conflicting predictions with respect to vanilla QM for some macroscopic interference experiments. But they may be impossible to conduct.
Let's think about designing an experimentum crucis?..

P. 8
"The particles in Bohmian mechanics move along the flow lines of the quantum flux".
Two equations are needed: 1) Schroedinger for the guiding field \psi; 2) equation for positions (see eq. 1.4).
Quantum formalism in its most general form follows from that.
E.g. Heisenberg's uncertainty is also just a consequence.
Description of double-slit experiment.
Basically, the wave function sets up a map of paths which the particle can take, hence the interference pattern appears, because it is dictated by the wave function itself; it doesn't mean that the particle is not a particle at any moment in time.

P. 9
Ontology is not equal to determinism. Bohmian mechanics is about restoring ontology in the quantum realm.
"A realistic quantum theory is a quantum theory which spells out what it is about".
Determinism is only a pleasant bonus here.

____________________________________________

Chapter 2.
Classical Physics


P. 11
Difference between classical and quantum physics.
Best description: classical physics arises when the interference of Schroedinger waves (wave functions) can be neglected.

P. 12
Newtonian mechanics describes the motion of point particles.
Configuration space. For Newtonian mechanics it's just a set of all combinations of possible positions of particles.

P. 13
In order for this to work, we need to pre-set the initial conditions (positions and velocities).
Hamiltonian formalism.

P. 14
Configuration space is multidimensional and thus cannot be depicted.
But it plays a fundamental role in quantum theory.

P. 15
Phase space (introduced by Boltzmann): positions plus momenta. Thus it has twice the dimensions of configuration space.

P. 16
For many effective forces a potential function exists: the force being the gradient of this function.
Hamilton function.
Curiously, Hamilton himself was denoting the function as H in honor of Huygens.
Hamiltonian equations.

P. 17
The solutions of this system of equations are unique, i.e. there is no ambivalence given the same initial conditions.
"One possible evolution of the entire system is represented by one curve in phase space".
These curves are called flow lines.
Once we've properly written down the H function, we only need to solve the equations, and we'll find the curves.
"The fundamental properties of the Hamiltonian flow are conservation of energy and conservation of volume".

P. 18
NB. Trajectories in phase space can never cross each other.
H can be any function of positions and momenta.
Conservation of energy = H is constant along a flow line.
Poisson bracket - allows one to calculate the time derivative of any function if we know H.

P. 19
Hamiltonian vector field is divergence-free (Liouville's theorem).
I.e. it behaves like an incompressible fluid.
Hence the volume element (in phase space) that gets transported along the Hamiltonian flow is conserved.
The volume itself cannot be introduced straightforwardly, but has to be derived using the Lebesgue measure.

P. 20
Volume element = measure. Stationary measure = it doesn't change with time.
"The stationary measure plays a distinguished role in justifying probabilistic reasoning".
"... under the Hamiltonian flow, sets change their shape but not their volume".

P. 21
Changes in density of the measure (because of changing shape of sets) are governed by continuity equation.

P. 23
"Liouville's theorem implies that the Lebesgue measure (= volume) is stationary for Hamiltonian flows".
"... even in the case where the Hamiltonian is time dependent (energy is not conserved), the volume (Lebesgue measure of a set) remains unchanged under the flow".
Cantor set with a Lebesgue measure of zero.
The problem of initial conditions that lead to a Newtonian universe.

P. 24
Hamilton sought to formulate mechanics in a way analogous to optics (with Huygens' principle and Fermat's extremal principle).
Least action principle and Lagrange formulation.
Lagrange function and Hamilton function are Legendre transforms of one another.

P. 25
Hamilton-Jacobi function.
It is multivalued (not uniquely defined).

P. 26
Hamilton-Jacobi differential equation.
Fields versus particles.
Fields being fundamental is a great idea, but the problem is that particles also generate them.

P. 27
Introduction of relativity where configuration space (positions of particles) is no longer fundamental, because each particle has its own "time", not an absolute one.
Einstein's conclusion that this means fundamental locality of physics "in the sense that no physical effect can move faster than light". This has been proven wrong by Bell.
Minkowski's length (interval). Lorentz transformations are simply rotations in 4-dimensional spacetime.

P. 28
Switching between proper time and coordinate time.
Free particle moves according to extremal principle with minimal Minkowski length between the starting and final position.

P. 29
Introducing Lagrange formalism to identify the values with known Newtonian ones - need to multiply by some dimensional constants.
The resulting Lagrange function and momentum.
Energy-momentum relation.

P. 30
The force must be always orthogonal to the velocity in the sense of the Minkowski metric.
This could be achieved if we represent the force as an antisymmetric tensor of rank 2.
One way (my bold) to generate such tensor is through 4-vector of potential.
The resulting force tensor.
Potential equation for fields generated by particles themselves.

P. 31
Gauge invariance (the force tensor is not uniquely determined by the potential function).
Lorentz gauge.
The problem of introducing distributed charges: the distance between them changes due to Lorentz contraction. One consequence of this is that e.g. mass of the electron would be bigger than the observed one.
So we are forced to consider the electron as a point particle.
Current of a point charge.

P. 32
System of charges interacting via fields.
Basically impossible to solve the equations even for one particle.

P. 33
NB. Existence of retarded and advanced Green's functions. Retarded function corresponds to the signal emitted in the past.
But since advanced function is also viable (as well as any linear combination of the two), "signal from the future" is too?.. Backward and forward light cones are equipotent.

P. 34
We arrive at a solution and find the field function.
But it's infinite at the position of the emitting particle.
"This problem is well known by the name of the electron self-interaction".
The electron generates the field which acts on the electron itself, but since electron is a point, the interaction is infinite.
Hence field approach is unfit for description of point charges. Unless you introduce additional artificial procedures like renormalization or modify the fields on small scales.
Maxwell-Lorentz approach only works for charges with spatial extent (e.g. macroscopic charged bodies) or when fields are introduced "externally" by hand.

But "... why does one need fields at all?"
Why not let particles interact directly?
Supporters of this idea: Fokker, Gauss, Schwarzschild, Tetrode.
Also Weber and his school.
To a degree (?) Wheeler and Feynman.

P. 35
The simplest way to formulate this is to say that particles with Minkowsky distance = 0 are interacting.
That is, if we suppose that light speed is the limit.
We'd still end up with light cones forward and backward influencing us.
Fokker-Wheeler-Feynman action.

P. 36
No diagonal terms (compared to Maxwell-Lorentz), hence no self-interaction and no infinities.
One can still introduce "fields" for practical purposes, but they aren't fundamental here. Particles interact with each other, not with fields.
OMFG
"On the fundamental level, there is no radiation field and there is no radiation. Therefore, opposite charges may orbit each other without "losing energy" due to radiation".
OMFG
"Famous solutions of that kind are known as Schild solutions".

Emission versus absorption. E/m is time reversible, so it is unclear why we only take the retarded Green's function and focus on the emission.
"... the problem of the electromagnetic arrow of time".
It is the natural consequence of isolating the objects and ignoring all the other objects in the Universe. Otherwise we'd see the absorption (of outside energy) really well. Such is the inherent solipsism of materialistic approach.

Wheeler-Feynman: absorber condition (everything that is emitted is absorbed). This is still time-reversible.
Only if you assume special initial conditions (e.g. more emission than absorption), the asymmetry appears, and with it, the arrow of time.

P. 37
"... Wheeler-Feynman electromagnetic theory is a mathematically consistent relativistic theory with interaction. In fact, it is the only such ... theory existing so far".
It is experimentally indistinguishable from Maxwell-Lorentz whenever the latter is well defined.

P. 38
Symplectic geometry.

P. 42
"... integrability is atypical for Hamiltonian systems".

____________________________________________

Chapter 3.
Symmetry


P. 43
"... Bohmian mechanics is a Galilean theory of nature".
Symmetry of a physical law = its invariance under certain variable transformations.
"... there may be "strange" actions on secondary (or derived) variables in order for the law to be invariant".
Sometimes symmetry is a priori, when the law is deliberately formulated with respect to it.

P. 44
E.g. if a law is formulated in Euclidean space, it should be invariant with respect to rotations in this space because of the inherent symmetry of the space itself.
Noether's theorem: symmetries correspond to conservation laws.
QM: unitary symmetry - invariance with respect to choice of basis in Hilbert space.
NB but actual "position" of a particle breaks this symmetry, as it is a fundamental "preferential" variable in the real world.

P. 45
"... the fundamental symmetries are clearly those of spacetime".
Galilean symmetries.
An example for one particle.

P. 46
Newtonian gravitational potential.
If the dynamical law's symmetry is about velocity (instead of acceleration), this is problematic for Galilean relativity.
"... first order theories ... cannot be Galilean invariant".
NB Aristotle [apparently] said that any motion is only guidance towards a final destination.

P. 47
But Hamilton-Jacobi theory (first order) is. What's the deal?
Non-trivial character of time reversal transformation. E.g. in Maxwell-Lorentz we not only invert the velocity, but also the magnetic field.
Contradiction between time reversal symmetry and entropy. The difference may arise from scaling: microscopically the theory is time-symmetric, but macroscopically entropy rises.

____________________________________________

Chapter 4.
Chance


P. 48 is missing completely - ?..
P. 49
"Physical laws do not usually involve probabilities".
Theories are deterministic. What is the role of randomness then?
2 questions of Smoluchowski:
1) how can random causes lead to lawlike effects?
2) how can lawlike causes lead to random effects?

P. 50
"... the answer to both is typicality".
Boltzmann's probabilistic reasoning within deterministic physics.

P. 51
Problem of defining the object of probability theory.
Probability theory is the doctrine of typical behavior (of physical systems).
Example with a coin. Stirling's approximate formula for binomial coefficient.

P. 52
Real physics: gas molecules in a box.
Macrostate versus microstate. Microstate does change (even in equilibrium), but a macrostate doesn't.
"Boltzmann's statistical reasoning is superior to physical theories, in the sense that it does not depend on the particular physical law".

P. 53
NB Typical = overwhelmingly many.
However, the number of microstates is uncountable, since positions and velocities represent a continuum.
Hence we use the phase space "size" or "volume" to conduct a comparison.

P. 54
Amplification of small initial fluctuations.
"... the more random the sequence is, the more simply we find the lawlike behavior".
Example: Galton's board.

P. 56
The board essentially magnifies the initial small uncertainty. And also repeatedly randomizes the result on every next step.

P. 58
Can we divide the intervals infinitely? (To get the continuous probability distribution functions).
Rademacher functions.

P. 59
Chaotic behavior.
Independence of each next step.
Second question (where the initial randomness comes from).

P. 60
Another example: container with particles dropping them into the box machine.

P. 62
"The role of the statistical hypothesis is nothing other than to define typicality".
Difficulty of proving the law of large numbers arises because of difficulty of proving the statistical independence in any realistic system.

P. 63
The problem of where does the randomness come from in a deterministic system.
"In a typical universe, things may look random, but they are not".

P. 64
For this to work, we still need to prove the law of large numbers for the universe.
"Which measure defines a typical universe?"
Gibbs: measure of typicality is called an ensemble.
"Typicality is really an equivalence class notion", i.e. there may be different typicality measures, but they all work in a similar manner.

P. 65
Phase space volume as a good typicality measure.
We require it to be stationary (i.e. it doesn't depend on time).
Gibbs' view is more popular, but Boltzmann's is better.

P. 66
We introduce a function called the canonical ensemble.

P. 67
Microcanonical ensemble - similar function, but when energy is conserved.

P. 69
Boltzmann and Einstein: microcanonical ensemble is the typicality measure for the Newtonian universe, since in canonical ensemble there are energy fluctuations, and here there are none.

P. 71
Boltzmann's constant is the heat capacity of a single gas molecule.

P. 72
Connection between thermodynamics and Newtonian mechanics through equipartition theorem and molecular kinetic theory.

P. 75
"The measure of typicality for a subsystem of a large system, which is typical with respect to the microcanonical ensemble, should be the canonical ensemble, if interaction between the subsystem and the environment is small".

P. 76
Microscopic definition of Clausius' entropy.
Boltzmann's logarithmic version.
"Boltzmann never wrote this formula down, although it is inscribed on his tombstone. Planck wrote the formula".
Planck also introduced Boltzmann's constant.

The authors claim that these fomulas are valid in general, not only for an ideal gas. But they never prove it or even make a remark about how that generalization could be possible.

P. 77
Splitting of a system into a subsystem + environment.
Expansion of statistical sums into a series. We use log so it becomes convergent.

P. 78
Difference between Boltzmann's and Gibbs' entropy.

P. 79
"... the relative energy fluctuation is negligible when N gets large".
Hence we arrive at the equivalence of ensembles.
Gaussian approximation.

P. 80
Thermodynamic free energy relation.
Gibbs entropy. It's equal to Boltzmann entropy in equilibrium.
But there's a big difference with regards to non-equilibrium, irreversibility and second law.
Boltzmann's entropy is a function on phase space while Gibbs' is a functional of distributions on phase space: "... technically very abstract, but computationally useful tool".

P. 81
How can we compute anything precisely?
"... we do not know".
Our Universe is not typical. E.g. we can generate the "unlikely" situations that would not have occurred "by themselves".
The world of meaning.
"A typical universe is an equilibrium universe". In the sense of heat death. But ours is not.
"Our Universe is atypical or in non-equilibrium".
Second law.
Is this just a problem of scale again? Information on microstate is lost, because we can't discern it anymore. We're too big and slow.
NB. "In thermodynamics there is no argument to back this law, no further insight to make the law plausible".
Boltzmann's attempt to explain it through Newtonian mechanics.

P. 82
Atypical initial state.
Why do we always start with that? And then show how it degenerates into a typical state?..
We shall consider typicality under given external constraints (i.e. the ones that have led to atypical initial state).
Boltzmann equation describes the evolution towards an equilibrium state.
"... the second law can be reduced to and therefore explained by fundamental microscopic laws of physics".
"... the key here is special initial conditions".
Extensive variables.

P. 83
Entropy is extensive.
Boltzmann was the first to express it as a function of phase space.
NB. One needs to normalize the entropy by the number of particles [factorial] for it to become truly extensive.
But what about divisibility of matter? If one looks at hydrogen atoms, the number of particles would be N. Yet if one counts these as a proton + electron, the number of particles is 2N. Continuing that logic, if matter is infinitely divisible, entropy would always tend to zero (because we normalize it by number of particles, which is infinite) and would not have any meaning whatsoever - ?..

P. 84
Evolution of a system to equilibrium (illustration).

P. 85
Mathematical example to illustrate how the entropy actually increased.
I have no idea what they're doing.

P. 86
"How can time reversible dynamics give rise to phenomenologically irreversible dynamics?"

P. 87
More specifically, how can one do so rigorously in mathematical sense?
This is what Boltzmann claimed to have achieved with his equation and his H-theorem.
Note that if we reverse all velocities in our special case after it became not special anymore (the gas has spread over the chamber's volume), the gas collects in one half of the volume again.
I.e. the information about the specialness of this case is never truly lost.
Hence entropy rises only for typical initial conditions. And there exist "bad initial conditions".
And Boltzmann equation holds only for "good" initial conditions.

P. 88
Poincare's recurrence theorem.
If gas started in one half of the container, it will return there at some point.
Boltzmann: it would do so even if it started while filling the whole container - just by random fluctuations. But it may take a really long time.

P. 89
Boltzmann's estimation of recurrence time.
Ergodic theory.
"Whatever microscopic time scale one uses, the resulting time spent in equilibrium is ridiculously large".
Boltzmann: we are a non-equilibrium island in an equilibrium sea.

P. 90
We don't see any equilibrium in astronomy, however. What does that mean? Is the fluctuation larger than we assumed?
Or you just assume that physics is not the only world that exists and acts. Meaning.
Feynman: Boltzmann's fluctuation hypothesis is ridiculous.
"We are convinced that our initial condition is a very special one".

P. 91
Ergodicity is not as useful as we may think.
Birkhoff's ergodic theorem.

P. 92
"Ergodic stationary measures are special among all stationary measures".

P. 95
Definition of mixing.

P. 96
Kolmogoroff: axiomatic formulation of probability theory in 1933.
Lebesgue integral.

P. 97
Borel algebra.

P. 98
Image measure, measurability and random variables.
"... a horrible and misleading terminology".
Expectation value.

P. 99
It predicts the empirical mean.

P. 100
Independence of the random variables.

P. 102
True meaning of probability is inherently linked to typicality.
Example with a coin toss calculation.
So in an "atypical" universe like ours probability doesn't really mean anything.
Bernoulli map.

P. 103
Probability theory can never give us an estimate of the very next result.
We are facing absolute uncertainty.
Bernoulli sequence.

P. 104
"The prediction will be of Boltzmann-type certainty, i.e., not certain but typical".
Empirical mean = ergodic mean.

P. 105
Weak law of large numbers.
Typical value for the empirical distribution is its theoretical expectation.
"The latter is commonly referred to as "probability"."

____________________________________________

Chapter 5.
Brownian motion


P. 109
Proof that atoms exist. It also bridges a gap between microscopic and macroscopic worlds.
"... macroscopic motion can look totally different (diffusive and irreversible) from microscopic motion (ballistic and reversible)".
Brownian motion equation is the same as Schroedinger's, but with imaginary time.

P. 110
Using ideal gas law to describe Brownian motion.

P. 112
Einstein relation (between diffusion/fluctuation, friction/dissipation and temperature).
Heat equation (describes diffusion and transfer of temperature).

P. 113
Using these relations we can determine Boltzmann's constant or Avogadro number from observing Brownian motion.

P. 114
"... direct view of the molecular motion via the mesoscopic Brownian particle as mediator".
Brownian motion is irreversible, but follows from reversible microscopic motions.
Introducing a toy 1-dimensional model.
Infinite models do not demonstrate Poincare cycles.

P. 116
Displacement is proportional to square root of time.

P. 118
"It is only in the limit that the irreversibility becomes "true"."
The limit is: infinite time and size of the system and infinite number of particles.
This is technically known as the central limit theorem for random variables.

P. 119
Borel algebra.
Wiener process.
Feynman-Kac path integration.

P. 120 is missing - ?..

____________________________________________

Chapter 6.
The Beginning of Quantum Theory


P. 121
Blackbody radiation example.
Canonical distribution of radiation energy.

P. 122
E/m radiation is just a bunch of uncoupled harmonic oscillators.
Average energy of radiation at a given temperature (Rayleigh-Jeans distribution).
Wien's conjecture.

P. 123
Planck's interpolation of both by a single formula.
If we assume integer values of n, everything suddenly becomes simple and easy.
Does it necessarily mean that "photons exist"? Doubtful.
"... neither Einstein ... nor ourselves know today what "photons" really are".

P. 124
Do they have size? What are they in the first place.
Bohr and Sommerfeld: quantization within atoms.
De Broglie's hypothesis.
Wave packet: superposition of plane waves centered around some momentum value.
Hence the wave packet is concentrated in some limited area.
Dependence of the frequency on momentum (dispersion).
Dispersion governs the spreading of wave packets.
Light waves in a vacuum have a special dispersion law which ensures they do not spread.

P. 125
Expansion of the phase into series.
Stationary phase argument (when linear expansion term vanishes).
Group velocity definition.
Group velocity is the "phenomenological" velocity, which corresponds to the speed at which the point of maximum amplitude (which is determined by interference) moves.

P. 126
Photons ("if they exist") move along light cones with zero proper time.
Energy-momentum vector of photons.
De Broglie's matter waves.
Relativistic dispersion relation for a massive particle.

P. 127
De Broglie has found the equations in 1927, but was met with heavy criticism (especially by Pauli). Bohm rediscovered them in 1952.
"All mysteries evaporated under Bohm's analysis, except maybe one: where does quantum randomness come from?"

____________________________________________

Chapter 7.
Schroedinger's Equation


P. 129
Once more a page is missing.
How de Broglie matter waves lead to Schroedinger's equation.

P. 130
The equation itself.

P. 131
Schroedinger's wave function cannot be factorized into separate particles. And it is a function on configuration space. This is the biggest revolution of quantum mechanics with respect to classical one.
"... such a description involves all particles in the universe at once".
Entanglement is the source of this non-locality.
"Schroedinger equation is the simplest Galilean invariant wave equation one can write down".
The potential must be Galilean invariant.

P. 132
For time reversibility the potential has to be real (not complex).

P. 134
The potential, moreover, has to be gauged by a constant. I.e. wave functions which differ only by a constant phase shift would describe the same physics.
NB. Magnetic field changes sign under time reversal.
What is the physical meaning of wave function?

P. 135
Born: interpretation of wave function as probability amplitude.
The probability is irreducible, i.e. fundamental.
"In truth, quantum randomness is good old Boltzmannian statistical equilibrium, albeit for a new mechanics: Bohmian mechanics".

P. 136
Born: "The motion of the particles follows probabilistic laws, while the probability itself evolves according to a causal law".
He also mentions the possibility of introducing hidden variables (e.g. particle positions), but says it's not necessary and we should be happy with fundamental indeterminism.
Contrary to Born, the new mechanics doesn't necessarily obey Newtonian principles.

P. 137
Born, Einstein or Schroedinger could have discovered Bohmian mechanics by introducing the hidden variables. But they didn't.
Madelung and de Broglie have done it though.
"What is quantum mechanics really about?" Philosophical discussions instead of answering this simple question.

P. 138
Schroedinger and Einstein: quantum description is incomplete.
Bohr, Heisenberg, Dirac, Pauli etc.: we need new philosophical view on nature.
In contrast, Bohmian mechanics is surprisingly trivial and answers all the problems.
Usually people look for solutions of Schroedinger's equation in Hilbert space. But Bohmian mechanics also requires classical solutions of the equation - as a wave equation.

P. 139
"Bohmian mechanics ... is a complete theory where nothing is left open, and above all, it does not need an interpretation".

P. 140
Potential competitor to Bohmian mechanics: dynamical reduction theory (GRW - Ghirardi, Rimini, Weber). There Schroedinger's unitary evolution doesn't exist. NB one can formulate it in a Lorentz-invariant way.
Quantum flux equation.

P. 142
Reinterpretation of quantum flux as particle trajectories.

____________________________________________

Chapter 8.
Bohmian Mechanics


P. 145
"... the theory is not at all Newtonian".
Particle trajectories are integral curves along the vector field defined by the wave function.

P. 146
Wave function obeys Schroedinger's equation.
"Bohmian trajectories are nothing but the flux lines along which the probability gets transported. The velocity field is thus simply the tangent vector field of the flux lines. This is the fastest way ... to define a Bohmian theory".
"Bohmian mechanics follows from arguments of minimality, simplicity and symmetry".

P. 148
One-particle Bohmian mechanics.

P. 149
Comparison to classical mechanics: appearance of an additional term. Bohm called it quantum potential. If it is zero, Bohmian trajectories follow classical trajectories.

P. 150
"... in general there may be a huge difference between "measured" trajectories and "unmeasured" trajectories".

P. 151
"Force and acceleration are not elements of the new theory".
Heisenberg's uncertainty is a consequence of Bohmian mechanics.
Quantum equilibrium: wave function as a measure of typicality.

P. 152
Equivariance: generalized stationarity.
Analogy to the Liouville theorem.

P. 153
Quantum equilibrium hypothesis.
For an ensemble of systems with the same wave function the probability density is wave function squared.
"One has existence and uniqueness of Bohmian trajectories for all times for almost all initial conditions".
Calculation for electron in a ground state.

P. 154
Curiously, Bohmian mechanics says that electron in a ground state of hydrogen atom does not move at all.
Hence it doesn't radiate, and the atom is stable. Electron momentum is exactly zero.
But that seems to contradict Heisenberg.

P. 155
Higher energy stationary states correspond to periodic orbits.
Superposition states may produce very complicated motions.
"... possible trajectories cannot cross in extended configuration space ..., while this is possible in Newtonian mechanics".
Description of the double slit experiment.

P. 157
Tunneling through a barrier description.
"Bohmian mechanics forces us to realize that the stationary picture is an idealization".

P. 158
What is spin.
"Spin is in fact as quantum mechanical as the wave function".
NB "Spin plays a role only in connection to electromagnetic fields".
Stern-Gerlach experiment description.

P. 160
"Bohmian mechanics for spinor wave functions is simple".
Pauli equation.

P. 161
SO(3) group has a double-valued representation: SU(2).
SO(3) is a Lie group, i.e. a manifold. Thus it is not simply connected. This is the topological reason for the double-valued covering.

P. 162
"Spinors are now introduced as a kind of square root of the matrix".
Hence one needs 4*Pi turn instead of 2*Pi to reach identity.
Dirac equation is the simplest equation which relates two Pauli spinors.
Four-spinor (Dirac spinor) is the two spinors combined.
"Spinors ... carry a representation of the rotations".
"Bohm-Pauli theory arises as a non-relativistic approximation of the Bohm-Dirac theory".

P. 164
Expectation value of spin in Stern-Gerlach experiment.

P. 165
Thought experiment with double reversed Stern-Gerlach setup.
Symmetry axis is still a topological barrier. Hence the spin must change along the way.
Hence "... spin is not a property of a particle ... Spin is a property of the guiding wave function, which is a spinor. The particle itself has only one property: position".
In this example the particle is guided by "spin +1/2" wave and then by "spin -1/2" wave (or vice versa). Which one comes first is determined by the particle's initial position.
By adding more Stern-Gerlach magnets we arrive at a quantum mechanical analog of Galton board.

P. 166
"... the wave function for many particles is an element of the tensorial spin space. A typical N-particle wave function is a linear superposition of N-fold tensor products of one-particle spinor wave functions".
Schroedinger was convinced that the return to discrete particles was not possible.
Discussion of why indistinguishability of particles leads to Bohmian mechanics.

P. 167
Indistinguishable character of bosons naturally arises through the way we define our configuration space. What about fermions then?
Questionable character of Dirac's reasoning. For his theory (where electrons can have negative energy) to work one must assume that all the negative energy states are already occupied (by infinite "Dirac sea" of electrons) and hence an electron can't just infinitely radiate by falling lower and lower into negative energies.
Pauli's principle is equivalent to statement that the wave function is antisymmetric.

P. 168
Spin-statistics theorem.
NB reference to topological effects in Bohmian mechanics - must see!
Some topological reasoning to explain fermions.
Generally, a simple manifold when factorized is not simple anymore (e.g. it may be multiply connected).

P. 169
In our case we have a plane with holes in it.
To integrate we need to use Riemann sheets. Instead of a circle we'll have a spiral staircase where we move to the next sheet as we go around.

P. 170
Periodicity condition of the wave function.
"... the truth behind indistinguishability ... lies in the topology of the configuration space".
"The whole construction should be done for spinor-valued wave functions".

P. 171
Braid group. Anyons.

____________________________________________

Chapter 9.
The Macroscopic World


P. 173
Bohmian mechanics and the measurement problem. How do we arrive at classical behavior from Bohmian mechanics?
Limit of Planck's constant equal to zero is physically meaningless, as it's a constant which is not equal to zero.
"Since Bohmian mechanics is not about measurements but about ontology, namely particles, it has no measurement problem".
"Apart from having a position, particles have no further properties".
Doubtful, but we'll let it slide.
So the only things we can really measure are positions and wave functions of particles.

P. 174
"... momentum is not a fundamental notion in Bohmian mechanics", hence uncertainty of momentum is not an issue.
"... most of what can be measured is not real and most of what is real cannot be measured, position being the exception".
Example of a measurement experiment.
Considering the wave functions of pointer positions: they consist of [almost] disjoint superpositions of wave functions of all the particles that make up the pointer.

P. 176
Explanation of why measurement problem is not a problem in Bohmian mechanics: because particle positions are real. Wave function simply guides the particles, but doesn't determine where they are exactly neither before, nor after the measurement. Hence superposition of wave functions is not an issue.

P. 177
Criticism of the significance of observer.

P. 178
For all practical purposes, macroscopic interference is not possible.

P. 179
Aside from Bohmian mechanics, the only other option of resolving the measurement issue is by assuming a non-linear equation instead of Schroedinger's. Thus macroscopic interference would be removed by the very nature of wave function dynamics. This is what GRW theory (dynamical reduction model) does.
Wave function cannot be measured in QM.

P. 180
"Bohmian mechanics provides us with a collapsed wave function".
Except for impossible cases where all the particles become coherent or a random fluctuation returns us into phase (Poincare recurrence), which takes nearly infinite time.
Other example of effectively disjoint wave functions.
What is Bohmian take on delayed choice experiments?

P. 181
Measurement of particle position with light.

P. 182
We arrive at an effective collapse of the wave function through decoherence.

P. 183
At this point it is impossible to experimentally distinguish between Bohmian mechanics and spontaneous localization theory.
Spreading of the wave packet with time. The lighter the particle, the more it spreads.

P. 185
Newtonian equations in the mean.
Limitation on the width of the wave function if we want classical behavior of the particle.

P. 186
Measurement (e.g. interaction with other particles) counters the spreading. It collapses the wave function and therefore we have the classical behavior.
But if this is the case, how can we use Schroedinger in the first place?
Obviously, the measurement must happen at the appropriate time scale for this.
Under what physical conditions are Bohmian trajectories close to Newtonian ones?
"In the long run the Bohmian trajectories of a freely moving wave packet become classical".

P. 187
It is the potential that defines the scale parameter of the problem. I.e. what we consider macroscopic.

P. 188
Stationary phase argument (again).

P. 189
Macroscopic Bohmian velocity field. Asymptotically the trajectories become straight lines.

P. 190
Heisenberg's uncertainty deals with the connection between distributions of position and velocity (momentum).

P. 191
"The classical trajectories simply move through each other, while Bohmian trajectories simply cannot do that".
But it's fine, since decoherence will fix the wave function.
Spontaneous localization models.

P. 192
Density matrix.
Dirac formalism.
Coordinate representation.

P. 193
Schroedinger equation in Dirac notation.

P. 194
Von Neumann equation: a quantum mechanical analogue of Liouville's equation.
Pure and mixed states.

P. 195
Reduced density matrix.
It is almost diagonal, but not quite.

P. 196
Time evolution of the reduced density matrix.

P. 198
Poincare recurrence "also appears in quantum mechanics".

____________________________________________

Chapter 10.
Nonlocality


P. 201
"... any theory which aims at a correct description of nature must be ... nonlocal".
I.e. FTL action between spacelike separated events must be possible.
However, no information transfer should be possible at FTL speeds.
Why?..
Newtonian theory is non-local, and so is Bohmian, but in a different way. Here the guiding wave depends on all the configuration space. That's where non-locality is coming from.

P. 202
"Bell's proof of the nonlocality of nature".
EPRB (for Bohm) as a modification of original EPR thought experiment.

P. 203
Experiment scheme.

P. 204
Bell's inequality.
Nonlocality is experimentally proven.

P. 205
Singlet wave function.
Total spin in the singlet state is zero.

P. 207
"EPR argument yields, by the locality assumption, local hidden variables for spin".
"... local hidden variables cannot reproduce the experimental correlations".
Bell's inequalities are for, not against Bohmian mechanics.
Bell's article "Bertlmann's Socks ..." where he argues against the "learning at a distance" interpretation.
But then why do you guys claim that information is not transferred with FTL speed? I still don't get it.

P. 208
"Why can one not send signals faster than light? One cannot, because of the quantum equilibrium hypothesis. The action at a distance which the wave function mediates is randomized in such a way that it is unusable. If quantum equilibrium were false, superluminal signalling might perhaps be possible". Hence non-equilibrium is also being researched.
I still don't get it.
"The collapse is the nonlocal effect which could be the source for nonlocal signalling".
Sure, but have you proven that it is the only effect that could be the source for nonlocal signalling?.. Rhetorical question, really.

____________________________________________

Chapter 11.
The Wave Function and Quantum Equilibrium


P. 211
"Bohmian mechanics thus recommends itself as the paradigm for Boltzmann's view of chance in physics".

P. 212
"Equivariance generalizes stationarity and defines the quantum equilibrium measure, the measure singled out by the dynamical law itself, and which defines typicality. The equivariance property ensures that typicality is time independent".
Stationarity itself is not enough because wave function is time dependent.
Wave function of the Universe. Is it stationary?..
Does it even exist?
"One reason to think that the wave function of the Universe is not stationary is macroscopic irreversibility, so that the wave function can be held responsible for the global non-equilibrium character of the Universe".

P. 213
"... the assumption of a non-equilibrium universal wave function is plausible, but not necessary".
In vanilla QM a stationary wave function would mean a stationary world, but this is not the case in Bohmian mechanics.
How does that relate to relational mechanics/Mach's principle?..
The authors say that one has to include even the most distant bodies into consideration, which essentially means Mach.
Look at possible scale dependence of psi factor in Weberian gravity. Maybe it's different at scales of solar radius (light bending), Mercury's orbit and galactic rotation.
Wide binaries as an intermediate? Stellar clusters?
Also maybe it's not 2gamma, i.e. what if gravity is not conservative?..
Also how it relates to Kozyrev's velocity?..


P. 214
Conditional wave function of a system.

P. 215
Conditional measure.
"That is what we believe to be the case in our world, i.e., that subsystems sometimes behave autonomously".

P. 216
Bohmian subsystem: a subsystem which has its own wave function, its own Schroedinger evolution and hence its own law for Bohmian trajectories.
"It is unreasonable to assume that the universal wave function has product structure. Generically, it will be a superposition of products, i.e., a bona fide entangled function".
Effective wave function.

P. 217
Conditional wave function always exists, and it is equal to effective wave function under certain conditions.
"... effective wave function is a mathematically precise concept of the collapsed wave function of orthodox quantum theory".

P. 218
Example showing how stationary universal wave function might lead to time-dependent wave function in a subsystem.

P. 220
"... we acquire good information about the effective wave function, at least about its modulus squared, via the empirical statistics".
"... quantum equilibrium hypothesis ... is ... the only link between theory and experience".

P. 222
"... equivariance is also technically crucial!"
Now we try to prove the ergodic hypothesis. We note that the times when experiment is done are random.

P. 223
"The initial randomness of the particle position translates into the randomness of the particle's asymptotic velocity".
This distribution spreads as the particle position distribution becomes smaller. This is how Heisenberg's uncertainty arises. It "... is a direct consequence of the quantum equilibrium distribution, i.e., Born's statistical law".
Quantum equilibrium restricts our knowledge about particle position to wave function squared probability.

P. 224
Problem of irreversibility once more.
"There is no need for a second law for the configurations in Bohmian mechanics. Quantum equilibrium ... is fortunately all we need".

____________________________________________

Chapter 12.
From Physics to Mathematics


P. 227
"Readers who know quantum mechanics from textbooks will find this chapter to be a revelation".
Well, let's see then.
"It is natural to think that an observable is a variable that can be observed". But this is often not the case in vanilla QM.

P. 228
"... the operator observables of quantum mechanics are book-keeping devices for effective wave function statistics".

P. 230
A special operator, "... quantum Swiss army knife, containing all that we need", from which everything can be found.
It's made of a family of orthogonal projectors.

P. 231
"... the book-keeping operator".

P. 232
The source of confusion (about what an observable is) is the fact that an operator has eigenvalues which correspond to a given experimental situation. An operator thus can be "measured".
"... one can associate operators with experiments as book-keeping devices for the statistics".

P. 233
The space of wave functions is a linear space (because Schroedinger's equation is linear) with a scalar product. E.g. Hilbert space.

P. 235
Projection-valued measure (PVM): it combines discrete values with continuous values.
I have no idea what they are trying to do here.

P. 236
Completeness relation.

P. 237
Example: asymptotic velocity and its distribution.

P. 238
PVM here is the spectral resolution of a self-adjoint operator.

P. 239
"The move from classical variables to operator observables was Heisenberg's invention to explain the discreteness of atomic spectra".
Generally speaking, in a measurement experiment nothing really gets measured.
"The role of the observables is just a book-keeping role for the statistics of an experiment".
"But sometimes something is measured: the Bohmian positions".
"... Heisenberg's uncertainty relation is wholly unsurprising once the quantum equilibrium has been understood".
POVM - ?
"PVMs are special cases of POVMs".

P. 241
"Bohmian mechanics contains variables which are not measurable in the sense of the experiment".
Namely, wave functions and Bohmian velocity.
If quantum interference is at play, velocity cannot be measured.

P. 242
"One can, however, measure the Bohmian velocity in a so-called weak measurement".
And we sometimes know the velocity even without measuring it. E.g. in a hydrogen ground state the velocity of an electron is zero.
"... it is the theory that decides what is measurable".
Joint probabilities.

P. 243
NB commutation of the spin operators in an EPR setup means that FTL comms are not possible.

P. 244
"... two Hermitian matrices have a common eigenbasis if and only if they commute".
"Only commuting observables have joint probabilities".
Hidden variables discussion.
The no-go theorem. It is "... nothing but a simple fact".

P. 245
"Trivial it is, but food for mysticism nevertheless".
"On the one side observables, contextual properties, "non-classical" logic, complementarity, wave-particle duality, uncertainty, intrinsic probability, cat paradox, no-go mysticism. And on the other side, the two equations defining Bohmian mechanics, governing the whole of the (non-relativistic) world. Which side should physics be on?"
Existence and uniqueness of solutions of Schroedinger's equation.

P. 247
"The problem of boundary conditions, which we need for the quantum equilibrium hypothesis to hold true at all times, goes hand in hand with the existence and uniqueness of solutions of the Schroedinger equation".

P. 248
What if we solve Schroedinger for a hydrogen atom with a Weber potential? How do we even parametrize velocity? Do we just find it from kinetic energy and assume to be constant?..
"... boundary conditions are therefore needed".
Correspondence between PVMs and observables is called spectral theorem.

____________________________________________

Chapter 13.
Hilbert Space


P. 251
Schroedinger equation is linear, so superpositions of solutions are also solutions.

P. 252
Bessel inequality.
Most of the elements of the space of wave functions are abstract constructs with no possible physical connotation. They're not differentiable and not continuous.

P. 253
If we pick Lebesgue integral as the norm, we arrive at Hilbert space.

P. 254
Riesz-Fischer theorem.

P. 255
"For us ...only so called separable Hilbert spaces are interesting, because they admit countable orthonormal bases".

P. 256
Parseval equality (basically a Pythagorean theorem).

P. 257
Tensor product structure.

P. 259
Plancherel equality.
It's just an unending stream of high level mathematics.

P. 263
Convolution.

P. 267
The functions do not decay at infinity.

P. 268
Self-duality of Hilbert spaces.

P. 271
How to describe entangled wave functions of many particles.
Tensor products.

P. 275
Fubini's theorem.

P. 276
In general the wave function evolves under Schroedinger in such a way that it cannot be represented as a product.

P. 278
"From now on, we shall say that the wave function is an element of a Hilbert space ... We know that only the nice smooth functions in that space, the real wave functions, are physically relevant".

____________________________________________

Chapter 14.
The Schroedinger Operator


P. 279
"In Bohmian mechanics, the dynamics of the wave function is determined by the Schroedinger equation and the dynamics of the particle positions is determined by the guiding equation".
The wave function must be differentiable, i.e. we look at classical solutions of the equation.
How to set up the equation so that it has unique solutions for any initial conditions?
We need the conservation of total probability, but it's not enough e.g. for singular potentials (Coulomb) or bounded configuration spaces.
Hence we need to introduce boundary conditions.

P. 280
The concept of self-adjointness might solve all the problems. We just need to select the physically correct self-adjoint version of the equation.
Definition of a strongly continuous unitary one-parameter group.

P. 281
Schroedinger operator.

P. 283
Schroedinger operator must be the generator of a unitary group. Then we'll have conservation of norm, existence and uniqueness of solutions.

P. 285
Stone's theorem.

P. 293
Von Neumann's theory of self-adjoint extension.

P. 294
Schroedinger operator for atomistic Hamiltonian with Coulomb potential.

____________________________________________

Chapter 15.
Measures and Operators


P. 299
Operator calculus. It allows us to do efficient and concise computations within quantum equilibrium.
Spectral theorem for self-adjoint operators.

P. 301
Projectors - idempotent operators. Self-adjoint projections are called orthogonal.

P. 303
Statistics of any experiment are described by a POVM.
"Therefore it might be surprising to find that these POVMs play little or no role in theoretical physics. The reason is that PVMs give rise to an elegant mathematical formalism, while POVMs give rise to nothing".
PVMs are a special class of POVMs.
"The moral is that experiments are designed and conducted for a small class of special initial data".
Thus the general structure of POVMs or PVMs in full Hilbert space is usually physically irrelevant.

P. 304
Self-adjoint position operator.

P. 305
Position representation of the position operator.
Heisenberg position operator.

P. 306
"The position at time t is random because the initial position of the particle is random!"
Asymptotic velocity of a free particle.
"The Bohmian trajectories following the asymptotic wave function are straight lines".

NB Did anyone use Bohmian mechanics to calculate the nuclear structure? This could be very productive!

P. 307
Momentum operator of quantum mechanics.

P. 309
Deriving Heisenberg's uncertainty.
The more the support of the initial wave function is localized, the faster its distribution spreads under time evolution.

P. 310
Uncertainty with time and energy.
Ehrenfest's theorem (Heisenberg's equation of motion).
"In Bohmian mechanics, the particle has no momentum, while in Newtonian or Hamiltonian mechanics it has momentum".
What about angular momentum then? Spin in particular.
Measurement of the position operator ("a nonsensical statement anyway") is not always a measurement of the position.

P. 313
Generalized eigenfunction.
Spectrum (= set of eigenvectors), spectral family (= PVM) and spectral measure.
Position representation and momentum representation. They are connected via Fourier transform.

P. 314
Spectral theorem.

P. 320
Riesz-Markov theorem.

P. 322
There is a one to one correspondence between bounded self-adjoint operators and compactly supported PVMs.

P. 324
"Life would be easier if we could avoid this topic altogether".
Damn, I love these guys.
How to deal with unbounded operators (e.g. Schroedinger operator): several different possibilities.

P. 329
PVM version of the spectral theorem.

P. 332
Self-adjoint operators are the generators of unitary groups.

P. 333
"... any strongly continuous unitary group is generated by a self-adjoint operator".
"... but that is enough abstract mathematics for now. Let us return to quantum mechanics".
Thank God!
What is the PVM for a free particle hamiltonian perturbed by external potential?
"In general we do not know, but..."

P. 334
Free propagator.

P. 337
Stationary phase method away from the stationary point.

P. 341
A self-adjoint operator can be written as a sum or an integral over a family of pairwise orthogonal projections, i.e., in terms of its associated PVM.

P. 344
"Now back to physics".
Promise me it's for real this time.
Scattering theory. Scattering states.

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paladin17
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Re: Bohmian Mechanics (AKA Pilot Wave Theory)

Unread post by paladin17 » Sun Dec 10, 2023 8:42 pm

____________________________________________


Chapter 16.
Bohmian Mechanics on Scattering Theory


P. 345
Quantum equilibrium distribution shows us the probability of a system being in a certain configuration at a certain time.
From this follows the formalism of POVMs, PVMs and self-adjoint operators in Hilbert space.
Born's 1926 papers where he applies Schroedinger's equation to scattering. Quantum equilibrium distribution is the distribution of the random position of a particle after scattering.
During the scattering experiment the time at which the detector clicks is also random.
Bohmian mechanics works great here, since we have actual trajectories in it.

P. 346
Figure of a scattering experiment.
"This problem is not the common textbook problem of quantum mechanics. The reason is that time is not an observable".
We are only interested in special wave functions here. The ones with physical meaning.

P. 347
Another figure.
Interaction with the detector might also matter, since it alters the wave function. So we need to prepare everything wisely so that trajectory is not significantly changed because of that.
"... we shall follow the common practice of quantum physics and henceforth not worry about the presence of detectors".

P. 348
Distribution of time when the particle leaves some area of interaction for the first time (expressed assuming quantum equilibrium).
"We have introduced the position PVM ... to make it look like advanced quantum mechanics".
KEKW
Quantum flux equation. Net flow through the spacetime surface.

P. 349
Illustration.
Bohmian trajectories are randomly distributed according to the distribution of the initial positions (determined by the square of wavefunction at t=0).
Indicator function of Boltzmann's collision cylinder.

P. 350
Boundary crossing probabilities.
They aren't additive while the expected values are.

P. 351
Probability itself.
Bohmian mechanics is good because it straightforwardly establishes that the crossing probability depends on quantum flux.

P. 352
4-current version of quantum flux equation.
"... the crossing probability is preserved when the current is divergence-free".
"... the expression always has the same form, a property which we called equivariance".

P. 353
(3N+1)-dimensional current in case of N particles. We integrate it over hypersurfaces with constant time and get the quantum equilibrium distribution.
Asymptotic situation.

P. 355
Flux-across-surfaces theorem.
"The formula ... is basic to scattering theory".
When can we assume that the interaction process is over and the particle moves freely? It's quite arbitrary, as in reality the wave function will always partly overlap with the potential.

P. 356
The wave approaches the scattering potential from the infinity, then gets transformed by the scattering potential and goes away to infinity once more.
"Such wave functions will be called scattering states".

P. 357
Illustration.
Scattering-into-cones theorem.

P. 358
"... in many-particle scattering, the quantum flux loses its meaning".

P. 359
Wave operator which encodes the potential.

P. 360
Continuous spectrum splits into two spectra.
Intertwining property.

P. 361
Generalized eigenfunctions.

P. 362
Generalized Fourier transform.

P. 363
Lippmann-Schwinger equation.

P. 364
Abel limit.

P. 365
"There is no need to appeal to wave operators: simply use generalized eigenfunctions".

P. 366
The incoming side of the scattering process is perhaps even more important than the outgoing side, since the former is under the control of experimenter.
"... the generalized Fourier transform has become something of a wallflower in mathematical scattering theory".

P. 367
S-matrix: the mapping of incoming to outgoing wave functions.

P. 368
We subtract the unscattered part (identity) from the S-matrix. The result is called the T-matrix.

P. 369
Scattering cross-section.

P. 370
We shoot "almost identical" particles with plane wave functions onto the target.

P. 372
Why does a quasi-stationary picture emerge.

P. 373
Illustration.

P. 375
Born approximation.

P. 376
"This is so simple that one must wonder what it means".

P. 377
No-spreading condition.
In realistic setups one has to take the number of scatterers into account.

____________________________________________

Chapter 17.
Epilogue


P. 379
"God keep me from ever completing anything".
Bohmian mechanics is a quantum theory without mysticism, paradoxes and superstition and in which observation does not play any fundamental role.
"Bohmian mechanics is an utterly straightforward completion of quantum mechanics".
What still needs to be done is to adapt it to Lorentz invariance.
"... never give up ontology!"

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JP Michael
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Re: Bohmian Mechanics (AKA Pilot Wave Theory)

Unread post by JP Michael » Thu Dec 21, 2023 11:09 am

Sweet as cheers for that mate!

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paladin17
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Re: Bohmian Mechanics (AKA Pilot Wave Theory)

Unread post by paladin17 » Wed Feb 14, 2024 9:35 am

I've made a video summarizing the theory: https://youtu.be/ktb8eJGa9Wg

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