Vanknee said:
First a disclaimer. I don't agree with all of DGZ arguments, some puzzle me and I think they are wrong on the exact interpretation to give to the wave function. Nevertheless, IMHO, monistic interpretation–interpretation which do not take the wave function as referring to a physical entity–do not have to give a special meaning to the wave function. The main problem of the dualistic interpretations is that they have to introduce a special status for the wave function, sometimes as a new physical entity (the information for instance). Moreover, these interpretation ususally stick to a Newtonian world picture: forces are needed.
My first question concerns the physical field behind the Hamiltonian. Tell me if my memories of classical physics are wrong, but the Hamiltonian is not the representation of a physical field. In some particular cases–equations of the generalized coordinates time independant–it represents the energy, but only in this condition, what if it is not fullfilled?
When you ask the question of the physical field/variable are you asking for some kind of force? I perhaps interpret you point of view in the way Belousek object to DGZ, he argues
that the monistic vision does not give a good explanation because it does not give a precise enough description of what is really going on in the motion of the particle. This requirement is asking for a causal explanation (and his solution involve forces as a material entity being represented by the wave function).
But I wonder if this requirement is necessary. You say that it is 'physically weird' to work with the guidance equation as equation of motion. In a way I agree, said like this there is no direct link between the apparatus (the external potential) and the motion of the particle, the wave function does all the job. On the other hand, the principle of least action works in a similar way. The action S is the source of the movement, you can derive it from the potential, but it needs some mathematical manipulation.
Typically, the problem I have with this kind of explanation is that it is not really physical. How can the particle be a boat guided by the radar of the wave function. Is this kind of explanation satisfactory? To be a bit provocative, it sounds to me like little omniscient faeries whispering to the hear of the corpuscle...
Finally, I am aware of the problem bohmian/deBB caracterization. I personally talk about 'corpuscle interpretation of quantum mechanics', which is neutral. I don't want to take part in the dispute...
Vanknee,
My first question concerns the physical field behind the Hamiltonian. Tell me if my memories of classical physics are wrong, but the Hamiltonian is not the representation of a physical field. In some particular cases–equations of the generalized coordinates time independant–it represents the energy, but only in this condition, what if it is not fullfilled?
Recall the classical Hamiltonian is a function on phase space and is composed of the sum of the kinetic energies for each particle in an N-particle system, plus the external potential energy (could be gravitational PE or electrostatic PE). Yes, the Hamiltonian is not itself a real field 'out there' in the world, as it lives in the R^6N dimensional phase space, as opposed to physical 3-space where all particles and physically (meaning experimentally) measurable fields live in. What the Hamiltonian, and more precisely, the gradient of the Hamiltonian or
grad H(x,p,t) = grad ( Sum[N; p^2/2m_i] + V_ext ),
says is that at the location of the ith particle, there is a force (F = m*d^2/dx^2) on the particle caused by a gradient in the rhat direction of the ith particle's own kinetic energy (in any Galilean frame), plus a gradient in the rhat direction of the ith particle's potential energy (remember potential energy, whether gravitational or electric, is a relational property in every reference frame). The Hamiltonian is an abstract, nonlocal, mathematical encoding of these N-particle properties. So the physically real things are the local
gradients in the potential energy (remember that potential energy itself is not physically real because its magnitude is entirely conventional) of the ith particle, plus the kinetic energy of the ith particle. So the physically real things that exert forces on the ith particle at a point in space are the gradient of the intrinsic kinetic energy of the ith particle, plus the gradient of the relational potential energy of the ith particle.
Now imagine if the Hamiltonian was just given by the potential energy so that H = V(x). Well, if the potential energy was, let's say, electric potential energy given by
V(x) = Sum[N; k*qi*qj/|ri - rj|],
Then it is obvious why the Hamiltonian is not a physically real field in the world, because the electric potential energy is a conventional object, and only the gradient of the electric potential energy, which gives the local electric field at a point in space, is what is physically real and what is really exerting a force on the particle, and is what actually exists in 3-space, and independently of the particle. Indeed we could add an arbitrarily large constant to V(x), and the gradient would give the same value for the electric field.
Now imagine if the Hamiltonian was just given by H = 0. This means the particle in some Galilean frame is at rest and has zero kinetic or potential energy. OK, now suppose at a later time t, the Hamiltonian now is given by H = p^2/2m. This means that some external force was exerted on the particle (either by collision with another particle with nonzero velocity, or by an external gravitational or electric (if the particle has charge) field). This also means that the particle has been given an additional physical property that it originally did not always have (unlike its rest mass which is a constant and intrinsic property of the particle).
In all these cases, we know that although the Hamiltonian is not physically real on its own, and rather is an abstract, nonlocal, mathematical encoding of the sum of kinetic and potential energies of N particles, we can clearly identify what physically real objects it is describing, in addition to and independently of the particles. This is not the case in the nomological interpretation of the wavefunction given by DGZ. They deny that there are additional fields/variables that actually induce a velocity or force on the Bohmian particle, even though they want to say the wavefunction is nomological in the same sense as is the classical Hamiltonian. So my point is that if one wants to claim the wavefunction is comparable to the Hamiltonian, then the wavefunction miust be encoding physical information about other physically real variables
in addition to the particles. Otherwise the comparison totally fails to be valid.
When you ask the question of the physical field/variable are you asking for some kind of force? I perhaps interpret you point of view in the way Belousek object to DGZ, he argues
that the monistic vision does not give a good explanation because it does not give a precise enough description of what is really going on in the motion of the particle. This requirement is asking for a causal explanation (and his solution involve forces as a material entity being represented by the wave function).
It could be a force or an impulse. I agree with Belousek's objection. Think of it this way: If are only particles in the physical world like DGZ claim, then if we ask how a single Bohm particle will move through spacetime when it collides with another Bohmian particle, it will still be the case that the Bohmian particle will move according to a nonlocal,
nonclassical law of motion prescribed by the wavefunction. If we then ask the question of how the Bohmian point particle differs from the classical mechanics point particle, mathematically there is no difference. They are both point masses and nothing more. On the other hand, if we ask what in the physical world makes the Bohmian point particle moves differently after collision with another Bohmian particle, than when a classical mechanics point particle collides with another classical mechanics point particle, we can say in the latter case because some external but locally propagating field initially gave the two point particles some momentum, and they took the path of least action towards their collision point (where they actually “touch” each other) and afterwards; and before they collided, their trajectories were totally uncorrelated. Of course, even here in classical mechanics, there is still a mystery about why the two point mass particles can scatter off each other, as opposed to stick together (i.e. why there are elastic and inelastic collisions). Because in classical mechanics these are only postulates, if we want an explanation of these postulates, we are obliged to then add more properties to the point particles, than just masses – we have to add electric charge. Then we have a less ad-hoc explanation of how the particles undergo either elastic or inelastic collisions. But according to DGZ, because the Bohmian point mass particles are the only physically real things in the world comprising matter, both of the spatially separated Bohmian point particles must somehow have nonlocal knowledge of each other's motions so as to conspire to move in a nonlocally correlated way that is totally nonclassical, and without any additional mediating field; but there is no way to understand how these Bohmian particles could have this knowledge if they are just point masses, as the mathematics of the guiding equation under their interpretation implies. To make this even more explicit, consider the scattering of two Bohmian point particles. If the DGZ interpretation of the wavefunction is correct, then two Bohmian point particles moving towards each other will eventually scatter at a certain time when, mathematically, their two wavefunctions with opposite energy and momentum sufficiently overlap in configuration space, even if the Bohmian particles in physical 3-space never actually directly contact each other. If there are ONLY Bohmian point mass particles, how do they know to scatter away from each other at a particular time, even when they are spatially separated?
But I wonder if this requirement is necessary. You say that it is 'physically weird' to work with the guidance equation as equation of motion. In a way I agree, said like this there is no direct link between the apparatus (the external potential) and the motion of the particle, the wave function does all the job. On the other hand, the principle of least action works in a similar way. The action S is the source of the movement, you can derive it from the potential, but it needs some mathematical manipulation.
Well consider another example of the weirdness of the DGZ wavefunction interpretation. In the hydrogen ground state, the Bohmian particle has zero velocity because the action S is a constant and grad S = 0. So the Bohmian particle is stationary at a point in space. Now, when you perform a measurement interaction with some pointer apparatus that is spatially very far away from the stationary Bohmian particle in the hydrogen atom, the latter begins to move exactly when its purely nomological wavefunction entangles with the purely nomological wavefunction of the pointer apparatus, so that the action of the Bohmian particle in the hydrogen atom is not constant in spacetime. But in the physically real world, since, according to DGZ, the wavefunction is not a real field and isn’t an approximation to any underlying physically real fields, the Bohmian particle in the hydrogen atom ground state somehow knew at a certain time that a pointer apparatus was coming in close proximity to it, and suddenly decided to have a nonzero action and to start moving towards the pointer apparatus! So the implication of the DGZ interpretation is that the particle in the real world somehow sensed the pointer apparatus (in an "action at a distance" sort of way), and started moving on its own accord, even though there was no actual physical field or medium between the Bohmian point mass particle and the pointer apparatus to allow them to indirectly interact with each other. Seems like far too much of a coincidence don't you think?
Typically, the problem I have with this kind of explanation is that it is not really physical. How can the particle be a boat guided by the radar of the wave function. Is this kind of explanation satisfactory? To be a bit provocative, it sounds to me like little omniscient faeries whispering to the hear of the corpuscle...
I think you’re taking the analogy either too literally or misunderstanding it. They just mean that the quantum potential force on a particle depends instantaneously on the positions of all the other particles in the world (the source of the nonlocality in the theory), and is independent of the magnitude of the wavefunction, instead only depending on the form of the wavefunction. More precisely, if R = |psi| and Q = -(hbar^2/2m)*Laplacian[R]/R, then C*R = C*|psi| leave Q unchanged. So all that matters is the form of |psi|, not its magnitude.
OK on keeping out of the deBB/Bohmian mechanics terminology controversy.
