On Fri, 9 Aug 1996, Jack Sarfatti wrote:

Lawrence B. Crowell wrote:

The point is that the quantum potential contains information that concerns the quantum corrections to a classical path,Start out with the wave function psi = Re^iS , where R = e^r. The point of doing this is to write the amplitude according to a generator, or rapidity (if that word is appropriate).

Yes, I understand that analogy here.

so that r and S are on the same footing. r and S are generators of the wave function.

OK, I understand that motivation.

When you work with psi =Re^iS you have part of the wave function on a group level and the other on the algebra level.

Cool!, Ok I understand that. :-)

It occurred to me some time back that this is really a rather inconvenient way of doing things.

Ok now put psi = e^(r + iS), with hbar = 1, into the Schrodinger equation and you get

-S_t = (1/2m)((grad S)^2 - (grad r)^2 - laplacian r) + V, ()_t = &()/&t

-r_t = (1/m)(grad r * grad S - laplacian S)

Now the quantum potential is expressed as

Q = (grad r)^2 - laplacian r

Let me check this. "I will a little think." To quote Einstein. :-)

but,

Therefore, I get

So I understand your two terms now, but I get a different relative sign and, of course, the coefficient 1/2m.

Whose algebra is right? I think mine is. I used to sit beside Hans Bethe in his office at Newman Laboratory for Nuclear Studies in 1960 watching him doing really complex calculations like Paginni doing "The Devil's" violin cadenza - that WAS amazing! So now I will think like Bethe. :-)

and is the same quantum potential as before. If you let r = lnR it is easy to show.

Now the momentum operator, p_op = -i grad, when it acts on psi gives

p_op psi = (grad S - igrad r)psi

which is the momentum eigenvalue for the wave function.

OK, I don't follow this last remark. If psi is a momentum eigenfunction |p>, then you get the eigenvalue multiplied by psi. In general you cannot make that assumption. psi is definitely not an eigenfunction of momentum for interesting problems. So, decomposing psi into a superposition of momentum eigenfunctions we get all possible momentum eigenvalues, i.e.,

Let this value be {p}. expected value.

I don't understand this remark either. The Copenhagen (Bohr-Born) expected value is

then

{p} = p -igrad r

Wait, we have to be very careful here with notation.

OK, let me try to do this.

Therefore,

It follows that, the Copenhagen expectation is

Since {p} is real, unless I made a mistake in algebra, we have a curious constraint that

This is the hermiticity condition in a new form. I have never seen this done in any text book this way.

OK, now I am getting something like what you are getting, but you were a bit too quick for my older semi-senile :-) 56 year old brain mit der formalism und der physicalischte interpretation. :-)

Then,

{p} = [p] + i[grad r]

and so,

[grad r] = 0

Wait, like Frank Sinatra sings let's "walk a little slower" and be more precise, because something interesting physically is coming from your intuition here. In Bohm's hidden-variable or "beable" theory

is the non-statistical individual actual momentum of the beable if it is actually at the given point in configuration space where S is evaluated. It is independent of the statistical ensemble averaged expectation value {p} which, according to the correspondence principle (Ehrenfest's theorem) you can call the classical value. Remember the beable surfing the pilot-wave in configuration space has a definite position and momentum at any instant. The uncertainty principle does not apply down at this level. The uncertainty principle only applies for the statistical ensemble root mean square deviations when strong canonical von Neumann measurements are made.

So, OK define a "Bohm smear" [...], I am not sure if we should call it a statistical average yet, as

where S is a function over all configuration space even though the beable is only at one point in that space at any instant. Similarly, for grad r

So OK now I have a physical picture, of sorts, for the formal expression

That is, the orthodox Copenhagen expectation splits into two Bohmian pieces. Your i[gradr] term looks like the non-transverse part of the electromagnetic vector potential in p -> p + (e/c)A for minimal coupling to a gauge field. You know this is exactly what I was looking for in my Ph.D. thesis on critical speed for superfluids. So my intuition was correct then after all. My Ph.D. idea was a remote-precognition of this event over 25 years later!

Look, what you have found is the back-action gauge field that Feynman first told me about in 1963 when we went for the ride in my Jaguar. I don't think Feynman had the math then either. It was Feynman who planted the seed of my back-action idea that led to my Ph.D idea which was incomplete. But I knew I would come back to it someday. My idea was that there was some kind of a minimally coupled Galilean (not Lorentz) gauge field responsible for the breakdown of frictionless superfluid flow in small tubes. This is because Feynman used the words "back-action" to me in the Jag in 1963 and also before as he was talking to a bunch of people in the archways at Cal Tech. I had come up from Ford Philco Aeronutronics in Newport Beach. It must have been an APS meeting or something. I forget exactly which.

OK, now I see what's turning you on. This is a fabulous synchronicity. Orthodox quantum mechanics has no back-action here because hermiticity, i.e., unitarity, local conservation of quantum probability currents in configuration space demand that = 0. The i is the nonunitary dissipative back-action term. That's what the damn i means, right! And this back-action term sort of resembles the minimally coupled Galilean group (nonrelativistic) sub-quantal effective gauge field I was looking for in superfluid helium (which is basically a giant thermally-protected pilot-wave).

and as such is identified as a quantum fluctuation where

{p} = [p] + &p

OK , so in orthodox quantum theory this "fluctuation" must be zero because of the i.

That is, you should write

where

and

in orthodox reversible quantum mechanics which corresponds to the zero back-action regime of sub-critical superfluid flow.

Therefore, your &p "fluctuation" is the back-action term here, and if we locally gauge it, to keep the action invariant, we should get a kind of transverse roton or vortex turbulence field - which is what I was looking for in my Ph.D. research over 25 years ago. Your grad r term is, of course, not transverse, but is longitudinal. I am still; looking for the transverse Galilean roton field responsible for the breakdown of superfluidity. i.e., quantized vortex generation as a back-action effect which was the idea of my Ph.D. in 1969 from my discussion with Feynman in my Jaguar.

It is from this that I come to write Q as

Q = (&p)^2 - lap r

By doing this transformation the quantum potential is telling us much more than before, and it all stems from expressing the wave function totally according to group generators so that all variables come from the algebra rather than from a group and an algebra.

The next step ....

OK later. I have to sit back and digest this little session which was totally psychic -some might say "psycho" or "wacko" - but this stuff is beginning to sound crazy enough to be true! :-)

To, summarize, using my expression for the quantum potential Q, which has opposite relative sign between the two terms than yours

Where-(hbar^2/2m)(grad r)^2 is the square of the new non-orthodox back-action term grad r, for the imaginary momentum fluctuation, not found in Bohm's original theory at the level of ensemble statistical averages that you have come up with. That is, the expectation value of grad r must be zero in orthodox quantum mechanics though not in the new mechanics. This is the term that enforces the feedback-control loop between Q-mind and brain beable. The amazing synchronicity is that this term is the square of grad r , which, I think, is the longitudinal part of a massive back-action Galilean gauge field that has a transverse vorticity part that should show up when we locally gauge this theory. That is, I want a massive Galilean collective superfluid mode that looks like what Maxwell's field would be if the universe really was Galilean instead of locally Lorentzian. The speed of these collective modes is the speed of sound since we are in a medium with a fundamental rest frame.

The result for the Copenhagen statistical ensemble average is

Note, the first term is the expectation of the square of the momentum back-action fluctuation grad r which need not be zero in orthodox quantum mechanics even though the expectation of the back-action fluctuation itself must be zero.

What is the meaning of r?

where rho is the Born quantum probability density in configuration space.

But log rho is essentially the Shannon entropy or information I divided by Boltzmann’s constant k^3N for N point particles in translational motion. Extra degrees of freedom, like rotation and vibration of molecular structures, will increase 3. This number decreases for 2D and 1D systems. What about quantum dots?

So, in the simplest case,

showing that the quantum pilot wave really is a wave of fundamental information in the higher-dimensional configuration space of classical chaos theory.

Therefore, the full quantum potential in proper units whose gradient is directly felt by the beable, prior to the Copenhagen statistical averaging is

where hbar is Planck’s constant, k is Boltzmann’s constant, and I is the Shannon information in configuration space. This should convince David Chalmers.

To which Crowell, on Aug 10, 1996, added:

Well &p is such that <&p> = 0. It is much like a delta function correlated term in a Langevin equation. Back action, and presumably quantum chaos, is a situation where this fluctuation can promote itself. In the case of quantum chaos this fluctuation has a Lyapunov exponent associated with it and so its statistics are no longer Markovian. For back action this needs to be the case. Then you get this nonhermitian term propagating itself forward so that fluctuation at one time are correlated to those at a subsequent time. The crucial difference between back action and chaos is that with back action these fluctuations carry information that feed back on the system in a control loop. With pure chaos, certainly for the case of Hamiltonian chaos, these fluctuations are just wild.

On Aug 11, 1996

Lawrence B. Crowell wrote:

Sarfatti wrote

What is the meaning of r?

r = log R = log rho^1/2 = (1/2) log rho

where rho is the Born quantum probability density in configuration space.

But log rho is essentially the Shannon entropy or information I divided by Boltzmann’s constant k^3N for N point particles in translational motion. ...

Close, but this is an interesting idea. If you have an alphabet of characters {n} and a symbol string comes at you where any given character has a probability p = p(n) of occurence then the Shannon formula for the entropy is

S = -k SUM P(n)log(P(n)).

Yes, I was well aware that the entropy and Shannon information is actually the expectation value of the log of the probabilities. This is when there is a mixture of pure states. Here, if we have only one pure state the entropy is zero. I was speaking loosely. I should define your r as a new kind of "quantum information density field" in configuration space, motivated by the above Shannon formula. I am thinking of what David Chalmer's is looking for in his Dec 1995 Scientific American article. The Shannon formula is for "outer" objective information which is zero for a pure quantum state. This new thing based on your r is fundamental "inner" subjective "sentient" information density inside the pure state. There may be a flow between the two types of information having something to do with the black hole horizon problem.

We then have the integral of r e^2r as the total quantum information inside the pure state. So this is a new kind of "hidden information" if you like.

As I read further down Crowell's message, I saw that he had the same thought, but took it farther mathematically.

Crowell wrote:

Now let's see... We know about the Von Neumann formula for the entropy of a statistical ensemble of wave functions

S = - kTr(rho log rho),

where in this case rho = |psi> S = k int d^3x(2r)e^(2r), r = (1/2)log rho, for rho = |psi|^2

where the integration over configuration space assumes the same role as the trace. Then the differential of the entropy on some small region of R^3 space is

In general this is R^3N configuration space, and rho = |psi)(psi| is in the configuration space representation, i.e., (R^3N|psi)(psi|R'^3N).

dS = k d^3x (2r)e^(2r) = kd^3x de^(2r)/dr

Now r = r(x) so d/dr = (dx/dr)d/dx, and so

dS = kd^3x (grad r)^{-1}*grad(e^(2r)), * = dot.

Now the (grad r)^{-1} might look odd, but such thinks show up in QED. so we will let this go for now. ZAP!!

This might not be the best way to go. Why don't we instead think of e^(r + iS) in terms of a partition function. Then a partition funciton looks like

Z[psi] = int d[psi]e^(r + iS)

The the energy is

E = dlog(Z)/d(kT) = int d[psi]dr/d(KT)e^(r + iS)

What does T mean here inside a pure quantum state? Is it inverse imaginary hidden time?

and

S = (kT)E + logZ = int d[psi](r + iS) + kTdr/d(1/kT))e^(r + iS)

So what to do with this? Well the system is on an energy surface in phase space whose topology is a torus.

Why must it be a torus? Can't it have many wormhole handles in general?

So then the modes of the system are much like a quantum particle in a box. Then let us write a plausible wave function that includes these modes plus modes that pertain to an exponential separation of momentum. The fluctuation in mometum is associated with a Lyapunov exponent

Do you mean it has a Lyapunov exponent only when there is back-action beyond orthodox quantum mechanics?

l = lim(t--> inf)log(sqrt(&p^2))

Is I the total inner subjective information integrated over all configuration space for the given single pure state psi? That is, the inner information is the information of the hidden-variable beables which is absent from the orthodox Copenhagen interpretation where the wavefunction is everything. How dumb, all the big shots were and are! Those Emperors have no clothes. :-)

so &p ~e^(lt) with time. So write r = A*prod_j(j^s)F(w), so now r is related to modes tied to the Lyapunov exp. I also have written r this way since I am quite convinced we can write this as a zeta function, where the integrations of the partition function can be performed simply on the poles of zeta. To be looked at later. Then our thermodynamic quantities become

E = dlog(Z)/d(kT) = int d[psi]zeta dF(w)/d(1/KT)e^(r + iS)

and

S = (kT)E + logZ = int d[psi](r + iS + ktdF(w)/d(1/kT))e^(r + iS)

In effect F(w) is a polynomial on the "fractal mode" w, and treat w as a temperature, or some power there of. We might then have F(w) = T^3, and it this is so then the energy goes as

E ~ T^4 (Wein's law)

and the entropy as

S ~ T^3

Wait! Hold on. I just saw what this means in a flash, like Mozart seeing the whole cantata with the enchilada! :-) 2r = log rho

Is the inner information density field of the hidden-variable beables. Your Wein's law etc formulas are assuming a black body thermal equilibrium inside the outer Copenhagen pure state of the inner beables. This is like Valentini's thesis. This will be true only in orthodox quantum mechanics when the Born probability rule rho = e^2r is obeyed. Back-action means a departure from this thermal equilibrium. So at some point we need to perturb this e^2r transform. In general we need something like e^2(sum cnr^n) maybe, where orthodox QM is c1 = 1 and all the other c's = 0? This will change the Schrodinger equation??? We need something dramatically different here.

which conform to well known thermodynamic quantities. Now if this is obtained then

r ~ T^3 and S ~ T^3

which still gives something to the effect that r ~ I that you derived. so looks as if we are on the right track.

OK, well the T^3 dependence must only be for orthodox quantum mechanics on the outer Copenhagen level. You got a sub-quantum theory here. I am still not sure what T is mathematically. Back-action must be "non-thermal" in the sense you have here.

I did some of this zeta function stuff some time back, and if memory serves me the partition function is essentially Z = const*T^3. Indeed I have thought that the proof for the zeros of the Reimann zeta function might lie with quantum mechanics.

On the other hand, I may be all wrong. Maybe this inner information has nothing to do with the beables, but is a property of the pilot-wave itself in configuration space where it is a real field.

-- Jack Sarfatti, sarfatti@well.com, http://www.well.com/user/sarfatti/index.html "Perhaps I did use such a philosophy earlier, and even wrote it, but it is nonsense all the same." Einstein on Bohr's positivism to Heisenberg in 1926.

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