July 1, 2015

Demonstration of necessity of the Born Rule

The Born Rule in quantum mechanics states that probability of detection of a particle etc. is proportional to the square of the absolute value of the net wavefunction at that place and time. Despite inviting comparison to energy density being proportional to field amplitude squared, the BR is often presented as mysterious--as if it were a free parameter of nature rather than something that makes logical sense. I came up with a simple way to show that the known form of the BR is necessary, if we neglect complicated and unusual alternatives. We also reasonably assume simple additive superposition of amplitudes, basic linearity (e.g. of filter response), and that exponents must be positive or zero (to avoid the zero-amplitude crisis.)

My proof derives from the need to conserve the total number of particles transiting a Mach-Zehnder interferometer with asymmetrical beamsplitters. The total is normalized as unity. An ABS splits an incoming beam into unequal outputs. Hence a ≠ b, where a is transmitted amplitude and b is reflected amplitude. These may have different phases and thus complex values, but the proof can proceed because of the equal phases that combine for the fully constructive output from the relevant channel. This demonstration may not show universal applicability of squared moduli, but it does rule out alternatives.

We know from the BR that the corresponding intensities equal a² and b², and hence in the idealized case of no absorption used for modeling: a² +  b² = 1 . But did that have to be true, instead of say,  cubed amplitudes; such that a³ +  b³ = 1? If we simplify by considering one-term exponent laws, then consistency says "yes." (Further exploration is welcome, but the case implies any alternative would be contrived.) So, consider an MZI with asymmetrical BS at each end. The first, ABS₁, has transmitting amplitude a, and reflecting as b. Considering simple exponents (which don't have to be integers), we need aⁿ + bⁿ = 1. So far, we have no way to narrow that. These beams are recombined in ABS₂. This latter follows typical practice of outputting maximum constructive interference (no phase difference) in the lower, "A" Channel. However, it reverses transmission/reflection amplitudes relative to ABS₁. So: the originally transmitted beam is reflected at ABS₂ into Ch. A for a final output amplitude there of a². The originally reflected beam is transmitted at ABS₂ into Ch. A for a final output amplitude there of b². Superposition gives the total as a² +  b².

That already looks promising but we aren't done yet. First, we have to ensure that the output at Channel B must be zero. We can: since the Ch. A output is already the maximum output, pairing it with other than zero amplitude would be a contradiction. Suppose zero output was paired with less than the maximum possible amplitude. If so, then pairing the maximum with any value zero or over, would produce a larger total than before. But the totals must always be the same, so zero and maximum are paired. (It may seem obvious, but it's good to show the formal necessity.)

Now, we can proceed to satisfy the following equation:

(a² +  b²)ⁿ  =  1

a² +  b²  =  1⁻ⁿ  =  1

That is basically it. If the rule had been say, the amplitude itself or the cube; it could not be so that
 a³ +  b³  =  1 and a² +  b²  =  1, as well. Note: this whole argument only makes sense if we assume or accept, that there really is a number of particles output according to some rule, and not just two "branches" of arbitrary relative amplitudes. The whole idea of probability falls apart in the latter case, despite awkward attempts by MWI supporters to contrive an equivalence.

November 2, 2013

Is this an electromagnetic paradox?

Because of relative motion of sources of B (regardless of how one imagines their source), there should be an E field near the poles of magnets spinning around their polar axes. (Compare to separate magnets held in a rotating cylinder and their relative motions.) It would be a similar configuration to the B field from a solenoid (even more so if we used a hollow cylindrical magnet to isolate magnetization as close to the rim.) It seems rather obscure.

  This seems rather prosaic until you start thinking about "sources of E" in terms of what projects potential and magnetic vector potential A. The usual view of sourcing E is to write, such as from Feynamn Lectures II:
E = -Del phi - dA/dt,
and the source of phi is retarded 1/r^2 from dV of rho, and source of A is 1/r projection from current density j.

   There is a problem here
  1. Because this is a stable configuration there can't be a continuing dA/dt, and
  2. magnets aren't really current loops (most of the B comes from axial electron quantum spin, not orbital motion anyway) so there isn't really a shifting of current density to project the changed phi.
We need to treat the spinning magnet poles as actual "pole currents" since it is misleading to pretend there are actual little loops with increased relativistically altered current densities in them. Yeah, amazing this wouldn't have been hashed out before, but there it is. Your thoughts?

May 23, 2013

Discussion thread for "The Light of Paradox"

This is the post for discussing "The Light of Paradox."

Greetings, readers of Analog Science Fiction and Fact Magazine, and others interested in the unexpectedly paradoxical physics of "wigglers" and undulators. This is my blog post dedicated to discussing issues raised in my [proposed, waiting on updates] Analog article "The Light of Paradox." I presume it is acceptable to pre-publish the Abstract for the article, (which may or may not appear in the published version if Analog accepts the article.)

We discuss three paradoxes deriving from interactions in devices (such as undulators) that simulate illumination by electromagnetic radiation. The major cause of the paradoxes is the lack of actual photons striking the targets exposed to this simulated light (SL.) The first paradox develops from the problematical nature of the additional momentum correction required when energy from SL is absorbed in a compound target. The second paradox concerns the light pressure from SL differing from that exerted by ordinary light. The third paradox concerns the difficulty of accounting for all momentum and energy when SL interferes with ordinary light. Attractive solutions are not evident.

As readers here can imagine, I can't provide or send you the whole article until finding out whether my submission was accepted. Best.

My new FQXi essay is available




March 8, 2013

To not boldly go anywhere?

This piece is depressing yet very well thought out and well put. I want to believe it's better, but at least consider the apt arguments. The author could have put some attention to advanced details like ion propulsion, not just classical rocket exhaust, but this is thought-provoking:

The Recline and Fall: To not boldly go anywhere.: I wrote this some time back after reading Tom Murphy's blog. I'll publish this as is because it is still worth saying even though it...

How about this, for some simple and reasonably fair Social Security reform:

Remove the cap on earnings subject to FICA, but don't institute any "means testing" other than just making SS benefits taxable by combination with other income. That would get some of the money back from the more wealthy folks, and in a simple way (ie, by not adding a new complex formula to decide how much one is paid SS to start with), without "messing with the system" by changing overall payment protocols, or switching to the odious chained CPI, etc.

September 5, 2012

My new FQXi essay is available

My entry for the latest FQXi Essay Contest is finally in, it was delayed due to some IT issues. The Contest orienting topic was:
Questioning the Foundations
Which of Our Basic Physical Assumptions Are Wrong?

Well here it is:
Can Repeated Interactions Show More About a Photon Than Current Theory Allows?

Essay Abstract

We explore whether it is possible in principle to find the "circularity" (amount of circular polarization) of a single photon to a degree not allowed in conventional quantum theory. The thought experiment involves passing the same photon many times through a half-wave plate (with intermediate correction) so the tiny "spin" interaction of the photon is amplified enough to transfer measurable angular momentum to the detector HWP. HWPs invert coefficients for RH and LH states instead of "collapsing" the photon into a circular basis. Because passing one photon many times through a HWP should be like passing many photons once each though the plate, the transferred angular momentum would be revealed on a continuum. Such a measurement would violate the projection postulate (which says that only yes/no answers to probabilistic detection questions can be found for a single particle).


Folks, I need your votes as much as anybody else "running" for some accomplishment that depends on them. Check out the proposal, remember it doesn't have to prove the point, just to be a thought-provoking attempt and exploration. All I ask is for what you think is fair. Thanks.

August 21, 2012

Older and why-sore ...

A Facebook Friend asked me awhile ago if I felt any wiser on my birthday ("fifty-something" will suffice.) I told him:
Well, somehow both wiser and more foolish. Certainly, I am why-sore: worn out from asking "why" so much!
Heh, any of you other seekers, delvers (I love that word) and paradoxers feel why-sore? Let us know.

June 21, 2012

Dirty Higgsy says:

I know what you’re thinking: "Did we find five sigma, or only four?" Well, to tell you the truth, in all this excitement, I’ve kinda lost track myself. But being this is the LHC, the most powerful collider in the world, and would blow your mind clean off, you’ve got to ask yourself one question: "Do I feel lucky?" Well do ya, punk?

March 17, 2012

Happy St. Patrick's Day!

Happy and peaceful St. Patrick's Day , everyone, and with all due respect to the ancient culture of the Celts. (I take as my faves, SPD and Halloween!) Don't get too tipsy-turvy .
 Try this great piece of classic Irish rock:


If the clip doesn't work directly, please go to http://www.youtube.com/watch?v=ojKoTjsSks8