The interesting bit is that solving the equations gives you a solution with a negative frequency. Generally those negative frequencies are ignored because for most people a negative light frequency doesn't make sense. But if you extend the analysis further you get a positive frequency light out of interactions with that negative frequency light. And the paper talks a bit about observing those second order effects:
"Here, we have shown how the same process generates a second, so-far-unnoticed peak that corresponds to resonant transfer of energy to the negative-frequency branch of the dispersion relation. "
which means the negative frequency light was something but what it means isn't clear. It could be the tip of a new way at looking at light, or it could be nothing. Some of the experiments it suggests with respect to gravity waves are interesting.
At the turn of the 19th century, physicists were quite sure that they had a complete model of the universe, with just two minor unsolved problems: The aberration of light / michelson-morley experiment (which required the theory of relativity to explain), black body radiation (which was eventually explained by quantum mechanics). Actually solving these unsolved problems turned physics on its head.
Negative frequency light might be just a weird artifact. Or it might turn physics on its head. I'd put money on the former, though I hope for the latter.
Those examples were deviations of reality from prediction/model. This is sort of the opposite -- the model predicts negative frequency light but we ignore this because it doesn't match what we thought reality was doing. Now we see that there is some tangible real-ness to these solutions. It may have interesting applications, but it wouldn't require us to change the theory (since the theory already predicts it correctly).
I love the way that math reveals ultimate truths. We consider math to be some abstract thing that we're applying to help explain reality. Instead, it often appears that the abstraction is somehow the True Reality(tm), and our reality is really just an imperfect view of Math(tm).
We view math about like we view electromagnetic waves with our eyes. We see effects of it and reflections and complex interactions of only small parts of the em spectrum.
It's interesting to wonder if we could develop a way to "see math" in its pure form.
(Sorry, Friday afternoon speculation... not currently taking any drugs.)
Math is but a medium for modeling. Any model can have predictive power.
The challenge is in determining which surprising aspects of the model have faithful correspondence to the domain and which are nonessential artifacts of the modeling medium.
The surprising thing is that the models generate predictions far beyond the domains they were designed for (and far beyond the original knowledge of the people making the models), and that the predictions are so mindbogglingly accurate that there seems to be Something Else going on.
See the Unreasonable Effectiveness of Mathematics link below.
> The surprising thing is that the models generate predictions far beyond the domains they were designed for (and far beyond the original knowledge of the people making the models), and that the predictions are so mindbogglingly accurate that there seems to be Something Else going on.
Yes -- and I can't resist citing antimatter as a perfect example. Dirac's Equation had two symmetrical results (sort of like a quadratic), and Dirac wasn't sure about the physical meaning of one of them. After antimatter was detected, Dirac was asked why he didn't simply predict antimatter himself based on a literal interpretation of his equation's implications. He replied, "Pure cowardice."
Yes, true, the term "antimatter" has always been somewhat of a misnomer. "Antimatter" differs from matter in a number of ways, but there's only one kind of gravitational mass-energy.
What happens is that we invent crazy math that is not supposed to have applicability, then some years go by and it's like, oh my God, quantum mechanics is somehow exactly all about the operation of unit Hermitian matrices... how crazy is that?? etc.
If it were some kind of model that we are able to successively refine, the progress of discovery would look something like a Taylor series, and it would be no surprise that we are eventually able to model phenomena within some tolerance epsilon.
But that is not what is happening! Rather, it's that we discover that some large and sophisticated piece of math, for which we had not thought of any particular applicability, turns out to exactly represent specific advanced physical phenomena. This happens over and over.
I'm confused by your statement that "Math's not the only modeling medium that can be Unreasonably Effective", because it's not clear to me what the notion of a "non-mathematical model" means. (That is to say, it's not clear to me that you're even correct in the weaker assertion that "Math's not the only modeling medium.") Can you explain?
Several of those are, in fact, math. (In particular, computer simulation.) The others are mostly phsyical models, which I must admit, I had not thought of. So that yields two; mathematical models (that is to say, formal models) and physical models (that is to say, real-world models). (The line can perhaps be a bit blurry in the case of e.g. digital electronic circuits, but, well, I'm not claiming there's a sharp line.)
That leaves "mythology", which I'm not convinced is a proper modeling medium at all.
But those are examples of modeling math, with a superficial layer of physical material. Another way to say it is the degree that these are useful models, is the same degree to which they reflect mathematics. With the possible exception of mythology.
Did you read the essay? Actually, both of them? The reason mathematics is considered Unreasonably Effective is because we have never seen anything with similar effectiveness. Ever.
Since math is just the formalization used to describe quantitative patterns observed, you're already seeing math as purely as it will ever get. Really, it's the other way around--since math is an abstraction of observation, what you observe with your senses day to day is a much more direct and fundamental reality. Math is a language, and reality is what it talks about. Similarly, people who speak English will find that the most effective visualization and perception of what they speak about is to just directly see and perceive the thing that they talk about.
That's why I've never thought that there's any "Unreasonable Effectiveness of Mathematics in Describing the World". Math is derived from reality and describes it insofar as it can be observed to be given a description, and so it's unsurprising that math is effective at describing what it's designed to effectively describe. It's like saying that English is unreasonably effective at talking about things.
The surprising bit is not that it explains the observations that it was modelled on, but that it explains further, unexpected observations that were unknown at that time.
It's not that maths is so great, or even that models are great, but that we were able to model some essence of a phenomenon, beyond what we witnessed.
This happens not because it's a model (you can have as many models/theories as you like), but because we selected that model, usually due to beauty/parsimony/elegance - which amounts to some version of Occam's Razor: to not make a model more complex than needed to explain the observations. (do not multiply elements unnecessarily; given different models that explain the observations, the simplest one is most likely the truest of them). http://en.wikipedia.org/wiki/Occam%27s_razor
The real puzzle (if it's a puzzle) is why Occam's Razor works so well...
This reminds me a lot of the observation that the double-slit experiment still produces interference patterns when you send particles instead of waves, and them even when you send those particles one-by-one in order to exclude the possibility of inter-particle interference.
My explanation of the effect is long a bit technical. I hope that it is intelligible.
(To keep this simple, I will ignore the phases of the waves.)
* * * Complex notation:
First, the equation for the electric field of the light is
E = A cos(kz-wt)
It's more convenient to write it as a sum of complex exponentials
E = A [exp(i(kz-wt)) + exp(i(-kz-(-w)t))] /2
E = Re( A [exp(i(kz-wt))])
By the linearity of the equations, the exp(i(kz-wt)) part and the exp(i(-kz-(-w)t)) have the same behavior, so you usually write simply
"E" = A exp(i(kz-wt))
and solve everything as if it where a complex function, but just before writing the final version or going to the laboratory you must remember that the other part was there, and that the real physical object is the real part of the function.
* * * Standard non linearity effects:
If the media is linear but not uniform, there appear other waves that travel in other direction z' (reflection/refractions). All of them have the same w.
"E_tot" = A exp(i(kz-wt)) + A' exp(i(kz'-wt))
Really all of them have two parts, one with w and the other with -w,
but usually you simply ignore that details, and only put a +cc (complex conjugate) or Re at the last minute.
If the media is no linear there can appear waves with a different frequency w'.
"E_tot" = A exp(i(kz-wt)) + A' exp(i(k'z-w't))
(There can appear more than two exponentials.)
Again they have two parts, and the real thing is the real part. It's more clear to choose w' as a positive number, because (-w') will appear in the hidden part of the equation.
E_tot = Re("E_tot")
* * * New non linearity effects in this article:
In this article they use a very sharp pulse in a very non linear material. So, from the
"E" = A exp(i(kz-wt))
part they get two new exponentials
"E_tot" = A exp(i(kz-wt)) + A' exp(i(k'z-w't)) + A'' exp(i(k_n''z-w_n''t))
where k' and w' are positive numbers as expected. But k_n'' and w_n'' are negative numbers!! They get this numbers from the same equation that has k' and w' as a solution, so all of them appear from the same mathematical term. They call this negative solution "NRR".
But it is important to remember that the original E has two exponentials, so you must repeat all the computations with the other part
"E*" = A exp(i(-kz-(-w)t))
everything is equivalent, so after some recalculations you get
"E*_tot" = A exp(i(-kz-(-w)t)) + A' exp(i(-k'z-(-w')t)) + A'' exp(i(-k_n''z-(-w_n'')t))
where every k and every w has an additional "-". They call this part "NRR* ". But now -k_n'' and -w_n'' are positive numbers. We can change the names, and call
k'' = - k_n''
w'' = - w_n''
and now k'' and w'' are positive numbers. So the first part of the solution is now
"E_tot" = A exp(i(kz-wt)) + A' exp(i(k'z-w't)) + A'' exp(i(-k''z-(-w'')t))
and the second part is
"E*_tot" = A exp(i(-kz-(-w)t)) + A' exp(i(-k'z-(-w')t)) + A'' exp(i(k''z-w''t))
And the real physical object is
E_tot = ("E_tot"+"E*_tot")/2
So we can regroup the term. We exchange the terms with k'' and w'' that have the wrong signs from on part to the other, because the sum doesn't change.
"E_totx" = A exp(i(kz-wt)) + A' exp(i(k'z-w't)) + A'' exp(i(k''z-w''t))
and the second part is
"E*_totx" = A exp(i(-kz-(-w)t)) + A' exp(i(-k'z-(-w')t)) + A'' exp(i(-k''z-(-w'')t))
and as before
E_tot = ("E_totx"+"E*_totx")/2
Now the interpretation of "E_totx" is straightforward. From the original field "E" you get three waves, with frequencies w, w' and w'', all of them positive. And in "E* _totx" is the complex conjugate part, so the final result is real.
And they can measure the three waves.
* * * Notes:
Usually, the A'' coefficient is so small that all this strange part can be ignored, but they were able to measure it in the laboratory.
One important detail is that k/w, k'/w' and k_n''/w_n''= k''/w'' are all positive, so they represent waves that travel in the same direction.
At times like this I think it's a shame that HN doesn't support TeX formatting. It's much easier to follow a mathematical argument with proper notation.
Which allow a pretty seamless interleaving of LaTeX and normal content within a page. Within a submission one need only enclose the LaTeX content in unique delimiters, like the often-used $$double-dollar-signs$$.
All this because W3C can't seem to agree on a standard way to render mathematics in the browser itself -- or more to the point, get the browser builders to agree to anything they've proposed.
This reminds me of P and N silicon; the negative frequency light is like hole current. (N silicon has "extra" electrons. P silicon has "missing" electrons that leave "holes", and you can have current where it's the holes that are moving.)
I still don't get what a negative frequency is. The frequency is the number of times something happens in a given amount of time. Would a negative frequency be the number of events that were expected to happen but didn't? If so, how could this be differentiated from an error in our expectations.
Physics often uses the sign of a frequency to indicate the direction of travel with regard to a reference frame.
E.g. 2 Hz is a wave oscillating two times per second and travelling to the right, -2 Hz is the same travelling to the left.
In some cases direction isn't really meaningful (standing waves, or the height or pressure of a medium as seen by a stationary observer). In others, the direction of propagation can be quite significant.
For me, this is easiest to see through the equation Velocity = Frequency x Wavelength. Negative frequency is algebraically equivalent to negative wave velocity (which already means opposite direction), or negative wavelength (which already means opposite direction). There are several different ways that you could try to interpret "negative frequency," but by shuffling the negative sign around it becomes clear it must be equivalent to a wave of the same absolute frequency, where the wavelengths point the other way.
Is a -2 Hz wave travelling to the right the same as a 2 Hz save travelling to the right, but going backwards in time?
Could there be something to John G. Cramer's transactional interpretation of Quantum Mechanics? Instead of probability waves collapsing instantaneously, could everything be mediated by photons going backwards in time?
"Is a -2 Hz wave travelling to the right the same as a 2 Hz save travelling to the right, but going backwards in time?"
Yes, all other things being equal. It's a bit less confusing if you leave the signs off and just say that "a wave travelling to the right at f in t is the same as a wave travelling to the left at f in -t.
This is also very much in the "spherical cow" sense of physics - it's true of many ideal cases on chalkboards, but it gets fuzzy in practice due to thermodynamics and other nonidealities. Plus, you can only observe things from the perspective of time moving forwards.
"Could there be something to John G. Cramer's transactional interpretation of Quantum Mechanics?"
There's nothing I'm aware of which is implicitly wrong with a transactional interpretation - you just have no way to empirically prove that it's a valid "why." In physics, we are very often limited to "how" and "what."
It looks to have some ideas in common with Feynman's reaction diagrams and path integrals, so it may be an interesting approach if you're partial to those. The catch is that wavefunction collapse is equally true: it's a model which makes accurate predictions about observable experimental outcomes.
They found a solution with a negative w and negative k, so it travels forward. They "neglect" the solutions that travel backward (with negative w and positive k, or positive w and negative k). The more clear parts about this is the Fig 1(a) (page 2) and the paragraph below it.
The problem with such interpretations is that they violate causality.
Check out Wheeler-Feynman absorber theory for history.
Concepts such as collapse of a wave function aren't necessary when you stop treating the observer in a special way and start considering observer-particle as a big quantum system.
So is this saying that a portion of the waves of energy are reflected back towards the source, that this energy has a negative frequency, and that it has an effect that has been elusive because it has little chance to interact with the positive frequency waves? I'm curious what some of the implications might be.
This is not true. A wave propagates both in time and space. You're mixing these two. A negative frequency is like a wave travelling backwards in time, not left/right spatial direction.
Not quite. A wave going left forward in time is equivalent to a wave 'going right' but backwards in time. Since it is further right in the past than now, it is actually going left.
There's a lot of similar effects in Physics, especially with time and/or antimatter.
Save me the trouble an learn some basic physics.
Read Jackson's Classical Electrodynamics, Chapter 7.
If it's too complicated for you, read about waves. See how they propagate in time and space, and pay special attention to what happens to the equation when you change the sign of t. If it's too difficult to imagine, try plotting using a program.
And anti-matter is not matter travelling backwards in time. See my post below.
Are we talking about the same thing?
I'm talking about particle waves, or wave functions of particles, not a function which satisfies the wave equation (a photon happens to satisfy the wave function, but I'm trying to be general here) and with frequency, I mean energy. Try changing the sign of t in Schrodinger equation and see if it simply amounts to changing the sign of your momentum vector or not.
I guess not. I had an impression of you talking about particles from the start of your comment, but then you went into "read about waves" and said to make a plot, which made it sound like you were talking about basic math. I only understand a few of the effects of altering time on physics so I'll shut up.
Which is exactly why the book has that title. The book by Jackson is required reading for any physics graduate in a field where EM radiation plays a role.
I have seen a negative sign grouped with frequency. Which was used to distinguish directionality. Most of the time it is understood from context what is meant, if not the speakers fall back on to more something more formal to ensure understanding.
Don't push the idea of a wave in space too much, that's not how stuff works in quantum mechanics. See Feynman Lectures of Physics, Vol 3, Section 3-3 for a text at the introductory level.
How exactly would light at "negative green" frequency be perceived by an eye/camera, when mixed with an equal amount of regular green? Would it cancel out, would the negative light have no effect, or would the powers add constructively?
The article doesn't indicate what the physical manifestation of negative-frequency wave would be. It only reports the observation of a positive-frequency wave that seems to require the transient existence of a negative-frequency intermediate.
This is similar to how particles like the Higgs boson are not directly detected in particle accelerators, but are inferred from the directly-observable particles that they produce. The difference is that physicists are (I think) pretty sure that the Higgs actually physically exists, if only for a short time, while the same is not clear for the negative-frequency intermediate wave described here.
You don't need anything fancy like negative frequency light bulb for that, only a regular light bulb that emits light 180 degree phase shifted to the already existing light in the room. It's actually impossible to do with a light bulb because the light that emits is not coherent but you can do it with a laser if the incoming light is also from a laser, in fact negative and positive laser interference is the basis for several technologies such as holograms.
"Here, we have shown how the same process generates a second, so-far-unnoticed peak that corresponds to resonant transfer of energy to the negative-frequency branch of the dispersion relation. "
which means the negative frequency light was something but what it means isn't clear. It could be the tip of a new way at looking at light, or it could be nothing. Some of the experiments it suggests with respect to gravity waves are interesting.