Half of a Coin: Negative Probabilities [pdf] | Hacker News Reader

Half of a Coin: Negative Probabilities [pdf]

wilmott.com

60 pointsjmount38 comments

As the article hints toward the end, if you extend probabilities into the domain of complex numbers, you basically get quantum mechanics.

Scott Aaronson has a well-written and thorough explanation on the matter: http://www.scottaaronson.com/democritus/lec9.html

That's an excellent piece, but he leaves out an important detail: not only can complex numbers generate negative probabilities, they can also generate negative entropies. This observation can be used to answer the riddle of how unitary transformations can produce quantum randomness. See:

http://arxiv.org/abs/quant-ph/9605002

Or for a more accessible account:

http://www.flownet.com/ron/QM.pdf

particularly section 5.

if you extend probabilities into the domain of complex numbers, you basically get quantum mechanics

To me, the most fundamental question of quantum physics is this: Why not quaternions? As a mathematician, the distinguishing characteristic of C is that it is the algebraic closure of R; but I see no physical relevance to that.

Is there not a model of QM which uses quaternions probabilities? Is there a place where it breaks down?

Aaronson actually answers this question directly towards the end of the lecture. The short answer is that quaternions don't work for the same reason that real numbers don't work: the probabilities don't work out right for unitary transformations.

You can represent two-state systems (qubits) with quaternions.

In general, quantum mechanics is about using complex vectors and unitary operations on them. As it is enough to implement quaternions, they are silently used in quantum mechanics.

likely because the probabilities in QM need to be commutative (QM operators are not commutative, but the probabilities themselves are).

I remember reading things online where people claim that the natural world (probably including what's known in quantum mechanics) is better explained with quaternions.

It's not broadly true. Quaternions are mainly useful for math on the surface of a sphere.

(You must be careful because this probabilities use the normalization \sum(p_i)=1, but in quantum mechanics you must use \sum(|p_i|^2)=1. I think that it's an important difference, because it leads to interference and other interesting phenomena.)

The idea is that all the quantum values are complex, and the measurable things are of the form f.f*=|x|^2, you have a symmetry. If you multiply all the universe by i (where i ~= \sqrt(-1) :) ) nothing changes. In general, you can multiply the universe by any complex z such that |z|=1. The global phase is irrelevant. This is the global U(1) symmetry of the universe.

But this symmetry is local. You can choose the phase of each point of the universe independently. The problem is that to compensate the arbitrary picks you have to introduce a new field that compensate for the differences of the local election of the phases. This is just the electromagnetic field. (I'm hiding a technical detail here.) This is the quantum version of the old electromagnetic field of Maxwell. The particles of this field are the photons.

I can't find an "easy" information source (for someone with a math degree), but you may start reading: http://en.wikipedia.org/wiki/Quantum_electrodynamics#Mathema...

The U(1) group has dimension 1, so you have only one "type" of photons. (This "type" is not related to the usual color of the light.) I think it's possible to extend this to the quaternions group of module 1, but it's a 3 dimensional group, so you would have ¿3? photons "types". If all the new "types" of photons are massless, this should be measurable in the experiments. (If the new "types" of photons are heavy, you need to get a bigger particle accelerator to see them, or a theoretical physicist to prove that this idea is actually impossible.)

More technical details: The arbitrary election is called gauge theory, this is a local gauge theory because. The other symmetries of the universe is the SU(3) "color" symmetry that is behind the strong force. It's the color of the quarks inside the nucleus it's not abelian so it's more complicated. The associated particles are the gluons, and there are 8 "types". The other symmetry is the SU(2) that is the cause of the weak force, and the particles are the W+, Z and W-.

Actually the SU(2) symmetry is mixed with the U(1) symmetry. So the real U(1) group create the a version of the electromagnetic field that is related to the weak-hypercharge, and the photon and Z particles are a mix of the particles of the two fields. The mathematical idea is nice, because you have a copy of U(1) inside SU(2), so you can mix both U(1) groups and mix the particles. As an introduction, you can read http://en.wikipedia.org/wiki/Electroweak_interaction#Formula... but if you don't want to dive in the technical details you can think that the U(1) group is (almost) the electromagnetism an live more happily.

As a representation theorist who vaguely knows the words, I enjoyed your post for what I thought was a nice picture of how these concepts fit together; but:

> More technical details: The arbitrary election is called gauge theory, this is a local gauge theory because.

What comes after 'because'?

Just to be clear, I'm not asking how we know quantum amplitudes are in C; I'm asking why, in the same sense as the question "why do we have 3+1 dimensions".

A little late to reply...

I should ask a specialist (In Physics, I only have a 50% major.) IIRC it's possible to use any Lie group, but they usually like simple connected groups. In particular, they prefer SU(2) to SO(3) in spite they have the same Lie algebra. (It's actually not an arbitrary preference, to describe spin 1/2 particles like the electron, proton, ... you need to use SU(2). The group SO(3) is useful only with integer spin particles.)

I read another article some time ago, cannot think how to locate it, that suggested we might have had QM far earlier if someone had seriously pondered the question of what happens if one extends P to the range [-1,1].

I'm really looking forward to physics about 20 years from now, when a lot of the current work in mathematics and complexity theory and information theory has worked its way through the physics community. Valiant's evolvability and Deutsch's information constructor work and pretty much all of Aaronson's work will illuminate bizarre new paths.

Scott Aaronson seems to go in the same direction as Eliezer Yudkowsky did with his posts about quantum physics [0]. Since all of the books I know follow the "historical" / "QWERTY" path, does anyone know/can recommend a book that goes in details along the Aaronson/Yudkowsky path?

[0] - http://lesswrong.com/lw/r5/the_quantum_physics_sequence/

Oh yay, I'm working my way through that book so I'll look forward to reading that part.

« Every other number is negative »

I don't understand this part. Other than what? Is there a translation error or something?

PS. Ok, got it. http://dictionary.reference.com/browse/every%20other

"Every other" is an english idiom to indicate alternation. I did not know it.

No error, it means every other coefficient in the p.g.f. is negative, which is like saying that certain outcomes have negative probability. In the p.g.f. for the double coin example, the coefficient of the monomial for z^0 is 1/4, meaning that the probability of two coins flipping to both tails is 1/4. If the coefficient were negative there, it doesn't make sense intuitively but that's the "magic" when it comes to these magical half coins.

I'm sorry I was not clear. It's not the negative part I don't get, it's the "every other number". Other than what? I'm not an english native speaker so it might be a math-related english expression I don't know. Is it?

"every other number" == "every second number", i.e. the first, third, fifth ... coefficients are positive, while the second, fourth, sixth .. coefficients are negative. Hope this helps.

"every other X" usually means, every even or every odd X, like:

... 0 1 0 2 0 3 0 4 0 5 ...

Every other number is 0 above.

Thanks for this link.

Many investigators have been dissatisfied with the strong assumptions that conventional measure theoretic probability requires you to make. The community that I'm most familiar with that is stretching these bounds is the imprecise probability people, http://www.sipta.org/index.php?id=cs

They have a conference every other year. The simplest generalization is to interval valued probabilities, but there are several other interesting theories.

There are some phenomena, such as a failure of long term averages of some actual physical-world stochastic processes to converge, that seem to indicate that conventional mathematical probability does not predict their behavior correctly. I.e., it is a theorem that long term averages of stationary processes should converge, and these don't.

There are some phenomena, such as a failure of long term averages of some actual physical-world stochastic processes to converge

That's a general property of levy-stable probability distributions, which occur all over the place, and you don't have to resort to unconventional probability to explain most of these processes.

No, this is different. There is a theorem, which you can find in Loeve (and other places too I'm sure), that any stationary stochastic process with finite first moment will have long-term averages that converge. IIRC, you don't even need finite second moments. In any case, it has great generality, and basically all you need is stationarity.

Presumably a physical system that has stable components in a stable environment will be stationary. You can build a simple electrical circuit of a certain type and compute its long term averages and they do not, in this case, converge. Not slow convergence, like you would see in a process with long range dependence or heavy tails, but lack of convergence, despite averaging over very long times.

I have not done is my experiment myself, but my PhD advisor, who did some of the early work on unconventional axiomatizations of probability, did.

Theory is unfortunately silent on how many observations it takes to see convergence. If you mean that there are processes that "should" converge that do not seem to empirically—yes, there are, but that is not evidence for failure of the theory.

Is this unsatisfying? Yes, it is.

levy-stable distributions with an alpha less than or equal to 1, (e.g. cauchy distribution) don't have finite first moments.

The cauchy distribution is most simply explained with the blind archer analogy. An archer is blindfolded and placed in a random orientation. (Let's restrict it to [0-pi] radians for simplicity). She then shoots the arrow in the direction she's pointing until it hits an infinitely long, straight wall. What is the position along the wall where the arrow strikes?

The CDF is the normalized arctan(x) - should be easy to visualize, but the integral of x arctan x (the first moment) is undefined.

I'm aware that there are distributions without finite first moments; that's why I specified it. But a real electrical signal cannot have infinite expected value, so that's not an explanation.

Note that your archer analogy implicitly requires infinite expected energy input.

> I.e., it is a theorem that long term averages of stationary processes should converge, and these don't.

Maybe they're non-stationary then? Doesn't sound like a failure of probability, just a failure of a particular probabilistic model.

The real answer is, we don't know yet.

There are applications of this notion to financial modeling.

The author mentions two interpretations of probability, the frequentist and the bayesian one. With both interpretations, a negative probability does not make sense. The frequentist thinks of the ratio number of occurences in the limit of an infinite number of experiments. Negative probabilities would imply negative numbers of occurences (which does not really make sense).

In the bayesian interpretation, probabilities are somewhat of a generalization of boolean logic. Based on the assumption that there is True (1) and a False (0), probabilities represent the degree of reasonable believe that a proposition is true. I.e. probabilities are bound in (0, 1).

While in math it often turns out that one should not dismiss a certain extension of a calculus (complex numbers are the textbook example, they are not immediately recognizable as "natural" but they work flawlessly and give interesting insights), I really do not get the point here, why would we need negative probabilities? What could we gain from using them?

The example brings up an example from the theory of interest, it seems that he identifies an expectancy value-like equation with negative probabilities. I wonder if this could be interpreted as a quasi-discrete case of probability-densities, which can be indeed negative.

really the link posted by another commenter here http://www.scottaaronson.com/democritus/lec9.html is worth a read

The probabilities discussed in this paper seem to have an L1 flavour (Manhattan distance) while those in quantum mechanics come from L2 functions (Pythagorean distance).

On page 67, "According to the binomial theorem," ...

What follows on the right hand side, I'm not sure where that came from.

According wikipedia: (1+x)^n = \sum_{k=0}^n {n \choose k}x^k. If you plug in 1/2 for n, you get a sum from n=0 to n=1/2, not n=0 to n=infinity as in the paper. So, this is confusing :)

He meant the Binomial series (i.e. the Taylor series about the origin of (1+x)^n. https://en.wikipedia.org/wiki/Binomial_series

On second thought, it looks what the binomial theorem could be extended such that k (in the paper this is n) goes to infinity, though for values of k greater than n the value is zero.

I'm more intuitive than mathematical. The thing feels like taking a taylor series of a square wave.

I'm not sure you should be allowed to call "probability" a number that can fall outside the [0,1] range (Or even the ]0,1[ range).

Without even reading the paper, I'm pretty sure it doesn't talk about probabilities, but about some related notion.

If you read it, yes, it is a generalization.

Interesting example of a pdf which is borked in the Firefox built-in reader, the summations ("Sigmas") are vertically stretched, this not seen in Evince for example.

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