Complex Colors

[Surlethe]

Simon, I think it's not so clear-cut. We can posit, for example, that mass be permitted to take complex values instead of positive real values, assume conservation laws hold (with appropriate modifications), and see what happens to dynamical laws.

By the way, it's not so easy to get 2 and 2 to be 5, but stranger things are mathematically consistent --- 2 + 3 = 7 (mod 2).

[Kuroneko]

I have trouble imagining what 23+5i hydrogen atoms might mean.

I think the correct interpretation is that the complex wavefunction of the electron is a meaningful physical quantity, not an abstraction, and that potential is more physically meaningful than a field.

I don't think the Aharomov-Bohm effect adds much to the question of phase, since it is sensitive to differences in phase only.

I'm not following here. Do you mean multiplication?

My guess is that he's referring to both multiplication and conjugation, since the complex numbers can be constructed as pairs of reals equipped with an involution operation, and both of these operations introduce important algebraic structure. Although in terms of 2-vectors, there are corresponding operations as well, and U(1) is isomorphic SO(2) anyway.

No physical quantity is ever complex.
...
Similarly, unstable particles having complex mass is only a mathematical formalism. The actual mass of those particles - meaning the traditional concept of mass and not some abstract quality - is still a real value and the imaginary component just represents the decay rate.

Complex fields represent electric charge naturally, and while they can be decomposed into two real fields if one wishes, I'm can't see how this is useful either for them or for the cases of fields (rather than particles) with imaginary or complex mass. Actually, if it can always be so decomposed in principle, it makes the policy of insisting on real quantities even more dubious--if they're mathematically and empirically equivalent, what does it even mean to say that one represents reality and the other does not?

What I really had in mind was that observables (and I'm using that word in its physics context) are always real-valued - I can provide references for this if someone asks.

I think that's misleading. Every operator that commutes with its adjoint (normal operator) has all the important properties of observables except having real eigenvalues: diagonalizable by a unitary transformation, being the sum of eigenvalues by the projection operators to the corresponding orthogonal eigenspaces, the Heisenberg uncertainty principle involving the commutator of two operators, etc. The only advantage of having Hermitian operators is that they're a useful generalization of self-adjoint operators, but only slightly so, since the esential issue is whether or not the operator and its adjoint have the same domain. If we define an observable as any operator of the form A = B+iC where B,C are commuting Hermitian operators, over the intersection of their domains, then this covers both Hermitian and normal operators at once, and hence covers observables with arbitrary complex eigenvalues.

To be fair, there's little practical reason to do so, since commuting B,C means they're simultaneously measurable anyway, so it's a bit simpler to only implicitly include normal operators "without loss of generality." But the point is that there's nothing in the formalism of QM that actually requires real eigenvalues--normal operators share the essential formal properties--and we simply define them to be real-valued without any intrinsic reason to do so, for convenience's sake only.

A quantum wavefunction is not an observable, whereas the the square of the modulus of the wavefunction is an observable, and is also always real-valued.

By the Born rule, the square of the modulus is the probability density ρ≥0 for the observable, not the observable itself. But if you count that, what's wrong with the rest of the wavefunction--the phase? Given a particle with mass m with position-space wavefunction ψ(x,t) = ρ1/2 exp(iS/ℏ), is the probability density for position measurements. By the Schrödinger equation, the probability current is (ρ/m)∇S, as so also has physical meaning up to an overall unobservable additive constant. Though really, the modulus-squared ρ has a similar ambiguity, since it's only by convention that we normalize 〈ψ|ψ〉 = 1, and it is no worse than potentials.

[Baffalo]

Perhaps we're supposing this in a manner that is way more complicated than it needs to be. I propose you consider the complex number to be a functional additive to deal with the non-observable but none the less known quantity of dark matter. For example, the sun is estimated to have a mass of 2x1030 kg. We know the approximate mass of the Earth to be 6x1024 kg, so what does it mean if the models we generated appeared to be off? Would that mean we miscalculated the mass of the Earth and Sun? If the real numbers didn't add up, then we would need to add a component to make the math work correctly (which we do already in overly complex situations to make the math overall simpler).

So in this case, suppose the Earth and Sun interaction is off, and all the numbers refuse to answer the question. It's not the mass of Earth that's off because the other planets aren't reacting to Earth's mass the same as they would if it were the heavy body, so it must be the sun. Nothing can be shown to account for the discrepancy in mass, not even a heavy core or something explainable like that, then the mass must come from somewhere. If it were dark matter adding appreciable mass without an increase in volume or density, we would have to describe it's effects with a complex number, even if it's a simple addition tacked onto the end of all mass equations denoting the presence of dark matter. So if the sun had an additional 1x1030 kg in mass, we would write the mass as (2+i)x1030 kg.

[Purple]

But if you did that, would you not be just using the complex number notation for something completely different as opposed to actually having mass as a complex value? That IMHO would be missing the point. The idea I want is a universe where both mass and quantity are actual complex numbers and observables do behave as such. To give an example you could quite literally measure, see and touch 1+2i main battle tanks.

[Simon_Jester]

But what does i bricks, or buckets of paint, or whatever mean?

We define i in terms of its square: i squared equals negative (one squared). But you can't just arbitrarily 'square' a tank or a man or a pile of cotton candy.

[Purple]

But what does i bricks, or buckets of paint, or whatever mean?

We define i in terms of its square: i squared equals negative (one squared). But you can't just arbitrarily 'square' a tank or a man or a pile of cotton candy.

Actually you can. It just takes nature behaving very, very strangely. In particular, physics needs to modify its behavior to create a special case for the quantities of "mass" and "quantity". It further requires nature to understand the difference between squaring and addition. Don't ask me how but say that it does.

For example, say you have four battle tanks. You subtract one, and you have three. Subtract another one and you have two. Normal nature here. However, if you take the aforementioned four battle tanks and subtract two, two squared is four. And because the quantity we are modifying is one of the special case values that means nature in our case will with equal probability produce either 2 battle tanks or 2i battle tanks. Now you might ask how would we even measure that. After all since you can't physically measure, see or feel i. You can however reverse the process to see what you got. Namely, squaring works not only in reverse but forward as well. If you do what I did back there you would get either 2 or 2i battle tanks. But you can't really tell what you got. However you can now add two battle tanks to them thus "squaring" their "amount". And since your "two" battle tanks are either actually 2 or 2i (you can't tell) your end result can be any of these: "2 x 2 = 4, 2 x 2i = 4i, 2i x 2i = 4 x -1 = -4 (your tanks suddenly vanish)" So you can actually devise an experiment where "squaring" a "special quantity" provides you with a result that tells you what quantity it was.

Is it crazy? Yes sure. But that's the fun bit.

[Magis]

No physical quantity is ever complex.
...
Similarly, unstable particles having complex mass is only a mathematical formalism. The actual mass of those particles - meaning the traditional concept of mass and not some abstract quality - is still a real value and the imaginary component just represents the decay rate.

Complex fields represent electric charge naturally, and while they can be decomposed into two real fields if one wishes, I'm can't see how this is useful either for them or for the cases of fields (rather than particles) with imaginary or complex mass. Actually, if it can always be so decomposed in principle, it makes the policy of insisting on real quantities even more dubious--if they're mathematically and empirically equivalent, what does it even mean to say that one represents reality and the other does not?[/quo

What I really had in mind was that observables (and I'm using that word in its physics context) are always real-valued - I can provide references for this if someone asks.

I think that's misleading. Every operator that commutes with its adjoint (normal operator) has all the important properties of observables except having real eigenvalues: diagonalizable by a unitary transformation, being the sum of eigenvalues by the projection operators to the corresponding orthogonal eigenspaces, the Heisenberg uncertainty principle involving the commutator of two operators, etc. The only advantage of having Hermitian operators is that they're a useful generalization of self-adjoint operators, but only slightly so, since the esential issue is whether or not the operator and its adjoint have the same domain. If we define an observable as any operator of the form A = B+iC where B,C are commuting Hermitian operators, over the intersection of their domains, then this covers both Hermitian and normal operators at once, and hence covers observables with arbitrary complex eigenvalues.

To be fair, there's little practical reason to do so, since commuting B,C means they're simultaneously measurable anyway, so it's a bit simpler to only implicitly include normal operators "without loss of generality." But the point is that there's nothing in the formalism of QM that actually requires real eigenvalues--normal operators share the essential formal properties--and we simply define them to be real-valued without any intrinsic reason to do so, for convenience's sake only.

A quantum wavefunction is not an observable, whereas the the square of the modulus of the wavefunction is an observable, and is also always real-valued.

By the Born rule, the square of the modulus is the probability density ρ≥0 for the observable, not the observable itself. But if you count that, what's wrong with the rest of the wavefunction--the phase? Given a particle with mass m with position-space wavefunction ψ(x,t) = ρ1/2 exp(iS/ℏ), is the probability density for position measurements. By the Schrödinger equation, the probability current is (ρ/m)∇S, as so also has physical meaning up to an overall unobservable additive constant. Though really, the modulus-squared ρ has a similar ambiguity, since it's only by convention that we normalize 〈ψ|ψ〉 = 1, and it is no worse than potentials.

Okay.... so are you claiming that observables can have imaginary components or not?

Talking about "this operator can do this, and that operator can do that" I don't think has any relevancy to, you know, science. More generally, whether a "physical" quantity can be imaginary is an interesting thought experiment as I've said before, but remember that in science every theory or model at some point needs to be tested against measurements, and measurements are real-valued. If some equation involves complex numbers, and that equation is useful in solving problems, then great. But just because something works on paper (heck, you can insert an imaginary number into any equation you want), that doesn't mean that is must therefore have some real actual world meaning or significance.

Anyway, I don't actually know what you're claiming in your post (or if you're claiming anything at all).

"Complex roots cannot represent physically measurable quantities and must be disregarded if we are solving for a physically meaningful quantity." - R. G. Mortimer, Mathematics for Physical Chemistry, Academic Press, 2006, page 58.

"Negative numbers and imaginary numbers have no direct physical presence in the real word, yet both serve an essential role in solving problems about the real world." - Henle and Kleinberg, Infinitesimal Calculus, Courier Dover Publications, 2003, page 4.

"How is it possible to use a mathematics that has no connection with our everyday experience? ... we change the physical problem into a complex number form, manipulate the complex numbers, and then change back into a physical answer." - S. Smith, Digital Signal Processing, Elsevier, 2003, page 558.

"Although real numbers quantify physical quantities, complex numbers provide very convenient representations of many physical phenomena. In quantum mechanics, the wave function is a complex function. Two-dimensional, incompressible, irrotational flows are represented by a complex flow potential." L. Albright, Albright's Chemical Engineering Handbook, CRC Press, 2008, page 143.

[Kuroneko]

What I really had in mind was that observables (and I'm using that word in its physics context) are always real-valued

Okay.... so are you claiming that observables can have imaginary components or not?

Talking about "this operator can do this, and that operator can do that" I don't think has any relevancy to, you know, science.

You don't think quantum mechanics has any relevancy to science? That's a very curious point of view. Actually, I though you were talking about quantum mechanics, since that's the one place where the term 'observable' is formally defined as a certain kind of operator, which happens to have real eigenvalues, while measurement gives one of the eigenvalues and leaves the state in the corresponding eigenspace (that's what state/wavefunction collapse refers to).

I am claiming that the definition of physical observables as necessarily real-valued is purely conventional and for done for convenience's sake, and absolutely nothing of physical importance breaks if complex-valued observables are included. I am also saying that a complex-valued measurement is going to be formally equivalent to a simultaneous measurement of two commuting real-valued observables (though whether it is in practice is another matter), but this equivalence also means putting labels to one as representing actual reality and the other as not is simply not science. How would you even begin to try to falsify that kind of claim? It's metaphysics, and not even interesting metaphysics.

"Complex roots cannot represent physically measurable quantities and must be disregarded if we are solving for a physically meaningful quantity." - R. G. Mortimer, Mathematics for Physical Chemistry, Academic Press, 2006, page 58.

Why would the author say this? In this age, it's easy too look up most of those books in seconds to see what exactly they're talking about, and in the same paragraph two sentences prior: "For most [polynomial] equations arising from physical and chemical problems, ..." Not only is it qualified to be completely correct but irrelevant to this conversation, but the subject matter in the section has no relevance here to begin with--

"Negative numbers and imaginary numbers have no direct physical presence in the real word, yet both serve an essential role in solving problems about the real world." - Henle and Kleinberg, Infinitesimal Calculus, Courier Dover Publications, 2003, page 4.

That looks like complete and utter nonsense until one realizes that the context of the authors' statement was likely to be historical (since the whole section is about the historical development of calculus) and also more constrained than your claims are. And if we're going to learn anything from history that's relevant to our argument, it's that many such statements of physical irrelevance turned out to be very, very wrong.

"How is it possible to use a mathematics that has no connection with our everyday experience? ... we change the physical problem into a complex number form, manipulate the complex numbers, and then change back into a physical answer." - S. Smith, Digital Signal Processing, Elsevier, 2003, page 558.

Once again, you are taking statements which make sense in the context of DSP, i.e., of encoding a real signal into complex form, and applying it to places where it simply doesn't go. Before quantum mechanics, it was reasonable to regard complex numbers as nothing more than clever book-keeping, but anyone who seriously believes that today should get with the twentieth century.

Because in QM, wavefunctions are necessarilly complex-valued. If they were not, and were just repackaging of a real-valued wave, they would simply fail, not just empirically, but even in representing momentum sensibly: just as you pointed out previously that the complex magnitude has actual empirical meaning, a probability density, the complex phase has an essential role as the probability flux, and hence the momentum of the wave. The de Broglie relationship p = ℏk is fundamental to quantum mechanics and it simply cannot be realized with a single real wave.

I qualify "single" above, but it doesn't really matter if you choose to write things like as a complex number or as rectangular some components (a,b) or polar (r,φ) or, like Feynman, draw strange little spinning arrows on a blackboard--if you wish to calculate probabilities correctly, you will have to respect the algebraic rules of complex numbers. Complex numbers are that structure; everything else is just notation. Here, Starglider's point cuts exactly to the heart of the matter.

"Although real numbers quantify physical quantities, complex numbers provide very convenient representations of many physical phenomena. In quantum mechanics, the wave function is a complex function. Two-dimensional, incompressible, irrotational flows are represented by a complex flow potential." L. Albright, Albright's Chemical Engineering Handbook, CRC Press, 2008, page 143.

See above.

[Simon_Jester]

But what does i bricks, or buckets of paint, or whatever mean?

We define i in terms of its square: i squared equals negative (one squared). But you can't just arbitrarily 'square' a tank or a man or a pile of cotton candy.

Actually you can. It just takes nature behaving very, very strangely. In particular, physics needs to modify its behavior to create a special case for the quantities of "mass" and "quantity". It further requires nature to understand the difference between squaring and addition. Don't ask me how but say that it does.

So what is this 'squaring' operator? Seriously, what IS it, what would be the effect in real life? Can you 'square' a group of three men and suddenly (poof) a group of nine men appears? Can you 'square' a group of three imaginary men and not only do THEY disappear, but so do nine other entirely unrelated people? Does it spontaneously ignore conservation of energy?

The first problem here is that you're taking arithmetic and assuming that real objects can be manipulated by all the same methods as integers, even when some of the manipulations aren't physically meaningful. A square of nine men is not "three men, squared." It's just nine men. A nine-foot pole is not "a three-foot pole, squared." It's just a nine-foot pole.

Also, you're saying that an 'imaginary' object is indistinguishable from a real one UNTIL we apply this mysterious 'squaring' operator. That's not how complex numbers work; there is a concrete difference between 2 and 2i. If we worry only about magnitudes we can't tell them apart... but in that case we can't tell 4 and -4 apart either.

Is it crazy? Yes sure. But that's the fun bit.

There's no fun for me because I can't visualize how the hell this is supposed to work. And I don't mean "can't visualize it physically." I mean I can't visualize how the accounting works.

Heck, I can't even visualize accounting in this environment, because you can't square dollar bills. We'd just end up with two different, non-convertible forms of currency: Dollars and iDollars.

[Purple]

So what is this 'squaring' operator?

Physical contact between objects or groups of objects who share both of the "special" properties. In other words, when the appropriate "quantity" of objects with identical "mass" touch.

Seriously, what IS it, what would be the effect in real life? Can you 'square' a group of three men and suddenly (poof) a group of nine men appears?

No, of course not. However if three groups of three men (each group having the men hold hands already) suddenly come along and held hands, and each group has the same total mass as well interesting things start happening.

Can you 'square' a group of three imaginary men and not only do THEY disappear, but so do nine other entirely unrelated people? Does it spontaneously ignore conservation of energy?

By introducing negative reals. Basically in my battle tank example there is indeed a case where 4 tanks just disappear. Except that they don't really disappear. They still exist as negative value tanks. That is to say, they do not physically exist but the place they are in becomes a negative value and if you park further battle tanks on that exact spot they will vanish until the numbers even out at 0. And yes, conservation of mass and energy would likely have to undergo some fun transformations for this to work, like say accepting that for their purpose -1 = +1. But that is why it is alt-physics.

The first problem here is that you're taking arithmetic and assuming that real objects can be manipulated by all the same methods as integers, even when some of the manipulations aren't physically meaningful. A square of nine men is not "three men, squared." It's just nine men. A nine-foot pole is not "a three-foot pole, squared." It's just a nine-foot pole.

That's not a problem, it's an assumption. It's the unrealistic bit that we accept as axiomous in order to start branching off our analysis of the alternative end result.

Also, you're saying that an 'imaginary' object is indistinguishable from a real one UNTIL we apply this mysterious 'squaring' operator. That's not how complex numbers work; there is a concrete difference between 2 and 2i. If we worry only about magnitudes we can't tell them apart... but in that case we can't tell 4 and -4 apart either.

The idea here is not that they are indistinguishable but that our "real" senses and "real" instruments have no way of distinguishing them. In fact, there probably would be some way of doing so according to theoretical physics of the universe in question but most likely the only way the common man could tell them apart would be by conducting experiments on them.

The concept being that in this particular alt-physics scenario there is always uncertainty as to what happens when you "square" objects or even when you do anything with them. For all you know, that shopping basket might have -3 bread in it. And you can't know that until you start putting bread inside and realize you have to pay for 4 pieces and not the 1 you wanted. But at the time you realize that it's too late.

Heck, I can't even visualize accounting in this environment, because you can't square dollar bills. We'd just end up with two different, non-convertible forms of currency: Dollars and iDollars.

You can't square them but you can add them up. The only thing that changes is the number of pieces you can add to a pile at any given time. After all, as long as the amount added to the physically pile of bills is newer equal to the size of said pile (can be larger or smaller) you are fine. And sure, you might lose some bills on the 1's but that is the fun part.

[Magis]

You don't think quantum mechanics has any relevancy to science? That's a very curious point of view.

What I'm saying is that mathematical models have to conform to measurements, not the other way around.

I am claiming that the definition of physical observables as necessarily real-valued is purely conventional and for done for convenience's sake

So I said that observables are always real-valued and you're saying that it's only the case because observables are real by definition. So I guess the implication here is that you agree with me.

And naturally the quotes I provided were always in the context of their particular areas of inquiry, i.e. the chemistry book quotes were in the context of chemistry and so on. But the point is that there is a wide and general consensus that measurable quantities are always real-valued because such quantities having an imaginary components makes no sense and has no physical meaning. I.E. a particle having imaginary proper mass is nonsensical, and a pigment existing in a mixture at imaginary concentrations is also nonsensical.

[Simon_Jester]

Physical contact between objects or groups of objects who share both of the "special" properties. In other words, when the appropriate "quantity" of objects with identical "mass" touch.

So if I get two bricks and bang them together, they "square" to form one brick?

What happens if I use half bricks? Does banging two half-bricks together create 0.5*0.5 equals one quarter of a brick?

And if I get two imaginary bricks and bang them together, they both disappear and create negative one brick? What does that mean, the nearest normal brick spontaneously disappears on its own?

And yes, conservation of mass and energy would likely have to undergo some fun transformations for this to work, like say accepting that for their purpose -1 = +1. But that is why it is alt-physics.

That's not alt-physics, it's not physics at all. If -1=1, then 0=2, which means 0=(any other number). There are no conservation laws, things could just arbitrarily pop out of nowhere, or vanish into nowhere, and that would be just as plausible as things staying the same.

For that matter, if -1=1, then (i squared) = (-i squared) = 1 and there's no such thing as complex numbers, either. One and i become the same number.

I know in theory there's no such thing as a stupid question, but... I'm having a hard time thinking of anything to call this except "a stupid question."

That's not a problem, it's an assumption. It's the unrealistic bit that we accept as axiomous in order to start branching off our analysis of the alternative end result.

You're inserting random, arbitrary axioms into an existing system of mathematics, without worrying about what happens when your axioms contradict the existing ones. That just makes the question meaningless.

Do you get what I'm saying here? You're trying to preserve the definitions of words (what we mean when we say "X of something") while making assumptions that break the axioms that went into the definitions.

[Purple]

Physical contact between objects or groups of objects who share both of the "special" properties. In other words, when the appropriate "quantity" of objects with identical "mass" touch.

So if I get two bricks and bang them together, they "square" to form one brick?

Yes, sort of like that. The act of slapping them together has an effect on them that behaves like math that should not be there. In particular we can assume that you did not examined the bricks before hand and don't know if they are really 1-Bricks or 1i-Bricks. In that case you you have 1/4 {(1,1),(1,1i)(1i,1)(1i,1i)} of both of them actually being 1i bricks instead of 1 bricks and them vanishing and creating a -1 brick and a 3/4 chance of just getting two bricks touching each other with no noticeable effect.

One thing to remember is that for all intents and purposes a 1-object and a 1i-object would behave identically to one another. The only difference would be that they produce the above described effect under the above described circumstances.

What happens if I use half bricks? Does banging two half-bricks together create 0.5*0.5 equals one quarter of a brick?

Not quite. A half brick is not 1/2 of a brick. It is just a smaller brick. So touching two half bricks of the same mass together would produce the same effect as touching any other two discrete objects together.

And if I get two imaginary bricks and bang them together, they both disappear and create negative one brick? What does that mean, the nearest normal brick spontaneously disappears on its own?

Actually it's the next brick that occupies the same space.

And yes, conservation of mass and energy would likely have to undergo some fun transformations for this to work, like say accepting that for their purpose -1 = +1. But that is why it is alt-physics.

That's not alt-physics, it's not physics at all. If -1=1, then 0=2, which means 0=(any other number). There are no conservation laws, things could just arbitrarily pop out of nowhere, or vanish into nowhere, and that would be just as plausible as things staying the same.

For that matter, if -1=1, then (i squared) = (-i squared) = 1 and there's no such thing as complex numbers, either. One and i become the same number.

Perhaps I explained it badly. In essence, conservation requires that there be a finite amount of mass and energy in the universe. How that amount is achieved is irrelevant. So for example 3 battle tanks can be achieved by having 3 regular battle tanks, or by having 3 (-6)-battle tanks and 9 regular battle tanks total. So conservation would have to expand into including all the various combinations.

I know in theory there's no such thing as a stupid question, but... I'm having a hard time thinking of anything to call this except "a stupid question."

I prefer the term insane.

You're inserting random, arbitrary axioms into an existing system of mathematics, without worrying about what happens when your axioms contradict the existing ones. That just makes the question meaningless.

No, I am inserting random, arbitrary axioms into an existing system of physics and asking you guys to see what happens with the rest of the system.

Do you get what I'm saying here? You're trying to preserve the definitions of words (what we mean when we say "X of something") while making assumptions that break the axioms that went into the definitions.

That is kind of the point. By doing that I am attempting to prompt you to back engineer new axioms that will work with the changed outcome. It's no different than trying to figure out what kind of weapon a phaser is by looking at the burn marks.

[Simon_Jester]

So if I get two bricks and bang them together, they "square" to form one brick?

Yes, sort of like that. The act of slapping them together has an effect on them that behaves like math that should not be there. In particular we can assume that you did not examined the bricks before hand and don't know if they are really 1-Bricks or 1i-Bricks. In that case you you have 1/4 {(1,1),(1,1i)(1i,1)(1i,1i)} of both of them actually being 1i bricks instead of 1 bricks and them vanishing and creating a -1 brick and a 3/4 chance of just getting two bricks touching each other with no noticeable effect.

But by that logic- smacking things together is a multiplication operator.

So smacking 1 brick and i brick together should cause them to fuse into i brick.

What happens if I use half bricks? Does banging two half-bricks together create 0.5*0.5 equals one quarter of a brick?

Not quite. A half brick is not 1/2 of a brick. It is just a smaller brick. So touching two half bricks of the same mass together would produce the same effect as touching any other two discrete objects together.

But then what's a discrete object? Bricks are made of atoms.

Suppose I smack together two lumps of wet sand. Are the lumps discrete? The individual sand grains? The atoms? Which item gets multiplied?

And if I get two imaginary bricks and bang them together, they both disappear and create negative one brick? What does that mean, the nearest normal brick spontaneously disappears on its own?

Actually it's the next brick that occupies the same space.

Can I pick up this negative brick? Can I interact with it? Can anything except another brick interact with it?

What about a block of granite, which contains some of the same atoms as the brick but not in the same ratios or configuration? What about a lump of clay, with exactly the same composition, but with the atoms arranged differently?

And yes, conservation of mass and energy would likely have to undergo some fun transformations for this to work, like say accepting that for their purpose -1 = +1. But that is why it is alt-physics.

That's not alt-physics, it's not physics at all. If -1=1, then 0=2, which means 0=(any other number). There are no conservation laws, things could just arbitrarily pop out of nowhere, or vanish into nowhere, and that would be just as plausible as things staying the same.

For that matter, if -1=1, then (i squared) = (-i squared) = 1 and there's no such thing as complex numbers, either. One and i become the same number.

Perhaps I explained it badly. In essence, conservation requires that there be a finite amount of mass and energy in the universe. How that amount is achieved is irrelevant. So for example 3 battle tanks can be achieved by having 3 regular battle tanks, or by having 3 (-6)-battle tanks and 9 regular battle tanks total. So conservation would have to expand into including all the various combinations.

But that means that any one of those combinations can turn into another.

The conservation laws don't stop me from rearranging things. So in this case, there'd be nothing stopping me from 'spawning' a brick and a negabrick. Conservation laws are satisfied just as well... but where the hell did those bricks come from? There are different real-world consequences in your system; it makes a difference whether I have (3-1) bricks or (303-301) bricks. Those extra bricks will be heavy and take up space and so on.

Plus, you're also saying that we can turn two bricks into one at will by banging them into each other. Or negative one. Either way, there is no conservation law, because there is no conserved quantity. You didn't change the rule, you erased it.

I know in theory there's no such thing as a stupid question, but... I'm having a hard time thinking of anything to call this except "a stupid question."

I prefer the term insane.

No, I think that would be giving it too much credit. Basically, what you're doing is asking ungrammatical questions and being smug when you're misunderstood. The only thing that's different is that the language in which you're being ungrammatical in is math.

If I asked you "monkey fruit loop, Purple hop leftly banana bread?" You would probably not be able to give a coherent answer. For the same reason, I can't give you a coherent answer to "what would happen if 1=0?"

You want to know what happens to the rest of the system? Nothing. There is no system, any more than there can be structures without building materials. There is no logical structure that can encompass "true equals false" or "something is the same as nothing, which is also the same as any other entirely different something." Because that isn't logical and doesn't have any structure to begin with.

Meanwhile, the other guys are having fun with a meaningful discussion of the physical implications of complex numbers... largely by ignoring you.

[Kuroneko]

What I'm saying is that mathematical models have to conform to measurements, not the other way around.

That's both true and yet irrelevant to our discussion, because that's not in dispute.

So I said that observables are always real-valued and you're saying that it's only the case because observables are real by definition. So I guess the implication here is that you agree with me.

No. Observables are commonly defined as real-valued for convenience's sake only, not because that is a necessary condition of measurements. Extending the formalism of quantum mechanics to include complex-valued measurements is very easy. It's seldon needed to be done explicitly, and so it's more common not to do it.

And naturally the quotes I provided were always in the context of their particular areas of inquiry, i.e. the chemistry book quotes were in the context of chemistry and so on. But the point is that there is a wide and general consensus that measurable quantities are always real-valued because such quantities having an imaginary components makes no sense and has no physical meaning.

You have a collection of quotes from people who may be eminently qualified to speak of the uses of complex numbers in their field, but they are not so qualified regarding all of science (and at most if not all of them didn't even make such all-encompassing claims to begin with, taken in context). This would be decent evidence provided that there was no exhibition of a field of science where complex-valued measurements not only make sense but are perhaps even more fundamental than real-valued ones. I gave exhibited such an example of such a field of science, and some uses complex values in it, as well as reasons .

So it's simply fallacious to fall back on those quotes, because at best they don't represent all of science, and at worst they're just taken out of context or outright mistaken. If you really don't wish to actually discuss any reasons for conceptually allowing or disallowing complex-valued measurements or their use (and you haven't given any at all!), alright, I'll indulge you in quote mining chemists instead:

"From a modern point of view, the essential representative of an observable is a Boolean algebra. The representation of an observable by a particular generating self-adjoint operator may be convenient but contains inessential elements. Because any normal operator generates a Boolean algebra, normal operators are (in spite of their complex-valued spectra) perfectly legitimate observables." Primas, H. Chemistry, quantum mechanics and reductionism, Springer-Verlag, 1983, p. 64.

Which is just a slightly more abstract version of exactly what I have been saying.

I.E. a particle having imaginary proper mass is nonsensical, ...

That is not true. In relativistic kinematics that just means a particle has a spacelike worldline--though that is very probably forbidden by physical law, it is sensible in the sense that we can discuss how such a particle would have to interact with the universe. More fundamentally, physics is about fields first, rather than particles, and bosonic fields of imaginary mass are not only quite sensible, but are a dime a dozen in perturbative theory when one expands around certain types of metastable vacua--though they don't transmit information superluminally as one might expect from kinematics, and so are comparatively less pathological.

... and a pigment existing in a mixture at imaginary concentrations is also nonsensical.

I don't disagree with that, since I don't know what that could mean either. But note that your claims were about all possible measurements and physical quantities whatsoever, and that I'm not claiming that every physically relevant quantity has a sensible complex analogue.

[madd0ct0r]

The conservation laws don't stop me from rearranging things. So in this case, there'd be nothing stopping me from 'spawning' a brick and a negabrick. Conservation laws are satisfied just as well... but where the hell did those bricks come from? There are different real-world consequences in your system; it makes a difference whether I have (3-1) bricks or (303-301) bricks. Those extra bricks will be heavy and take up space and so on.
.

I'm guessing conservation of mass would look more like constant= mass -imass

This 'rearranging' idea: the rearranging is a mathematical operation, so it might not apply to the real world (same way I can't Multiply a tank by the number of parking bays to get a full platoon).

So the image I'm getting is a sort of weird universe, where you have matter, anti-matter, i-matter and anti-i matter.

Matter + anti-matter = annihilation and energy (photons)
Matter + i-matter = annihilation and no resultant energy (equal numbers of photons and i-photons)

so:

i-matter + anti-i-matter = annihilation and energy (i-photons)
anti-matter and anti-i-matter = annihilation and no resultant energy (equal numbers of photons and i-photons)

an i-photon in wave form is identical to a photon, but exactly out of phase. (results of the double split experiment will be fun to work out)

Obviously this model doesn't deal with matter in units of bricks or tanks - simply atoms. So purple's analogy with the bread in a shopping basket is trivial (in the same way putting a loaf of bread in the same basket as a loaf of anti-matter will destroy both - it's the atoms interacting really)

and like anti-matter, things don't have to be perfect mirror versions to negate each other, just bounce down emitting photons until there's nothing left to react.

[Purple]

But by that logic- smacking things together is a multiplication operator.

Not exactly. For some in universe reason the multiplication behavior only exhibits it self when you smack discrete objects with the same mass and quantity together. Where a discrete object is defined as one or more objects (each with identical mass in case it's multiple) touching one another. The universe treats everything else as addition.

So smacking 1 brick and i brick together should cause them to fuse into i brick.

Yes, come to think of it. But only because 1 = 1. If it were 2 bricks and 1 i-bricks than the result would still be two 2 bricks and 1 i-brick.

But then what's a discrete object? Bricks are made of atoms.

I defined that just above for you. But to go into further detail. Basically, a discrete object is any body above but not including those on the atomic and smaller scale that is composed of a material different from its surroundings or any collection of separate discrete objects of identical mass in physical contact with one another. So a battle tank standing on a concrete parking lot would be a discrete object because it is standing in air and concrete, materials that are different atomically from its own composition.

Suppose I smack together two lumps of wet sand. Are the lumps discrete? The individual sand grains? The atoms? Which item gets multiplied?

A very good question. Let us assume that for some unknown (for now but let's figure it out) reason the answers are:
The lumps - Yes if they are not surrounded or otherwise touching wet sand. And if you smack the two together you get multiplication because the real segments or multipliers* 1 and 1 are equal.**

The individual sand grains - No, because by being in a lump they are by definition touching at least one other grain of sand.**

The atoms - No, because the principals do not extend to the atomic scale.

* The number multiplying i.
** That is assuming all grains of sand in the sample are of identical mass. If the grains are not identical than we might actually get loads of discrete grains instead mixed in with loads of individual lumps. And each would be interacting with what ever it touches either forming discrete lumps*** or multiplying.
*** Bodies form discrete lumps through addition. So for example three sand grains of identical mass would form a 3 grain sized lump whilst 2 grains of identical mass multiply. Also, to further expand on this. Lumps of different masses would interact just like they do in the real world. Lumps of the same total mass but different component masses (different sized grains inside) would "add" as two discrete lumps while lumps of the same total mass and composed out of identically sized component grains to one another and internally would multiply if the number of lumps matches the requirement for squaring.

Can I pick up this negative brick? Can I interact with it? Can anything except another brick interact with it?

No, no and no. In that order. A negative brick is not a physical object but a mathematical way of describing the state of the universe where conservation laws demand and will enact that a brick placed in the same volume or touching the volume in question will vanish to even out conservation. And yes, in this case conservation can delay its effects.

What about a block of granite, which contains some of the same atoms as the brick but not in the same ratios or configuration? What about a lump of clay, with exactly the same composition, but with the atoms arranged differently?

This is why I said the principal does not extent do the atomic level and lower. As a rule, you want to look at the total molecular structure. So not only what atoms form the molecules of the body but what ratios they have and even how they are arranged. So a stick of coal is not the same as a diamond even thou they both contain the same atoms and even in the same ratio since their structure is different.

But that means that any one of those combinations can turn into another.

The conservation laws don't stop me from rearranging things. So in this case, there'd be nothing stopping me from 'spawning' a brick and a negabrick. Conservation laws are satisfied just as well... but where the hell did those bricks come from? There are different real-world consequences in your system; it makes a difference whether I have (3-1) bricks or (303-301) bricks. Those extra bricks will be heavy and take up space and so on.

Yes, if you can figure out a way to do it nothing is stopping you from transforming one state into another. Ain't that fun? It's sort of like those magic setting universes where there is always an equivalent exchange but you newer see the paying.

Plus, you're also saying that we can turn two bricks into one at will by banging them into each other. Or negative one. Either way, there is no conservation law, because there is no conserved quantity. You didn't change the rule, you erased it.

But quantity is conserved. It's just done in a much more entertaining way.

No, I think that would be giving it too much credit. Basically, what you're doing is asking ungrammatical questions and being smug when you're misunderstood. The only thing that's different is that the language in which you're being ungrammatical in is math.

If I asked you "monkey fruit loop, Purple hop leftly banana bread?" You would probably not be able to give a coherent answer. For the same reason,

I was making a bad joke. You know, I prefer X but with X being even more offensive. And yes, it was a very bad joke so I can see how you did not understand. But I could not help it.

I can't give you a coherent answer to "what would happen if 1=0?"

Actually you can and probably will in a month or two when that thread gets its turn to be posted.

You want to know what happens to the rest of the system? Nothing. There is no system, any more than there can be structures without building materials. There is no logical structure that can encompass "true equals false" or "something is the same as nothing, which is also the same as any other entirely different something." Because that isn't logical and doesn't have any structure to begin with.

But I explained in detail that what you just said is not the case.

Meanwhile, the other guys are having fun with a meaningful discussion of the physical implications of complex numbers... largely by ignoring you.

Sadly. And besides WE are having THIS discussion. What THEY are doing is THEIR to do. What you just said is the equivalent of two people in a coffee shop talking about what the other people in the venue are talking about.

EDIT: In hindsight I also like madd0ct0rs idea. So chose which ever you prefer for the fallowing posts. Just make sure to notify me at the beginning of it.

[Surlethe]

I have trouble imagining what 23+5i hydrogen atoms might mean.

I think the correct interpretation is that the complex wavefunction of the electron is a meaningful physical quantity, not an abstraction, and that potential is more physically meaningful than a field.

I don't think the Aharomov-Bohm effect adds much to the question of phase, since it is sensitive to differences in phase only.

Well, it does imply that phase (or at least differences of phase) are physically meaningful since they can be measured.

I'm not following here. Do you mean multiplication?

My guess is that he's referring to both multiplication and conjugation, since the complex numbers can be constructed as pairs of reals equipped with an involution operation, and both of these operations introduce important algebraic structure. Although in terms of 2-vectors, there are corresponding operations as well, and U(1) is isomorphic SO(2) anyway.

That would make sense. C is not just R^2, it's R^2 equipped with multiplication and conjugation. Thinking in terms of 2-vectors, you're just representing the vector space R^2 in its own general linear group - that should be equivalent to simply defining a multiplication operation on R^2.

[Kuroneko]

Well, it does imply that phase (or at least differences of phase) are physically meaningful since they can be measured.

That's true, but what I mean is that the phase differences are physically meaningful from the start, as the modulus-squared of a wavefunction is a probability density and its phase determines the probability flux/current. Probability flux can be measured statistically just as probability density can be measured statistically. Aharomov-Bohm already has another surprise for us by demonstrating that the phase couples to electromagnetic four-potential even when the electric and magnetic fields themselves are zero.

By the way, since I think you've learned QM from Griffiths, there is a relevant passage in discussion of Berry's phase as a precursor to Aharomov-Bohm: "[w]e are accustomed to thinking that the phase of the wave function is arbitrary--physical quantities involve |Ψ|², and the phase factor cancels out." Once we express the state in the eigenbasis of the observable we're looking for, of course that's true. But what if we don't or look for an observable in an eigenbasis of another? Not that strange, considering that the motivation for momentum operator way back in Chapter 1 is completely equivalent to this...
[1.30] d‹x›/dt = -(iℏ/2m)Int[ Ψ^* ∂Ψ/∂x - ∂Ψ^*/∂x Ψ dx ] = (ρ/m)Int[ ∂S/∂x dx ]
where Ψ = ρ1/2 exp(iS/ℏ) and the last equality has a mathematically equivalent expression. It seems like it would have been a good place to connect to problems which are otherwise left fairly useless, particularly 1.14 and 4.41a. In terms of classical mechanics, the phase S takes the role of Hamilton's principal function, and so has a significant role in taking the classical limit, though the classical limit is probably going out of scope for Griffiths.

[Kuroneko]

Note: ρ = |Ψ|² needs to be inside the intergral; sorry.

[Ziggy Stardust]

I am claiming that the definition of physical observables as necessarily real-valued is purely conventional and for done for convenience's sake, and absolutely nothing of physical importance breaks if complex-valued observables are included.

Could you expand on this? I was under the impression that the converse was true. That is, complex/imaginary numbers only exist at all for convenience's sake, because it makes the math more sensible.

Also, going back to Purple's hypothetical universe (which makes my head hurt to try and rationalize), how would polar coordinates work? That is, circles and angles and trigonometry would begin to operate differently, wouldn't they? At least relative to the Cartesian system.

[Kuroneko]

Could you expand on this? I was under the impression that the converse was true. That is, complex/imaginary numbers only exist at all for convenience's sake, because it makes the math more sensible.

There are two issues here, one regarding complex-valued observables and another regarding use in wavefunctions.

In QM, physical states are vectors in a complex vector space and observables are represented by a certain kind of linear operator on that space; roughly, you can think of those operators as mapping vectors to other vectors in the following manner: do a rotation, scale the vector components by some numbers, and do the inverse rotation. This is a slightly oversimplified description of a spectral decomposition of that operator, and the numbers you scaled by correspond to the possible results of measurement (eigenvalues). Conventionally, they are taken to be real, but remember that we're dealing with a complex vector space, so this is an arbitrary restriction.

Thus, there is a larger class of operators we're artificially restricting to having real eigenvalues ("self-adjoint" operators), for no intrinsic reason whatsoever. Complex values are more natural because we're in a complex vector space and there is nothing about this that makes it conceptually or mathematically problematic. Although the restriction of being real-valued is typically harmless, because every complex-valued measurement is formally identical to two simultaneous real-valued measurements with no uncertainty principle between them, and it does simplify some calculations.

Ok, but does that wavefunction have to be complex (i.e., why is that vector space complex)? Unlike some examples from classical physics, where complex-valued quantities are tend to be clever repackaging of real quantities in order to manipulate them better, but this is not the case for quantum mechanics. A real-valued wave just doesn't work in even representing momentum correctly, much less anything else. That's not to say it can't be written in terms of real quantities: most obviously, one can separate the real and imaginary parts and make a pair of real-valued waves, and in framing QM in this way one gets real-valued matrix equations instead for matrices of the form [a,b;-b,a]. However, they are still related to each other in a complex-like way, and the algebra of such matrices is isomorphic to the complex numbers; e.g., instead of canonical commutator relation between position and momentum operators, xp-px = iℏ, you'll have instead of i a matrix operator whose square is the negative of the identity matrix. I wouldn't call that as truly getting rid of complex numbers--it's just a notation change.

Another way to decompose it is in polar components, from which one gets a continuity equation and a slightly deformed Hamilton-Jacobi equation. Under standard QM interpretation, they refer to probability density and probability flux. One could make other interpretations as well--e.g., for Bohm, they were the particle state and the "pilot wave", although that interpretation is broken for other reasons. In any case, those waves still mimic components of a complex number, and since anti-commuting of observables is both responsible for the Heisenberg uncertainty principle and finds the most natural expression in terms of complex numbers (as x,p above), doing so does little but conceptually obfuscate the conceptual structure of quantum mechanics.

--

In short, the view that there are no complex numbers in nature cannot be reasonably supported. Well, literally speaking there no numbers of any kind in nature, but presumably we mean something less trivial than that. So, rather: there is no reason that cuts complex numbers out that does not do the same to real numbers as well. Reality has a complex structure according to our currently most fundamental theory, the only way to cut them out of it is to make a pair-of-reals that mimic complex numbers, and the less-transparent that mimicry is, the more obfuscated conceptually important principles like the HUP become.

[Ziggy Stardust]

So, rather: there is no reason that cuts complex numbers out that does not do the same to real numbers as well. Reality has a complex structure according to our currently most fundamental theory, the only way to cut them out of it is to make a pair-of-reals that mimic complex numbers, and the less-transparent that mimicry is, the more obfuscated conceptually important principles like the HUP become.

Thanks for that post, it was an interesting read.

So if I am understanding this correctly, QM views physical observables as a spectrum in an abstract vector space, as opposed to unique values in some multidimensional space? I realize this is a bit of an oversimplification, but am I on the right track, here?

[Kuroneko]

So if I am understanding this correctly, QM views physical observables as a spectrum in an abstract vector space, as opposed to unique values in some multidimensional space? I realize this is a bit of an oversimplification, but am I on the right track, here?

The spectrum is just the set of all eigenvalues--the set of numbers we scaled by above. What that is depends on the observable, and it may be discrete or continuous, finite or infinite, depending on the observable and the physical system.

The observable is more than just the spectrum. A measurement of an observable gives one of the eigenvalues, but the state post-measurement is in its eigenspace (i.e., all vectors x such that the operator T acting on them gives the same vector scaled by that particular eigenvalue λ, Tx = λx). That's what "collapse of the wavefunction" means. If two observables share the same eigenvectors, then one can measure them simultaneously; otherwise, there will be an uncertainty principle at work.

[Ziggy Stardust]

I see. Thanks for the explanation; this is a subject I have always found fascinating, but unfortunately have not yet dedicated any time to really studying in detail.