Graduate Complex Numbers Not Necessary in QM: Explained

[PeterDonis]

There are no closed alphabets

If you mean there aren't any finite alphabets, this is obviously wrong. The alphabet we are using to write these posts is finite. Even if you want an alphabet to represent all human languages instead of just one, it's still finite.

But even leaving that aside, alphabets are certainly countable. Alphabets aren't continuous. They're discrete. Letters don't continuously change from one to another. They're like integers, not real numbers.

 

[A. Neumaier]

at least for the moment I having deep skepticism that there are countably many formulas - languages are a real as the Reals... just running in their shadow.

An alphabet is by definition finite. It distinguishes a finite set of images as being the characters of a particular language. If you extend the alphabet you change the language.

 

[Jimster41]

My reaction was/is confused. It seems clear to me that the process of communication is served by both that immutability and the mutability of it. We try to not change the alphabet but it is invented. Utility motivates its constraint. But to the point of countability of the “Definable Reals” based on the enforced finitude of an invented formal system... that just feels like bootstrapping a solution. I keep thinking that you could at some point give a computer the problem of coming up with a mathematical formula for every Real. Unless you very carefully constrained it wouldn’t it dutifully go blur the line between language, alphabet and the Reals. You could say, “I can’t understand what you mean by this symbol or word - string thing”. But then it could have a friend who is also a computer that is has been working with and they might just shake their heads and with shared clarity of their co-invented system, “that’s the formula for the Real Number r of course”. So what does “Definable” really mean? Isn’t it just interpretation.

This is a great thread I need to go continue reading. After years of just being totally baffled so far my (very incomplete) cartoon is that they (complex numbers) are symbols that allow for compact and expressive descriptions of wavy things, and especially helical screwy things. Their invention has naturally shaped what has thereafter been expressed.

 

[stevendaryl]

My reaction was/is confused. It seems clear to me that the process of communication is served by both that immutability and the mutability of it. We try to not change the alphabet but it is invented. Utility motivates its constraint. But to the point of countability of the “Definable Reals” based on the enforced finitude of an invented formal system... that just feels like bootstrapping a solution. I keep thinking that you could at some point give a computer the problem of coming up with a mathematical formula for every Real. Unless you very carefully constrained it, Wouldn’t it dutifully go blur the line between real language, alphabet and the Reals. You could say, “I can’t understand what you mean by this symbol or word - string thing”. But then it could have a friend, who is also a computer, that is has been working with and they might just shake their heads and with shared clarity of their co-invented system, “that’s the formula for the Real Number r of course”. So what does “Definable” really mean? Isn’t it just interpretation.

This is a great thread I need to go continue reading. After years of just being totally baffled so far my (very incomplete) cartoon is that they are symbols that allow for compact and expressive descriptions of wavy things, and especially helical screwy things. Their invention has naturally shaped what has thereafter been expressed.

I'm not sure exactly what the question is. With the usual notion of "alphabet", there are finitely many different characters in the alphabet. The extreme case is binary, where there are only two characters, 0 and 1. If the alphabet is finite, and the length of any particular formula is finite, then that means there are only countably many formulas.

Since computers use binary, they are intrinsically limited to countable languages. So whatever notion you have of a language, if it can be communicated through the internet in a forum such as this one, it's expressible using a countable language.

You could imagine that maybe in person, people could communicate using continuously many different symbols. If you idealize a character as a zero-thickness curve, then there are theoretically uncountably many different characters possible. However, actual communication requires error-correction. What I mean by that is that if I'm writing something on a chalkboard with a piece of chalk, I might make the letter "S". You can recognize that I have written the letter "S". But when you get down to the details, no two people draw an "S" in exactly the same way. Even with one person, no two instances of drawings of an "S" are exactly like. So for you to be able to reliably recognize that I have drawn an "S", you have to be forgiving of small variations. So the letter "S" is not a single curve. If you imagine the mathematical space of all possible curves, the letter "S" would not be a point in that space, but a region, containing all possible variations of "S" that would still be recognized as the same letter.

If you have a bound on the size of a letter (you're not going to consider a letter that doesn't fit on a sheet of paper, for example) and you have a bound on the resolution (how different must two letters be and still be considered the same letter) then there are only finitely many different letters possible.

 

[Jimster41]

I was thinking the extreme case is more the alphabet of all defined characters.

Not sure I get the 0,1 argument. We are made of atoms but I don't think you'd argue they are the building blocks of our formal system or that the limited number of atom types makes the manifestations of physical reality finite. Also, all those 0's and 1's are made of atoms (and photons I guess) - same stuff as us. What's the difference?

I'm not disagreeing with your third paragraph. Kind of what I was trying to say. After thinking about my initial knew jerk reaction to the proposal that a map from the Reals onto some definitely countable formal system can help tame the Reals my observation was just that the infinite mutability of your zero thickness curve and our formal agreement on what is an "S" in the space of all shapes of that curve (learned by the way) are both important. I don't see a natural law that says where to draw a line between those two poles, between infinitely mutable symbols and a single forever fixed perfectly detailed alphabet. Though you could argue that because we naturally do draw that line there is a natural law that says a line must be drawn.

Your last paragraph outlines some potentially useful constraints to character building but aren't they sort of arbitrary. I mean who uses paper? And what if we moved in 100,000 years to a fully animated sign-language/emojii like symbol space (traded by our virtual avatars to and from our visual cortex's via our implanted vision augmentation systems of course) - what is the resolution of an animation?

Probably you already know BTW there is a whole category of Art that plays with characters - fonts and symbols. They are being pushed to the limit all the time. Some of it is pretty amazing in the way it plays with our methods of making and breaking agreements on recognizably useful symbols. I've seen some in the past that really blew my mind - when I was putting of off my engineering studies to wander the art library at my school. Just now I googled "Art of symbols and alphabets" /images. It definitely gives me pause to then say - oh, yeah, that's a countable set.

 

[Auto-Didact]

I was thinking the extreme case is more the alphabet of all defined characters.

Not sure I get the 0,1 argument. We are made of atoms but I don't think you'd argue they are the building blocks of our formal system or that the limited number of atom types makes the manifestations of physical reality finite. Also, all those 0's and 1's are made of atoms (and photons I guess) - same stuff as us. What's the difference?

I'm not disagreeing with your third paragraph. Kind of what I was trying to say. After thinking about my initial knew jerk reaction to the proposal that a map from the Reals onto some definitely countable formal system can help tame the Reals my observation was just that the infinite mutability of your zero thickness curve and our formal agreement on what is an "S" in the space of all shapes of that curve (learned by the way) are both important. I don't see a natural law that says where to draw a line between those two poles, between infinitely mutable symbols and a single forever fixed perfectly detailed alphabet. Though you could argue that because we naturally do draw that line there is a natural law that says a line must be drawn.

Your last paragraph outlines some potentially useful constraints to character building but aren't they sort of arbitrary. I mean who uses paper? And what if we moved in 100,000 years to a fully animated sign-language/emojii like symbol space (traded by our virtual avatars to and from our visual cortex's via our implanted vision augmentation systems of course) - what is the resolution of an animation?

Probably you already know BTW there is a whole category of Art that plays with characters - fonts and symbols. They are being pushed to the limit all the time. Some of it is pretty amazing in the way it plays with our methods of making and breaking agreements on recognizably useful symbols. I've seen some in the past that really blew my mind - when I was putting of off my engineering studies to wander the art library at my school. Just now I googled "Art of symbols and alphabets" /images. It definitely gives me pause to then say - oh, yeah, that's a countable set.

You are mixing up two distinct semiotic ideas: the sign which can be (dis)continuously varied, with the symbol which is discrete.

As others have already stated each alphabet is finite, ultimately because each letter is a sign which represents a discrete object: a symbol. Changing the signs - i.e. the representation of the symbols - a little bit, doesn't change the symbol itself.

There need not be a bijective relationship between representations and objects and generally speaking, there isn't. The meaning of the sign in the form of concatenated signs i.e. as words is of course strongly context dependent.

 

[A. Neumaier]

So what does “Definable” really mean?

In the present context it has a precise meaning. Pick a formal language in which you have a specification of what the concept of a real number means. Say HOL light. In this language you can create certain formulas that define particular real numbers. The collection of all these numbers is the set of definable reals in this language. It is countable.

Note that like all mathematical notions, this notion depends on the formal specification and hence the language used, in the same way as the notion of a set is dependent on its precise specification.

 

[Jimster41]

In the present context it has a precise meaning. Pick a formal language in which you have a specification of what the concept of a real number means. Say HOL light. In this language you can create certain formulas that define particular real numbers. The collection of all these numbers is the set of definable reals in this language. It is countable.

Note that like all mathematical notions, this notion depends on the formal specification and hence the language used, in the same way as the notion of a set is dependent on its precise specification.

HOL and ML are two things I had not heard of. Pretty interesting.

 

[*now*]

I missed much adding to the btsm bibliography thread and one of the papers I missed adding there seems relevant here too,

https://arxiv.org/abs/1902.03026

Natural discrete differential calculus in physics

We sharpen a recent observation by Tim Maudlin: differential calculus is a natural language for physics only if additional structure, like the definition of a Hodge dual or a metric, is given; but the discrete version of this calculus provides this additional structure for free.

 

[*now*]

Given that link, an earlier paper might also be worth mentioning here too,
https://arxiv.org/abs/1508.00001
Michelangelo's Stone: an Argument against Platonism in Mathematics

If there is a "platonic world" M of mathematical facts, what does M contain precisely? I observe that if M is too large, it is uninteresting, because the value is in the selection, not in the totality; if it is smaller and interesting, it is not independent from us. Both alternatives challenge mathematical platonism. I suggest that the universality of our mathematics may be a prejudice hiding its contingency, and illustrate contingent aspects of classical geometry, arithmetic and linear algebra.

 

[bhobba]

I suggest that the universality of our mathematics may be a prejudice hiding its contingency, and illustrate contingent aspects of classical geometry, arithmetic and linear algebra.

That is philosophy which we do not discuss here. There are all sorts of views about eg conventionalism. On this forum we simply note that the fundamental science as far as we can tell today is written in the language of math. Why it's like that we, by forum rules, do not go into.

As far as the original question goes I have stayed out of it mostly, but mathematically QM is a generalized probability model. These have what are called pure states. In ordinary probability theory pure states are the outcomes of what probabilities are assigned to. There is no way to continuously go from one pure state to another, so if we want to model some situation and be able to do that to use the powerful methods of calculus you need to go to complex numbers. See:
https://www.scottaaronson.com/democritus/lec9.html
https://arxiv.org/abs/quant-ph/0101012
It is strange that nature is accommodating like that. Gell-Mann thinks its part of the self similarity we see as we delve deeper and deeper into different layers of nature:


Thanks
Bill

 

[SSequence]

There was some discussion regarding definability in the first few pages of this topic. I have posted a question about it (in-case someone might be interested).

 

[A. Neumaier]

If there is a "platonic world" M of mathematical facts, what does M contain precisely? I observe that if M is too large, it is uninteresting, because the value is in the selection, not in the totality

Your argument is not cogent.
You can ask the same question about the physical world, and find that it is far too large but most interesting, because the value is in the selection, not in the totality!

 

[microsansfil]

It is strange that nature is accommodating like that. Gell-Mann thinks its part of the self similarity we see as we delve deeper and deeper into different layers of nature:

Why would the world we perceive through our consciences be the world as it is in itself ? And so why would "aliens" with totally different sensory experiences than ours (color, sound, touch,...) build the same theoretical models as ours ?

/Patrick

 

[Auto-Didact]

Why would the world we perceive through our consciences be the world as it is in itself ? And so why would "aliens" with totally different sensory experiences than ours (color, sound, touch,...) build the same theoretical models as ours ?

/Patrick

Even with different sensors, most or at least some of the objects that they are detecting will have the same underlying characteristics and dynamics; this is what is captured by our mathematical models.

There are very strong arguments based on mathematical theorems, based on the unity of mathematics, as well as based on metrological constraints, that these models will be therefore quite similar, possibly approximate subsets of each other, or even different approximations from different angles to the same underlying mathematical structure.

Any key mathematical differences at a superficial stage of theorisation will be based on the different idealizations of the different assumed key mathematical properties which underly the differently constructed/evolved methods of mathematics.

 

[microsansfil]

Even with different sensors, most or at least some of the objects that they are detecting will have the same underlying characteristics and dynamics; this is what is captured by our mathematical models.

I'm not talking about different sensors, but different first-person experiences. First-person experiences by which we perceive the world before any act of objectification, by inter-subjectivity between us, as human beings.

For example about color.

http://www.ler.esalq.usp.br/aulas/lce1302/visao_aves.pdf : It is true, as many youngsters learn in school, that objects absorb some wavelengths of light and reflect the rest and that the colors we perceive “in” objects relate to the wavelengths of the reflected light. But color is not actually a property of light or of objects that reflect light. It is a sensation that arises within the brain.

you can explain to Tommy Edison, who has been blind since birth, the theory behind the operation of a color sensor, He still won't be what a color is, i.e living the experience of the feeling that colour is.

in addition, it should be noted that Chromesthesia or sound-to-color synesthesia is a type of synesthesia in which heard sounds stimuli automatically and involuntarily evoke an experience of color. And therefore not to confuse the cause with the effect. We, as humans being, have no choice but to start from the effects (first-person experiences) and seek to do "reverse engineering".

/Patrick

 

[Auto-Didact]

I'm not talking about different sensors, but different first-person experiences. First-person experiences by which we perceive the world before any act of objectification, by inter-subjectivity between us, as human beings.

That is not a question about physics but a question about physiology, specifically the distinction of sensation and perception. It is a widely studied phenomenon, with much more known than just thought experiments like Mary's room, e.g. colour blind people already satisfy all the criteria.

Even worse, there are people in Africa who can easily distinguish far more shades of green than the rest of the world can due to having grown up around many trees, even causing them to have names for those different greens. We are all colour blind to their many greens.

This hardly invalidates any physics, since in terms of physiology it can be explained more or less; the difficult part is to accurately reduce this purely to biophysics and natural selection, but there is little doubt this can be done apart from an explanation for consciousness itself.

 

[microsansfil]

This hardly invalidates any physics

There is no question in my remarks of seeking to invalidate physics I use every day indirectly. Just to become aware that consciousness is the starting point of any inquiry.

I don't remember who wrote this: Lived experience is where we start from and where all must link back to, like a guiding thread.

Human Being can question his beliefs, but not the one in which he believes deeply.

/Patrick

 

[Jimster41]

Your argument is not cogent.
You can ask the same question about the physical world, and find that it is far too large but most interesting, because the value is in the selection, not in the totality!

I think he was pretty much just quoting Rovelli... in the paper... so. Cute as the Gell-Mann Ted Talk is Rovelli seems to disagree pretty directly, giving examples of how basic geometric axioms we take to be universal aren't that obviously so.

Rovelli is a bit of a whacko tho.

 

[*now*]

Your argument is not cogent.
You can ask the same question about the physical world, and find that it is far too large but most interesting, because the value is in the selection, not in the totality!

Although Bhoba said this sort of discussion isn’t wanted, just there could be some confusion. The words I quoted are the abstract of the paper I linked in order to provide a wider impression than just linking the previous paper. So, the argument belongs with that second paper, not with me. Also, given that possible misunderstanding, I’m not sure if the comment concerns more of the sentence and argument of the second paper linked, which I think argues that an independent world of mathematical truths may be reduced to something “trivial, or contradictory”, and I think giving an alternative example.

I think he was pretty much just quoting Rovelli... in the paper... so. Cute as the Gell-Mann Ted Talk is Rovelli seems to disagree pretty directly, giving examples of how basic geometric axioms we take to be universal aren't that obviously so...

I’m not sure about disagreement, Jimster, there seems to be at least some agreement between the essay and the clip like solidity of mathematical theorems and a similar lack of requirement for humans/mind, and there might not be further disagreement either. Although the notions involved vary somewhat, Gell-Mann’s definition of fundamental law - a law unifying fundamental particles and forces, “is not a theory of everything”. It would apply to some but wouldn’t apply to the many chance outcomes that occur. So, it continues, assuming the law exists it only applies to some part excluding much that provides a huge amount of information. As notions of universality here seem to be with respect to some part and not all, perhaps there isn’t disagreement with the paper.

 

[Tendex]

As for QT, the case for the necessity of complex numbers seems to be much clearer than the case for complex numbers in SR. Orthodox QM is patently laden with complex numbers, complex vector spaces and holomorphic functions. There are tonnes of equations in QFT with explicit complex notions such as the gamma function.

The positive frequency condition of QFT - i.e. positive energy - can be deeply demonstrated to be dependent on the underlying complex structure. Moreover, the scale invariance of renormalization group theory is also deeply dependent on the notion of complex numbers.

You touch several interesting points here. At some time I worked on the holomorphicity of functions in the context of QM when showing its unitarity and I believe it had to do with the Stone-von Neumann theorem and the unitary map it allows in Bargmann-Segal space and its transforms for adjoint operators but it's been a while and don't recall all the details. When extending to relativistic quantum fields the Stone-von Neumann no longer holds and indeed one has to separate positive and negative energies with a procedure that involves complexification and (for the interacting case) analytic continuation to determine iepsilon prescription of Feynman propagators(LSZ normal ordering and all that stuff). And then one has not only commuting but anticommuting operators (i.e. we have both holomorphicity and antiholomorphicity) unlike in nonrelativistic quantum mechanics.
So I would say that somewhat subtly there is a clear need for complex objects at least when dealing with more than one particle or beyond the semiclassic approximmation since in the latter one never has to mix the holomorphic unitarity of the Schrodinger evolution with the non-holomorphicity of quantum measurement using the Born rule.

Hope I didn't introduce more confusion, maybe @Auto-Didact can further comment on these points.

 

[Demystifier]

I don't think that it is such a big problem. The Wilson method removes the doublers by a rather simple method. Essentially, one adds to the Lagrangian a discretized version of
$$a\partial^{\mu}\bar{\psi} \partial_{\mu}\psi$$
where $a$ is the lattice spacing. Sure, it violates the chiral symmetry, but so what? Lattice violates also the Lorentz, the rotational and the translational symmetry, and yet nobody gets too excited about it.

Now after reading the excellent review http://de.arxiv.org/abs/0912.2560 I understand it much better. The Lagrangian density mentioned above really takes the form
$$ra\partial^{\mu}\bar{\psi} \partial_{\mu}\psi$$
where $r$ is a free dimensionless parameter. Through the loop corrections, this term generates a change of fermion mass of the order
$$\delta m\sim \frac{r}{a}$$
The problem is that this correction is big when $a$ is small, if $r$ takes a "natural" value $r\sim 1$. To get right phenomenology one must take a much smaller value for $r$, of the order of
$$r\sim ma$$
or less. But where does such a small number come from? This shows that the problem of chiral fermions on the lattice (with the Wilson term) is really a problem of naturality, known also as a hierarchy problem. The Standard Model of elementary particles has naturality/hierarchy problems even in the continuum limit (e.g. the scalar Higgs mass), and we see that lattice regularization by the Wilson term creates one additional problem of this sort.

But is naturality really a problem? The principle of naturality is really a philosophical problem, based on a vague notion of theoretical "beauty". Some physicists and philosophers argue that it is not really a problem at all
https://www.amazon.com/dp/0465094252/?tag=pfamazon01-20
https://link.springer.com/article/10.1007/s10701-019-00249-zSo if one accepts the philosophy that parameters in the Lagrangian which are not of the order of unity are not a problem, then there is really no problem of chiral fermions on the lattice with the Wilson term. @atyy I would appreciate your comments.

 

[Tendex]

On rereading my post #201 above where I was referring to the following assertion by @Auto-Didact :"unitary evolution is a completely holomorphic notion", from a closed thread, I realize this doesn't seem right in quantum theory, the wave function is not required to be analytic(this is for instance explained here https://physics.stackexchange.com/questions/158432/must-the-wavefunction-be-analytic ) so I wonder if maybe something else was meant by this assertion.

 

[A. Neumaier]

On rereading my post #201 above where I was referring to the following assertion by @Auto-Didact :"unitary evolution is a completely holomorphic notion", from a closed thread, I realize this doesn't seem right in quantum theory, the wave function is not required to be analytic(this is for instance explained here https://physics.stackexchange.com/questions/158432/must-the-wavefunction-be-analytic ) so I wonder if maybe something else was meant by this assertion.

The spectrum of a Hamiltonian is real and bounded below. Hence $e^{izH}$ is a bounded operator-valued holomorphic function of the complex variable $z$ on the upper half plane. All the tools of complex analysis are available for this function.

 

[Tendex]

The spectrum of a Hamiltonian is real and bounded below. Hence $e^{izH}$ is a bounded operator-valued holomorphic function of the complex variable $z$ on the upper half plane. All the tools of complex analysis are available for this function.

Sure, but the wave function is not required to be of complex variable in quantum theory, self-adjointness of the operator is enough for unitary evolution. Auto-didact appeared to be saying that unitary evolution required holomorphicity.

 

[A. Neumaier]

Sure, but the wave function is not required to be of complex variable in quantum theory, self-adjointness of the operator is enough for unitary evolution. Auto-didact appeared to be saying that unitary evolution required holomorphicity.

In what you quoted, he didn't refer to wave functions but (among others) to the properties of functions of the Hamiltonian. That's where complex analysis is most essential - for example in time-independent scattering theory.

 

[Tendex]

In what you quoted, he didn't refer to wave functions but (among others) to the properties of functions of the Hamiltonian. That's where complex analysis is most essential - for example in time-independent scattering theory.

Hope he can answer himself what he exactly meant. By the way I'm already convinced about the importance of complex analysis in quantum theory, no need to stress it, I know it for a fact.

 

[Tendex]

More specifically, the claim by Auto-didact(https://www.physicsforums.com/threa...erpretations-of-qm.971782/page-5#post-6181930) was to justify a purported mathematical inconsistency between Schrodinger's time evolution and the Born rule based on a non-holomorphicity of complex conjugation as opposed to the "holomorphicity" of quantum unitary evolution. And yet the latter, through hermiticity of operators, rests upon the notion of complex conjugation as much as the Born rule does, so I fail to see how this is "holomorphic" in the stated sense, which appears to not be related at all to what A. Neumauer's mentions above.

 

[A. Neumaier]

quantum unitary evolution. And yet the latter, through hermiticity of operators, rests upon the notion of complex conjugation

Not through Hermiticity, which would reduce to symmetry in the case of a real ground field, but through the occurrence of the imaginary unit in the Schrödinger equation and the exponential expression $e^{-itH/\hbar}$ in the expression for the unitary time evolution operator.

 

[Tendex]

Not through Hermiticity, which would reduce to symmetry in the case of a real ground field, but through the occurrence of the imaginary unit in the Schrödinger equation and the exponential expression $e^{-itH/\hbar}$ in the expression for the unitary time evolution operator.

That is what I'm saying, and even without quantum considerations, purely mathematically which was my point, complex unitarity is about preserving a Hermitian form which includes the non-holomorphic notion of complex conjugation that was mentioned in the other thread as incompatible with "holomorphic" evolution of unitarity whatever that is. Glad finally the penny dropped.