Respected Thomas Krantz,

Thanks for your valuable comments on my answer.

Actually, I believe that describing nature by our language of mathematics or art or sculpture, whatever way it may be is nothing but a " mimesis " , as described by Plato and then Aristotle in his book " Poetics".

So, to feel the nature is far away from our all methods of describing nature.

All the way of describing nature that we use are nothing but a mimesis.

So, if we think about Physics, we describe the Nature by some Mathematics designed by us.

Initially, only natural numbers were used by Human beings for there daily life requirements. Later , with the advancement of requirements and man's query to know the nature, gradually , the number system are expanded by our own way and Mathematical rules and Algebras are designed.

In that journey, the extension of number system was needed to be expanded.

But, I say that , it could be , may be , possible, if we formulate those rules by some other ways to reach the same level of describing the Nature.

Thanks and Regards

N Das

The Davisson-Germer experiment performed for the first time that interference can happen for material particles like electrons. This means that when an electron of reasonably well-defined momentum enters a crystal and then leaves again, there can be maxima and minima seen between electron waves arriving at a given point via different paths. Amplitudes may be added constructively or destructively. We have said already that to even describe waves for material particles, we need complex quantities. Since interference involves superposition, adding such quantities must give a quantity of the same type. More formally it is essential that when we add objects representing states, we get other objects representing states, and here superposition involves not just addition

but addition and multiplication with complex numbers.

For more details read the following article:

https://courses.physics.illinois.edu/phys580/fa2013/susy_v2.pdf

Dear Mandeep Kaur ,

Thank you for your answer and also the reference to the interesting lecture.

Experienced in mathematics, I am much less in physics.

As such I share your argumentation to a large extend, but only partly your conclusions:

In every field you have an addition and multiplication with scalars (I give these examples: finite fields, field of rational numbers, field of reals, field of complex numbers, field of fractions of polynomial functions, field of surreal numbers, etc.)

In all(?) these fields you can superpose in a mathematical sense waves. I agree with you that commutation rules of location and of impulse operators involve the constant square root of -1. But on one hand there are other fields admitting square root of -1 e.g. the field of fractions of polynomials with complex coefficients, on the other hand it might be possible that there are different physical laws according to the field.

It would be interesting to find out why the physicists chose the field of complex numbers at the beginnings of quantum mechanics.

Did they exclude other fields or did they choose complex numbers for practical or physical reasons?

(I am not in favour of mathematical realism...)

If you are skilled enough, you do things via matrix method a la Heisenberg.

(and bypass the wave function)

Dear Juan Weisz ,

can you give a short explanation about this method and in how far it bypasses the wave function?

Best,

TK

Well, between operators, you have well known situations where they do not commute, some definite set of commutation relations between them. So you find a set of matrices, which satisfy the same commutation relations as the operators

in accordace with the specific problem, and then from thhe constructed matrix for

the Hamiltonian find the eigenvalues and the problem is solved.

Dear Juan Weisz ,

Very interesting, so we have to figure out how the commutation relations are affected when changing the field. This does not seem to be too complicate...

But then I wonder how the notion of manifold is affected by a field change. Can one use also a finite field for instance?

Fields are rather limited in number, the real R, the complex C are about the only ones you use in practice. Quaternions are non commuting and complicated to use. The matrices would have elements in either R or C.

The thing about waves is that you need phase and magnitude, the Complex has that. The wave function is in general a complex function. (psi1, psi2)

There is the very active domain of field theory (Galois theory etc.)

You would have both magnitude and phase for many extensions of the field of complex numbers.

Would it make sense to have parameters in the field C(X) of fractions of polynomials with complex coefficients for instance?

What is the physical meaning of the magnitude. Has it to be a positive number in R or could it be in another field? I mean it could take for instance a finite number of values in a certain interpretation.

What about coefficients in a ring like C[epsilon] with epsilon^2=0 ?

Would there be a physical situation this would correspond to?

Thomas

I realize you are in math.

Would urge you to read a bit of the Quantum Physics, to have some understanding of what may be usefull there.

Im just mentioning the usuall lines, Galois sounds very strange in this context.

You can if you like go from Poisson bracket in Classical Physics, to Commutator in the quantum, and then it is usefull to know the Dirac ket and bra language for the states, Hilbert space.

A lot of functional analysis is usefull, special functions and so on. Matrix theory of course.

regards, Juan

From the complex wave function, you form Psi* Psi, the wave function times its Complex conjugate.

This is a probability density to find the quantum

particle.

Personally Im fond of the ring of dual numbers, but do not know of its use in the quantum.

Eigenvalues of the hermitian operators are supposed to be real. Those of the Hamiltonian mean the energy levels to find the quantum particle.

On the other hand nilpotent elements or matrices are often used in optics and quantum field theory.

See quantum waves fromthe exclusion principle

on my RG page.

The number epsilon in the ring of dual numbers is an infinitesimal, like in Leibniz derivation theory. In abstract differential calculus, (or synthetic differential geometry) it is very useful and its coefficient represents in the most common e.g. 'the slope of a tangent line'. There are also generalized infinitesimal numbers corresponding to higher derivatives like in jet theory (due to Ehresmann if I remember well).

In physics I would see a use of epsilon as an 'infinitesimal time variable' describing what happens at the first order.

Since it has not been mentioned yet, it seems to be the case that every division algebra with a norm over the real numbers is the real numbers, complex numbers, quaternions or octonions . I am not sure what happens if we add synthetic infinitesimals and (by multiplicative inverse) infinitely large synthetic elements. It appears to me that the norm of (a+e*b) where a and b are reals and e is an infinitesimal is a^2+2*e*a*b which is not real valued (since e^2=0 in synthetic differential geometry). I suppose you could treat an infinitesimal (or infinite element) like the imaginary number i, and define the norm of a+e*b as (a+e*b)(a-e*b)=a^2, or define the norm of a+e*b as a^2 directly (which is easier if you have a+e*b+(1/e)*c, since the norm can then be defined as a^2+(1/e)^2*c^2). But then you would have a the norm of a complex number plus nilpotent infinitesimals representing a probability when the norm is a real number in the closed internal 0 to 1.

Dear Andrew Powell ,

Thank you for the hint on normed division algebras over the real numbers.

There are also results characterizing uniquely the field of real numbers, in a different way from the usual Dedekind- or Cauchy-completion of the field of fractions of integers. (I would have too look it up, it appears in literature on surreal numbers). But yet let me question the (largely uncontested) role of the real numbers in physics, why do we need the archimedian property, the Cauchy-completeness and so on. Are there "philosophical" or "empirical" reasons for this?

Best regards,

Tom Krantz

Dear all

In classical mechanis, there is a single phenomenon at each occupied position, which obeys the dynamical principle: Particels are found over extreme action trajectories (or, particles go along with extreme action trajectories).

In quantum mechanics, there are two phenomena at each occupied position:

The imaginary part of the action ensures that each trajectory in the equilibrium volume (e.g., hydrogen orbitals) has a single position randomly occupied at a time; reflecting the fact that quantum particles don't go along with trajectories.

One can find explanations why conventional quantum mechanics, using algebraically formal imaginary unit, is a no-work-around obstacle for feasible interpretation and, what is much more important, for practical implementation of quantum computing and quantum cryptography. Many references are available at https://soiguine.com/technology.htm. The story begins with replacement of qubits by geometrically unambiguous g-qubits and ends up in demonstration of state fields instantly propagating through the whole three dimensions and all, past and future, parameters of what is called time.

No. The use of the field of complex numbers is NOT essential to quantum mechanics.

Dear Thomas Krantz

Non-Archimedean totally ordered fields are not complete under least under bounds of subsets of the field, typically because the upper bound is not unique due to the ability to change the bound by an infinitesimal. Infinitesimals are difficult to quantify. Personally I think some clarity can be given by using infinite binary sequences (where the length of the sequence could be any infinite cardinal number below a certain bound) to represent any number (real numbers or higher), which can in turn be thought of an infinite sequence of real numbers. But developing that would take us too far off track.

Dear Andrew Powell ,

Alright, but then : Why do we need completeness of the field of reals in physics (if this is the case) and not another property?

Dear Thomas Krantz

The real numbers are used in physics because they are convenient, can be used to formulate a lot of physics (imagine starting with the rational numbers each time), and are unambiguous (the unique complete totally ordered field).

Quantum computing requires us to keep track of positive and negative probabilities of outcomes, that sum to unity. Imagine having half the options without complex numbers.

Dear Jeffrey Cohen ,

Imagine having different options with a different field like K[X] where P(X)=0 and P is an irreducible polynomial...

Dear Jeffrey Cohen ,

Mathematically what I said makes sense but my question is precisely if there are some examples which also physically have a meaning. It would need some expertise in field theory I don't have. Without requiring total order there might be more fields that are e.g. smooth in a certain sense if this should be necessary.

All and TF: I suggest, before quantum mechanics, because it will play a role in QM, the book by the well-known Melvin Schwartz, "Principles of Electrodynamics", of 1972. Electromagnetism and QM are deeply affected by SR, E and B are not independent, and must have the same units (already have, in SI CGS). The book shows that, likewise, complex numbers are not necessary in physics.

To repeat partly, read review below:

Unlike most textbooks on electromagnetic theory, which treat electricity, magnetism, Coulomb's law and Faraday's law as almost independent subjects within the framework of the theory, this well-written text takes a relativistic point of view in which electric and magnetic fields are really different aspects of the same physical quantity. Suitable for advanced undergraduates and graduate students, this volume offers a superb exposition of the essential unity of electromagnetism in its natural, relativistic framework while demonstrating the powerful constraint of relativistic invariance. ...

There is no real substitute for the use of real numbers. For complex nos.

there are a variety of expressions, including matrix, ordered pair and so on.

Complex analysis is not always that nice, with cuts and branches.

Their usefullness in analysis depends on the kind of problem, but vector analysis

is usually much more usefull.

Specifically in QM the complex field is rather important, along with functional analysis.

Ive heard of a Schwartz in math but not physics, not that well known I would say.

In separate threads I aalready explained why E;B sould not have the same units.

JM and all: Melvin Schwartz does not need my help, he received a Nobel prize in Physics, and he died. He was noteworthy for his insigths, not just classes. This question is about the complex numbers, if they are neccesarily used in QM.

No, they are not, and not in physics either. They also introduce artificial problems if used, see the book cited, by Melvin Schwartz.

Well, if you say that Complex numbers are not INDISPENSIBLE I probably agree.

However they are extensivly used in the practice of Quantum theory, so this statement is sensless rather than false.

You never give a coherent reason that E and B have the same units.

I say the Coulomb force and the Lorenz force give them different units.

Your reason is they both integrate the same tensor??

Notice that E and B are often accompanied by constants epsilon and mu

that change the units.

When real quantities occur in QM they are said to be measurable, while complex quantities are not. The use of complex field of quantities is often helpful and was originally a short cut calculation method.

Imaginary part of complex does not necessarily describe something that does not exist. The imaginary part represents something that may or may not exist in some dimensions other than our 4-D, thus making it not measurable in 4-D. A real problem can sometimes be solved with complex quantities to get a real answer that is difficult but not impossible to get by other means.

The sequence of real to complex to real can be thought of as a set of coordinate transformations from 4-D to complex plane and back to 4-D. Sometimes a problem cannot be solved in 4-D, but can be solved in 6-D, or 10-D, or 12-D, or 26-D, or some other frame that we cannot immediately specify.

Roger Penrose did much work with complex calculations, and suggested in his Road To Reality that the imaginary part represents the missing dimensions without saying how many there are and what they represent physically.

Researchers in recent decades have done much work in higher dimensions, where real problems can often be solved by real methods, giving a no answer to the present question.

A criticism of higher dimensions has been that they are not constructible or measurable. In other threads I showed that such higher dimension frames can be constructed and tested when they are based on velocity components instead of positions and distances, removing the objections.

Complex methods are still useful, but not always correct, and not essential in recent times.

Dear Ed Gerck ,

Thank you for the hint about electromagnetism.

Sorry, I cannot remember precisely what the link between 'probability waves' and 'electromagnetic waves' are. We used to speak about electrons then about magnetism finally about Maxwell equations. When introducing elementary particles we presupposed already the representation using complex numbers.

I believe that Electric field and Magnetic field are adjoint or dual in a certain sense (see adjointness in category theory).

I once read how to derive the equation of the electron in lagrangian mechanics ( David Bleecker, Gauge Theory and. Variational Principles, 1981) I should look up this again.

TF and all: Back to the topic of this thread, complex numbers can be very, very useful in physics and engineering, even in art rendering, also for visualization, but are not necessary in QM, physics, engineering, or art rendering. No physical value is complex. See more at

Preprint Overview of Complex Analysis and Applications

PS: Just beware of the "vector cross product" : Gibbs was wrong and it is not a vector.

JM and all: You asked, "You never give a coherent reason that E and B have the same units."

Well, I cited more than once the Nobel prize winner, well-known, 1972 Schwartz book, for you to read there yourself. The cause is that they are related by the same Lorentz transformation. The B field source becomes the Lorentz transformation of the E field, only. I walked through the example, in another thread. I suggest you read the Preface of the Schwartz book, you may find it at amazon or online.

It was explained also by the SI folk, when CGS accepted the unit called "gauss" to designate both E and B, and not because they preferred centimeters to meters...

They, E and B, are related by v/c. Nowadays, c (the speed of light in vacuo) is, also therefore, not a measurement anymore, but a fixed number, an integer. In Planck units, c = 1, exactly, also an integer.

And, finally, you read it as an RG preprint:

Preprint There Must Be Light!

In Physics, the mathematics used may be different, as a bookkeeping mechanism, it may even be non-euclidean, for example, or use complex numbers --- while everything we see is representable by finite rational numbers.

What matters in Physics, is that the physical facts are the same. It does not matter if we obtain them through a trip in the complex plane, which remains hidden, in the conclusions. We may even take longer to derive the results, simpler, faster, or according to other principles.

For example, the multiplication of two negative numbers can be seen as a rotation in the complex plane, visualizing more easily that the product of two negative numbers is a positive number. As Titchmarsh said, people balk at the square root of minus one, but readily accept the square root of two; however the former can be easily visualized, while the latter can not.

This is in opposition to the geometric view, which includes a model a priori. Mathematics and Geometry are all made up, made by humans, granted. Further, there is no Euclidean geometry (a geometry in which Euclid's fifth postulate holds, sometimes called parabolic geometry) or non-euclidean geometry (the complement, sometimes called hyperbolic geometry). There is just ageometry -- meaning, the absence of geometry -- from the Greek suffix "a-". The insistence that nature must obey geometry has been a straight-jacket for ideas since the times of Euclid and Nikolai Lobachevsky. Caratheodory, for example, did not follow into this trap, but provided us with a global framework in differential forms.

Dear Tiffany Gerck

I am not sure all physical quantities are real numbers. A simple example is the many types of physical quantities that are vectors (or tensors). It is also possible (although slight strange and heterodox) to think about the imaginary number i as a physical quantity. In geometric algebra in 2 dimensions the geometric product of two unit orthogonal basis vectors has the property that its square is -1. One might say that vectors are not real things; they are just useful products of the human mind. But we know that nature exhibits geometric patterns (crystalline and fractal patterns for example) and differential geometry is central to human theories of space and time. Personally I am a realist about geometry, and in fact about spatio-temporal structures. A friend of mine suggested a theory of the real numbers in which the smallest interval has the size of the Planck length; anything below that would have to be a probability of measuring an interval. This seems reasonable, and one could use the limit case to inform the topological and geometrical properties of the theory of the real numbers with a smallest interval size.

Dear all,

A physical meaning have also finite fields: At each time a particle can only be in a list of p^n states (with p prime) it is enough to use this description!?

Probability might be a real number in the interval [0,1] but to fix if something is true or not it is enough to take a value in {0,1}.

One could imagine a more rudimentary physics without reals and only truth values, as well as one could imagine a physics finer than traditional one, taking values in extensions of the real number field.

Dear Andrew Powell ,

I would add to your observation of before: tensors can also live in infinite dimensional spaces like smooth vector fields on a sphere etc.

AP, TK and all: Physics and Maths are all made up, but based in logic laws, not what one thinks, or quotes (what others say, no matter who). This avoids errors, where Nature is the arbiter in Physics, not humans. My previous answer is correct, and you can source it to Kronecker in maths, a logical science, or consult any reputable Physics text. No infinity in physics.

There is no infinite precision in Physics, hence the rationals would suffice, but most people would balk -- however, computers could do it, just with 0 and 1.

Vectors, phasors, or tensors are Platonic models; in higher formalisms, such as Lagrangian, they are not used at all, in any physical process, but generalized coordinates, with reals. Vectors are actually only used in very simplified applications, of 3D or less, because vector algebra does not form a complete space, not even in 3D.

Complex numbers are very useful, as an abstraction (aka Platonic model). Other abstractions can be used, such as quaternions and octonions, but they remain abstractions -- physicists don't use Platonic models any more, since the times of Copernicus.

It is not necessary to consider anything but real numbers in physics, but can be useful. Look for metric spaces and positive-definite functions. Therefore, keep in mind that everything we see is representable by real numbers, but just the rationals would suffice, or better yet, just the integers (e.g., agreeing with Kronecker) -- or , 0, 1 and X, tristate logic.

Dear Andrew Powell ,

When you say that you are realist, to which mathematical setting do you refer:

set theory with the axiom of foundation or without?...

Dear all,

Maxwell equations admit in the semi-riemannian setting a formulation as a single equation: "the covariant derivative of something is zero (or J according to the case)". I think it appears in the book "Eichfeldtheorie" by H. Baum and also in the book by D. Bleecker. (Hope this is correct.)

Progressively memory comes back: Magnetic differential form is really dual to the electric field.

Sorry to the physicists who know this very well, would you be so kind to recall this and also to explain how the norm of a complex number is linked to a probability.

Thanks,

TK

Dear Thomas Krantz,

My position is a little strange on realism. I am not a realist about pure sets in general (because consistency of large cardinal axioms for example does not imply truth), although I do think that the cumulative hierarchy of well-founded pure sets is a very natural model for much of mathematics. I am a realist about geometry and spatio-temporal structures (and if you push me that means the real numbers and Lie groups over the real numbers, including structures such as complex numbers and quaternions). I actually think that P. Lorenzen did give a constructive foundation (in his sense, not Brouwer's) to Euclidean geometry. I am enough of a Kantian to believe that you cannot have knowledge without intuition, which for me means spatio-temporal structures.

Dear all,

Sorry again, memory only comes back slowly:

In fact I was thinking of Yang-Mills theory:

https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory or

https://fr.wikipedia.org/wiki/Th%C3%A9orie_de_Yang-Mills

where Maxwell-equations take a nice form.

Impedance in an electric circuit is complex in general, that is the AC voltage is not in phase with the AC current.

JW and all: You realize that by writing "the AC voltage is not in phase with the AC current" you are using only real numbers.

There is no situation where a complex number is necessary in a physical quantity, by definition -- a physics value must be measurable.

Complex numbers can very much help, as in your example, but are neither necessary nor truthful to the physics. It is just a bookkeeping model, it has no ontology.

Ed,

Do not get you at all

I just finished giving you an example of the complex impedance.

Phase is a measurable quantity in terms of angle, a wave has amplitude and phase, both measurable. Why shouldent I use complex numbers?

You know about the polar representation of a complex number?

JW

JW: you wrote, addressed to me, "You know about the polar representation of a complex number?" Yes, and you too, in terms of real numbers, quantities we can measure. The Platonic model is NOT measurable. I could use a different Platonic model, can't you?

Yet, the physics has not changed, just the complex numbers are not used anymore. Write the Lagrangian, for example. Or, the Hamiltonian. Use PLA. The model is not the phenomena.

All: To agree with a common objection, to treat time as a complex number in Minkowski SR, we recall that there is no situation where a complex number is necessary in a physical quantity, by definition -- a physics value (such as time) must be measurable, but complex numbers are not measurable. Time cannot be complex.

Complex numbers can be very helpful though, as in Minkowski SR, but are neither necessary nor truthful to the physics. It is just a bookkeeping model, it has no ontology. It is arbitrary. One may not use it.

What has ontology is the 4D model, as one may find in situations using tensors. One should not confuse complex numbers with a 4D model of spacetime, when encoding 4D in simple complex numbers for time, without tensors. For example, Cl(3,1) encodes 3D + 1D in multivectors, using reals, but keeps space apart from time, and is therefore useful but not covariant.

Tensors are covariant because if we write A = A in any coordinate system X using tensors, still A = A in any system with constant velocity relative to X. The principle here is physical -- stated first by Einstein, with Galileo applying approximately, to low speeds.

In a similar way, one should not confuse a complex reactance in electronic circuits with a single circuit element, such as a capacitor.

A capacitor can be represented also without any complex numbers, exists, is physical, it is a physical fact.

However, the use of a complex reactance brings the association of circuit elements to a ready calculation in lower frequency, simple algebra. However, it cannot represent that same circuit element, in higher or optical frequencies.

Personally I am not opposed to or in favor of complex numbers, I just wanted to question the strictness of the formal setting of this field.

I rather tend to using symmetric monoidal closed categories when possible.

TK: Yes, the strictness of the formal setting of the complex number field was arbitrary. There is NO God-given Platonic model to obey, neither by Nature, but by humans only.

Other planets, or other civilizations, might not use complex numbers at all, and yet be much more advanced. A complex number has human-based constraints, as we know -- for example, every complex function is also a solution of the Laplace equation, which further constrains using them.

"A solution to the Laplace equation has the property that the average value over a spherical surface is equal to the value at the center of the sphere (the harmonic function theorem, by Gauss). Solutions have no local maxima or minima. Because the Laplace equation is linear, the superposition of any two solutions is also a solution." [op.cit]

Who is talking about the platonic model or a platonic model.? The model is quite concrete., if you know something about inductors, capacitors or resistances.

Do not know if you do or not.

If you say its not a perfect model, models rarely are.

I insist that you can measure both amplitudes and phase differences of AC

signals, as anyone who has spent some time in front of osciloscopes.

It is V=ZI where all three are taken complex.(Z instead of R)

Well, in SR you can either use x,y,z,ict

or you can use the diagonal metric +++ -, whatever one wishes.

I have no idea of what a symmetric monoidal closed category is.

About measurability, I can measure a table, how can I even measure a number,

complex or not. The remark may be true, but nonsensical. If anything the number is a measure in itself, with units of course.

Complex numbers not used any more? I can only laugh at that.

Mathematically the complex numbers form a field as the real numbers, even algebraically closed field: all polynomials split uniquely into linear factors.

The only difference is that the real number field is totally ordered as was said before, whereas the complex number field is not: One cannot compare in this sense two complex numbers, only compare their amplitude.

To form a ratio total order is not necessary. One can imitate in any field the geometric cross-ratio construction for instance. A complex line is two dimensional with respect to the real number line.

Juan Weisz

A symmetric monoidal closed category is roughly a sort of "generalized tensor calculus" or "generalized multilinear algebra".

https://en.wikipedia.org/wiki/Symmetric_monoidal_category

https://en.wikipedia.org/wiki/Closed_monoidal_category

This is NOT to be answered in maths. From a physics point of view, all one can ever have as a measurement is a rational number, which is all a ruler, computer, oscilloscope, or any device can produce.

This is well-known, and infinities are nonphysical. QM and SR must provide a finite, rational number, and they do. One way to do so is to define the real numbers, because of other properties. Then, the outcome of any experiment is a real number.

Mathematically, however, it is convenient to be able to speak of an "actual outcome" as a single object in a nice set of measurements, using real numbers, complex numbers, matrices, and tensors. They are not physically necessary, however, and may be a cause for nonphysical results. Further, every number in the set, if it is genuinely the result of a measure process, is always a rational.

Are all measurable physical quantities rational numbers? I don't think so. Recall Zeno's dichotomy paradox that motion is impossible because to move anywhere at all you would have to go through a completed infinity of half way points. Nowadays we resolve this paradox by allowing that some infinite sums of rational numbers (the geometric progression of half way points of a finite interval) converge to a (in general) real number. While this may not sound like physics, it is because analogue computers can be built to compute functions which are convergent infinite sums of rational numbers (i.e. real numbers) that solve differential equations that model some physical phenomenon. Now it is true that any finite computer will output a rational number, but is also true that increasingly accurate measurements will have a real number as their limit. So the question is: what is the actual value of the quantity measured? Is it the value of the best possible experimental result? I would say that the actual quantity is the real number limit value, even if we cannot ever measure it exactly.

The other thing to note is that vectors actually exist because we have all experienced the effects of momentum for example. If direction were not important, then all moving objects might crash into each.

I tend to think we can measure also algebraic quantities like square root of 2 etc.

For transcendent reals it might be true as long as they are computable.

But for non computable transcendent reals it might take an infinite time to present the result of the measurement, I would rather disagree.

Dear Andrew Powell ,

The computable real numbers form an field, which is closed to many respects, is this true?

The computable complex numbers form an algebraically closed field?

Dear Thomas Krantz,

I would expect that, whatever sense of computability, the computable reals and complex numbers will be fields under the standard addition and multiplication operators. A computable set can mean a number of different things (such as membership definable in an algebra if the algebraic operations are computable functions; membership computable via a recursive operation, which power series are; the set being enumerable by a computable function; or computability relative to a set which is assumed to be computable). For the discussion here I would be happy just to include real and complex numbers that are computable by means of a finite inductive definition over the natural numbers (such as the square root of 2, pi and e).

TK and AP: You guys just "jumped the shark" by TK suggesting here that one can measure in Physics the square root of 2, and AP suggesting here that one can measure in Physics the complex numbers. Misinformation will, eventually, destroy itself.

Physical possibility must come from Nature (the physics-defined independent channel), as an empirically-defined world, which can be at odds with a mathematically-defined world.

Please do not confuse measurement in Mathematics, with what is possible in Nature, in Physics, as https://arxiv.org/html/physics/0102047 did. And it is well-known that the Heine-Borel theorem contradicted Zeno's dichotomy "paradox". I suggest reading https://hbr.org/2019/05/how-to-overcome-the-bias-we-have-toward-our-own-ideas

EG: If you can measure ratio's by a geometrical construction, you can also measure geometrically constructible quantities like square root of 2.

Measuring a complex number is grosso modo localizing a point on a plane. That you can do by a scanner, a camera etc.

The previous remark was more about which numbers can be represented by an automatic device. Any real number cannot be, when the decimal development can not be reproduced by an algorithm with finite information.

Anyhow this is only a theoretical aspect, you cannot go below a lower limit. (Planck distance?)

I know what Ed. means, we hardly ever agree but we will here.

In early science courses you study error in measurement, precision and so on.

In practice it means that due to limitation of measurement processes, you only have the result within a few decimal points, plus or minus some error estimated.

That is why he sais all the results are rational.

A few things like frequencies can be measured with great accuracy these days,

say 1 part in ten to the power ten, but still rational nos.

Ed Gerck "Time is not complex."

In Minkowski SR math of 4-D, time is not complex, it is imaginary. The square of time is real in the 4-D representation (1,1,1,-1). This is just convenient math to treat time as a distance dimension using light speed to convert units.

Physically time is not a distance dimension, and not imaginary. The 4-D is complex math, but with possibility to derive a real metric related to Physics. Minkowski is helpful for SR to emerge from the math rather than having SR arbitrarily inserted.

One of my greatest objections to published material is that the math has usually been mistaken for the physics. More objectionable is that generations of researchers don't seem to understand the difference after it has been explained, as EG has found in this thread.

Dear Ed Gerck , Juan Weisz , Jerry Decker ,

The main problem with saying that the only quantities we can measure (in physics or elsewhere) are what a calibrated instrument reads out (up to a margin of error), which can be represented as a finite interval of rational numbers, is that the role of theory is ignored. I assume that experimental readings are useful because they can be used to test (support or refute) the predictions of a scientific theory. We can then ask what the actual value of the variable that we are measuring. Is it a rational number or it is what the scientific theory predicts? If we consider a prediction that involves a real number like pi, say the time period of a simple pendulum for small angular amplitudes of swing, then we see that could approximate pi by conducting a series of experiments by varying the mass and length of the pendulum (assuming g, the gravitational acceleration, is constant and the angle of swing is small). The formula for time period of a simple pendulum is a limit case, but we can correct for errors. I think this argument suggests that the time period is a real number rather than a rational number.

Yes,

If you measure everything to two decimal digits, it makes no sense to use many more digits of pi than that. I have not ever used any more than 8 digits of pi.

Dear all,

Referring also to the initial question, I would like to ask assistance in understanding two aspects relating non-locality of quantum physics to the non-commutative nature of quantum operator algebras.

1) The Bell theorem

https://plato.stanford.edu/entries/bell-theorem/

2) The Kochen Specker theorem

https://plato.stanford.edu/entries/kochen-specker/

Can their mathematical content be made more precise?

Best,

Tom Krantz

Dear all and in particular the experimental physicists,

I would like to recommend this book that made me understand that quantum physics is not limited to mathematics alone.

Best,

Tom

The principle of uncertainty related to the noncommutative operators,

has in principle nothing to do with nonlocality aspects. They are two different strange aspects of the quantum.

Of course its not math alone, you have to learn all the experiments that forced this on us, starting from Black body radiation and Max Planck, low temperature specific heat, diffraction experiments, the Hydrogen atom lines, spin experiments an so on.

Particle physics is even more involved.

Not so easy.

Dear Juan Weisz ,

You can trust me, I'm mathematician ;-)

I would like to see out of the mathematical formalism, what it means that a phenomenon is local or non-local. I don't understand yet. Non local could be related to a position operator non commuting with an impulsion operator.

Combinatorially, and simplifying as much as possible, if space consists in two points A and B, non-local could mean that the wave function f(A) in A is linearily related to the wave function f(B) in B. Practically one would have f(A)=-f(B). Right?

Dear all,

I got some partial answers to my preceding question in Chapter 3 by Chris Isham in the following book. Very interesting.

The book traces a summary of the recent research on categories applying to quantum theory. Monoidal categories become a subject of study in themselves and a bit disconnected from physical reality though.

Best,

TK

Thomas

Very well,

local means the usual in normal physics, that if you want to move something,

you apply the force on whatever you want to move, and not 2 miles away. So

you call it a local force.

In nonlocal, you complain as Einstein did, about spooky actions at a distance,

in a famous article, he gives the example of two spins corresponding to two particles going off in opposite directions, coming from a comon source with spin zero, and if you just measure the direction of one of the spins, you automatically know the direction of the other spin, even though it is very far away.

There are two main positions about this problem. One of these positions holds that

the two spins always keep the two particle wave function in some continuous way

at all times, so they are always correlated even though far away, so the result does not surprise.(That is the non local action)

The other position states that this would be some inexplainable mystery.(a real spooky action)

Usually people say quantum phenomena are holistic (unseparable) so you have to address the whole, not each part (the case of the two particles)

Position operator not commuting with momentum operator along the same direction, leads to the uncertainty principle,an inability to measure both at the same time. This princile has nothing to do with nonlocal, as I said before.

The nonlocal actually means that the total wave function of the two particles is not factorizabe as the simple product of independent wave functions, each one for a single particle. If it is factorizable, it would mean local...relate to probability ideas.

Regards, Juan

Dear Juan Weisz ,

Thank you for the developments and the explanations you are giving.

Best,

Tom

Dear Juan Weisz , dear all,

I jhust saw a chapter on this topic in Halvorson's book:

Chapter 8 - Einstein Meets von Neumann: Locality and Operational Independence in Algebraic Quantum Field Theory by Miklos Redei.

Best,

Tom

The statistics of the particles is also vital in this discussion.

Spin 1/2 particles, such as electrons, have Fermi-Dirac statistics,

so if you interchange the positions of the particles, the wave function changes sign (it is antisymmetric). The two particle wave function is antisymmetric.

For Bosons (integer spin) the wave function is symmetric.

Yet another, very well written book, which explains the axioms on which Quantum mechanics is founded.

In the article below, I argue that the fact that the complex plane has real dimension 2 plays a fundamental role in the Born rule. Hence other fields might not give the right probabilities.

Preprint Quantum Fractionalism: the Born Rule as a Consequence of the...

Dear Friends,

There was a related discussion that may be interesting if not too amateurish for the experts here

https://www.researchgate.net/post/Are_Universality_and_Complex_Numbers_Enough_for_Quantum_Computing

It doesn't appear that Scott Aaronson's writing on the role of complex numbers in quantum theory is linked to above but anyone interested in this question would want to read what he has to say

https://www.scottaaronson.com/blog/?p=4021

It seems like the question raises the point of departure taken by Feynman and the early theorists in quantum computing. When we say "use of complex numbers" it begs the question: Who/What exactly is using complex numbers? Or to be more formal: What is it that is realizing (storing and computing with) information in the form of complex amplitudes? For physicists this would be a "quantum system" or "quantum particle".

Moore and Mertens point out that "complex numbers", per se, aren't necessary but negative numbers are - because, of course, we can represent the structure of information found in complex numbers using only real numbers and matrices.

But that seems arguable since we would still really be using the structure of information we call a complex number - a vector, or object with magnitude and phase structure that is visualized as a point in the plane.

So, yes, it could be argued that using real matrices is not really using complex numbers. Well ... then why the extra structure?

But, Scott Aaronson makes a pretty good case that we get squeezed into a corner with using complex numbers. Yes, we can use real or even quaternions if we force them to act like complex numbers (and we wish to torture ourselves ...)

Tom, it would be interesting to hear how you, as a mathematician, see the link to the fundamental structure of measures on a plane (stereographic projection of a sphere of course). Then requiring that kind of measure to respect the rules of a probabilistic model of computing and we seem to get quantum mechanics (by Gleason's theorem). Maybe the fundamental thing here is the plane (projection of sphere).

Last year I heard Vaughan Jones give his talk What Is It About The Plane in which he says, "The plane is completely adequate for representing information in 3 or more dimensions. When you jump from 1 to 2 dimensions you get a huge gain ..." Here recorded at UCLA

https://www.youtube.com/watch?v=B2G3FseJteQ

Regards

Dear C. Jones ,

Thank you for your nice contribution to the topic. The references you give make visible new aspects of the question, and I need to take the time to consult your sources.

Beyond technical aspects of one or the other field that might be used or their representation (e.g. representing quaternions by a pair of complex numbers) I am also interested what forces the admittance of certain axioms von Neumann well puts in evidence in his Mathematische Grundlagen-book.

Which are the reasons for these axioms (like separability of the field)? Are they of practical or experimental nature, of philosophical, of esthetic and so forth. As far as reason is concerned one can or not require them. Why did the early quantum-theorists choose them?

Bests,

Tom

Tom,

I am no expert in early quantum history. But, from what I recall of the papers and books I've read on the subject, it would be generally correct to say that the early quantum theorists did not choose the structures of the quantum model so much as they were forced into them by the nature of their experiences (experiments/evidence) and had to patch together the necessary pieces without knowing really where it was all going in the beginning. A commonly heard sentiment from across the early founders of quantum theory is how surprising the outcome was to them. No one started from some kind of assumptions or first principles and those principles led them to Hilbert spaces, complex numbers, the measurement postulate, etc. Some prominent founders (Einstein perhaps) were not at all convinced that things had really ended up where they had. And then Von Neumann was on scene with the mathematical chops to nicely formalize the postulates as they were being used - which nevertheless include fundamentally incompatible processes (evolution and measurement).

Some argue that we are forced into the structures of quantum theory because it is the structure of physical reality (the ontic view) while others argue that we are forced into the structures of quantum theory because it is the structure of how we can know physical reality (the epistemic view). (Of course, these could both be true if physical reality is a model that contains the quantum model and if our model of cognition is quantum or contains the quantum model.)

The question: "Why the quantum?" or "Why does quantum theory (or physical theory) have to have this particular feature?" is a wide open question in quantum foundations research. Obviously that it is still open means that the founders of the theory did not have "reasons" for choosing the postulates beyond that it agreed with their experiences (experiments/evidence).

Regards

Most of the time you can get away with purely real eigenfunctions, but in certain situations (tunneling or magnetic field) you really need a complex part of the eigenfunction. That is why the saying is that in general a wave function is a complex function of space and time.

Dear C. Jones ,

You raise some good points. Scott Aaronson is always an interesting read. I can believe that (to use Aaronson's phrase) the complex numbers are the Goldilocks field for use in quantum mechanics, that the quaternions impose too much structure, while the real numbers do not impose enough. Does this mean though that quantum mechanics is somehow two dimensional? Clearly in one sense not, because in terms of spatial co-ordinates, the wave function is a function of the three dimensions of space and one of time, but also yes because the co-domain of the function is a two dimensional domain of waves the square of the amplitude of which represents a probability (in the case of a pure state at any rate). Personally (and this is not an orthodox opinion) I think that the complex numbers should be interpreted in terms of a geometric algebra on (usually spatial) unit vectors. The reason why I think that is that otherwise is that (sorry Thomas Krantz) abstract algebra does not provide an ontology by itself, and the complex numbers lack an obvious ontology (other than as field closed under roots of -1).

Dear @Andrew Powell and dear @C. Jones,

Being quite busy these days It will take me some more time to measure fully the content of your observations which sound very interesting. Let me jhust say this: Complex numbers are principally for electromagnetism the most important field. It seems to me that when other particles are concerned then other "fields' play a role: quaternions or even octonions etc.

Agree with Christian, you can always use Matrix mechanics,

it is just a little tougher to do things that way, but not at all

impossible

.

When complex numbers are replaced with geometrically more advanced objects - even elements of geometric algebra, where bivector planes can be explicitly defined, we get a chance to work in a frame, deeper than conventional quantum mechanics.

Physics is not math, and you lay your hands on any reasonable mathematical scheme that does not betray

essential physical rules.

You can repeat the same physics by using different mathematics, as has often been done with Maxwell equation

or the equivalence between Schrodinger and Heisenberg mechanics.

In Classical mechanics you get essentially the same physics using Newtonian Mechanics, Lagrangian Mechanics or Hamiltonian Mechanics.

Dear Juan,

It seems to me you did not realize that the frame I was talking about is another theory, deeper than conventional QM known to brainwashed physicists. The results of conventional QM can be lifted to that deeper theory giving new insight of what is going on in reality.

Physics can be expressed more clearly , or less so depending

on the mathematics used.

for example the Maxwell equation have been beautifully

summarized in very to notched mathematics of Elie Cartan,

but an engineer looking at it would have no clue of how to

use this knowledge in practice.

You get deeper in math but bury yourself in Physics.

That is my point of view.

Sorry, we live in different worlds.

regards, juan

Dear Juan Weisz ,

Physicists should learn from mathematics in doing things properly!

But I agree that 'nature' studied by physicists has 'swamps' that it is not easy to deal with. If concepts are good, a theory has much more chance to be reliable.

maybe the spectrum of a observable $a$ is not empty set (in Mackey's scheme) and in real Banach algebra we consider its complessification algebra to avoid this situation (see Bingren li - real operator algebra- pag 7).

This should direct the use of complex algebras .

Dear Thomas,

Complex numbers and variables can be useful in classical physics. However, they are not essential. To emphasize this, recall that forces, positions, momenta, potentials, electric and magnetic fields are all real quantities, and the equations describing them, Newton’s laws, Maxwell’s equations,etc. are all differential equations involving strictly real quantities.

Quantum mechanics is different. Factors of i = √ −1 are everywhere, for example from Heisenberg we have his famous commutation relation,

QP − P Q = ih, ¯

and from Schr¨odinger his equally famous equation

h/i ∂tΨ = HΨ.

The wave function Ψ is complex and so is practically every other quantity needed to formulate quantum mechanics. Granted that classical mechanics fails at distances of nanometer or less, why is it suddenly so necessary that everything be complex? But even more intriguing is the question of what happens on much bigger scales, centimeters, meters, and larger. Does quantum mechanics still govern things, or does it somehow break down?

Returning to the question of why quantum mechanics requires complex quantities, one reason becomes apparent upon trying to describe wavelike phenomena for material particles, electrons, neutrons, etc. To see this point, start with a classical electromagnetic wave and understand the necessity of quantum mechanics in term of complex numbers.

I think it is yes. In case of wave functions, for bigger scales complex numbers are essential for quantum mechanics.

Hope it is helpful.

Ashish

But bohr's heisenberg, schrodinger has fully explained how the quantum theory is strict to the complex number. I may be wrong but with interpretation it seems that we must be strict to the complex number provided scale is large rhather than using nano level scale. With large scale particle weight will be high and in that case classical mechanics is complex in nature.

Ashish

In fact the question is do we need algebraic structures richer than the structure of "ordinary numbers". Yes, we do, but that doesn't mean that it is essential to do so. In the end, the results of any measurement are "ordinary numbers". However, the calculations leading to the "ordinary numbers" can be made faster if we adopt certain richer and more complex algebraic rules.

That is what I say. Thank you very much. Complex number is essential tool to diagnose the wave functions and hence it adhered to the quantum mechanics.

Ashish

VD, AT, and all: The calculations leading to the "ordinary numbers" ARE THOUGHT to be made faster if we adopt certain richer and more complex algebraic rules, but actually it depends on the tools used. Nothing else is truly essential, except "ordinary numbers". Quantum mechanics (QM), for example, doesn't depend on complex numbers, which cannot be measured. QM can be done with "ordinary numbers", as any computer will tell you.

So-called “real numbers” are not needed either. They are used for their “nice”, err imagined properties, but they are not actually real — the reality that we can see and measure is limited to finite rationals, which can be mapped to integers -- "ordinary numbers" -- i.e., a digital number is all we get out of a calculator, a computer.

Ed Gerck

OK. I think I know what you mean. Let us be practical and answer the following question:

In your opinion, should we say that is equal to Dirac's delta(x-x'), or is it Kronecker's delta symbol deltaxx' ?