Somehow individual measurements are physical but tabulating them makes them incapable of being physical. — fdrake

How you tabulate results is a matter of convention, for communication, or performance. You can represent classical waves as sine waves: completely real. Or, for ease, you can represent them as complex waves. For the exact same wave. The latter adds information that is related to the former (because imaginary sines are related to real cosines and vice versa), but we're not transforming the physical wave from a real field to a complex field. It's still real. So how we tabulate things can't change their physical reality.

Nature can "know" about the vector elements but not the vector — fdrake

Yes. Vectors are a way of dealing with multiple similar quantities which transform in similar ways. Some of those quantities may be related to one another, but there's nothing I can think of that makes the interpretation of the vector as anything more than a notational convenience.

EG: if the criterion for a theory (as a whole) being physical is successful prediction of experimental results ("manifesting as real"), it's silent on theory elements... — fdrake

Beyond falsification, which depends on the element and how it works inside the theory, yes.

I won't quote the rest, just sum up. A theory is tested empirically, not it's individual elements. If the theory as a whole (or a subset of elements, catering for irrelevancies to a particular experiment) yields otherwise inexplicable or more accurate predictions for experimental outcomes, it's a good theory.

What we see in QM is that it's a good theory, but contains various combinations of elements to which experimental outcomes are insensitive. It might be that you cannot remove or replace or augment just one element, but the whole combination.

This is evident in the various interpretations of QM. in Copenhagen, the electron is a complex wave, the field acts linearly, there is one measurement outcome and spontaneous collapse. In MWI, the electron is a complex wave, the field acts linearly, there are an infinity of outcomes and the wave evolves forward in time deterministically. In Bohm, the electron is a real, classical particle, the field is nonlinear, there is one outcome, and the particle evolves forward in time deterministically. In transactional QM without my edits, the electron is a complex wave, the field is linear, there is one outcome, the wave evolves forward and backward in time but probabilistically.

They're not all right, yet they all, in principle, yield the same predictions of the same experimental outcomes. Nature cannot care that much how we represent it.

Yes. Vectors are a way of dealing with multiple similar quantities which transform in similar ways. Some of those quantities may be related to one another, but there's nothing I can think of that makes the interpretation of the vector as anything more than a notational convenience. — Kenosha Kid

To clarify: let's imagine that someone's repeated a measurement of a mass twice. They've written both down. The first measurement I'll call "a", the second measurement I'll call "b". Which of these (if either), do you mean:

(1) The process of aggregating both measurements into a vector (a,b) is not physically meaningful (I agree with that).
(2) Representing things as vectors is nothing more than a notational convenience.

I'll assume the premise:

(3) Things adopted for the sole reason of notational convenience are not physically meaningful.

I think that's justified from the remarks you've made about complex representations of mathematical objects being convenient tricks.

So, I don't think you mean (2) if you also think (3), as that rules out vectors from being physically meaningful. So that goes for all vector quantities! And then typical objects of physics are no longer physically meaningful by that rule. So I'll assume you mean (1).

I won't quote the rest, just sum up. A theory is tested empirically, not it's individual elements. If the theory as a whole (or a subset of elements, catering for irrelevancies to a particular experiment) yields otherwise inexplicable or more accurate predictions for experimental outcomes, it's a good theory. — Kenosha Kid

If you mean (1), I don't think it applies to the context I meant. I took something which was not physically meaningful (the complex number x+yi) because it had an imaginary component, fed that number through an isomorphism of structures into a real valued matrix which could not be ruled out of being physically meaningful on that basis. The two structures are equivalent, but the criterion of "must not contain an imaginary number" rules the first out of physical meaningfulness but not the second.

I also don't believe you have applied this doctrine: "A theory is tested empirically, not its individual elements" consistently, though there is a lot of ambiguity between going from talk of whether an element in a theory is physically meaningful and whether the whole theory is good. Regardless, I see a few cases:

(A) You are happy to declare that a whole theory is unphysical if it relies upon complex numbers.

I don't see that as plausible since you've said you see the wavefunction as ontic in some regard, and it relies upon complex numbers. If the criterion of whether a theory is physically meaningful or meaningless is determined by the accuracy and precision of its predictions (rephrasing your quote), it won't care whether the theory contains complex numbers anyway - so declaring a theory non-physical on the basis of it requiring complex numbers is an equivocation. Up to suspicions about needing to remove complex quantities from physics, anyway.

(B) You are happy to declare that an element of a theory is unphysical if it relies upon complex numbers.

I see that as plausible, as there are examples of you doing it in the thread. Which goes against the other idea (up to ambiguities) that a theory is physically meaningful iff it makes accurate predictions, and seems in tension to me with having ontic commitments to the wavefunction.

My remarks in thread have been in the context of (B), and I don't really want to get into a Motte and Bailey situation. Motte: complex numbers are unphysical. Bailey: the mark of a good theory is its capacity to produce accurate predictions. The issue in the Motte is to my mind about how one manages ontological commitments within theories (maintaining an ontic commitment to the wavefunction while claiming complex quantities are non-physical). The issue in the Bailey is to my mind the claim that good theories come with accurate predictions. I'm not picking a bone with the Bailey, I'm picking a bone with the Motte.

The general perspective I'm coming at this from is some sort of scientific realism, I'm ontologically committed to the existence of entities in scientific theories. I strongly agree with this:

Nature cannot care that much how we represent it. — Kenosha Kid

Which is why I'm pressing the issue; if nature doesn't care how we represent it, why would whether something could be physical or not vary with an isomorphism of structures? Why would a criterion to decide whether a structure is physical decide differently depending upon which representation of a structure you choose?

I'll give you a few examples of why the complex wavefunction might be complex because of representation-specific factors, which I think will answer your question.

1. The wavefunction contains unphysical information

The wavefunction doesn't just encode the dynamics of the particle, but of similar particles in similar scenarios. The actual particle is described entirely by real quantities, but we cannot know them and the minimal representation that covers them all is complex. This is the case in Bohmian mechanics and can be taken as an extreme case of the OP in which only real trajectories are ever tried.

In which case the best, minimal representation of the particle is not an isomorphism of the complex wavefunciton, avoiding the problem you see,

2. The wavefunction contains physics not about the particle

This is where I'd hedge my bets the most, which is essentially that, due to our choices about how to represent particles, the minimal quantity we can deal with has to be complex in order to yield good experimental predictions. In this case, the truest minimal representation of the particle (i.e. in a better representational framework) would again not be isomorphic with the complex wavefunction.

This is the case with special relativity, where to get the right answers out in simpler representational frameworks, some of the four-vector elements have to be complex. This is avoided by a more sophisticated framework involving metrics, which is not typically done in QM except in attempts to do QM in curved spacetime (as far as I am aware).

3. The wavefunction contains only information about the particle but in a suboptimal representation

Here we might not gain or lose information about the particle as we hop from one representation to another, and in fact the relationship might be isomorphic. However, in our choice of representation, that information must be encoded as a complex field. This is, I think, the one you have in mind.

An example might be that space is 4D, containing an additional compactified dimension, and that the true meaning of the complex phase is a coordinate along this angle. So what we represent as $\psi(\mathbf{x}) = f(\mathbf{x})e^{iA(\mathbf{x})}$ (such that $n = f^2$) might better be something like $\psi(\mathbf{x}, \theta) = f(\mathbf{x}) cos(\theta)$, which yields all of the desired interference effects.

This is an isomorphism to and from the complex wavefunction and would be physically meaningful. However the complexity of the wavefunction would then simply be a non-physical artefact of an incorrect choice of representation, i.e. a representation that does not correspond to the physical universe. Of course, we could only say that if there was some test for it. Maybe some future test is possible, but for the time being the wavefunction is not an observable, and, were it observable, QM says we would not observe it as having a complex value*, which is sufficient (I think) to state that it is non-physical as represented.

*The condition of real numbers for observable quantities is a postulate of quantum mechanics, manifest as the insistence on Hermitian matrices to represent physical operators. This is all I really had in mind when I referred to the complex wavefunction as nonphysical, i.e. the minimalist thing we can do with the wavefunction is to act on it with the Hermitian density operator. That said, I think I'll defend the broader point you're taking me to task on.

Which is why I'm pressing the issue; if nature doesn't care how we represent it, why would whether something could be physical or not vary with an isomorphism of structures? Why would a criterion to decide whether a structure is physical decide differently depending upon which representation of a structure you choose? — fdrake

Note that electric impedance is a complex variable. So there exist classic physical variables described by complex numbers.

Note that electric impedance is a complex variable. So there exist classic physical variables described by complex numbers. — Olivier5

That is also merely a convenient representation. Voltage and current are real, but in AC currents they are sinusoidal. Complex exponentials are easier to manipulate mathematically than sinusoidal functions, and one can simply dismiss either the real or the imaginary part at the end. Impedance is then the ratio of the complex voltage to the complex current. This does not mean that the voltage or the current are truly complex; it is merely convenient to treat them thus.

If complex numbers fit the bill better than real numbers to describe a particular phenomenon, maybe it means something...

If complex numbers fit the bill better than real numbers to describe a particular phenomenon, maybe it means something... — Olivier5

It does mean something. It means that complex exponentials are easier to do calculus with than sine waves.

Well, if complex numbers are nothing more than a mode of computation, there's no reason to worry about their use in the wave function.

Well, if complex numbers are nothing more than a mode of computation, there's no reason to worry about their use in the wave function. — Olivier5

Thanks, I'll try not to lose sleep over it.

l suppose the reason why complex numbers are good at calculating sinusoidal functions is that they are good at modeling rotations. An alternative current is created by the rotation of a dynamo, reason for which it's sinusoidal. Maybe that's the reason they are used in the wave function as well: it's a wave after all so Fourier's methods must apply?

it's a wave after all so Fourier's methods must apply? — Olivier5

Kenosha seems undecided on this. On the one hand Kenosha seems to insist that the wave function represents real waves. On the other, Kenosha asserts that the wavefunction is the representation of a particle. Someone, other than me, ought to tell Kenosha Kid that particles and waves have completely different spatial-temporal representations. And, the point which Kenosha refuses to acknowledge is that there is an incompatibility between the two representations which renders them as incommensurable.

The obvious problem is that the medium ('ether') of the electromagnetic waves has not been identified. Therefore the real properties of the waves cannot be observed or determined. Instead, these waves are represented by ideals such as sine waves. However, since these waves are not ideals, but real physical entities, with real spatial-temporal constraints, there is a degree of uncertainty in application. So until the real medium is identified, and the real waves are studied, the incompatibility between the two distinct spatial-temporal representations will remain.

As intriguing as complex representations in physics, for me, is how linear operators are so effective. One would think nature to be complicated and non-linear; linearity is a very stringent condition, while simplifying the math. However, it is a seasoned trick in the profession to approximate the non-linear by linear constructs, and, of course, ordinary differentiation and integration are linear operators. The simplest of functions on C, such as f(z)=az+b, are - according to most definitions - non-linear, although f(x)=ax+b is a linear function on R since its graph is a line. The word linear has several interpretations depending on contexts. The elementary function f(z)=e^iz is non-linear.

The Kid's real problem is with indeterminacy, which seems to give him a lot of trouble, so he is looking for a determinist interpretation of QM. Call that Newtonian nostalgia if you will. Apparently one such interpretation would involve the postulation of an infinite number of parallel universes. My limited understanding of it is that if one flips a coin, two universes get created, one in which the result was tail and the other where it was head.

I think that's pushing risk aversion a little too far.

As intriguing as complex representations in physics, for me, is how linear operators are so effective. One would think nature to be complicated and non-linear; linearity is a very stringent condition, while simplifying the math. — jgill

i think what is at issue here is the geometrical shape of a fundamental unit of space. If we propose a fundamental unit of space, as an infinitesimal, then the shape of the infinitesimal will influence the way that we conceive of the possibility of motion through space. Notice that the two fundamental representations of 2d (and consequently 3d) space, the square and the circle, are fundamentally incompatible. Straight lines can never be reconciled with curved lines. The two representations will not merge, and incompatibility is demonstrated every time we try. Though we have developed many ways to work around this issue, when we get to infinitesimals the difference becomes significant.

Consider the difference between representing an area of 2 dimensional "space" with cubes, and with circles. The cubes can be placed side by side, and all the "space" will be covered. This does not work with circles. Side by side circles will not cover the "space". So now we need to overlap the circles, and the process of representation becomes very complex. Is there a fundamental circumference size, or do they vary? If we assume that all fundamental, infinitesimal circles (spheres in reality), are the exact same size, then there is the complicating factor of position their centers, (points). The relationship between centers must be represented, and now we tend to fall back on the square representation.

But if we want to maintain the status of "fundamental unit" assigned to the infinitesimal circles, we cannot undermine this assignment by relating the circles to each other with squares, because that places the square representation as more fundamental than the circle. Therefore we need to disassociate "the point" from the dimensional representations (lines, squares, cubes), the point being non-dimension in essence anyway, and reconstruct a representation of 'real space' using curved lines, which is completely independent from, and not influenced by that faulty dimensional representation.

This would create fundamental circles, but then we still need to determine what type of relationship one point has to another, and this is where the real difficulty lies. The curved lines would represent the essence of space, but the relationship between non-dimensional points would represent the essence of time, being prior to space and therefore non-dimensional.

As intriguing as complex representations in physics, for me, is how linear operators are so effective. One would think nature to be complicated and non-linear; linearity is a very stringent condition, while simplifying the math. However, it is a seasoned trick in the profession to approximate the non-linear by linear constructs, and, of course, ordinary differentiation and integration are linear operators. — jgill

Yes, perturbation theory being an obvious example. Quantum electrodynamics is usually treated with perturbation theory, with each term in the perturbation series being a Feynman diagram.

the square and the circle, are fundamentally incompatible. — Metaphysician Undercover

Although topologically the same.

Straight lines can never be reconciled with curved lines — Metaphysician Undercover

I have done quite a few investigations into linear fractional transformations, and one feature that makes them important is they transform Circles into Circles, where the capital C is in recognition of the fact that a straight line is simply a circle with infinite radius. This has to do with the Riemann sphere.

It appears the rest of your post goes into the hyperreals, where others on TPF have greater competence.

I have done quite a few investigations into linear fractional transformations, and one feature that makes them important is they transform Circles into Circles, where the capital C is in recognition of the fact that a straight line is simply a circle with infinite radius. This has to do with the Riemann sphere. — jgill

A circle with an infinite radius is an incoherency. This is exactly the problem I am talking about, the faulty attempts by mathematicians to make circles compatible with straight lines. It necessarily results in incoherency.

The logical thing to do when faced with this glaring incompatibility is to address the nature of reality, and attempt to determine the reason for that incompatibility, rather than to attempt to veil it, or cover it up with such incoherent principles. It's no secret that mathematical axioms may contain incoherency. If the incoherency is veiled, and the axioms are useful, they will be accepted by convention. To bridge a gap of incompatibility, such as the relation between the non-dimensional point, and the dimensional line, we can use whatever means is proposed by mathematicians, and proves useful to that end, but unless the real nature of that divide is understood, there is no truth provided by the application of the axioms.

We have learnt over thousands of years of application, to measure distances between objects with straight lines and 3 dimensional representations. However, we've come to conceptualize force, which is the driver of motion, as based in torque, and this is rotational. Naturally, we fall back onto the only means of measuring which we have, the three dimensional straight lines which we know and love, employing things like vectors to represent rotational force. So the incompatibility rears its ugly head. Why do we insist on developing all sorts of convoluted and complex mathematical axioms designed to bridge this gap of incompatibility between the measuring technique (straight lines). and the actual reality (curved force), instead of dismissing that 3d measuring technique altogether, as inadequate, and delving into the true reality of curved existence?

The first principle of the curved space is "the point", which represents the center of the circle. We often even imagine a point as a tiny sphere. But the point is non-dimensional, so a spherical representation is unjustified. The second principle, is that if we model a three dimensional space as surrounding that point equally, we have irrational ratios (incoherency), known as pi and the square root of two. So the first logical conclusion is that equality in the dimensions of space is a false premise, irrational. We cannot represent space as having equal dimensions. Therefore a rotational force such as torque, cannot act equally in each of the dimensions which it is represented as acting in. Such a representation is an ideal, based in the necessary spatial equality of eternal circular motion (Aristotle), which is not a reality. The reality of such circular motion was dismissed when the orbits of the planets were discovered to be ellipses

So let's say that a force, which is derived from a non-dimensional point, must have a start. And, that start must be directional, it cannot be equal in all directions. The start cannot be equal in all directions because this is denied by irrationality. (Of course one might argue that reality is irrational, but that's pointless, and contrary to the vast evidence we have of our capacity to understand reality.) So we ought to dismiss the irrational approach of spatial equality surrounding a point, and start with the premise that a force emerging from a non-dimensional point is necessarily directional. The direction is not a straight line though, perhaps like a spheroid without symmetry because it necessarily has a starting direction.

It appears the rest of your post goes into the hyperreals, where others on TPF have greater competence. — jgill

Thanks for the heads-up, but obviously I don't accept the reals as an acceptable solution to the incompatibility described above, so the hyperreals are completely irrelevant to me. Therefore i would not adhere to any such principles. I approach infinitesimals from the metaphysical perspective of the problem in establishing a relationship between spatial extension and temporal continuity, like Peirce, not from the perspective of how mathematicians attempt to deal with this problem. The mathematical approach, of reducing rotational principles to straight line 3d representations, I reject as incoherent, just like your claim that a straight line is a circle with infinite radius.

Maybe that's the reason they are used in the wave function as well: it's a wave after all so Fourier's methods must apply? — Olivier5

They do. Generally, Fourier transforms of real functions are complex. The Fourier transform of a wavefunction of position and time is the wavefunction of momentum and frequency. Any wavefunction with nonzero momentum yields a complex wavefunction of space, thus the ground state wavefunction of a stationary system is generally real. This gets back to the fact that it is motion -- things changing position with time -- that introduces complexity. This is true also in special relativity, where the four-current in the vector representation is complex for nonzero current (for the + - - - convention... The charge is complex in the - + + + convention). This goes away in the tensor formulation of relativity because the spacetime metric handles the manipulation of real four-vector elements to yield real observables. Metrics are not used in QFT afaik, except in attempts to derive a QFT of gravity. Perhaps they should be, and perhaps if they were all wavefunctions would be real.

As an illustration, the four-momentum in vectorial relativity is: $P = [ E, i p_x, i p_y, i p_z ]$. The magnitude is then $E^2 - p_x^2 - p_y^2 - p_z^2 = E^2 - p^2 = m^2$ where units have c=1. This gives us the Einstein equation: $E^2 = p^2 + m^2$. The imaginary number is required to get minus square of momentum into the equation.

In tensor notation, we instead have the metric encode the relationship between time and space coordinates. R is a square matrix with diagonal (1, -1, -1, -1), 0 elsewhere. $P = [E, p_x, p_y, p_z]$. The above is then $P^aR^a_bP_b = m^2$. Everything is now real because the relationship between time and space has been removed from the quantity under consideration (the four-momentum) and placed in a tensor that solely handles that relationship.

Hey, just to let you know, I want to think on this some more, don't want to reply just for the sake of saying something.

A circle with an infinite radius is an incoherency. This is exactly the problem I am talking about, the faulty attempts by mathematicians to make circles compatible with straight lines. It necessarily results in incoherency. The logical thing to do when faced with this glaring incompatibility is to address the nature of reality, and attempt to determine the reason for that incompatibility, rather than to attempt to veil it, or cover it up with such incoherent principles. — Metaphysician Undercover

Go right ahead and spin your metaphysical web. Like the flock of sparrows now sitting on my fence, the peanut gallery awaits your penetrating views. :cool:

Like the flock of sparrows sitting on your fence, the peanut gallery has no interest in the true nature of space and time. The truth about reality is just too far removed from what they believe about reality, so they are unprepared to even set their bearings in the right direction.

Hey, just to let you know, I want to think on this some more, don't want to reply just for the sake of saying something. — SophistiCat

No worries Cat. I look forward to your challenging responses.

Like the flock of sparrows sitting on your fence, the peanut gallery has no interest in the true nature of space and time. — Metaphysician Undercover

But nonetheless can be counted upon to show up anyway.

In case you haven't noticed, TPF is full of them. So you may feel right at home here with your far fetched idealism.

The truth about reality is just too far removed . . . — Metaphysician Undercover

I suspect I will not understand the truth about reality when you reveal it, but I'll give it a try. :up:

I will never reveal such a thing, because it is not understood by anyone. But I can often tell when a theory takes us in the wrong direction.

I will never reveal such a thing, because it is not understood by anyone. — Metaphysician Undercover

Ha ha haaaaaa

Although, seriously, don't. Start a thread, by all means.

I already showed you how your thesis, which is a turning away from the vast array of evidence that energy is transmitted as waves, towards a theory which treats this transmission as a movement of particles, is a turn in the wrong direction.

I already showed you how your thesis, which is a turning away from the vast array of evidence that energy is transmitted as waves, towards a theory which treats this transmission as a movement of particles, is a turn in the wrong direction. — Metaphysician Undercover

I thank you for your input. I disagree with your analysis and do not see it as consistent with QM. As I said on page 1, whatever alternative theory you have outside of the QM framework might make an interesting thread in its own right.