Understanding the Complexity of Wavefunctions in Quantum Mechanics

[naqo]

Hi there, i have been studying a bit about QM, but ther's one fundamental question
about the wavefunction i can't understand: why is the wavef. defined complex? I mean,
couldn't one work from the beginning with a real wave?

Thanks

 

[Ben Niehoff]

One explanation is to consider a time-invariant potential, such as the infinite square well. Because the potential does not change over time, we know that the probability of finding an electron of a single energy eigenstate must also be constant in time. Therefore, we know that

\frac{\partial}{\partial t}|\psi|^2 = 0

Expanding in terms of the real and imaginary parts,

\frac{\partial}{\partial t}\left(\Re(\psi)^2 + \Im(\psi)^2\right) = 0

2\, \Re(\psi) \frac{\partial}{\partial t} \Re(\psi) + 2\, \Im(\psi) \frac{\partial}{\partial t} \Im(\psi) = 0

Now, if \Im(\psi) = 0, then this directly implies that

\frac{\partial}{\partial t} \Re(\psi) = 0

which would mean that \psi is not changing at all! This is a bit problematic, because then \psi is not a wave; it's just the square root of a probability distribution. It would have no frequency, because it would just be a fixed function in space.

The theory, if based on real waves only, would lose its ability to talk about the time dependence of energy eigenstates. Which means if a particle in an energy eigenstate ever had to interact with anything, the theory would be unable to model it. This would be a bit of a problem. You will run into a similar problem in situations with other conserved quantities; i.e., any situation where the probability distribution is unchanging in some space, such as momentum, time, etc.

If we allow the wavefunction to be complex, then it can have a constant squared magnitude, but still oscillate like a wave (by traversing a circle in the complex plane). This way, there is time-dependence even in time-invariant states, and we can model what happens when those states interact with some outside influence.

 

[akhmeteli]

As I mentioned in some of my earlier posts, quantum mechanics does not necessarily needs complex wavefunctions (real wavefunctions may be enough). This is not just my personal opinion. Shroedinger demonstrated this for a Klein-Gordon field interacting with electromagnetic field. The reasons offered in the preceding post do not work there as the probability density does not equal \psi^2 in that case

 

[Gerenuk]

Basically quantum mechanics works if you use complex wavefunctions and that's all we want.
One reason is that there are phenomena where for example three objects exist at some point individually with equal probability (amplitude), but if you combine them at this point (adding) they vanish (interfere destructively). That only works with complex numbers (exp(2pi*i*n/3) in this case).
Formally one could of course consider the real and imaginary part of the wavefunction. Then you have only real functions, but now two of them.

 

[akhmeteli]

Basically quantum mechanics works if you use complex wavefunctions and that's all we want.
One reason is that there are phenomena where for example three objects exist at some point individually with equal probability (amplitude), but if you combine them at this point (adding) they vanish (interfere destructively). That only works with complex numbers (exp(2pi*i*n/3) in this case).
Formally one could of course consider the real and imaginary part of the wavefunction. Then you have only real functions, but now two of them.

Again, this reasoning is based on the assumption that the probability density is \psi^2. This is not so, for example, for a Klein-Gordon field interacting with electromagnetic field (and this theory includes a lot of what quantum electrodynamics includes, the exceptions being such important phenomena as spin). In this case just one real field (not two) is enough to describe the charged Klein-Gordon field. This is achieved by choosing a gauge, as Schroedinger showed (there is a reference in one of my earlier posts).

 

[Gerenuk]

Where can I read up about the Klein-Gordon field that has only one real values function? I've only heard about the 4 dimensional Dirac stuff so far.
Actually I didn't assume that the probability density is \psi\psi^*. I only assumed that it is zero for \psi=0 and it is non-zero for \psi\neq 0

 

[akhmeteli]

Where can I read up about the Klein-Gordon field that has only one real values function? I've only heard about the 4 dimensional Dirac stuff so far.
Actually I didn't assume that the probability density is \psi\psi^*. I only assumed that it is zero for \psi=0 and it is non-zero for \psi\neq 0

E. Shroedinger, Nature, V. 169, P.538 (1952). There is also a discussion in my paper http://arxiv.org/abs/quant-ph/0509044
Actually, the assumption that the probability density is non-zero for \psi\neq 0 does not hold for the model I mentioned (you may check it in textbooks, in Shroedinger's short article, or look at expression (9) for the current in my paper; I am afraid I am too lazy to rewrite it here:-) )

 

[Peter Morgan]

Hi Akmeteli,
Mathematically, you can certainly use a real Hilbert space for generating probabilities, but when you want to represent position and momentum as noncommuting observables (which if you're doing QM, you do; if you're doing not-QM, you might not), it's convenient to introduce complex numbers. The elementary paper on this that I like best is Leon Cohen, Foundations of Physics, Volume 18, page 983, 1988, but I'm afraid it's behind a paywall. I believe myself that the use of complex structure in quantum mechanics is not anything to worry about much, partly because you can always write a dimension n complex Hilbert space as a dimension 2n real Hilbert space. What you lose by going real, however, is the fundamental theorem of algebra (for which see wikipedia, say), which mathematically is a huge convenience.

 

[mendocino]

Hi there, i have been studying a bit about QM, but ther's one fundamental question
about the wavefunction i can't understand: why is the wavef. defined complex? I mean,
couldn't one work from the beginning with a real wave?

Thanks

I think it related to the phase of wavef.
You can't use one real number to represent wave

 

[akhmeteli]

I think it related to the phase of wavef.
You can't use one real number to represent wave

Sorry, I'm afraid I did not quite get it. Are you implying that one complex number can be used to represent a wave?
If, however, you meant "You can't use one real function to represent wave", I tend to disagree, as it seems you can use a sine or a cosine function to represent a wave.

 

[akhmeteli]

Hi Akmeteli,
Mathematically, you can certainly use a real Hilbert space for generating probabilities, but when you want to represent position and momentum as noncommuting observables (which if you're doing QM, you do; if you're doing not-QM, you might not), it's convenient to introduce complex numbers... I believe myself that the use of complex structure in quantum mechanics is not anything to worry about much, partly because you can always write a dimension n complex Hilbert space as a dimension 2n real Hilbert space. What you lose by going real, however, is the fundamental theorem of algebra (for which see wikipedia, say), which mathematically is a huge convenience.

I have no problems with complex numbers in quantum mechanics. However, the original poster asked:"why is the wavef. defined complex? I mean,
couldn't one work from the beginning with a real wave?". I was just trying to answer that question, and my answer was "it might be possible". I would like to emphasize once again, I am not talking about using two real functions instead of one complex one. I might agree that "it is convenient to introduce complex numbers". However, whether something is more convenient or not, depends on what you want, on your agenda. The reason I am interested in formulating quantum mechanics in terms of real functions is such reformulation might be critical for interpretation of quantum mechanics. For example, it can lead to natural exclusion of the matter (the wavefunction). If you are interested in the Bohmian interpretation, the electromagnetic potential can play the role of the quantum potential after such reformulation.

 

[Xeinstein]

Sorry, I'm afraid I did not quite get it. Are you implying that one complex number can be used to represent a wave?
If, however, you meant "You can't use one real function to represent wave", I tend to disagree, as it seems you can use a sine or a cosine function to represent a wave.

You can use a sine or a cosine function to represent a wave with 0 or 90 degree phase, But you need both sine "and" cosine function to represent a wave with arbitrary phase

 

[akhmeteli]

You can use a sine or a cosine function to represent a wave with 0 or 90 degree phase, But you need both sine "and" cosine function to represent a wave with arbitrary phase

I had in mind something like \sin(k x+\alpha) or \cos(k x+\alpha), where \alpha is an arbitrary phase.

 

[Xeinstein]

I had in mind something like \sin(k x+\alpha) or \cos(k x+\alpha), where \alpha is an arbitrary phase.

Basically, what you did is just shift the origin,
But if you have many wave functions with different phases,
what can you do about them?
Are you going to shift origins for all of them?
Then you end up with No common origin of them
I think what we want is just one common origin for all of them

The sine and cosine functions are just the basis vectors/function in Hilbert space
Then any function is just linear combination of the basis function

 

[akhmeteli]

Basically, what you did is just shift the origin,
But if you have many wave functions with different phases,
what can you do about them?
Are you going to shift origins for all of them?
Then you end up with No common origin of them
I think what we want is just one common origin for all of them

The sine and cosine functions are just the basis vectors/function in Hilbert space
Then any function is just linear combination of the basis function

I am afraid you've lost me. Of course, you may use the sine and cosine functions as a basis. Two things elude me, though. How your remarks relate to the statement that I disagreed ("You can't use one real function to represent wave") and how the situation with real numbers is significantly different from the situation you have when using complex numbers. Maybe, one more thing. What do you call "common origin" and why do we need it?

 

[Xeinstein]

I am afraid you've lost me. Of course, you may use the sine and cosine functions as a basis. Two things elude me, though. How your remarks relate to the statement that I disagreed ("You can't use one real function to represent wave") and how the situation with real numbers is significantly different from the situation you have when using complex numbers. Maybe, one more thing. What do you call "common origin" and why do we need it?

I think I should call it offset, so you add offset to sine or cosine function
My point is this, we need both sine "and" cosine as basis functions in Fourier transform/analysis, You can't do Fourier transform with just sine "or" cosine only for arbitrary functions

 

[akhmeteli]

I think I should call it offset, so you add offset to sine or cosine function
My point is this, we need both sine "and" cosine as basis functions in Fourier transform/analysis, You can't do Fourier transform with just sine "or" cosine only for arbitrary functions

I agree. Furthermore, I appreciate that the exponential function is more convenient than sines and cosines. Nevertheless, I believe that a real function, say, a sine, can represent a wave, contrary to what mendocino stated.

 

[reilly]

There's no mystery about the fact that sin(kx) and cos(kx) are solutions to any linear wave equation -- k = wave number. But, the algebra and calculus of sin and cosine are somewhat cumbersome, not so for the algebra of complex exponentials. Complex variables bring a particularly convenient and powerful approach to dealing with wave functions, and many other topics as well.

So why complex wave functions? It's strictly pragmatism. We use complex variables, by active choice, because they work, and work well -- complex variables are used in virtually all branches of applied math, strictly for the sake of convenience. Complex variables provide some very powerful tools -- contour integration, for example -- for certain classes of mathematical problems, particularly those related to linear second order differential equations. Special functions and complex variables go hand-in-glove

Complex variables? No big deal, and with absolutely no conceptual bearing on physics.
Regards,
Reilly Atkinson

 

[jtbell]

Nevertheless, I believe that a real function, say, a sine, can represent a wave, contrary to what mendocino stated.

Can you replace the time-dependent Schrödinger Equation with a differential equation that does not have an i in it, has a real (not complex) wave function as its solution, and describes all the phenomena that the SE and its solutions can?

 

[akhmeteli]

Can you replace the time-dependent Schrödinger Equation with a differential equation that does not have an i in it, has a real (not complex) wave function as its solution, and describes all the phenomena that the SE and its solutions can?

Actually, yes.

Let me explain it (though you may have an impression that I am not explaining, but "caveating" it).

First, as I said, this is not my achievement and claim, but Shroedinger's (I gave a reference in this thread; sorry for "oe" instead of "o umlaut" in his name).

Second, saying "yes", I assume that the Klein-Gordon equation describes all the phenomena that the SE and its solutions can (actually, KGE is better than SE in the relativistic limit).

Third, the possible potentials are limited to such electromagnetic 4-potentials A^\mu that their source (the right-hand-side of the Maxwell equations) is the standard Klein-Gordon current (so we do not ignore the electromagnetic field generated by the Klein-Gordon particle, but why should we?) plus arbitrary conserved external currents (strictly speaking, Shroedinger considered a self-consistent theory, where the Klein-Gordon field interacts with electromagnetic field, but the generalization through addition of external conserved currents seems obvious). That does not look like a strong limitation.

Actually, the main idea is exremely straightforward (Shroedinger wrote: "That the wave function ... can be made real by a change of gauge is but a truism, though it contradicts the widespread belief about 'charged' fields requiring complex representation." And he did not have in mind a replacement of a complex function by two real ones!). Indeed, if you have a solution (\psi, A^\mu) of the Klein-Gordon-Maxwell equations (Klein-Gordon field interacting with electromagnetic field), then a gauge transform would give you an equivalent solution (\phi, B^\mu), where \phi is real, \phi^2=(\psi*)\psi, and the electromagnetic field is the same for A^\mu and B^/mu (Careful in thread https://www.physicsforums.com/showthread.php?p=830490#post830490 draw my attention to the fact that B^mu can have singularities, but, on the other hand, even the Coulomb potential is singular, let alone quantum field theory).

 

[Fra]

I've got a feeling that the apparent "mystery of complex amplitudes" has two faces, and only one part is dicusssed.

One is the mathematical relations between complex valued functions vs real valued ones and how a complex valued DE can be transformed into a set of real value DE's. How a system of two real fields obeying the KG equation can be transformed into the dirac equation by some transformations. One complex field or two real fields? which is "better"?

My impression is that was the focus here.

But the other side of the issue IMO which is less clear but more interesting, is from the point of view of probability interpretations and how to understand the concept of superposition and how that is all induced from the history of the system.

It's very easy to picture how collected data induces a real probability (relative frequency) by storing say "counts" in bins labeled by event ID, and how our expectations are based on our history.

But how do we continue this clear reasoning, to involved complex numbers? Clearly we are transforming our own records in a way that introduces "complex numbers" (or more real numbers). Where does this transformation come from (fourier transformatin) and is there any physical significance behind this? (regardless of wether we choose to talk about two real numbers a or b, or one complex number c = a +ib (it's equivalent))

As we know adding information of x and p, does not work like normal event additions. The instead work via the superposition principle. The question is, from the physical and statistical point of view, how this is induced from first principles, and in the spirit of the scientific method couple this to a history of observations and allow it to induce our expectations based on the partly retained history also in a general system rather than the custom axiomatisation.

My only point here is that I think there is still things here where we can expect to find more satisfactory answers.

/Fredrik

 

[reilly]

Why would you want to do get rid of the "i" What's the problem?
Regards,
Reilly Atkinson

Can you replace the time-dependent Schrödinger Equation with a differential equation that does not have an i in it, has a real (not complex) wave function as its solution, and describes all the phenomena that the SE and its solutions can?

 

[Peter Morgan]

Why would you want to do get rid of the "i" What's the problem?
Regards,
Reilly Atkinson

I think the problem is that mathematical isomorphisms between mathematical structures are not as generally appreciated as they should be. For a mathematical physicist, constructing an isomorphism between mathematical structures makes it possible to use either structure without worrying about which is "real". You have to work at a moderately sophisticated mathematical level to feel the force of this; which I think is beyond the mathematical level, for example, where such an argument is understood.

If mathematical structures are not isomorphic, there is still a question whether they are empirically equivalent or not at various levels of experimental sophistication. Calculating the empirical predictions of different theories well enough to allow experiments to distinguish them to be designed and implemented is often much more time-consuming.

It appears that the discussion on this thread is not about mathematically different theories. A proof that the mathematical structures suggested on this thread are different from the conventional Hilbert space mathematics of QM would be surprising, because that mathematics is remarkably general, but probably only publishable even then if it looked plausible that it would be possible to construct models for experiment that are empirically more successful than QM models for experiment. Even then, the need to learn the new approach and how to apply it to a wide range of experimental and technological applications, probably a steep learning curve, would rule out the wide adoption of any equally empirically adequate alternative to QM unless it were to have other, significant advantages.

 

[jtbell]

Why would you want to do get rid of the "i" What's the problem?

I don't have any problem with the i. I was addressing the claim that one can use a real-valued QM wave function instead of the complex-valued \Psi. It seems to me that in order to do that you have to use a wave equation that isn't manifestly complex, unlike the Schrödinger equation. (Or the Dirac equation, if you want to go relativistic.)

 

[akhmeteli]

I was addressing the claim that one can use a real-valued QM wave function instead of the complex-valued \Psi. It seems to me that in order to do that you have to use a wave equation that isn't manifestly complex, unlike the Schrödinger equation. (Or the Dirac equation, if you want to go relativistic.)

I am not sure, at least when I try to read your statement literally. Indeed, you can start with a Schroedinger equation or a Klein-Gordon equation for a particle in electromagnetic field described by potentials A^\mu. These equations will be manifestly complex. However, using a gauge transform, for every complex solution \psi of such equation you can get a real solution \phi of the same equation, but for different electromagnetic potentials B^\mu, which differ from A^\mu by a gradient, so the pairs (\psi, A^\mu) and (\phi,B^\mu) are physically equivalent (subject to Careful's remark that I referenced above). I agree, though, that you can then take the real and the imaginary parts of the equation, and you'll get two real equations for the real wavefunction \phi. One of these equations will be the equation of continuity for the current. However, if you do the same for a self-consistent Klein-Gordon-Maxwell system (maybe adding external conserved currents), you get the continuity equation from the Maxwell equations, so you can replace the Klein-Gordon equation with just one real equation for the real wavefunction.

 

[akhmeteli]

Apparently, several different questions were raised in this thread.

The original poster, naqo, asked: "couldn't one work from the beginning with a real wave?". On the basis of the Schroedinger's article, I gave a (qualified) positive reply in post #3 (mind you, I was not trying to answer the question "shouldn't one...?"). I have not seen an objection to that in this thread.

Then there was a discussion of whether a wave can be represented by one real function. I expressed my point of view (it "can"; whether it "should", is a completely different question). I don't want to speculate whether there is a consensus now.

jtbell asked: "Can you replace the time-dependent Schrödinger Equation with a differential equation that does not have an i in it, has a real (not complex) wave function as its solution, and describes all the phenomena that the SE and its solutions can?" Again, using the results of the Schroedinger's article, I gave a qualified positive reply in post #20 and have not seen an objection to that in this thread. Again, I was not trying to answer the question "Should you...?"

Finally, several knowledgeable people asked just that: Why do you need to replace a complex wavefunction with a real one, although it is much more convenient to work in the complex domain? Let me give my reasons here (the details may be found in my article quoted in my post #7).

I believe this opens a way for a different interpretation of quantum mechanics. Namely, in the Klein-Gordon-Maxwell system of equations you can naturally exclude the wavefunction describing the matter and obtain independent evolution of the electromagnetic field. Therefore, in the Bohmian interpretation of quantum mechanics, the electromagnetic field, not the quantum potential, plays the role of the guiding field (unfortunately, the extension of this conclusion to the Dirac-Maxwell system in the article is not satisfactory, and I hope I'll be able to correct this in a few days). Is this worth the trouble? Is the Bohmian interpretation itself worth the trouble? I think so, but I suspect most people in this forum will disagree

 

[Fra]

Why do you need to replace a complex wavefunction with a real one, although it is much more convenient to work in the complex domain? Let me give my reasons here (the details may be found in my article quoted in my post #7).

I believe this opens a way for a different interpretation of quantum mechanics. Namely, in the Klein-Gordon-Maxwell system of equations you can naturally exclude the wavefunction describing the matter and obtain independent evolution of the electromagnetic field. Therefore, in the Bohmian interpretation of quantum mechanics, the electromagnetic field, not the quantum potential, plays the role of the guiding field (unfortunately, the extension of this conclusion to the Dirac-Maxwell system in the article is not satisfactory, and I hope I'll be able to correct this in a few days). Is this worth the trouble? Is the Bohmian interpretation itself worth the trouble? I think so, but I suspect most people in this forum will disagree

I've got a feeling that once you've got that different interpretation, you also got to have some ideas in your sleeves how to take this beyond "interpretation only" and perhaps give a new angle to solve bigger problems in the context of unifications?

Is that possibly what you are after? :) Assuming your main objective is not just to restore classical realism at all cost, but also to bring the theory forward, I may hint a point and sense a remote connection to my perferred thinking.

Are you trying to restore classical objective realism, like my perception is many bohmians are after? or what is the ultimate purpose of your attention to this?

/Fredrik

 

[akhmeteli]

I've got a feeling that once you've got that different interpretation, you also got to have some ideas in your sleeves how to take this beyond "interpretation only" and perhaps give a new angle to solve bigger problems in the context of unifications?

Is that possibly what you are after? :) Assuming your main objective is not just to restore classical realism at all cost, but also to bring the theory forward, I may hint a point and sense a remote connection to my perferred thinking.

Are you trying to restore classical objective realism, like my perception is many bohmians are after? or what is the ultimate purpose of your attention to this?

/Fredrik

Thank you for your questions.

Initially, I just was not happy with the Copenhagen interpretation. Restoring classical realism at all costs? I don't quite see the point, because, if you ask me, the standard Bohmian interpretation already restored it (few people accept this interpretation, but for those who want realism at all costs it presents a solution). It seems to me that the cost is too high, but that's just my opinion. However, I hope that the interpretation may seem more attractive when it is the electromagnetic field, not the quantum potential, that guides the particle, so you are not "multiplying entities without necessity".

As for "a new angle to solve bigger problems in the context of unifications", actually, it might be possible, although so far this is pure speculation. But I don't know why I should not answer your question.

So it goes like this. What I am doing is only possible because the Klein-Gordon-Maxwell is a gauge-invariant theory. For the Dirac-Maxwell theory you can do pretty much the same by imposing the following constraint: "the axial current is zero". The resulting theory is limited, but meaningful, and you need just Majorana spinors instead of Dirac ones, and Majorana spinors are an analogue of real numbers for spinors. Whether such a theory is enough to describe all experiments, I don't know, but anyway, it is at least an interesting toy model of quantum mechanics, which allows a completely different interpretation, so it may be useful for discussions of interpretations. As for the Standard Model or unification theories, they are also gauge-invariant theories, so one can speculate that a similar mechanism can be applied to them, so real representations of the relevant groups, rather than complex, may be sufficient for fermions. However, I am not sure I am qualified to develop anything like this.

 

[Fra]

Thanks for your response.

To me my personal choice of interpretation is the one who provides me with the best expected stance for extending things, so interpretations are selected by the "power" of extrapolation IMO. If it wasn't for the extensions, a minimalist interpretation would be my choice.

I still don't quite understand your objective, just almost. As for who is qualified to do this or that, I don't know who that would be? I am not sure I am qualified either but I don't think that ever stopped anyone before

/Fredrik

 

[Fra]

However, I hope that the interpretation may seem more attractive when it is the electromagnetic field, not the quantum potential, that guides the particle, so you are not "multiplying entities without necessity".

I see, so electromagnetic field qualifies as a necessity and quantum potential does not?

Do you consider the necessity to be objective? or could it be that what is a necessity to one observer, is not to another? And how are necessities induced from experience? An from the particle view, how does a particle induce this necessity from experience?

If you are into bohm, did you ever try to merge the bohmian view with the subjective bayesian view? Sort of suggesting that the bohmian speculated degrees of freedom rather represents subjective estimates. They aren't real hidden variables in the objective classical sense?

I am not into bohm, but his thinking is not totally off chart IMO. But I think there is another way of seeing the bohm formalism, that does not make use of the deterministic philosophy.

I tried to ask demystifier who has posted a lot about bohmian views but it seems he does not acknowledge this association. I would be curious to hear a bohmian view of this, but since it in one sense may have similarities to the bohmian thinking, it is even farther away from it than the copenhagen thinking since it introduces even more fundamental uncertainty.

If I were to induce my thinking onto the bohmian stuff, i would described the bohmian degrees of freedom (here extending it to general degrees of freedem and leave unsaid to interpret as "particles" sets of particles or whatever) as part of the identity of the system, but it determines the expected action relative the system. And I think it's the fact that the expected action relative different systems is different, that gives rise to non-trivial dynamics.

So from outsider, the bohmian degrees of freedom are not hidden variables in the sense of a definite structure with an unknown state, this is wrong because the identification of the microstructure itself! even with a completely unknown (random) state DOES contain information. And this information doesn't exist on the outside.

So IMO, a "bohmian like" interpretation might be kind of possible without the notion of hidden variables.. because it gives the impresion that the varible structure is known it's just that their values are not. I rather see it that the not only is the variable value hidden, the variables themselves re hidden, and effectively doesn't exists - from the outside. which connects to a subjective reality interpretation of QM.

Is this anything in your taste? I ask this out of general curiosity since you are into bohm.

/Fredrik

 

[Fra]

To comment to the expected objection how this subjective reality can be compatible with scientific model, then the answer is that one would still expect the objective reality to be emergent as a collective equilibration and evolution. In line with any learning ideas. We learn and our baseline also evolves along with it.

/Fredrik

 

[akhmeteli]

Fra,

Thank you for your posts and interest.

I am not sure I'll be able to give short answers, and right now I have some deadlines to meet, so I'll try to answer by Monday. Sorry.

Anyway, I am not sure I'll be able to offer anything meaningful on philosophical issues.

 

[Fra]

Fra,

Thank you for your posts and interest.

I am not sure I'll be able to give short answers, and right now I have some deadlines to meet, so I'll try to answer by Monday. Sorry.

Anyway, I am not sure I'll be able to offer anything meaningful on philosophical issues.

I am just curious. Don't waste too much time to make up a response unless you can relate to my questions. Even a lack of response or an unexpected response is a kind of response too.

Good luck with your deadlines.

/Fredrik

 

[akhmeteli]

To me my personal choice of interpretation is the one who provides me with the best expected stance for extending things, so interpretations are selected by the "power" of extrapolation IMO. If it wasn't for the extensions, a minimalist interpretation would be my choice.

I still don't quite understand your objective, just almost. As for who is qualified to do this or that, I don't know who that would be? I am not sure I am qualified either but I don't think that ever stopped anyone before

/Fredrik

I think I already speculated enough on possible extrapolations of what I do

As for my objective... Initially I was motivated by a firm belief that if two similar experiments produce different results, we should be able at least to indicate the difference between them that makes the "mileage vary". May be I just was a fatalist back then While I still keep that belief, I don't feel I'm in the driver's seat any more as far as my research is concerned, I rather tend to follow the logic and the results of that research. For example, I could not even dream that it would be possible to eliminate the wavefunction from the Klein-Gordon-Maxwell theory and have deterministic equations of motion for the electromagnetic field. Applications of this result to an electron in a hydrogen atom or two a two-slit interferometer are fascinating, but I have not had time to study them in more detail.

 

[akhmeteli]

I see, so electromagnetic field qualifies as a necessity and quantum potential does not?

Do you consider the necessity to be objective? or could it be that what is a necessity to one observer, is not to another? And how are necessities induced from experience? An from the particle view, how does a particle induce this necessity from experience?

I guess it's a matter of simplicity. We tend to prefer a simpler theory, all other things being equal. Of course, observers may differ in what they call simple, but there are also objective criteria of simplicity/complexity. As for necessities and experience... If a simpler theory fails to describe the experimental results, we have to look for a more complex one. I like Einstein's "slogan": "as simple as possible, but not simpler". The same about particles, I guess.

If you are into bohm, did you ever try to merge the bohmian view with the subjective bayesian view? Sort of suggesting that the bohmian speculated degrees of freedom rather represents subjective estimates. They aren't real hidden variables in the objective classical sense?

I am not into bohm, but his thinking is not totally off chart IMO. But I think there is another way of seeing the bohm formalism, that does not make use of the deterministic philosophy.

I tried to ask demystifier who has posted a lot about bohmian views but it seems he does not acknowledge this association. I would be curious to hear a bohmian view of this, but since it in one sense may have similarities to the bohmian thinking, it is even farther away from it than the copenhagen thinking since it introduces even more fundamental uncertainty.

If I were to induce my thinking onto the bohmian stuff, i would described the bohmian degrees of freedom (here extending it to general degrees of freedem and leave unsaid to interpret as "particles" sets of particles or whatever) as part of the identity of the system, but it determines the expected action relative the system. And I think it's the fact that the expected action relative different systems is different, that gives rise to non-trivial dynamics.

So from outsider, the bohmian degrees of freedom are not hidden variables in the sense of a definite structure with an unknown state, this is wrong because the identification of the microstructure itself! even with a completely unknown (random) state DOES contain information. And this information doesn't exist on the outside.

So IMO, a "bohmian like" interpretation might be kind of possible without the notion of hidden variables.. because it gives the impresion that the varible structure is known it's just that their values are not. I rather see it that the not only is the variable value hidden, the variables themselves re hidden, and effectively doesn't exists - from the outside. which connects to a subjective reality interpretation of QM.

Is this anything in your taste? I ask this out of general curiosity since you are into bohm.

/Fredrik

Very generally speaking, I don't have problems with subjective bayesian view. For example, I very much like Jaynes' information theory approach to statistical physics. It is important to understand, however, whether such an approach is necessary at the fundamental level or at some higher level (statistical physics may be an example of such higher level). As for the fundamental level, I have yet to be convinced that the bayesian view simplifies the matter or, although adding complexity, is just necessary.

 

[Fra]

Very generally speaking, I don't have problems with subjective bayesian view. For example, I very much like Jaynes' information theory approach to statistical physics.

I'm glad to hear this.

It is important to understand, however, whether such an approach is necessary at the fundamental level or at some higher level (statistical physics may be an example of such higher level).

I agree.

As for the fundamental level, I have yet to be convinced that the bayesian view simplifies the matter or, although adding complexity, is just necessary.

I understand your skepsis. I am personally now quite confident that this is the way, but it wasn't straightforward. I've spent some efforts analysing this from the POV of philosophy of science as I prefer it, and I've arrived at a pretty confident personal position of the foundations. I judged this as absolutely necessary as I could not justify the high degree of speculation that I ended up with as a student.

I think the bayesian view alone is insufficient to explain everything, but it is a step in the right direction. I even think that our inability to understand how the "quantum world" scales from an information theoretic point, is correlated to our lack of understanding how QM and GR can be unified. I think separating them, is part of the problems also to understand the QM foundations.

This traces down to the philosophy of science.

IMO statistics, inductive inference and even probability theory really touches the essence of science. What is knowledge? How _confident_ are we in the "knowledge" we think we have? How do we _measure_ confidence? All these questions touches foundations of statistics and probability theory. Many of these things have issues, that are non-trivial, but by tradition are treated as philosophy by many physicists. I could never accept such attitude, but science also has social dimensions. And if this behaviour is accepted in the scientific society, because everybody does it, well then there you go. And from group dynamics it sure is going to take more than one opinion to change the group behaviour. And reflecting over this, I see deep connections to physical interactions and information theory.

This leads me to a new idea of simplity. Risk taking and speculation.

Sometimes you have to speculate and take a risk based in incomplete information, beucause that's life. Because choosing to not take any actions at all may actually be more risky, than taking one of a set of possible smaller risks.

I interpret "as simple as possible, but not simpler" in a more specific interpretation as:

One should not take unjustified risks, and should only take the justified risks necessary to optimise your self-preservation. The "no risk" options usually never exists. When you connect this game thinking, to probability theory and it's foundations many interesting things appear. But I think it would take that someone works out something explicit and applies to to make new predictions and accomplish at least part unification before the collective pays any interest to it. For some reason very few people seems attracted to this.

/Fredrik

 

[Xeinstein]

As I mentioned in some of my earlier posts, quantum mechanics does not necessarily needs complex wavefunctions (real wavefunctions may be enough). This is not just my personal opinion. Shroedinger demonstrated this for a Klein-Gordon field interacting with electromagnetic field. The reasons offered in the preceding post do not work there as the probability density does not equal \psi^2 in that case

Here is a related thread about "complex wavefunctions" that you might be interested

https://www.physicsforums.com/showthread.php?t=207417

 

[akhmeteli]

Here is a related thread about "complex wavefunctions" that you might be interested

https://www.physicsforums.com/showthread.php?t=207417

Thank you,

actually, I follow the thread that you started, but have not had time to participate in the discussion.

I still stand by what I have written in this thread. The arguments to the contrary in your thread did not convince me.

 

[reilly]

Hi there, i have been studying a bit about QM, but ther's one fundamental question
about the wavefunction i can't understand: why is the wavef. defined complex? I mean,
couldn't one work from the beginning with a real wave?

Thanks

It all started, in earnest, with Euler's exp(2πi) = 1. (At least I think it was Euler.)The rest is history; practical history. We use complex variables in all manner of sciences, engineering, and, yes, even economics. Why? Complex variables, through contour integration, analytic continuation, conformal mapping, etc. give us enormous mathematical power. (Not much different than the choice of Arabic vs. Roman numerals.)

Why do we let electric current be complex in RLC circuits? Convenience. Nothing more, nothing less. Think about the various phase relationships in circuits -- complex variables allow a particularly simple way of dealing with phases.

It's no big deal. If you don't like it, try QM without i. Why do we use 3 or 99, or x= something?

For the record, when I taught QM I required first that my students knew basic atomic and nuclear physics, particularly the key early experiments -- like Davisson Germer --And I also required students to have at least an undergraduate course in complex variables, and an E&M course that covered basic partial differential equations including special functions, radiation, etc. and advanced mechanics, including contact transformations and the Hamilton -- Jacobi EQ. I thought then, and still do, that without this basic background, it will be agonizingly difficult to understand and cope with QM,
Regards,
Reilly Atkinson

 

[danime]

If the wave-function represent a confined particle, there's allways some way to write it as a real function. See Landau-Lif****z volume 3.

 

[reilly]

Why use Arabic numerals rather than Roman numerals in physics?
Regards,
Reilly Atkinson