Is the quantum wave function a real object or a mathematical tool?

[selfsimilar]

When speaking about reality as a concept, I like to keep in mind the first few chapters of Feynman's Lectures. Here he talks about a flat table. About as real as you can get, right?. But let's zoom in. What we find is that table particles evaporate and are replaced by air particles. Where the table even starts is uncertain, let alone if it's flat. That's why, in physics, we don't pin concepts like that down precisely, because they've proven problematic. We use it in a colloquial common-sense way. The wisest words I have heard on it are from physicist Victor Stenger: reality kicks back.

Thanks
Bill

I agree. That is the sense I was talking about, nothing controversial. You can complicate it like what is mass but mass is equivalent to energy, what is energy( calculational accounting tool!) ...etc.

 

[javisot]

https://arxiv.org/abs/2602.09397 "historical debates over the physical reality of the wave functions", J.Barandes

 

[pines-demon]

https://arxiv.org/abs/2602.09397 "historical debates over the physical reality of the wave functions", J.Barandes

Lovely, this is history thread material.

 

[Demiurge]

If 'real' means to be perceived, then - No. The wavefunction is not real

 

[pines-demon]

I am sure the wavefunction is not real but complex-valued

 

[javisot]

I am sure the wavefunction is not real but complex-valued

Isn't there a 1960 work by Stueckelberg where he constructs a version with real numbers?

 

[akhmeteli]

I am sure the wavefunction is not real but complex-valued

This is not obvious. I usually cite Schrödinger's Nature, 169:538 (1952). He noticed that the wave function for the Klein-Gordon equation in electromagnetic field can be made real (rather than complex) by a gauge transformation. His comment: "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."

 

[PeterDonis]

the widespread belief about 'charged' fields requiring complex representation

You're conflating two separate issues here. The representation of charged fields (btw, the Klein-Gordon equation does not describe the electromagnetic field, which is spin 1, not spin 0--it's rather typical of Schrodinger to gloss over that issue) is a separate issue from needing complex probability amplitudes in QM in order to represent the fact that different possible ways something can happen can negatively interfere with each other. The latter is what @pines-demon was referring to.

 

[akhmeteli]

You're conflating two separate issues here. The representation of charged fields (btw, the Klein-Gordon equation does not describe the electromagnetic field, which is spin 1, not spin 0--it's rather typical of Schrodinger to gloss over that issue) is a separate issue from needing complex probability amplitudes in QM in order to represent the fact that different possible ways something can happen can negatively interfere with each other. The latter is what @pines-demon was referring to.

I did not say that the Klein-Gordon equation describes electromagnetic field, I was talking about the Klein-Gordon equation in electromagnetic field: $$

(\partial^\mu+ieA^\mu)(\partial_\mu+ieA_\mu)\psi+m^2\psi=0.$$ It describes the matter field $\psi$ in electromagnetic field $A_\mu$.

Note that you can make this matter field real (at least locally) by a gauge transformation $$

\varphi=\psi\exp(i e \omega), B_{\mu}=A_{\mu}-\omega_{,\mu}.$$ So if the original Klein-Gordon equation (which is gauge-invariant) can describe negative interference, then the gauge-transformed equation $$

\Box\varphi-(e^2 B^\mu B_\mu-m^2)\varphi=0$$ also can describe negative interference, although the matter field is real in the transformed equation and the gauge is fixed in this equation. So I am not sure what exactly I conflated or what Schrödinger glossed over.

@pines-demon talked about "wave function", Schrödinger used the word "field", but I fail to see how this can be problematic in the context of this thread. Note that one can make the wave function real by a gauge transformation in the case of the original Schrödinger equation as well. Is there some nuance of the difference between "amplitude" and "wave function"? But again, @pines-demon talked about wave function.

 

[PeterDonis]

I did not say that the Klein-Gordon equation describes electromagnetic field, I was talking about the Klein-Gordon equation in electromagnetic field

Ah, I see, the K-G equation is for the matter.

@pines-demon talked about "wave function", Schrödinger used the word "field", but I fail to see how this can be problematic in the context of this thread.

It makes a huge difference. In the context of QM, "fields" refers to quantum field theory. "Wave function" refers to ordinary non-relativistic QM. @bhobba has already posted in this thread about how those are not the same thing, and how non-relativistic QM is not necessarily even correctly described as an approximation to QFT.

In QFT, the "fields" are not even numbers to begin with--they're operators. The gauge transformation you're describing from Schrodinger is for a classical field equation where the fields are numbers. If the fields are operators, the question of whether they're complex or real numbers obviously doesn't even make sense. QFT represents negative interference as particular properties of the operators. Charged fields in QFT are simply different operators.

 

[akhmeteli]

It makes a huge difference. In the context of QM, "fields" refers to quantum field theory. "Wave function" refers to ordinary non-relativistic QM. @bhobba has already posted in this thread about how those are not the same thing, and how non-relativistic QM is not necessarily even correctly described as an approximation to QFT.

In QFT, the "fields" are not even numbers to begin with--they're operators. The gauge transformation you're describing from Schrodinger is for a classical field equation where the fields are numbers. If the fields are operators, the question of whether they're complex or real numbers obviously doesn't even make sense. QFT represents negative interference as particular properties of the operators. Charged fields in QFT are simply different operators.

Again, I don't see how the difference between "field" and "wave function" is problematic in the context of this thread. Obviously, Schrödinger wrote about a non-second-quantized wave function (which is often regarded as a classical field that nevertheless has quantum properties). He explicitly wrote: "the wave function ... can be made real by a change of gauge".

Do you dispute this statement? If not, then it is indeed not obvious that "the wavefunction is not real but complex-valued", as pines-demon wrote, and I am not sure what I conflated.

 

[PeterDonis]

Do you dispute this statement?

As a mathematical fact about the equation, of course it's true. Nobody is disputing that.

As a physics claim about wave functions, again, I'm not even sure it applies. The Klein-Gordon equation is not used in non-relativistic QM--the Schrodinger Equation is. The Klein-Gordon equation is used in QFT, but there, as I said, the fields aren't numbers, they're operators, so it doesn't even make sense to ask whether they're real or complex numbers.

Obviously, Schrödinger wrote about a non-second-quantized wave function (which is often regarded as a classical field that nevertheless has quantum properties).

Sure--and then he glossed over the fact that, as I said above, the Klein-Gordon equation is not the equation that's used in non-relativistic QM for the dynamics of wave functions, and in QFT, it's not even an equation for wave functions to begin with. He was talking like it was still the mid-1920s, when everybody thought that relativistic QM would turn out to look just like non-relativistic QM, but with a different field equation, like the Klein-Gordon equation, replacing the Schrodinger Equation. But that's not how things actually turned out--and he knew that quite well in 1952, when the paper you referenced was published. He just chose to ignore it.

 

[akhmeteli]

As a mathematical fact about the equation, of course it's true. Nobody is disputing that.

As a physics claim about wave functions, again, I'm not even sure it applies. The Klein-Gordon equation is not used in non-relativistic QM--the Schrodinger Equation is. The Klein-Gordon equation is used in QFT, but there, as I said, the fields aren't numbers, they're operators, so it doesn't even make sense to ask whether they're real or complex numbers.


Sure--and then he glossed over the fact that, as I said above, the Klein-Gordon equation is not the equation that's used in non-relativistic QM for the dynamics of wave functions, and in QFT, it's not even an equation for wave functions to begin with. He was talking like it was still the mid-1920s, when everybody thought that relativistic QM would turn out to look just like non-relativistic QM, but with a different field equation, like the Klein-Gordon equation, replacing the Schrodinger Equation. But that's not how things actually turned out--and he knew that quite well in 1952, when the paper you referenced was published. He just chose to ignore it.

I don't see why it is important that the Klein-Gordon equation is not used in non-relativistic QM, because the gauge transformation that Schrödinger used can be straightforwardly used for the non-relativistic Schrödinger equation in electromagnetic field as it is also hauge-invariant.

 

[PeterDonis]

the gauge transformation that Schrödinger used can be straightforwardly used for the non-relativistic Schrödinger equation in electromagnetic field as it is also hauge-invariant.

That reference seems odd--it talks about the Schrodinger Equation, but it claims to be discussing local gauge invariance in QED--which is a QFT and does not use the Schrodinger Equation. The local gauge invariance in QED is gauge invariance of the photon field, not the matter field.

 

[PeterDonis]

Schrödinger's Nature, 169:538 (1952).

Is this what you're referring to?

https://www.nature.com/articles/169538a0

 

[akhmeteli]

That reference seems odd--it talks about the Schrodinger Equation, but it claims to be discussing local gauge invariance in QED--which is a QFT and does not use the Schrodinger Equation. The local gauge invariance in QED is gauge invariance of the photon field, not the matter field.

Yes, there seems to be a conflict between the title and the content of the reference, but the content clearly includes a proof of gauge invariance of the non-relativistic Schrödinger equation in electromagnetic field. Moreover, I don't think you dispute that this non-relativistic equation is gauge-invariant.

And I don't agree that "The local gauge invariance in QED is gauge invariance of the photon field, not the matter field." Even if only the electromagnetic field is present in the gauge condition (such as the Lorenz gauge or Coulomb gauge), each gauge transformation changes both the electromagnetic field and the spinor field (it changes the phase of the spinor field). And theoretically, one can choose a gauge condition expressed in terms of the spinor field.

 

[akhmeteli]

Is this what you're referring to?

https://www.nature.com/articles/169538a0

Yes, this is the Schrödinger's article that I referred to (looks like it is in open access).

My previous post (#43 in this thread) is about https://www.hep.phy.cam.ac.uk/~chpotter/particleandnuclearphysics/AppendixE.pdf , where they prove gauge invariance of the non-relativistic Schrödinger equation in electromagnetic field (just to avoid confusion about what I meant by "reference" in that post).

 

[bhobba]

The Klein-Gordon equation is used in QFT, but there, as I said, the fields aren't numbers, they're operators, so it doesn't even make sense to ask whether they're real or complex numbers.

Schrodinger initially looked at the KG equation, but it had difficulties. From Wikipedia:

'The Klein–Gordon equation was first considered as a quantum wave equation by Erwin Schrödinger in his search for an equation describing de Broglie waves. The equation is found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. Yet because it fails to take into account the electron's spin, the equation predicts the hydrogen atom's fine structure incorrectly, including overestimating the overall magnitude of the splitting pattern by a factor of ⁠4n/2n − 1 for the n-th energy level.'

Due to this problem, in January 1926, Schrödinger submitted for publication a non-relativistic approximation that predicts the Bohr energy levels of hydrogen but without its fine structure.

Of course, once the cat was out of the bag, so to speak, the KG equation was soon discovered again. However, this time there was a more detailed analysis of its issues:



A new approach was required. The wave function cannot be interpreted as a quantum state used to calculate probabilities because it does not have a positive-definite probability density. It must be interpreted differently.

Dirac had published his theory of the quantum EM field. He used the idea of expanding the field into its Fourier components and considers these components as harmonic oscillators. It is easy to quantise a harmonic oscillator (with its creation and annihilation operators), and the quantum theory of the EM field was born. So we had two views of quantum physics: a field view and a particle view. This was obviously a problem.

So let's take it at face value: the KG equation is a field equation rather than a wave function that yields probabilities (leading to the misleading name "second quantisation"). This may seem problematic for quantum physics, but conceptually, it is simpler if everything is a field, and particles, such as photons of the EM field, are bundles within the field. This is the view of QFT. The field is modelled using creation and annihilation operators, as in the EM field. The KG equation was written in this way and was found to describe spin-zero particles, which, of course, the electron is not. Also, lo and behold, the issues disappeared. The field is not an ordinary classical field, but rather mathematically described by operators. For gory details, see:

https://nicf.net/articles/qft-free-fields/

From Ballentine, if we assume the axioms of ordinary QM and Galilean invariance, as shown in Chapter 3, Schrödinger's equation naturally follows. However, for Lorentz invariance, things could be different, and in fact, as the KG equation shows, they are different.

Thanks
Bill