Applying the Principle of Relativity to Quantum Mechanics

[S Steinhauer]

I read the forum rules, I hope I am not breaking them as these principle is generally accepted and I am not contradicting mainstream science.

"The principle of relativity, according to which the laws of physical phenomena should be the same, whether for an observer fixed, or for an observer carried along in a uniform movement of translation; so that we have not and could not have any means of discerning whether or not we are carried along in such a motion."

— Henri Poincaré, 1904 (from Wikipedia "Principle of relativity")

I noticed, that the same thing can be said for orientation and position in a space without forces. (Gravity doesn´t cause problems as long as the systems are small.)
For example if I was in an elevator and fell asleep there would be no way of telling where i am and which way I am facing after waking up (by looking at the local laws of physics). Change of orientation is of course not included as I could tell if I was spinning.

I will now attempt to apply this to quantum mechanics:

I am in a laboratory with an electron and when I measure its orientation, there is a 50% chance it is oriented one way and a 50% chance it is oriented in the opposite way. It would be the same to take the electrons frame of reference (which of course now has a constant orientation relative to itself) and state that now the laboratory has a 50% chance to be oriented each way upon interaction, since it only matters how these systems are oriented relative to each other.

If this principle applies to quantum mechanics this means, that I never have to consider a systems own wavefunction in its own coordinatesystem (for example no matter which way my head faces relative to the rest of the world it always faces the same way relative to itself, this seems kind of redundant but is in the context of quantum mechanics important).
Also I would expect a particle to behave the same relative to another particle, as it does relative to a laboratory as both should measure the same laws of physics. (Mass does not obey the principle so it could be that because of their mass difference they measure different laws of physics, however I will assume gravity does not play a role).

Example, spin entanglement: If I am in a system and an electron interacts with me in such a way as to tell me its orientation relative to me, it will keep that orientation even when I stop interacting with it. When a third system then interacts with me and knows my orientation, it also immediately knows that of the electron.
This is true whether the system I choose my coordinate system in is an electron or a laboratory.
(There are other differences to be found such as spin entanglement between electrons falling apart due to interaction with the enviroment, however I would not consider this a difference in the laws of physics, as I could simply look at spin entanglement in empty space. I will admit that most of my knowledge on this comes from wikipedia though, so please correct me if I´m wrong.)

This causes problems with transforming (position) between systems/particles of different mass: I am in a laboratory and a particle is a plane wave relative to me, I now transform into the particles reference frame and predict the laboratories plane wave. I will predict wildly different wavelengths for the two systems. This is not acceptable as consistency between coordinate transformations has to be given.
However: Nowhere in the principles is it required, that systems/particles that don´t interact with each other have to have the same space-time coordinates. So I will simply assume they have different ones, such that when transforming consistency is archieved. (Knowing that upon interaction they predict the same delta functions for each other giving them the same coordinates).

This has become so long I doubt anybody is going to read it so I will simply show the four equations that define the coordinate transformation. If I didn´t make an obvious mistake so far and you want to see me derive it I will do so in a comment.

$$\frac{\partial }{\partial x^\mu}f_2 =\frac{\partial y^\nu}{\partial x^\mu}\frac{\partial }{\partial y^\nu}f_1$$

With f₁(y^μ) exponent of the wavefunction of system 1 in system 2 coordinates
and f₂(x^μ) exponent of the wavefunction of system 2 in system 1 coordinates.

It looks a lot more complicated than it turns out to be, however I was still unable to solve it for an x-boosted particle (non-relativistic). I can write all the math in the comments, however I am tired and don´t want to put in the effort if this was wrong all along or if nobody reads it. Feel free to ask any questions and correct me.

 

[mfb]

I am in a laboratory with an electron and when I measure its orientation, there is a 50% chance it is oriented one way and a 50% chance it is oriented in the opposite way. It would be the same to take the electrons frame of reference (which of course now has a constant orientation relative to itself) and state that now the laboratory has a 50% chance to be oriented each way upon interaction, since it only matters how these systems are oriented relative to each other.

An electron cannot "measure itself".

Example, spin entanglement: If I am in a system and an electron interacts with me in such a way as to tell me its orientation relative to me, it will keep that orientation even when I stop interacting with it. When a third system then interacts with me and knows my orientation, it also immediately knows that of the electron.

If the third system knows the result of your measurement, yes.

This causes problems with transforming (position) between systems/particles of different mass

It does not.

and predict the laboratories plane wave

There is no such thing. You can consider a massive particle instead of the lab, then you can get a wavelength, in general it will be different. So what? If I look at your posts I see a different username than you see when looking at my posts. Why? Simply because we are different users.

However: Nowhere in the principles is it required, that systems/particles that don´t interact with each other have to have the same space-time coordinates. So I will simply assume they have different ones, such that when transforming consistency is archieved.

What does that mean?Quantum field theory (QFT) is based on special relativity, which is derived from the assumption that physics is the same in all reference frames. QFT obeys the principle of relativity by construction.

 

[bhobba]

"The principle of relativity, according to which the laws of physical phenomena should be the same, whether for an observer fixed, or for an observer carried along in a uniform movement of translation; so that we have not and could not have any means of discerning whether or not we are carried along in such a motion."

— Henri Poincaré, 1904 (from Wikipedia "Principle of relativity")

Thats not the modern view.

To state it you need first the modern definition of an inertial frame - its not the common one - the only book I know that defines it this way is Landau - Mechanics - but it has the virtue of being beautiful, elegant and correct.

Inertial frames are conventional standards of rest (ie on which experiments etc can be performed) where the laws of physics are the same regardless of where you are, what time it is and what direction you are facing.

The principle of relativity state the laws of physics are the same in any inertial frame.

That alone believe it or not is sufficient to derive relativity:
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

As MFB said there is no contradiction between this and QM - in fact if you read Ballentine - QM A Modern Development you will see how from the POR alone (and the principles of QM) you can derive, not postulate, but actually derive Schrodinger's equation, the momentum operator etc etc ie the dynamics (see chapter 3)

And again, not using Schrodinger's equation and that other stuff you can derive the Principle of least action of classical physics. From that alone, and symmetry principles you can derive classical mechanics - see Landaus book mentioned previously. You can even prove, and Landau does it in the book, mass exists and is positive.

This is one of the most profound, beautiful, deep, and striking developments of 20th century physics.

So there is not only no contradiction, both QM and relativity together are essential to modern physics. Its part of the deep connection between symmetry and physics started by Noether with her famous theorem.

Thanks
Bill

 

[S Steinhauer]

I will read up on the things you said. I am stopping to pursue my physics degree and just wanted to share this idea instead of letting it die.

Thank you for taking your time.

Edit: I meant there was a contradiction because the lab predicts different values for the particle relative to the lab than the particle does for the same thing. Anything that I said changes nothing in the laboratory frame of reference. I never ment to say there was a contradiction with QM and relativity, I was thinking there is one when attemting to take a particles frame of reference. (To the point where I only consider the lab´s wavefunction and the particle simply has a constant position.)

Imagine I as a human was shot through a double slit, what would I perceive. I can only imagine that I myself would still look normal in the sense that my position wouldn´t be suddendly undefined, but instead the lab´s position would be from my point of view undefined (which wouldn´t be a problem as my wavefunction relative to the lab can only remain uncollapsed if I do not interact with it.) I don´t think I am doing the best job explaining that idea.

When I was considering the lab´s wavefunction I considered it simply a massive particle, I should have said that.

Thank you for the bit on relativity and the comment on Schrödinger´s equation. I will read the pdf.

 

[S Steinhauer]

I was not able to edit my last post further so I am writing another comment.

So what? If I look at your posts I see a different username than you see when looking at my posts. Why? Simply because we are different users.

Imagine I had an unknown position relative to the Earth from the Earth's frame of reference. Assume it predicts 50% chance for me to "show up" in italy and a 50% chance for me to "show up" in germany.

Now I switch to my frame of reference. My position relative to the Earth is still unknown. However I predict a 70% chance for the Earth to be in a position such that I am in italy and a 30% chance for me to be in germany. This has to be a contradiction I think. When a lab predicts a different wave function for a particle relative to the lab, than the particle predicts for the lab relative to the particle.

What does that mean?

At this point I attempted to solve this by saying, that the particle has different coordinates and that that difference has to be such that both predict the same wavefunction. In other words space and time of the lab would be "distorted" from particle´s point of view. This would only be the case as long as there is no interaction as explained in my original post, so the particle would never know.

The example I gave above could not be resolved like that.

if you read Ballentine - QM A Modern Development [...] (see chapter 3)

I will read some of that book and the landau book. Not sure how much since they seem long. Thanks for the recommendations.

Edit: I found the modern definition of the inertial reference frame in the book, It is much "cleaner" than the quote I used. I don´t think this changes much if I am not mistaken. Although it is definitely worth the read.

 

[mfb]

Imagine I had an unknown position relative to the Earth from the Earth's frame of reference. Assume it predicts 50% chance for me to "show up" in italy and a 50% chance for me to "show up" in germany.

Now I switch to my frame of reference. My position relative to the Earth is still unknown. However I predict a 70% chance for the Earth to be in a position such that I am in italy and a 30% chance for me to be in germany. This has to be a contradiction I think. When a lab predicts a different wave function for a particle relative to the lab, than the particle predicts for the lab relative to the particle.

The wavelength has nothing to do with a spin orientation.

At this point I attempted to solve this by saying, that the particle has different coordinates and that that difference has to be such that both predict the same wavefunction. In other words space and time of the lab would be "distorted" from particle´s point of view. This would only be the case as long as there is no interaction as explained in my original post, so the particle would never know.

I don't understand that explanation either.

 

[bhobba]

I will read up on the things you said. I am stopping to pursue my physics degree and just wanted to share this idea instead of letting it die.

Well if you are pursuing a physics degree you will have your own textbooks. I will simply give some papers and note some books that IMHO will really help in understanding some of the issues you seem to be grappling with. BTW it's really great you are actually thinking about this stuff, far too many doing degrees don't and just want a bit of paper at the end to get some mega-buck (well they like to think so anyway) job at the end. Certainly the path of least resistance is what many do. But don't go too much the other way - when I did my degree all those many moons ago I did - I was considered a good thinking student, but as one lecturer said to me - Bill I can give you books that will answer all the questions you ask but you won't read them - they are so boring. He was right - there is a happy middle way we all must find.

OK here are some papers I would like you to read (in order of increasing mathematical sophistication):
https://www.scottaaronson.com/democritus/lec9.html
https://arxiv.org/pdf/quant-ph/0101012.pdf
https://arxiv.org/abs/1402.6562

And two books that explain the modern view of physics from symmetry - again in order of what you should read:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20
http://physicsfromsymmetry.com/

And last, but far from least THE book (IMHO) on QM:
https://www.amazon.com/dp/9814578584/?tag=pfamazon01-20

After going through those books you will have a much better idea of what is really going on in physics.

Don't let it get in the way of your degree, but when time permits go through them and I think you will find your understanding will likely be better.

Thanks
Bill

 

[bhobba]

Edit: I found the modern definition of the inertial reference frame in the book, It is much "cleaner" than the quote I used. I don´t think this changes much if I am not mistaken. Although it is definitely worth the read.

It changes nothing - of course. But don't dismiss the much cleaner observation you have made. Its what keeps your thinking clearer and you get yourself into less trouble. Note how its based on symmetry - like much of Landau's book as you will find when you hopefully read the whole book - that will sink into you mind so to speak as the unifying principle of classicist physics (along with the Principle of Least Action - which actually comes from QM - so QM is the real basis of classical physics - but I will go no further than mention it here - that is a revelation best discovered for yourself - QM is really the foundation of even classical physics). Landau doesn't mention Noether where symmetry really comes into its own but having studied Landau you are ready for that 'revelation'. The second book I mentioned after Landau above goes into Noether in a big way - after that you are well and truly prepared for many of the ideas of modern physics. For example in QM the concept of spin seems to confound many - but as that book explains its just a consequence of symmetry - although it requires some degree of mathematical sophistication to see it.

Thanks
Bill

 

[S Steinhauer]

The wavelength has nothing to do with a spin orientation.

It wasn´t about spin, it was about position. I am clearly not doing the best at explaining my idea in general so I think I might just drop it for now. Even if it would have been correct it would almost never make a difference as long as I stay In the lab frame (by that I mean: The lab doesn´t have a wavefunction).

Specifically it would make a difference if I was to measure the average time to decay for a particle twice with two different time transformations. From my point of view (lab) I would expect the time to flow differently for the particle in each case, meaning a different average decay time. (Imagine an observer looking at a clock close to a black hole and a clock that is further away from it.) This would be hard to archieve without applying a force which would make the priciples inapplicable as it only works for inertial frames of reference. Who knows what would happen with forces...

Don't let it get in the way of your degree, but when time permits go through them and I think you will find your understanding will likely be better.

As I said: I am stopping to pursue my degree, I found the teaching so rushed and focused on calculating specific cases (hydrogen atom and such) that I never understood the basic priciples and am now going to go somewhere else for making money and will keep physics as a hobby where I can focus on what I find interesting. I will slowly work my way through the books you recommended, thank you.

 

[mfb]

The decay time depends on the reference frame - but only via relativistic effects (time dilation). This is well-studied with muons from cosmic rays and with unstable particles in accelerators and particle detectors.

 

[S Steinhauer]

After your recommendations and questions I kept thinking about this idea and now want to provide a cleaner explanation that is hopefully more understandable.

For a clean definition of the inertial reference frame I recommend to follow bhobba´s recommendation "Landau - Mechanics" §3.

Assume there was a laboratory just floating in empty space, there are no forces and no other objects. In this lab there is a human conducting experiments to measure the laws of physics. The human will always measure the same laws in their simplest form as he is in an inertial frame of reference. It does not make sense to define the position, orientation or speed of the lab in space as none of these make any difference to the laws of physics and there is no other object to define these values relative to. In other words: None of these values matter.

Imagine a second lab was introduced into this space and both of the humans can see each other. Now lab_1 can define its position, speed and orientation relative to lab_2. Lab_1 can draw up a coordinate system and define a time dependant vector x(t) that contains all the information regarding the position of both lab´s. (Consider here that x(t) is the only relevant information regarding position, since position of both lab´s together in space does not make a difference just like in last paragraph. I will focus on position from now on and ignore velocity and orientation.)

As there is no reason to value anyone perspective over the other this has to be equivalent to lab_2 defining the vector from its position. It defines the vector y(t) which has to fulfill $y(t)=-x(t)$ as both of the lab´s have to agree on the vector defining their position. (Assuming both of them use the same units and the same orientation of their coordinate systems. If two humans measure the distance between them it cannot be that one of them measures 5 meters and the other one measures 3 meters and both are correct as $5 \neq 3$).

Superscripts are indices, subscripts are not!
Now let us assume lab_1 loses contact to lab_2 however it can write down a function predicting the probability of possible positions of lab_2 $\psi_x (x^\mu)$. By the last paragraph this function contains all the information of the position of these two lab´s and it has the equivalent perspective of lab_2 with $\psi_y (y^\mu) = \psi_x (x^\mu)$. I will leave out the orientation of the y vector (x is not negative) because I am using four-dimensional coordinates and time doesn´t get negative. This should give correct results as well but is less intuitive, this means lab_2 chooses a coordinate system that has its spatial axes mirrored relative to lab_1´s coordinate system. (As the two perspectives predict a vector that contains all the information of the position of the labs relative to each other, they have to agree about how likely a given vector is giving the equation above)

Let us now assume that these predictions are made according to quantum physics. Let us also assume both lab´s have different mass. If both lab´s have the same laws of physics (but different mass) they predict different functions for each other $\psi_y (y^\mu) \neq \psi_x (x^\mu)$. (I will assume the lab´s can be treated as super massive elementary particles regarding their behaviour relative to each other, as mfb pointed out. One can also drop the the laboratory example and simply consider two different elementary particles I think.)

This appears to be a contradition, with what we said earlier. I did find two possible but unlikely solutions to this:

1. Rest mass of a system is not relative (in contrary to position etc.) so theoretically the rest mass of the lab could change the laws of physics. In other words plank´s constant and the speed of light could be dependant on the lab´s own rest mass. Even in this simple case (no forces, two particles) this could cause problems as there are four coordinates but only two constants that can be changed. However I myself can´t think of an explicit example that creates a problem like that so I will include this option.
2. The two lab´s have different space-time coordinates. I said earlier that it would be a contradiction if there are two humans measuring the distance between them and both of them get different results and both are correct at the same time. This would be exactly that, however the argument here is that this could happen as long as the two systems are not interacting because they would never know. As the time-developement is always different there would be permanent difference in how time progresses. Even if I assume this doesn´t work for lab´s as they are not elementary particles this idea seems ridiculous, however these are the only two solutions I was able to come up with.

("Option 2" would have effects on the constants as well as both perspectives will disagree on what consitutes a second/meter. Again I will reference the earlier example where in one perspective the distance is 5 meters and in the other perspective the distance is 3 meters but both are correct with their measurement. With normal coordinate transformations of this kind, constants (and units) would be transformed as well, however here the same laws of physics are required, keeping the constants the same with different scales. Imagine x=2x´ normally if c=1 in the x-frame, then c=2 in the x´-frame. Here c=1 always, so the behaviour of light is different depending on perspective as light effectively travels half the distance in the x´-frame assuming t=t´.)

I would be lying if I would say I remotely understand the repercussions of either of these to options however they were the only possible solutions I could think of. They still both seem ridiculous to me.

Subscripts are not indices, superscripts are!
Deriving the equation: $\psi_y (y^\mu) \equiv \psi_x (x^\mu)$ I assume the two observers are allowed to disagree on distance and $\triangle t$ here.
Since there are no forces, I will assume both of these are normalized exponential functions.
$$N_y exp(f_y(y^\mu)) = N_x exp(f_x(x^\mu))$$
I take the logarithm
$$log(N_y) + f_y(y^\mu) = log(N_x) + f_x(x^\mu)$$
Since this has to be true for all points in space-time I take the partial derivative for all coordinates $\frac{\partial }{\partial x^\mu}$. This gives four equations:
$$\frac{\partial y^\nu}{\partial x^\mu}\frac{\partial }{\partial y^\nu}f_y(y^\sigma)=\frac{\partial }{\partial x^\mu}f_x(x^\sigma)$$

It might be possible to use the same equation for "Option 1" and simply replace x with k=(c,h) and y with k´=(c´,h´) (speed of light and plank´s constant). Replace derivatives as well I think.

I find this much cleaner and hope it is easier to understand. While the solutions of the apparent contradiction seem way off, I am more convinced than ever that I am onto something with attempting to take the perspective of a superpositioning particle. I do not have the understanding of quantum physics to immediately discard the solutions. I will keep on thinking about this and might update this if I find a solution that seems like it causes fewer issues. Sorry for the long post, I tried to keep it short and clear.

Thanks to bhobba and mfb. Anybody reading this feel free to ask questions and correct me.

 

[Nugatory]

If two humans measure the distance between them it cannot be that one of them measures 5 meters and the other one measures 3 meters and both are correct as $5 \neq 3$).

It can be, and it is. If they are moving relative to one another, then in general they will measure different values for the distance between them. In fact, a relative velocity of .8c leads to exactly what you describe: one of them correctly measures three meters separation and the other just as correctly measures five meters separation.

I did find two possible but unlikely solutions...
(2) The two lab´s have different space-time coordinates. I said earlier that it would be a contradiction if there are two humans measuring the distance between them and both of them get different results and both are correct at the same time.

That is pretty much what happens. However, the relationship (which is given by the Lorentz transformations) between the two sets of coordinates is such that all the laws of physics work no matter which one you use, so there is no contradiction. In particular, the speed of light is the same using either coordinate system.

This stuff is the essence of special relativity, so if it seems to you impossible or implausible you will want to solidify your understanding of special relativity before you take on the question of how it works with quantum mechanics. A textbook at the level of Taylor and Wheeler's "Spacetime Physics" would be a good starting point.

You also have to be aware that just about everything you've learned about QM so far will be the simpler non-relativistic form of the theory. Thus, the answer to the question "How does <this part of QM> work with special relativity?" is nearly always "It doesn't, and it's not expected to." To reconcile QM and SR you need quantum field theory,as mfb said above. Lancaster and Blundell's "Quantum field theory for the gifted amateur" is about the simplest honest treatment of QFT you'll find.

 

[S Steinhauer]

It can be, and it is. If they are moving relative to one another, then in general they will measure different values for the distance between them. In fact, a relative velocity of .8c leads to exactly what you describe: one of them correctly measures three meters separation and the other just as correctly measures five meters separation.

That is true, can´t believe I wrote it down like I did. All of what I did was assuming non-relativistic speeds of the particles relative to each other, I should have definitely mentioned this.
During non-relativistic speeds however what I said should be true, although I don´t really have a strong argument for it.

Superscripts are indices, subscripts are not.
Example, particles have no momentum relative to each other: $\psi_x(x^\mu)=exp(-iE_xx^0)$ with x⁰ = t and E_x=m_x and from the other perspective: $\psi_y(y^\mu)=exp(-iE_yy^0)$

This gives (with the equation from my post): $$-iE_y \frac{\partial y^0 }{\partial x^0} = -iE_x$$
$$\Leftrightarrow \frac{\partial y^0 }{\partial x^0} =\frac{E_y}{E_x}$$
$$y^0=\frac{E_y}{E_x}x^0$$

This would mean if $E_y \neq E_x$ time runs differently from each perspective (for no relative motion) as $t \neq t´$ which is why I said this seams way off. I should have included this example but was attempting to keep the post short.

I cannot imagine us not noticing something like this and on top of that I can only assume there is a host of other problems coming with it. I am not capable of solving the equation for an x¹-boost becuase the transformation of the momentum is given by the transformation that I am trying to find out:
$$y^1=\frac{p_y}{p_x}x^1$$
$$y^0=\frac{E_y}{E_x}x^0$$
With $p_x=m_x v_x$, $p_y=m_y v_y$ and $E=\sqrt{m^2+p^2}$ However the transformation of the velocity is given by the space and time transformation and I don´t think I can assume I know the velocities from both perspectives. That would be assuming I already know the transformation.

Clearly I failed at explaining my thoughts more clearly, I might just be entirely off with my approach as well, I would delete the post or edit in corrections however neither of that seems possible. Thanks for your time.

 

[vanhees71]

Where did you get all these equations from? To me they don't make any sense :-(.

 

[mfb]

Let us now assume that these predictions are made according to quantum physics. Let us also assume both lab´s have different mass. If both lab´s have the same laws of physics (but different mass) they predict different functions for each other $\psi_y (y^\mu) \neq \psi_x (x^\mu)$. (I will assume the lab´s can be treated as super massive elementary particles regarding their behaviour relative to each other, as mfb pointed out. One can also drop the the laboratory example and simply consider two different elementary particles I think.)

As mentioned before, systems with different mass are different. There is no reason to expect them to have the same quantum-mechanical (or even classical) description. They also have a different momentum in classical physics. So what?

Rest mass of a system is not relative (in contrary to position etc.) so theoretically the rest mass of the lab could change the laws of physics.

There are no laws changed. You consider two different systems. They are different. I don't see how this could be surprising in any way.

In other words plank´s constant and the speed of light could be dependant on the lab´s own rest mass.

They are not.

 

[S Steinhauer]

I was trying to find were I am going wrong as both options clearly seem wrong.

I think I found it: Quantum physics describes things from the perspective of a system where the classical limit can be assumed and describes particles where this is not the case, however in my example I treated both as quantum physics particles which simply might have a different description. I do not think I made a mistake saying that they (would) have to predict the same probability for a given vector as I explained. I do not know how to further explain this than I did but it seems kind of irrelevant now, considering there was a "hole" in it anyway. I should have noticed this way earlier, sorry for wasting your time.

Where did you get all these equations from? To me they don't make any sense :-(.

It is kind of irrelevant now. I will explain it to you anyways if you care for some reason, you will have to be more specific though: Do you mean in the example in the post right above yours?

 

[vanhees71]

Don't worry. I just wanted to express that your equations don't make any sense (at least not to me).