Galileo and Lorentz transformation

[meopemuk]

If it really implies what you described here (the same bomb explodes in one frame, but doesn't in other frames) then it contradicts the principle of relativity, which allows you to pick any frame for calculation and yet arrive at the same conclusion about the bombs fate. Therefore I have doubts if that is what the properties if the Poincare group really imply.

This is not how I see the essence of relativity. In my opinion, a relativistic theory must provide transformation rules, which allow one to calculate states of the system in all reference frames as soon as one knows the state in one particular frame. What I am suggesting is not different. The only difference is that the rules are slightly more complicated than usual length contraction and dilation of time intervals.

Quantum physics has many-worlds-interpretations. But in classical relativity there is only one bomb, that explodes or not. And every frame has to agree on that. Contrary to what the name "Relativity" suggests, the key of this theory are the absolute (frame invariant) quantities. Actually Einstein preferred the name "Invariantentheorie" (theory of invariants).

Apparently, the property that you call "fate" does not belong to the list of relativistic invariants. This is true in both quantum and classical physics. The many worlds interpretation has nothing to do with it.

Eugene.

 

[Al68]

This is not how I see the essence of relativity. In my opinion, a relativistic theory must provide transformation rules, which allow one to calculate states of the system in all reference frames as soon as one knows the state in one particular frame. What I am suggesting is not different. The only difference is that the rules are slightly more complicated than usual length contraction and dilation of time intervals.

The term "reference frame" is normally used to describe a system to assign space and time coordinates to an event, not a system to determine whether or not the event occurred. That's more than "slightly more complicated". That's a whole new conceptual definition of reference frame, a definition that simply isn't the one used by everyone else.

How can the coordinates of an event be transformed from one reference frame into a different reference frame unless it's the same event?

 

[Fredrik]

There is a long (logical) distance between two postulates of special relativity (the equivalence of all inertial frames and the invariance of the speed of light) and the idea of Minkowski space-time. To fill this gap one needs to make a few logical steps.

As I have argued many times before in this forum, there are no logical steps that can accomplish this, for the simple reason that the "postulates" are ill-defined. What you can do is to assume that inertial frames, whatever they will eventually be defined as, must be such that the functions representing a change of coordinates between two frames, take straight lines to straight lines, and form a group. This is only consistent with the Galilei and Poincaré groups. The second postulate is then interpreted as "Let's go with Poincaré".

The postulates are clearly insufficient to define a theory. They are just loosely stated guidelines that can help us guess what theory we should be using. It turns out that the natural choice is to define the theory by a set of axioms that tells us how to interpret the mathematics of Minkowski space as predictions about results of experiments, axioms like "A clock measures the proper time of the curve in Minkowski space that represents its motion".

Are you actually suggesting that this definition of SR is wrong? What then, is the proper definition of SR?

D'oh, I have more to say, but I have to go. Later.

 

[meopemuk]

The term "reference frame" is normally used to describe a system to assign space and time coordinates to an event, not a system to determine whether or not the event occurred. That's more than "slightly more complicated". That's a whole new conceptual definition of reference frame, a definition that simply isn't the one used by everyone else.

How can the coordinates of an event be transformed from one reference frame into a different reference frame unless it's the same event?

You are right, I have a different definition of "reference frame". In my opinion, a better term is "observer" or "laboratory". In my definition, a reference frame is a laboratory fully equipped with measuring devices for all basic observables, which include position, momentum, energy, mass, spin, etc. Time is also measured by the laboratory's clock. Assigning space and time labels to events is just a small part of functions performed by the observer/laboratory. There are many other observables that determine the state of the observed system. It is not a big deal if different observers see different events. A viable relativistic theory must provide transformation laws for connecting system's description between different observers. If these laws are more complicated than relabeling of space-time coordinates, then so be it.

Eugene.

 

[hamster143]

I haven't heard about such an observable as "coincidence" and I can't say what is the corresponding transformation law.

Would you agree that the difference of clock readings is a valid observable? I.e. time shown by clock B when clock A reads 12:00 PM. This difference has to transform as a representation of Lorentz group under boosts and rotations. The principle of relativity tells us that there can't be any preferred reference frames, which greatly constrains the choice of representations.

 

[meopemuk]

Are you actually suggesting that this definition of SR is wrong? What then, is the proper definition of SR?

I think that in order to have a fully operational relativistic theory it is sufficient to adopt three postulates:

1. All inertial frames are equivalent (the principle of relativity).
2. Transformations between inertial frames form the Poincare group.
3. Postulates of quantum mechanics.

The most relevant references are

E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group", Ann. Math.,
40 (1939), 149.

P. A. M. Dirac, "Forms of relativistic dynamics", Rev. Mod. Phys., 21 (1949), 392.

Eugene.

 

[A.T.]

But in classical relativity there is only one bomb, that explodes or not. And every frame has to agree on that.

Apparently, the property that you call "fate" does not belong to the list of relativistic invariants.

Yes it does. The 'fate' of the bomb is determined by the proper-times of the two clocks, which are frame invariant.

You are right, I have a different definition of "reference frame". In my opinion, a better term is "observer" or "laboratory"

Or a "parallel universe", where things happen, that don't happen in other frames.

 

[meopemuk]

Would you agree that the difference of clock readings is a valid observable? I.e. time shown by clock B when clock A reads 12:00 PM. This difference has to transform as a representation of Lorentz group under boosts and rotations. The principle of relativity tells us that there can't be any preferred reference frames, which greatly constrains the choice of representations.

I think we can agree that positions of clock arms \mathbf{r}_A and \mathbf{r}_B are good observables. Wigner-Dirac relativistic quantum theory allows us to find these positions in the moving reference frame


\mathbf{r}_A (\theta) = e^{-iK^A_x \theta} \mathbf{r}_A e^{iK^A_x \theta}
\mathbf{r}_B (\theta)= e^{-iK^B_x \theta} \mathbf{r}_B e^{iK^B_x \theta}

where K^A_x and K^B_x are total boost operators characteristic for the two clocks A and B, and \theta is the rapidity of the boost. Since two clocks are complex interacting systems, operators K^A_x and K^B_x depend not only on observables of the clocks' arms, but also on observables of other clocks' parts. Therefore, above transformations are complex functions, which cannot be written explicitly without the detailed knowledge of interactions acting inside the two clocks. So, it is not possible to conclude (as you suggest) that from \mathbf{r}_A =\mathbf{r}_B it must follow that \mathbf{r}_A(\theta) =\mathbf{r}_B (\theta).

Eugene.

 

[Fredrik]

1. All inertial frames are equivalent (the principle of relativity).
2. Transformations between inertial frames form the Poincare group.
3. Postulates of quantum mechanics.

In what way are the frames equivalent when they (according to you) don't even agree about measurement results?

How do you define classical SR? #2 is enough to imply Minkowski space, so what does #1 add? And didn't you just argue against Minkowski space?

 

[Dale]

Meopemuk, I think you may be confusing some concepts here, or I am misunderstanding your point. You seem to be erroneously applying a many-worlds interpretation to special relativity where every inertial reference frame is a separate world which can disagree about the existence of physical events.

In MWI if an event happens in this world then it will happen in all reference frames in this world. There may be another world where it does not happen, and in that world it does not happen in any frame. There is no world where it happens in one frame but not another.

If I am misunderstanding your position could you please clarify?

 

[Al68]

You are right, I have a different definition of "reference frame". In my opinion, a better term is "observer" or "laboratory". In my definition, a reference frame is a laboratory fully equipped with measuring devices for all basic observables, which include position, momentum, energy, mass, spin, etc. Time is also measured by the laboratory's clock. Assigning space and time labels to events is just a small part of functions performed by the observer/laboratory. There are many other observables that determine the state of the observed system. It is not a big deal if different observers see different events. A viable relativistic theory must provide transformation laws for connecting system's description between different observers. If these laws are more complicated than relabeling of space-time coordinates, then so be it.

Eugene.

I can agree with all of that. I would note that, by definition, if the events observed actually happened (not illusions), then if two observers observe two different events, then two different events occurred.

Which means that while the events can be related, it obviously must be possible for both to occur if both are observed.

 

[meopemuk]

In what way are the frames equivalent when they (according to you) don't even agree about measurement results?

Well, in "normal" special relativity different frames also disagree about measurement results (for distances and time intervals). However, this does not make them inequivalent.

#2 is enough to imply Minkowski space,

I strongly disagree about that. There is no way you can deduce the existence of Minkowski space-time from the Poincare group alone.

Eugene.

 

[meopemuk]

Meopemuk, I think you may be confusing some concepts here, or I am misunderstanding your point. You seem to be erroneously applying a many-worlds interpretation to special relativity where every inertial reference frame is a separate world which can disagree about the existence of physical events.

In MWI if an event happens in this world then it will happen in all reference frames in this world. There may be another world where it does not happen, and in that world it does not happen in any frame. There is no world where it happens in one frame but not another.

If I am misunderstanding your position could you please clarify?

No, I am not a follower of the many-world interpretation. Everything I'm talking about happens in our good old single world.

Eugene.

 

[Dale]

Well, in "normal" special relativity different frames also disagree about measurement results (for distances and time intervals).

No, they do not. If a clock measures a given time in one reference frame then it will measure that time in all reference frames. Similarly with the measurement obtained from any rod. Different reference frames may disagree about whether or not that measurement corresponds to that specific frame's coordinate time or distance, but they will all agree on the value that the clock or rod measures.

I think you have a gross misunderstanding of the implications of the first postulate.

 

[atyy]

E. V. Stefanovich, "Violations of Einstein's time dilation formula in particle decays", http://www.arxiv.org/abs/physics/0603043

I guess the deviation appears in the numerical simulations in Figure 2, which is based on Eq 60. But Eq 60 is an approximation - do you think the apparoximation could be responsible for the deviation?

Naively to me I wouldn't think time translation should do anything. On the other hand, it does seem that an "event" doesn't really exist in relativistic QFT - since the Newton-Wigner operator is kinda weird, as you point out.

 

[meopemuk]

I can agree with all of that. I would note that, by definition, if the events observed actually happened (not illusions), then if two observers observe two different events, then two different events occurred.

Which means that while the events can be related, it obviously must be possible for both to occur if both are observed.

I don't quite understand what you mean here. Nevertheless let me repeat my most "outrageous" claim, so that it would be easier for you to attack my position.

Let's say we have one physical system - a piece of explosive - which is observed from different reference frames. If the two frames are moving with respect to each other, then standard SR predicts that their observations are different. For example, the two observers disagree about the length of the object in the direction of movement. I am saying that, in addition to that, the two observers may disagree about more substantial things. For example, one observer may find the bomb unexploded while the other may see the explosion.

This claim may look less outrageous if you replace in the previous example the pair of relatively moving observers (i.e., observers related by a boost) by the pair of observers related by a time translation. Then the statement like "the bomb does not explode today, while it will explode tomorrow" looks completely normal.

According to the Poincare group ideology, different types of inertial transformations must bear some similarity (as transformations from the same group). So, if we accept non-trivial dynamical effects of time translations, then we must accept (at least in principle) the possibility of non-trivial dynamical effects of boosts.

Eugene.

 

[meopemuk]

No, they do not. If a clock measures a given time in one reference frame then it will measure that time in all reference frames. Similarly with the measurement obtained from any rod. Different reference frames may disagree about whether or not that measurement corresponds to that specific frame's coordinate time or distance, but they will all agree on the value that the clock or rod measures.

I think you have a gross misunderstanding of the implications of the first postulate.

Let's say we have a one-meter rod viewed by two different observers. The observer at rest with respect to the rod will find its length to be 1 meter exactly. The observer moving with respect to the rod will find its length to be shorter than 1 meter. This is called "length contraction". This is what I mean by saying that different observers disagree about measurement results.

Eugene.

 

[meopemuk]

I guess the deviation appears in the numerical simulations in Figure 2, which is based on Eq 60. But Eq 60 is an approximation - do you think the apparoximation could be responsible for the deviation?

Yes, eq. (60) is an approximation. However, this is a pretty good approximation, as discussed in the beginning of section 11. I believe that the error associated with this approximation is much less than the magnitude of the obtained effect.

Naively to me I wouldn't think time translation should do anything. On the other hand, it does seem that an "event" doesn't really exist in relativistic QFT - since the Newton-Wigner operator is kinda weird, as you point out.

I agree that "localized events" is an ill-defined concept in quantum mechanics. Quantum particles don't stay localized for long - their wave packets spread. Moreover, a particle localized in one frame may be seen as not localized in another moving frame.

On the other hand, the decay law (considered in the paper) is a well-defined quantum-mechanical concept. In order to measure the decay probability (in any frame) you just calculate the ratio of decayed particles to the total number of unstable particles in the batch. If the batch is big enough, statistical deviations can be reduced to zero.

Eugene.

 

[Al68]

I don't quite understand what you mean here. Nevertheless let me repeat my most "outrageous" claim, so that it would easier for you to attack my position.

Let's say we have one physical system - a piece of explosive - which is observed from different reference frames. If the two frames are moving with respect to each other, then standard SR predicts that their observations are different. For example, the two observers disagree about the length of the object in the direction of movement. I am saying that, in addition to that, the two observers may disagree about more substantial things. For example, one observer may find the bomb unexploded while the other may see the explosion.

That claim isn't outrageous at all.

But "device has exploded" and "device hasn't exploded" are not a single event with different coordinates assigned by different observers. And they obviously don't contradict each other, unless an observer observes the former precede the latter instead of the other way around.

Now if the events are "device exploded when the attached clock read noon" and "attached clock goes past noon but device hasn't exploded", if each event is observed by different observers, they are two different events, not a single event observed differently, unless one is an illusion.

 

[Fredrik]

Well, in "normal" special relativity different frames also disagree about measurement results (for distances and time intervals). However, this does not make them inequivalent.

They certainly don't disagree in the way that we've been talking about so far. When two clocks disagree, it's because they're displaying the proper times of two different curves in spacetime. When two (inertial) rulers disagree about the length of an object, it's because they measure the proper distances of two different curves in spacetime. This is all completely standard, but it's not what we've been talking about. What you've been claiming is the equivalent of claiming that a single clock can display two different times at a single point in spacetime, in two different coordinate systems. This is extremely non-standard, so I find it strange that when I ask you about it, you're replying with a comment about trivial standard stuff.

You didn't explain how you can consider two inertial frames that can't even agree about what happens at a single event to be equivalent in any way.

 

[Fredrik]

Regarding the first "postulate": I have said this many times before, and I guess I'll have to say it many times again. It isn't a well-defined statement that you can use as the starting point of a derivation! It's often stated in the form "The laws of physics are the same in all inertial frame". The most generous interpretation of this that we can make, is that this represents a set of well-defined statements, that has one member for each definition of "inertial frame", each definition of "law of physics" and each definition of what it means for two laws of physics to be "the same".

 

[meopemuk]

What you've been claiming is the equivalent of claiming that a single clock can display two different times at a single point in spacetime, in two different coordinate systems. This is extremely non-standard, so I find it strange that when I ask you about it, you're replying with a comment about trivial standard stuff.

I didn't say that. First, I have never referred to a "single point in spacetime". I even don't understand what this phrase means. I don't know which experimental measurements can tell two observers whether the events they see occur "at the same spacetime point" or at different spacetime points. I prefer not to use the "spacetime" language at all. Second, I don't like the term "coordinate system". It suggests the presumption that the only difference between moving reference frames (or observers) is in assigning different labels (or coordinates) to events, while all observers must agree on the presence/absence/nature of the events. I think that the terms "inertial observer" or "inertial laboratory" are more appropriate, because they do not exclude the possibility that different observers may actually see different events happening.

Eugene.

 

[meopemuk]

Regarding the first "postulate": I have said this many times before, and I guess I'll have to say it many times again. It isn't a well-defined statement that you can use as the starting point of a derivation! It's often stated in the form "The laws of physics are the same in all inertial frame". The most generous interpretation of this that we can make, is that this represents a set of well-defined statements, that has one member for each definition of "inertial frame", each definition of "law of physics" and each definition of what it means for two laws of physics to be "the same".

I disagree with that. In my opinion the first postulate is the most powerful, deep, and non-trivial statement in all physics. It just tells you that idential experiments in different laboratories produce identical results. It does not matter whether the laboratory is in Paris or in London. It does not matter whether the experiment was made today or a century ago. It does not matter whether the experiment was made in a spaceship standing still or in a spaceship moving with high velocity. Without this relativity principle, it would be impossible to compare results obtained by different researchers. Entire physics would be impossible.

Eugene.

 

[Al68]

Second, I don't like the term "coordinate system". It suggests the presumption that the only difference between moving reference frames (or observers) is in assigning different labels (or coordinates) to events, while all observers must agree on the presence/absence/nature of the events.

If their observations are contradictory, logic dictates that at least one of them is simply wrong.

 

[Dale]

Let's say we have a one-meter rod viewed by two different observers. The observer at rest with respect to the rod will find its length to be 1 meter exactly. The observer moving with respect to the rod will find its length to be shorter than 1 meter. This is called "length contraction". This is what I mean by saying that different observers disagree about measurement results.

This is standard SR length contraction, but even here every frame agrees about the outcome of all measurements. The moving observer agrees that the stationary observer measures 1 m, and the stationary observer agrees that the moving observer measures less than 1 m.

I am saying that, in addition to that, the two observers may disagree about more substantial things. ... like "the bomb does not explode today, while it will explode tomorrow" looks completely normal.

This is just standard relativity of simultaneity. Is there something more to your position or are you just trying to state standard SR in a provocative manner?

 

[meopemuk]

If their observations are contradictory, logic dictates that at least one of them is simply wrong.

Two inertial observers may see quite different events (not just the same events with different space-time labels, as usually postulated). This does not contradict any law of logic or physics.

Eugene.

 

[Dale]

For example?

 

[meopemuk]

I am saying that, in addition to that, the two observers may disagree about more substantial things. ... like "the bomb does not explode today, while it will explode tomorrow" looks completely normal.

This is just standard relativity of simultaneity. Is there something more to your position or are you just trying to state standard SR in a provocative manner?

This has nothing to do with the relativity of simultaneity.

In standard special relativity two moving observers may disagree about such "kinematical" properties as the length of an object or the duration of a time interval. However, they always agree about "dynamical" properties, like whether the bomb is exploded or not.

I am saying that these views must be generalized. Dynamical properties should be considered relative as well. I.e., different observers may disagree about them. It is possible that observer at rest does not see any explosion, while the moving observer (in the same location, at the same time) sees the explosion of the same object.

Eugene.

 

[Dale]

It is possible that observer at rest does not see any explosion, while the moving observer (in the same location, at the same time) sees the explosion of the same object.

This is completely contrary to the first postulate. It is also not appropriate for this forum.

 

[Al68]

Two inertial observers may see quite different events (not just the same events with different space-time labels, as usually postulated). This does not contradict any law of logic or physics.

Of course they may see quite different events, as long as the observations don't contradict each other.

Logic only dictates that two mutually exclusive events don't both happen, so if both are observed, one of the observations is in error.

But this still has nothing to do with transforming the coordinates of a single event between reference frames.

 

[meopemuk]

For example?

I keep repeating the example of a bomb, where two observers disagree about whether the explosion has occurred or not. I use this dramatic and unrealistic example just to make the general point absolutely clear.

More realistic examples concern observations of decays of unstable particles. In the post #5 I've cited a few references in which decay laws of moving particles have been studied in a rigorous relativistic quantum-mechanical setting. It follows, for example, that the particle that is seen as yet undecayed by the observer at rest (at time 0) has a non-zero decay probability from the point of view of the moving observer (at the same time). This is a more realistic analog of the unexploded/exploded bomb discussed above.

Of course, for known unstable particles and realistic observer speeds the "boost induced decay probability" is extremely small and cannot be presently observed. So, the whole issue is rather academic, but I think it is important nevertheless.

Eugene.

 

[meopemuk]

This is completely contrary to the first postulate.

The first postulate tells that experiments in different laboratories yield the same results. This means that each laboratory studies its own copy of the physical system. The first postulate does not say how measurements performed by different observers on the same object are related to each other. In order to find these transformation laws one needs a full dynamical description of the system, i.e., the representation of the Poincare group in the Hilbert space of the system. In the instant form of Dirac's dynamics, this description demands the non-trivial dynamical character of boosts. See, for example

S. Weinberg "The quantum theory of fields" vol. 1. section 3.3.

It is also not appropriate for this forum.

I am ready to stop if you think so.

Eugene.

 

[Fredrik]

I didn't say that.

Actually you did. I said this:

You seem to be saying that a person can get shot and killed at age 20 in one coordinate system and die of old age at the age of 125 in another.

You replied:

Your example is rather extreme. Calculations show that the dynamical effects of boost are rather weak (there are no experiments capable of seeing these effects today). However, as a matter of principle, I would answer "yes".

In SR, a person's entire existence is represented by a set of curves in Minkowski space. For our purposes, we can ignore the spatial separation between these curves and describe a person's existence approximately using only one curve. The endpoints of the curve represent the beginning and the end of the person's life. If a person gets shot and killed at the age of 20, then the endpoint of the curve that has the higher time coordinate (in all inertial frames) is the mathematical representation of his death in the real world. All of the other points on the curve are mathematical representations of events earlier in his life. Every one of those points represents an event where his age is 20 or less. And you said that there are points on this guy's world line at which his age is 125.

So you have clearly said (possibly without realizing it) that there's a point in Minkowski space where this particular "clock" (a person is a clock too) is displaying 125 years in one coordinate system and 20 or less in another.

First, I have never referred to a "single point in spacetime". I even don't understand what this phrase means.

How can you not? You must know that each point in Minkowski space is supposed to be a representation of an event in the real world, or rather in the universe described by the theory. (I prefer to think of a descriptive theory as SR as an exact description of a fictional universe that resembles our own, than as an approximate description of our universe).

I prefer not to use the "spacetime" language at all. Second, I don't like the term "coordinate system".

Special relativity is by definition a theory that uses a manifold called Minkowski spacetime to represent events. The definition of a manifold includes a bunch of stuff about coordinate systems, and a Lorentz transformation is a transition function between coordinate systems. So if you don't like those things, you must hate special relativity.

Edit: I have now read the posts where you talk about how the principle of relativity says that certain laboratories get the same results. I see what you mean now about laboratories vs. coordinate systems.

It suggests the presumption that the only difference between moving reference frames (or observers) is in assigning different labels (or coordinates) to events, while all observers must agree on the presence/absence/nature of the events. I think that the terms "inertial observer" or "inertial laboratory" are more appropriate, because they do not exclude the possibility that different observers may actually see different event happening.

I haven't completely ruled out that something like what you're suggesting might actually be valid, but the way you're talking about it is really strange. It's like you don't even see that what you're saying is something extremely different from anything that most of us have ever heard of in the context of SR. If it hadn't been you saying this (I've seen threads where you're the only one who gets it right), and if I hadn't read Leonard Susskind's claim that if you fall into a black hole, you pass through the horizon unharmed in one coordinate system and get incinerated by radiation in another, I would have dismissed it as crackpot nonsense right away.

I don't understand Susskind's example either, but at least there's an event horizon in his example to make things more complicated

 

[Fredrik]

It is also not appropriate for this forum.

Normally I'd agree, but since Meopemuk is a competent poster (at least in the quantum physics forum ), I hope he gets a chance to explain his position.

So if some moderator is thinking about closing the thread, please don't. At least not yet.

In my opinion the first postulate is the most powerful, deep, and non-trivial statement in all physics. It just tells you that idential experiments in different laboratories produce identical results. It does not matter whether the laboratory is in Paris or in London. It does not matter whether the experiment was made today or a century ago. It does not matter whether the experiment was made in a spaceship standing still or in a spaceship moving with high velocity. Without this relativity principle, it would be impossible to compare results obtained by different researchers. Entire physics would be impossible.

All of this is true, but it's still ill-defined. It doesn't unambiguously identify which laboratories produce the same results.

I'm going to get some sleep, so I won't read any answers for at least 8 hours.

 

[meopemuk]

Special relativity is by definition a theory that uses a manifold called Minkowski spacetime to represent events.

You and I are talking about two rather different approaches to relativity. They are based on different sets of postulates. Your (as well as Einstein's and lot of other people's) approach assumes the following postulates:

1. The principle of relativity (never mind that you don't like it, this principle remains true nevertheless).
2. The invariance of the speed of light.
3. The Minkowski space-time manifold in which all events are "embedded" (this postulate can be derived from 1. and 2. if you add the "coincidence condition" that we discussed earlier).

One problem with this logic is that it is not easily compatible with quantum mechanics. One example is the difficulty of defining the "time operator" (which, according to your approach, must exist, because space and time coordinates must be "interchangeable"). This difficulty was discussed in one of recent threads in the "Quantum physics" forum.


I begin from a different set of postulates:

1. The principle of relativity.
2. The Poincare group structure of transformations between different inertial observers.
3. Postulates of quantum mechanics.

In my approach I can also describe events with their space and time coordinates. However, these events are not regarded as points in the Minkowski space-time. The whole idea of the Minkowski space-time is just absent. Transformations of space and time labels of events between different observers can be calculated from quantum laws, and these transformations do not necessarily agree with Lorentz formulas, which are assumed exact and universal in your approach. Moreover, if one observer sees an event (e.g., an explosion or a collision of particles), another observer may not see it. The "coincidence condition" is not valid.

As far as I can tell, experimental consequences of the two approaches are pretty close. I couldn't find experiments, where predicted differences can be measured with modern tools. So, which approach is better should be decided on the basis of logic and consistency. You can make your own judgement.

Normally I'd agree, but since Meopemuk is a competent poster (at least in the quantum physics forum ), I hope he gets a chance to explain his position.

So if some moderator is thinking about closing the thread, please don't. At least not yet.

Thank you, Fredrik. I appreciate that.

Eugene.

 

[Fredrik]

You and I are talking about two rather different approaches to relativity. They are based on different sets of postulates. Your (as well as Einstein's and lot of other people's) approach assumes the following postulates:

1. The principle of relativity (never mind that you don't like it, this principle remains true nevertheless).
2. The invariance of the speed of light.
3. The Minkowski space-time manifold in which all events are "embedded" (this postulate can be derived from 1. and 2. if you add the "coincidence condition" that we discussed earlier).

It's not that I don't like the principle of relativity. It's just that it's ill-defined, and therefore useless as a mathematical axiom. 3 can't be derived from 1 and 2. It can be guessed from 1 and 2. Alternatively, we can interpret 1 as a set of well-defined statements and then determine which members of the set are consistent with the assumptions a) that spacetime is the set \mathbb R^4 with the standard manifold and vector space (or affine space) structure, and b) that functions that represent a change between coordinate systems are smooth and take straight lines to straight lines. These assumptions can be weakened, but it's definitely never correct to start with an ill-defined axiom.

Both of the approaches I described lead to 3, but the steps that take us from 1 and 2 to 3 (a guess, or a derivation based on clarifying and supplementary axioms) are not a part of the definition of the theory. The theory is defined by axioms that tell us how to interpret the mathematics of Minkowski space as predictions about results of experiments. (But there's of course a rigorous version of 1 and 2 that's implied by the definition of Minkowski space).

One problem with this logic is that it is not easily compatible with quantum mechanics. One example is the difficulty of defining the "time operator" (which, according to your approach, must exist, because space and time coordinates must be "interchangeable").

Not true. The only operators that must exist because of what I've said are the ones that can be constructed from the Poincaré algebra. The fact that Minkowski space has a non-trivial group of isometries make it obvious (once you understand the math, as I think you already do) that we should change the axioms of quantum theory to include the axiom that there's a group homomorphism from that group (the Poincaré group) into the group of probability preserving bijections on the set of unit rays. Wigner's theorem takes care of the rest.

I begin from a different set of postulates:

1. The principle of relativity.
2. The Poincare group structure of transformations between different inertial observers.
3. Postulates of quantum mechanics.

It's hard to see a difference between this and my approach, other than that you choose not to mention Minkowski space. I think the proper way to do that is to use the algebraic approach to QM. As I said in #42, according to this article (which I have only skimmed...I intend to return to it later), it's actually possible to reconstruct Minkowski spacetime from the Poincaré algebra (and the axioms of the algebraic approach).

 

[Dale]

The first postulate does not say how measurements performed by different observers on the same object are related to each other.

Yes, it does, all of the experimental results must be the same regardless of which coordinate system is used.

The bomb either explodes or does not explode based on the action of some set of physical laws from some set of boundary conditions. The coordinate system that you use to express those laws and those boundary conditions must lead to the same experimental outcomes in all cases. Otherwise you have violated the first postulate.

 

[Dale]

More realistic examples concern observations of decays of unstable particles. In the post #5 I've cited a few references in which decay laws of moving particles have been studied in a rigorous relativistic quantum-mechanical setting. It follows, for example, that the particle that is seen as yet undecayed by the observer at rest (at time 0) has a non-zero decay probability from the point of view of the moving observer (at the same time).

As long as the rest observer also predicts that the moving observer measures a non-zero probability and the moving observer also predicts that the rest observer measures a zero probability then this is standard fare for SR.

 

[Dale]

Normally I'd agree, but since Meopemuk is a competent poster (at least in the quantum physics forum ), I hope he gets a chance to explain his position.

I also have a nagging suspicion that this is a large miscommunication. Usually I am better at understanding what is being proposed than in this thread.

 

[atyy]

Yes, eq. (60) is an approximation. However, this is a pretty good approximation, as discussed in the beginning of section 11. I believe that the error associated with this approximation is much less than the magnitude of the obtained effect.

Yes, it seems like a good approximation, but why do you think the associated error is less than the obtained effect?

 

[meopemuk]

...it's actually possible to reconstruct Minkowski spacetime from the Poincaré algebra (and the axioms of the algebraic approach).

That's exactly the point where we disagree. Poincare group plus quantum mechanics does not imply (and does not need) the Minkowski spacetime.

Eugene.

 

[meopemuk]

As long as the rest observer also predicts that the moving observer measures a non-zero probability and the moving observer also predicts that the rest observer measures a zero probability then this is standard fare for SR.

Yes, in my approach (which is also the approach used by Wigner, Dirac, and Weinberg, though, unlike me, they never questioned the usefulness of the Minkowski spacetime) there are well-defined rules that connect system's descriptions by the two observers. However, in contrast to standard SR, these rules are more complicated than simple linear Lorentz transformation formulas (x,t) -> (x',t'). The exact boost transformation rules are different for different physical systems, they depend on interactions acting in the system and on the system's state. If F is operator of observable in the reference frame at rest, then operator of the same observable in the moving frame is obtained by formula

F(\theta) = e^{-iK_x \theta} F e^{iK_x \theta}

where K_x is the total boost operator which (similarly to the total Hamiltonian) contains interaction-dependent terms. These terms cannot be avoided in any relativistic interacting theory. The presence of these terms is responsible for the difference between exact transformation laws and (approximate) Lorentz formulas.

Eugene.

 

[meopemuk]

Yes, eq. (60) is an approximation. However, this is a pretty good approximation, as discussed in the beginning of section 11. I believe that the error associated with this approximation is much less than the magnitude of the obtained effect.

Yes, it seems like a good approximation, but why do you think the associated error is less than the obtained effect?

The reason for my belief is that different parameters control the accuracy of formula (60) and the size of the effect (the violation of the Einstein's time dilation formula).

The approximation used in the derivation of (60) is given in eq. (58). In words this means that the position-space wave function of the unstable particle is localized better than the distance passed by light during the particle's lifetime.

On the other hand, the size of the effect predicted by (60) is controlled by the ratio \Gamma/m, where Gamma is the width of the mass distribution and m is the particle's mass.

So, if we choose a particle with a sharply localized wave packet, small mass, and wide mass distribution (=short lifetime), we will minimize the error in formula (60) and we will maximize the effect of violation of the time dilation formula. So, it is possible to have situations in which the magnitude of the effect exceeds the error.

Eugene.

 

[Fredrik]

That's exactly the point where we disagree. Poincare group plus quantum mechanics does not imply (and does not need) the Minkowski spacetime.

I checked the article I mentioned again, and it seems that I remembered it wrong. What they claim to be able to do is something more complicated, and I neither have the time nor the knowledge to fully understand what their doing at this time, so let's drop that part of the discussion.

 

[atyy]

The reason for my belief is that different parameters control the accuracy of formula (60) and the size of the effect (the violation of the Einstein's time dilation formula).

The approximation used in the derivation of (60) is given in eq. (58). In words this means that the position-space wave function of the unstable particle is localized better than the distance passed by light during the particle's lifetime.

On the other hand, the size of the effect predicted by (60) is controlled by the ratio \Gamma/m, where Gamma is the width of the mass distribution and m is the particle's mass.

So, if we choose a particle with a sharply localized wave packet, small mass, and wide mass distribution (=short lifetime), we will minimize the error in formula (60) and we will maximize the effect of violation of the time dilation formula. So, it is possible to have situations in which the magnitude of the effect exceeds the error.

Eugene.

If the time dilation formula is not exact, then does that mean the speed of light is not the exactly the same in all inertial frames?

Edit: I guess the speed of light being constant is usually given by the dispersion relation in free space, ie. no interaction. But your point is that interaction modifies stuff? Also, even in classical SR there are processes where the time dilation formula doesn't apply just because they are not localized in any frame (I think), is the decay process analagous or not in your view? I know I'm being somewhat dense here, thanks for taking the time to answer questions!

 

[meopemuk]

If the time dilation formula is not exact, then does that mean the speed of light is not the exactly the same in all inertial frames?

No, these are totally unrelated issues. The speed of light is always c and this value is observer-independent. To prove that the speed of light is c, I note that light particles - photons - are massless, therefore their energy E is related to their momentum P as E=Pc. From the definition of relativistic speed I then obtain

V=Pc^2/E = c

The frame independence of this value can be proven by applying the unitary operator of boost transformation to V. For simplicity I consider the case in which the photon is moving along the x-axis, and the boost is apllied along the x-axis as well

V_x(\theta)=e^{-iK_x \theta} \frac{P_xc^2}{E} e^{-iK_x \theta}= \frac{(P_x \cosh \theta - E/c \sinh \theta)c^2}{E\cosh \theta -Pc \sinh \theta} = \frac{(P_x c^2(\cosh \theta -\sinh \theta)}{E(\cosh \theta - \sinh \theta)} =\frac{P_x c^2}{E} = c

Eugene.

 

[Al68]

I begin from a different set of postulates:

1. The principle of relativity.
2. The Poincare group structure of transformations between different inertial observers.
3. Postulates of quantum mechanics.

If that's the case, then it is inappropriate for the SR/GR forum, since SR/GR does not use those postulates.

But I think it should be obvious that some of your claims are clearly incompatible with #1, which says effectively that every reference frame agrees on what does or doesn't physically happen.

 

[atyy]

I begin from a different set of postulates:

1. The principle of relativity.
2. The Poincare group structure of transformations between different inertial observers.
3. Postulates of quantum mechanics.

These are absolutely standard, aren't they? So if there is a mistake, it's not at this point.

 

[meopemuk]

But your point is that interaction modifies stuff?

That's exactly my point. Formulas of special relativity are perfectly OK for systems not involving interactions, e.g, in the time clock where a free photon is bouncing between two mirrors. However, if interactions are present (as in the case of unstable particles), then Lorentz transformations and other SR formulas (such as the time dilation law) must be modified to take this interaction into account.

Also, even in classical SR there are processes where the time dilation formula doesn't apply just because they are not localized in any frame (I think), is the decay process analagous or not in your view?

I am not sure what you are talking about? I thought that the time dilation formula is always valid in SR independent on localization.

Eugene.

 

[meopemuk]

But I think it should be obvious that some of your claims are clearly incompatible with #1 [the principle of relativity], which says effectively that every reference frame agrees on what does or doesn't physically happen.

I disagree. The principle of relativity says that two different observers get exactly the same results for experiments with systems confined to their respective laboratories. The principle of relativity says absolutely nothing about how views of different observers on the *same* system are related. For example, it is not possible to derive the length contraction formula from the principle of relativity alone. You need an additional postulate. Usually, the invariance-of-the-spped-of-light postulate is chosen.

Eugene.