Planck constant is Lorentz invariant?

[keji8341]

It is widely recognized in physics textbooks that Planck constant is a "universal constant". But I nerver see a proof. As we know, in the special theory of relativity, c is a universal constant, namely a Lorentz invariant, which is Einstein's hypothesis. But How do we know the Plack constant h is also a Lorentz invariant?

Suppose E=hv in the lab frame, while E'=h'v' in the moving frame.
Usually, one assume that photon's momentum and energy forms a momentum-energy 4-vector, generalized from a real particle, like an electron which has non-zero rest mass and of which the velocity is less than c. However the derivation of electron's 4-vector is not valid for a photon. (k, w/c) is Lorentz covariant from the invariance of phase, but we don't know if h is Lorentz invariant. [Of course, if h is Lorentz invariant, (h_bar*k, h_bar*w/c) is a Lorentz covariant momentum-energy 4-vector.]

Therefore, the Lorentz invariance of Planck constant is only an artificial assumption. Am I right?

 

[vanhees71]

Planck's constant is assumed to be a Lorentz scalar, and quantum theory can be built in an explicitly Poincare-covariant way with this assumption. The resulting theory (which is relativistic quantum field theory) is one of the most successful scientific results ever, and thus we can be pretty sure that our assumption of \hbar being a scalar universal constant is good. That's the nature of any model building in the natural sciences: You make assumptions and look where they lead you in terms of observable predictions. Then you do an experiment to check, whether these predictions are correct and within which limits of physical circustances they are valid etc.

Of course, the momentum-four vector of a photon is Lorentz covariant. Otherwise it would not be a four vector to begin with! How do you come to the conclusion, it's not?

 

[PeterDonis]

However the derivation of electron's 4-vector is not valid for a photon.

Can you elaborate on this? I'm not sure what you mean by "the derivation of electron's 4-vector".

 

[keji8341]

Can you elaborate on this? I'm not sure what you mean by "the derivation of electron's 4-vector".

The electron's momentum-energy 4-vector is set up based on the 4-velocity V_4=(d/dt0)(x,ct) where t0 is the proper time and it is a Lorentz invariant. Since V_4 is Lorentz covariant and the electron's rest mass m0 is a Lorentz scalar from the principle of relativity, we conclude m0*V_4 = (p=m0*gamma*v, E/c=m0*gamma*c**2/c) is Lorentz covariant, and defined as momentum-energy 4-vector. But the derivation of V_4=(d/dt0)(x,ct) is not valid for a photon, because the photon's gamma-->infinity, a sigularity; in addition, the photon's rest mass is defined to be zero. Thus mathematically speaking, the derivation of electron's 4-vector is not applicable to a photon.

 

[keji8341]

Planck's constant is assumed to be a Lorentz scalar, and quantum theory can be built in an explicitly Poincare-covariant way with this assumption. ...
Of course, the momentum-four vector of a photon is Lorentz covariant. Otherwise it would not be a four vector to begin with! How do you come to the conclusion, it's not?

It is found that, Einstein’s Doppler formula is not applicable when a moving point light source is close enough to the observer; for example, it may break down or cannot specify a determinate value when the point source and the observer overlap. From this, we judge that (k,w/c) for a moving point light source is NOT Lorentz covariant, and if \hbar*(k,w/c) IS still Lorentz covariant, then \hbar must NOT be a Lorentz invariant.

I think to experimentally demonstarte the invariance of Planck constant is never easier than to demonstrate the invariance of one-way light speed.

 

[PeterDonis]

The electron's momentum-energy 4-vector is set up based on the 4-velocity V_4=(d/dt0)(x,ct) where t0 is the proper time and it is a Lorentz invariant. Since V_4 is Lorentz covariant and the electron's rest mass m0 is a Lorentz scalar from the principle of relativity, we conclude m0*V_4 = (p=m0*gamma*v, E/c=m0*gamma*c**2/c) is Lorentz covariant, and defined as momentum-energy 4-vector. But the derivation of V_4=(d/dt0)(x,ct) is not valid for a photon, because the photon's gamma-->infinity, a sigularity; in addition, the photon's rest mass is defined to be zero. Thus mathematically speaking, the derivation of electron's 4-vector is not applicable to a photon.

Ah, I see. I'm not sure that what you have given is an actual "derivation" of the electron's momentum-energy 4-vector; you are assuming that 4-velocity is primary, which I'm not sure is a valid assumption. For one thing, it involves that "position vector" (x,ct), which, while it is Lorentz "covariant", as you note, is not, IMO, a good candidate for a "fundamental" quantity.

An alternative would be to take 4-momentum as primary; after all, that's what we actually measure in experiments (we measure energies and momenta of particles like electrons, as well as photons; we don't measure positions, proper times, or velocities directly). You still have two "categories" of objects, those with nonzero rest mass and those with zero rest mass; but "rest mass" is now simply the invariant length of the energy-momentum 4-vector, which we obtain by the formula:

E^{2} - p^{2} = m^{2}

So if the energy is *larger* than the momentum, we say the particle is "massive" (like the electron), and we define the "mass" as what's "left over" when we've subtracted out the momentum from the energy (physically, this corresponds to being at rest with respect to the object, hence the term "rest mass"). Whereas if the energy is *equal* to the momentum, we say the particle is "massless" (like the photon). We don't have to use 4-velocity at all in any of this; 4-velocity becomes a derived quantity, which we obtain for massive objects by dividing the 4-momentum by its length, so it's kind of a "unit vector" in the direction of the 4-momentum. For massless objects, like the photon, we obviously can't do the division, but we can still define their "4-velocity" as being a "unit" null vector in the direction of the photon's momentum, which does make sense mathematically.

And (to get back to the question in the OP) if 4-momentum is primary, then it's obvious that Planck's constant has to be Lorentz invariant, because the formula you gave for photons, E = hbar w/c, p = hbar k, holds for *both* massive and massless objects; it's just that now that formula serves to define w and k, which are also derived quantities in this model. But since we can actually measure frequencies and wavelengths (which we can do for electrons as well as photons, for example in an electron diffraction experiment), we can check, as vanhees71 points out, to see that the formulas actually hold, and we find that they do, to high accuracy, for *both* electrons *and* photons (and every other quantum particle).

 

[PeterDonis]

It is found that, Einstein’s Doppler formula is not applicable when a moving point light source is close enough to the observer; for example, it may break down or cannot specify a determinate value when the point source and the observer overlap.

What formula are you calling "Einstein's Doppler formula"?

 

[keji8341]

Ah, I see. I'm not sure that what you have given is an actual "derivation" of the electron's momentum-energy 4-vector; you are assuming that 4-velocity is primary, which I'm not sure is a valid assumption. For one thing, it involves that "position vector" (x,ct), which, while it is Lorentz "covariant", as you note, is not, IMO, a good candidate for a "fundamental" quantity.

An alternative would be to take 4-momentum as primary; after all, that's what we actually measure in experiments (we measure energies and momenta of particles like electrons, as well as photons; we don't measure positions, proper times, or velocities directly). You still have two "categories" of objects, those with nonzero rest mass and those with zero rest mass; but "rest mass" is now simply the invariant length of the energy-momentum 4-vector, which we obtain by the formula:

E^{2} - p^{2} = m^{2}

So if the energy is *larger* than the momentum, we say the particle is "massive" (like the electron), and we define the "mass" as what's "left over" when we've subtracted out the momentum from the energy (physically, this corresponds to being at rest with respect to the object, hence the term "rest mass"). Whereas if the energy is *equal* to the momentum, we say the particle is "massless" (like the photon). We don't have to use 4-velocity at all in any of this; 4-velocity becomes a derived quantity, which we obtain for massive objects by dividing the 4-momentum by its length, so it's kind of a "unit vector" in the direction of the 4-momentum. For massless objects, like the photon, we obviously can't do the division, but we can still define their "4-velocity" as being a "unit" null vector in the direction of the photon's momentum, which does make sense mathematically.

And (to get back to the question in the OP) if 4-momentum is primary, then it's obvious that Planck's constant has to be Lorentz invariant, because the formula you gave for photons, E = hbar w/c, p = hbar k, holds for *both* massive and massless objects; it's just that now that formula serves to define w and k, which are also derived quantities in this model. But since we can actually measure frequencies and wavelengths (which we can do for electrons as well as photons, for example in an electron diffraction experiment), we can check, as vanhees71 points out, to see that the formulas actually hold, and we find that they do, to high accuracy, for *both* electrons *and* photons (and every other quantum particle).

I think you understand the invariance of Planck constant from the viewpoint of experiments.

1. For a plane wave in free space, (k,w/c) is Lorentz covariant. There are two methods to develop the photon's momentum-energy 4-vector: (1) Directly assume h_bar*(k,w/c) is a Lorentz covariant 4-vector, then we conclude that h_bar must be a Lorentz invariant mathematically. (2) Directly assume that h_bar is a Lorentz invariant, then we conclude that h_bar*(k,w/c) is a Lorentz covariant 4-vector. The two methods are equivalent, and so the invariance of Planck constant is only an artificial assumption. I think you like method-(1), which actually artificially assumes the invariance of Planck constant.

2. E^{2} - p^{2} = m^{2} ---Mathematically, even m is a Lorentz scalar, (p,E) is NOT necessarily Lorentz covariant. For example, suppose E is the electric field vector and |E| is its absolute value; (E, |E|) --->|E|**2-E**2=0 but (E, |E|) is not a Lorentz covariant 4-vector, because E is transformed in terms of EM-field strength tensor.

 

[keji8341]

What formula are you calling "Einstein's Doppler formula"?

Einstein's Doppler formula is the Doppler formula for a plane wave: w'=w*gamm*(1-n.beta), which can be seen in university physics textbooks. If it is applied to the moving point light source, when the observer and the point source overlap, n.beta is an inderterminate value, because the angle between n and beta can be arbitrary.

 

[PeterDonis]

Einstein's Doppler formula is the Doppler formula for a plane wave: w'=w*gamm*(1-n.beta), which can be seen in university physics textbooks. If it is applied to the moving point light source, when the observer and the point source overlap, n.beta is an inderterminate value, because the angle between n and beta can be arbitrary.

If the point source is moving, the observer and the source will only overlap for an instant. At any other instant, the angle between n and beta is not arbitrary, so obviously by continuity that angle must be the same, at the one instant where the two overlap, as it is at neighboring instants where they don't. So the instant of overlap doesn't invalidate the Lorentz covariance of the 4-vector (k, w/c).

 

[PeterDonis]

I think you understand the invariance of Planck constant from the viewpoint of experiments.

Yes, exactly. You don't *assume* that a given set of values forms a Lorentz covariant 4-vector, or that a single value is a Lorentz scalar; you *observe* whether the values behave in the appropriate ways when measured in different frames. Planck's constant is measured to be the same in different frames; therefore it is a Lorentz scalar. (For how we measure it, see next comment.)

1. For a plane wave in free space, (k,w/c) is Lorentz covariant. There are two methods to develop the photon's momentum-energy 4-vector: (1) Directly assume h_bar*(k,w/c) is a Lorentz covariant 4-vector, then we conclude that h_bar must be a Lorentz invariant mathematically. (2) Directly assume that h_bar is a Lorentz invariant, then we conclude that h_bar*(k,w/c) is a Lorentz covariant 4-vector. The two methods are equivalent, and so the invariance of Planck constant is only an artificial assumption. I think you like method-(1), which actually artificially assumes the invariance of Planck constant.

No, I like method (3): directly measure the energy and momentum of photons, and show that they transform as a Lorentz covariant 4-vector (p, E/c), *independently* of any measurements of frequency and wavelength. Then we know, physically, that *both* (p, E/c) *and* (k, w/c) are Lorentz covariant 4-vectors, from which it follows that their ratio, Planck's constant, must be a Lorentz scalar. So by measuring energy and momentum, *and* measuring frequency and wavelength, we can measure Planck's constant, and show that it is the same in all frames, as a Lorentz scalar must be.

2. E^{2} - p^{2} = m^{2} ---Mathematically, even m is a Lorentz scalar, (p,E) is NOT necessarily Lorentz covariant. For example, suppose E is the electric field vector and |E| is its absolute value; (E, |E|) --->|E|**2-E**2=0 but (E, |E|) is not a Lorentz covariant 4-vector, because E is transformed in terms of EM-field strength tensor.

What do the electric field vector and its absolute value have to do with the energy-momentum 4-vector, (p, E/c)? The latter is the 4-vector that appears in the formula you gave; E in that formula is energy, not electric field, by definition. And as I noted in my previous post, we don't claim that (p, E/c) is a Lorentz covariant 4-vector based on m being a Lorentz scalar; that's backwards. We verify directly, by experiments measuring E and p in different frames, that (p, E/c) transforms as a Lorentz covariant 4-vector. Then it follows that the frame-invariant length of that 4-vector, which is m, must be a Lorentz scalar.

 

[Dale]

An alternative would be to take 4-momentum as primary

I agree with this approach. The 4-momentum is a valid 4-vector and obeys all of the usual vector rules such as addition or scalar multiplication, regardless of whether it is a null vector or a timelike vector. The 4-velocity, on the other hand, does not follow vector rules like addition or scalar multiplication.

we can still define their "4-velocity" as being a "unit" null vector in the direction of the photon's momentum, which does make sense mathematically.

A vector cannot have a norm of 1 and 0, so I have a hard time seeing how a unit null vector makes sense mathematically.

 

[keji8341]

If the point source is moving, the observer and the source will only overlap for an instant. At any other instant, the angle between n and beta is not arbitrary, so obviously by continuity that angle must be the same, at the one instant where the two overlap, as it is at neighboring instants where they don't. So the instant of overlap doesn't invalidate the Lorentz covariance of the 4-vector (k, w/c).

The longitudinal Doppler effect from the Einstein's Doppler formula has a frequency-shift jump between the left-approaching overlap-point and the right-approaching overlap-point. I take the overlap-point as an example to show that Einstein's formula is not the exact formula for a moving point light source, although it is a good approximation when the observer is far away from the point source.

 

[keji8341]

Yes, exactly. You don't *assume* that a given set of values forms a Lorentz covariant 4-vector, or that a single value is a Lorentz scalar; you *observe* whether the values behave in the appropriate ways when measured in different frames. Planck's constant is measured to be the same in different frames; therefore it is a Lorentz scalar. (For how we measure it, see next comment.)



No, I like method (3): directly measure the energy and momentum of photons, and show that they transform as a Lorentz covariant 4-vector (p, E/c), *independently* of any measurements of frequency and wavelength. Then we know, physically, that *both* (p, E/c) *and* (k, w/c) are Lorentz covariant 4-vectors, from which it follows that their ratio, Planck's constant, must be a Lorentz scalar. So by measuring energy and momentum, *and* measuring frequency and wavelength, we can measure Planck's constant, and show that it is the same in all frames, as a Lorentz scalar must be.

You suggest a very good, original method to show the invariance of Planck constant. But I never see such experiments reported.

 

[PeterDonis]

A vector cannot have a norm of 1 and 0, so I have a hard time seeing how a unit null vector makes sense mathematically.

You're right, that was a poor choice of words. I was trying to express the idea that a photon's 4-momentum does have a "direction", which is simply the direction of its 3-vector spatial component. I'm sure there's a proper mathematical term for what I'm getting at, but I'm not sure what it is.

 

[PeterDonis]

The longitudinal Doppler effect from the Einstein's Doppler formula has a frequency-shift jump between the left-approaching overlap-point and the right-approaching overlap-point.

I assume that what you really mean is "left-approaching" and "right-receding" (or the reverse), as a description of the relative motion before and after the overlap. The frequency shift does change in this case from a blue shift (approaching) to a red shift (receding). However, this has nothing to do with any discontinuity in the *angle* between n and beta; it has to do with a change in the sign of beta (from positive, approaching, to negative, receding). I agree the change in sign happens, but it's more a matter of definition than physics; the "beta" in the Doppler formula is defined *differently* than the "beta" in the formulas for energy and momentum as they are used in the photon's energy-momentum 4-vector. There is no discontinuity in the actual motion of either object, or in the energy-momentum 4-vector of the photon.

You suggest a very good, original method to show the invariance of Planck constant. But I never see such experiments reported.

Well, Compton scattering experiments have clearly shown that photons can exchange momentum with electrons since the 1920's:

http://en.wikipedia.org/wiki/Compton_scattering

The photoelectric effect has been known to demonstrate that photons have energy since Einstein's paper on it was published in 1905:

http://en.wikipedia.org/wiki/Photoelectric_effect

(Compton scattering also shows that photons have energy, since they exchange energy as well as momentum with electrons.)

I'm sure there are other more recent experiments as well, but these show that measuring photon energy and momentum is nothing new, and is certainly not "original" with me.

I take it that measuring photon frequency and wavelength experimentally is not an issue either.

 

[keji8341]

... this has nothing to do with any discontinuity in the *angle* between n and beta; it has to do with a change in the sign of beta (from positive, approaching, to negative, receding). ...

angle-change and sign-change are the same thing; angle change = pi ---> cos(pi)=-1

 

[keji8341]

...I agree the change in sign happens, but it's more a matter of definition than physics; the "beta" in the Doppler formula is defined *differently* than the "beta" in the formulas for energy and momentum as they are used in the photon's energy-momentum 4-vector. There is no discontinuity in the actual motion of either object, or in the energy-momentum 4-vector of the photon.

The "beta" in the Doppler formula is the velocity of the moving point light source. In the photon's energy-momentum 4-vector (k,w/c), there is no "beta"; k is the wave vector. The observation angle has a discontinuity, jumping from zero to pi.

 

[keji8341]

Well, Compton scattering experiments have clearly shown that photons can exchange momentum with electrons since the 1920's:

http://en.wikipedia.org/wiki/Compton_scattering

The photoelectric effect has been known to demonstrate that photons have energy since Einstein's paper on it was published in 1905:

http://en.wikipedia.org/wiki/Photoelectric_effect

(Compton scattering also shows that photons have energy, since they exchange energy as well as momentum with electrons.)

I'm sure there are other more recent experiments as well, but these show that measuring photon energy and momentum is nothing new, and is certainly not "original" with me.

I take it that measuring photon frequency and wavelength experimentally is not an issue either.

Compton-effect experiment is a strong support to the Einstein's light-quantum hypothesis: A light wave has wave-particle duality. Taking the light wave consisting of particles, using energy- and momentum-conservation laws to have successfully explained the changes of scattered light-wavelengths and electron's momentums. But this experiment has nothing to do with the invariance of Planck constant, because, at least, all physical quantities are observed and measured in the same lab frame, with nothing to support the Lorentz invariance of Planck constant.

 

[Dale]

You're right, that was a poor choice of words. I was trying to express the idea that a photon's 4-momentum does have a "direction", which is simply the direction of its 3-vector spatial component. I'm sure there's a proper mathematical term for what I'm getting at, but I'm not sure what it is.

OK, I get that. I also don't know of a correct term for that, so there may not be a succinct way to say it.

I guess you could divide a null 4-momentum by the time component. That would give a null four-momentum whose spacelike components are a unit 3-vector. I don't know what you would call it though, and it would have all of the bad-behavior properties of the 4-velocity plus it wouldn't transform right even on its own. But it would have the desired spatial direction.

 

[PeterDonis]

angle-change and sign-change are the same thing; angle change = pi ---> cos(pi)=-1

Hm. Okay, I see better where you're coming from now.

The "beta" in the Doppler formula is the velocity of the moving point light source. In the photon's energy-momentum 4-vector (k,w/c), there is no "beta"; k is the wave vector. The observation angle has a discontinuity, jumping from zero to pi.

Two things: first, I was talking about the energy-momentum 4-vector, not the frequency-wavelength 4-vector. The two can be measured independently, as I said before.

Second, the "beta" in the Doppler formula you gave refers to the moving light source, not the photon. And that "beta" is *not* the same as the "beta" in the energy-momentum 4-vector of the *light source*, which is the one I was talking about. The beta in the energy-momentum 4-vector does not change sign when the moving source passes the stationary observer.

Compton-effect experiment is a strong support to the Einstein's light-quantum hypothesis: A light wave has wave-particle duality. Taking the light wave consisting of particles, using energy- and momentum-conservation laws to have successfully explained the changes of scattered light-wavelengths and electron's momentums. But this experiment has nothing to do with the invariance of Planck constant, because, at least, all physical quantities are observed and measured in the same lab frame, with nothing to support the Lorentz invariance of Planck constant.

I wasn't giving the Compton effect, by itself, as support for the Lorentz invariance of Planck's constant; I was giving it as an example of directly measuring the energy and momentum of photons, independently of their frequency and wavelength. It looks like you agree that it does that.

So, given that we can independently measure two 4-vectors, (E, p/c) and (k, w/c), and confirm that they both transform as Lorentz-covariant 4-vectors, it follows that their ratio, which is Planck's constant, *must* be a Lorentz scalar. *That* is what shows the Lorentz invariance of Planck's constant. But you have to combine the results of multiple experiments to show this.

 

[keji8341]

Hm. Okay, I see better where you're coming from now.



Two things: first, I was talking about the energy-momentum 4-vector, not the frequency-wavelength 4-vector. The two can be measured independently, as I said before.

Second, the "beta" in the Doppler formula you gave refers to the moving light source, not the photon. And that "beta" is *not* the same as the "beta" in the energy-momentum 4-vector of the *light source*, which is the one I was talking about. The beta in the energy-momentum 4-vector does not change sign when the moving source passes the stationary observer.



I wasn't giving the Compton effect, by itself, as support for the Lorentz invariance of Planck's constant; I was giving it as an example of directly measuring the energy and momentum of photons, independently of their frequency and wavelength. It looks like you agree that it does that.

So, given that we can independently measure two 4-vectors, (E, p/c) and (k, w/c), and confirm that they both transform as Lorentz-covariant 4-vectors, it follows that their ratio, which is Planck's constant, *must* be a Lorentz scalar. *That* is what shows the Lorentz invariance of Planck's constant. But you have to combine the results of multiple experiments to show this.

1. In the photon's energy-momentum 4-vector hbar*(k,w/c), there is no "beta". Photon's speed is taken to be c.
2. In the Compton-effect experiment, only the light wavelength (frequency) and the electron's velocity are directly measured (instead of photon's energy and momentum), and then use Einstein's light-quantum hypothesis to explain the scattered wavelength changes. If you could measure both photon's energy and frequency, then you could derive the Planck constant. Measuring Planck constatnt is different experiment; see http://www.sheldrake.org/experiments/constants/ .

 

[PeterDonis]

1. In the photon's energy-momentum 4-vector hbar*(k,w/c), there is no "beta". Photon's speed is taken to be c.

Please re-read carefully what I posted. I was referring to the "beta" in the Doppler formula you quoted:

w'=w*gamma*(1-n.beta)

The "beta" in that formula refers to the moving light source, *not* the photon. That light source also has an energy-momentum 4-vector (p, E/c) in the frame of the stationary observer, which is different from the 4-vector of the photon. Writing E and p in terms of the standard SR beta and gamma, we have the 4-vector (in units where c = 1, and where m is the rest mass of the moving source):

(E, p) = m (gamma, gamma * beta)

So we have two formulas referring to the moving light source that both have a "beta" in them, but the definitions are *different* for the two betas. The first beta (the one in the Doppler formula) changes sign when the moving source passes the stationary observer, as you note. The second beta does *not*. So the behavior of the first beta does *not* prevent (E, p), which involves the second beta, from being a genuine Lorentz-covariant 4-vector.

2. In the Compton-effect experiment, only the light wavelength (frequency) and the electron's velocity are directly measured (instead of photon's energy and momentum), and then use Einstein's light-quantum hypothesis to explain the scattered wavelength changes.

I'm not sure I would put it this way. We measure the change in the electron's momentum, and the change in the photon's frequency. But we relate the two using conservation of energy-momentum, not just the light quantum hypothesis. If energy-momentum were not conserved, we would have no way of relating the electron quantities to the photon quantities at all. See further comment below.

Also, the light quantum hypothesis is validated by other experiments too (e.g., the photoelectric effect), so it's not brought in solely to interpret Compton scattering.

If you could measure both photon's energy and frequency, then you could derive the Planck constant.

Or if you can measure its momentum and frequency, since momentum and energy are related, and frequency and wavelength are related. Or if you can measure the photon's frequency change and the momentum change of something it interacts with, since energy-momentum are conserved, as I noted above.

You don't have to get everything out of a single experiment. You can do one set of experiments to show that energy-momentum are conserved; another set of experiments to show that energy-momentum transform as a 4-vector; another set of experiments to show that frequency-wavelength transform as a 4-vector; and then yet another set of experiments which uses the results of the first three sets to, for example, relate the change in frequency of a photon to the change in momentum of an electron that it interacts with.

The full picture comes from putting together the results of multiple experiments and finding a consistent theory that accounts for them all. Each experiment individually is going to be missing things that have to be filled in by "interpretation", but you can test your interpretation by looking at other experiments. So when people say that Planck's constant is Lorentz-invariant, they mean that it's a feature of the consistent model that accounts for *all* of the experiments. They don't mean that there's a single, slam-dunk experiment, or a single, slam-dunk theoretical argument, that "proves" that Planck's constant is Lorentz invariant.

Measuring Planck constatnt is different experiment; see http://www.sheldrake.org/experiments/constants/ .

Interesting link, but I think it's talking about a separate issue from the one we're discussing. Whether Planck's constant can vary over cosmological time periods, or whether it can be different in different universes, is a separate question from whether Planck's constant, whatever it may be in a particular local region of spacetime, is Lorentz invariant. The latter is a local question and can be answered using a local model that combines the results of multiple local experiments, of the kind I described above. (In other words, a Lorentz-invariant scalar can still change with time; more generally, it can still assume different values at different events in spacetime, or in different universes. It's just that, whatever its value at any particular event, that value can't depend on the state of motion of the observer.)

 

[keji8341]

...That light source also has an energy-momentum 4-vector (p, E/c) in the frame of the stationary observer, which is different from the 4-vector of the photon. Writing E and p in terms of the standard SR beta and gamma, we have the 4-vector (in units where c = 1, and where m is the rest mass of the moving source):

(E, p) = m (gamma, gamma * beta)

So we have two formulas referring to the moving light source that both have a "beta" in them, but the definitions are *different* for the two betas. The first beta (the one in the Doppler formula) changes sign when the moving source passes the stationary observer, as you note. The second beta does *not*. So the behavior of the first beta does *not* prevent (E, p), which involves the second beta, from being a genuine Lorentz-covariant 4-vector.

In my understanding, you mean the total energy and the total momentum of the moving point light source. I don't know why you need them for Doppler effect analysis. The point light source with infinite energy and momentum is a physical model, just like a point charge of which the total energy is infinite.

 

[keji8341]

I'm not sure I would put it this way. We measure the change in the electron's momentum, and the change in the photon's frequency. But we relate the two using conservation of energy-momentum, not just the light quantum hypothesis. If energy-momentum were not conserved, we would have no way of relating the electron quantities to the photon quantities at all. See further comment below.

Also, the light quantum hypothesis is validated by other experiments too (e.g., the photoelectric effect), so it's not brought in solely to interpret Compton scattering.

You'd better check with physics textbooks about Compton effect. Experimentally measured quantities: scattered light wavelengths and electron's velocities. Theoretical analysis: Einstein's light-quatum hypothesis, energy conservation law, and momentum conservation law are used, and calculation results are in agreement with the experimentally measured quantities. Strictly speaking, this experiment is a support to the light wave-particle duality, but not a complete verification. Any single experiment cannot completely verify a theory. If all experiments support a theory, then the theory is well recognized.

 

[keji8341]

Interesting link, but I think it's talking about a separate issue from the one we're discussing. Whether Planck's constant can vary over cosmological time periods, or whether it can be different in different universes, is a separate question from whether Planck's constant, whatever it may be in a particular local region of spacetime, is Lorentz invariant. The latter is a local question and can be answered using a local model that combines the results of multiple local experiments, of the kind I described above. (In other words, a Lorentz-invariant scalar can still change with time; more generally, it can still assume different values at different events in spacetime, or in different universes. It's just that, whatever its value at any particular event, that value can't depend on the state of motion of the observer.)

I gave the link to show that measuring Planck constant is different thing. These experiments were done in different periods, but I cannot judge whether the Planck constant depends on the periods or there are some experimental errors.

 

[PeterDonis]

In my understanding, you mean the total energy and the total momentum of the moving point light source. I don't know why you need them for Doppler effect analysis.

It's true that you don't need the moving source's full energy and momentum just to analyze the Doppler shift; the moving source's beta is enough. But to claim that a photon's 4-vector (k, w/c) is not Lorentz invariant, it's not enough just to analyze the Doppler effect formula in isolation. You have to think about what it actually means, physically.

Here's the formula again:

w'=w*gamma*(1-n.beta)

As I understand your definitions of variables, w' is the Doppler-shifted "moving" frequency, and w is the "stationary" frequency that would be observed if the source was at rest. With these definitions, n.beta has to be *negative* when the source is approaching the observer (so w' is larger than w), and *positive* when the source is receding from the observer (so w' is smaller than w).

You are correct that this sign change can be explained as a change in the angle between n and beta, rather than just a change in the sign of beta itself. But consider what this means: it means that n itself changes direction; when the source is receding, n points in the *opposite* direction from when the source is approaching. (This changes the angle between n and beta by 180 degrees, which flips the sign of n.beta.) But that means that, as I said before (though you're right that I didn't do a good job of capturing this before), that the "discontinuity" you are claiming in the Doppler shift formula is because you're switching definitions in mid-stream, so to speak. (You're just modeling that shift as n pointing in the opposite direction, instead of beta.)

Physically, n is not an attribute of the photon itself; it's just a unit vector in the direction pointing from source to observer, and the change in n reflects, not a change in the properties of any single photon, but a *change in which photon is being described*. What is actually happening is that the two regimes, "source approaching" and "source receding" refer to *different sets of photons*. In other words, the Doppler formula when n.beta is negative (approaching source) describes a *different* photon than the Doppler formula when n.beta is positive (receding source). One and the same photon can only be emitted under one condition (approaching or receding). So the "discontinuity" you are claiming, as I said before, only applies to the definitions you are using in the formula; it does *not* apply to any actual photon, because each individual photon only has one version of the formula apply to it. So no single photon ever sees any discontinuity, and each individual photon's 4-vector (k, w/c) remains Lorentz covariant.

(As I said, I didn't capture this very well before, because I was thinking about the relative motion of the source and the observer, instead of the motion of the photon itself. But I was also thinking about other situations, like the Compton effect, where massive particles and photons interact. As I noted, a full understanding has to account for all experiments, not just one.)

 

[PeterDonis]

You'd better check with physics textbooks about Compton effect. Experimentally measured quantities: scattered light wavelengths and electron's velocities. Theoretical analysis: Einstein's light-quatum hypothesis, energy conservation law, and momentum conservation law are used, and calculation results are in agreement with the experimentally measured quantities. Strictly speaking, this experiment is a support to the light wave-particle duality, but not a complete verification. Any single experiment cannot completely verify a theory. If all experiments support a theory, then the theory is well recognized.

None of this is in disagreement with what I said. I agree that the actual measurements in the Compton experiments are light wavelengths (or frequencies, since vacuum dispersion, w = k, is assumed) and electron velocities (which are easily converted to momenta since we already know the electron's rest mass). I said that it takes multiple experiments to validate a theoretical model. I agree with everything you said in the above quote.

 

[PeterDonis]

These experiments were done in different periods, but I cannot judge whether the Planck constant depends on the periods or there are some experimental errors.

I lean towards experimental errors, but as I noted, that's a separate subject from what we're discussing in this thread.

 

[keji8341]

...w'=w*gamma*(1-n.beta)

...So the "discontinuity" you are claiming, as I said before, only applies to the definitions you are using in the formula; it does *not* apply to any actual photon, because each individual photon only has one version of the formula apply to it. So no single photon ever sees any discontinuity, and each individual photon's 4-vector (k, w/c) remains Lorentz covariant.
...

Taking light as a wave, the Doppler effect of wave period actually describes the relation between the time interval in which one moving observer emits two δ-light signals and the time interval in which the lab observer receives the two δ-signals at the same place. The period should be a measurable physical quantity. The lab observer cannot know the period before he receives the second δ-light signal.

Taking light as consisting of photons, a single photon has the information of frequency. But when using a single photon to derive Doppler formula, the photon's momentum and energy is supposed to form a momentum-energy 4-vector, which exactly corresponds to a plane wave, and the Doppler formula (namely Einstein's plane-wave formula) is only applicable to a plane wave.

Back to your question. How about to fire two photons, one left-approaching overlap-point and one right-approaching overlap point? The lab observer received two photons at the same time, which have different frequencies! You got "discontinuity".

 

[PeterDonis]

Taking light as a wave, the Doppler effect of wave period actually describes the relation between the time interval in which one moving observer emits two δ-light signals and the time interval in which the lab observer receives the two δ-signals at the same place. The period should be a measurable physical quantity. The lab observer cannot know the period before he receives the second δ-light signal.

Taking light as consisting of photons, a single photon has the information of frequency. But when using a single photon to derive Doppler formula, the photon's momentum and energy is supposed to form a momentum-energy 4-vector, which exactly corresponds to a plane wave, and the Doppler formula (namely Einstein's plane-wave formula) is only applicable to a plane wave.

True. See below.

How about to fire two photons, one left-approaching overlap-point and one right-approaching overlap point? The lab observer received two photons at the same time, which have different frequencies! You got "discontinuity".

So what? It's two different photons, with two different (k, w/c) 4-vectors. There's no discontinuity in either one individually.

In fact, you're not even picking the toughest example. Let's go back to your first proposed model above, where light is a wave, and in order to measure its frequency I need a wave train of finite length, over a finite interval of time. Suppose that finite interval of time includes the instant at which the moving source passes the stationary observer? It would seem in that case that we *would* indeed have the discontinuity in a single wave train!

However, even here the discontinuity is an illusion. What is actually happening is this: the moving source is emitting *two* wave trains, one we'll call A (for "approaching") in the positive x-direction (the same direction as its motion), and one we'll call R (for "receding") in the negative x-direction. Suppose the moving source passes the stationary observer at the instant t = 0, and suppose we look at the time interval -T to T in order to measure the frequency of the light. What the observer will see is that, at time t = 0, he abruptly stops receiving wave train A and starts receiving wave train R. If he includes both wave trains in a single measurement, then yes, it will look like there's a discontinuity in frequency, but that's because he's mixing together measurements from two separate wave trains. If instead he does the measurement right, measuring wave train A from -T to 0, and wave train R from 0 to T, then he will correctly conclude that wave train A's frequency is blueshifted and wave train R's frequency is redshifted, and there is no discontinuity in either wave train. The only discontinuity is that he stops receiving one wave train and starts receiving another at t = 0, but that has nothing to do with the Lorentz invariance of any 4-vectors involved.

 

[keji8341]

So what? It's two different photons, with two different (k, w/c) 4-vectors. There's no discontinuity in either one individually.

1. The energy and momentum of photons of a PLANE WAVE constitute a 4-vector. Einstein proved that (k,w/c) for a plane wave in free space is a Lorentz covariant 4-vector, then by imposing light-quantum hypothesis we have the momentum-energy 4-vector hbar*(k,w/c). No one gives a proof that (k,w/c) for a moving point source in free space is Lorentz covariant, and thus the covariance of hbar*(k,w/c) for a moving point source is questionable.

2. The two photons are emitted at the same time and the same place, but the momentums are opposite, which is not easy to understand. Suppose that a photon is emitted just when the point source and the observer overlap, how about the photon's momentum measured by the observer?

 

[keji8341]

In fact, you're not even picking the toughest example. Let's go back to your first proposed model above, where light is a wave, and in order to measure its frequency I need a wave train of finite length, over a finite interval of time. Suppose that finite interval of time includes the instant at which the moving source passes the stationary observer? It would seem in that case that we *would* indeed have the discontinuity in a single wave train!

However, even here the discontinuity is an illusion. What is actually happening is this: the moving source is emitting *two* wave trains, one we'll call A (for "approaching") in the positive x-direction (the same direction as its motion), and one we'll call R (for "receding") in the negative x-direction. Suppose the moving source passes the stationary observer at the instant t = 0, and suppose we look at the time interval -T to T in order to measure the frequency of the light. What the observer will see is that, at time t = 0, he abruptly stops receiving wave train A and starts receiving wave train R. If he includes both wave trains in a single measurement, then yes, it will look like there's a discontinuity in frequency, but that's because he's mixing together measurements from two separate wave trains. If instead he does the measurement right, measuring wave train A from -T to 0, and wave train R from 0 to T, then he will correctly conclude that wave train A's frequency is blueshifted and wave train R's frequency is redshifted, and there is no discontinuity in either wave train. The only discontinuity is that he stops receiving one wave train and starts receiving another at t = 0, but that has nothing to do with the Lorentz invariance of any 4-vectors involved.

I don't think you are using the Einstein's Doppler formula in above analysis.
My proposition is: Planck constant is Lorentz invariant?
My argument for it is: Einstein's Doppler formula is not applicable to the case with a moving point light source===>(k,w/c) is not Lorentz covariant===>Plack constant is not Lorentz invariant.

 

[Dale]

Einstein's Doppler formula is not applicable to the case with a moving point light source

Yes, it is applicable.

 

[PeterDonis]

The energy and momentum of photons of a PLANE WAVE constitute a 4-vector. Einstein proved that (k,w/c) for a plane wave in free space is a Lorentz covariant 4-vector, then by imposing light-quantum hypothesis we have the momentum-energy 4-vector hbar*(k,w/c). No one gives a proof that (k,w/c) for a moving point source in free space is Lorentz covariant, and thus the covariance of hbar*(k,w/c) for a moving point source is questionable.

By a moving point source, I assume you mean a spherical wave emanating from a moving point source? A spherical wave can't be described, mathematically, by a single 4-vector.

2. The two photons are emitted at the same time and the same place, but the momentums are opposite, which is not easy to understand.

Why not? Bear in mind that I'm only considering a single spatial dimension; in the full 3 space dimensions you would have a spherical wave front being emitted, as I noted above. That's how point sources work.

Suppose that a photon is emitted just when the point source and the observer overlap, how about the photon's momentum measured by the observer?

My inclination would be to say that the observer wouldn't detect the photon at all in this case. Certainly that's what would happen in any real experiment.

 

[PeterDonis]

I don't think you are using the Einstein's Doppler formula in above analysis.

Sure I am; I'm just trying to give more details about what it actually means, physically. If my idealization of a single space dimension (so what is actually a spherical wave front being emitted by the moving point source is modeled as two plane waves emitted in opposite directions) disturbs you, I suppose we could work with the full mathematical apparatus of spherical waves in 3 space dimensions, but that seems like overkill.

 

[keji8341]

Yes, it is applicable.

Einstein’s Doppler formula is not applicable when a moving point light source is close enough to the observer; for example, it may break down or cannot specify a determinate value when the point source and the observer overlap. How to explain to it?

 

[keji8341]

By a moving point source, I assume you mean a spherical wave emanating from a moving point source? A spherical wave can't be described, mathematically, by a single 4-vector.



Why not? Bear in mind that I'm only considering a single spatial dimension; in the full 3 space dimensions you would have a spherical wave front being emitted, as I noted above. That's how point sources work.



My inclination would be to say that the observer wouldn't detect the photon at all in this case. Certainly that's what would happen in any real experiment.

Einstein proved that (k,w/c) for a plane wave is Lorentz covariant. Can you prove that (k,w/c) for a moving point light source is Lorentz covariant?

 

[Dale]

Einstein’s Doppler formula is not applicable when a moving point light source is close enough to the observer; for example, it may break down or cannot specify a determinate value when the point source and the observer overlap. How to explain to it?

So what? Lots of things break down in the limit as r->0. Coulomb's law, Schwarzschild spacetime, Newtonian gravity, angular momentum, etc. If that were a reason to reject something we would have to get rid of a lot of physics.

Einstein's Doppler formula is definitely applicable to a moving point source.

http://teachers.web.cern.ch/teachers/archiv/hst2002/bubblech/mbitu/wave_4.htm

 

[PeterDonis]

Einstein proved that (k,w/c) for a plane wave is Lorentz covariant. Can you prove that (k,w/c) for a moving point light source is Lorentz covariant?

No, because there is no such thing. Did you read the part where I said that a spherical wave, which is what a point source emits, can't be described by a single 4-vector?

What I *can* prove is that the spherical wave emitted by a point source is Lorentz invariant. That's simple: the spherical wave is just the future null cone (actually the term should probably be "null hypercone" since its spatial slices are 2-spheres, not circles) of the emission event. Null cones are always left invariant by Lorentz transformations. QED.

Edit: Perhaps I should expand on this more. If we are looking at the entire spherical wave emitted by the point source, there isn't really any "energy-momentum" object to compare it to in order to evaluate Planck's constant. So that case is really irrelevant to the question in the OP.

But we can decide to pick out a particular null ray from this spherical wave, by looking at a particular pair of events, the given emission event (the source of the entire spherical wave--this is some event on the source's worldline), and a particular reception event (some point further out on the future null cone, where the receiver's worldline intersects it). Then we can associate a particular null 4-vector (k, w/c) with the null ray from the emission event to the specified reception event.

The Doppler effect (more precisely, the "longitudinal" Doppler effect) is simply the observation that the actual value of k (or w/c, since the vector is null they are equal in magnitude) depends on the relative velocity beta of the source and the observer, by the Einstein Doppler formula. (The n in that formula is just the spatial direction of the null ray we specified, so n.beta is the angle between that ray and the moving source's spatial velocity.)

But we can also observe that, once we've chosen the reception event, for a given beta, the 4-vector (k, w/c) *is* Lorentz covariant. We could show this by modeling the chosen null ray as a plane wave. (If you want to say that the plane wave approximation breaks down when the events are too close together, I suppose that's true, but it has nothing to do with any "discontinuity" when the source passes the observer; it's simply due to the curvature of the actual spherical wavefront, which makes the plane wave approximation less accurate the closer the emission and reception events are in space.) However, we can show it even more easily by simply observing that, by definition, null rays and their associated null 4-vectors are always Lorentz covariant. (This is because Lorentz transformations always leave null cones invariant, so individual null rays can never be made non-null; they can only be conformally mapped into other null rays. Such a conformal mapping preserves inner products of null rays, which is the definition of "Lorentz covariant".)

The apparent "discontinuity" when the moving source passes the observer is due to switching null rays in mid-stream, so to speak, by switching the pair of events (emission, reception) that we are considering, which also means switching which particular null cone we are picking the events out of. This has to be the case, because at anyone particular emission event, the moving source cannot both be approaching and receding from the observer. So as soon as we pick a particular emission event, we have implicitly also picked a particular n.beta in the Einstein Doppler formula, and a particular 4-vector (k, w/c).

Only by looking at two *different* null rays, one with the source approaching and one with the source receding, and then inappropriately combining them into a single "measurement" of frequency or wavelength, can we see any discontinuity. But in doing that, we are combining two *different* 4-vectors (k, w/c) and (k', w'/c), that are associated with two different null rays between two different pairs of events on two different null cones. It's not surprising that such a combination is not well-behaved, and all this doesn't prove or disprove anything about Planck's constant.

In summary: for any case where there is actually a unique, valid 4-vector (k, w/c) for a photon, it is Lorentz covariant, and therefore is consistent with Planck's constant being a Lorentz scalar. For any case where there appears to be a photon "4-vector" that is not Lorentz covariant, it's because there is not one unique 4-vector involved; instead, information from multiple different 4-vectors is being inappropriately combined into a single "measurement".

 

[keji8341]

So what? Lots of things break down in the limit as r->0. Coulomb's law, Schwarzschild spacetime, Newtonian gravity, angular momentum, etc. If that were a reason to reject something we would have to get rid of a lot of physics.

Einstein's Doppler formula is definitely applicable to a moving point source.

http://teachers.web.cern.ch/teachers/archiv/hst2002/bubblech/mbitu/wave_4.htm

That I take the overlap-point as an example is just for clarity. Actually when the point source is close enough, Einstein's formula is not applicable. The closer, the bigger the error is. Theoretically, you don't have any grouds to say that Einstein's Doppler formula is applicable to the moving point light source, unless you can prove it, or you just set it as an artificial assumption.

In the classical electromagnetic theory, Coulomb's law is valid even in the limit as r->0, because it satisfies Maxwell equations.

 

[keji8341]

So what? Lots of things break down in the limit as r->0. Coulomb's law, Schwarzschild spacetime, Newtonian gravity, angular momentum, etc. If that were a reason to reject something we would have to get rid of a lot of physics.

Einstein's Doppler formula is definitely applicable to a moving point source.

http://teachers.web.cern.ch/teachers/archiv/hst2002/bubblech/mbitu/wave_4.htm

I think you must know Planck length physics.
In physics, the Planck length, denoted ℓP, is a unit of length, equal to 1.616252(81)×10**(−35) metres. The physical significance of the Planck length is a topic of research.
http://en.wikipedia.org/wiki/Planck_length

 

[keji8341]

No, because there is no such thing. Did you read the part where I said that a spherical wave, which is what a point source emits, can't be described by a single 4-vector?

What I *can* prove is that the spherical wave emitted by a point source is Lorentz invariant. That's simple: the spherical wave is just the future null cone (actually the term should probably be "null hypercone" since its spatial slices are 2-spheres, not circles) of the emission event. Null cones are always left invariant by Lorentz transformations. QED.

Sorry, I don't understand what the "null cone". Cerenkov cone?

 

[Dale]

That I take the overlap-point as an example is just for clarity. Actually when the point source is close enough, Einstein's formula is not applicable. The closer, the bigger the error is.

What are you talking about? Can you derive the error you think is there?

Theoretically, you don't have any grouds to say that Einstein's Doppler formula is applicable to the moving point light source, unless you can prove it, or you just set it as an artificial assumption.

Any EM field that has a definite phase obeys the Doppler formula. The phase is the Minkowski inner product between the position four-vector and the wave four-vector. Since the wave four-vector is a four-vector it transforms like any other four-vector.

In the classical electromagnetic theory, Coulomb's law is valid even in the limit as r->0, because it satisfies Maxwell equations.

And the Doppler formula is valid because it satisfies the Lorentz transform.

 

[keji8341]

What are you talking about? Can you derive the error you think is there?

Einstein's Doppler formula is the Doppler formula for a plane wave: w'=w*gamm*(1-n.beta), which can be seen in university physics textbooks. If it is applied to the moving point light source, when the observer and the point source overlap, n.beta is an inderterminate value, because the angle between n and beta can be arbitrary. From this we can deduce that it is not applicable when the point source is close enough to the observer.

 

[keji8341]

Any EM field that has a definite phase obeys the Doppler formula. The phase is the Minkowski inner product between the position four-vector and the wave four-vector. Since the wave four-vector is a four-vector it transforms like any other four-vector.

It is not necessarily. For example, the spherical wave has a phase function of phi=(wt -|k||x|) where k and x has a strong constraint and the Lorentz covariance of (k,w/c) is destroyed.

Note: For a plane wave, the phase function is given by phi=wt-k.x where there is no constraint between k and x, and from the invariance of phase, (k,w/c) must be Lorentz covariant as shown by Einstein in 1905.

 

[keji8341]

And the Doppler formula is valid because it satisfies the Lorentz transform.

Since (k,w/c) for a moving point light source is not Lorentz covariant as mentioned above, its Doppler formula cannot be obtained directly from the Lorentz transformation of (k,w/c).

 

[Dale]

It is not necessarily. For example, the spherical wave has a phase function of phi=(wt -|k||x|) where k and x has a strong constraint and the Lorentz covariance of (k,w/c) is destroyed.

All that means is that the wave four-vector is a function of position for anything other than a plane wave. In other words, it is a tensor field of rank 1. But there is ample experimental evidence that radiation from point sources Doppler shifts ala Einstein.

 

[Dale]

Since (k,w/c) for a moving point light source is not Lorentz covariant as mentioned above, its Doppler formula cannot be obtained directly from the Lorentz transformation of (k,w/c).

Sure it can. Look, the phase is a scalar and the four-position is the prototypical four-vector, so the object which is multiplied with a vector to get a scalar is a vector and transforms as a vector.

In other words given a scalar phi and a vector x how else can you get
\phi=f(x)
besides
\phi=x^{\mu}k_{\mu}

In this formula, given phi and x, k must clearly be a vector. That vector is called the wave four vector.

 

[PeterDonis]

Sorry, I don't understand what the "null cone". Cerenkov cone?

A null cone is just the set of all points in a spacetime that are at a null interval from a given point (the "source"). If we adopt coordinates such that the source is at (t, x, y, z) = (0, 0, 0, 0), then the null cone is just the set of points for which:

t^{2} - x^{2} - y^{2} - z^{2} = 0

The future null cone is the portion of this set for which t > 0.