Does the photon have a 4-velocity in a medium?

[PFfan01]

Regarding specific weaknesses of this paper:
...
Poor understanding of background literature
...

The author claims that the Poynting vector does not necessarily represent EM power flow. I think this is incorrect. There are no experimental results indicating that the standard Poynting vector is inadequate. One might feel it strange that the photon momentum in moving matter has not the same direction as the flow of energy, but there is nothing wrong with it physically. This is simply a characteristic feature of the theory.

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Questionable assumptions
...

This paper does not contribute anything worth publishing to the long going discussion on the Minkowski-Abraham problem. In particular, the critical analysis by the author of the PRL paper. It is obvious to all those that understand the meaning of special relativity that the Abraham photon momentum and energy in a medium cannot constitute a Lorentz four-vector. This fact, however, does not mean that the Abraham approach contradicts theory of relativity because the medium defines a preferred frame of reference. There is absolutely no reason why the motion of photons or other particles in a medium must look the same in all coordinate systems.

 

[Dale]

This fact, however, does not mean that the Abraham approach contradicts theory of relativity because the medium defines a preferred frame of reference. There is absolutely no reason why the motion of photons or other particles in a medium must look the same in all coordinate systems.

I agree here. What must be invariant is the outcome of any experiment.

In my mind this freedom to partition the total momentum into an EM part and a matter part is similar to the gauge freedom. The Lorenz gauge is indeed most convenient for relativity, but the Coulomb gauge is nonetheless a valid gauge which may be useful for certain cases.

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Poor understanding of background literature
...

The paper claims that a “classical mathematic conjecture” is shown to be flawed in Ref. 41 (http://dx.doi.org/10.1139/cjp-2015-0198); however the so-called “classical mathematic conjecture” turns out to be a solid, well-established result of tensor calculus in textbooks.

 

[vanhees71]

I'm not sure what of the here discussed issues have to do with the question asked in the title of this thread.

The question asked in the title of the thread is very simple to answer. It is just unclear, what you mean by "4-velocity of a photon in a medium". This seems to indicate the usual misinterpretation of the word "photon" as a kind of point-particle concept. This is always misleading.

To treat a photon in the medium you need relativistic many-body QFT and to calculate the photon polarization tensor, which is not a trivial thing. The closest quantity you can get which is close to something like a four-velocity is the dispersion relation for photon quasiparticles in the medium. For a detailed treatment, see e.g.,

C. Gale, J. Kapusta, Thermal Field Theory, Cambridge University Press.

The other here discussed papers are about the age-old question of the definition of a covariant total four-momentum of electromagnetic fields, and this has a long history of confusion. I'm pretty confident that von Laue got it right. The reason is that for the interacting electromagnetic field, you cannot simply integrate the Belinfante energy-momentum stress tensor over the entire space of an arbitrary observer and expect to get a four-vector, because for this to hold true the corresponding four-current must be conserved, i.e., for an arbitrary inertial frame (let's discuss only the special relativistic case first) a tensor field $T^{\mu \nu \rho\ldots}$ leads to a four-tensor of lower rank via
$$\mathcal{T}^{\nu \rho\ldots}=\int_{\mathbb{R^3}} \mathrm{d}^3 \vec{x} T^{0\nu \rho \ldots},$$
only if
$$\partial_{\mu} T^{\mu \nu\rho\ldots}=0.$$
E.g., for electric charge you always must have $\partial_{\mu} j^{\mu}=0$, i.e., local charge conservation, due to gauge invariance and thus the electric charge is always
$$Q=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} j^0$$
conserved and a scalar, i.e., independent over which space-like hypersurface you integrate provided you cover the entire charge of the system.

The Belinfante tensor is the correct gauge-invariant energy-momentum tensor of the electromagnetic field, because it also can be derived as the source of the gravitational field in Einstein's field equations from the electromagnetic field. However (even in special relativity), it is clear that in the case of an em. field interacting with charge-current distributions, it does not define an energy-momentum four-vector, because
$$\partial_{\mu} T^{\mu \nu} = -F^{\nu \rho} j_{\rho}.$$
Only the total energy-momentum tensor
$$\Theta^{\mu \nu}=T^{\mu \nu} + \Theta_{\text{matter}}^{\mu \nu}$$
is conserved, i.e., it fulfills
$$\partial_{\mu} \Theta^{\mu \nu}=0,$$
and the total energy-momentum
$$P^{\nu}=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \Theta^{0 \nu}$$
defines a proper (conserved) four-vector under Lorentz transformations.

The proof uses the four-dimensional Gauss integral theorem and was known for sure to Poincare and von Laue. There is a lot of confusion in the textbook literature about this, because often the authors forget this fundamental mathematics of tensor calculus and the integral theorems. Then you have all kinds of unnecessary nonsense debates about "hidden momentum" (there's no such thing but just mechanical and electromagnetic stress and the details of the famous formula $E=m c^2$, which implies that also stress adds to the total invariant mass of a composite system), the famous "4/3 problem" in the theory of charged extended bodies (charged classical point particles do not exist in the strict sense at all), etc. The only good thing about this is that you have nice examples to analyze within the correct machinery of tensor analysis, and von Laues books on relativity are masterpieces in doing right this.

Unfortunately, I cannot read the articles in the Can. J. Phys. because our University is not subscribed to this journal :-(.

 

[Dale]

The other here discussed papers are about the age-old question of the definition of a covariant total four-momentum of electromagnetic fields, and this has a long history of confusion. I'm pretty confident that von Laue got it right.

I find the position of Pfeiffer et al the most compelling.

http://arxiv.org/abs/0710.0461

They are not the first to recognize the correct resolution of this question, but I like their paper a lot.

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Cumbersome notation

And the level of English is poor; I fear that most readers would be irritated by the quality of the English. Example of incomprehensible sentences:

“Maxwell equations support various forms of momentum conservation equations, which is a kind of indeterminacy. However it is this indeterminacy that results in the question of light momentum.”

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Questionable assumptions
...

Questionable plane-wave model.

The article is to seek resolution of the question of light momentum in media. Frankly, I fail to see the problem - the "question" of what form to have for the momentum of a plane-wave in an (infinite) medium seems too academical to me. Since there are no plane waves, which are merely abstractions, a sometimes convenient linear basis to study wave propagation, how can I know if this "problem" is not coming from this mental construction which is an artifact?

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Poor understanding of background literature
...

Poor understanding of the work of Pfeifer et al

In his introduction, the author presents a list of earlier works and, rather than present a scholarly and balanced assessment of these, seeks merely to discredit them with unsubstantiated comments.

The work of Pfeifer et al is belittled in the phase "Clearly, it is an insufficiency of the Pfeifer-coworkers theory that the EM momentum in a medium cannot be uniquely defined". Why "clearly"? The fact that the total momentum includes a contribution with both medium AND EM fields makes it quite clear that this is indeed the case, indeed the whole Abraham-Minkowski problem may be understood in precisely these terms.

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Poor understanding of background literature
...

This paper relies on a major misunderstanding. Applying Abraham definition to a single photon propagating as a plane wave state through a dielectric medium, the author deduces that the result is not covariant and thus violates the relativity principle. But the momentum conservation law which follows from Maxwell equations, which are manifestly Lorentz covariant, certainly respects the relativity principle. The argument advanced by the author can at best show that Abraham momentum is frame dependent, but not that it is inconsistent with the relativity principle.

 

[Dale]

And the level of English is poor; I fear that most readers would be irritated by the quality of the English.

I didn't put that on the list because it is most likely due to not being a native speaker, and I try to make allowances for that. However, that is one of the things that is particularly problematic in single author papers. Even when they are native speakers, having additional people work on it together helps avoid bad English.

 

[PFfan01]

... Regarding specific weaknesses of this paper:
...
Poor understanding of background literature
...

The author claims “no experimental observations of light momentum are quantitatively in agreement with the formulation given by Abraham” (http://dx.doi.org/10.1139/cjp-2015-0167); however, Abraham pressure of light has been confirmed by a recent experimental study (https://dx.doi.org/10.1088%2F1367-2630%2F17%2F5%2F053035 ), although the validity of the analysis of the experiments is questioned (https://en.wikipedia.org/wiki/Abraham%E2%80%93Minkowski_controversy).

 

[PFfan01]

I find the position of Pfeiffer et al the most compelling.

http://arxiv.org/abs/0710.0461

They are not the first to recognize the correct resolution of this question, but I like their paper a lot.

I love the review paper by Pfeiffer et in Reviews of Modern Physics (http://dx.doi.org/10.1103/RevModPhys.79.1197), but I love the paper by Barnett more in Physical Review Letters (http://dx.doi.org/10.1103/PhysRevLett.104.070401), because Physical Review Letters has a higher journal Eigenfactor (https://en.wikipedia.org/wiki/Eigenfactor), more reliable and better quality, although Barnett's paper is a single-author paper.

 

[PFfan01]

...
The proof uses the four-dimensional Gauss integral theorem and was known for sure to Poincare and von Laue. There is a lot of confusion in the textbook literature about this, because often the authors forget this fundamental mathematics of tensor calculus and the integral theorems. Then you have all kinds of unnecessary nonsense debates about "hidden momentum" (there's no such thing but just mechanical and electromagnetic stress and the details of the famous formula $E=m c^2$, which implies that also stress adds to the total invariant mass of a composite system), the famous "4/3 problem" in the theory of charged extended bodies (charged classical point particles do not exist in the strict sense at all), etc. The only good thing about this is that you have nice examples to analyze within the correct machinery of tensor analysis, and von Laues books on relativity are masterpieces in doing right this.

...

Hidden momentum in a hydrogen atom and the Lorentz-force law

J. S. Oliveira Filho and Pablo L. Saldanha
Phys. Rev. A 92, 052107 – Published 12 November 2015
http://journals.aps.org/pra/abstract/10.1103/PhysRevA.92.052107

In this work we used perturbation theory to compute the hidden momentum of a hydrogen atom in the presence of an external electric field when the magnetic dipole moment is due to the orbital angular momentum of the electron. We used two different methods for computing this quantity and obtained the same results, evidencing the existence of hidden momentum in the system and the consequent validity of the Lorentz force law in quantum systems.

 

[PFfan01]

... for an arbitrary inertial frame (let's discuss only the special relativistic case first) a tensor field $T^{\mu \nu \rho\ldots}$ leads to a four-tensor of lower rank via
$$\mathcal{T}^{\nu \rho\ldots}=\int_{\mathbb{R^3}} \mathrm{d}^3 \vec{x} T^{0\nu \rho \ldots},$$
only if
$$\partial_{\mu} T^{\mu \nu\rho\ldots}=0.$$
...

However, “… it is shown that the divergence-less itself is neither a sufficient nor a necessary condition” (http://dx.doi.org/10.1139/cjp-2015-0198)
Note: $$\partial_{\mu} T^{\mu \nu\rho\ldots}=0$$ is called "divergence-less".

Based on the divergence-less, the energy-momentum tensor is traditionally required to be symmetric to meet the conservation of angular momentum. For example, the review paper by Pfeifer et al indicates:

“The electromagnetic energy-momentum tensor of Minkowski was not diagonally symmetric, and this drew considerable criticism as it was held to be incompatible with the conservation of angular momentum.” (http://dx.doi.org/10.1103/RevModPhys.79.1197)

A recent review paper by Bethune-Waddell and Chau also indicates that symmetry of the energy-momentum tensor is a necessary condition to satisfy conservation of angular momentum; coped below:

“Symmetry of the energy-momentum tensor (equation (15)) is satisfied by only the Abraham, Einstein–Laub, Amperian, and Chu postulates. The Minkowski postulates form an energy-momentum tensor that is asymmetric, which has been argued to exclusively satisfy relativistic transformations [73, 77]. On the other hand, the Minkowski postulates can be made to form an energy-momentum tensor that is symmetric—a necessary condition to satisfy conservation of angular momentum …” (Rep. Prog. Phys. 78 (2015) 122401; http://dx.doi.org/10.1088/0034-4885/78/12/122401)

However this traditional view is also questioned in the article (http://dx.doi.org/10.1139/cjp-2015-0198):

“In ref. 2, the book by Landau and Lifshitz, the divergence-less of a tensor is taken as a sufficient condition, as shown in Eq. (32.6) on p. 83 and Eq. (32.11) on p. 84. The symmetry of the tensor is claimed to be required by “the law of conservation of angular momentum” by repeating use of their version of Laue’s theorem; see Eq. (32.10) on p. 84. As shown in Sect. 4 of the present paper, however, the divergence-less is never a sufficient condition; thus the correctness of the requirement of the symmetry is also questionable.”

 

[vanhees71]

Hidden momentum in a hydrogen atom and the Lorentz-force law

J. S. Oliveira Filho and Pablo L. Saldanha
Phys. Rev. A 92, 052107 – Published 12 November 2015
http://journals.aps.org/pra/abstract/10.1103/PhysRevA.92.052107

In this work we used perturbation theory to compute the hidden momentum of a hydrogen atom in the presence of an external electric field when the magnetic dipole moment is due to the orbital angular momentum of the electron. We used two different methods for computing this quantity and obtained the same results, evidencing the existence of hidden momentum in the system and the consequent validity of the Lorentz force law in quantum systems.

Hm, I've to read the interesting paper in detail, but already in their classical analogy (which is widely discussed in textbooks like Griffiths's) it becomes very clear that there is no hidden momentum anywhere. It's just that you have to evaluate all momenta (both that of the matter/charges and the electromagnetic field) relativistically. That's it. There is conserved total momentum, but nothing is hidden anywhere. Of course, you get contradictions, when you evaluate one part of the momentum non-relativistically (in this example the momentum of the matter) and the other relativistically (using the Poynting vector for the em. field means to evaluate a relativistic momentum of the em. field). Of course, an approximation is not exact, which is why it is an approximation, but nothing indicates that momentum is in any way hidden.

I also couldn't agree more with the statements in Sect. IV. As far as I know, all this was known and contained already in papers by Poincare and von Laue in the 1910s or 20s.

As I said, I still have to understand, what's discussed for the quantum case. It's not clear to me, how you can resolve an apparent problem of the "hidden momentum" type using the nonrelativistic Hamiltonian of the object under consideration (here a hydrogen atom).

 

[vanhees71]

However, “… it is shown that the divergence-less itself is neither a sufficient nor a necessary condition” (http://dx.doi.org/10.1139/cjp-2015-0198)
Note: $$\partial_{\mu} T^{\mu \nu\rho\ldots}=0$$ is called "divergence-less".

Based on the divergence-less, the energy-momentum tensor is traditionally required to be symmetric to meet the conservation of angular momentum. For example, the review paper by Pfeifer et al indicates:

“The electromagnetic energy-momentum tensor of Minkowski was not diagonally symmetric, and this drew considerable criticism as it was held to be incompatible with the conservation of angular momentum.” (http://dx.doi.org/10.1103/RevModPhys.79.1197)

A recent review paper by Bethune-Waddell and Chau also indicates that symmetry of the energy-momentum tensor is a necessary condition to satisfy conservation of angular momentum; coped below:

“Symmetry of the energy-momentum tensor (equation (15)) is satisfied by only the Abraham, Einstein–Laub, Amperian, and Chu postulates. The Minkowski postulates form an energy-momentum tensor that is asymmetric, which has been argued to exclusively satisfy relativistic transformations [73, 77]. On the other hand, the Minkowski postulates can be made to form an energy-momentum tensor that is symmetric—a necessary condition to satisfy conservation of angular momentum …” (Rep. Prog. Phys. 78 (2015) 122401; http://dx.doi.org/10.1088/0034-4885/78/12/122401)

However this traditional view is also questioned in the article (http://dx.doi.org/10.1139/cjp-2015-0198):

“In ref. 2, the book by Landau and Lifshitz, the divergence-less of a tensor is taken as a sufficient condition, as shown in Eq. (32.6) on p. 83 and Eq. (32.11) on p. 84. The symmetry of the tensor is claimed to be required by “the law of conservation of angular momentum” by repeating use of their version of Laue’s theorem; see Eq. (32.10) on p. 84. As shown in Sect. 4 of the present paper, however, the divergence-less is never a sufficient condition; thus the correctness of the requirement of the symmetry is also questionable.”

That's why I talked about the Belinfante energy-momentum tensor which is (of course) symmetric and gauge invariant. You cannot draw easily conclusions from a gauge dependent quantity like the canonical energy-momentum tensor, which, however leads to the same total momentum, no matter whether it is divergence less or not, because it differs from the Belinfante tensor only by a total divergence.

 

[PFfan01]

Hm, I've to read the interesting paper in detail, but already in their classical analogy (which is widely discussed in textbooks like Griffiths's) it becomes very clear that there is no hidden momentum anywhere. ... There is conserved total momentum, but nothing is hidden anywhere. ..., but nothing indicates that momentum is in any way hidden.
...

However Prof. Griffths indicates in their recent work:

[9] W. Shockley and R. P. James, Phys. Rev. Lett. 18, 876 (1967); W. H. Furry, Am. J. Phys. 37, 621 (1969); V. Hnizdo, Am. J. Phys. 65, 515 (1997); ref. 5, Example 12.12. The term \hidden momentum" is perhaps unfortunate, since it sounds mysterious or somehow illegitimate. Elsewhere, Mansuripur calls it “absurdity" (M. Mansuripur, Opt. Commun. 283, 1997 (2010), p. 1999). But hidden momentum is ordinary relativistic mechanical momentum; it occurs in systems with internally moving parts, such as current-carrying loops. Thus a Gilbert dipole in an electric field, having no moving parts, harbors no hidden momentum. See D. J. Griffths, Am. J. Phys.60, 979 (1992), p. 985. In any event, hidden momentum is not a “problem" to be “solved," as Mansuripur would have it, but a fact, to be acknowledged.

See: David J. Griffths and V. Hnizdo, Comment on “Trouble with the Lorentz Law of Force: Incompatibility with Special Relativity and Momentum Conservation" ; http://arxiv.org/pdf/1205.4646v1.pdf

 

[PFfan01]

That's why I talked about the Belinfante energy-momentum tensor which is (of course) symmetric and gauge invariant. You cannot draw easily conclusions from a gauge dependent quantity like the canonical energy-momentum tensor, which, however leads to the same total momentum, no matter whether it is divergence less or not, because it differs from the Belinfante tensor only by a total divergence.

The Belinfante-Rosenfeld tensor is a modification of the energy momentum tensor that is constructed from the canonical energy momentum tensor and the spin current so as to be symmetric yet still conserved. http://www.digplanet.com/wiki/Belinfante%E2%80%93Rosenfeld_stress%E2%80%93energy_tensor

Namely, Belinfante-Rosenfeld energy-momentumtensor is symmetric and divergence-less. However the article (http://dx.doi.org/10.1139/cjp-2015-0198) concludes:

“It is found in the paper that, the Landau-Lifshitz version of Laue’s theorem (where the divergence-less of a four-tensor is taken as a sufficient condition) and the Weinberg’s version of Laue’s theorem (where the divergence-less plus a symmetry is taken as a sufficient condition) are both flawed, although they are widely accepted as well-established basic results of tensor calculus [2,3]. That is because the two versions of Laue’s theorem are directly negated by the specific examples of the charged meta sphere and the finite electrostatic equilibrium structure, for which the EM stress-energy tensor is both symmetric and divergence-less, but the space integrals of the time-row (column) elements of the tensor cannot constitute a Lorentz four-vector, as shown in Sec. 4 and Sec. 5.”

 

[vanhees71]

However Prof. Griffths indicates in their recent work:

[9] W. Shockley and R. P. James, Phys. Rev. Lett. 18, 876 (1967); W. H. Furry, Am. J. Phys. 37, 621 (1969); V. Hnizdo, Am. J. Phys. 65, 515 (1997); ref. 5, Example 12.12. The term \hidden momentum" is perhaps unfortunate, since it sounds mysterious or somehow illegitimate. Elsewhere, Mansuripur calls it “absurdity" (M. Mansuripur, Opt. Commun. 283, 1997 (2010), p. 1999). But hidden momentum is ordinary relativistic mechanical momentum; it occurs in systems with internally moving parts, such as current-carrying loops. Thus a Gilbert dipole in an electric field, having no moving parts, harbors no hidden momentum. See D. J. Griffths, Am. J. Phys.60, 979 (1992), p. 985. In any event, hidden momentum is not a “problem" to be “solved," as Mansuripur would have it, but a fact, to be acknowledged.

See: David J. Griffths and V. Hnizdo, Comment on “Trouble with the Lorentz Law of Force: Incompatibility with Special Relativity and Momentum Conservation" ; http://arxiv.org/pdf/1205.4646v1.pdf

Sure, Griffiths is usually right with his analysis (at least, I don't know any wrong statement in his very nice papers at Am. J. Phys.), but he still keeps the indeed very unfortunate notion of "hidden momentum". There's nothing hidden and nothing problematic, as long as one treats everything relativistically. It's due to the sad fact that even new textbooks follow the tradition to first treat the in-medium Maxwell theory in the non-relativistic approximation and then having to "repair" this flaw with an extra chapter on "relativistic electrodynamics", where they then talk about "hidden momentum", which is just momentum treated consistently with the field, i.e., relativistically. I never understood, why somebody intends to write such traditional textbooks about E&M, because there's already the comprehensive book by Jackson, which is very hard (if not impossible) to make better than it already is (or better was until he introduced the SI in the 3rd edition, but that's a minor flaw compared to the non-relativistic treatment of in-medium CED, and Jackson of course uses Gaussian units when discussing the relativistically covariant theory).

Mansuripur usually writes paradoxical papers, which I find a shame to be published in refereed journals. The best of those are the refutations by other authors like Jackson or Griffiths ;-).

 

[vanhees71]

I cannot read the Canadian Journal, because I've no access to it. This Wang, however, seems to come to very strange conclusions. How can the usual treatment by von Laue et al be wrong? It's a mathematical theorem. Maybe you can construct artificial fields/charge-current distributions which do meet the conditions for which the theorem is valid. Usually this is related with some idealization of a real physics situation. There is e.g., trivial trouble if you consider an infinite cylindrical wire or an everywhere homogeneous electrostatic field, etc. but these are unphysical idealizations/approximations to a real situation.

 

[PFfan01]

...on researchgate(http://www.researchgate.net/profile/Changbiao_Wang3/publications ). ...

To #81 vanhees71
From the above link given by DaleSpam, you have free access to this Wang paper. http://www.researchgate.net/publication/283709557_CanJPhys_93_p1470_(2015)

 

[PeterDonis]

Please note, physicsforum1's question about photon energy and the light quantum hypothesis has been spun off to a new thread in the Quantum Physics forum:

https://www.physicsforums.com/threads/photon-energy-and-light-quantum-hypothesis.847848/