A Simple Quantum Picture of the Relativistic Doppler Effect

Round 1

Reviewer 1 Report

In this article Daniel Hodgson, Sara Kanzi and Almut Beige report their theoretical results on the Doppler effect using a local approach to the relativistic Doppler effect based on relativity, spatial and time translational symmetries, and energy conservation. The authors also present the implications of the relativistic Doppler effect for the quantum state transformations of wave packets of light and show that a local photon is a local photon at the same point in the spacetime diagram in all inertial frames. The results are of considerable fundamental interest and practical implications and hence can be considered for publication in Symmetry. The authors are encouraged to address the following comments.

1.      The authors should discuss the implications of the results with regard to the established experiments like Sganac effect (linear or rotational) to test the validity of the relativistic formalism by citing the related updated literature.

2.      Correct Eq. no. in Line 265: Substituting Eq. (28) into this relation yields.

3.      Lines 274-75: This shift in frequency is consistent with previous derivations of the relativistic Doppler shift for light propagating in the s direction. Please mention clearly where is this proof?.

4.      How the reported results could be justified in view of the uncertainty principle.

by considering the invariance of the speed of light.

5.      Please cite the following papers and comment on the justification of the reported results in view of the relativistic Dopler effect (especially given in Ref.b) taken in a different prospect by the author in these referred papers. Especially the authors should comment on the ongoing debate on the predictions of theory of relativity.

a.      Mohammad Shafiq Khan “Unraveling the truth in experimental science” Michelson– Morley experiment: A misconceived & misinterpreted experiment, Indian Journal of Science and Technology Vol. 4 No. 10 (Oct 2011) ISSN: 0974- 6846.

b.      Mohammad Shafiq Khan “Fallacy of space-time concept and a new theory against it” Experimental & theoretical evidences of fallacy of space-time concept and actual state of existence of the physical universe, Indian Journal of Science and Technology Vol. 5 No.3 (2012).

6.      Are the reported results also valid for the monochromatic light? Discuss and justify in the revised version.

7.      The authors discussed the relativistic connection of E and B in view of simple relativistic context. I encourage the authors to also justify the validity of the results in light of gauge invariance symmetry

Comments for author File: Comments.pdf

Author Response

We would like to thank the referee for their careful consideration of this manuscript and for their constructive suggestions and comments. Please see a reply to all comments below.

1.-5. To better explain the relevance of our results, text has been added to both the Introduction and the Conclusions. We also added relevant references.  In the Introduction we now comment that tests of the isotropy of the speed of light may have been misinterpreted and cite the articles suggested by the referee.  In addition, we point out that experimental tests of the Sagnac effect may be best understood when the speed of light is not invariant. However, for simplicity, we assume in this paper that all meters and clocks are calibrated such that light moves at the same speed in all reference frames. In the Conclusions we now point out that our results, being based on the principle of relativity, may aid experimental verifications of the predictions of the special theory of relativity.

2. The typographical error in Line 265 has been corrected. Thanks for pointing this out.

3. Relevant references which illustrated that our result is consistent with previous derivations of the frequency shift have been added to support the statement in the text.

4. To show the consistency of our results with the well-known uncertainty principle, we added Subsection III.C which discusses the Fourier representation of localised blip excitations. For light, uncertainty applies between the frequency and the position of wave packets in the same way as there is uncertainty between the frequency and position of classical waves. 

5. The suggested citations have been added to the Introduction.

6. To answer the question of the reviewer, we extended the discussion at the end of Section IV. There we calculate how frequencies and wavelength of monochromatic light changes (cf. Eq. (52)). In agreement with the classical Doppler effect, whilst blips maintain their position, monochromatic field excitations change their frequency in agreement with the literature.

7. In reply to this comment, text has been added to the Introduction regarding Eqs. (3) and (4).  It mentions that, as our quantisation deals only with the gauge invariant electric and magnetic field observables, no gauge fixing procedure is required. As a consequence, the field transformations given are sufficient to determine a relativistic transformation between blips.

Reviewer 2 Report

The manuscript deals with a local approach to the relativistic Doppler effect based on spatial and time translational symmetries and energy conservation. Probably, the primary outcome concerns the locality of the photons in all inertial frames.

 The paper is well-organized and well-written.

The calculations seem to be correct and consistent.

Here are some minor issues to be considered.

1- Eq. 19 There seems to be a missing factor, 1/c, multiplying the last term in the RHS.

2- Eq. 20 A factor c is missing in the RHS.

3- Eq. 24: Not sure about the validity: equal integrals do not imply equal integral functions. The result seems to be correct but can be derived more formally.

4- Line 265. " Substituting Eq. (28) into this relation yields", replace Eq.(28) with Eq. (26).

After these minor comments, the paper should be accepted for publication.

Author Response

We would like to thank the referee for their supportive comments and for pointing out typographical errors. These errors have now been corrected. The discussion around the topic of energy conservation has been completely re-written and the issues mentioned by the referee have been clarified.

Reviewer 3 Report

In the present manuscript, the Authors study the relativistic Doppler effect from a Quantum standpoint. They start with an introduction on special relativistic transformations, focusing on how light signal are emitted and detected by two observers in relative motion. They thus derive the classical relativistic Doppler effect. Thus, in Sections 3 and 4, the authors present a quantum description of electromagnetic fields and the relativistic Doppler effect.

If the intent was to have a pedagogical treatment of the topic, the Author surely accomplished it. If not,  most of the work is standard knowledge in special relativity and quantum optics. For example, Sections 1 and 2 can be greatly reduced by employing Lorentz transformations rather than deriving them. The content of Sections 3 and 4 is mostly covered by any book on quantum optics, although, to the Author convenience, the focus was mostly on a position representation rather than the standard momentum representation.

Thus, besides clearly stating the objective of the work and adapting its content, I believe the manuscript can be considered for publication.

Author Response

Until recently, the quantum optics community believed that it is impossible to quantise light in terms of locally-acting field annihilation operators a(x). However, in Refs. [25-28] we succeeded in overcoming previous no-go theorems and promoted a possible quantisation of the electromagnetic field in position space. Our approach differs from previous field quantisation schemes (cf. e.g. Ref. [30] and references therein) and requires for example a doubling of the standard Hilbert space of the electromagnetic field.  The main purpose of this manuscript is to verify the consistency of our generalisation of standard quantum optics approaches with the well-known Doppler effect [3,5–8].

To explain this more clearly and to adjust the focus of our manuscript, we now included an additional paragraph in the Introduction. We also shortened the derivation of the classical relativistic Doppler effect. For example, we removed the Lorentz transformation and instead highlight the relation between the natural coordinates chi_A = x_A -sc t_A and chi_B = x_B - sc t_B of Alice and Bob. We also removed one figure.

In addition to demonstrating consistency, we try to offer some new insight. For example, we show that a local approach can accommodate spatial and time translational symmetries in a straightforward way. We would like to thank the referee for pointing out that our treatment of the effect is pedagogical and for recommending publication.

Round 2

Reviewer 1 Report

I appreciate the efforts of the authors who have revised the manuscript incorporating the comments of the referee. The quality of the manuscript is enhanced considerably and hence can be accepted as it. 

Best Regards