A Consistent Theory of Tachyons with Interesting Physics for Neutrinosâ€
Round 1
Reviewer 1 Report
The author reviews his series of studies on a mathematical formalism of tachyons in this paper, which starts with a clarification of how a tachyon is defined. While many other authors start with the definition that tachyon is a particle described by a wave equation with an imaginary mass, and are usually led to broken Lorentz symmetry in the consideration of canonical quantization, the author instead constructs a tachyonic theory that retains the full Lorentz symmetry.
The author then reviewed in section 1 how tachyons can be counted taking into account the subtlety that the velocities of tachyons are unbounded. This consideration leads to a notion of momentum flow that differs from conventional slow-particle theory, as well as a modified commutation relation. As a result, this leads to a formalism of tachyonic quantum field theory that differs from the usual treatment. The author is rather succinct on how such a quantum field theory can be constructed, but relevant references have been given, where concrete discussions can be found. This might be fine due to the review nature of this part of the article, but it would be very useful if the author could elaborate the quantization procedure further instead of only giving the reference in the paragraph below Eq. (1.6). There are some more discussions on quantizing tachyons much later in section 2.6, but its connection to the relevant discussion in section 1 does not appear to be very clear at the first sight. Moreover, it would be useful for the author to include at least some references in section 1 on how tachyons are treated by other authors in the literature, instead of vaguely referring to upcoming articles in the special issue of Symmetry (see reference [9]).
The author then summarized related applications to neutrinos as tachyonic particles, and sketched how this proposal may be relevant to understanding dark energy and dark matter. After clarifying that which particles are incoming and outgoing is not a Lorentz invariant concept for tachyons, the author argued that the stress energy tensor associated with tachyons gains a new minus sign. It is then suggested that the tachyons may form a pervasive gas that contributes a negative pressure, in connection with the dark energy problem. Finally, the author showed that tachyonic particles and anti-particles contribute opposite signs to the stress energy tensor, and claimed that this "may explain both Dark Matter and Dark Energy." While these claims appear to be plausible, much more details seem to be needed to ultimately justify them. Some future questions are listed in section 2.7 in a rather schematic way. The style of this section looks more like some notes for a talk (as the author implied at the beginning) instead of part of a formal paper. Although the materials (including the more conjectural parts on dark energy and dark matter) presented in the section are quite interesting, I think its presentation needs to be improved significantly before publication (especially for section 2.7).
In section 3, the author stated that, for tachyons, the action of "sending" a signal does not have Lorentz invariant meaning. This is important for establishing a principle of causality for tachyons. The author then returned to the argument in section 2 of the paper on page 14 and stated that the problem negative energy states for tachyons is resolved by recognizing that the labeling of in and out states is not Lorentz invariant. I think it would be useful to make connection to the relevant discussions in section 2 of the present paper here.
Finally, the author discussed an interesting theory for tachyonic neutrinos and indicated that such neutrinos naturally exhibit parity violation, which might be useful for phenomenology. The author also further elaborated how the helicity of the tachyonic states tells between particles and anti-particles.
A side remark: it is not always clear to tell which part is original research and which part is review of previous works in this paper. It would be useful if the author could clarify.
The paper stresses on potentially important conceptual differences between the author's approach towards tachyons and the more studied formalisms in the literature. The overall style is on the qualitative side, but I think the paper clarifies important issues and is a useful addition to the special issue on tachyons. I therefore recommend the paper for publication in Symmetry if the remarks noted above may be addressed.
Author Response
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Author Response.pdf
Reviewer 2 Report
Here is a report on paper of Professor Schwartz from Berkeley. I recommend
acceptance of paper provided some concerns could be addressed.
First, Professor Schwartz uses a somewhat colloquial style in his paper. This
is in principle OK, but it might be that sometimes, he goes a little overboard.
For example, I would recommend that the opening lines of Section 2 be revised
so that they follow normal English grammar as opposed to a set of keywords. The
same applies to the keywords preceding Eq. (2.12). Again, some freedom in the
choice of style is appropriate, but the author may have gone overboard, also in
regard to grammatically incomplete sentences. A little too much personality
has also been included in other lists of keywords, and in the sentence
preceding Section 3: "For a static system, including tachyons. arXiv:1902.03621
[gr-qc]."
In Section 2.1, maybe, I would recommend to add the following citation:
V. L. Dadykin, G. T. Zatsepin, V. B. Karchagin et al., “Detection of a rare
event on 23 February 1987 by the neutrino radiation detector under Mont Blanc,”
JETP Letters, vol. 45, pp. 593–595, (1987).
The background of this citation is well known, and it could serve the submitted
paper better than the reference to the OPERA experiment.
I would strongly recommend to supplement Eq. (2.12) with an explanation
regarding the transformation of the field annihilation and creation operators
under Lorentz transformations. In view of a possible change of sign of the
tachyon energies under Lorentz transformation, some more explanations on the
point of view of the author on this point would definitely benefit the reader.
In particular, the field operator in Eq. (2.12) is formulated only in terms
of annihilation operators (for particles, not creation operators (for anti-
particles), as it would otherwise be the case for an ordinary Dirac field.
In view of the possible sign change of the energy variable under a Lorentz
transformation, Eq. (2.12) may be permissible, but a further short explanation
would benefit the readability of the paper.
I would also recommend to supplement a few additional remarks on the point
following Eq. (2.19):
"This says that particle and anti-particle contribute to the energy-momentum
tensor with opposite signs. Thus we may explain both Dark Matter and Dark
Energy with the proposition that Cosmic Background Neutrinos are tachyons, with
a mass in the neighborhood of 0.1 eV/c^2."
Just, some more explanation would be beneficial.
The list of references is a bit one-sided in terms of the authors involved in
the cited publication, which seem to have strong overlap with the author of the
submission itself. The Special Issue of Symmetry gives an opportunity to cite
authors working in the field. Incidentally, there might be another paper which
could be of interest:
Ya. B. Zel'dovich, "Singularities of vacuum decay and remarks on tachyons",
JETP Letters, vol. 20, pp. 151–152, (1974).
The contents could be interesting for the author, as it pertains to cosmology.
Maybe, some more inclusive citations could be included in the final version of
the paper.
Again, I heartily recommend acceptance after the modifications are made.
Author Response
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Reviewer 3 Report
In this article, in special and general relativity, a consistent mathematical
description of tachyons are explored. In particular, it is recognized that low energy tachyons will create large gravitational fields through the space-components of the energy-momentum tensor and leads to studying properties of the cosmic neutrino background. Furthermore, it is explained that there is a discourse on the various arguments that have been given against the very possibility for tachyons to exist.
The discussions could be interesting and the manuscript is well described. Moreover, considerations and explanations would be useful for the related future works. Hence, this article could be published in Symmetry.
Before acceptance, since this article would be an original paper,
it seems it would be better that the parts of ``Contents ...'' of Section 2 on page 5 will be written in ordinary sentences.
Author Response
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