Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and c → ∞ Limit
Round 1
Reviewer 1 Report
The manuscript considers a Lorentz-covariant formulation of quantum mechanics based on a pseudo-hermitean representation. The problem is well motivated though the investigation remains purely formal.
Author Response
Practical applications of `relativistic' quantum mechanics is
generally tricky. Even with dynamics for the electron, quantum
field theory is usually preferred. That is the main reason we
have been thinking about addressing the aspect only after we
have finished a systematic study of a corresponding spin 1/2
theory under our framework. As explicitly prompted by another
reviewer, we now include some comments about the case in the
extended last paragraph.
Reviewer 2 Report
In the manuscript "Group Theoretical Approach to Pseudo-Hermitian Quantum
Mechanics with Lorentz Covariance and c-> infinity Limit" the authors have presented a complete and comprehensive formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg-Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. In their study they have also considered the classical limit. The formulation of "covariant quantum mechanics as an irreducible component of the regular representation of the HR(1, 3) (quantum) relativity symmetry", with a pseudo-unitary inner product obtained based on earlier studies on the covariant harmonic oscillator problem identified as a representation of the same symmetry has been introduced thoroughly. The manuscript reads very well, and all aspects from the scientific content to the writing style introduced professionally. A short appendix has been added to the manuscript which I believe can be extended to include some details of the calculations in the main text. In addition, it would be great if the authors can also include clear physical interpretations of there results, this can potentially broaden the readers from various backgrounds as it contains fundamental concepts. My only major comment is that the authors have not included the implications of their work as regard to Dirac's equation. Since relativistic limits have been studied in this work, adding a few sentences from Dirac's equation perspective is recommended. Over all I have enjoyed reading this article and I strongly recommend publication after minor revision and some language check.
Author Response
For including the "implications" as regard to Dirac's equation,
short of having finished a systematic study of a corresponding
spin 1/2 theory under our framework, what we could say at this
point would be quite speculative. So, instead of doing that,
we present some comments more about the backgrounds especially
in relation to the unitary but otherwise covariant symplectic
formalism of 'relativistic' quantum mechanics initiated by
Stueckelberg early the 1940's, in the now extended last paragraph.
That, we believe, could give a good idea on the potential implications.
After all, without the backgrounds, it would be difficult to
address the implications. And that is quite beyond a few sentences
already. We will be happy to consider further supplements to the
manuscript if there are specific suggestions.
Reviewer 3 Report
This paper describes new results for a novel formulation of covariant quantum mechanics. It is quite well written and describes contractions which are of interest to explore relativistic classical, as well as non-relativistic quantum, limits as systematic approximations.
I have only one suggestion for the authors. They touch upon the relationship of their model with noncommutative spacetime in the introduction when they state that "..the wavefunctions on coherent state basis .... can be seen as operator coordinates of the quantum phase space...". Perhaps they would consider adding some sentences regarding the status of covariance in such noncommutative systems when generalizing to curved spacetimes. Related references (e.g. arXiv:1606.00769, arXiv:1707.00885 and arXiv:1712.07413) would also be helpful for the readers to get a good idea about the status of the field when considering full gravity.
Other than this one discussion which is missing in the draft, I find this paper to be interesting and, therefore, recommend publication.
Author Response
We thank the reviewer for prompting us to include more discussions on
the noncommutative geometric perspective behind the work and have now
added in a new (second last) paragraph in the conclusion section to
address the issue somewhat, focusing more on the unique aspect of our
approach compared to others in the literature, that the spacetime is
a phase space. A full picture of the subject matter and its development
in the literature of course cannot be presented in anything less than
a big review paper in itself.
Reviewer 4 Report
I think that the manuscript symmetry-1006379 is a typical example of modern scholastic. Jorge Hirsch (well known as the author of the Hirsch index https://en.wikipedia.org/wiki/H-index ) uses in the recent publication [1] of Andersen's fairy tale ‘The Emperor’s New Clothes’ in order to explain the devotion of most physicists to the conventional theory of superconductivity:
“I believe that much of the explanation for this unconditional devotion to the conventional theory of superconductivity can be found in Andersen’s little tale [15], that I will paraphrase here.
‘Many years ago there was an Emperor so exceedingly fond of new clothes that he spent all his money on being well dressed.’
Many years ago there were physicists so enamored with their mathematical abilities to deal with complicated field theories that they forgot about physical reality” [1].
The conventional theories of superconductivity, especially the Ginzburg-Landau theory, are perfect in some aspects. But these theories cannot describe the Meissner effect and irreversible thermodynamic processes observed in superconductors since they were created in the framework of equilibrium thermodynamics. Therefore Hirsch is right when he points out the inconsistency of the conventional theory of superconductivity in [2] and other publications.
The inconsistency of the relativistic quantum theory is more obvious. Einstein drew attention on the contradiction between quantum mechanics and the relativity theory as far back as 1927 during the discussion at the Fifth Solvay Conference [3]. John Bell fifty-seven years later said in his talk "Speakable and unspeakable in quantum mechanics" at Naples-Amalfi meeting 1984 about experimental evidence of violation of Bell's inequalities obtained by Alain Aspect with coauthors [4]: "For me then this is the real problem with quantum theory: the apparently essential conflict between any sharp formulation and fundamental relativity. That is to say, we have an apparent incompatibility, at the deepest level, between the two fundamental pillars of contemporary theory... and of our meeting", see p. 172 in [5].
Most people, including the authors of the manuscript, do not understand, unlike Einstein and Bell, that quantum mechanics and relativity are incompatible, because they do not understand that the orthodox quantum mechanics describe the observer's knowledge about the probability of the results of upcoming observations rather than reality.
The authors of the manuscript write about 'the language of pseudo-Hermitian quantum mechanics', 'the wavefunctions', 'quantum relativity symmetry' but they don't understand that their mathematical abilities has nothing to do with physical reality. Their 'quantum relativity symmetry' can't even make sense because of the contradiction between quantum mechanics and relativity.
Unfortunately, there are many publications of such scholasticism, the authors of which do not understand what they are writing about. I do not think that the number of such publications should be multiplied. Therefore I do not recommend publishing this manuscript. The authors have to understand, instead of engaging in meaningless scholasticism, that we do not have a consistent theory of quantum phenomena, even non-relativistic ones. At least the authors of the manuscript should know about the numerous publications the authors of which refute local realism because of its contradiction with quantum mechanics.
[1] J. E. Hirsch, Superconductivity, what the H? The emperor has no clothes. APS Forumon Physics and Society Newsletter, January 2020, p. 4-9; arXiv: https://arxiv.org/abs/2001.09496
[2] J. E. Hirsch, Inconsistency of the conventional theory of superconductivity. EPL (Europhysics Letters) 130, 17006 (2020); https://arxiv.org/abs/1909.12786 .
[3] A. Einstein, Electrons et photons. Rapports et discussions du cinquieme Gonseil de physique- Bruxelles du 24 au 29 octobre 1927 sous les auspices de 1' Institut International de physique Solvay, p. 253-256. Paris, Gautier-Villars et Gie, editeurs (1928).
[4] A. Aspect, Dalibard J., Roger, G.: Experimental Test of Bell's Inequalities Using Time - Varying Analyzers. Phys. Rev. Lett. 49, 1804-1807 (1982).
[5] J.S. Bell, Speakable and unspeakable in quantum mechanics. Collected papers on quantum philosophy. Cambridge University Press, Cambridge, 2004.
Author Response
We are well aware and much concerned about the difficulty of putting quantum
theories and gravitation and relativity theories under one consistent
framework of quantum gravity. However, we sure do not agree that without,
or rather before, having such a framework available, research efforts in
the related areas are all "meaningless scholasticism". In our case, we
actually work on a grand project, to which the current paper is a small
part of, with exactly building a consistent approach to quantum gravity
in mind, but can only take a small step at a time. Our perspective of
quantum mechanics as particle dynamics on a quantum, noncommutative,
model of space(time) is to be seen exactly as on the way to develop a full
notion of "local realism" in terms of noncommutative geometry with the aim
to eventually formulate a theory of quantum gravity as geometrodynamics
of quantum spacetime. Within that picture, quantum realism has to be
about the noncommutative space(time), not the classical one to which
the standard discussions related to quantum experiments refer. Each
observable has a definite value for a state, only that such values
cannot be represented by real numbers. They are to be represented
by some 'noncommutative numbers', which carry the full information the
mathematical formulation of the theory actually contains. Such information
is experimentally accessible, at least in principle. Our idea of having
the pseudo-unitary metric on the space of states comes mostly from the
intuition on the need to take the pseudo-unitary Minkowski metric seriously
as a quantum notion, for the noncommutative position and momentum operators
as coordinates for the space. We have added in a new (second last) paragraph
in the conclusion section to address the issue somewhat. A full theory of
quantum gravity along the line is of course still far away.
