Quantum Molecular Dynamics
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe letter of Riccardo Fantoni proposes a generalization of continuous-representation theory introduced by J.R. Klauder for reformulation of classical mechanics equations as those arising from extremal of quantum-mechanical action functional with respect to a restricted set of unit vectors whose c-number labels become the dynamical variables to the case of many-body systems. This work also presents the Wick rotational transformation of time which allows to mathematically connect quantum mechanical equations to statistical mechanics equations within the proposed theory.
The presented generalized theory is a powerful tool that can help to resolve complex quantum physics problems within the established statistical physics methods by treating imaginary time as a periodic coordinate. This may especially be useful for understanding thermal properties of matter within quantum field theory.
I think that the letter of Riccardo Fantoni reports on important theoretical results in the field of quantum physics and will be interesting for readers. The proposed theory is akin to the path integral Monte Carlo (PIMC) method for quantum fluid which interpolates between the classical fluid and the ground state of the quantum fluid. This theory proposes a novel kind of a many-body simulation that holds for a general quantum fluid and, like PIMC, can interpolate between high and low temperatures. In this respect, the paper can be regarded as original because there are indeed no similar simulation algorithms in the community.
The text is clearly written and well-structured. Overall, I recommend the manuscript for publication in Quantum Reports after minor revision. A few unclear issues are mentioned below and recommended for clarification.
- The functional form of φ is set by Eq. (5). If I am not mistaken, such form has been presented by Klauder in one of his works. This functional allows to reduce classical action to the convenient form given by Eq. (7). It indeed represents a useful mathematical maneuver, but it is not clear what physical idea lies beneath it? What were the prerequisites (except for mathematical convenience) for choosing that particular functional form?
- In Eq. (10), it is not clear why among the kinetic terms in the additional O-term, there is only the matrix of P2 Should it not be something like (φ0, [(P + Ꝓ)2 - P2]φ0)/2m?
- It is not clear why fiducial vector is chosen to be a Dirac delta function in dN spatial dimension, as given by Eq. (11)?
- I suggest to improve stylistics and check for grammatical and orthographical errors. For example, the phrase “We are not aware of any similar method able to treat any many body system irrespective of the statistics ruling the particles being them distinguishable” is not grammatically correct. I suspect that in the case of “any many body” the authors unintentionally omitted the hyphen in the “many-body,” but “being them distinguishable” is not correct grammatically. Another mistake is the “calssical action” instead of “classical action” in the paragraph before Eq. (7). I assume that there may be more such misprints or incorrectly build phrases throughout the text, thus, I urge the author to search for them carefully.
Author Response
Dear Editor,
I would like to thank the two Referees for their careful reading of the manuscript
and for their comments/suggestions for its improvement. I list below each of the 2
Referee reports and then my reply to each of them.
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First Referee Report
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The letter of Riccardo Fantoni proposes a generalization of continuous-representation theory introduced by J.R. Klauder for reformulation of classical mechanics equations as those arising from extremal of quantum-mechanical action functional with respect to a restricted set of unit vectors whose c-number labels become the dynamical variables to the case of many-body systems. This work also presents the Wick rotational transformation of time which allows to mathematically connect quantum mechanical equations to statistical mechanics equations within the proposed theory.
The presented generalized theory is a powerful tool that can help to resolve complex quantum physics problems within the established statistical physics methods by treating imaginary time as a periodic coordinate. This may especially be useful for understanding thermal properties of matter within quantum field theory.
I think that the letter of Riccardo Fantoni reports on important theoretical results in the field of quantum physics and will be interesting for readers. The proposed theory is akin to the path integral Monte Carlo (PIMC) method for quantum fluid which interpolates between the classical fluid and the ground state of the quantum fluid. This theory proposes a novel kind of a many-body simulation that holds for a general quantum fluid and, like PIMC, can interpolate between high and low temperatures. In this respect, the paper can be regarded as original because there are indeed no similar simulation algorithms in the community.
The text is clearly written and well-structured. Overall, I recommend the manuscript for publication in Quantum Reports after minor revision. A few unclear issues are mentioned below and recommended for clarification.
The functional form of φ is set by Eq. (5). If I am not mistaken, such form has been presented by Klauder in one of his works. This functional allows to reduce classical action to the convenient form given by Eq. (7). It indeed represents a useful mathematical maneuver, but it is not clear what physical idea lies beneath it? What were the prerequisites (except for mathematical convenience) for choosing that particular functional form?
In Eq. (10), it is not clear why among the kinetic terms in the additional O-term, there is only the matrix of P2 Should it not be something like (φ0, [(P + Ꝓ)2 - P2]φ0)/2m?
It is not clear why fiducial vector is chosen to be a Dirac delta function in dN spatial dimension, as given by Eq. (11)?
I suggest to improve stylistics and check for grammatical and orthographical errors. For example, the phrase “We are not aware of any similar method able to treat any many body system irrespective of the statistics ruling the particles being them distinguishable” is not grammatically correct. I suspect that in the case of “any many body” the authors unintentionally omitted the hyphen in the “many-body,” but “being them distinguishable” is not correct grammatically. Another mistake is the “calssical action” instead of “classical action” in the paragraph before Eq. (7). I assume that there may be more such misprints or incorrectly build phrases throughout the text, thus, I urge the author to search for them carefully.
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My Reply to first Referee
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I think that the Referee has a shift of -1 in the Equation numbers he is referring
to in his/her report.
Accordingly I have added the physical idea behind Eq. (6) in the following paragraph:
\red{It is constructed so that the three exponential factors are
such that the first one is just a phase, the second is a unitary operator that
builds a translation in the positions according to
$\exp(i\calq\cdot P/\hbar)f(Q)\exp(-i\calq\cdot P/\hbar)=f(Q+\calq)$
for any infinitely differentiable function $f$
\footnote{This is a consequence of Hadamard lemma. \label{foot:Hadamard}},
and the third builds a translation in the momenta according to
$\exp(-i\calp\cdot Q/\hbar)g(P)\exp(i\calp\cdot Q/\hbar)=g(P+\calp)$ for any
infinitely differentiable function $g$ \footref{foot:Hadamard}.}
The Referee is correct but (φ0, [(P + \calp)^2 - \calp^2]φ0) reduces to (φ0, P^2φ0)
since (φ0, Pφ0)=0 by Eq. (7b) and (φ0, \calp^2φ0) is just a constant. I do not
feel necessary to add anything in the current version of the manuscript on this
regards.
Regarding the fiducial vector choice of Eq. (12) I added:
\footnote{\red{It is rather gratifying to see how the choice of Eq. (\ref{eq:ic})
that is here the necessary initial condition (\ref{eq:Bic}) to the Bloch
equation for the coordinate representation of the many body density matrix was
also found necessary by Klauder in his pioneering paper
\cite{KlauderJCP1963b} at page 1064 of his section
``Canonical Transformations and Inexact 'Classical' Action Functionals''.
Quoting his writing ``{\sl By choosing $\Phi_0$ sharp in $Q$ space about
zero we can make $v(q)$ arbitrarily small}''.}}
I thank the Referee for his careful reading of the manuscript!
I also adjusted the English here and there.
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I hope that the revised manuscript will be suitable for publication on Quantum Reports.
kind regards,
Riccardo Fantoni
Reviewer 2 Report
Comments and Suggestions for AuthorsThe manuscript proposes a “Quantum Molecular Dynamics” approach based on Klauder’s continuous representation to evolve the quantum density matrix in imaginary time using classical-like Hamiltonian dynamics. The manuscript contains an interesting formal idea; however, I have a few questions
1. How is the imaginary-time Hamiltonian dynamics discretized and stabilized for large β?
2. How does the method recover correct fermionic behavior in the low-temperature (𝑇→0) limit?
3. How do these divergences affect interacting systems beyond the ideal gas case?
Author Response
Dear Editor,
I would like to thank the two Referees for their careful reading of the manuscript
and for their comments/suggestions for its improvement. I list below each of the 2
Referee reports and then my reply to each of them.
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Second Referee Report
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The manuscript proposes a “Quantum Molecular Dynamics” approach based on Klauder’s continuous representation to evolve the quantum density matrix in imaginary time using classical-like Hamiltonian dynamics. The manuscript contains an interesting formal idea; however, I have a few questions
1. How is the imaginary-time Hamiltonian dynamics discretized and stabilized for large β?
2. How does the method recover correct fermionic behavior in the low-temperature (?→0) limit?
3. How do these divergences affect interacting systems beyond the ideal gas case?
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My Reply to second Referee
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I thank the Referee for his/her interest in our new computational method.
Regarding his/her questions I can say the following:
1. I don't see any difficulty in the discretization necessary in the imaginary time
evolution. The algorithm will depend on a imaginary-time-step \tau. At low temperature
one needs to evolve the density matrix on a long imaginary time. For a fixed large \beta
one has to find the optimal balance beween the maximum accuracy in the final density matrix
and the minimum number of necessary time-steps.
2. Our formal computational algorithm solves the fermions sign problem for the measurement
of observables diagonal in their position representation. For non diagonal observables
like the momentum distribution or dynamical properties some more work is necessary.
3. Unlike PIMC our method will not suffer from the requirement of dealing with interactions
bounded from below since the interaction is not ``integrated'' but is ``evolved'' in the
imaginary time dynamics.
I added these comments in ending paragraphs of the conclusions.
I hope that the revised manuscript will be suitable for publication on Quantum Reports.
kind regards,
Riccardo Fantoni
