Scalar Product for a Version of Minisuperspace Model with Grassmann Variables

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

 

Repeatedly, the authors are returning, in their work, to the

currently most challenging problems emerging when one wants to

quantize gravity.

 

In these efforts, their key technical tool lies in a constructive

analysis of various versions of minisuperspace model. This time,

their attention is concentrated upon the underlying Hilbert-space

problem, i.e., on the problem of a consistent definition of the

inner product which is needed, in quantum theory of unitary systems,

for a quantification of the predictions, i.e., for the evaluation of

the mean values of the operators of relevant observables.

 

The paper is a progress report and, thus, one could have a lot of

critical comments concerning its form. One of them would concern an

absolutely insufficient coverage of the related relevant literature

on the subject. At the same time, regretfully, the Journal's rules

of game do not allow the recommendation of one's own work by the

referees, so the authors should revise without a more constructive

assistance. They must guess by themselves.

 

Also the section of Conclusions is far from satisfactory. For

example, while in the text two methods were compared and found to

disagree (cf. the deeply dis-motivating Figure), the resulting

problem itself has been left unresolved while merely noting that the

methods ``may have different Hilbert spaces''. This would precisely

be the space for an explicit inclusion of the results of the recent

progress in the field.

 

The readers must also feel annoyed when reading that ``It could be

possible that an effect of that kind plays a role in the picture

with the Grassmann variables producing an additional dispersion

[when] compared to the etalon picture'' and that ``possible ways to

correct the picture ... need further investigation''. By my opinion,

similar ``future research'' statements belong to the mass media and

tabloids rather than to serious scientific Journals and papers.

 

This being said, everybody knows that the full consistency of the

theory is still just a fairly remote dream. At the same time, the

sections of References and Conclusions (where the basic message has

to be specified and summarized clearly) would certainly deserve a

thorough revision and, perhaps, substantial extension.

 

Author Response

The authors are grateful to the reviewer for the valuable report. Citations from the report are marked by italic tape.

The paper is a progress report and, thus, one could have a
lot of critical comments concerning its form. One of them would
concern an absolutely insufficient coverage of the related
relevant literature on the subject. At the same time, regretfully,
the Journal's rules of game do not allow the recommendation of
one's own work by the referees, so the authors should revise
without a more constructive assistance. They must guess by
themselves.

This is rather a research paper, than a status report. To enhance the context, we added the relevant references in the revised version.

Also the section of Conclusions is far from satisfactory. For
example, while in the text two methods were compared and found to
disagree (cf. the deeply dis-motivating Figure), the resulting
problem itself has been left unresolved while merely noting that
the methods ``may have different Hilbert spaces''. This would
precisely be the space for an explicit inclusion of the results of
the recent progress in the field. 

Thank you for this note. Actually, there are a lot of discussions
about the method with the Grassmann variables [17-22], but, unfortunately, we found no clear answer to the question about the feasibility of this method. Here, we consider some concrete tests for this method but without a general discussion.

The readers must also feel annoyed when reading that ``It
could be possible that an effect of that kind plays a role in the
picture with the Grassmann variables producing an additional
dispersion [when] compared to the etalon picture'' and that
``possible ways to..

As we understand,  this remark concerns a ``Zitterbewegung.'' In the present version, we have considered this topic in Appendix.

Best regards

The Authors

Reviewer 2 Report

Comments and Suggestions for Authors

In the paper, two quantization methods of QG have been analyzed: the etalon picture (that should be elucidated in the presentation, maybe a short introduction would be welcome) without and with the Grassman variables (ghosts for quantization with constraints). The authors search for a scalar product to define observables in the Hilbert space. 

I don't think the paper is recommendable for publication for the following reasons: 

1) the presentation is very poor lacking logic and a minimal background to be understandable 

2) Some equations are not sound, for example, the set of eqs 8. There the equations for N and its conjugate are missing. In addition, it is not clear why one should add the constraint (10), this is a solution to the equation of motion, why do they need to add it to the theory since the EOM implies it?  

3) However, the main problem with the definition of the scalar product in eq. (37). This is not a scalar product since it is not positive definite and therefore the complete quantum interpretation of the system is missing. The authors compare with KG scalar product, but as is well-known this is also not a scalar product, and that has led to the Dirac equation.  

4) The problem of a scalar product with the Grassman variables is well-known but not solved in the present paper. To define a scalar product one needs two copies of Grassman variables in order to be able to define a positive definite scalar product. 

Comments on the Quality of English Language

The quality of English is mediocre 

Author Response

The authors are grateful to the reviewer for the report, which is very
valuable for us. Citations from the report are marked by italic tape.

1) Some equations are not sound, for example, the set of eqs
8. There the equations for N and its conjugate are missing. In
addition, it is not clear why one should add the constraint (10),
this is a solution to the equation of motion, why do they need to
add it to the theory since the EOM implies it? 

If a canonical gauge fixing condition is used, as in Section 2
then there is no equation for N: a variation of the action over
N gives the Hamiltonian constraint. Both constraint Φ₁ and
Φ2 are integrals of motion of EOM.  Otherwise, the differentiation of these constraints over time will give new constraints. If the non-canonical gauge fixing is used as in Section 3, one could write an equation for N, but it seems simpler to integrate over N in the functional integral as it has been done in this section.

2) However, the main problem with the definition of the
scalar product in eq. (37). This is not a scalar product since it
is not positive definite and therefore the complete quantum
interpretation of the system is missing. The authors compare with
KG scalar product, but as is well-known this is also not a scalar
product, and that has led to the Dirac equation.

Thank you for this note. We have considered only a half-space of
solutions of the Wheeler-DeWitt equation, where the Klein-Gordon scalar product gives a positive norm of a state. In the present version of the paper, we express that through both formulas and the additional comments.

3)The problem of a scalar product with the Grassman variables
is well-known but not solved in the present paper. To define a
scalar product one needs two copies of Grassman variables in order
to be able to define a positive definite scalar product.

It is a very interesting note. We hope that the approach mentioned by the reviewer waits for further comprehensive analysis to give a method for calculating the operator mean values. We included the reference [22] concerning this subject.

Best regards,
Authors

Reviewer 3 Report

Comments and Suggestions for Authors

In their manuscript, the authors suggest an explicit formula for the
scalar product which is related to the Klein-Gordon scalar product. This is
important for calculating expected values of operators.

The proposed model involves the Friedmann-Robertson-Walker (FRW) geometry  and a scalar field as material content,  minimally coupled with gravity.  There are two points that I believe need clarification.

The first point is that the equation that arises from the equation $\hat{H}\Psi=0$, by Dirac's formalism,  does not take the form of a Schrödinger equation. There is, in fact, no dependence on time differential operators.

Therefore, I believe a better explanation on how equation (13) of  the manuscript leads to $\hat{H}\Psi=i\frac{\partial}{\partial \eta}$ is needed.

The second point is that since the scale factor variable is in the positive
semi-axis of real numbers, according to the literature [J. Math. Phys. 37
(1996) 1449] wave functions must be such that their inner product
has a weight function $p(x)$, thus ensuring that the Hamiltonian operator is
self-adjoint. My question is, does the inner product given by (33) ensure
that the Hamiltonian operator in the form
$H=\frac{1}{2}g^{ij}p_{i}p_{j}+\pi_{\theta}\pi_{\bar{\theta}}$

is self-adjoint? If so, I believe the authors should demonstrate this in
their manuscript.

Author Response

Authors are grateful to reviewer for the report. In the revised version of the paper we rearrange the formulas and  explain that  a principle of  the derivation is
obtaining of an expression for functional Z without any
preexponential factors, and then using H in the Schrodinger
equation. This holds as for obtaining physical Hamiltonian (see
(20)), so for the Hamiltonian with the Grassmann variables (see
(26)). Some general argumentation in favor of Schrodinger
equation could be found in Ref.[42].

Weight function in the scalar product depends on the operator
ordering. We explain in the text more carefully that we use
Laplacian operator ordering and the measure equals to the  square
root of determinant of the supermetric, so that Laplacian is
self-ajoint operator. No any boundary terms arise in the scalar
product, because one of the functions contain Dirac
delta-function.

As for the final result, we ourself are disappointed, because there is
no full correspondence between the methods in calculation of the
mean values. In principle, it could be related with the operator
ordering, but the Laplacian ordering seems preferable, otherwise a
dependence on a way of choosing variables in α,φ superspace
arises.

As for WDW equation, it is another way of quantization, which
tells nothing without definition of the scalar product. To obtain
time evolution in this picture one has to supplement WDW by
Klein-Gordon scalar product with time-dependent integration plane
[40].

 

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript "Scalar product for a version of minisuperspace model with the Grassmann variables" considers a quantum cosmological model with a starting action consisting of standard General Relativity minimally coupled to a dynamical scalar field. The authors find the related Hamiltonian and solve the Schroedinger equation, thus comparing the results with those coming from the application of the Wheeler-DeWitt equation.

 

The paper is interesting and self-consistent, though in my opinion there are some minor points that the authors should address:

 

On p. 2, line 37, it should be indicated that greek indexes run from 0 to 3, that G is the Newton constant, g^{\mu \nu} the metric tensor with g being its determinant.

 

New symbols appearing in Eqs. 2, 3, 4 should be introduced for the benefit of the reader

 

The authors should better explain why to use the Schroedinger equation to the physical Hamiltonian, instead of the WDW equation H \psi = 0, considering that the Hamiltonian constraint has been used a few lines above. In this regard, the authors only write "The most simple and straightforward way to describe a quantum evolution is to formulate the Schroedinger equation" but I believe that the discussion should be expanded and cannot be liquidated in one sentence.

 

Several equations are written but not commented from a physical point of view. For instance, in Eqs. 18 and 19 the authors compute the mean value of a^2, but no physical interpretation of the resulting quantity is provided. I suggest to expand the related discussion in this direction also for other equations throughout the manuscript.

Comments on the Quality of English Language

The English style should be improved, as several sentences are either written in a colloquial language or not properly for a scientific paper.

 

For instance, the first typo occurs in the abstact when the authors write: "The Grassmann variables are used to formally transform a system with constraints into an unconstraint system". The word "unconstraint" should be replaced with "unconstrained". Several other linguistic errors are spread out throughout the manuscript

Author Response

We are grateful to the reviewer for the considerable notes. We are
trying to take them into account in the present version. In
particular, we derive the physical Hamiltonian and the Hamiltonian
with the Grassmann variables in a one style throughout.

A footnote about physical interpretation of the mean value of the
scale factor is added.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

-

Author Response

We are grateful to reviewer for the report. In the present version we have added some references and rewrite the introduction. 

Reviewer 3 Report

Comments and Suggestions for Authors

Originally, the WDW equation does not contain time derivatives; it is the so-called "problem of time" in quantum cosmology.

From my point of view, the best way to reintroduce time is to use the Schultz formalism, i.e., by the introduction of a linear radiation-like fluid, which can be chosen as time variable. The model in discussion does not consider a fluid as its matter content but, rather, considers a scalar field. Hence the latter should be considered as the new "time", and it does not simply "complement" the equation with a temporal derivative term.

Another point is that, in the model, there is a factor 1/a^2 multiplying the derivative in phi.  I understand that this term serves as the weight function of the inner product, thus ensuring the orthonormality of the eigenfunctions, and also ensuring that the Hamiltonian operator be self-adjoint.

Finally, and of lesser importance, it seems to me that the wave packet (15), with C(k) given by (17), is not normalized.

Therefore, I do not recommend the work for publication.

Author Response

We are grateful to the reviewer again for the valuable remarks. Citations from the report are marked by italic tape.

Originally, the WDW equation does not contain time
derivatives; it is the so-called "problem of time" in quantum
cosmology.

Actually, the WDW equation does not contain time derivatives, but it
does not tell about the absence of time evolution, because the WDW equation tells nothing at all without scalar product. Evolution arises
from a time-dependent plane in the Klein-Gordon-type scalar product [37,41]. We should note that the WDW equation is not a theme of the present paper.

From my point of view, the best way to reintroduce time is to
use the Schultz formalism, i.e., by the introduction of a linear
radiation-like fluid, which can be chosen as time variable. The
model in discussion does not consider a fluid as its matter
content but, rather, considers a scalar field. Hence the latter
should be considered as the new "time", and it does not simply
"complement" the equation with a temporal derivative term.

Quantization actually depends on the choice of a clock.
Radiation-like fluid is one of the possible clocks. Your advice to
use the scalar field as a clock is reasonable for this simple model,
but it could be not convenient in a more general case when the scalar
field begins to oscillate and a time associated with the scalar
field begins to flow backward. We prefer to use some abstract time, which 
is not attached to some concrete fluid.

In the present version of the paper, we added an example of a
particle-clock in the appendix to emphasize, that  even for such
simple system the different interpretations are possible.

 Another point is that, in the model, there is a factor
1/a² multiplying the derivative in phi.  I understand that this
term serves as the weight function of the inner product, thus
ensuring the orthogonality of the eigenfunctions, and also
ensuring that the Hamiltonian operator be self-adjoint.

The minisuperspace model is formally analogous to a point particle in
curved space [45]. If the Laplacian operator ordering is chosen, a measure
√g has to appear in the scalar product.

Finally, and of lesser importance, it seems to me that the
wave packet (19), with C(k) given by (21), is not normalized.

Thank you for this note. Actually, σ must be in 5/2 degree.

 

 

Reviewer 4 Report

Comments and Suggestions for Authors

In the revised version, the authors have addressed some of my concerns, especially those related to the grammar style.

I appreciate the effort of the authors in reviewing the manuscript along the lines sketched in my previous review, but I believe that there are still some points that remain unsolved:

1) The physical meaning of several equations is still unclear; following my suggestion, the authors have added a footnote on p.3, with the aim to explain the mean value of a^2. In my opinion this in not enough and does not address the query raised in my review. 

2) The authors have included a detailed discussion (probably aimed at addressing part of my review)  regarding the physical Hamiltonian. However, my question was mainly focused on the reason why applying the Schroedinger equation, rather to the physical Hamiltonian.

I believe that the authors should address these two points before the manuscript can be accepted for publication.

Comments on the Quality of English Language

.

Author Response

We are grateful to the reviewer for the valuable remarks. Citations from the report are marked by italic tape.

1) The physical meaning of several equations is still
unclear; following my suggestion, the authors have added a
footnote on p.3, with the aim to explain the mean value of $a^2$.
In my opinion, this is not enough and does not address the query
raised in my review.

In the present version, a ``no-boundary'' proposal is also mentioned.

2) The authors have included a detailed discussion (probably
aimed at addressing part of my review)  regarding the physical
Hamiltonian. However, my question was mainly focused on the reason
why applying the Schroedinger equation, rather to the physical
Hamiltonian.

The principles of obtaining physical Hamiltonian and Hamiltonian
with the Grassmann variables are similar: obtaining the <in|out>
functional in the form without any pre-exponential factors and
using H(p,q) in the Schroedinger equation. Some argumentation in
favor of the Schroedinger equation could be found in [43]. In the
present version of the paper, we have discussed an example of the
"particle-clock" for completeness.

Round 3

Reviewer 4 Report

Comments and Suggestions for Authors

The authors have addressed the concerns pointed out in my previous review, then I recomment the manuscript for publication in the present form.