A Universe that Does Not Know the Time

Round  1

Reviewer 1 Report

This manuscript presents a well-motivated exploration of Chern-Simons (CS) time as a quantum mechanical operator that does not commute with a Cosmological Constant. The corresponding Heisenberg Uncertainty Principle is shown to allow for both classical and quantum phases of a universe depending on whether a parameter akin to h-bar is small (classical phase) or large (quantum phase). In the quantum phase, the uncertainty in CS time and the uncertainty in the cosmological constant cannot both be small. The physical implications of large uncertainty in CS time are explored. The possibility of cycling classical and quantum phases is studied and argued to provide an explanation for the big bang and present state of our universe. The concept of the evolution of a universe without time and the emergence of classical time form within this phase is intriguing. Section 8 (Outlook) provides an excellent summary of the implications explored in the paper.

The paper is very readable. The arguments are tight and well-framed. The conditions under which the arguments apply are explicitly stated. References to prior related publications are sufficient. 

I support publication of this manuscript as is, except for a final review to catch some minor typos (see lines 189/190, 257, 259, 271). The content of this manuscript is well-suited for this special issue of Universe.

Author Response

We thank the referee for her/his kind comments, and for spotting the typos referred, which have now been corrected.

Reviewer 2 Report

This paper presents a discussion of internal time choices motivated by the Chern-Simons action appearing in Kodama's state. The authors emphasize the property that their time does not commute with some other observables, which leads to quantum uncertainty in the knowledge of time.  Unfortunately, several puzzling aspects of the authors' scenario have not been addressed in the present version of the paper:

1. At several places, the authors point out that the operators they consider are kinematical. Nevertheless, they call these operators "observables" even though it is unclear how one can measure a quantity in the kinematical Hilbert space that does not take into account physical constraints. Moreover, working at the kinematic level means that the proposal of time not commuting with other quantities is not as radical as the authors present it: Any parameterized model in standard quantum mechanics has a kinematical Hilbert space in which time and energy are canonically conjugate operators, and therefore don't commute. Nevertheless, after imposing the constraint E-H(q,p)=0, the system is equivalent to standard quantum mechanics with a Schroedinger equation acting on wave functions. In this situation, one would not claim that a quantum particle does not know the time, even though, formally, the setting is identical with that constructed by the authors. 

2. Several statements about canonical relationships are very confusing. At the end of section 2, the authors list several properties of the constructions yet to come. In point 2, they say that "Lambda is in fact the operator conjugate to T_{CS}," but then continue in point 3, saying that "their commutator is in fact proportional to l_{Pl}^2 Lambda." Conjugate operators have a constant commutator, which contradicts the last statement. These claims summarize the discussion on page 7. It is not clear why there can be two different choices of expressions conjugate to the same object, given by (21) and (26). It seems that the authors are arbitrarily choosing what should be conjugate to their time choice. They try to justify the conjugate variable by writing the Kodama state as a plane wave (16), from which they deduce a conjugate pair. However, conjugate relationships are not determined by states; they are a property of the algebra of operators acting on the Hilbert space, or of the classical analog given by the Poisson bracket. Referring to the corresponding classical theory, one would infer that the Poisson bracket of T_{CS}=(Ha)^3 [Eq. (7)] with any function of Lambda, such as (21) or (26), is zero because kinematically, Lambda is independent of the canonical pair (a^3,H). Therefore, none of the two choices (21) or (26) can be conjugate to T_{CS}.

3. Several further statements are very unclear. For instance, the significance of the claim that "the standard Big Bang Universe is going backwards" and then "rewinds," discussed after Eq. (7), is questionable because, as the authors note on the next page, one could just change the sign of their choice of time. In this context, the question of how their picture can be consistent with unitary evolution in quantum mechanics is very relevant, which is discussed nowhere in the paper. After Eq. (27), it is not clear what the authors mean by "the most general state in the Hilbert space" and I found it very difficult to follow the discussion of a Lambda-dependent Planck constant after Eq. (41). As a minor comment, the no boundary proposal on page 12 is due to Hartle and Hawking, not just Hawking.

In summary, I think that the paper needs a lot of work to become publishable. By holding their main discussions at the level of metaphors, the authors have allowed the scientific content to suffer considerably. I have not been able to find a new statement about the issue of time in cosmology that would not just repackage well-known standard properties of parameterized models in a new language. Perhaps, a clear presentation would reveal a novel idea.

Author Response

We believe most of the referee's comments stem from lexical misunderstandings which we hope to clarify below and in the revised manuscript.

1. Clearly this is purely a language problem. We do not use the term "kinematical" in the sense used by the referee (which is indeed the sense it has acquired in a very specific community), but in a broader sense that preceded it: that of a structure which is not endowed with a dynamics, a Hamiltonian,
or a time-evolution driven by a Hamiltonian. The point is that, since there may be no time in the conventional sense, we should also be prepared to eschew the concept of conventional dynamics, viz. *time* evolution resulting from a Hamiltonian, as well as the idea that the commutator of an operator with the Hamiltonian generates its propagation in time. Ditto with the idea that (quantum) commutators arise from a (classical) Poisson bracket structure, with the latter generating a classical evolution in time.

This does not mean that non-trivial commutators cannot be postulated with non-trivial effects. For one thing they imply representations (e.g. diagonalizing in one variable we know how to represent the operator for the complementary variable), from which we can set up eigenfunctions of complementary variables in terms of the other (and the Kodama state can be reinterpreted in this way). More importantly for the purpose of this paper, the Heisenberg uncertainty principle follows from the commutator via a purely ``kinematical'', argument, i.e. a calculation that nowhere requires a Hamiltonian dynamics (it only requires an inner product).

The referee is confused because in the canonical QG/LQG worlds "kinematical" is indeed a code word for "unphysical, because it does not commute with the quantum constraints." Therefore, by definition within that Dirac quantization framework, "observables" are never 'kinematical" because observers commute with the constraints. As explained, we do not use the term "kinematical" in this sense here. Defining observables
requires an inner product, but not a Hamiltonian, time evolution or a Hamiltonian dynamics, let alone a system of Hamiltonian constraints.

In order to clarify this matter we added extensive material making the above points when the term ``kinematical'' first appears in the paper, in the Introduction.

2. The source of confusion here arises directly from the issue of point . In our approach ``conjugate'' variables do not have their origin in a classical phase space and a Poisson bracket structure (see clarification added to the footnote at the Introduction). They are purely quantum operators, and their commutators are postulated kinematically. This does (kinematically) imply uncertainty relations and the form of eigenfunctions in terms of representations diagonalising the complementary variable (of which the Kodama state is an example). In this sense the functions E and T are choices associated with the postulates of the theory, not something forced upon us by any classical phase space.

As we say in the paper the Kodama state provides inspiration to our proposal, but is superseded by our construction. For the Kodamam state CS time and 1/Lambda are indeed the conjugates, and we have corrected that part of the ms (note that Section II is merely motivation for what is to come).

3. The sign of the CS time does not matter: the point is that its direction of flow is the opposite in the Big Bang phase and in whatever is added on to it to solve its horizon problem (such as acceleration). Indeed the existence of such a loop in CS time is a remarkable way to rephrase the horizon problem and its solution. We already discuss these matters very carefully in the opening paragraphs of Section 3. There, the discussion is at the classical level and therefore the issue of unitary evolution has no bearing upon it.

When later in the paper we do discuss quantum rewinding this is done in terms of transitions, given that there is
no timeline or dynamical evolution. The issue of unitarity is therefore irrelevant there too, because there is no time and so no time evolution.

Eq. 41 would be part of the postulates of the theory, such as the functions E and T (see above). One can always accuse a postulate of being ad hoc, but clearly any postulate is motivated by its consequences.

The "most general state in the Hilbert space" refers simply to the fact that some states will lead to Universes like ours (sharp Lambda, long rewind time), others not. In the absence of a measure of probability we have nothing to say other than what is already said in 199 and following and 255 and following.

Hartle is now mentioned in tandem with Hawking where appropriate.  

Reviewer 3 Report

The article proposes a new series of ideas on time in cosmology. They are interesting, though speculative, and therefore it would be good to get them published so the community can chime in. 

Author Response

We thank the referee for her/his kind comments.

Reviewer 4 Report

The paper discusses an interesting proposal about the change of the notion of cosmological time and their consequences on some related conceptual problems. I would like to add a few questions that could be used by the authors to improve the presentation:

1) It could be interesting to discuss the implications (if any) on the cosmological constant problem of the proposal to consider cosmological time and the cosmological constant as non-commuting quantum operators.    

2) The paper discusses the notion of cosmological time but nothing is said about space. Are the fluctuations to the homogeneity affected by the proposal of cosmological time ?

3) Is it possible to imagine some cosmological observations that could be used to contrast the solution to the horizon (and flatness) problem based on a transition from a quantum phase to a classical phase with the solution based on an inflationary primordial phase ?

4) A couple of typos: footnote 3 on pg4. from the (the) unavoidable; line 228 trigger could a combination ... --> trigger could be a combination ...

Author Response

We thank the referee for deep questions asked. Here are some replies.  

1 This is work in progress. In this paper all we can say is that there are many states for our Universe and that we appear to be in a squeezed state sharply peaked around Lambda=0, with a spread of the order of the measured value of Lambda. In turn this provides the delocalization in time capable of recycling the Universe on a time scale associated with the life of our Universe. Other states exist, but as we say after 199 and 255, it is enough for this state to exist with non-zero probability for an explanation of our Universe to follow. At this level, therefore, we do not solve the Lambda problem in any of its guises: we merely relate it to the age of our Universe and the necessary ebb and flow required to solve the horizon problem for something as large and as old as the world we live in.

We are, however, working on a further extension of our model which could go further than this.

2. Time and space are ontologically different in our scenario, but the assumption of homogeneity is not needed. Indeed the CS functional can still be evaluated for perturbed Universes, with a measure of time provided in this case too. Regarding the density fluctuations and tensor modes (which we presume is what the referee has in mind) we have regrettably nothing to add at this stage. It is a very important matter, no doubt, and we have barely started to scratch the surface.

3. Not really at the moment. The eschatology is spectacularly different and one might argue that is an experimental prediction, but not a very practical one. It serves to make the point that the scenario is physically different. On a more practical level the issue of inhomogeneities is probably our best bet for a test such as that suggested by the referee. This is coupled to point 2 above.

Our responses to points 1, 2, 3 have led to a new paragraph added to the very end of the paper, where our responses to the referee are briefly sumarised.

4. Typos corrected. Thanks!

Round  2

Reviewer 2 Report

The authors have made some corrections, but the main questions remain and should be addressed. On the positive side, some of the clarifications have given me the impression that there is indeed a new proposal behind the authors' constructions. However, I still find it difficult to reconcile some of their statements with one another and with established methods.

1. The authors now state explicitly that they work at a kinematical level because they are interested in hypothetical situations in which time and Hamiltonians may not exist. I am willing to accept this hypothesis, but the problem is that the authors still talk about concepts that are usually associated with time or Hamiltonians. For instance, what is the meaning of the Hubble parameter H if there is no time? And why does a conservation law such as (43) refer less to time than an equation of motion (such as the continuity equation it replaces)? The conserved quantity is, after all, conserved in time, such that time is still required to have a meaningful conservation law. At a more technical level, equation (43) is valid only in a temporal situation in which one has time-dependent trajectories, because the parameter C is constant only on a given trajectory (in time) but can take different values on different trajectories. Saying that C is constant, as far as I can see, is therefore meaningful only in a situation in which there is time, or something else that characterizes a flow along trajectories, which in turn are solutions of equations of motion. The authors might dismiss these comments as a "language problem." However, these issues indicate an underlying conceptual problem which, at the very least, makes it difficult for some readers (like myself) to follow the presentation.

In the same context, it is not clear how the authors reconcile their claim that they work at the kinematical level without Hamiltonians, but then mention that their variables are subject to constraints such as (24) or (42). How do they intend to implement the constraints at the quantum level, if not through a procedure analogous to the established method they seem to dismiss in footnote 3? It is true that there is no Hamiltonian in the cosmological models analyzed here, but there are Hamiltonian constraints such as (24) or (42) which imply, in the usual understanding, that staying at the kinematical level is not sufficient. Again, I am willing to accept the authors' hypothesis that there may be situations in which there is no time or Hamiltonians, but they should first reconcile their own discussion with this claim. 

2. Here is what I believe the authors have in mind in their discussion of different commutation relations: The classical Poisson bracket would indicate that their T and E in (41) should commute. It is common to seek a direct correspondence between classical Poisson brackets and quantum commutators in order to make sure that the resulting quantum theory has the correct classical limit. As the authors seem to observe, however, they are free to violate this correspondence if their intention is to discuss a situation in which classical physics is simply inadequate and the classical limit should not be taken. A non-zero commutator of T and E in such a situation is not inconsistent, but they still need to make sure that they can reach a classical limit once they get out of their timeless situation. They do so by making hbar dependent on the energy density in (41), such that the commutator indeed goes to zero at sufficiently small densities. The non-zero commutator at high densities could perhaps be interpreted as or motivated by some new kind of non-commutative geometry that refers to the cosmological constant and its related time instead of space-time coordinates. I think that all this would make sense, but some such arguments should be given in the paper. (It should also be made more clear what Omega_- and Omega_+ are after equation (41).) I do believe that there should be a more efficient way of arriving at this result than by continually modifying one's commutation relations from section to section.

3. As far as I can see, the authors have not addressed my concern that the same arguments could be applied to time and energy of a parameterized quantum system, in which case one would not claim that the particle does not know the time. Perhaps, a motivation via non-commutative geometry as mentioned above could help to distinguish the two cases of a particle and a universe from each other.

Author Response

We thank the referee once again, and provide replies and what we hope will be the final iteration on this review process (having already addressed a total of 4 referee reports).

Point 1.
We suggest the referee consults reference 29, where a distinction is made between the space of the "constants", and the "base" space where the latter act as parameters. It may happen that the first space is kinematical, in the sense we have defined, whilst the latter has a Hamiltonian (dependent on those parameters) and is therefore dynamical. This is indeed the set up of the various models in Ref 29, where the point is that non-trivial results follow if the kinematical parameters or constants do not commute. We may, or may not quantize the base space, so the issue of the quantization of the constraints on the base space is not central.

The situation here is slightly different because one of the "kinematical" quantities is itself a measure of time. This does not preclude the existence of other definitions of time, namely the `t' of the equations in the base space. As we say in Section 2 there may be different definitions and versions of time, valid in different contexts and circumstances. Here we use two definitions: the formal `t` which appears in the base space equations, and the CS time, which we claim survives a quantum phase of the universe, where delocalization in time may happen. During a classical phase, they are related. This is the meaning of equations 10 and 11. But it could be that the connection is lost and the two times have entirely different natures.

In the revised ms we make this point where the relevant issues appear (in footnote 3, and after Eqs.11, 24 and around eqs 42-43).

We stress that we only use the various concepts referred by the referee when the system has entered a classical phase. There is of course an issue about the transition between the classical and quantum phase in our sketchy model of Section VI, but as we warn the reader in the ms "The fact that the transition between the two is fuzzy should not bother us. After all, this model is all about losing and then regaining a conventional `time' line, defined in terms of a classical gravitational clock time."

We have excised the comment about first integrals, because we now understand this may be confusing. We have defined \Omega_\pm. They refer to the triggers defined in words in the previous paragraph.


Points 2 and 3
We understand the referee's rephrasing of our work, but that is not how we see it, nor do we feel it is appropriate in an opening paper to aspire to that level of refinement. We do not feel that non-commutative geometry is required for an explanation of our ideas. Also, the classical regime may indeed be brought about in a better way (as admitted in our paper), and it could be that this could be achieved using the referee's suggestions. We encourage the referee to write a paper explaining this rephrasing and developing these suggestions, but we do not think it suitable to do so in a paper sketching out an idea that can be interesting to others, as already confirmed by 3 other referees.

Our main motivation here was to illustrate how a quantum time could avoid the ubiquitous time-like tower of turtles into which nearly all the cosmological models fall. We believe we have made our point.

(Parenthetically we are also puzzled by the referee's "concern that the same arguments could be applied to time and energy of a parameterized quantum system, in which case one would not claim that the particle does not know the time."  But there is an important difference between a parameterized particle and the universe, which is that only the former has a non-dynamical, fixed time to consult, which is a time coordinate (in some inertial frame) on Minkowski spacetime. Again, we do not think this is relevant for the discussion in this paper, although it might be used as a motivation for the paper we encourage the referee to write.)

In addition to the above we have corrected a few typos and updated and complemented the references.

Round  3

Reviewer 2 Report

For the most part, the paper is now clear at least at a technical level. It is usually important to motivate all steps undertaken in the construction of models and in the derivation of results, and I think that the authors could still do a better job in this regard. Nevertheless, I understand that they prefer to preserve an exploratory flavor in their paper and to avoid pinning it down in a specific approach by providing explicit motivations.