Note on the problem of time in quantum gravity

D. G. Chakalov
35 Sutherland Street
London SW1V 4JU, UK


An informal discussion paper addressed to the community of theoretical physicists, in which a number of striking similarities between the problem of psychological time arrow and the problem of time in quantum gravity have been underlined. Hence the task of this paper is to clarify whether we could bridge the two issues, and how. Ensuing from the solution to the first problem (Ulric Neisser's cognitive cycle), an effort has been made to adjust the problem of time in quantum gravity to a particular situation that could resemble the one with the human brain and its psychological time arrow. Should a discussion prompted by this paper confirm the custom-made 'diagnosis' of the problem of time, the author would be greatly encouraged to make the next step of suggesting concrete ideas potentially applicable for a background-free quantum gravity endowed with some 'universal time arrow' matching the psychological time arrow.

Revision date: August 15, 1999

1. Introduction

It has been known since the first days of quantum mechanics (QM) that what we call 'time', as measured by clocks, is an external parameter that cannot in principle fit into QM formalism -- there are no time operators in QM. A consistent physical interpretation of QM formalism can be given only if we divide the physical world into two parts: the system under study, represented by vectors and operators in a Hilbert space, and the observer placed in the rest of the world, for which a classical description must be used [Peres, 1977]. We must impose this 'cut' (Schnitt) dividing the world into two parts, or else QM will be simply meaningless [von Neumann, 1955].

However, there is no such 'cut' in the case of canonical quantum gravity, for there is no physical reality external to the Universe, and perhaps the first striking result is the infamous problem of time in Wheeler-DeWitt equation -- there is no external global time parameter, and our time is "frozen" [Isham, 1993]. We in fact lose both the time and the space, due to the full reparametrization invariance of general relativity [Parentani, 1997]. We know from special relativity theory that can lose the spacetime only if we travel with the speed of light [Tegmark, 1997], so it is natural to ask what, if any, would be the link between those two mysteries, and why the Nature has chosen to supply "our" world with an "unphysical" copy that we label with 'tachyonic world' [Tegmark, 1997].

Many efforts have been made to address the problem of time in quantum gravity (e.g., by trying to 'isolate time', cf. [Ashtekar 1992], particularly in quantum cosmology, by prescribing some custom-made initial conditions such that the evolution of the Universe would trace some 'cosmological time arrow'. There are two very popular suggestions known as quantum tunneling proposal (A. Vilenkin) and Hartle-Hawking no-boundary proposal, but in both cases the problem with the alleged cosmological time arrow is that it is born "along with the Universe" -- if we run the cosmological time backwards, we shall inevitably hit some totally non-physical state. This problem was best illustrated by Yakov Zel'dovich: "There was a time during which there was still no time at all" (Y. Zel'dovich, private communication, 1986). I think there is a genuine paradox in this born-along-with-the-Universe approach to the cosmological time arrow: should the Universe has ever been in that totally non-physical "initial" state, there would be no physical reason whatsoever to get out of it. To illustrate this, imagine some super-powerful vacuum cleaner that could suck absolutely every physical stuff. Run it, and at some critical point, the vacuum cleaner would suck itself and would simply disappear, being sucked by itself. There would be neither time nor space, not even Wheeler-DeWitt equation to show us the problem of time in quantum gravity.

Perhaps it's a matter of taste, but I don't like such a backdoor in the Universe, through which some "Great Supervisor" could easily operate with that "initial" non-physical state, making a miraculous tune-up of the "future" fundamental constants (the anthropic principle) and providing some very special 'quintessence' needed to explain both an accelerated expansion of the Universe [Cohn (1998); Carroll (1998); Rosenberg (1999); Huterer and Turner (1998); Zlatev et al., (1999)] and extremely powerful gamma-ray bursts [NASA Space Science News. May 6, 1998; BATSE Gamma-Ray Burst Sky Map].

Apparently, we in fact have a very clear notion of what we call psychological time arrow: we remember the past, not the future, and anticipate the future, not the past. We feel the time flying from the past toward the future. There is a simple and beautiful model of psychological time arrow, suggested by Ulric Neisser in 1976 [Neisser, 1976], and it would be interesting to see if we could apply it to the problem of time in quantum gravity.

The plan of the paper is as follows. In Section 2 I will try to demonstrate that the human brain does not have any 'background', first, and secondly -- that our brain does act on itself, for we think about that with which we think, by that with which we think. In Section 3, a custom-made 'diagnosis' of the problem of time will be suggested, such that it could resemble the case of the human brain and its psychological time arrow. Finally, in Section 4 I will offer an outline to the ideas presented here, along with a simple recipe for how one could reject everything written in this paper: there is a well-known problem of time in Einstein's general relativity (GR) -- no preferred timelike direction [cf. Note 1]. I will consider this paper totally unneeded iff someone could suggest an idea about how we could solve this very old problem. It seems painfully clear to me that without such a 'common denominator' QM and GR can never be reconciled. I don't know, for instance, if Ashtekar variables for classical and quantum gravity, as suggested in 1986, could give us a hint for a preferred timelike direction, to justify the efforts for building quantum gravity [Ashtekar, 1992]. If there is such a possibility, in Ashtekar program or in any other approach [e.g., Kitada, 1998 and Kitada and Fletcher, 1996], this paper would be violating Occam's razor and I will immediately delete it.

Should a discussion prompted by this paper confirm the custom-made interpretation of the cause (or 'diagnosis') for the problem of time, I would be greatly encouraged to make the next step of suggesting concrete ideas potentially applicable for a background-free quantum gravity endowed with some 'universal time arrow' matching the psychological time arrow. If we could describe that universal time arrow to cover all interactions, by a mechanism resembling Neisser's cognitive cycle [Neisser, 1976], then perhaps we could reach a classical limit (by letting a particular parameter to approach zero) at which classical GR and QM are applicable on a 'classical background' such as the continuous spacetime manifold. Perhaps this could lead to a brand new theory of quantum gravity [cf. Note 1], involving a timeless emergence of probability current.

To begin with, let's think of the Universe as a giant 'brain', and see if some similarities between our brain and the Universe could be found in today's quantum gravity. To resemble the human brain and its psychological time arrow, the Universe should possess some non-physical state (similar, but not identical to the unspeakable Noetic world) from which the probability current could emerge, and a cut dividing the Universe from that non-physical state. I will try to place this cut 'between' the elementary increments of time (chronons) along some universal time arrow resembling the psychological one, and will suggest a mechanism for the elementary step of the Universe along this universal time arrow, borrowed from Neisser's cognitive cycle. (The latter will be interpreted, in Section 3, as a timeless interaction of two non-physical universes, usually interpreted as material and tachyonic  [Tegmark, 1997], in the putative non-physical state 'between' the elementary increments of time.) Hence the assumed non-physical state of the whole Universe, which could provide us with a non-physical absolute reference frame, would be placed 'between' the two 'edges' of the elementary increment of time (chronon). Physically, we cannot measure the duration of the chronon, because what we call 'physical reality' is only at the two 'edges' of the chronon, but not "during" the chronon. The reason why we observe these increments of time post factum only, due to the so-called "speed" of light, is a great mystery to me. I believe that there should be a limit on the minimal time interval separating any two events, because of that mysterious "speed" of light. If so, this limit would be the 'duration' of the chronon. It should also mark the 'edge' of the physical reality, much like the the gravitational uncertainty principle [Adler and Santiago, 1999].

2. The Universe viewed as a human brain.

Talking about some non-physical state of the whole Universe, I do not imply some mental entity (like the human mind) but a superposition of all possible states of the Universe, placed in the potential future of some 'universal time arrow'. What makes this superposition of all possible states non-physical is the assumptions that they are to be connected via some non-trivial topology resembling the 'structure' of our unspeakable Noetic world, and also that, when the Universe chooses one and only one of these possible states by jumping from one 'edge' of the chronon to the other 'edge', the rest of the possible states do not disappear but remain stored for future steps in the 'memory' of the Universe. Pretty much like the human memory: when we see an object that we recognize as 'apple', this act does not eliminate all the other memory traces that we have for different shapes or colors of apples. They all remain in our memory. Let me try to explain.

Psychologically, a Platonic idea of an 'apple per se' is a superposition of infinite possible explications (not reductions) as "this apple, here-and-now". The explication of the Platonic idea does not crash/collapse neither the Platonic idea of 'apple per se' nor any other Platonic idea entangled with it. This is neither first nor second type process [von Neumann, 1955], but some third-type continuous process, some smooth, knot-free transition from the Platonic idea of an 'apple per se' to some concrete explication as "this apple, here and now", and back to the initial Platonic idea of an 'apple per se'. Any time we do this cognitive cycle, our Platonic idea of 'apple per se' is gaining information: yet another possible explication of 'apple per se' as concrete apple. Hence the possible explications of the Platonic idea become invariant under a new transformation embedding all memorized cases of already perceived apples with the new case of "this apple, here-and-now", regardless of their different shapes, size, etc. The Platonic idea does not collapse nor crash, it just becomes 'richer' and more abstract.

Mathematically, I think there is a great problem in modeling these possible explications of the Platonic idea: they simply do not form a set, being sort of 'opened' for new ones. They could be regarder as 'set of elements' only when we do not add a new element, so that the rest of the elements could be enumerated. However, the cognitice cycle (the so-called third-type continuous process, cf. above), creating the instants 'now' of the psychological time arrow, is constantly adding new elements to this, say, pseudo-set. If we time this truly creative process, we could do this only by timing the moments of adding new elements of the pseudo-set, while their 'common denominator', the Platonic idea itself, is definitely 'outside' its possible explications.

This is an information gain through a non-unitary transformation of the explications of the Platonic idea, before and after our cognitive cycle. It's an irreversible gain of information, stored in the Platonic idea. This is our psychological time arrow. Any time we do this, we enrich our Platonic idea of an 'apple per se' and simply got smarter (as far as apples are concerned).

Well, if we can do it, perhaps the Universe can do it. Maybe even better.

[To be continued]

References and notes

1. A. Peres (1997). Critique of the Wheeler-DeWitt equation.

2. A. Peres (1999). Classical interventions in quantum systems. I. The measuring process.

3. J. von Neumann (1955). The Mathematical Foundations of Quantum  Mechanics. Translated from the German edition by Robert T. Beyer. Princeton University Press: Princeton (NJ), pp. 417-445. 

John von Neumann: "And were our physiological knowledge more precise than it is today, we could go still further, tracing the chemical reactions which produce the impression of this image on the retina, in the optic nerve tract and in his brain, and then in the end say: these chemical changes of his brain cells are perceived by the observer" (p. 419).

Following this trend to its logical extreme, von Neumann wrote that in the final case the observing system is "his abstract ego" (p. 421).

4. C.J. Isham (1993). Prima Facie Questions in Quantum Gravity.

5. R. Parentani (1997). The notions of time and evolution in quantum cosmology.

6. J.D. Cohn (1998). Living with Lambda.

Joanne D. Cohn: "From the theoretical point of view, the unnatural (as currently understood) size of the cosmological constant Lambda currently suggested by the data gives rise to the question of whether the data supports some other form of energy density as well."

7. S.M. Carroll (1998). Quintessence and the Rest of the World.
Journal-ref: Phys. Rev. Lett. 81 (1998) 3067-3070.

8. D.E. Rosenberg (1999). Quintessence, Quantum Gravity and the Big Bang.

9. D. Huterer, M.S. Turner (1998). Revealing Quintessence.

10. I. Zlatev, L. Wang, P.J. Steinhardt (1999). Quintessence, Cosmic Coincidence, and the Cosmological Constant.
Journal-ref: Phys. Rev. Lett. 82 (1999) 896-899.

11. NASA Space Science News. Blasts From the Past. May 6, 1998.

12. Today in Space: The BATSE Gamma-Ray Burst Sky Map.

13. U. Neisser (1976). Cognition and Reality. Principles and Implications of Cognitive Psychology. Freeman: San Francisco, Fig. 2 and Chs. 2 and 4.

Note 1. Let me quote  [J. Butterfield and C.J. Isham, 1999], section Time in General Relativity:

J. Butterfield and C.J. Isham: "When we turn to classical general relativity, the treatment of time is very different. Time is not treated as a background parameter, even in the liberal sense used in special relativity, viz. as an aspect of a fixed, background spacetime structure. Rather, what counts as a choice of a time (i.e. of a timelike direction) is influenced by what matter is present; (as is, of course, the spatial metrical structure). The existence of many such times is reflected in the fact that if the spacetime manifold has a topology that enables it to be foliated as a one-parameter family of spacelike surfaces, this can generally be done in many ways -- without any subset of foliations being singled out in the way families of inertial reference frames are singled out in special relativity. From one perspective, each such parameter might be regarded as a legitimate definition of (global) time. However, in general, there is no way of selecting a particular foliation, or a special family of such, that is 'natural' within the context of the theory alone. In particular, these definitions of time are in general unphysical, in that they provide no hint as to how their time might be measured or registered."

And another excerpt:

"4. Start ab initio with a radically new theory.
The idea here is that both classical general relativity and standard quantum theory emerge from a theory that looks very different from both. Such a theory would indeed be radically new. For recall that we classified as examples of the second type of approach above, quantisations of spatial or spatiotemporal structure other than the metric; for example, quantisations of topology or causal structure. So the kind of theory envisaged here would somehow be still more radical than that; presumably by not being a quantum theory, even in a broad sense -- for example, in the sense of states giving amplitudes to the values of quantities, whose norms squared give probabilities."

"Of course, very little is known about potential schemes of this type; let alone whether it is necessary to adopt such an iconoclastic position in order to solve the problem of quantum gravity."

14. C.J. Isham and J. Butterfield (1999). On the Emergence of Time in Quantum Gravity.

C.J. Isham and J. Butterfield: "The difficulty of finding a buried time in the Wheeler-DeWitt equation (and the related difficulty of finding an 'internal time' before quantisation) prompts the idea that geometrodynamics, and perhaps quantum theory in general, can -- or even should -- be understood in an essentially 'timeless' way.

"Suppose we are given a solution to the Wheeler-DeWitt: a functional Psi of 3-geometries. How should it be interpreted?

"This means, incidentally, that in seeking a time within Psi (in accordance with the 'time after quantization' strategy), one is in fact thrown back to the same kind of technical and conceptual problems that beset the strategy of seeking a time before quantizing. In short, we are back in murky waters!"

15. J. Butterfield and C.J. Isham (1999). Spacetime and the Philosophical Challenge of Quantum Gravity.

J. Butterfield and C.J. Isham: "Though this construal of 'quantum gravity' is broad, we take it to exclude studies of a quantum field propagating in a spacetime manifold equipped with a fixed background Lorentzian metric.
That is to say, 'quantum gravity' must involve in some way a quantum interaction of the gravitational field with itself."

16. J.C. Baez (1999). Higher-Dimensional Algebra and Planck-Scale Physics.

17. R.H.A. Farias, E. Recami. Introduction of a Quantum of Time (chronon), and its Consequences for Quantum Mechanics.

R.H.A. Farias and E. Recami: "There are three equations, -- retarded, symmetric, and advanced Schrödinger equations, all of them transforming into the (same) continuous equation when the fundamental interval of time (that now can be called just tau) goes to zero."

"The introduction of a fundamental interval of time in the description of the measurement problem makes possible a simple but effective formalization of the state reduction process. Such behaviour is only observed for the retarded case."

"We could consider the system being described simultaneously by the three equations. However, QM works with averages over ensembles what is a description of a purely mathematical reality. The question is that, if we accept the ergodic hypothesis, such averages over ensembles are equivalent to averages over time."

18. Max Tegmark. On the dimensionality of spacetime.
Journal-ref: Classical and Quantum Gravity 14 (1997) L69-L75.

M. Tegmark, Footnote 4: "The only remaining possibility is the rather contrived case where data is specified on a null hypersurface. To measure such data, an observer would need to "live on the light cone", i.e., travel with the speed of light, which means that it would subjectively not perceive any time at all (its proper time would stand still)."

19. Ronald J. Adler, David I. Santiago. On Gravity and the Uncertainty
Journal-ref: Mod. Phys. Lett. A14 (1999) 1371.

R.J. Adler and D.I. Santiago: "We have shown, using Newtonian and general relativistic gravity, that the position-momentum uncertainty principle of quantum mechanics is modified by an additional term. In both theories it is clear that the extra term must be proportional to the energy or momentum of the photon, so on purely dimensional grounds the order of magnitude of the extra term is uniquely determined. As a consequence there is an absolute minimum uncertainty in the position of any particle such as an electron. Not surprisingly the minimum is of order of the Planck distance."

20. Abhay Ashtekar. Mathematical Problems of Non-perturbative Quantum General Relativity.

A. Ashtekar: "At a fundamental level, since there is no background metric, there is no a priori notion of time either. What does dynamics and evolution even mean if there is no background space-time? How is time born in the framework? Is it only an approximate concept or is it exact? If it is approximate, is the notion of unitarity also approximate? Why then do we not perceive any violations? These questions naturally lead one to the issue of measurement theory. Constructing a mathematical framework is only half of the story. We need a suitable framework to discuss the issues of interpretation as well. In absence of a background space-time geometry, the Copenhagen interpretation is of very little use. What should replace it?
The probabilities for an exhaustive set of mutually exclusive alternatives should add up to one. In quantum mechanics, this is generally ensured by using an instant of time to specify such alternatives. What is one to do when there is no time and no instants? These are fascinating issues."

"As emphasized earlier, however, there is no background space-time and therefore, in particular, no a priori notion of time. How can one speak of dynamics and time-evolution then? The idea is that a suitable component of the argument q_{ab} of the wave function is to play the role of time and the Wheeler-DeWitt equation is to tell us how the wave function evolves with respect to that time. The counting goes as follows. q_{ab} has six components. Roughly speaking the condition of (3-dimensional) diffeomorphism invariance tells us that a physical state can depend on only 3 of the 6 components of q_{ab}. Two of these are the true, dynamical degrees while the third represents "time". The Wheeler-DeWitt equation is thus to tell us how the dependence of the wave function on the true degrees of freedom changes as the variable representing time increases, and these changes are to be interpreted as time evolution. Thus, time, in spite of its name, is to be an "internal" variable, not an external clock. Until we isolate time, the Wheeler-DeWitt equation is just a constraint on the allowable wave functions. Once time is isolated, the same equation can be interpreted as providing evolution; dynamics is then born."

21. Hitoshi Kitada (1998). The Problem of Time.

22. Hitoshi Kitada, Lancelot R. Fletcher (1996). What is and What should be Time?

H. Kitada, L.R. Fletcher: "What the observer can see is the local classical (general) relativistic phenomenon. On the contrary, the total universe is quantum mechanical, and its quantum mechanical nature cannot be seen directly by any observer."

"Our basic assumption as an alternative for Newton's absolute time is that the total, entire universe has no time. Namely, contrary to Newton's absolute time that flows equably throughout the entire universe, we assume that the total universe is static and stationary. More specifically, the total universe is assumed to be a quantum mechanical bound state, i.e. an eigenstate of a Hamiltonian, denoted H, of infinite degrees of freedom, in a certain sense (see axiom 1 in Kitada 1994a, 1994b). (We should state that our notion of the eigenstate of H with infinite degrees of freedom is a rather weak one than the usual meaning of eigenstate.)"

"If we try to be rigorous in philosophical sense, we should remark that the  non-existence of time in our context should be interpreted as the "eternity" in Spinoza's sense (Spinoza, The Ethics, in Descartes, Spinoza, and Leibniz 1960)  rather than as the conventional meaning of the eternal continuance that lasts forever without a beginning or end."

"Our axiom 1 which asserts that the total universe, which will be denoted  [phi], is an eigenstate of a total Hamiltonian H, means that the universe  [phi] is an eternal truth, which cannot be explained in terms of continuance or time. In fact, being an eigenstate contains no notion of time as seen from its definition:  [H\phi =\lambda \phi] for some real number [lambda]. The reader might think that this definition just states that the entire universe [phi] is freezing at an instant which lasts forever without a beginning or end. However, as we will see, the total universe [phi] has an infinite degrees of freedom inside itself, as internal motion of finite and local systems, and never freezes. Therefore, as an existence itself, the universe [phi] does not change, however, at the same time, it is not freezing internally. These two seemingly contradicting aspects of the universe [phi] are possible by the quantum mechanical nature of the definition of eigenstates."

"Thus the universe itself does not change. However, inside itself, the universe can vary quantum mechanically, in any local region or in any local system consisting of a finite number of (quantum mechanical) particles. Therefore, we can define a local time in each local system as a measure or a clock of (quantum mechanical) motions in that local system."

August 15, 1999