How can the past and future be when the past no longer is and the future is not yet? As for the present, if it were always present and never moved on to become the past, it would not be time but eternity.

St. Augustine



White Paper
On the time parameter in the Schrödinger equation

D. Chakalov, IAM*
Wednesday, 22 October 2003
Last updated on Wednesday, March 10, 2004


Let me begin with a quotation:

In rebus mathematicis errores quan minimi non sunt contemnendi.

This quote is from Bishop George Berkeley. My Latin is not very good (especially on Wednesdays), but perhaps his statement can be translated in the following way:

"You may think that you know what you mean when you talk about the time parameter in the Schrödinger equation, but you're deadly wrong."

Or something similar. The point of Bishop Berkeley is that any errors in mathematics, no matter how small, can not be tolerated. This warning has been issued in the context of Newton's infinitesimal calculus, the central notion of which is the so-called point. Hence the notion of time emerges, as moving from one point to the nearest one. The mathematical apparatus does work flawlessly for calculation tasks, but if it includes an error, then we're in deep murky waters, or something similar.

Where could the error be? In eliminating the phenomenon of transition from one point to the nearest one, with speed that can not exceed the speed of light in vacuum. Why do we need the phenomenon of transition? Because in order to have a continuum of transitive points, there should be something that links them, the atom of Lucretius, which is a brand new (to mathematical physics) object. It is not present in differential and tensor calculus, as Bishop Berkeley would have probably said today. You pick up something and instruct it to run straight into infinity, hence obliterating the atom of Lucretius and the nature of time and space. Huge error. You can never discover what is the physical meaning of that unique upper bound on speed in spacetime, where it comes from, and why it has a numerically finite value.

Two examples follow, from the famous experts in mathematical physics Prof. Sir Roger Penrose and Prof. Chris Isham.

On Fri, 19 Apr 2002 13:18:05 +0100, Roger Penrose wrote:

"Ordinary unitary evolution is "local" simply because it is described by a partial differential equation (the Schrodinger equation), albeit in configuration space rather than in ordinary space.  By contrast, state reduction (an essential feature of EPR effects) is essentially non-local, as can be understood as the reduction in one part of an entangled state affecting the other part, which could refer to something distant."

Every engineer or astronomer using partial differential equations knows what is local. In one sentence: you have an object which moves along a trajectory comprised of infinitely many and infinitely small "points", such that you can calculate the instantaneous state (e.g., instantaneous velocity) of the object at the "points" at which that same object exists 'out there' when you don't look at it, thanks to which you can apply the rules of Special Relativity Theory (SRT). Example: the Sun.

That's local, really. The hypothetical "unitary evolution" of [dead cat & alive cat] isn't. None of these requirements for local hold for standard Quantum Mechanics (QM). Why? Because of the "state reduction". (More on it from Chris Isham below.)

Next question: Is the so-called state reduction "essentially non-local", as claimed by Roger Penrose? The necessary prerequisite for answering this question is to pinpoint the alleged state reduction somewhere on the light cone. That's how you begin to answer it. All possible options have been rigorously explored in the past seventy-three years, but we still don't know where is that "state reduction" located, hence cannot even approach this truly fundamental question. All we know is that (i) some 'peaceful co-existence' of QM and SRT does hold, and (ii) everything we observe has been already complied with the laws of SRT, being already placed in our past light cone. This is the realm of facts. But a definite understanding of the nature of the "state reduction" -- if any -- is still out of sight.

Hence we have to keep our mind open for all possibilities, one of which is to abandon the "obvious" alternative for the duration of the putative state reduction, either instantaneous or of finite duration, and to embrace the two options as complementary sides of the things in the quantum realm. The wavefunction does not refer to some object 'out there', but gives rather (on squaring its modulus) the density of probability. Probability of what, exactly? Not of some object being there-and-now, like the Sun, but of some propensities for being found there, provided its position is "prepared" and then "measured". Hence we have a legitimate case for abandoning the alternative 'either ... or' above. It would have been an alternative  iff  the propensities of the quantum state were living somewhere on the light cone.

"When you have eliminated the impossible, whatever remains, however improbable, must be the truth", says Arthur Conan Doyle. Fine, but how do we identify 'impossible'?

Firstly, if you examine the time read by a physical clock, it is impossible to talk about time spanned over more than one instantaneous state, which we model geometrically with a "point" (pretty drastic approximation, much like 'a spherical cow').

Think of time as an adjective, say, blue. Time is not an independent observable, just like the quality of a blue car being 'blue' cannot exist without a blue car. If the car disappears, its color goes away with it. (I will not delve here on the so-called information loss in the case of some hypothetical black hole, in which you can speculate about the fate of the 'blue' without the car, if the latter is being swollen by the near-by black monster.) "There is no such thing as an empty space, i.e., a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field", says John Stachel.

Besides, it is a great puzzle why we should ever consider time as observable, as noted by John Baez on March 11, 2000, since it is impossible to have a clock that would work perfectly, regardless of what its state is. In the example provided by John Baez, you may have a perfect clock only if you set  d/dt = 1  for the instantaneous state you're interested in. But then your clock will freeze, and there is nothing in modern mathematical physics that could give you a hint how this frozen clock would move to the nearest point along a trajectory (unless the clock at its instantaneous state has the faculty of self-acting). There is no room for the phenomenon of transience in present-day mathematical physics, as stressed by Abner Shimony. Quantum interactions "take place" on null surfaces, as explained by Kevin Brown, and "during" a special pseudo-time, as in Cramer's interpretation of QM. A physical clock will inevitably read the proper time of these interactions as zero.

Secondly, it is impossible to talk about classical time parameter if the latter refers simultaneously to more than one state of an object.

In the case of quantum physics, we should never employ the notion of classical time parameter, in any circumstances. We are in a completely different world of conditional propensities (plural) for one potentially observable quantum state (singular). If a car can exist in some superposed state(s) of all possible colors, then its classical time parameter -- which depends on the exact pinpointed color of the car here-and-now -- does not make any sense whatsoever. Thus, we eliminate the possibility of 'quantum time parameter' being fixed by one color only, and "whatever remains, however improbable, must be the truth", namely, that the time parameter in the Schrödinger equation is impossible to match the good old time parameter from Newtonian mechanics and SRT. It is a brand new case of things existing in a brand new kind of spacetime, such that the two options from classical mechanics -- things being either instantaneous or having finite duration -- should be considered complementary, not alternatives. This is because in the quantum realm the distinction between 'one' and 'many' can not be made with notions taken from classical physics, and if we want to understand the quantum world, the only way to proceed is to apply the general notion of complementarity. All we have to bear in mind is that the quantum world is strictly UNspeakable: the truth-values of statements about the quantum state cannot be set to 'either true or false'. If the car above was a quantum system, any statement about its "intrinsic" color will be 'both true and false'. It might sound like a headache, but this new kind of reality can be easily demonstrated with the human brain. Besides, it is not at all complicated (please see below). 

I have great difficulties in communicating with Roger Penrose and don't know what he is working on, but I believe it is highly unlikely that all this would be new to him, hence I'm deeply puzzled by his bold statement that the unitary "evolution" were [quote] local. Math is a language that we have to use with great caution, not like "simply because it is described by a partial differential equation (the Schrodinger equation)." Anyway, my first effort to contact Sir Roger was in 1988, by requesting a preprint from a paper on energy non-conservation in General Relativity (GR), then have sent him many letters and email, but the sole feedback received in the past sixteen years was the above-quoted email of Friday, 19 April 2002.

Let's move to the next story about the time parameter in the Schrödinger equation, as told by Prof. Chris Isham.

Eleven years ago, in September 1993, Chris Isham wrote a fascinating paper, entitled: Prima Facie Questions in Quantum Gravity, gr-qc/9310031. Regarding the parameter  t  in the time-dependent Schrödinger equation, Eq. 3 therein, he wrote (p. 14):

"The background Newtonian time appears explicitly in the time-dependent Schroedinger equation (3), but it is pertinent to note that such a time is truly an abstraction in the sense that no physical clock can provide a precise measure of it [UW89]: there is always a small probability that a real clock will sometimes run backwards with respect to Newtonian time."

I'm afraid the first part of this sentence is wrong: the background Newtonian time does not appear in the "time-dependent" Schrödinger equation, since this background Newtonian time, as read by a physical clock, pertains to a perfect continuum of facts, in perfect agreement with SRT.

There are many reason why the background Newtonian time is an abstraction, as elaborated above, although we can indeed employ this static, frozen, instantaneous (local) time to describe a trajectory, thanks to the global mode of time. The latter enables the transition from one point to the nearest one, hence creating a perfect continuum of facts in the local mode of time (I was very much eager to explain this issue in November 2002, but it didn't work out). All this is well-known and very easy to explain. But let me quote from the same page 14:

"One of the central requirements of the scalar product on the Hilbert space of states is that it is conserved under the time evolution (3). This is closely connected to the unitarity requirement that probabilities always sum to one."

Trouble is, the scalar product on the Hilbert space of states can be conserved only "over" one point from the Newtonian time, the same local time that can be read by an inanimate physical clock, hence the unitarity requirement does require the global time. I believe this is far more important reason to reject the background Newtonian time and trash it as "truly an abstraction".

The main reason is the reduction of the state-vector: it makes the whole task of explaining the dynamical transition from quantum to classical regime, and from classical to quantum, truly impossible. If you believe the math is complete and there is no need to bring up the global mode of time, nothing could bridge the gap between these two worlds. The state-vector reduction "process" is non-unitary, which is nothing but an absurd, plain and simple. Here's the detailed explanation by Chris Isham.

On Sat, 13 Mar 1999 19:50:40 +0000, Chris Isham wrote:

"The `non-unitary' refers to the fact (i) any genuine time-evolution in quantum theory that is determined by an actual physical process is implemented by a unitary operator (the exponential of itH where H is the associated Hamiltonian operator); (ii) the scalar product between a pair of vectors is maintained by a unitary transformation; and (iii) the transformation associated with state-vector reduction does *not* preserve scalar products. Hence the state-vector reduction process can not, even in principle, be construed as a normal dynamical process."

If the state-vector reduction does not preserve scalar products, then what could be the link of the quantum world to the macro-world of tables and chairs, in which we can easily calculate, say, the circumference of a circle? Again, the link goes only "through" one single point only. This is the "window" which an inanimate physical clock can use to take a glimpse at the quantum world.

Does that mean that there is no quantum world? Not at all. There is a quantum world, and there is a macro-world for which the Newtonian mechanics and SRT apply perfectly well. Only the math says that both worlds "exist" over one single time instant. This is what the math says, again. If it was a complete description of what we observe around us, there would be nothing but one single instant of time, an instantaneous snapshot of a frozen stuff, as noted by St. Augustine of Hippo (354-430 AD). He specifically wrote about two kinds of time, 'past' and 'to come' (Augustine's Confessions, Book 11, translated by E.B. Pusey):

11.14.17: "Those two times then, past and to come, how are they, seeing the past now is not, and that to come is not yet?"

This is the global mode of spacetime, which is needed for the very existence of the instant 'now' from the local mode of spacetime. Contemporary physics deals only with the local mode of spacetime. But it is the global mode of spacetime that binds the 'past that is not' with the future that is 'not yet'.

If we ignore this vital requirement for the instant 'now', by shrinking the global mode of spacetime into a mathematical point (cf. Bishop Berkeley above), the instant 'now' will become eternity and the time will stand still:

"But the present, should it always be present, and never pass into time past, verily it should not be time, but eternity." (Ibid.)

Briefly, if the treatment of time in present-day theoretical physics was correct, we would have to live, along with St. Augustine, in one single snapshot of eternity, some transcendental tachyon which is absolutely everywhere in no time -- instantaneously. No physical interaction would ever occur, simply because its proper "time" would be that of a photon. Zero. Zilch. In order to move in this block universe, you would need a second time parameter modeled with an axis orthogonal to the 4-D spacetime. But you can't introduce such extra-parameter in your 4-D spacetime, since it won't be gauge-independent, and therefore cannot qualify as an observable. You're buried in a single timeless instant from the global mode of spacetime taken en bloc, which is rooted on the so-called "cosmic singularity" where all known physical laws break down.

What could be more ridiculous? Consequently, can we crack this paradox in any way other than the solution proposed above? We simply introduce a brand new kind of spacetime, which pertains to the quantum world, and in which things are -- with respect to the time read by a physical clock -- both 'here' and 'there', and have both instantaneous and finite duration of existence. Only the global mode of spacetime is utterly UNspeakable, again. The "device" that apparently can "log in" to the quantum world is the human brain.

No inanimate physical clock or interferometer, such as LIGO, can connect to the quantum world, however. It can only "filter" one collapsed snapshot from it. If we want to map the quantum world to the classical one, but insist on bypassing the so-called collapse, then we would have to literally kill the quantum world. Why? Because the quantum observable that has to be mapped to the time read by an inanimate physical clock must be shrunk to one single value, just like we have one color for one car, in the example above. But it is physically impossible to shrink the position of an electron to a truly point-like state, and even if you use some magic to achieve this, its complementary observable will be totally unphysical, being stretched straight to infinity. No way. That is why we need the so-called collapse. But in order to understand it, we need new ideas about time and space. The collapse, by itself, does not make any sense. Why not?

Recall that there are no time operators in QM, and we have to use the notions of 'time' and 'probability' from classical mechanics. If we cast a dice, the concept of probability from classical mechanics works perfectly well, since in each and every instantaneous point from the trajectory of the dice, up to the final instant in which it lands on the table with some side up, all sides of the dice do exist 'out there'. The meaning of 'out there' is that, throughout the entire trajectory, all sides of the dice have a probability of one of being somewhere in the Universe at a given time. For all six possibilities corresponding to the sides of the dice, there are instants of time from their trajectories, in which they all exist with certainly, that is, with probability unity. Once again: all six sides exist simultaneously and independently in each and every instant from their trajectories, and in each and every instant the probability current is conserved, and each side at each and every instant is uniquely defined with probability of one of being in one point from its trajectory at a given instant of time. That's how things exist 'out there' in classical physics, in line with the local mode of time.

Under these fixed conditions, we can safely apply the notion of probability, in relative-frequency presentation, and make a strictly counterfactual (and utterly metaphysical) claim: if we had been able to make infinite (Sic!) series of independent trials, then, at the "point" at which the number of these trials does indeed reach infinity, there will be 1/6 probability for each side of the dice, and the unitarity requirement will be satisfied. We tacitly presuppose the notion of actual infinity, again. Just like the sum-over-infinite-number-of-infinitesimals which does produce a finite thing, say, the circumference of a circle, after Archimedes. Or a finite time interval of 2 min, as read by a clock.

 This is well known, of course, only here we somehow stop thinking and do not ask the next question: what kind of time could correspond to the "point" at which the number of trials does indeed reach infinity? Certainly not the parameter time from classical mechanics nor the parameter time in General Relativity. Here comes the global mode of time, as you might have already guessed.

Now, compare this situation with the case of relativistic "collapse". Everything said above applies only, and exclusively only, to the instants of observation. We cannot say anything about the 'gaps' between two observations. The "time parameter" in the Schrödinger equation can only relate one instant of measurement that has been already performed to another instant of measurement. It says nothing about the gap, since this gap cannot be read by any inanimate physical clock.

Hence a theory of Lorentz-invariant nonlocality and a relativistic "collapse" of an entangled EPR state are impossible with the local mode of time of the Special Relativity Theory (SRT). The latter is totally incompatible with the "instantaneous" annihilation of |dead cat> in the case of observing |alive cat> (or the other way around). This so-called collapse is being considered "instantaneous" because the state |alive cat> can be observed only and exclusively only post factum, in our past light cone. As a fact. This factual, from SRT, state has to "instantaneously" absorb the rest of the probability current left for |dead cat>, hence reaching certainly, i.e., probability unit. In what reference frame does this miracle "happen"? In all reference frames, like a ghost, instantaneously. For zero time.

I tried to suggest some basic ideas for ameliorating this paradoxical situation in my email of May 8, 2001. The norming of "probabilities" like '50% for |alive cat> + 50% for |dead cat>' makes sense only if we introduce the global mode of time, in which 'the cat per se' exists in its Holon state. The "duration" of this Holon state, if read by an inanimate physical clock, will be inevitably zero, since such clock can read only the local mode of time. Hence we need, again and again, the global mode of time. It pertains to a new kind of potential reality: things existing in their non-unitary Holon state(s), due to which they can cast either a state |alive cat> or a state |dead cat> at the local mode of time, each of these states with probability unit, in our past light cone, as a fact. Otherwise we have to delete either QM or SRT.

The price to pay for the 'peaceful co-existence' of QM and SRT is to "attach" the ultimate 'chooser' -- the rest of the whole universe in its Holon state -- to each and every instantaneous point from the trajectories modeled with the local, Newtonian time. At the scale of tables and chairs, the effect of this global mode of time is vanishing small, and so is the quantum wave of our tables and chairs. See Karel Kuchar for details, he has spent a lot of time discussing these issues with Chris Isham.

I too had the rare privilege of talking with Chris Isham on numerous issues related to this new (to contemporary theoretical physics) kind of reality. The need for it can be derived from his papers and textbooks. Regrettably, Prof. Chris Isham never explained to me what/where my errors are, firstly, and secondly -- decided to explore a very esoteric approach, without looking back at the fundamental tasks of reconciling QM with SRT and solving the paradox of the so-called collapse.

It seems to me that the great advantage of the new kind of reality proposed here is that it can be explained and understood very easy. All you need is a brain.

There is a well-known paradox of the dynamics of human brain: it does not run with discrete steps, jumping over some time gaps during which we should have a total blackout. This is what we should experience, if the Newtonian time were applicable to brain's dynamics.

Consider this. You have a continuous trajectory of brain's states, along one-dimensional timeline. Take a physical clock and start timing the incoming flow of stimuli. At the point zero, you mark the entering flow of stimuli in the brain, from all five channels. To begin with, you have some 80 msec of total blackout, during which the brain makes the preliminary binding of all stuff that has already came into it. Then you need another 250-300 msec of blackout, before the brain can assemble everything, check out your memory, expectations, and current chemical needs, and finally deliver his homework up to the level of consciousness, at which point you proudly say -- oh yes, that's a car. A blue one. Suppose you say this 500 msec after time zero. Fine, but then you have to experience a huge gap of total blackout, because the poor brain, as running along the Newtonian time, needs to prepare for the next binding of stimuli, interpretation, and final delivery at the level of consciousness.

The simple and elegant solution to this paradox was proposed by Ulric Neisser, many years ago. Everything said here about the nature of spacetime in the quantum world is just a translation of his ideas into the language of modern physics. The human brain and the quantum world utilize the same global-and-local spacetime, and a universal time arrow that matches the psychological time arrow. Only the solution requires a non-trivial topology: the topology of cognitive states space is both "closed" and "open", since the brain somehow runs simultaneously both along a straight line and along a closed circle. That's the cognitive cycle of Ulric Neisser, which I use to suggest the universal time arrow and the two modes of spacetime, global and local. What is the topology of the universe? Same solution.

Last but not least, the global mode of spacetime is a perfect candidate for the absolute reference frame needed to explain (i) the ubiquitous dark smooth stuff and (ii) the Cosmic Background Radiation (CBR), as explained by George Smoot. They are isotropic and their values remain invariant in all reference frames. Like some transcendental tachyon that can be absolutely everywhere for zero time, hence cannot move and cannot change, because there is no "next" place left to go to. The nature of this unique object is known since the time of Aristotle; recall his First Cause or Unmoved Mover.

If you stick to the current ideas, however, you have to use the format 'either ... or', just like Max Tegmark and Neil Cornish. Moreover, you can never understand the nature of continuum, gravity, and the 'chooser' in the quantum world: you've missed the atom, as explained by Lucretius some 2060 years ago. Last but not least, your brain would not work the way it does. It would have to be some parallel-computing device and will crash before your blink.

Finally, let me quote a famous statement known as Wilson's Law: If you want to make enemies, try to change something.

Neither of the two is the aim of this White Paper. I cannot change anything. The verdict of a highly respected theoretical physicist and leading expert in quantum gravity does not allow me, even if I wanted, to make enemies: "You do not know enough theoretical physics to help with any research in that area." Read about it here.

All I can do is to print my CD ROM "Physics of Human Intention" and set up an Institute for Applied Metaphysics, IAM. In PHI, we use the human brain as a black box. It works, even without a complete theory of quantum gravity. I always carry my Institute for Applied Metaphysics with me; it is right above my neck.

If you, my dear reader, want to join IAM, please don't hesitate. There is plenty of room for everyone. Just please bear in mind the warning of Bishop Berkeley. Perhaps you too can translate it in many different ways.

Acknowledgements: I am very grateful to Lucretius, St. Augustine, Bishop Berkeley, Ulric Neisser, and Henry Margenau for their invaluable support and leadership. God bless them.


*Institute for Applied Metaphysics, Box 13, Dragalevtsy, BG-1415 Sofia, Bulgaria; email


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