Forward:
The following is a set of four essays written by me c1999-2000. They are true to the original form with the exception of hand written corrections having been applied. They are written as a one sided conversation, me trying to explain an idea I had in graduate school to my advisor or other interested party. The nature of the essays is highly speculative. While I know this work to be original it was definitely inspired by things I was reading at the time so any similarity to other work is merely evidence of that inspiration and not an intentional copy of said work.
In 1999-2000 I was invited by the humanities department at LTU, where I taught physics, to give a lecture on the nature of space and time from a physicist’s point of view. Naturally the discussion lead to questions about what quantum time would mean and what the philosophical implications of this would be. Not being one to put the cart before the horse I felt compelled to define, to the best of my ability, what time actually is. After all this was the point of the talk.
The original title of the group of essays is:
Everything that modern physicists know about the nature of time and causality is fundamentally classical
Essay 1:
Imagine watching water flow steadily down a stream or river. Under reasonably well-behaved conditions (not turbulent), one only observes smooth, steady, uniform motion in one direction. If a leaf falls in the water it is carried away with this steady flow. Now imagine that you can see the water moving at the micro level, that is microscopic but not yet quantum. Chances are that you would see small section of “water” (that substance which constitutes what we call water at the macro level) moving turbulently in all directions. The steady flow is really an average over a countless number of chaotic, turbulent trajectories that do not appear to have any preferred order (furthermore the “average” could lead to no flow). Now imagine the leaf. It sails comfortably, gently along with the flow never apparently affected, never aware that the chaotic, random motion of countless molecules is responsible for creating a steady flow. (The leaf is actually affected by the turbulent motion in the sense that this turbulent motion is responsible for the overall smoothness of the flow). The idea of a smooth continuum emerging from a discrete spectrum is a feature often associated with a quantum system or the passing from micro physics to macro physics.
Causality in the classical world is seen as a natural ordering of events. In the large-scale world there exists a certain intuitive causality and we are taught to use it to our advantage when solving problems. In fact it could be said that predicting the future outcome of a system based on initial data is where the real power of physics lies. However, it becomes difficult to interpret the meaning of quantum time when one does not have a sense of time against which to compare it. The main problem lies in the fact that we try, naturally, to use our intuition to afford an explanation for quantum phenomenon. This is still true even today in atomic physics and particle physics where quantum results and their “exotic properties” are interpreted against a classical intuitive reality. This is precisely what makes results “exotic” in quantum mechanics. In fact, what is truly exotic is that of all the real possibilities that quantum mechanics affords, systems seem to choose this particular classical reality as the preferred possibility and that our senses are designed in such a way that we are like the leaf floating down a stream unaware of what lies underneath.
Macro causality is a factual part of physics. It is however at best a classical part of physics. We are all familiar with scenarios like the following.
“There is a pot of water on the stove. Turn on the burner and in time the water will boil.”
Suggest that the water might boil without turning on the burner and you would immediately be accosted with heckles.
“That’s impossible, there has to be a cause, a reason for the water to boil.”
The thoughtful scientist (or any person) would not be so quick to make this criticism. For we all know that there could be many reasonable explanations, not statistical in nature, for finding a pot of water boiling even though the burner was never turned on (sunlight, lenses, etc may all be at work here). But no one would find themselves quick to defend the following claim.
“The boiling water caused the burner to turn on.”
Or better yet.
“In fact all that really needs to happen is that for every pot of water found boiling in history one must turn on a burner to account for the boiling. You see it is simply a matter of book keeping. Each pot of boiling water needs a burner. They do not however need to be temporally connected. If the books don’t balance just flick a switch and the universe will sigh a sigh of relief.”
While the notion of intuitive causality makes this claim seem absurd in the kitchen, I propose that it is far from absurd when applied to the micro scale of the space time continuum. In fact, it may be that the only rule time and space obey is one of simple book keeping.
But is it not already the case that the classical sense of causality was made to seem murky with the Copenhagen Interpretation of the Quantum Theory of Particle Dynamics? Not really, although we cannot track the exact trajectory (state) of a particle with arbitrary precision, it is not true that the relation between one particle’s motion and the interaction of that particle with other particles is non-causal. Causal structure is still very much alive and well in the quantum world. We always restrict the possible outcomes of consecutive measurements to those that are causally related. But how can we judge which events cause which other events to occur in a world to which we do not have direct access? In this case causality is governed by the simple division of events into two classes, those that are separated by a time like curve and those that are not. Those that are can communicate by the use of a photon, and could therefore likely be causally related while those that are not could never communicate, thus forbidding any causal relation, or any other relation for that matter, from existing.
When we speak of quantizing time [that is, you and I specifically. Quantizing time is not a consequence of quantum gravity and I will point out later] we are possibly speaking of two things that need to be separated before we proceed or we will find ourselves confused. Time has an apparent mathematical structure, i.e. that of a straight line. While events that occur in time have an apparent causal structure or a preferred sequencing in time (along this line). A question that naturally arises is whether time can have the structure of a line without events to define that structure and similarly whether events can have a preferred ordering without the concept of time having a particular mathematical structure (i.e. open set topology, as ordering points endows the set with a mathematical structure).
I do not believe that these issues can be dealt with independently of one another. Time, as we experience it, also has a measure. The duration of a cycle is used to define a unit of time, just as size only has meaning when two things are compared (my arm is shorter than my leg) time intervals only have meaning when two cycles are compared. In ordinary quantum mechanics as well as quantum gravity (without the ADM split) we see quantum effects applied to the measure of time. So far (as far as I can tell) quantum phenomena and its interpretation have never been applied to the concept of event sequencing. Two popular views have existed throughout history. One states that time is an eternal universal structure that passes or flows from “past” to “future” or in a “definite direction”. This eternal structure allows us to give meaning to concepts like motion, cycles and causality. The other view holds that time is defined by the existence of cycles, motion and preferred ordering of events, without cycles of motion there is no passing of time. One thing is clear, an object cannot causally affect another object without interaction. Without interaction there is no change, no causality to speak of and hence no passing of time.
Essay 2:
Can you imagine a time (no pun intended) when you were in a dark room, or had your eyes closed and were trying to fall asleep, so dark in fact that you could not make out any sense of depth or dimension at all? If not, close you eyes and ask yourself the following question as you stair into the backs of your eyelids.
Do I have the sense that I am seeing an infinite void or that there is no space or distance beyond the edge of my skin?
Can you tell the difference between these two situations? If you ever find yourself in a dark room, so dark that you cannot tell the difference between having your eyes closed or open take the opportunity to ponder the meaning of distance. If space exists independently of objects moving within it then concepts like distance and dimension should also exist. While this may make sense in the abstract world of mathematics, in a practical sense we as observers only acquire knowledge of distance and dimension through the observation and measurement of the relative positions of external bodies (bodies other than our own) and their shapes and sizes, and this information comes to us because of the electromagnetic field as Einstein pointed out in The Meaning of Relativity. Objective definitions of size require comparison to some standard. Yet it does seem that nature provides us with size scales, atomic, nuclear, terrestrial, astronomical etc.
But what about time? A scientist once said that “… time must be different from space because when I close my eyes I do not have a sense of distance but I have a sense of enduring”. Does this truly mean that time is different (different to our senses)? Or is it evidence that we are still aware of, or feeling, the electromagnetic forces within our bodies and therefore acquiring data on the cycles within our bodies? With this data we are observing time by comparing one cycle to another. Through the practice of meditation some people claim to be able to still the mind to the point where the sense that time is passing disappears. Is this an illusion or the true equivalent to closing our eyes to time? When experienced musicians perform a musical piece there is sometimes a feeling within the performers that time has not passed at all during the performance. This effect is not from nervousness but from a manipulation of “time cycles” by defining and keeping a steady beat (based on my own experience). In fact, in many cases where an individual experiences a sense of timelessness rhythm is involved. One could obviously protest any objective relation between these experiences to the passing of true time in the physics sense but it is important, I feel, to try to isolate and quantify how the human mind constructs a time flow from sensory input if we are to gain any depth of understanding on this topic. After all “true sense” in science comes from observation with our senses. During meditation there are two large-scale cycles that are always present, the beating of the human heart and breathing. The nerves in the body can feel both, and both can be manipulated by the mind. But of these two cycles only the breathing can be slowed to a point where it is difficult to “keep time” with in the musical sense. It is during these durations of time, when breathing is very slow, that the person meditating cannot seem to tell how long they have been in meditation (or so they claim).
In a laboratory setting we do not rely on our minds to define or track time. Instead, we compare events to one particular and reliable cycle. Our realization of the ordering of events and the liner flow of time emerge from this procedure. I believe there is no difference between this procedure and an individual feeling their own heart and breath.
Essay 3:
Now on to the “Quantum Nature of Time”, if there is any. Recall in the first essay the use of the term Macro Causality and the example presented there. We use this same type of reasoning in every area of physics. Quantum field theoretic models of relativistic interactions and all other types of systems are always restricted a priori to have many of the qualities that are seen in nature on the macro scale. This is the best we humans can do, and rarely do we feel compelled to explore other possibilities until the models fail to match experimental results. To begin we need to understand what actually is being quantized and how it is related to time (the time we think we know).
The general theory of relativity provides us with a theory of the gravitational force field that obeys the same laws of covariance that are obeyed by the electromagnetic field. The theory has been accepted as successful with respect to planetary phenomenon, has provided us with a model for the creation of the universe, and is now a standard tool in cosmology and astrophysics. However, with this success came a somewhat disturbing and still reluctant shift of paradigm. Space and time are seen to be on equal footing in relativity (even in the special theory) and the mixing of spatial and temporal measurements is a salient feature of the special and general theories. Furthermore, the mathematical language used to construct the general theory is differential geometry, the geometry of curved surfaces. One interprets the flux of gravitational force as the intrinsic curvature of the space-time continuum and a geometrical field called the metric tensor describes the equivalent of potential energy stored in the gravitational field. This metric tensor contains information needed to calculate the lengths of curves and the angle between intersecting curves in the neighborhood of the point of intersection. Except in exotic cases the metric contains all relevant information about the geometry of a surface. The paradigm shift comes when one realizes that we are not speaking of the surface of a material body or any similar thing embedded in a Euclidian space (like the Earth), but the space and time to which we have always associated the mathematical structure of four simple straight lines.
Even though the structure of space and time have been generalized and the description of a particle’s behavior is now due to the curvature of space-time, the two features that were originally present remain, the structure of a continuum (a property of straight lines and manifolds in general) and the “causal ordering” or sequencing of events (light cone structure, at least locally). There are however, solutions to Einstein’s equations that contain what are called closed time-like curves. The temporal equivalent to an equator on a globe, these weird creatures are not part of a quantum version of reality but a very classical part of gravity. Typically, these closed time curves (ideal for time machine construction) are argued out of existence because they require exotic matter and energy with properties that defy our intuition. This leaves us comfortably safe, living in a bent rubbery world with nothing special happening.
Now what are those quantities that we call space and time and how do they fit into the structure of general relativity? There are two things being discussed when one does differential geometry. First is the set of points that we call a space, a topological space to be precise. Then there is the choice of measure on that set, referred to as a metric. Once the measure has been chosen we are speaking of a metric space. Gravity is associated with the choice of metric structure. Different types of metrics lead to different gravitational force fields. The structure of space as a point set and the properties that space-time acquire due to its being a topological space are not at issue. The assumption that space-time be a topological space is crucial. One cannot have a metric space without first having a point set with a topological structure. So the properties of continuity and well ordering are fundamental to the structure of space-time and it would seem that nothing could change this (except a postulate).
When a system is quantized, in the modern sense, the fundamental degrees of freedom and their associated momentum are elevated to operators, which obey a postulated algebra. The formal definitions of fundamental degrees of freedom and associated momentum come from classical mechanics. Quantization occurs when the postulated algebra is imposed on these operators. From this formal structure all of the familiar concepts like uncertainty relation, wave function, occupation numbers etc emerge. In gravity (space-time geometry) the fundamental degree of freedom is the metric. So quantizing gravity means quantizing the metric field and dealing with things like “uncertainty in metric structure”. Since the metric defines measure we have quite naturally an uncertainty in distance and time measure. This does not change the fact that the underlying structure is smooth and well ordered. It simply means that an observer could not tell with infinite precision both the distance between two events in space-time and the direction of steepest decent (my interpretation of the momentum associated with the metric). Quantum gravity is a very tricky business and a self consistent non-trivial picture has eluded theorists for decades. But recently (past 12 years) progress has been made. This new picture (The Loop Space Representation) leads to such quantities as distance, area and volume operators, which have in many cases a discrete spectrum. This means that at the expectation value level certain properties of space-time that we normally think of as being smooth might appear choppy, or wrinkled at the quantum level, only to smooth out in the limit of large quantum numbers. Yet once again I stress that the apparent properties of these measurements does not change the fact that the original point set has a certain degree of smoothness. The underlying open set topological structure is not cracked, discrete or uncertain. Furthermore non-causal structure is intrinsically absent, by construction, in these models.
There is one last issue related to modern quantum gravity to be discussed. Due to a certain prejudice (or arrogance) that we humans have it is generally assumed that the “topological” structure of the space-time continuum is fixed before one even begins to apply to it the prescription of quantum mechanics. The particular structure, known as the ADM split, is described as follows.
“… a three dimensional space of any particular but fixed topology and a time like direction which is similar to the real number line”. Personally, I have never liked this.
[String and Membrane theorists are now (c1999-2000), in fact, anticipating the emergence of space-time structure as a particular result or solution of some as yet unknown equation supplied by an as yet unknown unified field theory. Although I do not like string theory per say I do agree with this expectation. I have always personally felt that things like the shape, structure and dimensionality of our space-time should never be assumed but rather “given to us” by the inevitable Theory of Everything.]
This assumed structure of space-time leads to some difficulties upon quantization. First of all the equations that describe the “space” part of space-time are completely time independent. The theory turns out to have a symmetry in is called diffeomorphism invariance. As a result of this symmetry it turns out that any choice of time scale is just as good as any other. Time is no longer part of the dynamic picture and once again finds its home on a very high horse. A second result is that the quantum equations for the state function have no time dependence and the quantum states are all annihilated by the Hamiltonian operator (the most important operator in quantum mechanics). This means that all states and Null states. Contrary to the suggestive name, not all Null states are trivial. However, if memory serves no one has been able to find a solution to this problem other than the trivial Null state (this would be the state that yields zero probability to find anything). These last comments are based on old information (c1994). Without the ADM split time measure would be quantized since the whole theory is no longer diffeomorphism invariant.
Essay 4:
In the previous essay some light was shed on exactly what is meant by a quantum measure of time. In particular, by definition, to quantize Einsteinian gravity means to quantize measure in the geometric sense. So, for example, calculation of time duration based on observations of the metric (gravitational potential energy) will have a quantum uncertainty to them. This does not in any way require or suggest an uncertainty in the ordering of events. Furthermore, many assumed restrictions (which are reasonable based on our sensory experience) have been placed on the system leading us along a particular path. The quote “God does not play dice” is attributed to Einstein. The response to that statement “don’t tell God what to do” belongs to either Heisenberg or Bohr, I forget. Einstein found the accepted interpretation of Quantum theory distasteful mainly because he thought that it ruined the deterministic framework of physics. I prefer to think of quantum mechanics as a completely deterministic theory. The difference between quantum theory and classical mechanics is that different things are being determined. Schrodinger’s equation allows us to predict, with absolute certainty, the wave function of a system at any time given the precise wave function at some other (usually taken to be earlier) time. This is determinism. Of course, the wave function represents “the probability amplitude to find a particle at a given position in the laboratory”. Oh well, who believes in particles anyway?
When I think of time, space, matter and causality at a fundamental level I cannot help but think that any a priori structure is too much. I think of “events” not as positions on a (or of a) pre-determined space-time but as tiles all mixed up in a bag. Of course don’t attach volume and size to the bag and the tiles, it’s just a metaphor. Events are like elements of a pure point set, a set with no structure (or minimal structure). You don’t even have the right to say any two points are near each other in the traditional sense. Now here is where probability comes in. Rather than saying that time and space have a fixed topology and that gravity (geometry) is a degree of freedom to which the quantum postulates should be applied I would say that topology (the choice of smooth structure on the point set) is the degree of freedom. There is a potential catch. In the Schrodinger representation of Quantum Mechanics a smooth structure for space-time is required (or assumed) to exist since this is the only way that classical observables can be defined. When it comes to “choosing topology”, or more appropriately to answering the question “with what probability will observation of the universe yield topology T?” one must truly roll the dice. It almost seems childishly simple to replace wave functions and differential equations with raw probabilities but in the end it seems like the right thing to do.
Now let’s say that the tiles in the bag have a label attached to them. With this label one can naturally associate an ordering to the set, of course the choice of order (<, >, =) is up to you. For this discussion let’s assume that we are going to create a one-dimensional line out of these points, say a time line, and that the only structure we impose is ordering. With every choice of ordering you get a natural causal structure, e.g. {a<b<c<d…} Choose another, {c<z<w<b<a…} and you get an entirely different causal structure. The same would apply to higher dimensional spaces. Interestingly enough it does not require any more points to make a space of 500 dimensions (or N dimensions, where N is an integer) as it does to make a space of 1 dimension. So dimensionality could also be seen as random in this game. All this can be summed up by saying that I prefer to imagine the points of the space-time continuum as particles trapped in a bag in a gaseous state (metaphorically speaking). This is how I see time and space from a quantum point of view, as an abstraction. It is not how I experience or observe time and space. In thermodynamics macroscopic concepts such as temperature are related to the random motion of particles, a microscopic concept. Smooth space-time is to the bag of tiles as temperature is to the gas.
The only issue left to address is perhaps the most important one. How do we as macroscopic entities “observe” causality and how we may “observe” this alleged randomness of causal structure or ordering? Observation is very important in science, which is why I decided to mention meditation in the second essay. We must explore every possible avenue on our quest to understand the nature of time and meditation, music etc seem to be the most readily available methods (as subjective as they may seem). Yet it seems that a strange trend has emerged since the birth of quantum theory and relativity. In dealing with Newtonian Mechanics or Electrodynamics we usually develop some intuition about the world based on experience before we are given the intellectual treatment of these phenomena. It is then quite easy to use knowledge of the outcome to guide us in our mathematical treatment of problems (teachers always tell their students “Ask yourself if the answer makes nay sense. Do you think it could happen?”). However, nature did not construct our senses to be refined enough to feel the effects of the atomic and nuclear worlds or the effects of high speed relative motion (near the speed of light). By building extensions of our senses we have been able to see both of these worlds but in many cases these extensions did not come until after the mathematics gave predictions of what would be seen. By understanding the mathematical patterns of nature we construct a very powerful oracle, which allows us to search for things that our senses cannot experience. Many of these predictions are about the existence or non-existence of composite objects and their properties and have nothing to do with the future or fate of a system. Quantum theory and relativity make predictions that match experiment, but the interpretation of what these theories imply about the world in which we live has been too difficult for most people to accept (even scientists). For this reason modern theorists have lived by the wisdom “let the mathematics be your intuition”. I remember being taught early on in undergraduate school that experimentalists and theorists were different animals, motivated by different appetites. For the experimentalist the search is in their hands, eyes and other senses whereas, for the theorist the search is in the mind and logical patterns implied by observations. But ultimately they are on the same quest. Compare the situation to an archeological quest. By the use of maps, historical documents and eye witness accounts one could predict where something is or the most likely place to find it. But this is not as satisfying as getting you hands on it. To some degree modern physics does not satisfy the urge to “get your hands on it”. Most of our observation comes from indirect evidence. We do something to a system, look at the effect, and see if it matches the predictions given by a model. The more matching you get the stronger your belief in the model as a representation of the truth. Just as we can feel mechanics and not nuclear physics we are designed to feel macro causality and not whatever micro-causality may be. Furthermore this macro causality that governs pots of water and cats in boxes would be some weighted average over all possible causal structures of the elementary constituents of the water and the cat. Therefore it would not be proper to ask for a justification for this view of causality based on our intuition. Instead we must proceed with our maps and documents in the hopes that we may find buried treasure. The good news is that topology does have a measurable effect on many systems in nature so one should be able to construct an experiment to verify predictions that are based on this view of space-time.
Afterward:
As I wrote these essays I had a clear vision of what I was trying to say but hadn’t the language to express these thoughts. Years later it is clearer to me what I was getting at. First of all is the idea that time and space are observable by humans. That is to say the very nature of these quantities as topological point sets is observable. If I have not convinced the reader of this then our world views diverge at this point. However, if I have convinced the reader that time and space are indeed observable then the next point of the essays is to try and make sense of how the postulates of quantum mechanics would be applied to these observables. Within these essays I offer an interpretation of quantum space-time. I am not pushing this as a paradigm but as a natural evolution and, I believe, a logical conclusion of applying the Copenhagen interpretation of quantum theory to space and time. If this conclusion is unsatisfactory, baring glaring philosophical inconsistencies, we may be compelled to abandon the Copenhagen interpretation of quantum theory. That is really the direction in which I was going at the time (pun intended). We are used to being fed a paradigm in which space and time are absolute (even in relativity) and everything we observe evolves in this space-time arena. If we take space and time as objects to be observed we have no choice but to follow the trail to it’s natural conclusion. If our space and time points are drawn at random from moment to moment solving partial differential equations to determine the evolution of a particle or wave places the cart before the horse. We have essentially pre-chosen an open set topology and imposed the laws of ordinary calculus upon our world. All human observations support this paradigm. But this is at best a classical approximation.
If you asked me 15 years ago what time and space are I would have confidently answered they are real number lines! That is to say I would have asserted that there is a perfect equivalence between time and space and the usual open set topology of the real numbers. I now assert that time and space are things we experience. In our attempt to describe these experiences to others we have naturally used the real numbers as a representation of our experience. My intention is to start a dialog along these lines and either uncover the true nature of space, time and causality or alter the Copenhagen interpretation of quantum mechanics.
copyright 2014 (c1999) David R Bergman
