Understanding physical time

8/8/2019 Understanding physical time

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Understanding physical time

The formulation of a conceptual framework for understanding

physical time and temporal phenomena

Mathijs de Bruin

University of Amsterdam

July 25, 2010


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Thanks

While writing this thesis I had a lot of help and support from both my supervisor,Karel van der Leeuw, as well as from many of my friends who offered theirpatience and time by looking through my rough drafts as well as tolerating mysometimes antisocial behaviour during the time of writing. Special thanks goout to Wout Merbis, who lend me his skills as a theoretical phycisist to supportand critize my ideas. Further thanks go out to my girlfriend, Stephanie Samyn,who had to suffer my impatient desire to finish such a complex project in suchlimited time with sufficient delicacy.


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Abstract

This work is an attempt at conceptually understanding the way we conceive ofchange, as mathematically described in physical models of time. It will discusssome traditional conceptions of time as they are currently used in classical, rel-ativistic and quantum physics, and demonstrate severe inconsistencies betweenthese conceptions. As an attempt to solve (some of) these inconsistencies, a rad-ically new conception of time will be presented, in which change is presented asfundamentally probabilistic and nonlinear in nature. This novel model is shownto yield consequences for the directionality and scaling behaviour of temporalphenomena. It is also shown to pose fundamental limitations to our knowledgeof reality and this is suggested to lead to branching phenomena, similar to thosesuggested in some interpretations of quantum physics. Lastly a tentative con-

nection shall be made to some of the effects known from Einsteins theories ofrelativity.


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Contents

1 Introduction 2

2 Time in physics 3

2.1 The arrow of time dilemma . . . . . . . . . . . . . . . . . . . . . 3

2.2 Einsteins relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 The Many Worlds interpretation . . . . . . . . . . . . . . . . . . 6

3 An alternative view on time 7

3.1 Simple Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 State space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.3 Probabilistic timelines . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Consequences of a new model of time 11

4.1 Fundamental uncertainties . . . . . . . . . . . . . . . . . . . . . . 11

4.2 Situatedness and limits to observation . . . . . . . . . . . . . . . 12

4.3 Relativity of observation . . . . . . . . . . . . . . . . . . . . . . . 13

4.4 Directionality in state space . . . . . . . . . . . . . . . . . . . . . 14

4.5 Scaling behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Spatial extension 16

5.1 Spatial coordinates as properties in state space . . . . . . . . . . 16

5.2 Velocity in state space . . . . . . . . . . . . . . . . . . . . . . . . 16

5.3 Limits to velocity and time dilatation . . . . . . . . . . . . . . . 17

6 Conclusion 19

References 20

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1 Introduction

This thesis is to be seen as an attempt at conceptually understanding the waywe tend to conceive of change, as mathematically described in physical models oftime. In order to do so we will briefly discuss the traditional conceptions of timeas they are currently used in classical, relativistic and quantum physics. Basedin this we will demonstrate a severe inconsistency between the the conceptionof time in classical physics and that of quantum physics, by explaining thediscrepancies between them in concordance with the interpretative issues of thelatter.

After illustrating some of the issues with the traditional conceptions of time,we will provide a radically alternative conception of thereof. In order to limit thecomplexity of this thesis, initially only a limited model defined as Simple Time

will be considered, being essentially a notion of time without reference to spatialinformation. Our solution will suggest a fundamentally timeless perspective onthe universe, in which statistical ensembles of probabilistic (partial) timelinesdescribe the phenomena of change as we commonly experience them.

Consequently, the newly introduced model of time is shown to yield partic-ular consequences in for our conception of the world but also for the physicalreality we take part of. For one, the model implies a fundamental uncertaintywith regards to both our future as well as our past. This again is shownto yield fundamental limitations to the domain of reality which we can haveknowledge about.

Furthermore, the uncertainty within our model implies the loss of a naturalor pregiven direction of time. Rather, it will be suggested that a well-defined

directionality of time is an emergent phenomenon only at particular scales. Atvery small scales it is to be expected that temporal phenomenon will be perceivedas utterly random in nature while at extremely large scales a static universe isto be expected.

Lastly, preliminary suggestions are made regarding the spatial extensionof the presented theory, from which relativistic phenomena, as described byEinsteins theories, are expected to follow naturally.

At this point it is perhaps necessary to stress the tentative character of thisthesis. While the underlying physical picture might provide a strong basis forthe foundation or extension of our physical and metaphysical understanding ofreality, its main purpose is to inspire a broader conception of temporal phenom-ena at large.

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2 Time in physics

2.1 The arrow of time dilemma

In the modern Western world we have traditionally been conceiving of time asa linear phenomenon. From Newtons laws onwards, it has been a commonconception that time is best modelled as an ordered collection of infinitesimallyseparated points; a line.

The geometrical structure of the line has provided us with a model for thestructure of time; one wherein each moment has a uniquely determined past andonly one possible future. However, the direction of the movement along this linehas not been so clearly determined. While classical thinking often assumed thisdirection as what could be called an axiom, (modern) statistical mechanics, as

initiated by Ludwig Boltzmann, allows us to explain this as a natural striving ofany system for a large-scale or macroscopic state which has a maximum numberof corresponding small-scale or microscopic states.

This can perhaps best be explained by looking at a very simple classicalsystem, shown in Figure 1, for which the same statistics are applicable: a boxwith its bottom partly covered by marbles. As the box is (randomly) shaken,this system is, roughly equivalent to a flat slice of gas the marbles willexhibit random motion and will collide with one another elastically, much likethe (classical description of) atoms in a gas.

(a) All the marbles in onecorner of the box.

(b) Marbles spread outevenly over the box.

Figure 1: Distributions of marbles over the bottom of a box. Since there are manyways to spread out the individual marbles evenly and just a few ways of havingthem all in a particular corner, the likelihood of finding them spread out equally

when (randomly) shaking the box is enourmously higher.

When we start this experiment with all the marbles at one particular spotof the box we will, given sufficient time, always end up in a situation wherethe marbles are randomly spread out over the entire surface of the box. Thusthe system will show a natural preference for a particular state even thoughthe positions and velocities of individual marbles (the microstates) are merelydriven my random perturbations.

This is because there are many more ways of spreading marbles evenly overthe box, compared to the possible combinations available for spreading the sameamount of marbles in just a particular area. And since, in this case, we are

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merely looking at the amount of marbles within a particular area of the surface

(the macrostate, in which we disregard the position of individual marbles, whichwe regard as equivalent and exchangeable) we will conclude that the collectivesystem of the box with marbles has a natural tendency towards a homogeneousdistribution of the marbles over the boxs surface.

And this is, roughly, what we see happening in nature. It is not at allimpossible for a broken glass to spontaneously rematerialize in front of our eyes,we simply do not live long enough for it to actually happen. But on smallertemporal and spatial scales, phenomena like this do indeed appear to happen and do so without failing a single physical law.

This is because mathematically, all the laws of physics,1 from Newton, toquantum physics to general and special relativity, are symmetrical with regardsto the direction of time. Somewhere on the border between the macroscopic

and microscopic behaviour of systems, a directional preference seems to emergenaturally.

2.2 Time according to Einsteins relativity

At the beginning of the century physicists found out that, apparently, the speedof light is constant for all observers2.[1, 2] This is a very remarkable featureas our intuition would suggest that, as distinct observers move with differentspeeds relative to one another, they would observe a light speed equal to theone seen at the source with the velocity of the moving observer (relative to thesource) subtracted.

But this turns out not to be the case. In fact, the only proper way to describethe observations done from inertial (non-accelerating) frames of reference3 is togive each observer their own axes of space and time which can be translated intoone another using the Lorentz transformations. Because the Lorentz transfor-mations involve mixing up of space and time, the term spacetime was coinedto describe the geometrical structure, as introduced by Minkowski.

The consequences of this are very far fetching. For one: temporal and spa-tial ordering is not necessarily preserved in relativistic spacetime. Events thatare causally separated4 can be perceived to occur at a different order in time.Similarly, events that are causally connected can be perceived to occur on dif-ferent spatial locations depending on the inertial frame of an observer. Otherremarkable effects imply that moving timepieces move more slowly when viewedfrom an observer at rest, and that space is contracted in the direction of motion.

However, due to the complex geometrical structure involved in both generalas well as special relativity, combined with the limited scope and scale of this

1Although the second law of thermodynamics states that entropy can only increase in time,this depends entirely on the temporal and spatial scale relation between the microscopic andthe macroscopic states being considered.

2A peculiar role is played by the observer in relativity. In short, an observer is a trainedphysicist fully aware of the consequences and implications of (general) relativity.

3The limiting domain of special relativity is that of inertial frames. For a full description ofrelativistic phenomenon, Einstein introduced general relativity which is considerably harderto properly explain and understand. As the current work bears only a weak relation to evenspecial relativity we will refrain from a further reference to general relativity.

4Due to their distance and the speed of light as a postulated upper limit to velocity.

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thesis we shall refrain from giving a more thorough discussion of the issues

involved. In order to avoid this discussion, we will limit ourselves to what wedefine as Simple Time, which is basically what time would look like withoutany spatial dimensions. A more detailed definition of this concept will be givenin subsection 3.1.

2.3 Time in quantum physics

Quantum physics is a mathematical and conceptual framework that allows us tomodel the behaviour of nature at its smallest time, space and energy domains. [3]In classical mechanics we describe a system by functions that yield the exactposition, velocity and acceleration for a specific time which are (numericallyor exactly) solvable for specific initial conditions of that system. On smaller

scales it appears however that a radically different way of describing systems isrequired.

Instead of having separate functions describing the behaviour of (classically)separated particles, we can only describe a system as a whole and only approx-imate the behaviour of individual elements by assuming they are separated.5

Moreover, the relationship between our representation or description of a sys-tem and the actual physical state thereof is only one of probability; apparentlythe quantum formalism allows us just to describe a system in a statistical matter,resulting in inherent uncertainty with regards to our individual measurements.

As opposed to what many people have erroneously come to know aboutquantum mechanics, the time evolution as described by the Schrdinger equa-tion6 is perfectly deterministic, just as Newtons description of the universe. It

is therefore equally possible to represent the evolution of a quantum system as acollection of singular moments, ordered by time. However, the description of asingle quantum system can represent multiple, mutually excluding, measurablestates.

The aforementioned statement; the indeterministic relation between ourmodel of nature and the perception or measurement thereof, has perplexedscientists and philosophers from the very moment of its discovery. First of allit was incredibly hard for them to accept that there might be fundamental un-certainties in the way the universe works. Secondly, the particular role thatobservations or measurements play in this thinking aroused controversy as towhether they should be seen as fundamentally different from other physicalprocesses.[4]

Moreover, the resolution of these paradoxes was not helped by the discov-ery that the world, at quantum scale, cannot possibly be described as merelyanother classical system that is simply too small for us to measure and that wethus can only describe our knowledge about a system in a statistical matter. [5]Rather, it turns out that parts of quantum systems can have correlations calledentanglements that seem the violate the ordinary limits of both space and time.7

5We do this by assuming that they do not exchange any energy whatsoever.6The one formula governing the behaviour of any quantum system.7This is what Einstein called spooky action at a distance, and fiercely denied, for it would

mean that information, in this case, were transferred faster than the speed of light whichis an absolute maximum according to both general and special relativity.

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This discovery; that quantum physics is an inherently statistical theory that,

at best, describes a physics fundamentally different from all our classical theo-ries, has spawned the possibility for fundamentally diverging interpretations ofthe same mathematical formalism.[6] While these interpretations generally areconsistent with our observations (through measurements) of physical reality,they lead to fundamental differences in our understanding of reality.

2.4 The Many Worlds interpretation

One such interpretations, perhaps one of the most popular and the one specifi-cally discussed in this thesis, is the so-called Many Worlds Interpretation intro-duced by Everett and popularised by DeWitt.[7, 8] According to this interpre-tation, the current moment branches off to a distinctly existing but nonetheless

real alternate world for every possible distinguishable future situation. How-ever, a similar mechanism might as well occur the other way around, leadingto a collapse of worlds which would explain the particular workings of thequantum measurement.

Whereas the Many Worlds interpretation, by assuming the simultaneousand real existence of a plethora of alternate worlds, might provide a sufficientexplanation of part of the behaviour of quantum mechanical systems, it posesfundamental problems for the conception of time. This is so because in quantumphysics, as explained before, times arrow has no preferred direction. Yet, theMany Worlds Interpretation does indeed suggest a branching off of our timelinein a single temporal direction.

This discrepancy would force us to accept the bold assumption of a time ar-

row from within our interpretation of our physical model, just like we have donein classical physics before the introduction of Boltzmanns statistical mechanics.At the same time, in our modelling of the universe, we will always have to pickan artificial now from which our time tree branches, which will certainly neverbe the (implicitly) postulated real root of the time tree.

Furthermore, as in the Many Worlds interpretation the timeline continuouslybranches off for each and every possible future state of the universe, we wouldend up with an unforeseeable amount of future branches or worlds for everysingle moment in time. In a way, this consequence was contained in the last twoarguments stressing the problems of the Many Worlds interpretation: somehowwe would have to assume that there is a natural direction for the progressionof time, which has tremendous consequence for both our conception of the uni-

verse as well as the possibility to formally represent this conception (e.g. in amathematical, geometric or logical structure).

Moreover, this inherent time arrow is exactly what statistical physics haddone away with, for the classical world conception, showing that the impressionof time moving in a particular direction is merely a statistical phenomenon thatarises natural for systems above a certain scale, as explained in subsection 2.1.This leaves classical mechanics and the currently described Many Worlds in-terpretation irreconcilable, showing a paradox in our current conceptions oftemporal phenomena.

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3 An alternative view on time

As explained in section 2, the Many Worlds interpretation which is held valid bymany people in both physics and philosophy of science communities, solves thecollapse problem in the transition from quantum to macroscopic scales of theuniverse. However, it involves making assumptions that physicists and philoso-phers alike have historically been very happy to cut away using the belovedprecision of Ockhams razor: the explicit assumption (versus Boltzmanns natu-ral emergence) of an arrow of time in the interpretation of our physical universe.

The task set before us now is to find a description of time which unites thisnatural emergence of times arrow with the simultaneous reality of alternaterealities that helps us to interpret the world of quantum physics.

3.1 The assumption of Simple Time

In order for this thesis not to loose its explanatory power by assuming too largea scope we have chosen to attempt to explain only a very limited model of theworld, henceforward called Simple Time. Due to the usage of this simplifieddefinition, we can refrain from involving Einsteins special and general relativity,thus greatly enhancing the accessibility of the theory while presenting a clearpath for its expansion.

Simple time is what time would look like without any spatial dimensions.Thus the entire state description of the universe must be contained inside a singlepoint which can have any number of qualitative physical properties. From thisassumption it follows that time, regardless of the actual structure, will consist of

discrete (individually separate and distinct) moments we will thus forcefullyrule out the possibility of continuous change or flow in time. One furtherassumption we will make is that these qualitative physical properties can beapproximated quantitatively and that the numbers representing them form anordered collection, the significance of which shall be elucidated further on inthis thesis.

These assumptions, constituting the definition of Simple Time, can by nomeans be considered trivial nor should they be considered a complete descriptionof reality. Rather, they are purely pragmatic, instrumental assumptions carry-ing the sole purpose of making the current theory understandable and keepingthe scale and complexity within bounds. Nevertheless, the central philosophyof the presented theory might provide an interesting path towards a better un-

derstanding of reality, by providing a more powerful physical description of ouruniverse. It might well be possible to formulate the same or a similar theoryas the one exposed in the current thesis without making all or some of theseassumptions. This possibility shall be briefly discussed in section 5.

3.2 State space: time from a birds eye perspective

Absolute, true, and mathematical time, in and of itself and of its ownnature, without reference to anything external, flows uniformly andby another name is called duration. Relative, apparent, and common

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time is any sensible and external measure (precise or imprecise) of

duration by means of motion; such a measure - for example, an hour,a day, a month, a year - is commonly used instead of true time.

Sir Isaac Newton, Principia Mathematica [9]

Like in Newtons classical mechanics, we will uphold the existence of an absolutetime independent of anything external, which flows uniformly. And similarly tothe way physicists view a collection of consecutive moments as a line, we willview the collection of moments as a whole. Like in the Newtonian model, wewill make the deterministic assumption that, from this birds eye perspective,each and every possible state of the universe is real, existing and unchanging.

The fundamental difference lies in the fact that, unlike Newton, we will not

represent this birds eye perspective as a line. Rather, we will use what iscommonly called a phase space: a collection of all possible states of a system insuch a way that every degree of freedom (classically this is the tuple of spatialposition and impulse) or parameter of the system is represented as an axis of ahigher dimensional space. A schematic comparison of these two structures andthe Many World interpretations perspective is shown in Figure 2.

The single path connecting states of the universe in Newtonian timeline ismerely a subset of this phase space. It has been commonly interpreted by physi-cists and mathematicians alike as an exhaustive collection of possible states asystem can theoretically assume given any combination of influence the com-posing parts can have on one another. Given the actual mutual influences ofthese parts on one another we can thus derive the actual trajectory of a systemthrough phase space, which will be the Newtonian timeline. In classical dynam-ics, phase space is thus conceived of as a purely mathematical structure whichhas no interpretable meaning in our physical reality.

In this thesis we will make heavy use of this concept of phase space, butwith one very significantly differing assumption: we will attribute to this space asimilar absolute realism that Newton attributed to his timeline. To discriminatebetween the classical concept of phase space and our own we shall, from nowon, refer to it as state space.

State space, in short, is thus to be interpreted as the collection of all possibleand factually existing states of the universe. In this abstract space, all pointsare moments in Simple Time, which are ordered such that each quantifiableproperty of the point in Simple Time forms an additional dimension or axis in

state space.

3.3 Probabilistic timelines

Apart from attributing absolute reality to all points in state space (which isequivalent to saying that all possible states of the universe are factually existingstates), we will furthermore attribute a probability measure to all the pointsin state space. Hence, every possible state of the universe has a value associ-ated with it, defining the (relative) likelihood of its occurrence. This is shownin Figure 3, where opacities of the lines and dots of Figure 2(c) represent, re-spectively, the (relative) probabilities associated with paths and points. In this

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t t t

(a) Structure of lineartime.

t t t

(b) Many Worlds in-terpretation.

(c) Structure of statespace.

Figure 2: A schematic illustration demonstrating the geometrical structure of thediscussed time models. Figure 2(a) displays time as presented by Newton, whereeach moment has exactly one past and one future moment. Figure 2(b) exhibits themodel introduced by Everett, where a single root leads to multiple (equally valid)

future branches. Figure 2(c) shows the state space (in this case with two distinctproperties or degrees of freedom), in which any possible path between neighbouringpoints forms a valid timeline. For each additional degree of freedom an axis wouldbe added to this structure.

figure, darker dots lead to darker lines which signify a more likely transitionfrom one point in state space to another.

Similar to the single path (or: line segment) traced through classical phasespace, we will now consider a statistical ensemble, a large collection, of pathsthrough state space. As we have probabilities associated to all points in statespace, we can define a (relative) probability measure for the full path by takingthe product of the probability values for the points in state space connected by

the path.As also discussed in the caption of Figure 3, the absolute probability of

a particular path is given by dividing the relative probability by the sum ofthe relative probability of all equivalent paths, that is: paths with the samestarting and endpoint. The absolute probabilities are thus the only observableand empirically relevant elements of the presented theory.8

Given any particular point of reference in state space we can now considerthe ensemble of paths crossing it to define the likelihood for that point to bepart of any given path. Moreover, we can consider the subensemble of pathsstarting at a particular point to define the relative likelihood of possible futurestates and, similarly, take the subensemble of paths ending at a particular pointto define the possible pasts, relative to a particular point in state space.

Our knowledge about our current, past and future positions in state spacewill thus necessarily take on a probabilistic character. At each moment in timewe conceive of a plethora of possible pasts and, similarly, a multitude of possiblefutures.

A remarkable consequence of this concept of probabilistic timelines is thatthe probabilities of line segments decrease with their distance from a point ofreference. This essentially means two things: the uncertainty we mentioned in

8This leads to an invariant symmetry in the relative probability measure: we can multiplyall the probability values with the same constant value and still retain the same absoluteprobability values.

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Figure 3: Schematic display of state space displaying the (relative) probabilitiesof states through the opacity of the points. The opacities of the lines connecting thepoints represents the (relative) probability of traversing from one point to the nextand are given by the product of the probability of the two connected points. Theabsolute probability of a particular path is given by dividing the relative probabilityby the sum of the relative probability of all equivalent paths, that is: paths with

the same starting and endpoint.

the previous paragraph will increase as we progress further from our referencepoint and the odds of finding the equivalent of a full (mathematical) line,9

either closed or open, vanish.

Another remark worth making is that we find no reason for excluding therepetitious occurrence of points in state space within a single timeline. Thismeans that it might very well be that the exact same state of the universe willbe occurring again and again. However, as a point in state space, per definition,describes everything there is to know about a particular state in the universe,there is no way to discriminate (from within the universe) a repeated from an

original occurrence. Therefore, the repeated occurrence of the same point instate space remains utterly imperceiveable and thus emperically irrelevant.

Both of these properties and their profound implications shall be detailedupon in section 4.

9Mathematical lines have, per definition, no end and no beginning.

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4 Consequences of a new model of time

The model we have just described constitutes a possible model for characterizingtime as an absolute but inherently underdetermined phenomenon; there is noway to cognize the actual path we are taking through state space, because ofits probabilistic character. Nor can we observe or measure this absolute clockticking; there is no way to determine the speed with which we progress throughstate space.

The consequences of this are manifold. For one, it introduces a fundamentallimitation in our conception and perception of reality, as we become inherentlysecluded from an absolute perception of now, past and future. These aspectswill be described in subsection 4.1 and subsection 4.2.

Since we can only deduce physically relevant (measurable) properties using

the absolute probabilities, the determination of which requires a reference pointor path in state space, it follows that all our knowledge about reality within thismodel will be relative with regards to the observer acquiring this knowledge.This consequence will be discussed in subsection 4.3.

Furthermore, we are left with a fundamental uncertainty with regards to(our knowledge about) the direction of time. In other words: we do not know,of all possible timelines, which is the one actually being followed. Nor do weknow the specific direction a possible timeline is being followed in. This aspectshall be discussed in subsection 4.4.

If we extend our reasoning on the described probabilistic time model acrossseveral orders of magnitude in scale and take into consideration the underdeter-minicity of directionality and position in state space, we will show it to be verylikely for this time model to display an intricate scaling behaviour in state space.This means that, when looking at a probabilistic timeline on small scales wewould find a similar pattern as when looking at the same line at extremelylarge scales. Rather, the appearance of a definite and linear time direction isan emergent phenomenon only when we consider state space at intermediatescales. These features shall be discussed in subsection 4.5.

Moreover, due to the geometrically spatial nature of state space and the fun-damental parallelism assumed to exist in this theory, we would have to concludethat time as we understand it here is an inherent multidimensional phenomenon,as opposed to the classical Newtonian model of a line.10

4.1 Fundamental uncertainties

As we have assumed the progression of time from one moment to the next tobe probabilistic in nature, it is inherently uncertain what path is actually takenthrough state space. The consequences of this are far fetching.

Traditionally in physics, we can follow the behaviour of a system in time and

10This outcome is perhaps not too far from what has, more formally, been stated in [10].Yet this author has chosen to work from classical linear time towards Einsteins relativistictime and makes no reference whatsoever to quantum physics and the particular problemsassociated with it. Still, it is not unthinkable and possibly worth pursuing a unification of thecurrent model with the model in this reference.

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from thereon derive its properties and this yields us all of its future and past

states. In our model, however, the analysis of past states of a system helpsus only to roughly determine where we are in state space and thus singles outonly statistical collections or subensembles of possible paths taken.

As we mentioned, we will have to give up the ambition to measure time inany absolute sense, as this would require in a way a clock that ticks fasterthan the absolute master clock of the universe. Thus the rate of change withinone domain of a system can only be known relative to that of another. Thisconsideration leads, partly, to the implication of the fundamental relativity ofobservation, as discussed in subsection 4.3.

These two features lead us directly to the conclusion that there is an inherentuncertainty, noticeable as random noise, in our perception of all quantities.From this point of view it seems not unthinkable to relate this uncertainty with

the kind of uncertainties we might find in quantum physics.For example; if we were to take on Heisenbergs uncertainty principle, which

states that certainty about the spatial location of an object fundamentally leadsto a decreased certainty about its velocity, we could expect that (when relievedfrom the limitations of Simple Time and thus provided with spatial extension)the current theory might lead to a similarly formulated uncertainty.

4.2 Situatedness and limits to observation

Taking the current probabilistic theory at face value yields more than just theelementary uncertainties with regards to our own situatedness in state space.Rather, we will show that far fetching implications will follow for the status ofobservations and knowledge.

First of all, it follows from the vanishing of the probabilities of longer paths,as explained in the previous section, that some sort of locality emerges. Con-cretely, this means that objects (and therefore observers) have a fundamentalhorizon limiting the domain of state space which their observations allow themto have knowledge about.

But this has profound implications when we fully realise that every observer,much like in Einsteins relativity, occupies a position or more precisely: a col-lection of paths, an ensemble, through state space. Similarly, every observedobject or phenomenon has a distinct situatedness in state space of a similarform.

To put this in other words: when making an observation, this takes a certainamount of time; a certain number of ticks of the master clock of the universe.During these ticks both observer as well as the observed object trace overtheir respective collections of paths or ensembles through state space (whereeach path, as previously described, has a probabilistic measure associated withit, discerning the likelihood of its occurrence). In order to constitute a mea-surement, we have to assume (at the very least) a causal connection from theobserved to the observer and hence only the paths through state space thatlead from the observed to the observer count as eligible for the constitution ofmeasurements or observations.

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This already, once again, displays and strengthens the fundamental uncer-

tainty we previously stated: not only are we fundamentally unaware of our ownposition in state space and thus our current state of the universe, we are alsolimited with regards to the amount of knowledge we can have about other pointsin state space and thus measurable properties of the universe. Actually, mostparameters or properties of the universe are fundamentally beyond our reachas the probability values of the paths reaching from these particular points instate space to our own current position become negligible.

This consequence should not be entirely unfamiliar for a reader familiar withone or both of Einsteins theories of relativity, although in our current conceptionwe have intentionally left out all space-like properties and limit ourselves only tosome sorts of qualitative changes (which are assumed to be quantifiable). It is,however, likely that we will see essentially very similar phenomena, were we to

proceed to formulate a spatial extension of the current theory. Some tentativeconclusions about this shall be made in section 5.

4.3 Relativity of observation

Continuing our investigation in the description of the measurement or obser-vational process we observe another phenomenon entirely unknown within theframework of Einsteins relativity, namely, the natural occurrence of branch-ing phenomena. As previously mentioned in subsection 2.3, this is not at allan uncommon concept in the interpretation of quantum physical measurementprocesses. And even though there are slightly differing variants considering theactual structure and relations of branching and splitting processes, it is often

assumed to be related to specific events wherein the quantum world interactswith (typically) macroscopic objects (in the form of measurement devices).

As a truly quantum mechanical description of this problem lies beyond thescope of this (essentially argumentative) work, we will only make a description ofits equivalent within the current framework. A more (mathematically) thoroughdescription, accompanied by a tentative proposal of a solution, can be found in[11].

For now, let us proceed by formulating merely the conditions for what is oftencalled the measurement problem in quantum physics wherein a measurementof the same physical system can yield a different description of the systems state,suggesting diverging measurement results withing the same objective reality.Within our current framework, we will show that such is not the case and that

the mentioned branching phenomena can be seen as only a limiting case of amuch richer reality.

In order to do this we will take a situation where two observers, with theirrespective current situatedness in state space perform the same measurementof a property situated elsewhere in state space. This implies the existence ofensembles of paths tracing from the observed state towards both observers. Ifwe assume now that the number of steps along the (shortest) path between theobservers is relatively small, we necessarily conclude that the observers have a(mutual) causal relation and thus are able to communicate their measurementresults.

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As these paths are all assumed to have limited lengths, the resulting uncer-

tainty about the measurement will also be limited of magnitude. This meansthat it is (highly) likely for two observers and an observable to be closely to-gether in state space will perceive, roughly, the same results and therefore thesame reality. But as the number of steps between both observers increases,the amount of meaningful information they can have about one another andabout the observed phenomenon decreases. From this it follows that there isan increasing likelihood for the two observers to measure mutually exclusiverealities.

This conclusion can perhaps be illustrated more clearly by stressing that allobservations, necessarily, entail paths through state space from an ensemble, acollection, of points in state space towards collectionsof points associated withthe observers. If we take merely a small number of points between observer and

the observed, it is likely that multiple observers measure a smaller subensembleof points in state space. But as the amount of points between observer andobserved increases, it is very likely for different observers to measure the resultof diverging paths through state space.

The consequence of this is the appearance of branches mutually exclusivestates of the universe occurring simultaneously in the observation of (separated)observers looking at the exact same properties of a system. But within the viewthus presented, this branching is only an ephiphenomenon, in fact what weperceive to be branches are merely subensembles of paths through state spacerepresenting the particular outcomes of measurements.

4.4 Directionality in state space

Another feature of the view on time presented here entails the directionality oftime, meaning the way we move from now into the future. In other words:the way the current coordinate in state space is replaced by the next.

Classically, this kind of progress is simple. Once we have found our singlepath or timeline through phase space and pick out a particular point we baptisethe now, we simply move from one point to the next along the timeline in anarbitrarily chosen but consistently applied positive time direction. But in thecurrent model we do not deal with a single timeline rather we have a funda-mentally stochastic collection of intersecting timelines, a statistical ensemble,that governs phenomena of change indeterministically.

As each property in Simple Time defines an axis in state space and all thepoints along the axes have an associated probability measure, time evolutionimplies a (weighed) random walk through state space. The direction of thiswalk will, at best, have a bias towards a specific direction for which paths havea higher (absolute) probability. However, there is always a multitude of possiblepaths into the future and hence the temporal evolution of the universe couldnever be uniquely determined.

This means that, for time to actually move forward, we would require adelicate composition of the probabilities belonging to points in state space. Forif these probabilities were too homogeneous, we would expect our position instate space to hardly change at all: we would randomly fluctuate around our

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starting position.

But this need not be so strange. As mentioned in subsection 3.3 we explicitlydo not exclude the repetitious occurrence of the same point state space withina single path or timeline. Since there is no way to discriminate between aprimary, secondary or any later occurrence this would lead to a series of loopsin time. For an observer within such a loop it would not be experienced as such;similar to a goldfish, swimming loops in a fish bowl, the return to a previouslycrossed point in state space necessarily implies the previous erasure of all tracesand memories of future states.[12]

4.5 Scaling behaviour

If we think of state space on extremely large scales, it does not seem to us a

ridiculous assumption to suggest that the relatively small differences betweenthe probability values for points in state space average out. This means that,even though on a small scale time appears roughly linear, like we would expectit to do in classical and quantum dynamics, on a large scale it might well berepetitious and probabilistic in nature. As we explained in subsection 4.4, wewould simply have no way of telling whether such is the case.

In a similar manner we could reason that, even for a system where on large(but not extremely large) timescales probabilities in state space are set up suchthat there is an overall preference for a specific direction, on a smaller scalethis directional bias need not be visible from within the system. We mightwell perceive this as if time will not be move forward at all. All change we seeconsists of (reversible) random fluctuations.

From these consequences it might have occurred to the attentive reader thattemporal linearity is a scaling phenomenon: looking at a probabilistic randomwalk through state space with a large scale bias for a specific direction willstill look much like a fully random walk when small time scales are considered.At intermediate scales, however, this random path will look much like a lineas it consists of random fluctuations averaging out to form a bias in only onedirection. This impression of linear behaviour becomes even more obvious whenwe acknowledge that from the point of an observer within the system, repetitionsare utterly unobservable.

If we look at state space from an even wider perspective, that is, when wetake into consideration the full range of points therein, we are likely to find thatit will look to us (with regards to the associated probability values) perfectlyhomogeneous, given that differences in the probability values for points aresmall enough. This again implies that, on a larger scale, time does not seemto move forward at all. Moreover, on this large scale we will not perceive therandom fluctuations we found before on microscopic scales, as they cancel outcompletely. We would simply see a static universe full of change as we narrowour perspective.

This behaviour is similar to what we often see in statistical systems whenlooking across scales; where a system looks disordered as small scales, it mightexhibit order at an intermediate scale and look very homogeneous at the largestscales.

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5 Spatial extension

In the previous sections we have presented a radically new conception of physicaltime that allows us to understand both macroscopic, linear, time as it is usedin Newtonian physics as well as the branching model used in the Many Worldsinterpretation of quantum physics.

Yet in order to simplify the introduction of this novel conception, we haveintentionally left out all specificities of spatial structure. This has been doneby assuming the definition of Simple Time, wherein we assume a physical worldconsisting, spatially, of only a single mathematical point which is assumedto have qualitative properties nonetheless. But to give a proper account oftemporal phenomenology it stands clear that we have to include spatial structureand extension in our model of the universe.

5.1 Spatial coordinates as properties in state space

In order to explore the consequences of a possible approach to the spatial exten-sion of the discussed model of time, we will take the approach that lies closest athand: we will deem the spatial coordinates of an ordinary Euclidean space to bequantities associated with properties of points in state space. This means thatfor each spatial point in the universe we will have a complete set of non-spatialproperties associated with it. Or conversely; every set of non-spatial propertieshas unique spatial coordinates associated with it.

So even though the way spatial extension is introduced into state spaceis, mathematically, very similar to the way that it is used in the concept of

phase space in classical mechanics, there is a fundamental distinction: whereasphase space usually describes a limited set of objects with a given position andimpulse, in the current model we find a description of all possible properties forthe infinite amount of points in state space.

5.2 Velocity in state space

As change and thus motion itself is not incorporated in the description of statespace11 it becomes impossible to make impulse12 a property in state space.

Rather, it becomes possible for us to introduce motion as a measure forspatial change or displacement as a function of general change in state space.

But how could we formulate such a concept when it is not possible to rely, asdemonstrated in subsection 4.1, on linear and absolute progression of time?

The solution to this is both simple and elegant, but differs greatly from ourcommonsensical conceptions of velocity or impulse. As we have stated before,we could think of the passage of time as a universal master clock ticking awaydiscrete pulses of time. And since there is no single way to measure either theduration or the frequency of those ticks they should be conceived of as lyingoutside the empirical domain.

11For no reference is made to the actualpaths followed through state space, there is merelya statistical description.

12Classically defined as the product of an objects mass with its velocity.

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But if we assume that the magnitude of these steps are equal in size we

gain an import possibility for quantifying change, as in a comparative measure.Namely, for a single point in state space (this time with spatial position as aproperty), we can calculate the likelihood of paths that signify change in positionover ticks of time.

Moreover, we can introduce the measurement of time using clocks if we canidentify structures in state spaces for which repetitious changes of non-spatialproperties are likely to occur and correlate to some non-repetitious changesin some other property, similar to a watch measuring the repetitious passageof seconds in order to count the (on these scales) non-repetitious passage ofminutes or hours.

Thus having suggested a measure for both spatial displacement as well asnon-spatial displacement (as qualitative change) as a function of the ticking

of the universes master clock, we can state that we have successfully formu-lated the criteria for the measurement of velocity within the current temporalframework regardless of the absence any absolutely measurable time or thefundamental inability to specify direction therein.

Lastly, as we have defined change within our model only to be a probabilisticphenomenon, this implies that spatial position and change therein is also a prob-abilistic in nature. From this it follows that we should expect, macroscopically,change only to take place partly in the spatial dimensions; spatial displacementis something we perceive, together with other changes, on an average scale while at a microscopic scale changes take place with discrete steps in only asingle (but fundamentally random) direction in state space (hence only a singleproperty changes), as defined by the probabilities of the paths connecting the

current point in state space to its surroundings.

5.3 Limits to velocity and time dilatation

When taking the previously introduced measure of velocity in consideration, itbecomes feasible to suggest an absolute upper limit to the amount of measurableduration as a function of spatial displacement. In other words: it follows that,if change in the universe proceeds with equal and discrete steps, there should bea maximum amount of change with regards to spatial coordinates and thusalso to the measurement thereof.

Similarly, as we boldly assumed for the change in the universe to be equal inmagnitude for the discrete steps it is making, it follows that the (macroscopic)total amount of change per number of ticks is a constant property. This meansthat if our probability distribution is set up such that there is a 100% likelihoodfor change to occur only and repeatedly in a single spatial direction, an externalobserver as introduced in subsection 4.3 would observe13 a finite and limitedmaximum velocity.

Moreover, as in this case all change within a particular domain in state space

13As observing this velocity by itself requires a minimum amount of change or causal in-fluence other than that in this spatial dimension, the current example might seem to be apathological case. However, the measurement of velocity requires an observer only to measurethe spatial position at two instances in time, which should be of negligible influence on thecurrent thought experiment.

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takes place in a single spatial dimension, we can be sure that there is no change

in any other dimension and, consequently, in this domain in state space thereis to other perceptible change other than the spatial displacement. Likewise,if only a portion of the change in a domain of state space constitutes movingthis necessarily leads to a decreased probability of qualitative change and thus,it will seem (for a non-moving observer) as though time is moving slower.

Conclusively, within the discussed model, the assumption of homogeneouschange (the equality of the magnitude of steps through state space), leads di-rectly to two consequences traditionally associated with Einsteins Special Rela-tivity. For one: there is a fundamental maximum velocity, which is measurable.Secondly: as a greater amount of spatial displacement is assumed, the amountof non-spatial changes within that same area is necessarily decreased. This hap-pens in such a way that, whenever the probability distribution in a domain in

state space is set up such that this maximum velocity is reached, it will appear asthough time stands still for observers outside this area. In other words: it seemsas though we have, somehow, recovered the phenomenon of time dilatation.

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6 Conclusion

In this thesis we have first shown that some popular conceptions of time inas they are traditionally applied in classical, relativistic and quantum physicsknow some serious limitations. While they seem provide a consistent pictureof reality within their respective domains, any attempt at annexation of thesetime conceptions lead to severe conceptual (and mathematical) inconsistencies.

In the remainder of this thesis we have attempted to resolve these inconsis-tencies by providing a radically new conception of time that seeks to explain(some of) the observable implications of traditional physical models of timewithin the framework of a single theory. While not a mathematical theory,we believe that a proper mathematical backing is indeed achievable given theassumptions and structures presented here.

Some of the effects typical for quantum physics are found to exist in thisnew model, such as fundamental uncertainties with regards to the results ofmeasurements as well as what could be interpreted as branching phenomenon.At the same time, our model yields a maximum limit to velocity, from which it iseasily shown that relativistic effects such as time dilatation and a limited sphereof influence naturally arise. Furthermore, at scales where neither quantum norrelativistic physics are applicable, it is likely that statistical fluctuations cancelout in order to give rise to classical phenomenology.

Besides the recovery of these well known physical properties of the universewe also show several peculiarities of the presented model that are entirely un-familiar in any vested physical conception of reality. One of those features isan intricate scaling behaviour: at microscopic scales no temporal tendencies

appear to be present and only random fluctuations seem to occur, where atmacroscopic scales smaller temporal patterns cancel out to form a static pictureof the universe. Only at intermediate scales, a normal time arrow is to beexpected.

Another consequence of this particular theory is the frequent but inherentlyunobservable repeated occurrence of the same state of the universe. Whileit is argued that this might indeed be a common feature of our reality, thisparticular consequence has no causal or measurable implications whatsoever,since any repeated occurrence of a state requires the erasure of all traces of theprevious occurrence.

As not all of these consequences have direct empirical implications and themodel currently lacks any mathematical or formal representation, the presentedtheory should be interpreted as primarily a metaphysical exercise. Also, it istentative in character and many of its features are open for debate. However,it is the sincere hope of the author that the currently presented conceptionof temporal phenomena can actively contribute to a wider debate about ourconception of temporal phenomenon and the implications they have for our(physical) understanding of our reality.

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References

[1] Lorentz, H., Einstein, A., Minkowski, H. & Weyl, H. On the electrodynam-ics of moving bodies. In The principle of relativity: a collection of originalmemoirs on the special and general theory of relativity, 3565 (New York:Dover Publications, 1952).

[2] Griffiths, D. & Inglefield, C. Introduction to electrodynamics (Prentice Hall,1999).

[3] Griffiths, D. & Harris, E. Introduction to quantum mechanics (PrenticeHall, 1995).

[4] Allahverdyan, A. E., Balian, R. & Nieuwenhuizen, T. M. Dy-namics of a quantum measurement. Physica E: Low-dimensional

Systems and Nanostructures 29, 261271 (2005). URL http://www.sciencedirect.com/science/article/B6VMT-4GKW7HH-5/2/

822e8de5858d9268372aad68307575ae.

[5] Einstein, A., Podolsky, B. & Rosen, N. Can quantum-mechanical descrip-tion of physical reality be considered complete? Phys. Rev. 47, 777780(1935).

[6] Ballentine, L. E. The statistical interpretation of quantum mechanics. Rev.Mod. Phys. 42, 358381 (1970).

[7] Everett, H. Relative state formulation of quantum mechanics. Rev. Mod.Phys. 29, 454462 (1957).

[8] Everett, H. & DeWitt, B. The many-worlds interpretation of quantummechanics: a fundamental exposition (Princeton Univ. Press, 1973).

[9] Newton, S. & Chittenden, N. Newtons Principia: The mathematical prin-ciples of natural philosophy (D. Adee, 1848).

[10] Atkin, R. H. Time as a pattern on a multi-dimensional structure.Journal of Social and Biological Systems 1, 281 295 (1978). URLhttp://www.sciencedirect.com/science/article/B7GX4-4CC1PB6-5/

2/48f06671718cc3c10cf743ee9ab0e156.

[11] Rovelli, C. Relational quantum mechanics URL http://arxiv.org/abs/quant-ph/9609002v2. quant-ph/9609002v2.

[12] Maccone, L. Quantum solution to the arrow-of-time dilemma. Phys. Rev.Lett. 103 (2009).

[13] Rovelli, C. Forget time (2009). URL http://arxiv.org/abs/0903.3832v3. 0903.3832v3.

[14] Callender, C. Is time an illusion? Scientific American 302, 5865 (2010).

[15] Barbour, J. The nature of time (2009). URL http://arxiv.org/abs/0903.3489v1. 0903.3489v1.

[16] Zeman, J. Time in science and philosophy (Elsevier, 1971).

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http://www.sciencedirect.com/science/article/B6VMT-4GKW7HH-5/2/822e8de5858d9268372aad68307575aehttp://www.sciencedirect.com/science/article/B6VMT-4GKW7HH-5/2/822e8de5858d9268372aad68307575aehttp://www.sciencedirect.com/science/article/B6VMT-4GKW7HH-5/2/822e8de5858d9268372aad68307575aehttp://www.sciencedirect.com/science/article/B6VMT-4GKW7HH-5/2/822e8de5858d9268372aad68307575aehttp://www.sciencedirect.com/science/article/B7GX4-4CC1PB6-5/2/48f06671718cc3c10cf743ee9ab0e156http://www.sciencedirect.com/science/article/B7GX4-4CC1PB6-5/2/48f06671718cc3c10cf743ee9ab0e156http://www.sciencedirect.com/science/article/B7GX4-4CC1PB6-5/2/48f06671718cc3c10cf743ee9ab0e156http://arxiv.org/abs/quant-ph/9609002v2http://arxiv.org/abs/quant-ph/9609002v2http://quant-ph/9609002v2http://arxiv.org/abs/0903.3832v3http://arxiv.org/abs/0903.3832v3http://0903.3832v3/http://0903.3832v3/http://arxiv.org/abs/0903.3489v1http://arxiv.org/abs/0903.3489v1http://0903.3489v1/http://0903.3489v1/http://0903.3489v1/http://arxiv.org/abs/0903.3489v1http://arxiv.org/abs/0903.3489v1http://0903.3832v3/http://arxiv.org/abs/0903.3832v3http://arxiv.org/abs/0903.3832v3http://quant-ph/9609002v2http://arxiv.org/abs/quant-ph/9609002v2http://arxiv.org/abs/quant-ph/9609002v2http://www.sciencedirect.com/science/article/B7GX4-4CC1PB6-5/2/48f06671718cc3c10cf743ee9ab0e156http://www.sciencedirect.com/science/article/B7GX4-4CC1PB6-5/2/48f06671718cc3c10cf743ee9ab0e156http://www.sciencedirect.com/science/article/B6VMT-4GKW7HH-5/2/822e8de5858d9268372aad68307575aehttp://www.sciencedirect.com/science/article/B6VMT-4GKW7HH-5/2/822e8de5858d9268372aad68307575aehttp://www.sciencedirect.com/science/article/B6VMT-4GKW7HH-5/2/822e8de5858d9268372aad68307575ae

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Understanding physical time