On the Universe as a Process of
Unique Events
Notes from (and on) the article by Marina Cortês and Lee Smolin, "The
Universe as a Process of Unique Events" (
Physical Review, 2014). With a few corrections and observations.
In this important article, Cortês and Smolin (herafter C&S) describe a new class of
models of quantum space-time based on energetic causal sets and show
that under natural conditions space-time emerges from them. These are
causal sets whose causal links are labeled by energy and momentum and
conservation laws are applied at events. The models are motivated by
principles they propose govern microscopic physics which posit a
fundamental irreversibility of time. One consequence is that each event
in the history of the universe has a distinct causal relationship to
the rest. This requires a novel form of dynamics which can be applied
to uniquely distinctive events. They hence introduce a new kind of
deterministic dynamics for a causal set in which new events are
generated from pairs of progenitor events by a rule which is based on
extremizing the distinctions between causal past sets of events. This
dynamics is asymmetric in time, but they find evidence from numerical
simulations of a 1 + 1 dimensional model, that an effective dynamics
emerges which restores approximate time reversal symmetry. Finally they
also present a natural twistorial representation of energetic causal
sets.
Thus far the abstract. Only mathematicians and first-class theoretical physicists are advised to read the original paper in
Physics Review,
linked below. This is a mere attempt at popularization (a modest one,
as the paper has already won an award in cosmology). Although I have
detected several typos, the
mathematics of the paper is beyond my ken, so I will "skip the
equations" in one fell swoop and focus on C&S's account of the
import and significance of the operations. One must point out, though,
that:
- the theory in this paper is complemented by two companion papers by
C&S in which the mathematical model for fundamental irreversibility
in cosmology is further specified and developed, and some of its
implications for quantum foundations are drawn out.
- the mathematical treatment of this issue should be placed in the
context of
Lee
Smolin's evolutionary cosmology, as exposed in
Time Reborn and (with Roberto
Mangabeira Unger) in
The Singular
Universe and the Reality of Time. A
lively and accessible summary is available (in English and Spanish) in
Smolin's interview with Robert Sawyer linked in the works cited.
This theory includes not just a new conception of time and cosmology,
but also a "downgrading" of the status of mathematics, which is demoted
from its transcendental position dictating the underlying structure of
reality, beyond time, to a more modest instrumental role in a human
world. Mathematical models are
tools
which, though they do not uncover any fundamental realm transcending
the universe (such as the mind of God, etc.), are useful in modelling
for their manipulation a selection of aspects of physical reality, or
allowing operations within the much more localized realm of human
communication and research in scientific disciplines. That is, in spite of the phrasing used sometimes in the paper, the
mathematical model presented here is not
the formula which generates reality,
the way E=mc
2
is sometimes conceived as being inscribed in some constellation. It is a
mathematical set of formulas which are meant to account what other
theories leave out, i. e. the singularity of events in the universe
whose description is necessitated by the cosmological scope of the
theory.
The paper helps to provide a crucial theoretical foundation which
allows the (consilient) embedding of classical physics, and its
mathematical formulation of time-reversible dynamics, within the novel
paradigm of evolutionary cosmology based on the irreversibility and
the inclusive reality of time.
A fuller exposition of this paradigm, which amounts to a substantial
revision of the current paradigm of physical science and mathematics,
is available in the volume
The Singular Universe and the Reality of Time, by Roberto Mangabeira Unger and Lee Smolin.
The current scientific consensus about time, the one they reject, is
formulated thus by C&S:
"Time is an illusion. It emerges as we coarse grain, zooming out from
timeless fundamental physics. It's not a fundamental quantity of
nature. In searches for quantum gravity we are to seek an underlying
Plack length description, expressed in the form of timeless equations,
where time is not to play any role. When time emerges it is to
parametrize equations that are symmetric under time reversal. The
evident time-asymmetry of nature is then held to be an accident, due to
an improbable choice of initial conditions."
C&S complain that this view "twice diminishes our basic experience
of the world as an unfolding series of moments, to the realm of
accident and illusion (i) because the asymmetry of time is held to be
an accident, and (ii) because time itself is held to be emergent".
[Note, by the way, that the
appeal to "our basic experience of the world" may be misleading here,
because the whole issue of the experience of time in its
perceptual-animal sense, not to mention the peculiar dimensions of
human time, is outside the scope of this paper, which focuses
altogether on the basic processes giving rise to events in nature,
quite apart from the issue of how these events are construed by
sentient beings.]
"In this paper we will propose the diametrically opposite view. We
develop the hypothesis that time is both fundamental and irreversible,
as opposed to reversible and emergent. We will argue that the
irreversible passage of time must be incorporated in fundamental
physics to enable progress in our current understanding. The true laws
of physics may evolve in time and depend on a distinction between the
past, present and future, a distinction which is absent in the standard
block universe perspective.
The present instant is a primitive and part of fundamental processes,
and the laws of physics may refer to it preferentially. There is a
process continually acting in the present bringing into existence the
next moment. The present may code aspects of past states but the past
is no longer accessible, as could be argued to be the case in the block
universe picture. Along the same lines the future has yet to happen,
and aspects of it may even be open, i.e. not computable from a complete
knowledge of the present.
We'll argue that, in contrast to the time reversal symmetry which is
standard practice, time is fundamentally asymmetric and irreversible.
The future is different from the past: the process by which the present
becomes the past and gives rise to the future cannot be inverted to
allow the perfect reconstruction of the past.
Based on this view we propose four principles. The first will concern
the nature of time.
(i) Principle A Time is
a fundamental quantity; the agency of time is the most elementary
process in physics, by which new events are created out of present
events. Causality results directly from the irreversible agency of time.
(ii) Principle B Time
has a fundamental directionality. The future develops out of the
present constantly; there are no causal loops and no regions or
phenomena where time "evolves backwards." This implies that the
fundamental laws that evolve the future from the past are irreversible in the sense that they
have no inverse by which the past state can be reconstructed from the
present state.
A second pair frame the way that the dynamics of the world may be
expressed.
(i) Principle C We
choose a relational point of view, according to which the space-time
properties of an object or event arise from its relationship with other
objects or events. All space-time properties have a dynamical origin.
(ii) Principle D Energy
is fundamental. Energy and momentum are not emergent from space-tim,
rather the opposite is the case, space-time is emergent from a more
fundamental causal and dynamical regime in which energy and momentum
are primitives.
We chose to model these assumptions within a discrete framework. This
means that we envision the history of the universe as a set of events,
endowed with both a causal structure and intrincis energy-momentum
variables.
Before presenting this model, we not that the assumption that events
are described relationally has a strong consequence, which is the
(i) Uniqueness of events
The relational properties of each event in cosmological evolution make
it unique and
distinguishable from all others.
This is a consequence of the demand that each event be distinguishable
by its relational properties. Within a discrete causal structure an
event can only be distinguished by its causal past. Furthermore the
events that make up the causal past of a given event cannot have any
absolute labels, for those are only labeled by their causal pasts. But
those are part of the causal past of the given event. This implies that
any two events must have nonisomorphic causal pasts in that there is no
map that takes one to the other preserving their causal pasts. This can
be considered to be a consequence of Leibniz's principle of the
identity of the indiscernible.
Uniqueness need not contradict our standard scientific method which we
base on statistical inference from repeated experiments. Repeated
systems do occur—to a sufficient degree of approximation—if we are
considering subsets of the whole universe. Whenever we apply a boundary
to define the system under study in the laboratory, we are severing the
relations it has with the remainder of the system. Truncating these
connections makes it possible for the subsystem to appear approximately
similar to other subsets as well as to itself when subject to
repetitions in time of the initial conditions. Similarity is local, and
a result of truncation, and repetition is never exact when the whole
system is considered.
Uniqueness of events is a strong requirement as it demands each event
has a causal past that is complex enough to be district from the causal
pasts of all the other events. As a result, fundamental laws acting on
unique events need to take their complexity into consideration. This
would seem to imply large informational inputs and outputs, which
appears to contradict the idea that elementary events should be simple.
We will show below that this query has a simple answer, which is that a
space-time must emerge within which the network of complex historical
relations embeds. When an embedding of the history of events in a
space-time exists one can use coordinates on the space-time to uniquely
identify each event. this is a highly nontrivial requirement, but in
the next section we describe a model in which it is satisfied.
Having set out the physical picture which follows from our assertions,
we present in Sec.II a simple model which is useful to study them. In
this model a causal set [1] is generated by an event generator which
acts according to a rule satisfying the principles we laid out. The
events are endowed with energy and momenta which are propagated from
old to new events subject to conservation and energy-momentum
relations. We propose to call causal sets so endowed energetic causal sets. We find some
remarkable results:
(1) Space-time is not part of the fundamental description but emerges
as an arena for a statistical description of the fundamental process.
(2) The emergent equations of motion, which describe the embedding of
the causal processes in space-time, cannot always be solved
consistently but when they can there is a classical limit which is a
theory of interacting relativistic particles.
(3) Numerical studies of the model in 1+1 dimensions show that a
reversible effective dynamics can emerge from irreversible evolution
rules.
(4) In 3 + 1 dimensions the model can be reformulated in the language
of twistor theory."
(...)
[In this model,]
"the history of the universe is described as a set of N
events, EI, I = 1, ..., N.
Principle A is incorporated
here, in that each event is created as the result of a process acting
on the prior set of events. The process is the activity of time. In the model we
call it the events generator.
The event generator must make two decisions each time it creates an
event: First, which of the prior set of events are to be progenitors of
the new event. Second, how are the properties of the new event
determined from the properties of its progenitors.
Each event is endowed with energy and momenta, which are held to be
primitive properties. These are conveyed to an event from its
progenitors. This realizes Principle D."
(...)
"We can gain more insight into the physics of this model by introducing
a statistical formulation. Since the dynnamics which generates the
causal set is deterministic, but non-local, we have to define carefully
what ensemble the statistical averaging corresponds to. We also have to
respect the assumption that each fundamental event is unique when all
the information about its causal past is known. There is also a single,
unique causal history for the whole universe, Cuniverse.
A statistical ensemble can then only arise from the consideration of
subsystemss of intermediate scale, neither cosmological nor universal.
We propose then to introduce statistics by studyin ensembles composed
of subsystems which are characterized by incomplete information about
their causal pasts."
(...)
"The emergence of the zaI's
as coordinates of the embeddings of the events in a space-time neatly
solves the puzzle of how to give dynamics to unique events. As we posed
the question in the Introduction, the question was how to conceive of a
law or a rule for generating unique events when that rula had to be
simple, but what distinguishes the events in a big universe are the
intricacies of their histories. The answer is to invent space-time so
that each event has a unique embedding in that space-time."
(...)
Twistor formulation
"We
can solve the energy-momentum constraints for massless particles (3) in
terms of two component spinors. (...)
So we elegantly reconstruct the
embeddings of the causal processes in Minkowski space-time."
Description of the model
"We
will consider events that have two incoming photons and two outgoing
photons, in which one pair is left-movin, the other is right moving"
(...) "The spatial compactness is also helpful to get an
interesting model; without it the worldliness disappears as nothing
happens after a finite number of interactions" [
Garbled text; my hypothetical reconstruction].
(...)
"Each event is distinguished by its past, as well as its position and
time labels, and is stored in the past of the parent event that
contributed the left incoming ray. Hence all events belong to one and
only one past, and the intersection of all pasts is null."
However[,] even a moderately complex measure of each
[past] quickly becomes computationally unfeasible because the number of
differences of pasts to compute in the generation of each new [event]
increases exponentially with the number of intervening events. Since
our sole requirement for each past is that it be unique, the need for
complexity can be alleviated by taking a simple measure of each as the
average of the space-intervals of events in that past. This we know to
be unique since we have continuum space-time available. So the
emergence of space-time from the interaction momenta is at the same
time ensuring [the] uniqueness of each past. (...) Each event is
distinguished by its past, as well as its position and time labels, and
is stored in the past of the parent event that contributed the left
incoming ray. Hence all events belong to one and only one past, and the
intersection of all pasts is null.
This model is very simple[:] the momenta play a small role, which is
just to propagate themselves acoording to the conservation rules. But
it can be used to study the question of how time reversible approximate
laws might emerge out of time irreversible fundamental laws, and hence
address our central assumption, principle A."
(...)
Description of results
"We saw evidence for a simple characterization of the time evolution in
which the systems pass in time thorugh two distinct phases. The systems
begin in a disordered phase, followed by an ordered phase we call the
locked in phase. In the disordered phase, the time asymmetry of the
event generation rule manifests itself in a visible time asymmetry of
the pattern of space-time positions of the events. In this initial
phase, the events form a roughly random pattern in space-time,
characterized by a large variety of spatial positions, within an
envelope that is asymmetric under time reversal. In the [locked-in]
phase an approximate time symmetry emerges. This second phase is
dominated by persistent repeating patterns we can call quasiparticles."
[
Note the similarity to the account
of the origin of repetition, equilibration and partially stable systems
out of an original undifferentiated force in Herbert Spencer's theory of evolutionary cosmology, as expounded in his
First Principles.]
"Each run began with a period of seemingly chaotic behavior, manifested
by a disordered ensemble of events in which the time asymmetry of the
algorithm is evident. In this disordered phase all the past-sets
intervene with similar weights in the generation of new events. Variety
is maximal. Any two past-sets interact only very briefly—two times—just
enough for both paris of photons of the parent to be used and
discarded, and interaction moves on to another pair." (...) "In Fig. 1
we see a very different behavior emerging after a sufficient number of
events has been generated. As time progresses, we gradually begin to
see two past-sets interlocking, ever so briefly for a short succession
of events, and we start to see momentarily glimpses of stable
trajectories in the plotted evolution. The interlocking between pairs
of past-sets lasts longer each time it occurs. These give rise to
recognizable trajectories of quasiparticles." (...) "Quasiparticles
move in straight lines in space-time ,and as such have a dynamics which
appears to be unchanged under reversal of the time coordinate. We call
this the lock-in phase." (...
) "given enough time models eventually evolve towards the lock-in and
regular phase".
"The general behavior is thus an evolution from initial disordered
behavior, from which a few quasiparticles eventually set off and
interact only amongst themselves. The other pasts are left behind and
stop interacting. These results support the hypothesis that
approximately time reversal symmetric behavior can emerge from time
asymmetric evolution rules."
"The emergence of regularity always seems to be related to the loss of
novelty in the system. In our case the novelty element is the random
number of cycles added at each event generation. In the beginning all
pasts intervene in similar amounts, any pair interacts only twice and
interaction moves on to the next pair. The information in each past is
minimal and novelty, in the form of the random number of cycles added,
plays a more determining role in the generation of new events than the
interaction rule. However, with each event generation the information
in each past-set builds up, and begins to weigh increasingly more in
the selection of pairs. Correspondingly novelty plays progressively a
minor role, until there are no new positions created and the number of
possible space positions is constant. This is represented by the
emergence of quasiparticles, with regular trajectories, which we
observe in all four figures. This is the lock-in phase which is
identifiable by the regularity of structures in the network. We
observed that, quite surprisingly, this lock-in regular phase exists
[for each] of the interaction rules and initial conditions we
considered."
[
Note that the 'novelty' C&S
refer to here is exactly the reverse of 'the novelty of emergent
phenomena', as the novelty of emergent phenomena, such as
quasiparticles, is enabled precisely by the build-up of information
from the past, from the increasing complexity of locked-in phenomena,
and from the loss of the short-term randomness of the initial phase as
described here. This process has also been put in verse by Abraham
Cowley. See 'Chaos
and the Emergence of Order in Evolutionary Cosmology. Two evolutionary
accounts'.]
"So
there is an interplay between the random input and the regularity of
the rule we apply. The observed tendency is for the regularity to
eventually overcome the amount of diversity. Irreversible dynamics seems to have the
tendency to evolve toward predictable, reversible evolution."
"The general system is irreversible because each state may not have a
unique predecessor, even if determinism requires each state to have a
unique successor. But once a limit cycle is entered, each state has
both a unique predecessor and a unique successor. Hence, restricted to
the limit cycles, the system may be said to be governed by an effective
reversible dynamics. This will be discussed in more detail in
[Fundamental Irreversibility in Cosmology]."
Summary and conclusions
"We began this work as we began the paper, by a search for principles
that could guide the search for a physics that could be applied to the
whole universe. This must be a novel form of dynamics that can be
applied to a unique system—the universe as a whole, that is neither
quantum mechanics nor general relativity, from which quantum physics
and space-time emerge as approximate descriptions of systems that may
be regarded to a sufficient degree of approximation as isolated
subsystems."
A. Summary of ideas
"We formulated four principles, which we presented as Principles A to D
in Sec. I. The first two have to do with the nature of time and assert
that time, taken as the agency that produces new events from present
events, is truly fundamental and irreversible. This focuses attention
on an agent which creates novel events—what we call the event
generator[.] The third principle asserts the philosophy of
relationalism—that space-time emerges from the action of that agent as
a reflection of the network of causal relatins that agent creates. The
fourth principle limits the reach of relationalism to assert that in
addition to their relational, spatial-temporal properties, events are
endowed by intrinsic properties, and these include energy and momenta.
From these principles we draw several conclusions.
(1) We work within an ontology of events, and the causal processes that
create new events out of old events.
(2) Each event should be uniquely distinct from the rest. This is a
consequence of Leibniz's principle of the identity of the indiscernible
applied to an ontology of events and causal processes.
(3) The fundamental dynamics resides in the choices made by the events
generator: which pairs (or small sets of) present events will give rise
to a new event and how are the properties o those progenitor
events—both relational and intrinsic—transferred to the newly created
event. We should emphasize that no existing paradigm addresses the
question of why particular events are created. The existing causal set
of dynamics are either stochastic or based on a sum of stories approach
to a quantum physics. Another paradigm based on an events ontology—the
Feynman diagram approach to QFT—is
also based on a sum over stories.
(4) The event generator must make use of and create tha data that
uniquely identifies each event.
This emphasis on the uniqueness of events, and its role in the
fundamental dynamics, stands in strong contrast to the existing
paradigms for dynamics, which assume that microscopic systems are
simple and come in vast ensembles of identical copies. This puts the
emphasis on symmetry principles applied to identical elementary
particles. In contrast to this we assert that at the level of the
elementary events and the dynamics that generates them every event is
unique so there are no ensembles of identical events and no fundamental symmetries.
Consequently we assert that the kind of physics we have developed to
this point, based on identical properties classified by symmetries,
must emerge at an intermediate scale. We can say more specifically that
this emergence has to do with truncating the description within which
each event is unique, so as to give an effective description of
subsystems of the universe with minimal information.
It is also natural to
conjecture that this truncation is responsible for the statistical
nature of quantum physics. The stochastic description arises by
neglecting data that renders each event unique which results in a
description of emergent quasiparticles that fall into large classes of
nearly identical copies. It is at this emergent level that symmetry and
identity arise as aspects of a statistical and approximate description.
We can express this by saying that there are hidden variables which
reside in the data that makes each event unique and determines which
events take place.
Time
reversal invariance is amongst the symmetries that must emerge from the
coarse graining that neglects the unique identity of each event. From
our perspective fundamental physics is time asymmetric and the apparent
time symmetry of the laws of nature is approximate and emergent.
We explored the
implications of this new approach to fundamental physics by inventing a
new kind of model of space-time physics: an energetic causal set, which
differs from the usual causal sets by having events endowed with
intrinsic energy and momenta, which are transmitted by causal
processes. We discovered that under natural assumptions—a small enough
number of progenitors—space-time naturally emerges as an approximate
description based on stationary phase approximation that appears in a
statistical description of these systems. This addresses a long
standing difficulty of causal set models to generate emergent [low-dimensional] space-times.
We can offer a suggestion
of why space-time emerges, which is that it resolves a problem faced by
the event generator, which is how to uniquely label each event in a
manner that requires a small amount of information and so is
computationally efficient. At first, each event is distinguished by its
causal past, but to specify these in a large universe to the point they
render each event unique would take a vast amount of information. The
emergent [spacetime] summarizes this information in a small set of
numbers."
[
Note that the phrasing and
presuppositions here are curiously formalistic, and ascribe to the
universe processes which are actually relative to the mathematical
theory used to model it. This is particularly striking in view of
Smolin's instrumental view of mathematics; this may be attributed to
the mathematical views of Cortês, or perhaps it is a case of resolutely
adopting for the sake of the reasoning the mathematicians' view of
mathematics as literally generating the structure of reality.
I propose that this be considered a rhetorical licence.]
(...)
B. Future prospects
"A key goal of future work
will be to identify the limit where the science of the subsystems
breaks down and uniqueness of cosmological events must be taken into
account. Locally the structures in the universal network appear
identical, and we achieve the limit of repeatable experiments by
truncating the description to subsystems. There is however a scale
where this frequentist science breaks down, subsystems are no longer
similar, and we need to extend a subsystem to include its unique
history in order to obtain a consistent cosmological dynamics."
[
This might be the way to address such phenomena as quantum entanglement]
"In the model of unique
events, the present is defined as the set of open events, instants that
are in the process of completing. In the model we described here, these
are those events or instants that still have unused photons. So the
present is not a single instant, but a set of instants. This has a
parallel in the discussion of whether the present moment is thin or
thick: in this view the present is thick, made of those events that are
in the process of realizing themselves."
[
Note once again how the formal needs
of the model lead C&S to use a language that seems to contradict
their assumption of time as an irreducible primitive; mathematics
necessarily maps time and events into a simultaneous atemporal graph in
which some of the operations represent 'the present'.]
"One important question to
study is the nature of the set of present events. In the model we
studied in the last section, the present is continually generated from
interaction of open events. Even in the case of stationary
quasiparticles, that have static space-time trajectories, they [are in
fact interacting:] their trajectories are created mutually by
interacting with each other. There is no absolute stillness, the agency
of time is continual. Both in the emergence of quasiparticles and in
the irreversible phase, at each moment, one event or two events are
transpoted into the past, stop being part of the present, and a new
event is created, taking their place, and is brought into the present.
The amount of information
in the current live events set characterizes the potential for novelty
and diversity of the future network. There are multiple possibilities
for how to [quantify] the amount of latent novelty in a system. In our
model, novelty arises with the random number of cycles introduced at
each event generation. As argued above, when this novelty is exhausted
lock-in occurs and the regular phase is reached. In the regular phase
the diversity in the system is constant and limited. It will be an
interesting [experiment] to make the injection of novelty vary in time
and observe how the [system] reacts with regards to its reversible or
irreversible dynamics."
[
This sounds like a promising avenue
to further theorize the "locked-in" nature of such phenomena as strong
atomic force, the binding of quarks, etc.—
as
subsystems which partially escape the action of time, or dimensional
scales of nature where "time has stopped" at a local level]
"Our results suggest that
the irreversibility persists so long as diversity in the system is
abundant. If there is a fixed quantity of novelty, the tendency is for
it to be used up in the generation of irreversibility. When novelty is
exhausted, the amount of diversity (however we choose to quantify it)
becomes limited, and the dynamics regularizes, becoming patterned and
reversible."
[
One might argue that, at a
convenient emergent distance, this is the matter with my blog of late;
the influx of novelty is null, time has stopped or a locked-in phase
has been reached, and the dynamics has become patterned and reversible]
"It is notable to observe
that the lock-in and consequent emergence of reversible dynamics is
robust and occurs whether the interaction rule is local or nonlocal
i.e. whether the progenitors of now events are chosen to have past-sets
which are maximally similar or maximally diverse.
A goal for future work
will be to understand in more detail how fundamental irreversibility
gives rise to standard reversible laws; what is the quantity that
represents novelty; and how does the transition to the reversible
regime occur."
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Unger, Roberto Mangabeira, and Lee Smolin.
The Singular Universe and the Reality of
Time. Cambridge: Cambridge UP, 2015.
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