Event symmetry

Event symmetry

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The term event symmetry refers to invariance principles that have been used in some discrete approaches to quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

 where the diffeomorphism invariance of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

 can be extended to a covariance
Covariant transformation
In physics, a covariant transformation is a rule , that describes how certain physical entities change under a change of coordinate system....

 under any permutation of spacetime events.

What it means

Since general relativity was discovered by Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

 in 1915, observation and experiment have demonstrated that it is an accurate gravitation theory up to cosmic scales. On small scales, the laws of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 have likewise been found to describe nature in a way consistent with every experiment performed, so far. To describe the laws of the universe fully a synthesis of general relativity and quantum mechanics must be found. Only then can physicists hope to understand the realms where gravity and quantum come together. The big bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

 is one such place.

The task to find such a theory of quantum gravity is one of the major scientific endeavours of our time. Many physicists believe that string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

 is the leading candidate, but string theory has so far failed to provide an adequate description of the big bang, and its success is just as incomplete in other ways. That could be because physicists do not really know what the correct underlying principles of string theory are, so they do not have the right formulation that would allow them to answer the important questions. In particular, string theory treats spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

 in quite an old fashioned way even though it indicates that spacetime must be very different at small scales from what we are familiar with.

General relativity by contrast, is a model theory based on a geometric symmetry principle from which its dynamics can be elegantly derived. The symmetry is called general covariance
General covariance
In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations...

 or diffeomorphism invariance
General covariance
In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations...

. It says that the dynamical equations of the gravitational field and any matter must be unchanged in form under any smooth transformation of spacetime coordinates. To understand what that means you have to think of a region of spacetime as a set of events, each one labelled by unique values of four coordinate values x,y,z, and t. The first three tell us where in space the event happened, while the fourth is time and tells us when it happened. But the choice of coordinates that are used is arbitrary, so the laws of physics should not depend on what the choice is. It follows that if any smooth mathematical function is used to map one coordinate system to any other, the equations of dynamics must transform in such a way that they look the same as they did before. This symmetry principle is a strong constraint on the possible range of equations and can be used to derive the laws of gravity almost uniquely.

The principle of general covariance works on the assumption that spacetime is smooth and continuous. Although this fits in with our normal experience, there are reasons to suspect that it may not be a suitable assumption for quantum gravity. In quantum field theory, continuous fields are replaced with a more complex structure that has a dual particle-wave nature as if they can be both continuous and discrete depending on how you measure them. Research in string theory and several other approaches to quantum gravity suggest that spacetime must also have a dual continuous and discrete nature, but without the power to probe spacetime at sufficient energies it is difficult to measure its properties directly to find out how such a quantised spacetime should work.

This is where event symmetry comes in. In a discrete spacetime treated as a disordered set of events it is natural to extend the symmetry of general covariance to a discrete event symmetry in which any function mapping the set of events to itself replaces the smooth functions used in general relativity. Such a function is also called a permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

, so the principle of event symmetry states that the equations governing the laws of physics must be unchanged when transformed by any permutation of spacetime events.

How it works

It is not immediately obvious how event symmetry could work. It seems to say that taking one part of space time and swapping it with another part a long distance away is a valid physical operation, and that the laws of physics must written to support this. Clearly this symmetry can only be correct if it is hidden or broken. To get this in perspective consider what the symmetry of general relativity seems to say. A smooth coordinate transformation or diffeomorphism can stretch and twist spacetime in any way so long as it is not torn. The laws of general relativity are unchanged in form under such a transformation. Yet this does not mean that objects can be stretched or bent without being opposed by a physical force. Likewise, event symmetry does not mean that objects can be torn apart in the way the permutations of spacetime would make us believe. In the case of general relativity the gravitational force acts as a background field that controls the measurement properties of spacetime. In ordinary circumstances the geometry of space is flat and Euclidean and the diffeomorphism invariance of general relativity is hidden thanks to this background field. Only in the extreme proximity of a violent collision of black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

s would the flexibility of spacetime become apparent. In a similar way, event symmetry could be hidden by a background field that determines not just the geometry of spacetime, but also its topology.

General relativity is often explained in terms of curved spacetime. We can picture the universe as the curved surface of a membrane like a soap film that changes dynamically in time. The same picture can help us understand how event symmetry would be broken. A soap bubble is made from molecules that interact via forces that depend on the orientations of the molecules and the distance between them. If you wrote down the equations of motion for all the molecules in terms of their positions, velocities and orientations, then those equations would be unchanged in form under any permutation of the molecules (which we will assume are all the same). This is mathematically analogous to the event symmetry of spacetime events. The equations may be different, and unlike the molecules on the surface of a bubble, the events of spacetime are not embedded in a higher dimensional space, yet the mathematical principle is the same.

Physicists do not presently know if event symmetry is a correct symmetry of nature, but the example of a soap bubble
Soap bubble
A soap bubble is a thin film of soapy water enclosing air, that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact with another object. They are often used for children's enjoyment, but they are also...

 shows that it is a logical possibility. If it can be used to explain real physical observations then it merits serious consideration.

Maximal Permutability

American philosopher of physics John Stachel
John Stachel
John Stachel is an American physicist and philosopher of science.Stachel earned his PhD at Stevens Institute of Technology in Physics about a topic in General relativity in 1958...

 has used permutability of spacetime events to generalize Einstein's hole argument
Hole argument
In general relativity, the hole argument is a "paradox" which much troubled Albert Einstein while developing his famous field equation.It is incorrectly interpreted by some philosophers as an argument against manifold substantialism, a doctrine that the manifold of events in spacetime are a...

. Stachel uses the term quiddity
In scholastic philosophy, quiddity was another term for the essence of an object, literally its "whatness," or "what it is." The term derives from the Latin word "quidditas," which was used by the medieval scholastics as a literal translation of the equivalent term in Aristotle's Greek.It...

 to describe the universal qualities of an entity and haecceity
Haecceity is a term from medieval philosophy first coined by Duns Scotus which denotes the discrete qualities, properties or characteristics of a thing which make it a particular thing...

 to describe its individuality. He makes use of the analogy with quantum mechanical particles, that have quiddity but no haecceity. The permutation symmetry of systems of particles leaves the equations of motion and the description of the system invariant. This is generalised to a principle of maximal permutability that should be applied to physical entities. In an approach to quantum gravity where spacetime events are discrete, the principle implies that physics must be symmetric under any permutations of events, so the principle of event symmetry is a special case of the principle of maximal permutability.

Stachel's view builds on the work of philosophers such as Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

 whose monadology
The Monadology is one of Gottfried Leibniz’s best known works representing his later philosophy. It is a short text which sketches in some 90 paragraphs a metaphysics of simple substances, or monads.- Text :...

 proposed that the world should be viewed only in terms of relations between objects rather than their absolute positions. Ernst Mach
Ernst Mach
Ernst Mach was an Austrian physicist and philosopher, noted for his contributions to physics such as the Mach number and the study of shock waves...

 used this to formulate his relational principle, which influenced Einstein in his formulation of general relativity. Some quantum gravity physicists believe that the true theory of quantum gravity will be a relational theory
Relational theory
In physics and philosophy, a relational theory is a framework to understand reality or a physical system in such a way that the positions and other properties of objects are only meaningful relative to other objects...

 with no spacetime. The events of spacetime are then no longer a background in which physics happens. Instead they are just the set of events where an interaction between entities took place. Characteristics of spacetime that we are familiar with (such as distance, continuity and dimension) should be emergent
It may also mean:* Emergent , Neural Simulation Software* Emergent , a 2003 album by Gordian Knot* emergent plant, a plant which grows in water but which pierces the surface so that it is partially in air...

 in such a theory, rather than put in by hand.

Quantum Graphity and other random graph models

In a random graph
Random graph
In mathematics, a random graph is a graph that is generated by some random process. The theory of random graphs lies at the intersection between graph theory and probability theory, and studies the properties of typical random graphs.-Random graph models:...

 model of spacetime, points in space or events in spacetime are represented by nodes of a graph. Each node may be connected to any other node by a link. In mathematical terms this structure is called a graph. The smallest number of links that it takes to go between two nodes of the graph can be interpreted as a measure of the distance between them in space. The dynamics can be represented either by using a Hamiltonian
Hamiltonian may refer toIn mathematics :* Hamiltonian system* Hamiltonian path, in graph theory** Hamiltonian cycle, a special case of a Hamiltonian path* Hamiltonian group, in group theory* Hamiltonian...

formalism if the nodes are points in space, or a Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

 formalism if the nodes are events in spacetime. Either way, the dynamics allow the links to connect or disconnect in a random way according to specified probability rule. The model is event-symmetric if the rules are invariant under any permutation of the graph nodes.

The mathematical discipline of random graph
Random graph
In mathematics, a random graph is a graph that is generated by some random process. The theory of random graphs lies at the intersection between graph theory and probability theory, and studies the properties of typical random graphs.-Random graph models:...

 theory was founded in the 1950s by Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...

 and Alfréd Rényi
Alfréd Rényi
Alfréd Rényi was a Hungarian mathematician who made contributions in combinatorics, graph theory, number theory but mostly in probability theory.-Life:...

. They proved the existence of sudden changes in characteristics of a random graph as parameters of the model varied. These are similar to phase transitions in physical systems. The subject has been extensively studied since with applications in many areas including computation and biology. A standard text is "Random Graphs" by Béla Bollobás
Béla Bollobás
Béla Bollobás FRS is a Hungarian-born British mathematician who has worked in various areas of mathematics, including functional analysis, combinatorics, graph theory and percolation. As a student, he took part in the first three International Mathematical Olympiads, winning two gold medals...


Application to quantum gravity came later. Early random graph models of space-time have been proposed by Frank Antonsen (1993), Manfred Requardt (1996) and Thomas Filk (2000). Tomasz Konopka, Fotini Markopoulou-Kalamara
Fotini Markopoulou-Kalamara
Fotini G. Markopoulou-Kalamara is a Greek theoretical physicist interested in foundational mathematics and quantum mechanics. She is a faculty member at Perimeter Institute for Theoretical Physics and is an adjunct professor at the University of Waterloo.Markopoulou received her Ph.D...

, Simone Severini and Lee Smolin
Lee Smolin
Lee Smolin is an American theoretical physicist, a researcher at the Perimeter Institute for Theoretical Physics, and an adjunct professor of physics at the University of Waterloo. He is married to Dina Graser, a communications lawyer in Toronto. His brother is David M...

 of the Canadian Perimeter Institute for Theoretical Physics introduced a graph model that they called Quantum Graphity,. An argument based on quantum graphity combined with the holographic principle
Holographic principle
The holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon...

 can resolve the horizon problem
Horizon problem
The horizon problem is a problem with the standard cosmological model of the Big Bang which was identified in the 1970s. It points out that different regions of the universe have not "contacted" each other because of the great distances between them, but nevertheless they have the same temperature...

 and explain the observed scale invariance
Scale invariance
In physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...

cosmic background radiation fluctuations without the need for cosmic inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...


In the quantum graphity model, points in spacetime are represented by nodes on a graph connected by links that can be on or off. This indicates whether or not the two points are directly connected as if they are next to each other in spacetime. When they are on the links have additional state variables that define the random dynamics of the graph under the influence of quantum fluctuations and temperature. At high temperature the graph is in Phase I where all the points are randomly connected to each other and no concept of spacetime as we know it exists. As the temperature drops and the graph cools, it is conjectured to undergo a phase transition to a Phase II where spacetime forms. It will then look like a spacetime manifold on large scales with only near-neighbour points being connected in the graph. The hypothesis of quantum graphity is that this geometrogenesis models the condensation of spacetime in the big bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...


Event symmetry and string theory

String theory is formulated on a background spacetime just as quantum field theory is. Such a background fixes spacetime curvature, which in general relativity is like saying that the gravitational field is fixed. However, analysis shows that the excitations of the string fields act as gravitons, which can perturb the gravitational field away from the fixed background. So, string theory actually includes dynamic quantised gravity. More detailed studies have shown that different string theories in different background spacetimes can be related by dualities. There is also good evidence that string theory supports changes in topology of spacetime. Relativists have therefore criticised string theory for not being formulated in a background independent way, so that changes of spacetime geometry and topology can be more directly expressed in terms of the fundamental degrees of freedom of the strings.

The difficulty in achieving a truly background independent formulation for string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

 is demonstrated by a problem known as Witten's Puzzle. Ed Witten asked the question "What could the full symmetry group of string theory be if it includes diffeomorphism invariance on a spacetime with changing topology?". This is hard to answer because the diffeomorphism group for each spacetime topology is different and there is no natural way to form a larger group containing them all such that the action of the group on continuous spacetime events makes sense. This puzzle is solved if the spacetime is regarded as a discrete set of events with different topologies formed dynamically as different string field configurations. Then the full symmetry need only contain the permutation group of spacetime events. Since any diffeomorphism for any topology is a special kind of permutation on the discrete events, the permutation group does contain all the different diffeomorphism groups for all possible topologies.

There is some evidence from Matrix Models that event-symmetry is included in string theory. A random matrix model can be formed from a random graph model by taking the variables on the links of the graph and arranging them in a N by N square matrix, where N is the number of nodes on the graph. The element of the matrix in the nth column and mth row gives the variable on the link joining the nth nodes to the mth node. The event-symmetry can then be extended to a larger N dimensional rotational symmetry.

In string theory, random matrix
Random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems...

 models were introduced to provide a non-perturbative formulation of M-Theory
In theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...

 using noncommutative geometry
Noncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...

. Coordinates of spacetime are normally commutative but in noncommutative geometry they are replaced by matrix operators that do not commute. In the original M(atrix) Theory these matrices were interpreted as connections between instantons (also known as D0-branes), and the matrix rotations were a gauge symmetry. Later, Iso and Kawai reinterpreted this as a permutation symmetry of space-time events and argued that diffeomorphism invariance was included in this symmetry. The two interpretations are equivalent if no distinction is made between instantons and events, which is what would be expected in a relational theory. This shows that Event Symmetry can already be regarded as part of string theory.

Greg Egan's Dust Theory

The first known publication of the idea of event symmetry is in a work of science fiction rather than a journal of science. Greg Egan
Greg Egan
Greg Egan is an Australian science fiction author.Egan published his first work in 1983. He specialises in hard science fiction stories with mathematical and quantum ontology themes, including the nature of consciousness...

 used the idea in a short story called "Dust" in 1992 and expanded it into the novel Permutation City
Permutation City
Permutation City is a 1994 science fiction novel by Greg Egan that explores many concepts, including quantum ontology, via various philosophical aspects of artificial life and simulated reality. Sections of the story were adapted from Egan's 1992 short story "Dust" which dealt with many of the same...

in 1995. Egan used dust theory as a way of exploring the question of whether a perfect computer simulation of a person differs from the real thing. However, his description of the dust theory as an extension of general relativity is also a consistent statement of the principle of event symmetry as used in quantum gravity.

The essence of the argument can be found in chapter 12 of "Permutation City". Paul, the main character of the story set in the future, has created a copy of himself in a computer simulator. The simulation runs on a distributed network sufficiently powerful to emulate his thoughts and experiences. Paul argues that the events of his simulated world have been remapped to events in the real world by the computer in a way that resembles a coordinate transformation in relativity. General relativity only allows for covariance under continuous transformations whereas the computer network has formed a discontinuous mapping that permutes events like "a cosmic anagram". Yet Paul's copy in the simulator experiences physics as if it were unchanged. Paul realises that this is "Like […] gravity and acceleration in General Relativity — it all depends on what you can't tell apart. This is a new Principle of Equivalence, a new symmetry between observers."