Beyond the Standard Model
Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons
Standard Model
Evidence Hierarchy problem Dark matter Dark energy Quintessence Phantom energy Dark radiation Dark photon Cosmological constant problem Strong CP problem Neutrino oscillation
Theories Brans–Dicke theory Cosmic censorship hypothesis Fifth force F-theory Theory of everything Unified field theory Grand Unified Theory Technicolor Kaluza–Klein theory 6D (2,0) superconformal field theory Noncommutative quantum field theory Quantum cosmology Brane cosmology String theory Superstring theory M-theory Mathematical universe hypothesis Mirror matter Randall–Sundrum model N = 4 supersymmetric Yang–Mills theory Twistor string theory Dark fluid Doubly special relativity de Sitter invariant special relativity Causal fermion systems Black hole thermodynamics Unparticle physics Graviphoton Graviscalar Graviton Gravitino Massive gravity Gauge gravitation theory Gauge theory gravity CPT symmetry
Supersymmetry MSSM NMSSM Superstring theory M-theory Supergravity Supersymmetry breaking Extra dimensions Large extra dimensions
Quantum gravity False vacuum String theory Spin foam Quantum foam Quantum geometry Loop quantum gravity Quantum cosmology Loop quantum cosmology Causal dynamical triangulation Causal fermion systems Causal sets Canonical quantum gravity Semiclassical gravity Superfluid vacuum theory
Experiments ANNIE Gran Sasso INO LHC SNO Super-K Tevatron NOvA

The causal sets program is an approach to quantum gravity. Its founding principles are that spacetime is fundamentally discrete (a collection of discrete spacetime points, called the elements of the causal set) and that spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between spacetime events.

History

For some decades after the formulation of General Relativity, the attitude towards Lorentzian geometry was mostly dedicated to understanding its physical implications and not concerned with theoretical issues.^[1] However, early attempts to use causality as a starting point have been provided by Weyl and Lorentz.^[2] Alfred Robb in two books in 1914 and 1936 suggested an axiomatic framework where the causal precedence played a critical role.^[1] The first explicit proposal of quantising the causal structure of spacetime is attributed by S. Surya^[1] to Kronheimer and Penrose,^[3] who invented Causal spaces in order to admit structures which can be very different from a manifold. Causal spaces are defined axiomatically, by considering not only causal precedence, but also chronological precedence.

The program of causal sets is based on a theorem^[4] by David Malament, extending former results by E.C. Zeeman^[5] and Hawking, King, McCarthy.^[6][1] Malament Theorem states that if there is a bijective map between two past and future distinguishing space times that preserves their causal structure then the map is a conformal isomorphism. The conformal factor that is left undetermined is related to the volume of regions in the spacetime. This volume factor can be recovered by specifying a volume element for each space time point. The volume of a space time region could then be found by counting the number of points in that region.

Causal sets was initiated by Rafael Sorkin who continues to be the main proponent of the program. He has coined the slogan "Order + Number = Geometry" to characterize the above argument. The program provides a theory in which space time is fundamentally discrete while retaining local Lorentz invariance.

Definition

A causal set (or causet) is a set ${\displaystyle C}$ with a partial order relation ${\displaystyle \preceq }$ that is

• Reflexive: For all ${\displaystyle x\in C}$, we have ${\displaystyle x\preceq x}$.
• Antisymmetric: For all ${\displaystyle x,y\in C}$, we have ${\displaystyle x\preceq y}$ and ${\displaystyle y\preceq x}$ implies ${\displaystyle x=y}$.
• Transitive: For all ${\displaystyle x,y,z\in C}$, we have ${\displaystyle x\preceq y}$ and ${\displaystyle y\preceq z}$ implies ${\displaystyle x\preceq z}$.
• Locally finite: For all ${\displaystyle x,z\in C}$, we have ${\displaystyle \{y\in C|x\preceq y\preceq z\}}$ is a finite set.

We'll write ${\displaystyle x\prec y}$ if ${\displaystyle x\preceq y}$ and ${\displaystyle x\neq y}$.

The set ${\displaystyle C}$ represents the set of spacetime events and the order relation ${\displaystyle \preceq }$ represents the causal relationship between events (see causal structure for the analogous idea in a Lorentzian manifold).

Although this definition uses the reflexive convention we could have chosen the irreflexive convention in which the order relation is irreflexive and asymmetric.

The causal relation of a Lorentzian manifold (without closed causal curves) satisfies the first three conditions. It is the local finiteness condition that introduces spacetime discreteness.

Comparison to the continuum

Given a causal set we may ask whether it can be embedded into a Lorentzian manifold. An embedding would be a map taking elements of the causal set into points in the manifold such that the order relation of the causal set matches the causal ordering of the manifold. A further criterion is needed however before the embedding is suitable. If, on average, the number of causal set elements mapped into a region of the manifold is proportional to the volume of the region then the embedding is said to be faithful. In this case we can consider the causal set to be 'manifold-like'.

A central conjecture of the causal set program, called the Hauptvermutung ('fundamental conjecture'), is that the same causal set cannot be faithfully embedded into two spacetimes that are not similar on large scales.

It is difficult to define this conjecture precisely because it is difficult to decide when two spacetimes are 'similar on large scales'. Modelling spacetime as a causal set would require us to restrict attention to those causal sets that are 'manifold-like'. Given a causal set this is a difficult property to determine.

Sprinkling

A plot of 1000 sprinkled points in 1+1 dimensions

The difficulty of determining whether a causal set can be embedded into a manifold can be approached from the other direction. We can create a causal set by sprinkling points into a Lorentzian manifold. By sprinkling points in proportion to the volume of the spacetime regions and using the causal order relations in the manifold to induce order relations between the sprinkled points, we can produce a causal set that (by construction) can be faithfully embedded into the manifold.

To maintain Lorentz invariance this sprinkling of points must be done randomly using a Poisson process. Thus the probability of sprinkling ${\displaystyle n}$ points into a region of volume ${\displaystyle V}$ is

${\displaystyle P(n)={\frac {(\rho V)^{n}e^{-\rho V}}{n!}}}$

where ${\displaystyle \rho }$ is the density of the sprinkling.

Sprinkling points as a regular lattice would not keep the number of points proportional to the region volume.

Geometry

Some geometrical constructions in manifolds carry over to causal sets. When defining these we must remember to rely only on the causal set itself, not on any background spacetime into which it might be embedded. For an overview of these constructions, see.^[7]

Geodesics

A plot of geodesics between two points in a 180-point causal set made by sprinkling into 1+1 dimensions

A link in a causal set is a pair of elements ${\displaystyle x,y\in C}$ such that ${\displaystyle x\prec y}$ but with no ${\displaystyle z\in C}$ such that ${\displaystyle x\prec z\prec y}$.

A chain is a sequence of elements ${\displaystyle x_{0},x_{1},\ldots ,x_{n}}$ such that ${\displaystyle x_{i}\prec x_{i+1}}$ for ${\displaystyle i=0,\ldots ,n-1}$. The length of a chain is ${\displaystyle n}$. If every ${\displaystyle x_{i},x_{i+1}}$ in the chain form a link, then the chain is called a path.

We can use this to define the notion of a geodesic between two causal set elements, provided they are order comparable, that is, causally connected (physically, this means they are time-like). A geodesic between two elements ${\displaystyle x\preceq y\in C}$ is a chain consisting only of links such that

1. ${\displaystyle x_{0}=x}$ and ${\displaystyle x_{n}=y}$
2. The length of the chain, ${\displaystyle n}$, is maximal over all chains from ${\displaystyle x}$ to ${\displaystyle y}$.

In general there can be more than one geodesic between two comparable elements.

Myrheim^[8] first suggested that the length of such a geodesic should be directly proportional to the proper time along a timelike geodesic joining the two spacetime points. Tests of this conjecture have been made using causal sets generated from sprinklings into flat spacetimes. The proportionality has been shown to hold and is conjectured to hold for sprinklings in curved spacetimes too.

Dynamics

An ongoing task is to develop the correct dynamics for causal sets. These would provide a set of rules that determine which causal sets correspond to physically realistic spacetimes. The most popular approach to developing causal set dynamics is based on the sum-over-histories version of quantum mechanics. This approach would perform a "sum-over-causal sets" by growing a causal set one element at a time. Elements would be added according to quantum mechanical rules and interference would ensure a large manifold-like spacetime would dominate the contributions. The best model for dynamics at the moment is a classical model in which elements are added according to probabilities. This model, due to David Rideout and Rafael Sorkin, is known as classical sequential growth (CSG) dynamics.^[10] The classical sequential growth model is a way to generate causal sets by adding new elements one after another. Rules for how new elements are added are specified and, depending on the parameters in the model, different causal sets result.

In analogy to the path integral formulation of quantum mechanics, one approach to developing a quantum dynamics for causal sets has been to apply an action principle in the sum-over-causal sets approach. Sorkin has proposed a discrete analogue for the d'Alembertian, which can in turn be used to define the Ricci curvature scalar and thereby the Benincasa–Dowker action on a causal set.^[11][12] Monte-Carlo simulations have provided evidence for a continuum phase in 2D using the Benincasa–Dowker action.^[13]

See also

• Causal dynamical triangulation (CDT)
• Causal structure
• General relativity
• Order theory

References

Further reading

External links

• The causal set approach to quantum gravity a review article by Joe Henson on causal sets
• Space-time as a causal set - one of the first papers by Luca Bombelli, Joohan Lee, David Meyer, and Rafael D. Sorkin
• Geometry from order: causal sets - non-technical article by Rafael D. Sorkin on Einstein Online
• Focus Issue: The causal set approach to quantum gravity Classical and Quantum Gravity Vol. 35, 2018