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Wolfram Physics Project: A Discussion with Fay Dowker

Wolfram・170 minutes read

Interest in causal sets for physics and distributed computing applications; emphasis on direct handling of physical degrees of freedom in causal sets, showcasing various outcomes for a given state.

Insights

Interest in causal sets for physics projects and distributed computing applications.
Faye Alka's collaboration with Rafael Sorkin introduced the concept of causal sets as discrete histories in space-time.
Emphasis on direct handling of physical degrees of freedom in causal sets for physics and computing applications.
The multi-way causal graph showcases various outcomes and causal relationships akin to quantum mechanics histories.
Causal sets offer insights into the dynamics of time and space, emphasizing the reality of past events and their impact on the present.

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Recent questions

What are causal sets in physics?
Causal sets are discrete histories in space-time.
How do causal sets impact distributed computing?
Causal sets serve as a foundation for distributed computing.
What is the significance of frames and foliations in causal sets?
Frames and foliations in causal sets aid in physics and computing.
How are multi-way graphs utilized in causal sets?
Multi-way graphs showcase various outcomes and causal relations.
What challenges do causal sets pose in defining space-time singularities?
Causal sets offer insights into space-time singularities.

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Summary

00:00

"Causal Sets: Discrete Space-Time for Physics"

Livestream topic: causal sets, expert guest Faye Alka introduced
Team members introduced: Jonathan Gorod in Cambridge, Taliesin Ben in South Africa, Max with stylish backdrop
Interest in causal sets for physics projects and distributed computing applications
Faye Alka's entry into discrete space-time field through collaboration with Rafael Sorkin
Concept of causal sets as discrete histories in space-time, replacing path integrals
Spatial hypergraph leading to causal graph in causal set theory
Absence of underlying continuum in causal sets, reality defined by causal relations
Interest in frames and foliations in causal sets for physics and distributed computing
Use of causal graphs as discretization method for numerical relativity
Emphasis on direct handling of physical degrees of freedom in causal sets, challenging but crucial in physics and computing applications.

18:51

"Multi-way causal graph reveals quantum mechanics"

The multi-way graph displays possible rewritings based on rules, showcasing various outcomes for a given state.
Orange sections in the graph indicate causal relationships between updating events, showing the order of necessary updates.
The multi-way causal graph represents all possible sequences of updating events and their causal relations, akin to quantum mechanics histories.
The multi-way graph reveals both the path integral and Einstein equations, derived similarly but for different aspects of space-time.
The multi-way causal graph can encompass space-time and branch time causality, offering a broader view of causal relationships.
The classical sequential growth model mirrors the multi-way causal graph, suggesting multiple applications of rules lead to various evolution histories.
The spatial hypergraph serves as a tool to understand space in the causal graph, though ultimately the causal graph contains all essential information.
Defining non-time-like geodesics in causal sets poses challenges, with spatial hypergraphs aiding in understanding causal relationships.
Distributed computing primarily relies on the causal graph, with spatial hypergraphs serving as a stepping stone to reach causal insights.
Time in the model is distinct from space, viewed as a computational process versus a spatial construct, with the relationship crucial in deriving key physics concepts.

35:27

"Probabilistic Growth Model of Causal Relationships"

The classical sequential growth model involves probabilities in growing a single causal set element by element, with each new element choosing a set of past elements.
A stage in this model represents an update event where a causal set of cardinality n becomes n+1 via the birth of a new space-time atom.
The growth in this model only occurs towards the future, with specific probabilities assigned to each step.
The discrete general covariance condition ensures equal probabilities for different paths leading to the same outcome in the growth process.
The multi-way system constructed in this model exhibits a property of branching and merging, ensuring convergence despite branching paths.
The multi-way system operates on strings, replacing specific patterns to reach a final sorted state.
The multi-way graph in this model represents the evolving universe's state, growing spatial states that define causal relationships.
Events in this model signify the birth of a new space-time atom, simplifying the concept of time as inexorable computation.
The model's approach to time as progressive computation applied to a state differs from the traditional view of the entire space-time history at each moment.
Despite foundational differences, the model's causal relationships align with standard relativistic features, allowing for the derivation of Einstein equations from the causal graph's features.

52:48

Developing Energy Momentum Tensor in Causal Sets

A new version of a non-trivial result was developed, including the energy momentum tensor.
The discrete del invasion concept can define a Ricci curvature, particularly for a constant scalar field on a causal zone.
Most causal sets are not related to continuum space-time, posing an entropic problem when going discrete.
Dynamics are essential to identify nice manifold-like causal sets from the majority of garbage-like ones.
Different rules can lead to various outcomes, with some producing nice manifolds resembling spatial hypergraphs.
Dimension estimators can be used to measure the dimension of causal sets, akin to Lorentzian nature.
Maya dimension estimators efficiently estimate dimensions based on the volume and number of relations in a causal interval.
To generate interesting posets, one can use transitive percolation or grow a causal set by adding relations between existing elements.
In the Lorentzian case, the order of elements matters, with relations added based on probabilities.
A method to generate posets involves placing points in a Minkowski diamond based on random coordinates and ordering them according to specific criteria.

01:10:01

Mapping Points with 45-Degree Rotation Matrix

The process involves creating points and applying a 45-degree rotation matrix to map them.
A list and a graphics line are used to plot the points initially.
Points are connected with lines to visualize their order.
The order is determined by considering light cones and establishing a partial order.
A total order is selected from the partial order by joining points.
The process involves creating a Hasse diagram using directed edges.
A function is applied to pairs of points to determine directed edges.
A code is written to generate a new graph with specific probabilities.
The transitive reduction is computed for each graph to obtain Hasse diagrams.
Different probabilities result in various outcomes, with all non-zero probabilities leading to infinitely many posts in the causal set.

01:26:30

Space-time Volume Estimation and Dimension Calculation

The midpoint scaling dimension construction involves selecting an initial and final event to estimate the volume of space-time between them.
An asymmetric version of this construction results in a spacetime cone.
In the continuum, calculating the dimension involves projecting a space-time cone in a specific time-like direction and observing its volume growth.
Locally defining geodesic normal coordinates aids in determining the geometry of a space-like hypersurface.
The volume of a cone in ordinary continuum geometry is determined by the dimension of the unseen manifold and a correction term related to the space-time curvature.
The formula for the volume of a causal interval includes correction terms proportional to the Ricci scalar and curvature tensor projection.
Spectral dimension involves random walkers to determine dimension, but it's challenging with causal sets due to their Lorentzian nature.
The formula for the volume of a four-sphere in Euclidean space includes correction terms related to the Ricci scalar.
Constructing future and past light cones involves aligning them to reverse the cone construction effectively.
Understanding the trace part in formulas and the derivation of Einstein's equations involves showing the vanishing of certain terms to preserve space dimensions.

01:44:01

Causal edges, Lagrangian, and faithful embedding relations

The interpretation of causal edges involves energy flux through space and momentum flux through time.
The Lagrangian represents the divergence of causal edges, leading to the equivalence of the final path integral and the Einstein equations.
Jd6 in space-time are influenced by T minou, while paths in quantum space are affected by the Lagrangian in a multi-way graph.
Embedding a causal set into space-time faithfully involves uniform density of elements and respecting the space-time causal order.
The edges in the causal set are not physical lines in space-time but binary relations, not geometric or one-dimensional.
Faithful embedding requires the number of vertices in a region to correspond to its volume and preserving the causal partial order.
The faithful embedding transformation is not about precise metric correspondence but a volumetric correspondence and preserving the causal order.
The causal set is a skeleton of the Lorentzian manifold, capturing geometry at a scale between discreteness and continuity.
Skeletonization involves random sampling to generate causal sets, not guaranteeing isomorphism with Lorentzian manifolds.
The process of faithful embedding and skeletonization are inverse procedures, with random sampling leading to various causal sets.

02:01:03

Constructing Multi-Way Causal Sets from Lorentzian Space-Time

The text discusses randomly sampling a Lorentzian space-time to create causal sets that faithfully embed by definition.
There are multiple causal sets that can fulfill this task.
The piece of code being run is different from previous code and does not involve probabilities.
The code is a more elegant version of the previous one, growing causal sets in a multi-way manner.
A multi-way version of the construction involves considering all possible causal sets that can be grown.
The construction involves space-like edges and branch-like edges in a graph.
A transitive percolation process is mentioned in the discussion.
The failure of a two-chain to change is attributed to a bug in adding edges.
The concept of natural labeling and total order in causal sets is explained.
The text delves into the idea of inertial frames and foliations in causal sets, discussing their implications and challenges.

02:17:33

Causal Invariance and Lorentz Symmetry in Physics

The equivalence of different foliation x' is a consequence of causal invariance property.
Causal invariance implies that every multi-way evolution branch yields isomorphic causal graphs.
Different updating orders can be parametrized by different foliation x'.
Causal invariance implies Lorentz symmetry.
Proving Lorentz invariance is crucial for understanding the dynamics of causal sets.
The emergence of Navier-Stokes equations does not depend on microscopic details in fluid dynamics.
Two branches in exploring causal sets: proving manifold-like behavior and generic properties.
The physical universe's behavior in a weird limiting case may not resemble continuum space.
The density of the grid in causal sets is high enough to cover the universe's volume.
Estimates suggest an elementary length of 10^-91 meters and 10^400 elements in the universe's causal graph.

02:34:14

Causal Sets and Space-Time Measure Challenges

The hypothesis is that causal sets are sufficient, requiring no additional decorations, with finitely many elements in a finite space-time range.
Derivation of general relativity provides a way to define a natural measure using random walks in discrete metric measure spaces.
A transportation distance, Vashta stein distance, allows defining a naturally induced measure on space in both finite and discrete countable cases.
Confidence in statements about limits to continuum manifolds is due to derivations like the Einstein equations in both finite and countable cases.
The causal set program has been around longer, but challenges remain in defining a compatible measure and scalar field.
Transitioning from finiteness to accountability may pose issues beyond measure compatibility in the formalism.
Space-time singularities are understood through causal graphs, with features like event horizons and disconnected spacial hypergraphs.
Classical sequential growth models exhibit posts leading to effective post-dynamics and self-organized criticality.
Behavior after posts in classical sequential growth models tends towards transitive percolation, impacting cosmological fine-tuning problems.
The complexity of classical sequential growth models contrasts with simpler update rules in causal sets, with each update introducing new refinements.

02:50:29

"Exploring Dynamics and Singularities in Causal Sets"

Parameters exist for each stage in models, defining a dynamical space.
The space of dynamics is infinite-dimensional, like a projective space.
Renormalization acts on the space of dynamics, transitioning between different dynamics.
Accumulating a large causal set leads to a special property at certain points.
Singularities in causal graphs indicate unique properties in the causal set.
Time-like and space-like singularities have distinct characteristics.
Trouser universes show disconnected causal graphs, not disconnected space.
Dimensional perturbations in space-time can lead to fractional dimensions.
Finding a rule that exhibits all desired properties in models remains a challenge.
Computational zoology and studying causal set models require more experimentation.

03:06:46

"Structural Challenges in Causal Sets"

Causal sets struggle with anything related to space due to lacking structure.
An antichain in a causal circle lacks causal relations, offering only cardinality.
Thickening antichains by adding steps allows for extracting spatial topology information.
While antichains lack structure, they are essential for modeling space-like hypersurfaces.
Different updating strategies in causal graphs lead to distinct but isomorphic causal sets.
The multi-way causal graph represents all possible updating orders and resulting causal sets.
A natural foliation in causal graphs aids in understanding the order of events.
Thickening procedures smooth out wild antichains, revealing preferred foliations.
Thickening works by expanding anti chains to include a fixed number of elements in the future.
Posts in distributed computing act as bottlenecks, reducing parallel efficiency in operations.

03:23:19

"Rare Post Sets, Maximal Elements, Causal Graphs"

The number of possible post sets is exponentially rare, with only one out of the possible orderings being selected for a post.
The number of maximal elements is roughly one over P, with a new element attaching itself with probability P to one of the existing elements.
There is a finite probability that the next element will attach itself above all existing maximal elements, leading to a space-like singularity.
Causal sets offer potential insights into the dynamics of time and space, providing a unique perspective on the passage of time and event occurrences.
The focus on causal graphs highlights the influences of the past without needing to preserve every aspect of the past structure, emphasizing the reality of past events and their impact on the present.

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