Beyond Spacetime

By Amanda Gefter
New Scientist, August 8, 2011

Edited by Andy Ross

We live in an 8D phase space that merges spacetime and momenergy, say Lee Smolin, Laurent Freidel, Jerzy Kowalski-Glikman, and Giovanni Amelino-Camelia.

In 1938, Max Born saw that several key equations in quantum mechanics remain the same whether expressed in spacetime coordinates or in momentum space coordinates. He wondered about a union of general relativity and quantum mechanics. Born reciprocity suggests that if spacetime can be curved by mass then so can momentum space.

Smolin and colleagues applied standard rules for translating between momentum space and spacetime. They discovered that observers living in a curved momentum space will no longer agree on measurements made in a unified spacetime. For observers in a curved momentum space, even spacetime is relative.

Relative locality could shed light on the black hole information-loss paradox. Stephen Hawking discovered that black holes radiate and evaporate away. Relativity prevents stuff that falls into a black hole from escaping. But quantum mechanics says information cannot just vanish. Relative locality says that as you look back to the time when stuff fell in, you find that locations in spacetime are so fuzzy there is no way to tell how stuff fell in. The paradox dissolves.

To see if momentum space is curved, we can look at light from distant gamma-ray bursts. High-energy photons should arrive later than lower-energy photons from the same burst. NASA's Fermi space telescope collects light from gamma-ray bursts. The data shows the predicted correlation between arrival time and energy. But we need more data.

Shahn Majid, a mathematical physicist at Queen Mary University of London, showed that curved momentum space is equivalent to noncommutative spacetime. The uncertainty principle says it will be fuzzy. Curved momentum space is quantum spacetime in another guise. Momentum space is also quantum. Observers will not agree on momenergy measurements either. If both are relative, what is the true fabric of reality?

Smolin's hunch is that we live in an 8D phase space that represents all possible values of position, time, energy and momentum. In relativity, what one observer views as space, another views as time and vice versa. In Smolin's quantum gravity, what one observer sees as spacetime another sees as momentum space, and only the phase space is absolute and invariant to all observers.

Imperial College London physicist Joćo Magueijo says spacetime and momenergy may be entangled in quantum gravity.

The principle of relative locality

Giovanni Amelino-Camelia, Laurent Freidel, Jerzy Kowalski-Glikman, Lee Smolin

We propose a deepening of the relativity principle according to which the invariant arena for non-quantum physics is a phase space rather than spacetime. Descriptions of particles propagating and interacting in spacetimes are constructed by observers, but different observers, separated from each other by translations, construct different spacetime projections from the invariant phase space. Nonetheless, all observers agree that interactions are local in the spacetime coordinates constructed by observers local to them.
This framework, in which absolute locality is replaced by relative locality, results from deforming momentum space, just as the passage from absolute to relative simultaneity results from deforming the linear addition of velocities. Different aspects of momentum space geometry, such as its curvature, torsion and non-metricity, are reflected in different kinds of deformations of the energy-momentum conservation laws. These are in principle all measurable by appropriate experiments. We also discuss a natural set of physical hypotheses which singles out the cases of momentum space with a metric compatible connection and constant curvature.

Gamma ray burst delay times probe the geometry of momentum space

Laurent Freidel, Lee Smolin

We study the application of the recently proposed framework of relative locality to the problem of energy dependent delays of arrival times of photons that are produced simultaneously in distant events such as gamma ray bursts. Within this framework, possible modifications of special relativity are coded in the geometry of momentum space. The metric of momentum space codes modifications in the energy momentum relation, while the connection on momentum space describes possible non-linear modifications in the laws of conservation of energy and momentum. In this paper, we study effects of first order in the inverse Planck scale, which are coded in the torsion and non-metricity of momentum space. We find that time delays of order Distance * Energies/m_p are coded in the non-metricity of momentum space. Current experimental bounds on such time delays hence bound the components of this tensor of order 1/m_p. We also find a new effect, whereby photons from distant sources can appear to arrive from angles slightly off the direction to the sources, which we call gravitational lensing. This is found to be coded into the torsion of momentum space.

AR  This seems an eminently reasonable way to go for quantum gravity. The 8-space generalization of relativistic 4-space is more deeply motivated than the earlier 5-space Kaluza-Klein generalization for encompassing electromagnetism.

Computational Complexity

Physics arXiv Blog

The computational complexity of a problem measures how the resources needed to solve it scale with the problem size, say n. There are two main classes of complexity: P and NP. Solutions that scale slowly, such as polynomially in n, are in class P. Solutions that scale quickly, or exponentially with n, are in class NP. Problems in class P are easy. NP problems are hard.

MIT computer scientist Scott Aaronson says computational complexity theory will transform philosophy. Aaronson: "Think, for example, of the difference between reading a 400-page book and reading every possible such book, or between writing down a thousand-digit number and counting to that number."

Computational complexity theory can help philosophers decide whether computers will ever think like humans. One way to measure the difference between a human and computer is with a Turing test. If we cannot tell the difference between the conversational responses given by a computer and a human, then their thinking is practically the same.

Imagine a computer that records all conversations it hears between humans. This computer builds up a big database for making conversation. When asked a question, it looks up a human answer in its database. A fast enough computer can always converse like a human. But think about the computational complexity of the problem. A conversation of length n requires computational resources that grow exponentially with n. A database approach will fail for long conversations.

Computational complexity theorists have not yet proved that the classes P and NP are distinct. If someone proved P = NP, it would be game over for the theorists, but this seems most unlikely. Philosophers should accept the importance of complexity theory.

Why philosophers should care about computational complexity

Scott Aaronson

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory - the field that studies the resources (such as time, space, and randomness) needed to solve computational problems - leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume's problem of induction and Goodman's grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.

AR  Those are all topics I care about, and P = NP would be stunning. Must read.