By Amanda Gefter
New Scientist, August
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
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.
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
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.
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
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.
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
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
AR Those are all topics I care about, and
P = NP would be stunning. Must read.