Einstein's Unfinished Revolution
Page 33
* Wallace and Myrvold have since left Oxford; Deutsch, Greaves, and Saunders remain as of 2018.
* The idea that decoherence defines the branches in the Many Worlds Interpretation had been suggested earlier by others including Heinz-Dieter Zeh, Wojciech Zurek, Murray Gell-Mann, and James Hartle.
* Indeed, the holographic principle (which is defined on pages 259–260) requires that any system that can be fit into a box with walls of a finite area has a finite number of states. This is certainly the case with any system of the kind we are discussing here—an atomic system interacting with a measuring instrument. One reply that might be given is that we live in an ever-expanding universe, which may imply that the dimension of the state space is continually expanding, in which case there is no Poincaré recurrence. This raises several fascinating issues. But, for the moment, it is enough to note that this amounts to claiming that quantum mechanics makes sense only when applied to the universe as a whole.
* Note that this principle cannot be taken as an end of the story of making sense of probabilities, because it isolates something that we would really like to understand from first principles. What is missing is a convincing argument that would compel us to line up our subjective probabilities with the objective chances.
* Including Greaves, Myrvold, and Wallace. I should note that they introduce lines of argument I haven’t mentioned here, about which experts disagree, so the situation is somewhat more complex than the overview I have presented.
* My own view as to how science works and the role of individual judgments in forming a consensus of the whole scientific community is outlined in chapter 17 of my book The Trouble with Physics.
* Rather than just ruling out local hidden variable theories.
* At this point I have to make an aside, which the non-expert reader may skip.
An expert might object to my characterization of Shannon information by pointing out that that quantity is equal to the negative of the entropy of the message. Entropy, they would argue, is an objective, physical property of nature, which is governed (when the system is in thermodynamic equilibrium) by the laws of thermodynamics. Hence, by virtue of its connection with entropy, Shannon information must be objective and physical.
I would answer by making three points. First, it is changes in the thermodynamic entropy, not the entropy itself, that come into the laws of thermodynamics. Second, as Karl Popper pointed out years ago, the statistical definition of entropy which Shannon information is related to is not a completely objective quantity. It depends on a choice of coarse-graining, which provides us with an approximate description of the system. The entropy of the exact description, given in terms of the exact state, is always zero. The need to specify this approximation brings an element of subjectivity to the definition of entropy. This is seen in the dependence of the entropy of a quantum system on a splitting into two subsystems. Finally, the attribution of entropy to a message is a definition, which defines entropy in terms of Shannon information.
* This step of the argument is called “Wigner’s friend.”
* As I argue in Time Reborn, and in The Singular Universe and the Reality of Time, with Roberto Mangabeira Unger.
* Which I defined back in chapter 8.
* The phases of a wave function refers to the locations of the peaks and troughs, and the patterns they form.
* That is, a complex number.
* For much more on this point, see my books, Time Reborn, and The Singular Universe and the Reality of Time, with Roberto Mangabeira Unger.
* In technical terms, the observables algebra and inner product.
* String theory does not do this; instead, it fixes the total number of dimensions, including possible microscopic dimensions. That could be a good thing, if it didn’t give us infinite choices for the geometry and number of these hypothesized tiny dimensions.
* An event can be followed by a second event that reverses the action of the first, but then you have two events; this is not equivalent to a spacetime in which neither happened.
* This is not a new idea; as I noted in chapter 9, Roger Penrose mentioned it as motivation for his spin networks model in the early 1960s.
* Juan Maldacena and Leonard Susskind have since introduced a version of this idea they call ER=EPR, where ER stands for an Einstein-Rosen bridge, which is a wormhole connecting two points far from each other in space (“Cool Horizons for Entangled Black Holes,” arXiv:1306.0533).
* More precisely, the negative of the action.
* In this real ensemble formulation, the information in a wave function of a quantum system is spread throughout the universe, coded into the configurations of the copies. A key question is how many copies a system must have for the information coded into the copies to be adequate to reproduce the information in the wave function. That information increases exponentially with the number of particles in the quantum system. But the number of copies of a system the universe will likely contain decreases rapidly with the number of particles that make up the system. So there is a size of a system beyond which the information in the copies does not suffice, with the consequence that either quantum mechanics breaks down, or this approach is wrong. I suspect that even modest quantum computers will cross this line.
* I.e., those with the same preparation, evolution, and measurement.
* Some relativists point to the existence of solutions to the Einstein equations which have closed causal loops. I don’t think this has any force because the universe is described by at most one solution to general relativity, and that solution need not have every exotic behavior shown in other solutions. More definitively, those solutions which have closed causal loops (including one proposed by the great logician Kurt Gödel) are very special in that they have a lot of symmetry. If we impose the principle of the identity of indiscernibles, then solutions with symmetries are excluded. These solutions are also unstable and collapse to singularities at the faintest hint of a perturbation.
* These fundamental units of area are equal to the product of Newton’s gravitational constant and Planck’s constant.
* For more about the holographic hypothesis, see my book Three Roads to Quantum Gravity.
* In the Newtonian case, the momentum of a particle is proportional to its velocity. The constant of proportionality is the mass.
* And rotational symmetry implies the conservation of angular momentum.
* For the experts, a CPT transformation.
* A few years later we understood this two-phased behavior in terms of the dynamics of a general class of deterministic dynamical systems, with a finite number of possible states. Such systems evolve to cycles, and the two phases are the phase of convergence to a cycle followed by cyclic behavior. But a cycle is reversible, because each event has a single child and a single parent.
* I should warn the reader not to be misled by the colloquial understanding of a “view” in which it stands for the subjective opinion of an individual.
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