Reversible or not?
Newtonian mechanics is time reversible. In the absence of dissipation, that is. Roll a stone on the ground - it does not roll forever. And once stopped, it does not by itself start rolling all over again. 2d law. Being at rest is the natural state of things, Aristotle opined -- things that are not at rest undergo an irreversible transition to that state. Galileo had to labor mightily to disabuse people of this notion - in the absence of interactions, things move with constant velocities, he said. Interactions (forces), Newton said, cause velocities to change according to his equation of motion. Switching the direction of time in the equation just reveres the motions. Motions with time moving toward the future are just as acceptable as motions the other way around. So, how does irreversibility arise?
In his description of the discussion of the crumpled pieces of paper that were burnt, Paul argued that the ashes and smoke etc. contain all of the information present originally -- implying, like Laplace, that if some intelligence could determine the precise positions and velocities of each of these particles, and if all of their velocities could be reversed, then the initial state would be regained. A Newtonian scenario. The trick here, though, is in the accessibility of complete and precise information about the state of affairs after the burning. Which hints at a messier problem that we will have to address later - does information reside in the system independently of an observer?
In his derivation of the 2d law, Boltzmann used a "coarse-grained" description of the motions of molecules. That is, rather than try to account for the positions and velocities of each of a very large number of particles (impossible in practice, in any event), he used categories, each category defined by neighboring values of position and of velocity. In the paper-burning experiment, Boltzmann would allow knowledge only of the number of occupants of the boxes into which he has categorized the particles, with only uncertain information about the positions and velocities of the occupants, and no information whatsoever about which particle is to be associated with what position or velocity. This categorization, it is argued, is what introduced the irreversibility (see, e.g., Rothman, 1998; Stapp, 2004). If you don't coarse-grain, you get Newton. But coarse-graining is a subjective process. So was the "derivation" a swindle? Was it all just in Boltzmann's mind?
Quantum Mechanics to the rescue.
Boltzmann, in fact, anticipated quantum mechanics. More precisely, he anticipated the Heisenberg uncertainty principle. The coordinates of the mathematical space in which Boltzmann (and physicists in general) describes the motions of particles are positions and momenta, momentum = mv, where m is the mass of the particle, and v its velocity. The motion of a particle moving in one dimension can be represented by a point in a space one of whose coordinate axes is position, and the other, momentum. As the particle changes its position and its velocity, the point representing it moves in this two-dimensional space. Coarse-graining this space involves subdividing it into boxes. In a coarse-grained description, the precise position and momentum of the particle are no longer available, the only information available is the location of the box in which the representative point is located. The sizes of the box along the position and momentum axes are the uncertainties of the positions and momenta of the particles. In the Newtonian scheme, where the motions of points are described, both uncertainties can be zero, the box can be reduced to a point, both position and momentum can simultaneously be determined with infinite precision, and reversibility is possible in principle. In the coarse-grained scheme, the uncertainties of both are finite, and reversibility is lost. Before Heisenberg, coarse-graining was considered subjective and arbitrary. The Heisenberg principle, which requires that the product of the uncertainties in position and in momentum - the area of the box -- must exceed Planck's constant, not only demands coarse-graining, it also prescribes the smallest possible box size.
"In one sense there is no nontrivial objective second law in classical physics: a classical state is supposed to be objectively well defined, and hence it always has probability one. Consequently, the entropy is zero at the outset and remains so forevermore. Normally, however, one adopts some rule of "coarse graining" that destroys information and hence allows probabilities to be different from unity, and then embarks upon an endeavor to deduce the laws of thermodynamics from statistical considerations." (Stapp, 2004).
Quantum mechanics introduces irreversibility in yet another way. Like Newton's equation of motion, Schrödinger's equation, its quantum mechanical counterpart, is time reversible. Unlike Newton's equation, however, Schrödinger's equation does not describe the time evolution of something directly measurable, like position, or velocity. Rather, it describes the behavior of a wave function which contains within it the probabilities of finding the system in any one of its possible states - its position, say, or its velocity. In general, quantum theory does not presume that the system it describes is in a certain state even if subsequent measurement shows it to be in that state. This is unlike the situation in Newtonian physics where it is assumed that an observation reveals some property that already existed prior to the observation. Rather, "a quantum measurement changes the system into one of the possible new states defined by the measurement apparatus in a fundamentally unpredictable way, and thus cannot be claimed to reveal a property existing before the measurement is performed." (Brukner & Zeilinger, 2002). The result of the measurement provides no information about the state of the system before the measurement was made. Quantum measurement is an irreversible process.
It's not just T, it's TCP.
Physicists are not really all that enamored of time reversal or time symmetry by itself. Rather, it's a combination of three symmetries that they hold dear: time reversal (T), parity inversion (P, or mirror symmetry -- left becomes right, up becomes down, and forward becomes backward), and charge conjugation (C, particles become antiparticles). Given any process, that process obtained from the first by reversing time, inverting space, and changing particles to antiparticles will also occur. Any one of the symmetries may be violated provided one of the others also is. Experiments done in the 1960's first showed that the decay of certain particles (kaons, so called) violate the combination of C and P. Therefore, T must also be violated. Time reversal is not sacred. This violation of time reversal, in fact, is used to explain why there seem to be more particles than antiparticles in our universe. The small violation of time reversal invariance in the decay of the kaon is yet another arrow of time (see Hawking, 1988 or Rothman, 1998 for more arrows).
But if we are to base our discussion of things like information and entropy on quantum mechanical notions (which perhaps we should if the world is ultimately quantum mechanical), then we face other problems. One problem is that raised by Paul about whether an observer is necessary for information to exist. Another has to do with the meaning of a quantum mechanical observation mentioned above. Both are related to the claim that "the referent of quantum physics is not reality per se but, as Niels Bohr said, it is what can be said about the world." (Zielinger 2004). And beyond all these is the fact that there does not exist a universally accepted interpretation of quantum mechanics, although the "Copenhagen Interpretation" is probably as close to universal as one can get. When physicists start philosophizing about the meaning of quantum mechanics, they are usually enjoined by other physicists to shut up and compute! Quantum mechanical computations are non-controversial and in some cases have been verified by experiment with a precision of one part in ten billion. Discussions about the meaning of quantum mechanics, on the other hand, never seem to lead to a consensus. But this has never kept physicists, especially older ones, from speculating about what quantum mechanics really means.
Quantum of information?
One physicist who refused to just shut up and compute is Anton Zeilinger, an experimental physicist at the University of Vienna, where Boltzmann used to teach, and whose laboratory is located on Boltzmanngasse, just to make the connenctions complete. Inspired by some questions raised by John Wheeler (who coined the term "black hole" among many other accomplishments), questions which Wheeler called RBQ's - really big questions -- "Why the Quantum ? It from Bit? [does the universe (It) emerge from information (Bit)?]; A Participatory Universe? [does the observer participate in shaping the universe?], Zeilinger uses the notion of information to try to define an Urprinzip, a foundational principle, for quantum mechanics - a "pithy, comprehensible maxim that anchor[s] the formulae [of quantum mechanics] in the everyday world." (von Bayer, 2001).
Zeilinger analyzes the information content of quantum systems. Systems are described by sets of propositions whose truths are measurable. Complex systems ought to be decomposable to simpler systems described by fewer propositions. But this decomposition has got to end somewhere, and where it ends is an elementary [quantum] system: An elementary system represents the truth value of one proposition. The truth value of one proposition is a bit. So, An elementary system carries one bit of information (Zeilinger, 1999).
If an elementary system carries only one bit, it can exist only in one of two states - it is quantized. If complex systems are collections of elementary systems, then they, too are quantized. "The quantum is then a reflection of the fact that all we can do is make statements about the world expressed in a discrete number of bits." (Zeilinger, 2004).
Finite information content also leads to the uncertainty principle. In the case of an elementary system, only one yes-no question can lead to a definite answer. Asking another question will necessarily give a random answer - this is the version of the uncertainty principle for an elementary system. "Quantum physics," he and his student Brukner assert, "is an elementary theory of information." (Brukner & Zeilinger, 2002).
A Participatory Universe?
Another physicist who did not just compute is Henry Stapp of Lawrence Berkeley Laboratories who goes even further and probes connections between consciousness and the universe and the extent to which quantum theory demands that conscious choices [by human observers] participate in determining the evolution of the universe. The manuscript of a book in which he describes these musings is on the web (Stapp,2004)
Below are some random quotes from the manuscript:
in direct contrast to the ideas of classical physical theory, quantum theory is about "our knowledge." We, and in particular our mental aspects, have entered into the structure of basic physical theory. p. 13
the realities of prime importance in a scientific account of nature are the actions of agents who gather knowledge or information, and the knowledge or information that they gather. The physical world, as represented in the theory, brings the knowledge-acquiring actions of agents, and the knowledge garnered by those actions, into the theory in fundamental and non eliminable way. p. 21
Wigner (1961): "the laws of quantum mechanics cannot be formulated...without recourse to the concept of consciousness.''
Bohr (1934): "In our description of nature the purpose is not to disclose the real essence of phenomena but only to track down as far as possible relations between the multifold aspects of our experience."
Physics vs logic?
The issues that Paul raised about differences between physics on the one hand and logic and the other sciences, on the other, reflects arguments between the philosopher Carnap and von Neumann and Pauli, described in some detail in Stapp's manuscript:
Rudolph Carnap was a distinguished philosopher, and member of the Vienna Circle. He was in some sense a dualist. He had studied one of the central problems of philosophy, namely the distinction between analytic statements and synthetic statements. (The former are true or false by virtue of a specified set of rules held in our minds, whereas the latter are true or false by virtue their concordance with physical or empirical facts.) His conclusions had led him to the idea that there are two different domains of truth, one pertaining to logic and mathematics and the other to physics and the natural sciences. This led to the claim that there are "Two Concepts of Probability," one logical the other physical. That conclusion was in line with the fact that philosophers were then divided between two main schools as to whether probability should be understood in terms of abstract idealizations or physical sequences of outcomes of measurements. Carnap's bifurcations implied a similar division between two different concepts of information, and of entropy. In 1952 Carnap was working at the Institute for Advanced Study in Princeton and about to publish a work on his dualistic theory of information, according to which epistemological concepts like information should be treated separately from physics. Von Neumann, in private discussion, raised objections, and Pauli later wrote a forceful letter, asserting that "I am quite strongly opposed to the position you take." Later he adds "I am indeed concerned that the confusion in the area of the foundations of statistical mechanics not grow further (and I fear very much that a publication of your work in its present form would have this effect)." Carnap's view was in line with the Cartesian separation between a domain of real objective physical facts and a domain of ideas and concepts. But von Neumann's view, and also Pauli's, linked the probability that occurred in physics, in connection with entropy, to knowledge, in direct opposition to Carnap's view that epistemology (considerations pertaining to knowledge) should be separated from physics. pp. 58-59
Stapp summarizes von Neumann's view, which is the basis of his book, that there is a division between two types of processes in quantum physics. One, which von Neumann called Process 1, are those which involve choices made by an observer about how and what observations to make. Another, called Process 2 are those that proceed in accordance with quantized classical laws - Schrödinger's equation, for instance. The latter are continuous in time and presumably reversible, the former are discrete and irreversible. Process 1 events are observer-dependent, Process 2 events are not. The entropy associated with process 2 events, like that of classical physics without coarse-graining, does not change. The entropy associated with Process 1 events never decreases.
Because Process 1 events involve conscious decisions on the part of the observer, Stapp uses these events to connect the observer and the evolution of the system being observed (actually, Bohr et al. and von Neumann already did that), and uses it as an entry to a discussion of the mind-brain problem ... but that is for somebody else to pursue.
Hawking, S.(1988). A Brief History of Time. New York: Bantam Books.
Rothman, T. and Sudarshan, G. (1998). Doubt and Certainty. Reading, MA: Perseus Books.
Stapp, H. (2004). The Mindful Universe. (http://www-physics.lbl.gov/~stapp/MUI.pdf).
Von Baeyer, H. C. (2001). In the beginning was the bit. New Scientist Magazine, 17 February 2001.
Von Baeyer, H. C. (2004). Information. Cambridge, MA: Harvard University Press.
Zeilinger, A. (1996). On the Interpretation and Philosophical Foundation of Quantum Mechanics. "Vastakohtien todellisuus," Festschrift for K. V. Laurikainen, U. Ketvel et al., (Eds), Helsinki University Press. (available in http://www.quantum.univie.ac.at/zeilinger/philosop.html)
Zeilinger, A. (1999). A Foundational Principle for Quantum Mechanics. Foundations of Physics, 29, 631-643.
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