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ISIS Summit: The Information Society at the Crossroads

On Certain Questions Related to Information and Symmetries - In Physics From Certain View of Philosophy of Science

György Darvas

Published Jun 30, 2015• 0 citations
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Abstract

Introduction My presentation aims at answering a part of the questions posted in the invitation to the session. The questions arose in and originated from the FIS discussion on Physical informatics … in October 2014. No doubt, they were formulated in provocative manner. The goal was to challenge discussion. I plan to illustrate my personal answers with a few examples quoted from the history of 20th c. physics. My answers to the questions are not intended to be enunciations and to provide final solutions, rather they serve as arguments and indicate that nothing is closed, the discussion is open. Methods 1 What do we consider physical information? Can one speak about physical information when there is no live percipient to accept, evaluate and use it? Can one speak about physical information (e.g., signal exchange) between inanimate physical objects (cf., e.g., Feynman diagrams)? And if so, what is it for? Is (physical) information a passive phenomenon, or its existence presumes activity? Interpretation of ‘activity’ plays important role in the possible answers. One of the interpretations says that activity is an antropomorph phenomenon. It is a privilege of the human mind, which is able to perceive and process information, able to teleologically evaluate its possible consequences and (re)act accordingly. Another interpretation says that there is inanimate activity, that means, reception of information between physical agents and their reaction to it. I argue for the latter concept. Let us see the example of the interaction between two (electric) charges. There were two approaches in the classical age of developing these theories (1928-33). When two electric charges interact, there appear two types of interaction. One is a Coulomb-type repulsion/attraction (according to their mutual signs) governed by their scalar potential, and the other is a Lorentz-type one governed by their vector potential depending on their relative velocity to each other. One type of the theories considered first the interaction between the scalar potentials and calculated the effect of the vector potentials as perturbation. It considered two charges approaching to each other from the infinity, when in first approximation they have got information on the amount of the Coulomb charge of the other, but not its velocity and the caused Lorentz force. The latter was considered in the perturbation process. The other type of the theories considered first the effect of the Lorentz force of the approaching charges, and took into consideration the effect of the Coulomb force in course of the perturbation. Representatives of both types of theories agreed that the roles of the interacting electric charges must be symmetrical, but Christian Møller. Møller [7], who belonged to the latter type of theoreticians, applied scattering matrices (1931). He showed that there appeared a component among the matrix elements that was asymmetric in respect to the two interacting charges. H. Bethe (that time a doctoral student of E. Fermi, 1932) could not accept this asymmetry and ‘corrected’ Møller’s theory. He ‘symmetrised’ those matrix elements artificially [1]. That was a rough and unjustified involvement in Møller’s equation, but due to the later attained high authority of both Bethe and Fermi, the physics community accepted the apparently convenient symmetrisation of the theory without discussion and has not treated it until the recent years. Thus any possible distinction between (roles and properties of) interacting charges were unrevealed until the past decade. However, the question can be formulated so: how do the charges, ready to interact, get information from each other? All theories agree that interaction between two charges take place by the exchange of a boson. In case of electromagnetic interaction this boson is a photon. Which of them emits the first photon towards the other? Did this emitter receive any information from the direction of the partner charge prior to the photon emission? What else is this if not an asymmetry between the roles, and possibly between properties of the two interacting charges? The distinction between the interacting charges was introduced by the isotopic field-charge theory, and the notion of ‘isotopic field-charge’ made the distinction between the properties of the two charges [2],[3],[4],[5]. The latter led to the proof of the gauge invariance under rotation of a newly introduced property, the isotopic field-charge spin, in an abstract field, and its conservation. This conservation is a result of a symmetry. Neverthe-less, this mechanism argues for activity between physical objects without an animated (human) agent. 2 What are the limits between (closed and open) systems, from the aspects of information and of symmetries? Further, if so, how wide can we extend the meaning of activity to be still accepted for generating information? What are the roles of different appearances of symmetries in taking a stand in the mentioned questions? What kinds of symmetry (or their absence) may play a role in making decision in the listed problems? The ‘classical’ (20th c.) relativity theories demanded that all physical laws were invariant under the Lorentz transformation. This was established first in the special theory of relativity that was formulated for electromagnetic interactions [6]. Lorentz invariance of physical laws was in fact a symmetry principle (conservation of the form of the laws during reference frame change) [9]. This invariance proved to hold for many other physical laws, so later also the symmetry principle was extended to other physical laws as well. The Lorentz invariant relativity theory included another consequence: there is no distinct (odd) reference frame in nature. In other words, all reference frames are equivalent. Aren’t they? Based on Noether’s theorems [8], one can show that conservation laws hold in all reference frames. However, the quantity of the conserved property (e.g., mass, charge, etc.) may change in the different reference frames. E.g., the amount of mass of matter in a closed system depends on the velocity of the observer relative to that system where the mass is to be measured. One can always find a reference frame from which the amount of the measured mass is minimal. This fact contradicts to the absence of a distinct (odd) reference frame. And there are more. Is there any invariance that compensates this lost equivalence of all reference frames? The observer can be not only a human agent who measures with instruments and reads the records. It can be another inanimate mass that perceives information about the mass from the observed system. Its ‘activity’ is that this information can be obtained by experiencing a force. This force can be gravitational or inertial. According to the general theory of relativity the ‘observer mass’ is unable to make distinction whether the experienced force is of ‘inertial’ or ‘gravitational’ origin. At the same time, the inertial mass changes its value according to its relative velocity to the observer, while the gravitational mass does not. Thus the ‘observer mass’ in different reference frames will experience different inertial forces originating from the ‘observed mass’. Something similar distinction can be made between the Coulomb charges and the Lorentz-type (current) charges. They are sources of the Coulomb force and the Lorentz force, respectively, and similar to the two types of masses, originate from the scalar potential and from the vector potential of the Hamiltonian of the charged object, respectively. The two types of masses and the two types of electric charges are called isotopic field-charge pairs, respectively. They are subject of the same gauge invariance. As such, they can exchange their roles (switch into each other) by the exchange of a gauge boson (in addition to the graviton and the photon, respectively), called delta bosons in the theory. As a consequence of the additional invariance and the corresponding additional mediating gauge boson, the respective systems of the two interacting isotopic field-charges (masses or electric charges) are not subjects of the Lorentz invariance alone. They are subjects of a convolution of the Lorentz- and this additional invariance. One can conclude two things. First, along with the development of physics, there is no more enough to demand invariance under the Lorentz transformation. At extended conditions, one should demand the invariance under a combination of the Lorentz invariance and an additional invariance. In short, we demand invariance under (the applicable) transformations. Secondly, when two charges (let they be either gravitational, electric, or other field-charges) interact, they make a distinction between each other. The system, composed of the interacting two field-charges, follow the Pauli principle. That means, the two interacting field-charges must be in different quantum states. Since this state in which they differ cannot be characterized by any of the earlier known properties, it must be a characteristic of the newly introduced property. It must be one of the two stable positions of the isotopic field-charge that are rotated into each other by the SU(2) symmetry group in the isotopic field-charge field. These two stable positions are called, by an analogically given name, the isotopic field-charge spin (not identical either with the angular momentum spin or the isotopic spin). According to its proven invariance, the isotopic field-charge spin is a conserved property. When two field-charges interact, they must be in the opposite isotopic field-charge spin states. The information that they exchange about each other is about this state: they check whether the partner is in the opposite state. Otherwise they are ‘not allowed’ to interact (Pauli’s exclusion principle). The information exchange takes place by the exchange of a delta boson (called also dion) between them, in addition to the exchange of the traditional mediating bosons (like graviton, photon, weak charged and neutral bosons, or gluons). That delta boson switches the emitting charge from inertial to potential state, and the absorbing charge from potential to inertial state. Conclusions The asymmetry of the interacting charges has been explained. It was subject of information exchange between the interacting particle partners. In order to meet the Pauli principle, physical objects should exchange information about the (opposite) states of each other before getting into active interaction. The explanation led to the loss of an invariance property. However, this loss has been restored by introducing a new physical property (isotopic field-charge spin), by proving its conservation, and completing the Lorentz invariance with the respective invariance attributed to the newly proven conservation. References Bethe, H., Fermi, E., Über die Wechselwirkung von zwei elektronen, Zeitschrift für Physik, 77 (1932), 5-6, 296-306. Darvas, G., The Isotopic Field Charge Spin Assumption, International Journal of Theoretical Physics, 50 (2011), 10, 2961-2991.DOI: 10.1007/s10773-011-0796-9 Darvas, G., The Isotopic Field-Charge Assumption Applied to the Electromagnetic Interaction, Int J Theor Phys, 52 (2013a), 11, 3853-3869, DOI 10.1007/s10773-013-1693-1. Darvas, G., A symmetric adventure beyond the Standard Model – Isotopic field-charge spin conservation in the electromagnetic interaction, Symmetry: Culture and Science, 24 (2013b), 1-4, 17-40. Darvas, G., Electromagnetic Interaction in the Presence of Isotopic Field-Charges and a Kinetic Field, Int J Theor Phys, 53 (2014), 1, 39-51, DOI 10.1007/s10773-013-1781-2. Einstein, A., Zur Elektrodynamik bewegter Körper, Annalen der Physik, 17:891 (1905); English translation: On the electrodynamics of moving bodies, in: The Principle of Relativity, (1923), London: Methuen and Co. Møller, C., Über den Stoß zweier Teilchen unter Berücksichtigung der Retardation der Kräfte, Zeitschrift für Physik, 70 (1931), 11-12, 786-795. Noether, E. A., Invariante Variationsprobleme, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen: Mathematisch-physikalische Klasse, (1918) 235-257. Wigner, E.P. , Events, laws of nature and invariance principles, pp. 38-50, in: Wigner, Symmetries and Reflections, Bloomington: Indiana University Press (1967).

DOI
10.3390/isis-summit-vienna-2015-t4002

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Length contraction Consider a rod at rest in the primed system S' lying along the x-axis and having the length 12 xx ′−′=λ given by the distance between its ends 21 and xx ′′ measured at the same instant of time t'. The length in an inertial frame in which the rod is at rest is called its proper length. To define length in a frame S in which the rod is moving, an observer must locate the position of both end points, 21 and xx ′′ , at the same time From the inverse equations (4.11) we obtain 1 2 .t t t= = 1 1 2 1 x ct x ct x xx x x xβ ββ β β − −′ ′ ′ ′= = − = − − − − so that the apparent length 12 xxl −= is 21 βλ −=l (4.13) This equation is known as the Lorentz contraction and states that a rod parallel to the direction of its motion is shortened by the factor .1 2β− It results from the fact that simultaneous events in the rest frame of one rod are not simultaneous in the rest frame of the other. A rod transverse to its direction of motion is unchanged in length. 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Bits more basic for physics than fundamental particles

Bengt-Arne Vedin

There are several links between information theory — between »information as such« whatever that might mean — and the foundations of nature, i e, basic physics. Well, since physics a science still evolving (despite propositions that »the end of science« now in sight), some of the links between information and physics are hypothetical. One linkage fairly obvious — thermodynamics depends upon entropy or some measure of order or disorder, and that clearly has something to do with information. The other possible relationship might be between the uncertainty that a key characteristic of quantum mechanics. We can never know everything on the quantum scale; the Heisenberg uncertainty principle tells that the more information we gain in one direction the less we know in another. If we, on that scale, obtain certainty as to one factor, then another will stay entirely undecided — try to affix a position to a quantum particle, and the momentum will be out of bounds entirely. (This the established dogma currently, though there are some deviating ideas.1) Then the question arises: who does the actual measuring, with what — what particles, at what level, are involved in this act of, as it pronounced, forcing the wave function to collapse? This wave function is the indeterminate particle: the various statistics describing the probabilities that the particle (which simultaneously takes on aspects of a wave) may be in one state or another. We will now attempt to discuss these different — or are they really different? — interdependencies, starting with black holes.

Human IT•Jan 1, 1999•0 citations
Introduction My presentation aims at answering a part of the questions posted in the invitation to the session. The questions arose in and originated from the FIS discussion on Physical informatics … in October 2014. No doubt, they were formulated in provocative manner. The goal was to challenge discussion. I plan to illustrate my personal answers with a few examples quoted from the history of 20th c. physics. My answers to the questions are not intended to be enunciations and to provide final solutions, rather they serve as arguments and indicate that nothing is closed, the discussion is open. Methods 1 What do we consider physical information? Can one speak about physical information when there is no live percipient to accept, evaluate and use it? Can one speak about physical information (e.g., signal exchange) between inanimate physical objects (cf., e.g., Feynman diagrams)? And if so, what is it for? Is (physical) information a passive phenomenon, or its existence presumes activity? Interpretation of ‘activity’ plays important role in the possible answers. One of the interpretations says that activity is an antropomorph phenomenon. It is a privilege of the human mind, which is able to perceive and process information, able to teleologically evaluate its possible consequences and (re)act accordingly. Another interpretation says that there is inanimate activity, that means, reception of information between physical agents and their reaction to it. I argue for the latter concept. Let us see the example of the interaction between two (electric) charges. There were two approaches in the classical age of developing these theories (1928-33). When two electric charges interact, there appear two types of interaction. One is a Coulomb-type repulsion/attraction (according to their mutual signs) governed by their scalar potential, and the other is a Lorentz-type one governed by their vector potential depending on their relative velocity to each other. One type of the theories considered first the interaction between the scalar potentials and calculated the effect of the vector potentials as perturbation. It considered two charges approaching to each other from the infinity, when in first approximation they have got information on the amount of the Coulomb charge of the other, but not its velocity and the caused Lorentz force. The latter was considered in the perturbation process. The other type of the theories considered first the effect of the Lorentz force of the approaching charges, and took into consideration the effect of the Coulomb force in course of the perturbation. Representatives of both types of theories agreed that the roles of the interacting electric charges must be symmetrical, but Christian Møller. Møller [7], who belonged to the latter type of theoreticians, applied scattering matrices (1931). He showed that there appeared a component among the matrix elements that was asymmetric in respect to the two interacting charges. H. Bethe (that time a doctoral student of E. Fermi, 1932) could not accept this asymmetry and ‘corrected’ Møller’s theory. He ‘symmetrised’ those matrix elements artificially [1]. That was a rough and unjustified involvement in Møller’s equation, but due to the later attained high authority of both Bethe and Fermi, the physics community accepted the apparently convenient symmetrisation of the theory without discussion and has not treated it until the recent years. Thus any possible distinction between (roles and properties of) interacting charges were unrevealed until the past decade. However, the question can be formulated so: how do the charges, ready to interact, get information from each other? All theories agree that interaction between two charges take place by the exchange of a boson. In case of electromagnetic interaction this boson is a photon. Which of them emits the first photon towards the other? Did this emitter receive any information from the direction of the partner charge prior to the photon emission? What else is this if not an asymmetry between the roles, and possibly between properties of the two interacting charges? The distinction between the interacting charges was introduced by the isotopic field-charge theory, and the notion of ‘isotopic field-charge’ made the distinction between the properties of the two charges [2],[3],[4],[5]. The latter led to the proof of the gauge invariance under rotation of a newly introduced property, the isotopic field-charge spin, in an abstract field, and its conservation. This conservation is a result of a symmetry. Neverthe-less, this mechanism argues for activity between physical objects without an animated (human) agent. 2 What are the limits between (closed and open) systems, from the aspects of information and of symmetries? Further, if so, how wide can we extend the meaning of activity to be still accepted for generating information? What are the roles of different appearances of symmetries in taking a stand in the mentioned questions? What kinds of symmetry (or their absence) may play a role in making decision in the listed problems? The ‘classical’ (20th c.) relativity theories demanded that all physical laws were invariant under the Lorentz transformation. This was established first in the special theory of relativity that was formulated for electromagnetic interactions [6]. Lorentz invariance of physical laws was in fact a symmetry principle (conservation of the form of the laws during reference frame change) [9]. This invariance proved to hold for many other physical laws, so later also the symmetry principle was extended to other physical laws as well. The Lorentz invariant relativity theory included another consequence: there is no distinct (odd) reference frame in nature. In other words, all reference frames are equivalent. Aren’t they? Based on Noether’s theorems [8], one can show that conservation laws hold in all reference frames. However, the quantity of the conserved property (e.g., mass, charge, etc.) may change in the different reference frames. E.g., the amount of mass of matter in a closed system depends on the velocity of the observer relative to that system where the mass is to be measured. One can always find a reference frame from which the amount of the measured mass is minimal. This fact contradicts to the absence of a distinct (odd) reference frame. And there are more. Is there any invariance that compensates this lost equivalence of all reference frames? The observer can be not only a human agent who measures with instruments and reads the records. It can be another inanimate mass that perceives information about the mass from the observed system. Its ‘activity’ is that this information can be obtained by experiencing a force. This force can be gravitational or inertial. According to the general theory of relativity the ‘observer mass’ is unable to make distinction whether the experienced force is of ‘inertial’ or ‘gravitational’ origin. At the same time, the inertial mass changes its value according to its relative velocity to the observer, while the gravitational mass does not. Thus the ‘observer mass’ in different reference frames will experience different inertial forces originating from the ‘observed mass’. Something similar distinction can be made between the Coulomb charges and the Lorentz-type (current) charges. They are sources of the Coulomb force and the Lorentz force, respectively, and similar to the two types of masses, originate from the scalar potential and from the vector potential of the Hamiltonian of the charged object, respectively. The two types of masses and the two types of electric charges are called isotopic field-charge pairs, respectively. They are subject of the same gauge invariance. As such, they can exchange their roles (switch into each other) by the exchange of a gauge boson (in addition to the graviton and the photon, respectively), called delta bosons in the theory. As a consequence of the additional invariance and the corresponding additional mediating gauge boson, the respective systems of the two interacting isotopic field-charges (masses or electric charges) are not subjects of the Lorentz invariance alone. They are subjects of a convolution of the Lorentz- and this additional invariance. One can conclude two things. First, along with the development of physics, there is no more enough to demand invariance under the Lorentz transformation. At extended conditions, one should demand the invariance under a combination of the Lorentz invariance and an additional invariance. In short, we demand invariance under (the applicable) transformations. Secondly, when two charges (let they be either gravitational, electric, or other field-charges) interact, they make a distinction between each other. The system, composed of the interacting two field-charges, follow the Pauli principle. That means, the two interacting field-charges must be in different quantum states. Since this state in which they differ cannot be characterized by any of the earlier known properties, it must be a characteristic of the newly introduced property. It must be one of the two stable positions of the isotopic field-charge that are rotated into each other by the SU(2) symmetry group in the isotopic field-charge field. These two stable positions are called, by an analogically given name, the isotopic field-charge spin (not identical either with the angular momentum spin or the isotopic spin). According to its proven invariance, the isotopic field-charge spin is a conserved property. When two field-charges interact, they must be in the opposite isotopic field-charge spin states. The information that they exchange about each other is about this state: they check whether the partner is in the opposite state. Otherwise they are ‘not allowed’ to interact (Pauli’s exclusion principle). The information exchange takes place by the exchange of a delta boson (called also dion) between them, in addition to the exchange of the traditional mediating bosons (like graviton, photon, weak charged and neutral bosons, or gluons). That delta boson switches the emitting charge from inertial to potential state, and the absorbing charge from potential to inertial state. Conclusions The asymmetry of the interacting charges has been explained. It was subject of information exchange between the interacting particle partners. In order to meet the Pauli principle, physical objects should exchange information about the (opposite) states of each other before getting into active interaction. The explanation led to the loss of an invariance property. However, this loss has been restored by introducing a new physical property (isotopic field-charge spin), by proving its conservation, and completing the Lorentz invariance with the respective invariance attributed to the newly proven conservation. References Bethe, H., Fermi, E., Über die Wechselwirkung von zwei elektronen, Zeitschrift für Physik, 77 (1932), 5-6, 296-306. Darvas, G., The Isotopic Field Charge Spin Assumption, International Journal of Theoretical Physics, 50 (2011), 10, 2961-2991.DOI: 10.1007/s10773-011-0796-9 Darvas, G., The Isotopic Field-Charge Assumption Applied to the Electromagnetic Interaction, Int J Theor Phys, 52 (2013a), 11, 3853-3869, DOI 10.1007/s10773-013-1693-1. Darvas, G., A symmetric adventure beyond the Standard Model – Isotopic field-charge spin conservation in the electromagnetic interaction, Symmetry: Culture and Science, 24 (2013b), 1-4, 17-40. Darvas, G., Electromagnetic Interaction in the Presence of Isotopic Field-Charges and a Kinetic Field, Int J Theor Phys, 53 (2014), 1, 39-51, DOI 10.1007/s10773-013-1781-2. Einstein, A., Zur Elektrodynamik bewegter Körper, Annalen der Physik, 17:891 (1905); English translation: On the electrodynamics of moving bodies, in: The Principle of Relativity, (1923), London: Methuen and Co. Møller, C., Über den Stoß zweier Teilchen unter Berücksichtigung der Retardation der Kräfte, Zeitschrift für Physik, 70 (1931), 11-12, 786-795. Noether, E. A., Invariante Variationsprobleme, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen: Mathematisch-physikalische Klasse, (1918) 235-257. Wigner, E.P. , Events, laws of nature and invariance principles, pp. 38-50, in: Wigner, Symmetries and Reflections, Bloomington: Indiana University Press (1967).
Symmetries play a central role in physics. They dictate what one can change in a physical system without affecting any of its properties. You might have encountered symmetries like translational symmetry, where a system remains unchanged if it is spatially translated by an arbitrary distance. A system with rotational symmetry, however, is invariant under rotations. Some symmetries, like the ones mentioned above, give information about the structure of the system. Others have to do with the more fundamental physical framework that we adopt. An example for this is the invariance under Lorentz transformations in relativistic physics. Other types of symmetries can be even more subtle. For example, it is rather self-evident that physics should remain unchanged if we exchange two identical point-like particles. Nevertheless, this fundamental property that we call statistical symmetry gives rise to rich and beautiful physics. In three spatial dimensions it dictates the existence of bosons and fermions. These are particles with very different quantum mechanical properties. Their wave function acquires a +1 or a -1 phase, respectively, whenever two particles are exchanged. A direct consequence of this is that bosons can actually occupy the same state. In contrast, fermions can only be stacked together with each particle occupying a different state. When one considers two spatial dimensions, a wide variety of statistical behaviours is possible. Apart from bosonic and fermionic behaviours, arbitrary phase factors, or even non-trivial unitary evolutions, can be obtained when two particles are exchanged (Leinaas and Myrheim, 1977).
The laws of physics were not handed down from above. Neither are they rules somehow built into the structure of the universe. They are ingredients of the models that physicists invent to describe observations. Rather than being restrictions on the behavior of matter, the laws of physics are restrictions on the behavior of physicists. If the models of physics are to describe observations based on an objective reality, then those models cannot depend on the point of view of the observer. This suggests a principle of point-of-view invariance that is equivalent to the principle of covariance when applied to space-time. As Noether showed, space-time symmetries lead to the principles of energy, linear momentum, and angular momentum conservation--essentially all of classical mechanics. It also leads to Lorentz invariance and special relativity. When generalized to the abstract space of functions such as the quantum state vector, point-of-view invariance is identified with gauge invariance. Quantum mechanics is then just the mathematics of gauge transformations with no additional assumptions needed to obtain its rules, including the superposition and uncertainty principles. The conservation and quantization of electric charge follow from global gauge invariance. The electromagnetic force is introduced to preserve local gauge invariance. Although not discussed here, the other forces in the standard model of elementary particles are also fields introduced to preserve local gauge invariance. Gravity can also be viewed as such a field. Thus practically all of fundamental physics as we know it follows directly from the single principle of point-of-view invariance.
The concept of invariance relates to both the intrinsic symmetries of physical systems and the symmetry of the set of equivalent reference frames used to observe them. Standard algebraic expressions for electrostatic potentials and crystal-field effective operators display both types of invariance. The concept of a reference frame is generalized to that of an ‘observing system’, which can, for example, be the basis states of a quantum system. This idea is related to Racah’s mathematical machinery for evaluating the matrix elements of many-electron 4f open-shell states in lanthanide ions. It is argued, on the basis of computational flexibility and ease of interpretation, that all equations that represent physical processes be expressible in terms of invariants of the set of observing systems. This ‘Principle of Invariance’ is then applied to special relativity, leading to a simple geometrical interpretation of Maxwell’s electromagnetic field equations. The close relationship between Dirac’s relativistic wave equation and Maxwell’s equations is then exposed. This leads to the concept of an inner structure of space-time and the reinterpretation of particle spin. Finally, it is shown that the use of invariants in relativity theory identifies a set of observing systems with a higher symmetry than that of Minkowski space-time.
What is information? A short answer could be: Anything that changes our minds and states of thinking! Information is virtual. It can be carried by speech, an image, scratches on a stone, patterns in a photon field, the connections of neurons in our brains or even by the wavefunction of an electron. Information does not depend on the actual means of transmission nor does its identity change when changing the carrier. Information is physical. It needs to be sustained by a physical substrate and rearranging that substrate to contain the wanted information requires to spend energy or work. Without a physical world, information can not be stored, processed, or transmitted. Our world is permeated with different representations of information. Actually Information about the physical laws governing our Universe can be found everywhere and in everything. Could it then be that the real fundamental elements of this world are tiny bits of information? Could we get it from bit?1 A number of researcher investigate this possibility, and several of their attempts to recover physical laws from the principles governing information are reported about in this special issue on physics of information. The articles by Gregg Jaeger,2 by Ariel Caticha,3 and by John Skilling and Kevin Knuth4 address the reconstruction of quantum mechanics from information theory. Furthermore, Kevin Knuth and James Walsh5 show how to recover space-time from particles sending each others signals in an initially geometry-free setting. A central element of most works on information is the concept of a probability. Probabilities can be defined via various routes. Most familiar might be their definitions by the frequentists school as relative frequencies of events (in the limit of infinite amounts of such). A more general definition of probabilities is the Bayesians' one, which regards probabilities as quantifiers of statements, expressing how much those should be expected to be true. Although this definition looks subjective on the first glance, it rests on a solid mathematical theorem by Richard Cox6 that any extension of binary logic to a continuum of single valued truthfulness quantifier that respects a minimum of requirements must be isomorphic to probability theory. And this definition turns out to embrace the frequentists' definition in case that infinite numbers of comparable events are actually available and can be counted. The majority of authors of this special issues seem to favor the Bayesian perspective, which is probably a result of our own, the guest editors', preference and not necessarily a reflection of the prevailing position of contemporary scientists. The different views on probabilities of frequentists and Bayesians are addressed by the article of Allen Caldwell.7 He builds a bridge between these antagonistic camps by explaining frequentists' constructions in a Bayesian language. In particular, his argumentation why the omnipresent p-value in statistical and experimental literature might often capture a relevant signal of model misfit is reassuring, given that a large fraction of our medical system is based on decisions backed up by p-values. A Bayesian might add that it would be even more reassuring, if a system rigorously build on logic and its continuous extension would be used for decisions affecting live-or-death questions. The laws governing probabilities themselves seem to reflect that of more elementary mathematical structures. Kevin Knuth shows how these emerge from lattice theory, which deals with relations in partially ordered sets like that of collections of possibilities.8 The sum rule of probability theory is a direct consequence of the underlying lattice structure. Interestingly, a number of other theories are resting also on such lattices and therefore exhibit their own versions of sum rules: measures of sets, mutual information, the Euler characteristics of topological spaces, as well as quantum mechanical interactions that are quantified by pairs of numbers, the real and imaginary parts of complex probability amplitudes (see also4). The theory of information has very practical application in physics and elsewhere. Manfred Opper discusses how the solutions of stochastic differential equations can be inferred via variational methods, which have their origin in thermodynamics.9 Torsten Enßlin introduces information field theory (IFT), the information theory for fields, as a means to infer the configuration of a physical field from incomplete and noisy data.10 The mathematical apparatus for this is that of quantum field theory (QFT), just with ordinary probabilities instead of complex probability amplitudes as in the latter. Could it be that the difference between IFT and QFT is just be the same as between classical information theory and quantum mechanics, the transition from real valued to complex valued quantifier for statements? Anyhow, any practical inference machinery, be it for dynamical systems or for fields, requires sophisticated numerical methods. Michael Betancourt reviews Markov Chain Monte Carlo Methods to calculate expectation values over high dimensional probabilities.11 Theo Steininger and coworkers present the NIFTy3 software package for numerical information field theory calculations, which can handle very high dimensional field inference problems.12 Finally, in an article focused on the subtleties of the Monty Hall problem and mostly aiming at information theoretical entertainment, Torsten Enßlin and Margret Westercamp investigate the possibility that human intuition might be more rational than contemporary psychology thinks.13 A number of possible topics for an issue on physics of information are not addressed here. We would have loved to have articles on Landauer's principle, which states that erasing information has to cost energy, on QBism, a Bayesian version of quantum mechanics, on artificial and biological neural networks, as well as any other aspect of the information-soaked biological world. What has a reader of this special issue to be equipped with in order to follow the lines of thoughts? In addition to the usual physicist's education, the minimal additional equipment are the basics of probability theory and the concept of entropy. Most of the other concepts used in the articles rest on those and are introduced there. Let A, B, and C be propositions (or possibilities). The probability P ( A | B ) of A being true assuming that B is true is a number between 0 and 1, with P ( A | B ) = 1 expressing certainty of A under B ( B ⇒ A ) and P ( A | B ) = 0 expressing certainty of not A being the case under B ( B ⇒ ¬ A ).
Ψ∂ (4.1) The harmonic trial solution of this equation with respect to a fixed inertial frame ( is given by )tx, ( ) ( ) ( )ctxikectxktx −Ψ=Ψ−Ψ=Ψ 00 orcos, In another inertial frame( ),, tx ′′ considered to be in uniform translational motion with velocity as described by the Galilean transformation (2.13), with respect to the fixed frame, the wave is represented by a wave function ,v ( ) ( ), related to ,x t x′ ′Ψ Ψ t as 2 2i.e., x t x x x t x x x x ′ ′∂Ψ ∂Ψ ∂ ∂Ψ ∂ ∂Ψ ∂ Ψ ∂ Ψ= + = =′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ and 2 2i.e., 2 x t v v v t x t t t x t x tt x ′ ′∂Ψ ∂Ψ ∂ ∂Ψ ∂ ∂Ψ ∂Ψ ∂ Ψ ∂ Ψ ∂ Ψ ∂ Ψ= + = − + = − +′ ′ ′ ′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2t∂ ∂ ∂ where use has been made of known properties of partial derivatives. It follows that obeys the following wave equation, with respect to the moving inertial frame ( tx ′′Ψ , ) ( ) tx v tx vc ′∂′∂ Ψ∂−′∂ Ψ∂=′∂ Ψ∂− 2 22 2 (4.2) which is obviously different from the wave equation (4.1) that holds in the fixed inertial frame. In other words the electromagnetic wave equation is not invariant with respect to a uniform translation of the inertial frame, using a Galilean transformation. Inserting into Eq.(4.2) harmonic solution written as we obtain 2 2 2 2 2 2( ) ( ) 2 ( ) i.e., orc v k k w v k w w v c w c− − = − − = − ± = ± v (4.3) Hence Galileo's transformation predicts that the velocity of light should be different from c, if measured with respect to a moving inertial frame. We therefore have a choice to make before we can fully understand the electromagnetic phenomena. We can either accept Galileo's principle of relativity which states that all inertial frames are equivalent or the Galilean transformation (2.13). We can sacrifice Galileo's principle of relativity by saying that there is a privileged inertial frame, called the ether, in which it is supposed that the electromagnetic wave velocity is c as given by Maxwell's equations, although they made no essential reference to the ether. A long series of investigations has been carried out, in order to detect some effect of the motion of the earth through the ether, always with negative results. All of the experiments (the best known being that of Michelson-Morley) have indicated that the speed of light is always the same in all directions and is independent of the relative motion of observer, source and transmitting medium. Einstein has reaffirmed Galileo's principle of relativity on the equivalence of all the inertial frames and, in addition, has assumed as a basic postulate the experimental fact that the speed of light is always constant. As a consequence, the Galilean transformation had to be replaced by another, in order to preserve the same speed of light in all systems. Einstein's basic assumptions are known as the postulates of special relativity: 1. The laws of physics take the same form in all inertial systems 2. The speed of light in empty space is a universal constant c and is the same for all observers. It is independent of the state of motion of the emitting body Special relativity is only concerned with transformations between inertial frames, as indicated by the restriction contained in the first postulate. General relativity covers reference frames in arbitrary relative motion, and is based on the principle of equivalence, confirmed experimentally, which unifies gravitational and inertial forces. The first of Einstein's postulates implies that it is only possible, by means of physical measurements, to demonstrate that two gravity-free reference frames are moving relative to each other. There is no physical basis to infer that any such frame is intrinsically at rest or in uniform motion. The notion of absolute rest is not consistent with the first postulate. The second postulate is in contradiction with the Galilean transformation which would preserve the form of Newton's laws of motion, as illustrated by Eq.(4.3). Therefore, we must find a transformation between two frames moving with uniform relative motion, which is consistent with the second postulate. We must then generalize the laws of Newtonian mechanics in a form which is invariant under that transformation, as required by the first postulate. A fundamental requirement of special relativity is that the laws of Newtonian mechanics are valid in the low speed limit and must be obtained as limiting cases of relativistic formulae for .cv << This assumption plays, for special relativity, a role analogous to that of Bohr's correspondence principle in quantum mechanics (see Chapter 28). 4.2. THE LORENTZ TRANSFORMATION Einstein's procedure is to locate an event in a reference frame by its spatial Cartesian coordinates x, y and z and a temporal coordinate t. Consider two frames S and S' moving with uniform relative motion, such that the Cartesian axis system in coincides with that in S, Oxyz, at time ,S ′ zyxO ′′′′ .0==′ tt As illustrated in Figure 4.1, the x and axes are parallel to the direction of relative motion: in moves in the x direction with velocity v, and in x′ , the frameS ′S S, the frameS ′ moves with velocity in the direction. A pulse of light emitted from O at time v− x′ 0=t spreads as a spherical wave travelling with speed c, so that the equation of the wave front in S, at a time t will be (4.4) 22222 tczyx =++ 0 0' z' y' x,x' z y Figure 4.1. Cartesian axes of two inertial frames in uniform relative motion The second postulate requires that the wave front in S ′ can be represented by (4.5) 22222 tczyx ′=′+′+′ In other words an observer in also sees a spherical wave travelling from its origin if not, any change of the wave front could be used to infer that the system is moving uniformly instead of being at rest. The invariance of the expression S ′ ,O′ S ′ (4.6) 2222222222 tczyxtczyx ′−′+′+′=−++ is a basic requirement that the transformation between the two systems must satisfy. The Galilean transformation (2.13), which in our case reads ttzzyyvtxx =′=′=′−=′ ,,, is not consistent with the invariance property (4.6), as it gives xvttvtczyxtczyx 2222222222222 −+−++=′−′+′+′ The first postulate brings some simplification to the possible form of the desired transformation. Since a uniform rectilinear motion in S must go over into a uniform rectilinear motion in S', the transformation must be linear. A nonlinear transformation would predict acceleration in S' even if the velocity were constant in S. The choice of the origins in space and time has avoided constant terms, and the transformation will be homogenous. The directions perpendicular to the relative motion, which are effectively at rest, must be left unchanged by the transformation. We then take the general form BtAxx +=′ yy =′ (4.7) zz =′ FtDxt +=′ Further simplification comes from the argument that the origin O', which has the coordinate moves along the x axis with velocity ,0=′x ,/ vdtdx = that is ABvBtAx /or0 −=+= while the origin O, with coordinate ,0=x moves along x-axis with velocity Recalling that when , we can write .'/' vdtdx −= 0=x 0=t 00 ort D B Bx A B t v F F F ′ − ⋅′ ′= ⋅ + = − = so that Inserting now Eqs.(4.7) into Eq.(4.5), we obtain the equation .FA = ( ) ( ) 02 22222222222222222 =−+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −−++−=′−′+′+′ DcBAxt c BAtczyxDcAtczyx The identity (4.6) requires that 2222 =−=−=− DcB c BADcA Since ,B vA= we obtain 2222 /1 /, /1 , /1 cv cvD cv vB cv A − −= − −= − = (4.8) so that Eqs.(4.7) now read 22 /1 cv vtxx − −=′ yy =′ (4.9) zz =′ /1 / cv cvxtt − −=′ The four equations (4.9) are known as the Lorentz transformation, which in this case applies to the relative motion of inertial frames along the x-axis. It is a straightforward matter to show that Eqs.(4.9) are consistent with the invariance relation (4.6), which is thus a sufficient as well as a necessary condition for the Lorentz transformation. With the usual abbreviations ( )1 /1 1, 222 ≥ − = − == γ β γβ cvc v (4.10) the Lorentz transformation simplifies to ( ) ( )x x ct x x ctγ β γ β′ ′= − = + ′ a) yyyy ′==′ or b) (4.11) zzzz ′==′ c) ⎟⎠ ⎞⎜⎝ ⎛ ′+′=⎟⎠ ⎞⎜⎝ ⎛ −=′ x c ttx c tt βγβγ d) The right hand side shows the inverse transformations where the primed and unprimed quantities are interchanged and the sign of the relative velocity is reversed, because its direction is the only difference between the systems S and S'. In the low velocity limit the Lorentz transformation reduces to the Galilean transformation since the only difference consists in the presence of the factor γ and the term ,1< that, in Eq.(4.11a), γ indicates a change of measuring stick length and in Eq.(4.11d) a change in the clock rate in the Lorentz transformation. The term ( )xc/β− in Eq.(4.11d) shows that the time is dependent on the position of a given event. Therefore, the immediate consequences of the Lorentz transformation require the revision of the usual concepts of simultaneity, time and length. Simultaneity Consider two events occuring at the same time ttt == 21 at two different positions and in the unprimed system S. According to Eqs.(4.11) the events are recorded in the primed system S' at different times 1x 2x / /, 1 1 t x c t xt t cβ ββ β − −′ ′= = − − so that the apparent time interval in S' is given by ( ) 0 1 / 2 21 12 ≠− −=′−′=′∆ β β cxx ttt (4.12) and hence, it is a function of the relative velocity v. The events will not appear as simultaneous to observers in S'. We are then forced to abandon the intuitive idea of absolute simultaneity, which is based on our experience with objects of low velocity, and accept the relativity of simultaneity, which depends on the coordinate system and applies in the domain of high velocities. Length contraction Consider a rod at rest in the primed system S' lying along the x-axis and having the length 12 xx ′−′=λ given by the distance between its ends 21 and xx ′′ measured at the same instant of time t'. The length in an inertial frame in which the rod is at rest is called its proper length. To define length in a frame S in which the rod is moving, an observer must locate the position of both end points, 21 and xx ′′ , at the same time From the inverse equations (4.11) we obtain 1 2 .t t t= = 1 1 2 1 x ct x ct x xx x x xβ ββ β β − −′ ′ ′ ′= = − = − − − − so that the apparent length 12 xxl −= is 21 βλ −=l (4.13) This equation is known as the Lorentz contraction and states that a rod parallel to the direction of its motion is shortened by the factor .1 2β− It results from the fact that simultaneous events in the rest frame of one rod are not simultaneous in the rest frame of the other. A rod transverse to its direction of motion is unchanged in length. The proper length λ is independent of the orientation of the rod, as a manifestation of the isotropy of inertial rest frame S'. If the orientation of a rod is neither longitudinal nor transverse to the direction of relative motion, both the length and the inclination are changed in the laboratory frame S, as an effect of a contraction in the direction of the relative motion with no change of transverse dimensions.
There are several links between information theory — between »information as such« whatever that might mean — and the foundations of nature, i e, basic physics. Well, since physics a science still evolving (despite propositions that »the end of science« now in sight), some of the links between information and physics are hypothetical. One linkage fairly obvious — thermodynamics depends upon entropy or some measure of order or disorder, and that clearly has something to do with information. The other possible relationship might be between the uncertainty that a key characteristic of quantum mechanics. We can never know everything on the quantum scale; the Heisenberg uncertainty principle tells that the more information we gain in one direction the less we know in another. If we, on that scale, obtain certainty as to one factor, then another will stay entirely undecided — try to affix a position to a quantum particle, and the momentum will be out of bounds entirely. (This the established dogma currently, though there are some deviating ideas.1) Then the question arises: who does the actual measuring, with what — what particles, at what level, are involved in this act of, as it pronounced, forcing the wave function to collapse? This wave function is the indeterminate particle: the various statistics describing the probabilities that the particle (which simultaneously takes on aspects of a wave) may be in one state or another. We will now attempt to discuss these different — or are they really different? — interdependencies, starting with black holes.