Entanglement

“ It is for the sake of a somewhat commensurate appreciation of vitality, of high-wrought molecular organization, that it is necessary, again and again, to point out the might of potentiality intrinsic to matter, the vast and specific stores of force locked up in the peculiar molecular entanglements which constitute our different substances. ”

“ Wherever and whenever an attempt was made to show to the world that it was not true, however hostile and powerful the men engaged against us, we always—without a single exception—were brought through our difficulties in triumph, although, at times, we could not see how we were to escape the entanglements often thrown around us. ”

“ I cannot read any significance into a physical world when it is held before me upside down, as happened just now. For that reason I am interested in entropy not only because it shortens calculations which can be made by other methods, but because it determines an orientation which cannot be found by other methods.
The scientific world is, as I have often repeated, a shadow-world, shadowing a world familiar to our consciousness. Just how much do we expect it to shadow? We do not expect it to shadow all that is in our mind, emotions, memory, etc. ”

“ We are accustomed to regard as real those sense perceptions which are common to different individuals, and which therefore are, in a measure, impersonal. The natural sciences, and in particular, the most fundamental of them, physics, deal with such sense perceptions. The conception of physical bodies, in particular of rigid bodies, is a relatively constant complex of such sense perceptions. A clock is also a body, or a system, in the same sense, with the additional property that the series of events which it counts is formed of elements all of which can be regarded as equal. ”

“ In the case before us it has solved the entropy puzzle which has exercised physicists, as well as engineers, for decades, and it has thereby emancipated the Second Law from all anthropomorphism, from all dependence on human experimental skill. When we take the broadest possible view of its character, this Calculus enables us to read the present riddle of our universe, namely, why it is in its present improbable state. We have therefore in this Calculus an engine for investigation which is of great power and is likely to play a large part in the future in the ascertainment of physical truth. ”

“ Denoting the entropy at a particular starting state as S0, and the relative probability of a state with entropy S as W we have in general:
We now consider the following special case. Let a number (n) of movable points (for example molecules) be present in a volume v0, these points will be the subject of our considerations. Other than these, arbitrarily many other movable points can be present. As to the law that describes how the considered points move around in the space the only assumption is that no part of the space (and no direction) is favored over others. ”

“ We should have been compelled by logic to seek some law expressed in terms of “tensors,” and therefore independent of our choice of co-ordinates. [Pg 143] It is impossible without mathematics to explain the theory of tensors; the non-mathematician must be content to know that it is the technical method by which we eliminate the conventional element from our measurements and laws, and thus arrive at physical laws which are independent of the observer’s point of view. Of this method, Einstein’s law of gravitation is the most splendid example. ”

“ Now it is the main object of the Principle of Relativity to establish a connexion between the laws of a quiet system and those of a disturbed system. In order to establish such a connexion Einstein shows that it is only necessary to introduce a small number of additional fundamental propositions into his theory, and he shows also — it is this which makes his work so brilliantly ingenious — that the additional propositions which must be introduced do not concern the laws of any quiet system. ”

“ As soon as a science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a theory. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the "truth" of the theory lies. ”

“ The physical properties of space would not then be wholly independent, that is, uninfluenced by matter, but in the main they would be, and only in small measure, conditioned by matter. Such a dualistic conception is even in itself not satisfactory ”

“ Thus, just as the infinite universe through which the spheres roll is filled with invisible particles of a subtile matter to which physicists and astronomers give the name of ether, and which supplies the only key to cosmic phenomena, the finite universe in which organization unfolds itself is thronged with corpuscles no less invisible, forming what the illustrious Ehrenberg calls the milky-way of lower organisms, and no less essential for explanation of the processes of which we have traced the general course. ”

“ Evidently the greater the complexity of the molecular motions the less governable they are by any direct external force or influence we can bring to bear. This is because we are unable to act directly on the individual molecules and sway them to our special technical purpose. ”

“ It should rather be realized that a fiction containing an initial but only provisional contradiction serves as a starting-point for just those investigations that are most subtle and that have far-reaching consequences. We should have no Infinitesimal Calculus, no Algebra, no Atomic Theory, no Theory of Gravitation if, to avoid all initial contradictions, we surrender the fiction of differentials, of imaginary quantities, of the atom, of action at a distance. ”

“ First, notice that this account of dynamical complexity also gives us a neat way of formalizing the state of a system as a sort of message so that its Shannon entropy can be judged: the state of a system is represented by its position in configuration space, and facts about how the system changes over time are represented as patterns in how that system moves through configuration space. ”

“ Now up to this time no natural event has contradicted this theorem or its corollaries. The proof for it is cumulative, wholly experiential and therefore exactly like that for the law of conservation of energy.
Returning to irreversibility, the matter for immediate discussion, we premise that it will here clarify and simplify our ideas if we consider all the participating bodies as parts of the system experiencing the contemplated process. It is in this sense that we must understand the statement: Every natural event leaves the universe different from what it was before. ”

“ No one supposes the agreement of the phenomena of light with the theory of undulations to be merely fortuitous. It must arise from the actual identity of some of the laws of undulations with some of those of light; and if there be that identity, it is reasonable to suppose that its consequences would not end with the phenomena which first suggested the identification, nor be even confined to such phenomena as were known at the time. ”

“ Growth of entropy corresponds to an increase in the number of molecular complexions.
(6) Finally we give a mathematical concept which covers the whole domain of physics: "Any function whose time variation always has the same sign until a certain state is reached and is then zero, may be called an entropy function."
[18] This need cause no surprise, for it is only very recently that the conviction is gaining ground that the Second Law has no independent significance, but that its full content will only be grasped when its roots are sought in the Theorems of the Calculus of Probabilities. ”

“ We shall be true to the principle of relativity in its broadest sense if we give such a form to the laws that they are valid in every such four-dimensional system of co-ordinates, that is, if the equations expressing the laws are co-variant with respect to arbitrary transformations. ”

“ When we consider the behavior of aggregates we recognize at once a general, empirical law, which has also been called the one physical axiom, namely, that all natural processes are essentially irreversible. ”

“ Thus, if we admit the correctness of the kinetic theory, the principle of entropy becomes one of maximum probability, losing thereby its status of absolute validity and assuming a statistical significance. The representation of entropy in terms of probability permits an easy application of the principle to the problems of molecular physics and to atomistic physics in general. We may note that it was by following this method, or rather by combining the two definitions of entropy, that Planck established his quantum radiation formula, and was thus led to his quantum theory. ”

“ In a word, if we consider the matter, not within the ridiculously limited field of experiment in the laboratory, but in the vast field of celestial space, the real universe is not Euclidean, because in it light does not travel in a straight line. ”

“ Looking at it from another aspect—if the unity of a man’s consciousness is not an illusion, there must be some corresponding unity in the relations of the mind-stuff which is behind the pointer readings. Applying our measures of relation structure, as in chapter XI, we shall build matter and fields of force obeying identically the principal field-laws; the atoms will individually be in no way different from those which are without this unity in the background. ”

“ Now, after the principle of elementary disorder has been introduced and accepted by us as valid throughout nature, the fundamental question arises as to the calculation of the probability of a given state, and the actual derivation of the entropy therefrom. From the entropy all the laws of thermodynamic states of equilibrium, for material substances, and also for energy radiation, may be uniquely derived. ”

“ Each of these two sets furnishes three components of motion or a total of six components; then we say that the perfectly unconstrained motion of the body has six degrees of freedom. If a body moves parallel to one of the co-ordinate planes, we say it has two degrees of freedom. When [Pg 17] we come to consider molecular motion in general and the independence which characterizes the motion of each of the many molecules we see that altogether we have here an extraordinary number of degrees of freedom, and composed of such is the realm of our "elementary chaos." ”

“ All we can demand is that they be satisfied in the very restricted domain which science can explore. In the case of the principle of entropy, however, there is no reason to fear any conflict with the possibility of rebirth or re-creation. ”

“ Let us at the outset consider what denial of plan in the structure of the universe implies, and note, among other things signified, the following. The exact conformity on matter's part to Nature's laws, everywhere observable, and even more striking perhaps in minute details than in grandiose generalities, is purely accidental. ”

“ The world which we have built from the relation-structure is no doubt doomed to be pulled about a good deal as our knowledge progresses. The quantum theory shows that some radical change is impending. But I think that our building exercise has at any rate widened our minds to the possibilities and has given us a different orientation towards the idea of physical law. The points which I stress are:
Firstly, a strictly quantitative science can arise from a basis which is purely qualitative. ”

“ One would not be surprised if in the reconstruction of the scheme of physics which the quantum theory is now pressing on us, secondary law becomes the basis and primary law is discarded. In the reconstructed world nothing is impossible though many things are improbable. ”

“ If the Entropy theory is followed out to its conclusion, it results, firstly, that in theory all processes must be reversible—which is one of the basic postulates of dynamics and is reasserted with all rigour in the law of the Conservation of Energy—but, secondly, that in actuality processes of Nature in their entirety are irreversible. ”

“ This new calculus was to the previous integration what integration was to addition. Integration is an infinite summation of very small terms, and this new mathematics was an integration in an infinite number of dimensions. The beginner first learns to integrate in two dimensions. Then come three. Einstein had carried his mathematics to four. The machine seemed to work in an infinite number of dimensions, but the conditions of the problem really chose the four out of infinity that were under discussion. An infinite number of dimensions has no physical meaning. ”

“ In the physical sciences we are compelled to assume on the one hand the atomic or granular constitution of matter, in other words, manifoldness. On the other hand, if “action at a distance” is to be escaped, we are bound to assume a continuum of some sort like the ether. ”

“ What has never hitherto been fully felt, and what leads me to regard the Entropy theory (1850) as the beginning of the destruction of that masterpiece of Western intelligence, the old dynamic physics, is the deep opposition of theory and actuality which is here for the first time introduced into theory itself. ”