Next: The Euclidean Lattice Up: Conceptual Foundations Previous: Conceptual Foundations Contents
The first thing we must do here is to point out an important duality of nature, according to which all things that exist in nature can be classified either as belonging to what has been termed the world of atoms, or as belonging to what can be described as the world of bits. By the first of these two general classes what is meant here is really the set of all physical objects. To put it in a more precise and fundamental way, this consists of matter and radiation. The name “world of atoms” is too restrictive for our purposes, and maybe one should use “world of energy” instead, since the energy is the one concept common to all forms of matter and radiation. This world of energy is the set of objects which are the subject of the physical sciences, and comprises what one usually understands in physics as objective physical reality.
The world of bits, on the other hand, is better described, perhaps, as the world of information. The concept of information is much harder to define than the concept of energy, perhaps because it is a rather new concept, while the concept of energy has a long story behind it. However, it is rather easy to recall a few familiar examples, such as the contents of a book, the contents stored within a digital computer, and the contents of the nuclear DNA of the cells of a living being. Energy and information are very different concepts. While the most fundamental property of energy is its conservation, that is, the fact that it may be transformed into many different forms while its total amount remains constant in all physical processes, information can be easily destroyed, and can also be created, usually with some difficulty, as those involved in research activities know well.
The two concepts are of course related, and one may even go so far as to further classify information according to the nature of this relation. For example, one may consider active and passive forms of information, the active form being one that interacts more closely with some object capable of information processing, enabling it to do things it would not be otherwise capable of doing, while the passive form consists of stacks of data that may or may not be used by this information-processing enabled object. Examples of active information in the case of a digital computer are the programs that can run on it, while passive information consists of stacks of data that can be used by such programs. In the case of the human mind one can say that the acquisition of skills, such as playing the piano or performing differentiations and integrations in mathematical calculus, is as example of the existence of active information within the mind, while the memorization of lists of things and facts may be understood as the acquisition by the mind of passive information.
Our universal duality becomes then the duality of energy and information. What we must point out here is that physical theory exists in the interface between these two worlds. If, on the one hand, it must contain a representation of objective physical reality, its structure must also allow for reasonably easy manipulation by the human mind, which is an object living in the world of information. Note that just as one can think of the hardware and software aspects of a computer, so one can think of the brain as a hardware aspect belonging to the world of energy, and of the mind as a software aspect belonging to the world of information. It is the mind that matters here.
One very fundamental example of an object that deals with the interface between these two worlds is the concept of a physical measurement, which is one of the fundamental concepts of quantum mechanics. A physical measurement can be described as a process taking place in the world of energy that has as its end-result the production of a certain amount of information. It is therefore a process that starts in the world of energy and ends up in the world of information. We see therefore that both the experimental or measurement aspect and the theoretical aspect of physics exist in the same realm.
Since physical theory exists in the interface of the worlds of energy and information, it may be argued that its structure must cater as much to the facts of objective physical reality as to the characteristics and limitations of the mind it is built to serve. We will therefore not require the physical theory to contain exclusively the elements of objective physical reality, but rather require that it also make life easy for the mind. In other words, physical theory will be allowed to contain elements which are not mandated by objective physical reality, and that are allowed in for the convenience of the mind.
The next subject we must tackle here relates to the issue of the mathematical definition of the structure of the physical theory. At risk of stating the obvious, we must say here that the mathematical structure of the theory should be clearly and completely defined. Theoretical physics is a very difficult subject, and it leads to a strong tendency towards unbridled speculation. Although this may be a good and healthy thing, and can be an important tool of discovery, it should not become an end in itself. After all the speculations are proposed, examined, and possibly discarded, a well-structured mathematical theory should emerge. Too much wild speculation for too long can not only lead to loss of contact with physical reality, it may also lead to loss of contact with mathematical reality, and even to loss of contact with logic.
Besides requiring that the mathematical structure of the theory be stated clearly and completely, we must further require that this definition be constructive, that is, ultimately built without gaps from basic things such as the arithmetic of integer and real numbers. One way to interpret this requirement is to say that the definition should allow for an algorithmic realization. What this means is that, given a definite physical question within the theory, it should be possible to derive from its definition a set of rules and chained operations that would make it possible to answer that question, at least in principle, by the use of a program running on a digital computer. The proviso “at least in principle” is included because obtaining an infinitely precise answer might require the use of an infinitely powerful computer, being therefore impossible in practice. A less strict but sufficient requirement would be that the answer can be obtained within a finite and limited level of precision, given a sufficiently powerful computer and sufficient time to run the program on it.
Note that we do not regard as a requirement that it be possible to execute the necessary calculations with the unaided human mind, or by the use of the traditional analytical methods of mathematics. The ability to use a digital computer may be an essential element for the utilization of a physical theory as we understand it here. Although it is conceivable that the future may bring new analytical methods in mathematics, which may find use in the most important calculations in the theoretical physics described here, no such analytical methods are known at present, and it may even turn out that none exist. We do not, therefore, require that purely analytical methods be applicable to the theory, and regard direct numerical methods as sufficient.
The next subject to be discussed here has to do with the nature of the limits involved in the mathematical definition of the theory. Although the mathematical structures involved in the theory will, by the end of the definition process, become continuum mathematical structures, we will require that such continuum structures be obtainable by means of limiting procedures starting from finite mathematical structures. What we mean here is that the structures involved in the construction of the limits be not only discrete, but actually finite as well, in terms of the number of elements involved.
A simple and familiar object defined by a limiting procedure of this kind is the Riemann integral, which is defined as a limit from a set of finite sums, that is, sums with a finite number of terms. Note that there is no veto here to the existence of some other definition, equivalent to the original one, which may be formulated exclusively in terms of continuum quantities. What we impose here is a veto against continuum mathematical structures that cannot be formulated as limits from strictly finite mathematical structures. We will also regard the definition by such a limiting procedure as the most fundamental one, and choose it over any others in case any doubts arise regarding the equivalences among them.
This philosophical attitude, which some may regard as a preconception or prejudice, can be motivated by the fact that, ultimately, the set of all possible physical measurements and experiments that can be actually carried out by us within any possibly large but certainly finite amount of time is certainly a finite set. It is therefore not natural to think that the results of all these measurements and experiments can only be systematized and encoded within a mathematical structure which is intrinsically undefinable in terms of finite mathematical structures. One may also formulate this philosophical concept in terms of the number of particles of either matter or radiation that can be ever detected by us, which is also necessarily a finite number.
The next subject to be discussed is that of the definition of observables within the theory. As explained before, we do not expect the theory to consist only of observable quantities. Even non-relativistic quantum mechanics contains elements that, although playing important roles within the theory, are not themselves observable, such as the wave function. We expect the theory to contain elements that are there for the convenience of the mind, not because they are observables. One such object in quantum field theory is the fundamental field itself, because the value of the field at a given point of space-time is not an observable of the theory.
What is required of the theory regarding the physical observables is that they be clearly and concisely identified as such. The definition of the theory must include a clear and concise discrimination of the parts or aspects of the structure which are physical observables. The requirement of conciseness is included because it will not do to have a definition which has to be revised or modified each time a new quantity comes into consideration. There must be a global and fixed set of criteria that can be used to identify the quantities which correspond to physical observables.
Note that this leaves open the possibility that one may use a computer, or some other computational means, to perform mathematical probes into the structure of the theory, which may not necessarily correspond to physically realizable observations. This “inner look” into the mathematical structure of the theory may be useful for establishing a better understanding of the inner workings of the theory, which may have consequences for our understanding of the realm of physical observables, even if only in an indirect way. In the subsequent chapters we will be using this freedom constantly, and we will arrive at a fairly complete definition of the observables only very slowly, using our mathematical and computational probes into the inner workings of the theory along the way.
The last issue we would like to address here is that of the closure of the theory. By this we mean that, once it is defined completely in a concise way, all physical elements which are needed to describe objective physical reality must be contained within its structure, either directly as fundamental elements or as higher-level constructs derived from them. It should not be necessary to call on any external elements in order to complete the description. Although this may sound as somewhat self-evident, it is important to state it here, for clarity of definition. The role of this property will become evident in the subsequent chapters, but its most important consequences only come into play much later on, when one discusses the geometry of space-time, and are therefore outside the scope of this book.
The closure of the theory has a role to play in the very difficult issue of the physical measurement process. Measurement apparata are part of physical reality and are of course subject to the same physical laws they are meant to probe. The definition of the observables within the theory if of course closely tied to the definition of the process of physical measurement, by which information is created out of physical processes involving energy, and is therefore equally difficult. In the chapters that follow we will attempt to narrow down gradually the definition of an observable, but no complete description of the physical measurement process will be possible within the confines of the Gaussian model, which contains only free, non-interacting fields.
The complete description of the measurement process, including what is usually referred to as the reduction of the wave packet, and hence the complete definition of the physical observables, is probably only possible in the context of a complete and interacting theory, in which there are stable bound states. This is so because without bound states it would not be possible to store the bits of information generated by the measurement process. Such a complete and realistic theory is certainly a very difficult object to deal with, and we are not currently in a position to describe a complete model having all these properties.