Quantization, Space-Time

the common name for generalizations of the theory of elementary particles (quantum field theory), which are based on the hypothesis that there are finite minimum distances and time intervals. The construction of a noncontradictory theory in which all physical quantities would be finite is the immediate purpose of such generalizations.

The concepts of space and time that are used in modern physical theory are formulated most consistently in Einstein’s theory of relativity. These concepts are macroscopic, that is, they rely on experience acquired in the study of macroscopic objects, large distances, and large time intervals. In the construction of a theory describing the phenomena of the microworld—quantum mechanics and quantum field theory—the classical geometric picture, which presupposes the continuity of space and time, was transferred to the new area without alteration. Experimental verification of the conclusions of quantum theory does not yet directly indicate the existence of a boundary beyond which classical geometric concepts cease to be applicable. However, the theory of elementary particles itself contains difficulties that would suggest that the geometric concepts developed on the basis of macroscopic experimentation are invalid for the infinitesimal distances and time intervals characteristic of the microworld and that the concepts of physical space and time need to be reexamined.

These theoretical difficulties are connected with the problem of divergences: calculations of certain physical quantities lead to physically meaningless, infinitely large values, or divergences. Divergences appear since, in current theory, elementary particles are considered as “points,” that is, as material objects without dimension. In its simplest form, this was manifested as early as classical electromagnetic field theory (classical electrodynamics), in which there occurs a Coulomb divergence—an infinitely large value for the energy of the Coulomb field of a charged point particle [since at very small distances r from a particle (r → 0) the field increases without limit].

Not only is Coulomb divergence retained in quantum field theory, but new divergences appear as well (for example, divergence for an electrical charge) that are also ultimately connected with the point nature of particles. The condition of the point nature of particles in quantum field theory appears in the form of the requirement of what is called the locality of interactions: the interaction between fields is determined by quantities that describe the fields and that are taken at the same point in space and at the same instant in time. It would seem that divergence can be easily eliminated if particles are considered not as point particles but as extended particles that are spread over some small volume. But here the theory of relativity imposes significant limitations. According to the theory the speed of any signal (that is, the rate of transfer of energy or the rate of transfer of interaction) cannot exceed the speed of light c. The assumption that an interaction can be transferred at velocities greater than the speed of light leads to contradiction of the traditional concepts (confirmed by all human experience) of a temporal sequence of events connected causally: it seems that an effect can precede a cause. The finite nature of the rate of propagation of an interaction cannot be coexistent with the indivisibility of particles: in principle, such powerful momentum could very rapidly be imparted to some small part of an extended particle that the particle would fly off before the signal reached the part that remained.

Thus, the requirements of the theory of relativity and causality make it necessary to consider particles as point entities. But the concept of the point nature of particles is closely connected with the nature of the geometry used in the theory and, in particular, with whether this geometry is based on the assumption that it is possible in principle to make whatever precise measurement of distances (lengths) and time intervals as may be necessary. In ordinary theory this possibility is assumed explicitly or, more often, implicitly.

In all variant geometries a major role belongs to the fundamental length l, which has been introduced into theory as a new universal constant (along with Planck’s constant h and the speed of light c). The introduction of the fundamental length l corresponds to the assumption that the measurement of distances is possible in principle only with limited accuracy of the order of l (the measurement of time, with an accuracy of the order of l/c). Therefore, l is also called the minimum length. If particles are considered to be extended entities, then their dimensions fill the role of some minimal scale of length. Thus, the introduction of a fundamental (minimal) length in one sense conceals the extended nature of particles. This gives hope of constructing a theory free of divergences.

One of the first attempts to introduce a fundamental length was connected with the transition from the continuous coordinates x, y, z, and time t to discrete coordinates: x → n₁l, Y → n₂l, z → n₃l and t → (n₄l), where n₁, n₃ and n₄ are integers that may assume values from minus infinity to plus infinity. The replacement of continuous coordinates with discrete coordinates is somewhat reminiscent of Bohr’s rules of quantization in the original theory of the atom—hence the term “space-time quantization.”

If we examine large distances and time intervals, then every “elementary step” l or l/c may be considered infinitesimal. Therefore, “large-scale” geometry appears routine. However, on a “small scale” the effect of such quantization becomes significant. In particular, introduction of the minimal length l excludes the existence of waves with a length λ < l, that is, precisely those quanta of infinitely large frequency ν = c/λ and, consequently, of energies ε = hν which, as quantum field theory demonstrates, are responsible for the appearance of divergences. How the change in geometrical concepts entails important physical consequences is graphic here.

The introduction of “cellular” space (with “cells” of dimension) in this manner is associated with a disruption of the isotropy of space—violation of the equivalence of all directions. This is one of the significant shortcomings of this theory.

Just as Bohr’s theory (in which the quantization conditions were postulated) was replaced by quantum mechanics (in which quantization was seen as a natural consequence of the theory’s fundamental propositions), more advanced variations followed the first attempts at space-time quantization. Common to all of them is the consideration of coordinates and time as operators and not as ordinary numbers (here, too, there is an analogy with quantum mechanics, in which operators are posited to correspond to physical quantities). An important general theorem is formulated in quantum mechanics: if certain operators are not commutative (that is, if the sequence of the factors cannot be changed in multiplying such operators), then the physical quantities corresponding to these operators cannot be simultaneously determined with precision. Such, for example, are the operators of the coordinates ̂x and the momentum ̂p_x of a particle (operators are commonly designated by the same letters as the corresponding physical quantities but with a “cap” above them). The noncommutative nature of these operators is a mathematical reflection that the uncertainty relation

which shows the limits of the accuracy with which p_x and x can be determined simultaneously, obtains for the coordinates and momentum of a particle. A particle cannot have precisely defined coordinates and momentum simultaneously: the more precisely the coordinates are defined, the less precise is the momentum, and vice versa (the probabilistic description of a particle’s state in quantum mechanics is associated with this).

In space-time quantization, operators that are associated with the coordinates of points in space and instants in time are considered noncommutative. The noncommutative nature of the operators ̂x and ̂i, ̂x and ̂y and so on means that the precise value of the coordinate x, for example, at a given instant t cannot be determined, just as the precise value of several coordinates cannot be given simultaneously. This leads to the probabilistic description of space-time. The type of operator is chosen in such a way that the average values of the coordinates can assume only integral values that are multiples of the fundamental length l. The scale of error, or the uncertainty, of the coordinates is determined by the fundamental length. Some versions of the theory postulate the noncommutability of the coordinate operators and the operators that describe the field. This is equivalent to the assumption that it is impossible simultaneously to specify precisely the quantities that describe a field and the point in space to which these quantities refer (variants of this type often are called theories of nonlocalizable states).

In most known attempts at space-time quantization, postulates concerning the “microstructure” of space-time are introduced first; later the resultant space is “populated” with particles, the laws of whose motion are made to correspond with the new geometry. A number of interesting results have been obtained in this manner: certain divergences are eliminated (although new ones sometimes appear in their place) and, in some cases, the mass spectrum of elementary particles is obtained, that is, the possible masses of particles are predicted. However, radical progress has not yet been possible, although the methodological value of the work so far accomplished is indisputable. It seems likely that the difficulties that arise here attest to short comings in the very approach to the problem, in which the construction of new theory begins with postulates concerning “empty” space (that is, purely geometric postulates, independent of the matter populating the space).

A reexamination of geometric concepts is needed; this has been recognized almost universally. However, such a reexamination obviously should provide much greater consideration of the indivisibility of the concepts of space, time, and matter.