The Cellular Automaton Interpretation of Quantum Mechanics by Gerard 't Hooft

Author:Gerard 't Hooft
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

11.2 Notation

It is difficult keep our notation completely unambiguous. In Chap. 16, we are dealing with many different types of variables and operators. When a dynamical variable is an integer there, we shall use capitals  . Variables that are periodic with period , or at least constrained to lie in an interval such as , are angles, mostly denoted by Greek lower case letters  , whereas real variables will most often be denoted by lower case Latin letters  . Yet sometimes we run out of symbols and deviate from this scheme, if it seems to be harmless to do so. For instance, indices will still be for space-like vector components, for spinors and for Lorentz indices. The Greek letters and will be used for wave functions as well.

Yet it is difficult to keep our notation completely consistent; in some chapters before Chap. 16, we use the quantum numbers and of the representations to denote the integers that earlier were denoted as or , and later in Chap. 16 replaced by capitals.

As in Part I, we use a super- or subscript “op” to distinguish an operator from an ordinary numerical variable. The caret () will be reserved for vectors with length one, the arrow for more general vectors, not necessarily restricted to three dimensional space. Only in Chap. 20, where norms of vectors do not arise, we use the caret for the Fourier transform of a function.

Dirac’s constant and the velocity of light will nearly always be defined to be one in the units chosen. In previous work, we used a spacial symbol to denote as an alternative basis for exponential functions. This would indeed sometimes be useful for calculations, when we use fractions that lie between 0 and 1, rather than angles, and it would require that we normalize Planck’s original constant rather then to one, but in the present monograph we return to the more usual notation.

Concepts frequently discussed are the following: discrete variables are variables such as the integer numbers, whose possible values can be counted. Opposed to continuous variables, which are typically represented by real or complex numbers.

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