기호설명 Notation
(Not to be read until you are familiar with the concepts, but perhaps find the fonts confusing!)
I have tried to be reasonably consistent in the use of particular fonts in this book, but as not all of this is standard, it may be helpful to the reader to have the major usage that I have adopted made explicit.
Italic lightface (Greek or Latin) letters, such as in w/2, /pn, log z, cos θ, ei/θ/, or ex are used in the conventional way for mathematical variables which are numerical or scalar quantities; but established numerical constants, such as e, i, or π or established functions such as sin, cos, or log are denoted by upright letters. Standard physical constants such as c, G, h, ħ, g, or k are italic, however.
A vector or tensor quantity, when being thought of in its (abstract) entirety, is denoted by a boldface italic letter, such as R for the Riemann curvature tensor, while its set of components might be written with lightface italic letters (both for the kernel symbol its indices) as Rabcd. In accordance with the abstract-index notation, introduced here in §12.8, the quantity Rabcd may alternatively stand for the entire tensor R, if this interpretation is appropriate, and this should be clear from the text. Abstract linear transformations are kinds of tensors, and boldface italic letters such as T are used for such entities also. The abstract-index form Tab is also used here for an abstract linear transformation, where appropriate, the staggering of the indices making clear the precise connection with the ordering of matrix multiplication. Thus, the (abstract-)index expression SabTbc stands for the product ST of linear transformations. As with general tensors, the symbols Sab and Tbc could alternatively (according to context or explicit specification in the text) stand for the corresponding arrays of components---these being matrices---for which the corresponding bold upright letters S and T can also be used. In that case, ST denotes the corresponding matrix product. This ‘ambivalent’ interpretation of symbols such as Rabcd or Sab (either standing for the array of components or for the abstract tensor itself) should not cause confusion, as the algebraic (or differential) relations that these symbols are subject to are identical for both interpretations. A third notation for such quantities---the diagrammatic notation---is also sometimes used here, and is described in Figs. 12.17, 12.18, 14.6, 14.7, 14.21, 19.1 and elsewhere in the book.
There are places in this book where I need to distinguish the 4-dimensional spacetime entities of relativity theory from the corresponding ordinary 3-dimensional purely spatial entities. Thus, while a boldface italic notation might be used, as above, such as p or x, for the 4-momentum or 4-position, respectively, the corresponding 3-dimensional purely spatial entities would be denoted by the corresponding upright bold letters p or x. By analogy with the notation T for a matrix, above, as opposed to T for an abstract linear transformation, the quantities p and x would tend to be thought of as ‘standing for’ the three spatial components, in each case, whereas p and x might be viewed as having a more abstract component-free interpretation (although I shall not be particularly strict about this). The Euclidean ‘length’ of a 3-vector quantity a = (a/1,/a/2,/a/3) may be written /a, where 
, and the scalar product of a with b = (b/1,/b/2,/b/3), written a • b = /a/1/b/1 + /a/2/b/2 + /a/3/b/3. This ‘dot’ notation for scalar products applies also in the general /n-dimensional context, for the scalar (or inner) product α • ξ of an abstract covector α with a vector ξ.
A notational complication arises with quantum mechanics, however, since physical quantities, in that subject, tend to be represented as linear operators. I do not adopt what is a quite standard procedure in this context, of putting ‘hats’ (circumflexes) on the letters representing the quantum-operator versions of the familiar classical quantities, as I believe that this leads to an unnecessary cluttering of symbols. (Instead, I shall tend to adopt a philosophical standpoint that the classical and quantum entities are really the ‘same’---and so it is fair to use the same symbols for each---except that in the classical case one is justified in ignoring quantities of the order of ħ, so that the classical commutation properties ab = ba can hold, whereas in quantum mechanics, ab might differ from ba by something of order ħ.) For consistency with the above, such linear operators would seem to have to be denoted by italic bold letters (like T), but that would nullify the philosophy and the distinctions called for in the preceding paragraph. Accordingly, with regard to specific quantities, such as the momentum p or p, or the position x or x, I shall tend to use the same notation as in the classical case, in line with what has been said earlier in this paragraph. But for less specific quantum operators, bold italic letters such as Q will tend to be used.
The shell letters ℕ, ℤ, ℝ, ℂ, and 𝔽q, respectively, for the system of natural numbers (i.e. non-negative integers), integers, real numbers, complex numbers, and the finite field with q elements (q being some power of a prime number, see §16.1), are now standard in mathematics, as are the corresponding ℕn, ℤn, ℝn, ℂn, 𝔽
, for the systems of ordered n-tuples of such numbers. These are canonical mathematical entities in standard use. In this book (as is not all that uncommon), this notation is extended to some other standard mathematical structures such as Euclidean 3-space 𝔼3 or, more generally, Euclidean n-space 𝔼n. In frequent use in this book is the standard flat 4-dimensional Minkowski spacetime, which is itself a kind of ‘pseudo-’ Euclidean space, so I use the shell letter 𝕄 for this space (with 𝕄n to denote the n-dimensional version---a ‘Lorentzian’ spacetime with 1 time and (n — 1) space dimensions). Sometimes I use ℂ as an adjective, to denote ‘complexified’, so that we might consider the complex Euclidean 4-space, for example, denoted by ℂ𝔼n. The shell letter ℙ can also be used as an adjective, to denote ‘projective’ (see §15.6), or as a noun, with ℙn denoting projective n-space (or I use ℝℙn or ℂℙn if it is to be made clear that we are concerned with real or complex projective n-space, respectively). In twistor theory (Chapter 33), there is the complex 4-space 𝕋, which is related to 𝕄 (or its complexification ℂ𝕄) in a canonical way, and there is also the projective version ℙ𝕋. In this theory, there is also a space ℕ of null twistors (the double duty that this letter serves causing no conflict here), and its projective version ℙℕ.
The adjectival role of the shell letter ℂ should not be confused with that of the lightface sans serif C, which here stands for ‘complex conjugate of’ (as used in §13.1,2). This is basically similar to another use of C in particle physics, namely charge conjugation, which is the operation which interchanges each particle with its antiparticle (see Chapters 25, 30). This operation is usually considered in conjunction with two other basic particle-physics operations, namely P for parity which refers to the operation of reflection in a mirror, and T, which refers to time-reversal. Sans serif letters which are bold serve a different purpose here, labelling vector spaces, the letters V, W, and H, being most frequently used for this purpose. The use of H, is specific to the Hilbert spaces of quantum mechanics, and *H*n would stand for a Hilbert space of n complex dimensions. Vector spaces are, in a clear sense, flat. Spaces which are (or could be) curved are denoted by script letters, such as M, S, or T, where there is a special use for the particular script font J to denote null infinity. In addition, I follow a fairly common convention to use script letters for Lagrangians (L) and Hamiltonians (H), in view of their very special status in physical theory.