1.2. PHYSICAL PROLOGUE: DIMENSIONS, UNITS, CONSTANTS, AND PARTICLES 5
where δ denotes the delta-function (point mass at the origin) on
Rn.
This is the
Fourier inversion formula, stated in the informal language of distributions. Readers
who do not yet qualify as mathematical physicists by this criterion are advised to
spend some time playing with this formula until they understand how to make sense
of it and are convinced of its validity, for it will be used on many occasions in this
book. (The necessary tools from distribution theory can be found, for example, in
Folland [48].)
It is useful to remember that the factors of always go with the measure on
momentum space; that is, the standard measure on momentum space (or, more
abstractly, “Fourier space” as opposed to “configuration space”) Rn is dnp/(2π)n.
A corollary of this is that delta-functions on momentum space are normally ac-
companied by a factor of (2π)n so that their integral against this measure is still
1.
Symbolic homonyms. It is an unfortunate fact of life that some letters of
the alphabet have more than one conventional meaning, while others are used
sometimes conventionally and sometimes just as a convenient label for a variable.
In such cases one must rely on context for clarification. For example, all of us have
surely seen uses of the letter π that have nothing to do with the constant 3.14 . . .;
there are some in this book. Other notable examples: (1) e is both exp(1) and the
electric charge that functions as the coupling constant in quantum electrodynamics
(typically the charge of the electron or its absolute value); occasionally it is also
the symbol for the electron. (Fortunately, we never need it to be the eccentricity of
an ellipse!) (2) q is sometimes an electric charge and sometimes a momentum. (3)
αj is a type of Dirac matrix, but α is the fine structure constant. (4) γμ is another
type of Dirac matrix, but γ is the Euler-Mascheroni constant and occasionally the
symbol for the photon. (5) Z is sometimes the number of protons in a nucleus,
sometimes a field renormalization constant, sometimes the generating functional in
the functional integral approach to field theory, and sometimes the symbol for the
neutral vector Boson of the weak interaction.
Caveat lector.
1.2. Physical prologue: dimensions, units, constants, and particles
The fundamental aspects of the universe to which all physical measurements
relate are mass [m], length or distance [l], and time [t]. All quantities in physics
come with “dimensions” that can be expressed in terms of these three basic ones.
For example, velocity v = dx/dt has dimensions of distance per unit time, or
[lt−1].
The dimensions of some of the basic quantities of mechanics are summarized
in Table 1.1. The fact that angular momentum and action both have dimensions
[ml2t−1]
seems a mere coincidence in classical mechanics, but it acquires a deeper
significance in quantum mechanics, because
[ml2t−1]
is the dimensions of Planck’s
constant.
Units for electromagnetic quantities are related to those of mechanics by taking
the constant of proportionality in some basic law to be equal to one. For example,
the ampere is the unit of current defined as follows: if two infinite, straight, perfectly
conducting wires are placed parallel to each other one meter apart and a current
of one ampere flows in each of them, the (magnetic) force between them per unit
length is
2×10−7
newton per meter. The other everyday units of electromagnetism
(coulomb, volt, ohm, etc.) are then defined in terms of the ampere and the MKS
Previous Page Next Page