cloud chamber. It isn’t recorded whether, after the discovery of the posi-
tron, Dirac turned to Kapitsa and said, “There!” If Dirac could have re-
sisted the temptation, he would have been one of the rare people able to
do so. Dirac shared the Nobel Prize in 1933.
In mathematics, one way to avoid the dilemma posed by the existence of
gardens of negative width is to restrict the domain of the function (the set
of allowed input values) under consideration. Thus, when considering the
equations for the garden described at the outset of this chapter, one might
consider only those values of L and W (the length and width of the gar-
den) that are positive. Thus restricted, the quadratic equation we obtained
has only one solution in the allowed domain of the function, and the
problem of gardens of negative width is eliminated.
However, as in the example of Dirac’s equation, the physicist cannot
cavalierly restrict the domain of functions that describe phenomena. By
doing so, the restricted domain might describe phenomena that are
known—but in the part of the domain that was excluded might lurk
something unexpected and wonderful.
Complex Cookies
Mathematical concepts are idealizations. Some idealizations, such as
“three” or “point,” have close correspondences with our intuitive under-
standing of the world. Some, such as i (the square root of 1) have utilit y
without such close correspondence. A staple mathematical tool in quan-
tum mechanics is the wave function, which is a complex-valued function
whose squares are probability density functions. Probability density func-
tions are fairly easy to understand: I am more likely to be in Los Angeles
(my home) next Tuesday than I am to be in Cleveland, but there are cer-
tainly events that would necessitate a trip there. Low-probability events, to
be sure—but not impossible ones. The complex-valued function whose
square is a probability density function does not seem to have any corre-
spondence to the world—it is a mathematical entity that, when properly
manipulated, gives accurate results about the world.
But what have complex numbers to do with the real world? We cannot
buy 2 3 i cookies for 10 15 i cents per cookie—but if we could, we could
pay the bill! Using the formula that cost equals number of cookies times
the price per cookie, the total cost would be
(2 3 i) (10 15 i) 20 30 i 30 i 45 65 centsAnalogous situations frequently occur in physics—real phenomena have
unreal, but useful, descriptions. Does the utility of these descriptions
136 How Math Explains the World