8 The Most Accurate Theory in Physics
161
factors, equal to 1/(137 × 137 × 137), or 1/2,571,353. Orders higher than
this can be safely neglected.
Before concluding this Chapter we must draw attention to a very large
elephant that has been hulking around the offices and laboratories of physicists since the middle of the last century. So accustomed have they become
to its presence that they pursue their studies amidst the elephant’s trampling
limbs and swinging trunk with scarcely a moment of concern. They have even
given it a name: renormalisation.
The first indication of its presence came when considering the effect of
virtual electron–positron pairs created in the vicinity of a real electron. The
real electron should attract the virtual positrons towards it, resulting in it
appearing to have less electric charge than it actually has. QED enables us to
calculate how much this reduction of charge should be, and the answer turns
out to be infinite. Undaunted, physicists simply propose that the unshielded
electron has an infinite negative charge so that when the infinite positive
shielding charge is added to it, the result is just the small negative value that
the electron actually has. To sum up:
∞ − ∞ = −1.60 × 10
−19 Coulombs.
This procedure is known as renormalisation. The electron mass also
requires a similar sleight-of-hand to obtain the experimentally observed value.
We have discussed in Chap. 3 the difference between physicists and mathematicians in their attitudes to truth. No mathematician would consider
subtracting two infinities from each other as anything but nonsense. To be
honest, many physicists agree with them, but would be unwilling to confess
it in public. Nobel Laureates can afford to be more forthright:
“I must say that I am very dissatisfied with the situation, because this socalled ‘good theory’ does involve neglecting infinities which appear in its equations,
neglecting them in an arbitrary way. This is just not sensible mathematics.
Sensible mathematics involves neglecting a quantity when it turns out to be
small—not neglecting it just because it is infinitely great and you do not want
it!” – Paul Dirac, Nobel Laureate, 1933 [4].
“But no matter how clever the word (renormalisation), it is still what I
would call a dippy process! Having to resort to such hocus-pocus has prevented
us from proving that the theory of quantum electrodynamics is mathematically
self-consistent. It’s surprising that the theory still hasn’t been proved self-consistent
one way or the other by now; I suspect that renormalization is not mathematically
legitimate.” – Richard Feynman, Nobel Laureate, 1965 [5].
However, once the renormalisation procedure is carried out, the agreement
of experiment with the predictions of QED is so astonishingly good that there
161
factors, equal to 1/(137 × 137 × 137), or 1/2,571,353. Orders higher than
this can be safely neglected.
Before concluding this Chapter we must draw attention to a very large
elephant that has been hulking around the offices and laboratories of physicists since the middle of the last century. So accustomed have they become
to its presence that they pursue their studies amidst the elephant’s trampling
limbs and swinging trunk with scarcely a moment of concern. They have even
given it a name: renormalisation.
The first indication of its presence came when considering the effect of
virtual electron–positron pairs created in the vicinity of a real electron. The
real electron should attract the virtual positrons towards it, resulting in it
appearing to have less electric charge than it actually has. QED enables us to
calculate how much this reduction of charge should be, and the answer turns
out to be infinite. Undaunted, physicists simply propose that the unshielded
electron has an infinite negative charge so that when the infinite positive
shielding charge is added to it, the result is just the small negative value that
the electron actually has. To sum up:
∞ − ∞ = −1.60 × 10
−19 Coulombs.
This procedure is known as renormalisation. The electron mass also
requires a similar sleight-of-hand to obtain the experimentally observed value.
We have discussed in Chap. 3 the difference between physicists and mathematicians in their attitudes to truth. No mathematician would consider
subtracting two infinities from each other as anything but nonsense. To be
honest, many physicists agree with them, but would be unwilling to confess
it in public. Nobel Laureates can afford to be more forthright:
“I must say that I am very dissatisfied with the situation, because this socalled ‘good theory’ does involve neglecting infinities which appear in its equations,
neglecting them in an arbitrary way. This is just not sensible mathematics.
Sensible mathematics involves neglecting a quantity when it turns out to be
small—not neglecting it just because it is infinitely great and you do not want
it!” – Paul Dirac, Nobel Laureate, 1933 [4].
“But no matter how clever the word (renormalisation), it is still what I
would call a dippy process! Having to resort to such hocus-pocus has prevented
us from proving that the theory of quantum electrodynamics is mathematically
self-consistent. It’s surprising that the theory still hasn’t been proved self-consistent
one way or the other by now; I suspect that renormalization is not mathematically
legitimate.” – Richard Feynman, Nobel Laureate, 1965 [5].
However, once the renormalisation procedure is carried out, the agreement
of experiment with the predictions of QED is so astonishingly good that there