77
3.4. The negative-energy solutions
relative to the normal vacuum:
energy of ‘hole’ = −(E neg ) → positive energy
charge of ‘hole’ = −(q e ) → positive charge.
Thus the absence of a negative-energy electron is equivalent to the presence of
a positive-energy positively charged version of the electron, that is a positron.
In the same way, the absence of a ‘spin-up’ negative-energy electron is equivalent to the presence of a ‘spin-down’ positive-energy positron. This last point
is the reason for the subtlety in the choice of χ
s mentioned after (3.82): we
choose
( )
( )
0
1
χ
1 =
χ
2 =
(3.84)
1
0
the opposite way round from the choice for the positive-energy spinors (3.73).
Dirac’s brilliant re-interpretation of (unfilled) negative-energy solutions in
terms of antiparticles is one of the triumphs of theoretical physics
2 : Carl
Anderson received the Nobel Prize for his discovery of the positron in 1932
(Anderson 1932).
In this way it proved possible to obtain sensible results from the Dirac
equation and its negative-energy solutions. It is clear, however, that the theory
is no longer really a ‘single-particle’ theory, since we can excite electrons from
the infinite ‘sea’ of filled negative-energy states that constitute the normal
‘empty state’. For example, if we excite one negative-energy electron to a
positive-energy state, we have in the final state a positive-energy electron plus
a positive-energy positron ‘hole’ in the vacuum: this corresponds physically to
+
the process of e e
− pair creation. Thus this way of dealing with the negativeenergy problem for fermions leads us directly to the need for a quantum field
theory. The appropriate formalism will be presented later, in section 7.2.
3.4.4 Feynman’s interpretation of the negative-energy
solutions of the KG and Dirac equations
It is clear that despite its brilliant success for spin1 particles, Dirac’s inter2
pretation cannot be applied to spin-0 particles, since bosons are not subject to
the exclusion principle. Besides, spin-0 particles also have their corresponding
antiparticles (e.g. π
+ and π
− ), and so do spin-1 particles (W
+ and W
− , for
instance). A consistent picture for both bosons and fermions does emerge
from quantum field theory, as we shall see in chapters 5–7, which is perhaps
one of the strongest reasons for mastering it. Nevertheless, it is useful to have
an alternative, non-field-theoretic, interpretation of the negative-energy solutions which works for both bosons and fermions. Such an interpretation is due
2 At that time, this was not universally recognized. For example, Pauli (1933) wrote:
‘Dirac has tried to identify holes with antielectrons. . . we do not believe that this explanation
can be seriously considered.’
3.4. The negative-energy solutions
relative to the normal vacuum:
energy of ‘hole’ = −(E neg ) → positive energy
charge of ‘hole’ = −(q e ) → positive charge.
Thus the absence of a negative-energy electron is equivalent to the presence of
a positive-energy positively charged version of the electron, that is a positron.
In the same way, the absence of a ‘spin-up’ negative-energy electron is equivalent to the presence of a ‘spin-down’ positive-energy positron. This last point
is the reason for the subtlety in the choice of χ
s mentioned after (3.82): we
choose
( )
( )
0
1
χ
1 =
χ
2 =
(3.84)
1
0
the opposite way round from the choice for the positive-energy spinors (3.73).
Dirac’s brilliant re-interpretation of (unfilled) negative-energy solutions in
terms of antiparticles is one of the triumphs of theoretical physics
2 : Carl
Anderson received the Nobel Prize for his discovery of the positron in 1932
(Anderson 1932).
In this way it proved possible to obtain sensible results from the Dirac
equation and its negative-energy solutions. It is clear, however, that the theory
is no longer really a ‘single-particle’ theory, since we can excite electrons from
the infinite ‘sea’ of filled negative-energy states that constitute the normal
‘empty state’. For example, if we excite one negative-energy electron to a
positive-energy state, we have in the final state a positive-energy electron plus
a positive-energy positron ‘hole’ in the vacuum: this corresponds physically to
+
the process of e e
− pair creation. Thus this way of dealing with the negativeenergy problem for fermions leads us directly to the need for a quantum field
theory. The appropriate formalism will be presented later, in section 7.2.
3.4.4 Feynman’s interpretation of the negative-energy
solutions of the KG and Dirac equations
It is clear that despite its brilliant success for spin1 particles, Dirac’s inter2
pretation cannot be applied to spin-0 particles, since bosons are not subject to
the exclusion principle. Besides, spin-0 particles also have their corresponding
antiparticles (e.g. π
+ and π
− ), and so do spin-1 particles (W
+ and W
− , for
instance). A consistent picture for both bosons and fermions does emerge
from quantum field theory, as we shall see in chapters 5–7, which is perhaps
one of the strongest reasons for mastering it. Nevertheless, it is useful to have
an alternative, non-field-theoretic, interpretation of the negative-energy solutions which works for both bosons and fermions. Such an interpretation is due
2 At that time, this was not universally recognized. For example, Pauli (1933) wrote:
‘Dirac has tried to identify holes with antielectrons. . . we do not believe that this explanation
can be seriously considered.’