Download 7.2.4. Normal Ordering

7.2.4. Normal Ordering
In the last section, we get rid of an infinite vacuum energy on the ground that only
relative energies have physical meaning. However, if the structure of spacetime is to
be determined by matter distribution, the vacuum must be at zero energy. Therefore,
we must impose some rules in the construction of the Hamiltonian operator so that
this is achieved automatically. The standard procedure is to write Ĥ entirely in
normal ordering, i.e., with all creation operators to the left of all annihilation
operators. Using :  : to denote normal ordering for the product of operators  ,
eq(7.21a) becomes
Hˆ  
d 3k
 2  2k
3

k :  aˆ   k  â  k  ĉ  k  cˆ   k   :
d 3k
 2  2k
3
k  â   k  â  k  ĉ  k  ĉ  k  
Since the terms in the square bracket are simply the number of particles and
antiparticles with momentum k, the total energy is always positive.
Obviously, the technique should be applied to all “total” operators that involve
integration over all degrees of freedom.
defined by [see (7.4)],
For example, the number operator is

i
3
ˆ   0 ˆ :
d
x
:

N̂   d 3 x : ˆj 0 : 
2m 
After some tedious but straightforward manipulations, we get [cf §7.2.3],
N̂  
d 3k
 2  2k
3
 â   k  â  k  ĉ  k  ĉ  k  
(7.22)
As in the 1st quantization case, ĵ 0  x  is not non-negative so that it cannot represent
the probability density of finding a particle at x.
Now, we define an anti-particle as a “particle” whose attribute quantum numbers are
all equal but of opposite signs to those of its particle partner. Some examples of
attribute quantum numbers are lepton numbers, baryon numbers, charge, isospin,
strangeness, etc.
Thus, ĵ 0  x  is the net probability density so that the equation of
continuity denotes the conservation of net number of “particles”.
Finally, we mention that some neutral particles are identical to their anti-particles.
Notable examples are photons and neutral pions. Since one can call any given
photon a particle as well as anti-particle, the net number of “particles” is always
conserved as long as the total number of photons present is always odd (or even). In
other words, the number of photons is not conserved. Note that this is necessarily
the case if photons are to be interpreted as quanta of electromagnetic fields.