292
9
Quantum Mechanics Formalism
• The state

x
, α
would describe a particle located in
x
at a time
t
. Due to the
possibility, in relativistic processes, of creating new particles, provided the energy
involved is large enough, and in the light of Heisenberg’s uncertainty principle,
there is a physical obstruction in determining the position of a particle at a given
time with indefinite precision: The smaller the distances we wish to probe, in order
to locate a particle with a sufficiently high precision, the larger the momentum and
thus the energy we need to transfer to the particle and, if the energy transferred
is large enough to produce one or more particles identical to the original one, we
may end up with a system of virtually undistinguishable particles, thus making
our initial position measurement meaningless. This is also related to the problem
with interpreting the relativistic field
α
(
x
)
as the wave function associated with
a given singleparticle state, like we did in the nonrelativistic theory. We shall
comment on this in some more detail in the introduction to next chapter.
• The normalization (
9.95
), which guarantees that all the states of the basis have a
positivenorm,isnotLorentzinvariant.Indeed,while
δ
4
(
x
−
x
)
isLorentzinvariant,
δ
α,β
would be invariant only if the representation
D
were
unitary
. There is however
a property in group theory which states that
unitary representations of the Lorentz
group can only be infinite dimensional
(like the one acting on the infinitely many
independent quantum states of a particle). Being
D
finitedimensional, it cannot
be unitary, namely
D
†
D
=
1
. As an example, suppose the representation
D
is the
fundamental (defining) representation of the Lorentz group, that is
D
(
)
=
=
(
μ
ν
)
. If these matrices were unitary, being real, they would be orthogonal. We
have learned, however, that
are
pseudoorthogonal
matrices, namely they leave
the Minkowski metric
η
μν
,
rather than the Euclidean one
δ
μν
,
invariant. In other
words
T
=
1
.
13
These are some of the reasons why the coordinate basis
{
x
, α
}
is ill defined in
a relativistic context. We find it however pedagogically useful to use such states in
order to introduce the main objects and notations of relativistic field theory starting
from nonrelativistic quantum mechanics. From now on the main object associated
with a given particle will be the relativistic field
α
(
x
)
.
13
A problem related to the nonunitarity of
D
is the fact that if we defined
α
(
x
)
=
x
, α

a
,
as
we did in the nonrelativistic theory, it would no longer have the correct transformation property
(
9.99
) under Poincaré transformations. For spin 1
/
2 and 1 particles, we can however define a real
symmetric matrix
γ
=
(γ
αβ
)
squaring to the identity
γ
2
=
1
,
such that
γ
D
(
)
†
γ
=
D