Quantum Field Theory I

1.3. RELATIVITY AND QUANTUM THEORY 39
1.3 Relativity and Quantum Theory
Actually, as to the quantum fields, the keyword is relativity. Even if QFT is
useful also in the nonrelativistic context (see the previous section), the very
notion of the quantum field originated from an endeavor to fit together relativity
and quantum theory. This is a nontrivial task: to formulate a relativistic
quantum theory is significantly more complicated than it is in a nonrelativistic
case. The reason is that specification of a Hamiltonian, the crucial operator of
any quantum theory, is much more restricted in relativistic theories.
To understand the source of difficulties, it is sufficient to realize that to have
a relativistic quantum theory means to have a quantum theory with measurable
predictions which remain unchanged by the relativistic transformations. The
relativistic transformations at the classical level are the space-time transformationsconservingthe
intervalds = η µν dx µ dx ν , i.e. boosts, rotations,translations
and space-time inversions (the list is exhaustive). They constitute a group.
Now to the quantum level: whenever macroscopic measuring and/or preparing
devices enjoy a relativistic transformation, the Hilbert (Fock) space should
transformaccordingtoacorrespondinglinear 42 transformation. Itisalsoalmost
self-evident that the quantum transformation corresponding to a composition of
classical transformations should be equal to the composition of quantum transformations
corresponding to the individual classical transformations. So at the
quantumleveltherelativistictransformationsshouldconstitutearepresentation
of the group of classical transformations.
The point now is that the Hamiltonian, as the time-translations generator,
is among the generators of the Poincaré group (boosts, rotations, translations).
Consequently, unlike in a nonrelativistic QM, in a relativistic quantum theory
one cannot specify the Hamiltonian alone, one has rather to specify it within a
complete representation of the Poincaré algebra. This is the starting point of
any effort to get a relativistic quantum theory, even if it is not always stated
explicitly. The outcome of such efforts are quantum fields. Depending on the
philosophy adopted, they use to emerge in at least two different ways. We will
call them particle-focused and field-focused.
The particle-focused approach, represented mostly by the Weinberg’s book,
is very much in the spirit of the previous section. One starts from the Fock
space, which is systematically built up from the 1-particle Hilbert space. Creationandannihilationoperatorsarethendefinedasverynaturalobjects,namely
as maps from the n-particle subspace into (n±1)-particle ones, and only afterwards
quantum fields are built from these operators in a bit sophisticated way
(keywords being cluster decomposition principle and relativistic invariance).
The field-focused approach (represented by Peskin–Schroeder, and shared
by the majority of textbooks on the subject) starts from the quantization of a
classical field, introducing the creation and annihilation operators in this way
as quantum incarnations of the normal mode expansion coefficients, and finally
providingFockspaceasthe worldforthese operatorstolivein. The logicbehind
this formal development is nothing else but construction of the Poincaré group
generators. So, again, the corner-stone is the relativistic invariance.
42 Linearity of transformations at the quantum level is necessary to preserve the superposition
principle. The symmetry transformations should not change measurable things,
which would not be the case if the superposition |ψ〉 = c 1 |ψ 1 〉 + c 2 |ψ 2 〉 would transform
to T |ψ〉 ≠ c 1 T |ψ 1 〉+c 2 T |ψ 2 〉.