200 From Classical Mechanics to Quantum Field Theory. A Tutorial
helicity 1 to electromagnetic fields satisfying Maxwell equations. To be concise
we shall not analyze these higher spin field theories in these lectures and we shall
concentrate on scalar fields.
3.3 QuantumFieldTheory........................
The most interesting case of field theory is that which concerns relativistic fields.
The compatibility of quantum theories with the theory of relativity is not im-
mediate. The first attempts to formulate a quantum dynamics compatible with
the theory of relativity lead to puzzling theories full of paradoxes, like the Klein
paradox, which arises in the dynamics defined by Klein-Gordon or Dirac equa-
tions. The solution to those puzzles comes from the quantization of classical field
theories.
A quantum field theory is a quantum theory which is relativistic invariant
and where there is a special type of quantum operators which are associated with
classical fields.
In the case of a real scalar fieldφa consistent theory should satisfy the following
principles.
- P1Quantum principle:The space of quantum states is the space of rays
in a separable Hilbert spaceH. - P2Unitarity:There is a antiunitary representationU(Λ,a)ofthePoincar ́e
group inH, where time reversalTis represented as an antiunitary operator
U(T). - P3Spectral condition:The spectrum of generators of space-time transla-
tionsPμis contained in the forward like cone
V ̄+={pμ;p^2 ≥ 0 ,p 0 ≥ 0 }. - P4 Vacuum state: Thereisauniquestate| 0 〉∈H, satisfying that
Pμ| 0 〉 =0. - P5Field theory (real boson):For any classical fieldfin the spaceS(R^3 )
offastdecreasing smoothC∞(R^3 ) functions^3 there is a field operatorφ(f)
inHwhich satisfiesφ(f)=φ(f)†. The field operator can be considered
as the smearing byfof a fundamental field operatorφ(x)
φ(f)=∫
d^3 xf(x)φ(x).(^3) In the case of massless fields the classical field test functionsfmust have compact support,
i.e.f∈D(R^3 )=C 0 ∞(R^3 ).