If e0 ∈ R1+1 is a future-directed timelike unit vector, and if e1 is the unique spacelike unit vector with e0e1 = 0 that “points to the right,” then coordinates x0 and x1 on R1+1 are defined by x0(q) := qe0 and x1(q) := qe1. The partial differential operator
□x : = ∂2x0 − ∂2x1
does not depend on the choice of e0.
The Fourier transform of the Klein-Gordon equation
(□ + m2)u = 0 —– (1)
where m > 0 is a given mass, is
(−p2 + m2)û(p) = 0 —– (2)
As a consequence, the support of û has to be a subset of the hyperbola Hm ⊂ R1+1 specified by the condition p2 = m2. One connected component of Hm consists of positive-energy vectors only; it is called the upper mass shell Hm+. The elements of Hm+ are the 4-momenta of classical relativistic point particles.
Denote by L1 the restricted Lorentz group, i.e., the connected component of the Lorentz group containing its unit element. In 1 + 1 dimensions, L1 coincides with the one-parameter Abelian group B(χ), χ ∈ R, of boosts. Hm+ is an orbit of L1 without fixed points. So if one chooses any point p′ ∈ Hm+, then there is, for each p ∈ Hm+, a unique χ(p) ∈ R with p = B(χ(p))p′. By construction, χ(B(ξ)p) = χ(p) + ξ, so the measure dχ on Hm+ is invariant under boosts and does note depend on the choice of p′.
For each p ∈ Hm+, the plane wave q ↦ e±ipq on R1+1 is a classical solution of the Klein-Gordon equation. The Klein-Gordon equation is linear, so if a+ and a− are, say, integrable functions on Hm+, then
F(q) := ∫Hm+ (a+(p)e-ipq + a–(p)eipq dχ(p) —– (3)
is a solution of the Klein-Gordon equation as well. If the functions a± are not integrable, the field F may still be well defined as a distribution. As an example, put a± ≡ (2π)−1, then
F(q) = (2π)−1 ∫Hm+ (e-ipq + eipq) dχ(p) = π−1 ∫Hm+ cos(pq) dχ(p) =: Φ(q) —– (4)
and for a± ≡ ±(2πi)−1, F equals
F(q) = (2πi)−1 ∫Hm+ (e-ipq – eipq) dχ(p) = π−1 ∫Hm+ sin(pq) dχ(p) =: ∆(q) —– (5)
Quantum fields are obtained by “plugging” classical field equations and their solutions into the well-known second quantization procedure. This procedure replaces the complex (or, more generally speaking, finite-dimensional vector) field values by linear operators in an infinite-dimensional Hilbert space, namely, a Fock space. The Hilbert space of the hermitian scalar field is constructed from wave functions that are considered as the wave functions of one or several particles of mass m. The single-particle wave functions are the elements of the Hilbert space H1 := L2(Hm+, dχ). Put the vacuum (zero-particle) space H0 equal to C, define the vacuum vector Ω := 1 ∈ H0, and define the N-particle space HN as the Hilbert space of symmetric wave functions in L2((Hm+)N, dNχ), i.e., all wave functions ψ with
ψ(pπ(1) ···pπ(N)) = ψ(p1 ···pN)
∀ permutations π ∈ SN. The bosonic Fock space H is defined by
H := ⊕N∈N HN.
D := ∪M∈N ⊕0≤M≤N HN is called a finite particle space.
The definition of the N-particle wave functions as symmetric functions endows the field with a Bose–Einstein statistics. To each wave function φ ∈ H1, assign a creation operator a+(φ) by
a+(φ)ψ := CNφ ⊗s ψ, ψ ∈ D,
where ⊗s denotes the symmetrized tensor product and where CN is a constant.
(a+(φ)ψ)(p1 ···pN) = CN/N ∑v φ(pν)ψ(pπ(1) ···p̂ν ···pπ(N)) —– (6)
where the hat symbol indicates omission of the argument. This defines a+(φ) as a linear operator on the finite-particle space D.
The adjoint operator a(φ) := a+(φ)∗ is called an annihilation operator; it assigns to each ψ ∈ HN, N ≥ 1, the wave function a(φ)ψ ∈ HN−1 defined by
(a(φ)ψ)(p1 ···pN) := CN ∫Hm+ φ(p)ψ(p1 ···pN−1, p) dχ(p)
together with a(φ)Ω := 0, this suffices to specify a(φ) on D. Annihilation operators can also be defined for sharp momenta. Namely, one can define to each p ∈ Hm+ the annihilation operator a(p) assigning to
each ψ ∈ HN, N ≥ 1, the wave function a(p)ψ ∈ HN−1 given by
(a(p)ψ)(p1 ···pN−1) := CN ψ(p, p1 ···pN−1), ψ ∈ HN,
and assigning 0 ∈ H to Ω. a(p) is, like a(φ), well defined on the finite-particle space D as an operator, but its hermitian adjoint is ill-defined as an operator, since the symmetric tensor product of a wave function by a delta function is no wave function.
Given any single-particle wave functions ψ, φ ∈ H1, the commutators [a(ψ), a(φ)] and [a+(ψ), a+(φ)] vanish by construction. It is customary to choose the constants CN in such a fashion that creation and annihilation operators exhibit the commutation relation
[a(φ), a+(ψ)] = ⟨φ, ψ⟩ —– (7)
which requires CN = N. With this choice, all creation and annihilation operators are unbounded, i.e., they are not continuous.
When defining the hermitian scalar field as an operator valued distribution, it must be taken into account that an annihilation operator a(φ) depends on its argument φ in an antilinear fashion. The dependence is, however, R-linear, and one can define the scalar field as a C-linear distribution in two steps.
For each real-valued test function φ on R1+1, define
Φ(φ) := a(φˆ|Hm+) + a+(φˆ|Hm+)
then one can define for an arbitrary complex-valued φ
Φ(φ) := Φ(Re(φ)) + iΦ(Im(φ))
Referring to (4), Φ is called the hermitian scalar field of mass m.
Thereafter, one could see
[Φ(q), Φ(q′)] = i∆(q − q′) —– (8)
Referring to (5), which is to be read as an equation of distributions. The distribution ∆ vanishes outside the light cone, i.e., ∆(q) = 0 if q2 < 0. Namely, the integrand in (5) is odd with respect to some p′ ∈ Hm+ if q is spacelike. Note that pq > 0 for all p ∈ Hm+ if q ∈ V+. The consequence of this is called microcausality: field operators located in spacelike separated regions commute (for the hermitian scalar field).