Quantum field theory was founded by Heisenberg and Pauli in 1929.24From the physical point of view the following is crucial:
A quantum field can be treated as a system of an infinite number of quantum particles where creation and annihilation of particles are possible.
In particular, for studying the radiation of atoms and molecules, one has to consider the quantum field of photons. In quantum electrodynamics, one investigates the quantum field of electrons, positrons, and photons.
The second quantization of the Schr¨odinger equation.As a pro- totype, let us consider the quantum field corresponding to the Schr¨odinger equation. We will proceed in several steps.
• Step 1: Classical mechanics. We start with a classical particle on the real line. The principle of critical action reads as
t1 t0
L(q(t),q(t))dt˙ = critical!
along with the boundary condition “q(t) = given” fort=t0, t1. This leads to the Euler–Lagrange equation
d
dtLq˙(q(t),q(t)) =˙ Lq(q(t),q(t))˙
which describes the motion, q=q(t), of the classical particle on the real line.25 Let us consider the special case where
24W. Heisenberg and W. Pauli, On quantum field theory (in German), Zeitschrift f¨ur Physik56(1929), 1–61;59(1930), 108–190.
25The derivation of the Euler–Lagrange equation in classical mechanics along with symplectic and Poissonian geometry will be studied in Chaps. 4 and 5 of Vol. II.
L(q,q) :=˙ mq˙2
2 −κU(q).
We define the momentump:=Lq˙and the HamiltonianH :=pq˙−L.Hence p=mq, and˙
H = p2
2m+κU(q).
SetH(t) := p(t)2m2 +κU(q(t)).By energy conservation, we have H(t) =H(0) for all t∈R,
for each smooth solutionq=q(t) of the Euler–Lagrange equation.
• Step 2: First quantization by using Heisenberg’s particle picture. We want to describe a quantum particle on the real line.
To this end, we replace the classical trajectoryq=q(t)by an operator- valued function.
This implies the operatorsp(t) := mq(t) and˙ H(t) as given above. More precisely, for each timet, we have the commutation relation
[q(t), p(t)]−= iI and the following equations of motion26
iq(t) = [q(t), H(t)]˙ −, ip(t) = [p(t), H˙ (t)]−.
We will show in Volume II that this implies the Newtonian equation of motionm¨q(t) =−κU(q(t)). Furthermore, the energy operatorH(t) does not depend on timet. To simplify notation, this operator is denoted by the symbolH.
• Step 3: First quantization by using Schr¨odinger’s wave picture. Here, the quantum particle on the real line is described by the complex-valued wave functionψ=ψ(x, t) which satisfies the Schr¨odinger equation
iψ(x, t) =˙ −2
2m ψxx(x, t) +κU(x)ψ(x, t). (1.41) First of all we want to derive the Schr¨odinger equation by the principle of critical action of the form
t1 t0
x1
x0
Ldx
dt= critical! (1.42)
along with the boundary condition “ψ(x, t) = given” on the boundary∂Ω of the rectangleΩ:={(x, t)∈R2:x0≤x≤x1, t0≤t≤t1}.Explicitly, for the Lagrangian density,
26Recall that [A, B]−:=AB−BA.
L(ψ,ψ, ψ˙ x;ψ†,ψ˙†, ψ†x) := iψ†ψ˙− 2
2mψx†ψx−κU ψ†ψ
with the real potentialU =U(x).Recall that ˙ψdenotes the partial deriva- tive with respect to time t. Each classical solution ψ =ψ(x, t) of (1.42) satisfies the two Euler–Lagrange equations
∂
∂xLψx+ ∂
∂tLψ˙ =Lψ (1.43)
and
∂
∂xLψ†x+ ∂
∂tLψ˙† =Lψ†. (1.44) Equation (1.43) is precisely the Schr¨odinger equation (1.41), whereas equa- tion (1.44) is obtained from the Schr¨odinger equation by applying the op- eration of complex conjugation, that is,
−iψ˙†=−2
2mψxx† +κU ψ†.
Thus, equation (1.44) does not provide us any new information. Introduce the momentum
π:=Lψ˙.
Explicitly, π(x, t) = iψ†(x, t). Moreover, we introduce the Hamiltonian density
H:=πψ˙− L and the HamiltonianH:=∞
−∞Hdx.Explicitly, H= 2
2m ψ†xψx+U ψ†ψ.
Here,H represents the energy of the classical field ψ.
• Step 4: Second quantization of the Schr¨odinger equation and the quantum field. We now want to describe an infinite number of quantum particles on the real line including the creation and annihilation of particles.
To this end, we replace the classical wave functionψ=ψ(x, t) by an operator-valued function.
More precisely,ψ(x, t) is an operator which, for all positions x, y∈Rand all timest∈R, satisfies the so-called canonical commutation relations
[ψ(x, t), π(y, t)]− = iδ(x−y),
[ψ(x, t), ψ(y, t)]− = [π(x, t), π(y, t)]−= 0 along with the equations of motion
iψ˙ = [ψ, H]−, iπ˙ = [π, H]−.
It turns out that this implies the Schr¨odinger equation for the quantum fieldψ=ψ(x, t).27
The prototype of a quantum field and the method of Fourier quantization.Suppose that we know a system ϕ0, ϕ1, . . .of eigensolutions of the stationary Schr¨odinger equation,
Eϕn=−2
2m(ϕn)xx+κU ϕn, n= 0,1,2, . . .
where ϕ0, ϕ1, . . . represents a complete orthonormal system in the Hilbert spaceL2(R).The Fourier series
ψ(x, t) = ∞ n=0
ϕn(x)e−iEnt/an (1.45) with complex numbersa0, a1, . . .is then a solution of the Schr¨odinger equa- tion. Replace now the classical Fourier coefficients by operators a0, a1, . . . which, for alln, m= 0,1, . . .satisfy the commutation relations
an, a†m
−=δnmI, [an, am]−= a†n, a†m
−= 0.
The classical fieldψ from (1.45) passes then over to a quantum field which consists of an infinite number of particles having the energiesE0, E1, . . . We assume that there exists a state |0 which is free of particles. This state of lowest energyE0 is called ground state (or vacuum).The symbol
a†i
1a†i
2ã ã ãa†i
N|0
represents then a state of the quantum field which consists of precisely N free particles possessing the energiesEi1, . . . , EiN.Moreover, the symbol
ψ†free(x1, t)ã ã ãψfree† (xN, t)|0
represents a state at time t which is related to N free particles. Here, it is important to distinguish between
• the ground state|0 of the free quantum fieldψfreewithout any interactions,
• and the ground state|0int of the interacting quantum fieldψ.
The main trouble of quantum field theories concerns the investigation of interacting quantum fields in rigorous mathematical terms.
27The Dirac delta functionδ represents a generalization of the Kronecker symbol δij to infinite degrees of freedom. In particular, δ(x−y) = 0 if x=y. For the heuristic and rigorous definition ofδ, see pages 590 and 609, respectively.
Commutation relations for creation and annihilation operators.
In elementary particle physics, we have to distinguish between bosons (parti- cles of integer spin, e.g., photons) and fermions (particles of half-integer spin, e.g., electrons). The prototype of commutation relations for annihilation op- eratorsa(p) and creation operatorsa†(p) of bosonic particles of a momentum vectorpreads as28
a(p), a†(q)
−=δpqI, [a(p), a(q)]−=
a†(p), a†(q)
−= 0 for all momentum vectors p,q which lie on a fixed lattice of width ∆p in 3-dimensional momentum space. Here, we use the 3-dimensional Kronecker symbol defined by δpp := 1 and δpq = 0 if p = q. Physicists pass to the formal continuum limit. To consider this, let us rescale the annihilation and creation operators by setting
a(p) := a(p)
(∆p)3, a†(p) := a†(p) (∆p)3. Hence
a(p),a†(q)
−= δpq
(∆p)3 I, [a(p),a(q)]− =
a†(p),a†(q)
− = 0.
The formal continuum limit∆p→0 yields then a(p),a†(q)
− =δ3(p−q)I, [a(p),a(q)]− =
a†(p),a†(q)
−= 0 for all 3-dimensional momentum vectorspand q.The relation between the discrete Dirac delta function and its continuum limit is studied on page 673.
The rigorous mathematical approach to creation and annihilation operators for free quantum particles in terms of the so-called Fock space can be found in Volume II.
The fundamental role of correlations of a quantum field.The ex- perience of physicists in quantum physics shows that one should prefer the study of quantities which are related to measurements in physical experi- ments. From the physical point of view, we can measure
• cross sections of scattering processes for elementary particles, and
• masses of bound particles (like the proton as a bound state of three quarks).
It turns out that these quantities are related to correlations between different space-time points of the quantum field. According to Feynman, the basic quantity is the correlation function
G2(x1, t1;x2, t2) :=0int|Tψ(x1, t1)ψ†(x2, t2)|0int
28Recall that [A, B]−:=AB−BAand [A, B]+ :=AB+BA. For fermions, one has to replace the Lie bracket [., .]−by the Jordan–Wigner bracket [., .]+.
which is also called the 2-point Green’s function of the interacting quantum fieldψ.This function describes the correlation between the quantum field at positionx1at timet1and the quantum field at positionx2and timet2. Here, the symbolT denotes the chronological operator. Explicitly,
T(ψ(x1, t1)ψ†(x2, t2)) :=
ψ(x1, t1)ψ†(x2, t2) if t1≥t2, ψ†(x2, t2)ψ(x1, t1) if t2> t1. It turns out that
The 2-point Green’s function G2 of a quantum field is a highly sin- gular mathematical object.
This fact causes serious mathematical difficulties. Similarly, the 2n-point Green’s function is obtained by replacing the productψ(x1, t1)ψ†(x2, t2) by a product of 2n field operators. For example, the 4-point Green’s function G4 is given by
0int|Tψ(x1, t1)ψ(x2, t2)ψ†(x3, t3)ψ†(x4, t4)|0int .
The Green’s functionsG2, G4, G6, . . . of a quantum field are closely related to the moments of the quantum field which contain the information on the probability structure of the quantum field.