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Fock space can be constructed from the normalized r-particle
dkl ... dkrIF(kb ... ,krWrl/2
dkl dkrF(kl, ... ,kr)at(kl) at(kr)IO)
(3-45)
The wave function F(k 1 , .. , kr) is symmetric in the interchange of the fourmomenta k" all belonging to the upper sheet of the hyperboloid k 2 = m 2 , the so-called mass shell. A number operator N commuting with PI' is
dk at(k)a(k)
(3-46)
and, of course, acting on the states (3-45) :
Nlr)=rlr)
(3-47)
It is crucial to verify that relativistic invariance has been maintained. The above formulation uses a
particular frame, and we may wonder whether quantization in any other one obtained by a Poincare transformation would lead to an equivalent theory. If primes refer to the coordinates in this new frame, we want to know whether there exists a quantum canonical transformation relating the fundamental creation and annihilation operators a and at to the corresponding d and d t obtained when quantizing on the hyperplane t' = constant. This will be the case if to each transformation x -> x' = Ax + a we may associate a unitary operator U(a, A) in Fock space, such that
U(a, A)<p(x)Ut(a, A)
<p(Ax
+ a)
(3-48)
Assuming differentiability in the parameters a and A it is sufficient to study infinitesimal transformations of the type
with bwpv
+ bw v" =
O. Hence we want to find 10 hermitian operators MPv, (MPV
+ MVP =
0), and
QUANTIZATION-FREE FIELDS
P" such that U
I - - bw M"' 2"'
+ iba
(3-49)
(3-50)
The 10 generators P and M will have to fulfill the geometrical commutation rules [P", P'] = 0 [M"', PA]
i(g").P' - g').P")
(3-51)
where the commutator replaces (up to i) the classical Poisson bracket. As was already anticipated when deriving P", we expect that these generators are given by Noether's theorem (3-52) with 0"' given by Eq. (3-41), or more explicitly
(x) = ~ : (n 2 + (Vcp)2
+ m2cp2) :
(3-53)
0 J(x) = : nfYcp:
The expressions for P" coincide with (3-38) and (3-39) and are indeed such that i[P", cp(x)] As for M"' we find
8"cp(x).
'f-t (8
dk a (k) k j 8k l - kl 8k J a(k)
(3-54)
We verify that Eqs. (3-50) and (3-51) are satisfied. Since M"' contains only an angular contribution the particles created and destroyed by the field cp have no intrinsic spin.
Having satisfied ourselves with the covariance of the quantized theory we now want to explore the relationship between the description in terms of particles of mass m, spin zero, and the hermitian field <p which could belong to the set of measurable quantities. Even though at fixed time fields referring to various positions do commute, this is no longer the case when we compare them at different times. From canonical quantization, the commutator oftwo free fields <p(x) and <p(y) has, however, a simple structure as a c-number distribution which assumes the form
[<p(x), <p(y)] =
dk [e-ik'(x-y)
eik'(x-y)]
iil(x - y)
(3-55)
The real distribution il(x) can also be written with the help of the function
e(u) = u/I u I:
(3-56)
118 QUANTIJM FIELD THEORY
showing that it is an odd, Lorentz invariant, solution of the Klein-Gordon equation. From Ll(O, x) = and Lorentz invariance we find, in fact, that Ll(x) vanishes outside the light cone, i.e., in the region x 2 < 0. Measurements at spacelike separated points do not interfere, a consequence of locality and causality. Note also that canonical quantization entails
(3-57)
To construct coherent states which diagonalize the positive frequency part of the quantum fieldthe analogs of the minimal wave packets of the harmonic oscillator-we assume a normalizable function lI(k) to be given on the mass shell k Z = mZ, kO > 0:
Consider the state 111 > given by
(3-58)
which appears in Fock space as a coherent superposition of states with 0, 1, 2, ... particles. It can be used as a generating function to obtain the states with a fixed number of particles if we allow 11 to remain arbitrary. Let us compute the norm of 111> using the identity eA eB = eA+B+[A,BJ/Z valid for two operators A and B commuting with [A, BJ. We find
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