causality once interactions are allowed, with influence from what happens
at one point in space-time traveling to another at faster than the speed of
light.
To construct a multi-particle theory with thisH 1 along the same lines as the
non-relativistic case, we would need to introduce a complex conjugate spaceH 1
and then apply the Bargmann-Fock method. We would then be quantizing a
theory with dual phase space a subspace of the solutions of the complex Klein-
Gordon equation, those satisfying a condition (positive energy) that is non-local
in space-time.
A more straightforward way to construct this theory is to start with dual
phase spaceMthe real-valued solutions of the Klein-Gordon equation. This is
a real vector space with no given complex structure, but recall equation 43.11,
which describes points inMby a complex functionα(p) rather than a pair of
real functionsφ(x) andπ(x). We can take this as a choice of complex structure,
defining:
Definition(Relativistic complex structure).The relativistic complex structure
on the spaceMis given by the operatorJrthat, extended toM⊗C, is+ion
theα(p),−ion theα(p).
TheA(p) are then a continuous basis ofM+Jr, theA(p) a continuous basis of
M−Jrand
M⊗C=M+Jr⊕M−Jr
A confusing aspect of this setup is that after complexification ofMto get
M⊗C,α(p) andα(p) are not complex conjugates in general, only on the real
subspaceM. We are starting with a real dual phase spaceM, which can be
parametrized by complex functions
f+(p) =f(ωp,p), f−(p) =f(−ωp,p)that satisfy the reality condition (see equation 43.6)
f−(p) =f+(−p)Complexifying,M⊗Cwill be given by pairsf+,f−of complex functions, with
no reality condition relating them. The choice of relativistic complex structure
Jris such thatM+Jris the space of pairs withf−= 0 (the complex functions
on the positive energy hyperboloid),M−Jr is the space of pairs withf+= 0
(complex functions on the negative energy hyperboloid). The conjugation map
onM⊗Cis NOT the map conjugating the values off−orf+. It is the map
that interchanges
(f+(p),f−(p))←→(f−(−p),f+(−p))Theα(p) andα(p) given byα(p) =f+(p)
√
2 ωp, α(p) =f−(−p)
√
2 ωp