Quantum Mechanics for Mathematicians

A(α(p))∈M−Jr=H 1 →a(α(p))

HereA(α(p)) is the positive energy solution of the (complexified) Klein-Gordon
equation with initial data given byα(p) anda†(α(p)) is the operator on the
Fock space of symmetric tensor products ofH 1 given by equation 36.12.A(α(p))
is the conjugate negative energy solution anda(α(p)) is the operator given by
equation 36.13. All these objects are often written in distributional form, where
one has
A(p)→a†(p), A(p)→a(p)

which one can interpret as corresponding to taking the limit ofα(p′)→δ(p−p′).
The operatorsa(α(p)) anda†(α(p)) satisfy the commutation relations

[a(α 1 (p)),a†(α 2 (p))] =〈α 1 (p),α 2 (p)〉

or in distributional form

[a(p),a†(p′)] =δ^3 (p−p′)

For the Hamiltonian we take the normal ordered form

Ĥ=

∫

R^3

ωpa†(p)a(p)d^3 p

Starting with a vacuum state| 0 〉, by applying creation operators one can create
arbitrary positive energy multi-particle states of free relativistic particles, with
single-particle states having the energy momentum relation

E(p) =ωp=

√

|p|^2 +m^2

This description of the quantum system is essentially the same as that of the
non-relativistic theory, which seems to differ only in the energy-momentum re-

lation, which in that case wasE(p) = |p|

2
2 m. The different complex structure
used for quantization in the relativistic theory changes the physical meaning of
annihilation and creation operators:

• Non-relativistic theory.M=H 1 is the space of complex solutions of the
  free particle Schr ̈odinger equation, which all have positive energy.H 1 is
  the conjugate space. For each continuous basis elementA(p) ofH 1 (these
  are initial data for a positive energy solution with momentump), quanti-
  zation takes this to a creation operatora†(p), which acts with the physical
  interpretation of addition of a particle with momentump. Quantization
  of the complex conjugateA(p) inH 1 gives an annihilation operatora(p),
  which removes a particle with momentump.


• Relativistic theory. M+Jr =H 1 is the space of positive energy solutions
  of the Klein-Gordon equation. It has continuous basis elementsA(p)
  which after quantization become creation operators adding a particle of
  momentumpand energyωp.M−Jris the space of negative energy solutions
  of the Klein-Gordon equation. Its continuous basis elementsA(p) after
  quantization become annihilation operators for antiparticles of momentum
  −pand positive energyωp.