7.1 Quantization of the Free Scalar Field

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Let _(x) be a real scalar _eld satisfying the Klein-Gordon equation.

􀀀

D_@_ 􀀀m2_

_(x) = 0 (7.1)

Let f__g span the space S of solutions. We shall assume that the spacetime

is globally hyperbolic, i.e. that 9 a Cauchy surface _. A point in the space

S then corresponds to a choice of initial data on _. The space S has a

natural symplectic structure.

__ ^ __ =

Z

_

dS___

$@

_

__; (= 􀀀__ ^ __) (7.2)

where $@

is de_ned by

f $@

g = f@g 􀀀 g@f (7.3)

`Natural' means that ^ does not depend on the choice of _.

(__ ^ __)_ 􀀀 (__ ^ __)_0 =

Z

S

d4xp􀀀gD_

_

__

$@

_

__

_

(7.4)

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.......................................................................................................................................................................................................................................................................................................................................................

....

......

........

........

..........

..........

...............

............

..........

............

............

..........

. S ...

_0

_

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But

D_

_

__

$@

_

__

_

= __ (D_@___) 􀀀 (D_@___) __ (7.5)

= __

􀀀

m2__

_

􀀀

􀀀

m2__

_

__ = 0 ; (7.6)

using the Klein-Gordon equation in the last step.

The antisymmetric form __ ^ __ can be brought to a canonical block

diagonal form, with 2_2 blocks

_

0 1

􀀀1 0

_

, by a change of basis (Darboux's

theorem). Thus, real solutions of the Klein-Gordon equation can be grouped

in pairs (_; _0) with _ ^ _0 = 1. It then follows that the complex solution

= (_ 􀀀 i_0) =p2 has unit norm if we de_ne its norm jj jj by jj jj2 = _^_0

or, equivalently,

jj jj2 = i

Z

_

dS_ _ $@

_

: (7.7)

More generally, we can introduce a complex basis f ig of solutions of the

Klein-Gordon equation with hermitian inner product de_ned by

( i; j) = i

Z

dS_ _i

$@

_

j ; (7.8)

and we can choose this basis such that ( i; j) = _ij . This inner product

is not positive de_nite, however, because jj _jj2 = 􀀀jj jj2. In fact, we can

choose the basis f ig such that

0

BB@

( i; j) = _ij

_

i; _j

_

= 0

( _i ; j) = 0

_

_i ; _j

_

= 􀀀_ij

1

CCA

(7.9)

We could interpret the complex solution            =

P

i ai i as the wavefunction

of a free particle since ( ; ) is positive-de_nite when restricted to such

solutions, but this cannot work when interactions are present. It is also

inapplicable for real scalar _elds. A real solution _ of the K-G equation can

be written as

_(x) =

X

[ai i(x) + a_i _i (x)] (7.10)

To quantize we pass to the quantum _eld

_(x) =

Xh

ai i(x) + ayi

_i (x)

i

(7.11)

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where faig are now operators in a Hilbert space H with Hermitian conjugates

ayi

satisfying the commutation relations

[ai; aj ] = 0;

h

ai; ayj

i

= _ij (~ = 1) (7.12)

We choose the Hilbert space to be the Fock space built from a `vacuum'

state jvaci satisfying

ai jvaci = 0 8i (7.13)

hvacjvaci = 1 (7.14)

i.e. H has the basis n

jvaci ; ayi

jvaci ; ayi

ayj

jvaci ; : : :

o

h j i is a positive-de_nite inner product on this space.

This basis for H is determined by the choice of jvaci, but this depends on

the choice of complex basis f ig of solutions of the K-G equation satisfying

(7.9). There are many such bases.

Consider f 0ig where

0i =

X

j

􀀀

Aij j + Bij _j

_

(7.15)

This has the same inner product matrix (7.9) provided that

AAy 􀀀 BBy = 1

ABT 􀀀 BAT = 0

(7.16)

Inversion of (7.15) leads to

j =

X

k

A0jk 0k

+ B0j

k 0k_ (7.17)

where

A0 = Ay; B0 = 􀀀BT (7.18)

Check

0 = A

􀀀

A0 0 + B0 0_

_

+ B

􀀀

A0_ 0_ + B0_ 0

_

(7.19)

=

􀀀

AA0 + BB0_

_

0 +

􀀀

AB0 + BA0_

_

0_ (7.20)

=

_

AAy 􀀀 BBy

_

0 􀀀

_

ABT 􀀀 BAT

_

0 (7.21)

= 0 (7.22)

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But A0 and B0 must satisfy the same conditions as A and B, i.e.

A0A0y 􀀀 B0B0y = 1 (7.23)

A0B0T 􀀀 B0A0T = 0 (7.24)

Equivalently,

AyA 􀀀 BTB_ = 1

AyB 􀀀 BTA_ = 0

(7.25)

These conditions are not implied by (7.16); the additional information contained

in them is the invertibility of the change of basis.

In a general spacetime there is no `preferred' choice of basis satisfying

(7.9) and so no preferred choice of vacuum. In a stationary spacetime,

however, we can choose the basis fuig of positive frequency eigenfunctions

of k, i.e.

k_@_ui = 􀀀i!iui; !i _ 0 (7.26)

Notes

(1) Since k is Killing it maps solutions of the Klein-Gordon equation to

solutions (Proof: Exercise).

(2) k is anti-hermitian, so it can be diagonalized with pure-imaginary

eigenvalues.

(3) Eigenfunctions with distinct eigenvalues are orthogonal so

􀀀

ui; u_j

_

= 0 (7.27)

We can normalize fuig s.t. (ui; uj) = _ij , so the basis fuig can be

chosen s.t. (7.9) is satis_ed.

(4) We exclude functions with ! = 0.

For this choice of basis the vacuum state jvaci is actually the state of

lowest energy. The states ayi

jvaci are one-particle states, ayi

ayj

jvaci twoparticle

states, etc., and

N =

X

i

ayi

ai (7.28)

is the particle number operator.

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