130_notes.dvi

Of course the average value of the Electric or Magnetic field vectoris zero by symmetry. To get an
idea about the size of field fluctuations, we should look at themean square value of the field,
for examplein the vacuum state. We compute〈 0 |E~·E~| 0 〉.

E~ = −^1

c

∂A~

∂t

Aμ =

1

√

V

∑

kα

√

̄hc^2

2 ω

ǫ(μα)

(

ak,α(0)eikρxρ+a†k,α(0)e−ikρxρ

)

A~ = √^1

V

∑

kα

√

̄hc^2

2 ω

ˆǫ(α)

(

ak,α(0)eikρxρ+a†k,α(0)e−ikρxρ

)

E~ = −i^1

c

1

√

V

∑

kα

√

̄hc^2

2 ω

ˆǫ(α)

(

−ωak,α(0)eikρxρ+ωa†k,α(0)e−ikρxρ

)

E~ = √i

V

∑

kα

√

̄hω

2

ˆǫ(α)

(

ak,α(0)eikρxρ−a†k,α(0)e−ikρxρ

)

E~| 0 〉 = √i

V

∑

kα

√

̄hω

2

ˆǫ(α)

(

−a†k,αe−ikρxρ

)

| 0 〉

〈 0 |E~·E~| 0 〉 =

1

V

∑

kα

̄hω

2

1

〈 0 |E~·E~| 0 〉 =

1

V

∑

k

̄hω→∞

(Notice that we arebasically taking the absolute squareofE~| 0 〉and that the orthogonality of
the states collapses the result down to a single sum.)

The calculation is illustrative even though the answer is infinite. Basically, a term proportional to
aa†first creates one photon then absorbs itgiving a nonzero contribution for every oscillator
mode. The terms sum to infinity but really its the infinitesimally short wavelengths that cause this.
Again, some cut off in the maximum energy would make sense.

Theeffect of these field fluctuations on particlesis mitigated by quantum mechanics. In
reality, any quantum particle will be spread out over a finite volume and its the average field over
the volume that might cause the particle to experience a force. So we could average the Electric
field over a volume, then take the mean square of the average. If we average over a cubic volume
∆V= ∆l^3 , then we find that

〈 0 |E~·E~| 0 〉≈

̄hc

∆l^4

.

Thus if we can probe short distances, the effective size of the fluctuations increases.

Even theE and B fields do not commute. It can be shown that

[Ex(x),By(x′)] =ic ̄hδ(ds=

√

(x−x′)ρ(x−x′)ρ)

There is a nonzero commutator of the two spacetime points are connected by a light-like vector. An-
other way to say this is that thecommutator is non-zero if the coordinates are simultaneous.
This is a reasonable result considering causality.