Pictures Paths Particles Processes

98 November 7, 2013
we find that the only particle modes emitted by the source that have a chance
of propagating over distances much further than σ must satisfy
E ≈ √ |⃗p| 2 c 2 + M 2 c 4 , m = Mc
¯h . (3.36)
This is the mass shell condition, which prescribes the relation between the energy
E (in Joule), momentum ⃗p (in kg m/s), and mechanical mass M (in kg) of
a particle moving freely through spacetime. We recognize the quantity m that
we have been using so far as the inverse Compton wavelength of the particle 17 .
Given that the particle is emitted on its mass shell, the integral φ(x) is not
yet automatically large. The complex phase in Eq.(3.32) will lead to extremely
rapid oscillatory behaviour of the integrand, and an essentially vanishing result,
except for those regions where the phase of the integrand is stationary. This
happens if
∂
∂ ⃗ k
(
)
x 0 k 0 − ⃗x · ⃗k
= ∂ (
)
∂ ⃗ x 0 ω( ⃗ k) − ⃗x · ⃗k = ⃗ k
k
ω( ⃗ k) x0 − ⃗x = 0 . (3.37)
That is, φ(x) is appreciable on a line in spacetime given by
⃗x = t c ⃗p
p 0 : (3.38)
the particle moves along a straight line, with constant velocity c⃗p/p 0 . This is
Newton’s First Law.
A further remark is in order. It might be proposed that the source we have
used would become more æsthetically pleasing if also the time dependence were
Gaussian. However, in that case the k 0 contour cannot be simply closed since
exp(−k 02 ) diverges badly for arg(k 0 ) between −π/4 and −3π/4. Hence the
dispersion relation k 0 = ω( ⃗ k) does not hold, and the mass-shell condition on p µ
does not apply. In a sense, a Gaussian time dependence implies that the source is
‘switched on’ and ‘switched off’ too rapidly, so that energies are not well-defined.
In a similar spirit, one might feel uncomfortable with the pole at p 0 /¯h − i/σ 0 in
the complex-k 0 plane. Indeed, one can get rid of it by multiplying the source
of Eq.(3.30) by θ(x 0 < 0) so that the source is only active up to x 0 = 0 and
then stops. However, the absence of this pole means that after the k 0 integral
we have
not
1
(
p 0 /¯h − ω( ⃗ 2
k))
+ 1/σ0
2
but
1
p 0 /¯h − ω( ⃗ k) + i/σ 0
17 A particle is called on-shell if its momentum p µ satisfies Eq.(3.36) ; if not, it is called
off-shell. Off-shell particles are not exotic or improbable ; they are just not visible as the
result of any experiment since they cannot propagate well. In popular literature, off-shell
particles are often dicussed with a lot of mumbling about ‘uncertainty relations’, ‘borrowing
energy from the vacuum’, and so on. Do not be misguided ! When a theorist starts invoking
the uncertainty principle as a reason for something, keep your hand on your wallet. The
‘uncertainty principle’ is not a reason but a result.