148
5.8
5. Quantum Field Theory I: The Free Scalar Field
on the difference of coordinates x 1 − x 2 , consistent with translation
invariance. Show that D(x 1 , x 2 ) vanishes for t 1 = t 2 . Explain why
the right-hand side of (5.161) is Lorentz invariant (see the exercise
in appendix E), and use this fact to show that D(x 1 , x 2 ) vanishes
for all space-like separations (x 1 −x 2 )
2 < 0. Discuss the significance
of this result – or see the discussion in section 6.3.2!
5.7 Insert the plane-wave expansions for the operators φ ˆ and ˆ
π into the equation for H ˆ , (5.124), and verify equation (5.132). [Hint : note that ω is defined
to be always positive, so that (5.126) should strictly be written ω = |k|.]
˙
ˆ
(a) Use (5.117) and (5.124) to verify that ˆ
π(x, t) = φ(x, t) is consistent
with the Heisenberg equation of motion for φ ˆ (x, t). [Hint : write the
integral in (5.124) as over y, not x!]
(b) Similarly, verify the consistency of (5.141) and (5.121).
5.8
5. Quantum Field Theory I: The Free Scalar Field
on the difference of coordinates x 1 − x 2 , consistent with translation
invariance. Show that D(x 1 , x 2 ) vanishes for t 1 = t 2 . Explain why
the right-hand side of (5.161) is Lorentz invariant (see the exercise
in appendix E), and use this fact to show that D(x 1 , x 2 ) vanishes
for all space-like separations (x 1 −x 2 )
2 < 0. Discuss the significance
of this result – or see the discussion in section 6.3.2!
5.7 Insert the plane-wave expansions for the operators φ ˆ and ˆ
π into the equation for H ˆ , (5.124), and verify equation (5.132). [Hint : note that ω is defined
to be always positive, so that (5.126) should strictly be written ω = |k|.]
˙
ˆ
(a) Use (5.117) and (5.124) to verify that ˆ
π(x, t) = φ(x, t) is consistent
with the Heisenberg equation of motion for φ ˆ (x, t). [Hint : write the
integral in (5.124) as over y, not x!]
(b) Similarly, verify the consistency of (5.141) and (5.121).