Problems
35
In the following chapters our aim will be to lead the reader through the
mathematical formalism involved in giving precise quantitative form to what
we have so far described only qualitatively and to provide physical interpretation where appropriate. In the remainder of part I of the present volume,
we first show how Schr¨ odinger’s quantum mechanics and Maxwell’s electromagnetic theory may be combined as a gauge theory – in fact the simplest
example of such a theory. We then introduce relativistic quantum mechanics
for spin-0 and spin1 particles, and include electromagnetism via the gauge
2
principle. Lorentz transformations and discrete symmetries are also covered.
In part II, we develop the formalism of quantum field theory, beginning with
scalar fields and moving on to QED; this is then applied to many simple (‘tree
level’) QED processes in part III. In the final part IV, we present an introduction to renormalization at the one-loop level, including renormalization
of QED. The more complicated gauge theories of QCD and the electroweak
theory are reserved for volume 2.
Problems
1.1 Evaluate the integral in (1.26) directly. [Hint : Use spherical polar coordinates with the polar axis along the direction of q, so that d
3
r = r
2 dr sin θ dθ dφ,
and exp(iq · r) = exp(i|q|r cos θ). Make the change of variable x = cos θ, and
do the φ integral (trivial) and the x integral. Finally do the r integral.]
1.2 Using the concept of strangeness conservation in strong interactions, explain why the threshold energy (for π
− incident on stationary protons) for
π
− + p → K
0 + anything
is less than for
π
− + p → K ¯ 0 + anything
assuming both processes proceed through the strong interaction.
2
1.3 Note: the invariant square p of a 4-momentum p = (E, p) is defined as
2
p = E
2
− p
2 . We remind the reader that ħ = c = 1 (see Appendix B).
(a) An electron of 4-momentum k scatters from a stationary proton
of mass M via a one-photon exchange process, producing a final
hadronic state of 4-momentum p
′ , the final electron 4-momentum
being k
′ . Show that
′ 2
p = q
2 + 2M (E − E
′ ) + M
2
2
where q = (k − k
′ )
2 , and E, E
′ are the initial and final electron
energies in this frame (i.e. the one in which the target proton is
35
In the following chapters our aim will be to lead the reader through the
mathematical formalism involved in giving precise quantitative form to what
we have so far described only qualitatively and to provide physical interpretation where appropriate. In the remainder of part I of the present volume,
we first show how Schr¨ odinger’s quantum mechanics and Maxwell’s electromagnetic theory may be combined as a gauge theory – in fact the simplest
example of such a theory. We then introduce relativistic quantum mechanics
for spin-0 and spin1 particles, and include electromagnetism via the gauge
2
principle. Lorentz transformations and discrete symmetries are also covered.
In part II, we develop the formalism of quantum field theory, beginning with
scalar fields and moving on to QED; this is then applied to many simple (‘tree
level’) QED processes in part III. In the final part IV, we present an introduction to renormalization at the one-loop level, including renormalization
of QED. The more complicated gauge theories of QCD and the electroweak
theory are reserved for volume 2.
Problems
1.1 Evaluate the integral in (1.26) directly. [Hint : Use spherical polar coordinates with the polar axis along the direction of q, so that d
3
r = r
2 dr sin θ dθ dφ,
and exp(iq · r) = exp(i|q|r cos θ). Make the change of variable x = cos θ, and
do the φ integral (trivial) and the x integral. Finally do the r integral.]
1.2 Using the concept of strangeness conservation in strong interactions, explain why the threshold energy (for π
− incident on stationary protons) for
π
− + p → K
0 + anything
is less than for
π
− + p → K ¯ 0 + anything
assuming both processes proceed through the strong interaction.
2
1.3 Note: the invariant square p of a 4-momentum p = (E, p) is defined as
2
p = E
2
− p
2 . We remind the reader that ħ = c = 1 (see Appendix B).
(a) An electron of 4-momentum k scatters from a stationary proton
of mass M via a one-photon exchange process, producing a final
hadronic state of 4-momentum p
′ , the final electron 4-momentum
being k
′ . Show that
′ 2
p = q
2 + 2M (E − E
′ ) + M
2
2
where q = (k − k
′ )
2 , and E, E
′ are the initial and final electron
energies in this frame (i.e. the one in which the target proton is