Exercises
63
development in time of the bubble can be obtained by continuing in time from
Euclidean time to real time. Thus, whereas in Euclidean time the surface of the
bubble was at
p = Jtl + x 2 = PO
(2.197)
in real time the surface of the bubble is at
v'x 2 - c 2 t 2 = PO
(2.198)
(restoring the explicit speed of light c which we have been setting to I). The
quantity PO is, in general, on a sub-microscopic scale and so negligible. Thus, to
a good approximation, the surface of the bubble is at x 2 = c 2 t 2 • Consequently,
the radius of the bubble grows with the speed of light.
2.10 Exercises
I. Recast the effective potential of (2.62) in the form (2.77).
2. Show that the zero-temperature effective potential deriving from (2.94)
always has a minimum at lPc = 0 when B > 0 and et > O. Also, show
that when B < 0 there can only be a minimum at the origin when et < O.
3. Check that the SU(5) symmetric, SU(4) x U(I) symmetric and SU(3) x
SU(2) x U(I) symmetric minima of (2.123), (2.124) and (2.125) all have
V =0.
4. Derive the Euclidean action (2.173) and the equation of motion (2.174) when
lP is a function of the four-dimensional radial variable p alone.
5. Check that the bounce solution satisfies (2.178) and that, for Vo given by
(2.179), this leads to the explicit solution (2.180).
2.11 General references
The books and review articles that we have found most useful in preparing this
chapter are:
• Kolb E Wand Turner M S 1990 The Early Universe (Reading, MA:
Addison-Wesley)
• Olive K A 1990 Phys. Rep. 190 307
• Linde A D 1979 Rep. Prog. Phys. 42 389
• Bailin D and Love A 1993 Introduction to Gauge Field Theory (Bristol: lOP)
• Bailin D and Love A 1994 Super symmetric Gauge Field Theory and String
Theory (Bristol: lOP)
Bibliography
[I] Bailin D and Love A 1993 Introduction to Gauge Field Theory (Bristol: lOP) ch 17
63
development in time of the bubble can be obtained by continuing in time from
Euclidean time to real time. Thus, whereas in Euclidean time the surface of the
bubble was at
p = Jtl + x 2 = PO
(2.197)
in real time the surface of the bubble is at
v'x 2 - c 2 t 2 = PO
(2.198)
(restoring the explicit speed of light c which we have been setting to I). The
quantity PO is, in general, on a sub-microscopic scale and so negligible. Thus, to
a good approximation, the surface of the bubble is at x 2 = c 2 t 2 • Consequently,
the radius of the bubble grows with the speed of light.
2.10 Exercises
I. Recast the effective potential of (2.62) in the form (2.77).
2. Show that the zero-temperature effective potential deriving from (2.94)
always has a minimum at lPc = 0 when B > 0 and et > O. Also, show
that when B < 0 there can only be a minimum at the origin when et < O.
3. Check that the SU(5) symmetric, SU(4) x U(I) symmetric and SU(3) x
SU(2) x U(I) symmetric minima of (2.123), (2.124) and (2.125) all have
V =0.
4. Derive the Euclidean action (2.173) and the equation of motion (2.174) when
lP is a function of the four-dimensional radial variable p alone.
5. Check that the bounce solution satisfies (2.178) and that, for Vo given by
(2.179), this leads to the explicit solution (2.180).
2.11 General references
The books and review articles that we have found most useful in preparing this
chapter are:
• Kolb E Wand Turner M S 1990 The Early Universe (Reading, MA:
Addison-Wesley)
• Olive K A 1990 Phys. Rep. 190 307
• Linde A D 1979 Rep. Prog. Phys. 42 389
• Bailin D and Love A 1993 Introduction to Gauge Field Theory (Bristol: lOP)
• Bailin D and Love A 1994 Super symmetric Gauge Field Theory and String
Theory (Bristol: lOP)
Bibliography
[I] Bailin D and Love A 1993 Introduction to Gauge Field Theory (Bristol: lOP) ch 17