Phys 624 - Quantum Field Theory
University of Maryland, College Park
Fall 2017, Professor: Ted Jacobson
Questions
3. Where did the
scalar field action come from? If we look
for an action that is quadratic in the scalar field, so it
will yield a linear field equation, corresponding to a
noninteracting theory, this is the only possibility consistent
with Lorentz invariance and positivity of energy. I will
explain this more fully when we review special relativity.
2. What is it that Einstein inferred about the fluctuations of
the em field in thermal equilibrium containing a particle-like
and a wave-like term, and how is it that Jordan's analysis of
the quantized string explained this in terms of field
quantization? In 1909 Einstein evaluated
the energy fluctuations in a small frequency interval in
blackbody radiation according to the Planck distribution, and
found that the mean square fluctuation contains a term
proportional to the mean energy and a term proportional to the
square of the mean energy. These correspond to the Wien and
Rayleigh-Jeans limits of the Planck distribution, and to
particle-like and wave-like fluctuations of energy. For a
discussion, see Sections 1.4 & 1.5 of The
Conceptual Framework of Quantum Field theory, by
Tony Duncan.
1. Does the action always have a stationary point, for fixed
initial & final configurations? No.
For example, for a harmonic oscillator, if the time separation
is equal to one period, and the initial and final positions
are not the same, there is no solution, hence no stationary
point of the action. This means in particular that there is no
minimum action path, but, moreover, there is no saddle point.
Hence it must be possible to find a 1-parameter family of
paths that monotonically decreases the action, with a
nonvanishing derivative of the action with respect to the
parameter labeling the path. For a simple example with uniform
motion, consider a harmonic oscillator, with Lagrangian L =
1/2 (dx/dt)2 - 1/2 x2, and the paths
x(t)=vt for t from 0 to t', and x(t)=v(2t'-t) for t from t' to
2t'. The speed v is the parameter labeling these constant
speed paths. The kinetic energy contribution to the action is
then v2t' and the potential energy contribution is
-v2t'3/3. If t' > sqrt(3) the action
is negative, and always decreasing as v increases. I think the
minimum value of 2t' for the action to lack a stationary point
if the initial and final points at x=0 must be π...