Welcome to Quantum Field Theory 2.
Field quantization.
As the name implies, quantum field theory
is centered on the concept of a field.
A simple definition of a field is a physical
quantity, or property, continuously distributed
throughout space and time.
The electron wavefunction of quantum mechanics
is one example of a field.
In this case itís a complex, scalar function
of the coordinates.
Complex means that its values consist of complex
numbers, having both real and imaginary parts.
Scalar means that the field has only an amplitude.
The electric field of classical electromagnetic
theory is an example of a real, vector function
of coordinates.
Vector means that the field has both an amplitude
and a direction.
At a given time and at a given point, we can
represent this as an arrow based at that point.
The arrow length represents the field amplitude,
and the arrow direction the field direction.
At different points, the field has different
values.
The way the field changes through space and
time is governed by a field equation.
A challenge for developing a quantum theory
of a field is that at a given time the field
has an infinite number of values.
One, or more, at every point in space.
And, unlike the harmonic oscillator we considered
in the previous video, it can, typically,
oscillate at an infinite number of frequencies.
We are left with the problem of trying to
figure out what to quantize.
A foundational treatment of this problem was
presented in a series of papers titled, on
quantum mechanics, submitted in 19 25 by Born,
Heisenberg and Jordan.
Born and Heisenberg figure prominently in
our quantum mechanics series.
Max Born devised the Born rule.
The interpretation of the quantum mechanical
wavefunction as a probability amplitude for
finding a particle at a given position in
space and time.
Werner Heisenberg is famous for, among many
other things, the Heisenberg uncertainty principle.
This series of papers summarized, and made
new contributions to many aspects of quantum
theory.
Pascual Jordan appears to have been the author
primarily responsible for a section on field
quantization.
When faced with a new and complex problem,
physicists often formulate a simplified, even
cartoonish, version so they can focus on the
essence of the problem without getting sidetracked
by details.
This is what Born, Heisenberg and Jordan did
in this paper.
They wrote, to avoid calculational complications
which have no bearing upon the nature of the
case, we base ourselves on the simplest conceivable
case, namely a vibrating string fastened at
its ends.
Incidentally, all essential points of the
calculation can immediately be taken over
in more general instances.
The authors were not interested in developing
a quantum theory of violin strings, but in
exploring the fundamental problem of quantizing
an oscillating field.
Hereís the problem.
Let ëxí be the horizontal coordinate.
A string is fixed at ëxí equals 0 and ëxí
equals ëLí.
At time ëtí a point on the string with a
given ëxí coordinate has a vertical displacement
ëuí.
This defines the shape of the string through
space and time as ëuí of ëxí and ëtí.
In quantum mechanics we typically identify
a set of classical dynamical variables corresponding
to position and momentum.
We then transition to quantum theory by replacing
these with quantum operators.
So, for the string, what are the dynamical
variables?
We might answer that the position and momentum
of every point on the string is a dynamical
variable.
But, there are an infinite number of points
on a continuous string.
How can we ever hope to express, let alone
solve, a quantum system with an infinite number
of dynamical variables?
One approach is to approximate the continuous
system by a discrete one.
Letís replace the continuous string with
discrete masses connected by massless springs.
Conceptually, we think of the string as divided
into a number of segments.
Then collapse all the mass of each segment
to a point, and represent the stringís elastic
properties as massless springs connecting
these point masses.
With enough masses this approximation should
be very good.
Unfortunately, these masses do not oscillate
independently.
When one moves it pulls on both of its neighbors.
So.
their motion is highly coupled.
In the chemistry video of the quantum mechanics
series we saw how difficult it is to treat
a quantum system of many coupled particles.
So this seems like a dead end.
Before we explore this problem further, we
need to consider the equation of motion of
the string.
This will form our field equation.
The next two screens contain quite a bit of
math, but the take away is a single equation.
Itís not necessary to follow the details,
they are given for completeness.
Letís start with our discrete string model.
Assume the segment length, when the string
is at rest, is little ëlí.
Call the string tension ëFí.
Each mass feels a force ëFí to the left
and a force ëFí to the right.
The net force is zero, so the mass remains
at rest.
When the string vibrates, the vertical displacements
ëuí can be non-zero.
Letís suppose that the displacements are
small, and the string tension remains ëFí.
Consider masses ëní and ëní minus 1 . Their
positions form a triangle.
The horizontal side has length ëlí, and
the vertical side has length ëuí ëní minus
ëuí ëní minus 1 .The string forms the
hypotenuse, at an angle theta to the ëxí
axis.
The string segment will pull on the n-th mass
in the horizontal direction with force ëFí
times cosine theta.
For small theta the cosine is approximately
one, so this force component is just ëFí,
in the left direction.
The vertical force is ëFí times sine theta.
For small theta the sine is approximately
the ratio of the vertical to horizontal side.
So the vertical force is ëFí times ëuí
ëní minus ëuí ëní minurs 1 over ëlí,
in the downward direction.
Now consider the effect of mass ëní plus
1 . It will also produce horizontal and vertical
forces on mass ëní.
The horizontal force will cancel that due
to mass ëní minus 1 . But the vertical force
will, in general, not.
With the convention that a downward force
is negative, the net force on mass ëní is,
minus ëFí, u-n minus u-n-minus-1 over ëlí,
due to mass n-minus-1, minus ëFí, u-n minus
u-n-plus-1 over ëlí, due to mass n-plus-1
. By Newtonís second law this equals mass
times acceleration, ëmí times u-n double-dot.
We can divide through by ëmí and rearrange
To get the form shown here.
In the denominator weíve also multiplied
and divided by ëLí.
Now we go from the discrete approximation
back to the continuous case.
The acceleration of the discrete masses becomes
the slope of the slope in time of the continuous
function ëuí.
Which is essentially the curvature in time
of ëuí.
Represented by a curly ëdí sub ëtí squared
operator applied to ëuí.
For the expression in brackets, the numerator
is the difference of two terms.
Each term has the form, change in vertical
coordinate over change in horizontal coordinate.
That is the spatial slope of the string.
The entire expression represents the change
in slope over change in horizontal coordinate.
The slope of the slope, or essentially the
curvature in space of ëuí.
Letís set the factor ëFí over ëmí over
ëlí equal to a constant ëcí squared.
ëcí turns out to be the wave speed.
Eventually weíll apply these ideas to the
electromagnetic field in which case ëcí
will be the speed of light.
Then we obtain the one dimensional wave equation.
Essentially, curvature in time of the field
equals speed squared times curvature in space
of the field.
Hereís a brief illustration of the physical
interpretation of the wave equation.
Suppose at time t-0 we take a snapshot of
the stringís shape.
This defines the graph ëuí of ëxí t-0
. At a point x-0 we can find the circle which
passes through the graph and best represents
the graphís slope and curvature there.
The wave equation then tells us that if we
plot the field at point x-0 as a function
of time, and find the best-fit circle at time
t-0, then the radius of this circle is determined
by the radius of the previous circle.
Roughly speaking, the radius of the time circle
equals the radius of the space circle divided
by c-squared.
