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.