 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 wave function of quantum mechanics is one example of a field. In this case, it's 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. A 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 a 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 1925 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 Wave Function 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. Pascal 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, But in exploring the fundamental problem of quantizing an oscillating field, here's the problem. Let X be the horizontal coordinate. A string is fixed at X equals 0 and X equals L. At time t, a point on the string with a given X coordinate has a vertical displacement u. This defines the shape of the string through space and time as u of X and t. 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. Let's 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 string's 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 takeaway is a single equation. It's not necessary to follow the details. They're given for completeness. Let's start with our discrete string model. Assume the segment length when the string is at rest is little l. Call the string tension f. Each mass feels a force f to the left and a force f to the right. The net force is zero, so the mass remains at rest. When the string vibrates, the vertical displacements u can be non-zero. Let's suppose that the displacements are small and the string tension remains f. Consider masses n and n-1. Their positions form a triangle. The horizontal side has length l and the vertical side has length un-un-1. The string forms the hypotenuse at an angle theta to the x-axis. The string segment will pull on the nth mass in the horizontal direction with force f times cosine theta. For small theta, the cosine is approximately 1, so this force component is just f in the left direction. The vertical force is f times sine theta. For small theta, the sine is approximately the ratio of the vertical to horizontal side. So the vertical force is f times un-un-1 over l in the downward direction. Now consider the effect of mass n plus 1. It will also produce horizontal and vertical forces on mass n. The horizontal force will cancel that due to mass n minus 1, but the vertical force will in general not. With the convention that a downward force is negative, the net force on mass n is minus f un-un-1 over l due to mass n minus 1, minus f un-un-1 over l due to mass n plus 1. By Newton's second law, this equals mass times acceleration, m times un double dot. We can divide through by m and rearrange to get the form shown here. In the denominator, we've also multiplied and divided by l. Now we go from the discrete approximation back to the continuous case. The acceleration of the discrete masses become the slope of the slope in time of the continuous function u, which is essentially the curvature in time of u, represented by a curly d sub t squared operator applied to u. 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 u. Let's set the factor f over m over l equal to a constant c squared. c turns out to be the wave speed. Eventually, we'll apply these ideas to the electromagnetic field, in which case c 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. Here's a brief illustration of the physical interpretation of the wave equation. Suppose at time t zero, we take a snapshot of the string's shape. This defines the graph u of x t zero. At a point x zero, we can find the circle which passes through the graph and best represents the graph's slope and curvature there. The wave equation then tells us that if we plot the field at point x zero as a function of time and find the best fit circle at time t zero, 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.