The Foundations of Quantum Field Theory

Quantum Field Theory is an axiomatic concept that rests on a number of general assumptions. Every single thing you have learned so far, from the force of gravity to the spectrum of hydrogen, follows almost predictably from these 3 general principles. (To my knowledge, Julian Schwinger is the sole individual that has offered Quatum Field Theory in this axiomatic way, at least in the remarkable courses he presented at Harvard University in the 1950’s.).


1. The field principle. The initial pillar is the assumption that nature is composed of fields. These fields are implanted in what physicists consider flat or Euclidean three-dimensional space– the type of space that you intuitively believe in. Every single field includes a series of physical properties at each and every single point of space, with formulas which define just how all of these properties or field intensities influence each other and transform with time. In Quantum Field Theory there are no particles, no round balls, no sharp edges. One need to bear in mind, however, that the suggestion of fields that permeate space is actually not intuitive. It eluded Newton, who could not accept action-at-a-distance. It wasn’t before 1845 that Faraday, motivated by patternings of iron filings, initially envisaged fields. The use of colors is my attempt in order to make the field account more palatable.

2. The quantum principle (discretization). The quantum principle is the second pillar, following from Planck’s 1900 proposition that EM fields are comprised of discrete morsels. In Quantum field Theory, all physical properties are addressed as carrying discrete values. Even field strengths, whose values are constant, are deemed the limit of significantly finer discrete values.

The basic principle of discretization was uncovered experimentally in 1922 by Otto Stern and Walther Gerlach. Their experiment showed that the angular momentum (or spin) of the electron in a given path can have only 2 values: + 1/2 or– 1/2 Planck units.

The principle of discretization leads to an additional significant distinction separating quantum and classical fields: the principle of superposition. Due to the fact that the angular momentum throughout a particular axis will only possess discrete values, this means that atoms whose angular momentum has been identified throughout a different axis are actually in a superposition of conditions characterized by the axis of the magnetic field used by Stern and Gerlach. That very same superposition principle applies to quantum fields: the field strength at a point can be a superposition of values. And just like interaction of the atom with a magnet “selects” one of the values with comparable possibilities, so “measurement” of field intensity at a point will choose one of the possible values with comparable possibility (see “Field Collapse” in Chapter 8). It is discretization and superposition that brought about Hilbert algebra as the mathematical language of QFT.

3. The relativity principle. There is another fundamental assumption– that the field equations must be the alike for all uniformly-moving observer. This is referred to as the Principle of Relativity, notoriously proclaimed by Einstein in 1905 (see Appendix A). Relativistic invariance is built into QFT as the third pillar. QFT is actually the lone theory which brings together the relativity and quantum principles.


I’m tempted to put in one more principle, yet it’s actually more of a wish than a rule. I’m referring to Occam’s razor, which states basically, “All things being equal, the simplest explanation is best.” Einstein expressed it differently: “A physical theory should be as simple as possible, but no simpler.” The final phrase is important since, as Schwinger said, “nature does not always select what we, in our ignorance, would judge to be the most symmetrical and harmonious possibility” (S1970, p. 393). Supposing that the theory were as simple as possible, there might be just one field (or perhaps none!), and the planet would certainly be very dull– not to mention uninhabitable. I feel it can be said that the equations of Quantum Field theory are truly about as simple as possible, but no simpler.

The move from a particle description to a field description will be especially fruitful if the fields obey simple equations, so that we can calculate the future values of fields from the values they have now … Maxwell’s theory of electromagnetism, general relativity, and QCD [quantum chromodynamics] all have this property. Evidently, Nature has taken the opportunity to keep things relatively simple by using fields.– F. Wilczek (W2008, p. 86).


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