In this chapter we will look at the entire edifice of QFT. We will see that it is based on three simple principles. We will also list some of its achievements, including some new insights and understandings not previously mentioned.
THE FOUNDATION
QFT is an axiomatic theory that rests on a few basic assumptions. Everything you have learned so far, from the force of gravity to the spectrum of hydrogen, follows almost inevitably from these three basic principles. {To my knowledge, Julian Schwinger is the only person who has presented QFT in this axiomatic way, at least in the amazing courses he taught at Harvard University in the 1950's.}
1. The field principle. The first pillar is the assumption that nature is made of fields. These fields are embedded in what physicists call flat or Euclidean three-dimensional space-the kind of space that you intuitively believe in. Each field consists of a set of physical properties at every point of space, with equations that describe how these particles or field intensities influence each other and change with time. In QFT there are no particles, no round balls, no sharp edges. You should remember, however, that the idea of fields that permeate space is not intuitive. It eluded Newton, who could not accept action-at-a-distance. It wasn't until 1845 that Faraday, inspired by patterns of iron filings, first conceived of fields. The use of colors is my attempt to make the field picture more palatable.
2. The quantum principle {discetization}. The quantum principle is the second pillar, following from Planck's 1900 proposal that EM fields are made up of discrete pieces. In QFT, all physical properties are treated as having discrete values. Even field strengths, whose values are continues, are regarded as the limit of increasingly finer discrete values.
The principle of discretization was discovered experimentally in 1922 by Otto Stern and Walther Gerlach. Their experiment {Fig. 7-1} showed that the angular momentum {or spin} of the electron in a given direction can have only two values: +1/2 or -1/2 {Fig. 7-1}.
The principle of discretization leads to another important difference between quantum and classical fields: the principle of superposition. Because the angular momentum along a certain axis can only have discrete values {Fig. 7-1}, this means that atoms whose angular momentum has been determined along a different axis are in a superposition of states defined by the axis of the magnet. This same superposition principle applies to quantum fields: the field intensity at a point can be a superposition of values. And just as interaction of the atom with a magnet "selects" one of the values with corresponding probabilities, so "measurement" of field intensity at a point will select one of the possible values with corresponding probability {see "Field Collapse" in Chapter 8}. It is discretization and superposition that lead to Hilbert space as the mathematical language of QFT.
3. The relativity principle. There is one more fundamental assumption-that the field equations must be the same for all uniformly-moving observers. This is known as the Principle of Relativity, famously enunciated by Einstein in 1905 {see Appendix A}. Relativistic invariance is built into QFT as the third pillar. QFT is the only theory that combines the relativity and quantum principles.
THE FOUNDATION
QFT is an axiomatic theory that rests on a few basic assumptions. Everything you have learned so far, from the force of gravity to the spectrum of hydrogen, follows almost inevitably from these three basic principles. {To my knowledge, Julian Schwinger is the only person who has presented QFT in this axiomatic way, at least in the amazing courses he taught at Harvard University in the 1950's.}
1. The field principle. The first pillar is the assumption that nature is made of fields. These fields are embedded in what physicists call flat or Euclidean three-dimensional space-the kind of space that you intuitively believe in. Each field consists of a set of physical properties at every point of space, with equations that describe how these particles or field intensities influence each other and change with time. In QFT there are no particles, no round balls, no sharp edges. You should remember, however, that the idea of fields that permeate space is not intuitive. It eluded Newton, who could not accept action-at-a-distance. It wasn't until 1845 that Faraday, inspired by patterns of iron filings, first conceived of fields. The use of colors is my attempt to make the field picture more palatable.
2. The quantum principle {discetization}. The quantum principle is the second pillar, following from Planck's 1900 proposal that EM fields are made up of discrete pieces. In QFT, all physical properties are treated as having discrete values. Even field strengths, whose values are continues, are regarded as the limit of increasingly finer discrete values.
The principle of discretization was discovered experimentally in 1922 by Otto Stern and Walther Gerlach. Their experiment {Fig. 7-1} showed that the angular momentum {or spin} of the electron in a given direction can have only two values: +1/2 or -1/2 {Fig. 7-1}.
The principle of discretization leads to another important difference between quantum and classical fields: the principle of superposition. Because the angular momentum along a certain axis can only have discrete values {Fig. 7-1}, this means that atoms whose angular momentum has been determined along a different axis are in a superposition of states defined by the axis of the magnet. This same superposition principle applies to quantum fields: the field intensity at a point can be a superposition of values. And just as interaction of the atom with a magnet "selects" one of the values with corresponding probabilities, so "measurement" of field intensity at a point will select one of the possible values with corresponding probability {see "Field Collapse" in Chapter 8}. It is discretization and superposition that lead to Hilbert space as the mathematical language of QFT.
3. The relativity principle. There is one more fundamental assumption-that the field equations must be the same for all uniformly-moving observers. This is known as the Principle of Relativity, famously enunciated by Einstein in 1905 {see Appendix A}. Relativistic invariance is built into QFT as the third pillar. QFT is the only theory that combines the relativity and quantum principles.
( Rodney A. Brooks )
[ Fields of Color: The theory ]
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