As our most fundamental description of physical phenomena, quantum field theory (QFT) is the natural culmination of classical mechanics, classical field theory, and quantum mechanics. To understand what QFT is and what makes it so special, it is worthwhile to briefly look at each of these structures in turn.
Classical mechanics describes the motion of objects in space and time in terms of well-defined positions and momenta that, taken together, form the phase space of the system. The location of an object in phase space can, in principle, be determined at any time, and once known suffices to predict its location at all future times given a complete enough knowledge of the forces at work. There are two main formulations of classical mechanics: the Hamiltonian formulation, which concentrates on how to find quantities at later times in terms of their known values at earlier times, and the Lagrangian one, which derives the same information from variational principles. These principles state that an object’s trajectory in phase space extremizes a certain quantity called the action, which is the integral over the whole phase space history of a functional called a Lagrangian. The dimension of the phase space is usually finite, but can be enlarged beyond the usual 3+3 dimensions for the location and velocity of a point particle to allow for rotations of a rigid body or changes of shape of a deformable body. Relativity can be accommodated, but the ideas of quantum mechanics do not appear.
The move to an infinite number of phase space dimensions takes one from classical mechanics to classical field theory. Fluid mechanics, for example, allows for an infinite variety of shapes that a fluid can take making the phase space infinite. Sums that arose in classical mechanics are now integrals, and the general problems of analysis become much more challenging. Nevertheless, many problems are tractable, and theories based on classical fields with an infinite number of degrees of freedom have had great success. Notable among these is Maxwell’s electrodynamics, which describes the electromagnetic field in terms of electric and magnetic fields that can vary in space and time and are present at all points of space and time. The phase space of classical electromagnetism is then infinite dimensional, and this accounts for much of the richness of the physics that it can describe.
Another shift away from classical mechanics is to maintain a finite number of degrees of freedom but to allow that points in phase space cannot be arbitrarily well known. The Heisenberg uncertainty principle placed constraints on the accuracy with which position and momentum could be known simultaneously, and the points of the phase space acquired a certain fuzziness. For example, in place of a point with well-defined position x and momentum px in the x-direction, one had something like a fuzzy disk of area roughly equal to the square of the Planck length, and shape determined by whether one tried to accurately determine x or px—one could only know one with a consequent sacrifice of information about the other. A system cannot now be thought of as a well-defined classical point moving around in phase space. Rather, information about the system is encoded in a function on the phase space called the wave function Ψ, and the theory allows all that can be known about the system to be recovered by applying various linear operators to Ψ.
The step to QFT involves allowing an infinite number of degrees of freedom in quantum mechanics. Thus, one might say that QFT is to quantum mechanics what classical field theory is to classical mechanics. The shift is profound and carries a number of important implications.
The first is that QFT takes as its phase space classical fields upon which an uncertainty paralleling that in quantum mechanics has been imposed. For example, the role of x could be taken by a classical field and the role of momentum px by the rate of change of something (the Lagrangian) with respect to the time derivative of This quantity is called the momentum conjugate to and it is impossible to know both it and with arbitrary precision.
Time and space are now just dummy labels (parameters) in the theory and are integrated over, their main use being to enforce the notion of locality in a Lagrangian. This point is often misunderstood as people look for the “locations” of field quanta in physical space. In QFT, one looks at as a dynamical quantity as opposed to x. An excellent discussion of this point can be found in Schwinger (1970).
It turns out, perhaps somewhat tautologically, that one can often expand fields in Fourier series and identify each of the modes as a “particle” with momentum given by the mode numbers. It is a remarkable fact that experiments to detect fields in nature, when performed with sufficient resolution, always find them occurring in (or at least interacting as) discrete chunks or quanta. In this sense, QFT is a theory of particles, and the degree to which these quanta can be identified as isolated, discrete objects determines how well they can be thought of as particles.
As a field changes in time so too do the various Fourier modes and this is interpreted as saying that particles appear and disappear; in other words, particles can be created and destroyed in QFT. This is something foreign to both quantum and classical mechanics and was needed with the advent of high-energy physics experiments in which particles are routinely created and destroyed.
Two of many excellent texts describing how QFT is used in the calculation of physical processes are those by Peskin & Schroeder (1995) and by Weinberg (1995). The text by Itzykson and Zuber (1980) is also very good. A more philosophically inclined and less calculationally oriented reader will find much to think about in the book by Teller (1995). It is easy to suppose that QFT is intrinsically, or somehow must be, relativistic. In fact, the concepts can be very successfully applied to problems in condensed matter physics in nonrelativistic situations, and the interested reader will find the delightful introductory book by Mattuck (1992) and the more advanced book by Abrikosov et al. (1975) excellent places to start.
Several caveats are in order. First of all, in the relativistic case (the real world), these quanta do not generally admit a localization in the sense of having well-defined classical-like positions. That is, they do not behave like billiard balls. Second, if interactions between them are very strong they may not act as intuition might suggest. For example, while electrons and photons are in many ways like what one might expect particles to be, protons and neutrons seem to be comprised of particles called quarks, which cannot be removed. One says that the interactions between quarks are so strong that they are “confined.” Phonons are quantized vibrations in crystals that can be detected in neutron scattering experiments and act in many ways
Figure 1. A Feynman diagram representing a first approximation to the scattering of one electron from another (the two straight lines) via the exchange of a photon (the wiggly line). Each diagram like this corresponds to a physically clear picture as well as to a well-defined mathematical expression contributing to the quantum mechanical amplitude for the process to take place. Only the topology of the diagram is important. Bending the lines around represents processes involving electrons and their antiparticles as different aspects of the same basic phenomenon, and connects scattering in space with scattering forward and backward in time, while not violating causality.
like particles, but they are only there insofar as there is a crystal lattice to vibrate. Any attempt to isolate a phonon by pulverizing a crystal and looking for one among the fragments is doomed to failure.
When it makes sense to think of weakly coupled particles, there is a well developed calculational framework called “perturbative quantum field theory” (pQFT), which allows one to make systematic estimates of physical quantities, often using sketches called Feynman diagrams. Figure 1 shows a Feynman diagram contributing to the scatting of one electron by another (the straight lines) via the exchange of a photon (the wiggly line).
The mathematical procedures involved are fraught with difficulties, and the series that arise are often divergent on physical grounds. In addition, the individual terms also tend to be infinite, and several procedures have been developed for handling such problems. The general approach is to first make ill-defined formal expressions finite by changing them (“regularizing” them) in a way controlled by some parameter, then comparing such regularized expressions only to one another and linking only such comparisons to observations (“renormalization”). Despite the feelings of many of the founders that there might be something deeply wrong with these procedures, it turns out that many quantities calculated in this way have given answers accurate to parts per million or better.
For strongly interacting theories the state of the art is much less well-developed. Powerful computers can be used to treat theories that are based on restricting fields to live on a grid or lattice that is meant to approximate space-time, but with a finite number of points. Analytic techniques also offer some hope, but the problems are formidable.
One of the most dramatic consequences of QFT is that the vacuum becomes a very complex object. Recalling the notion of the uncertainly principle applied to fields, one cannot set both a field and its conjugate momentum equal to zero. This, in fact is the case for each possible mode of a field and leads to an infinite energy in the vacuum due to these fluctuating, uncertain fields. As mentioned earlier, infinities like this are basically swept under the rug by insisting that only comparisons are meaningful, but in this case there are remarkable physical consequences.
For example, there are more modes that can exist for the electromagnetic field between two parallel metal plates that are far apart than for ones that are closer together. The infinite energies between the plates in these two cases can be compared and the result is finite—there is less energy between two plates that are close together than two that are far apart. This implies that two parallel metal plates, in order to reduce the energy between them (in empty space), will pull together. This is called the “Casimir effect” and has actually been observed in the laboratory, making it clear that QFT and its strange vacuum are more than theoretical fantasies.
Although it is often not made clear, one final aspect of QFT that differs from quantum mechanics is that QFT admits an infinite number of distinct vacua, all of zero energy. None of these vacua can be reached from another by a unitary transformation, making them physically distinct. This richness of “empty” space is a critical part of what makes QFT able to describe so much. To do justice to this point would require more space than is available, but the interested reader would do well to start with the book by Umezawa (1993). One consequence of this existence of unitarily inequivalent vacua is the amazing fact that the vacuum of one observer can be devoid of particles, while that of another may actually have particles present.
Abrikosov, A.A., Gorkov, L.P. & Dzyaloshinski, I.E. 1975. Methods of Quantum Field Theory in Statistical Physics, revised edition, translated and edited by Richard A. Silverman, New York: Dover
Itzykson, C. & Zuber, J.-B. 1980. Quantum Field Theory, New York: McGraw-Hill
Mattuck, R.D. 1992. A Guide to Feynman Diagrams in the Many-Body Problem, 2nd edition, New York: Dover
Peskin, M.E. & Schroeder, D.V. 1995. An Introduction to Quantum Field Theory, Reading, MA: Addison-Wesley
Schwinger, J. 1970. Particles, Sources, and Fields, Reading, MA: Addison-Wesley
Teller, P. 1995. An Interpretative Introduction to Quantum Field Theory, Princeton, NJ: Princeton University Press
Umezawa, H. 1993. Advanced Field Theory, New York: American Institute of Physics Press
Weinberg, S. 1995–2000. The Quantum Theory of Fields, 3 vols., Cambridge and New York: Cambridge University Press
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and the role of momentum px by the rate of change of something (the Lagrangian) with respect to the time derivative of 
This quantity is called the momentum conjugate to 
and it is impossible to know both it and 
with arbitrary precision.
as a dynamical quantity as opposed to x. An excellent discussion of this point can be found in Schwinger (1970).
