## Quantum field theory

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## Quantum Field Theory

# QUANTUM FIELD THEORY

Quantum field theory is the widely accepted tool for describing systems such as the electromagnetic field, in accordance with the laws of quantum mechanics as well as with other restrictions depending on the context. In elementary particle physics, relativistic quantum field theory, whose predictions respect the principle of relativity, is used. Quantum field theory is also used extensively in other areas such as superconductivity, the quantum Hall effect, and statistical mechanics.

Knowledge of some ideas from mechanics and quantum mechanics is necessary in order to understand quantum field theory. Consider a single mass, coupled to an anchored spring and vibrating along the *x* -direction on a frictionless surface. This oscillator is a system with one degree of freedom, since only one coordinate *x* is needed to describe it. A system of *N* masses moving along the *x* -axis would have *N* degrees of freedom. A field is a system with infinite number of degrees of freedom. For example, if one had an infinite array of masses in a line, coupled to their neighbors by springs, the displacements of each mass from equilibrium would constitute a field. Another example is the electromagnetic field whose degrees of freedom are the values of the electric and magnetic field at each point in space. Notice that the latter has an infinite number of degrees of freedom within any finite volume, and this causes many problems. The aim of quantum field theory is to describe these degrees of freedom according to the laws of quantum mechanics.

The field can be arrived at by starting with the simplest system: a single mass *m* attached to a spring of force constant *k* . In classical mechanics this mass will vibrate with angular frequency ω = 2πƒ = √*k* /*m* It can have any finite amplitude *x*_{0}, and the corresponding energy will be In the quantum version the oscillator can only have energy *E* = ħ*w* (*n* + ½), where *n* = 0, 1, 2, …, and ħ = 1.05 · 10^{-34} J/s is Planck's constant. Note that the lowest energy is not zero but ½ħ*w* , which is referred to as zero-point energy. The uncertainty principle in quantum mechanics, which forbids a state of definite location and momentum, does not allow the classical zero-energy state in which the mass sits at rest in the equilibrium position. Next, the fact that the levels are equally spaced means that instead of saying the oscillator is in a state labeled by the quantum number *n* , one can say that there are *n* quanta of energy ħ. This seemingly semantic point proves seminal, as shall be seen.

Now consider two masses *m* attached to springs of force constant *k* as in Figure 1. The coordinates *x*_{1} and *x*_{2} are coupled to each other, and the motion is quite complicated. But consider the following combination of coordinates called normal coordinates: These behave like independent oscillators of frequency ω = √*k* /*m* , √3*k* /*m* , respectively. To see this, imagine starting off the masses with equal displacements, so that *X*_{1} is nonzero and *X*_{2} = 0. Since the middle spring is undistorted, the masses will begin moving in response to the end springs. Since these are identical, the condition *x*_{1} = *x*_{2} (the same as *X*_{2} = 0) will be preserved for all times, and the coordinate *X*_{1} will vibrate atω = √*k* /*m* , On the other hand, if both masses are given equal and *opposite* displacements, then *X*_{1} = 0 initially. The middle spring is distorted twice as much as the end springs (so the effective force constant felt by the masses is 3*k* ), and the masses vibrate with equal and opposite displacements at ω = √3*k* /*m* , That is, *X*_{1} = 0 for all times, and *X*_{2} vibrates at ω = √3*k* /*m* , If some arbitrary initial displacements are given so that both *X*_{1} and *X*_{2}are nonzero, one can compute their future values (which is easy since they behave like independent oscillators)

**FIGURE 1**

and go back to *x*_{1} and *x*_{2} at the end. The same strategy used for the two-mass problem also works for any number of masses.

Thus, consider *N* such masses coupled to their neighbors by *N* such springs of equilibrium length *a* arranged around a circle of circumference *L* = *Na* . (units *m* = *k* = 1.) Let Φ(*n* ) be the displacement of mass numbered *n* , where 1 ≤ *n* ≤ *N* . The system can once again be reduced to those of decoupled oscillators. The normal coordinates are just the Fourier coefficients: The requirement that the points numbered *n* and *n* + *N* are one and the same implies whereas the fact that *K* and *K* + 2π/*a* are indistinguishable in the exponential factor limits *r* to the range of size *N* and *K* to an interval of width 22π/*a* , which is chosen to be -π/*a* ≤ *K* ≤ π/*a* . The system is equivalent to *N* oscillators whose frequencies can be shown to be *w* (*K* ) = 2[1 - cos(*Ka* )]. If one lets *N* → ∞, the allowed values of *K* become continuous in the interval [-π/*a* , π/*a* ]. The field *Φ* is a classical field.

The quantum version will be obtained by treating the decoupled oscillators in terms of quantum mechanics. Each oscillator of frequency *w* can only have energy *E* = *ħω* (*n* + ½), where *n* 0, 1, 2,. . . . This means that even in the ground state the field has zero-point energy ½ħ*w* per oscillator, which can have observable effects. (The zero-point energy of the electromagnetic field leads to the spontaneous decay of atoms.) Next, the fact that the levels are equally spaced means that instead of saying the oscillator is in a state labeled by *n* , one can say that there are *n* quanta of energy ħ*w* and momentum ħ*K* , where the identification of ħ*K* with momentum comes from examining the interaction of the system, described in terms of the decoupled oscillators, with any external probe. Thus, the quantum state of the field is specified by saying how many quanta there are at each *K* . In the present problem of vibrating atoms the phenomenon is just sound, and the quanta are called phonons.

If these methods are applied to the electromagnetic field, which has degrees of freedom at each point in space, that is, *a* = 0, the allowed *K* values will go from -∞ to ∞, a consequence of which soon follows. The quanta of this field are referred to as photons.

Both quanta above come from bosonic oscillators, for which the quantum number *n* is not restricted. When a macroscopic number of bosonic quanta, say, photons, are present in a state with some *K* , it is perceived as a classical (electromagnetic) field at that wave number and the corresponding energy density. To describe fermions like electrons as quanta, one needs to quantize a fermionic oscillator that cannot support more than one quantum. This reflects the Pauli exclusion principle, which says that no two electrons can have the same quantum numbers. There is no classical manifestation of such fermionic fields, which is why they are so unfamiliar.

In the cases considered, the number of quanta in each oscillator stays fixed, a result of the fact that the total energy is a quadratic function in the coordinates and velocities, which, in turn, is why normal coordinates that evolve independently of each other. Upon adding higher-order interaction terms, a change in these numbers may occur. Since the quanta correspond to particles, this means either that particles change their energy or momentum values or that new particles are created. Consider quantum electrodynamics (QED), in which a term quadratic in the electron field and linear in the photon field is added to the total energy with a coefficient *e* , which is the electron's charge. This cubic term describes a process in which an electron emits or absorbs a photon. This can cause, among other things, two electrons to scatter from their original momentum states to new ones.

The results are computed in a perturbation series in *e* or, equivalently, α = *e*^{2}/ħ*c* ≃ 1/137, called the fine structure constant. Various contributions to the series are represented by Feynman diagrams. For example, Figure 2(a) shows a process second order in *e* (or first order in ') in which an electron emits a photon and recoils, while another captures the photons and recoils the other way. *Thus, the quanta of the field (photons) also mediate interactions between other quanta (electrons).* If one goes to higher orders in the expansion, more complicated scattering diagrams may be obtained, say, a diagram where one of the electrons emits two photons, and then either both are absorbed by the other, or one is absorbed by the emitter itself and one is absorbed by the other. The Feynman diagrams go on to infinite order, and one usually stops after a few terms and obtains excellent numbers since is so small. In a general theory, there may be no such small parameter. But even with the small α, there is a problem extracting predictions in QED due to the following divergence problem.

Consider a lone electron that emits a photon and reabsorbs it, so that for a while one has an electron and a photon, as illustrated in Figure 2(b). This can be shown to change the observed mass of the electron from the value *m*_{0} to

*m = m*_{0}+ α*I*

where *m* is the observed mass, calculated to order in perturbation theory, and *I* is an integral that sums over the various electron-photon states that can occur between the emission and reabsorption. This sum diverges because of the infinite range of momenta for the quanta. (This does not happen in Figure 2(a)

**FIGURE 2**

since the photon's momentum is fixed by those of the electrons and hence not summed over. In Figure 2(a), the *sum* of the momenta of the electron and photon is fixed by that of the incoming electron but not their individual pieces, one of which can be chosen at will.)

Since the observed mass of the electron is finite, one resorts to renormalization, in which all energy and momentum sums are cut off at some large value Λ. Thus, *m = m*_{0} + α*I* (Λ), but one now requires that *m*_{0} itself be Λ dependent in such a way that *m* , which is the observed mass, is finite and Λ independent. Next, one finds that at higher orders the scattering rate of two electrons is infinite as well. One now says that the coupling is not given by α ≃ 1/137 but by α0(Λ), chosen so that the scattering rate turns out to be finite and cut-off independent and corresponds to the measured value of α ≃ 1/137. *Remarkably enough, once these parameters are thus chosen, no new infinities arise,* a feature called renormalizability. Renormalizability has been a guiding principle in arriving at the theory of strong interactions (called quantum chromodynamics or QCD) and in the unified theory of electromagnetic and weak interactions, the *Glashow-Weinberg-Salam (GWS) model.* In recent years, a far deeper understanding of the renormalizability of quantum field theories has emerged by casting them in terms of the mathematically equivalent problem of phase transitions in statistical mechanics.

An important guide in searching for the right field theory, that is, the right set of interaction terms, is symmetry. In particle physics the predictions have to be invariant under space-time symmetries such as translations, rotations, and Lorentz transformations, or internal symmetries, such as isospin symmetry, in which a proton and neutron are exchanged. Invariance under translations or rotations means that an experiment will give the same answer if all the relevant parts of the experiment are translated (shifted) to a new location or rotated. Each symmetry implies a conservation law. For instance, translation symmetry leads to the conservation of momentum, whereas rotational symmetry leads to the conservation of angular momentum. Local gauge symmetry, which is the invariance of the theory to a particular redefinition of the fields that varies from point to point in space-time, is enjoyed by QED, QCD, and the GWS model.

Typically, the lowest-energy state or vacuum state is invariant under all the symmetries of the interactions. For example in a theory invariant under rotations, the lowest-energy state will look the same when rotated. In a magnet, however, the ground state may be magnetized in some direction and will therefore look different when rotated. Now, the direction of magnetization is chosen randomly from all possible directions, none of which is favored intrinsically by the microscopic interactions. This is referred to as a spontaneous breakdown of symmetry and can occur in the field theories of particle physics. According to Goldstone's theorem, for every spontaneously broken symmetry, there is one massless excitation or particle, called a Goldstone boson. The nearly massless pion is the Goldstone boson corresponding to the nearly exact chiral symmetry. There is one way out of Goldstone's theorem. If the variable that breaks the symmetry (like the magnetization) interacts with a massless gauge field, then instead of a massless Goldstone boson, one ends up with a massive gauge field. This Higgs mechanism plays a crucial role in the unified theory of the GWS whereby the massive gauge bosons that mediate weak interactions start out massless and become massive due to the Higgs mechanism.

All quantum field theories are formulated in terms of point particles like electrons and photons. In the last two decades of the twentieth century an alternate view, in which the building blocks are strings, came into prominence. It is being vigorously studied and the correct answer is not yet known.

*See also:*Dirac, Paul; Fermi, Enrico; Gauge Theory; Pauli, Wolfgang; Renormalization; Salam, Abdus; Schwinger, Julian; Tomonaga, Sin-Itiro

## Bibliography

Abrikosov, A. A.; Gorkov, L. P.; and Dzyaloshinski, I. E. *Methods of Quantum Field Theory in Statistical Mechanics* (Dover, New York, 1963).

Green, B. *The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory* (Vintage Books, New York, 2000).

Itzykson, C., and Zuber, J. B. *Quantum Field Theory* (McGraw Hill, New York, 1980).

Mahan, G. D. *Many Body Physics* (Plenum, New York, 1981).

Peskin, M. E., and Schroeder, D. V. *Introduction to Quantum Field Theory* (Perseus Press, New York, 1995).

Shankar, R. *Principles of Quantum Mechanics* (Plenum, New York, 1994).

Weinberg, S. *The Quantum Theory of Fields* (Cambridge University Press, Cambridge, UK, 1995).

*Ramamurti Shankar*

## Quantum Field Theory

# Quantum Field Theory

Quantum field theory is obtained by combining special relativity and quantum mechanics. Until 1981 this was the primary tool for the understanding of elementary particles of matter and the nongravitational forces of matter. However, such theories were known to possess deficiencies and many calculations of observable quantities led, formally, to infinite answers. Yet, by the application of well defined rules these infinities could be removed to leave finite answers that agree with observation to as many as fourteen decimal places of precision. It was then discovered that these deficiencies could be avoided by replacing their theories of pointlike particles by string theories that treated the most fundamental entities in nature as lines or loops of energy (*strings* ) possessing a certain symmetry (*supersymmetry* ).

String theories avoid the infinities and paradoxes of quantum field theories and are a promising candidate for a complete theory of all elementary particles and forces of nature. The stringlike loops of energy in these theories possess a tension that increases as the temperature of the environment falls. Thus at very high temperatures, for example in the first moments of the expansion of the universe, they would have behaved in an intrinsically stringy manner. As the universe expanded and cooled, the string tensions would increase and the loops of string would behave more and more like single points of mass and energy. As a result, in the low temperature world all the predictions of the earlier quantum field theories are expected to be obtained, in agreement with experiment.

*See also* Physics, Quantum; Field Theories; String Theory

*Bibliography*

barrow, john. theories of everything. london: vintage, 1992.

greene, brian. the elegant universe. new york: norton, 1999.

john d. barrow

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