Quantum field theory

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. The current Standard Model of particle physics is based on QFT.

Despite its extraordinary predictive success, QFT faces ongoing challenges in fully incorporating gravity and in establishing a completely rigorous mathematical foundation.

History

Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory—quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure. A second major barrier came with QFT's apparent inability to describe the weak and strong interactions, to the point where some theorists called for the abandonment of the field theoretic approach. The development of gauge theory and the completion of the Standard Model in the 1970s led to a renaissance of quantum field theory.

Theoretical background

thumb|200px|[[Magnetic field lines visualized using iron filings. When a piece of paper is sprinkled with iron filings and placed above a bar magnet, the filings align according to the direction of the magnetic field, forming arcs allowing viewers to clearly see the poles of the magnet and to see the magnetic field generated.]]

Quantum field theory is the result of the combination of classical field theory, quantum mechanics, and special relativity.

Fields began to take on an existence of their own with the development of electromagnetism in the 19th century. Michael Faraday coined the English term "field" in 1845. He introduced fields as properties of space (even when it is devoid of matter) having physical effects. He argued against "action at a distance", and proposed that interactions between objects occur via space-filling "lines of force". This description of fields remains to this day.

The theory of classical electromagnetism was completed in 1864 with Maxwell's equations, which described the relationship between the electric field, the magnetic field, electric current, and electric charge. Maxwell's equations implied the existence of electromagnetic waves, a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed, which turns out to be the speed of light. Action-at-a-distance was thus conclusively refuted. Max Planck's study of blackbody radiation marked the beginning of quantum mechanics. He treated atoms, which absorb and emit electromagnetic radiation, as tiny oscillators with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as quantum harmonic oscillators. This process of restricting energies to discrete values is called quantization. Building on this idea, Albert Einstein proposed in 1905 an explanation for the photoelectric effect, that light is composed of individual packets of energy called photons (the quanta of light). This implied that the electromagnetic radiation, while being waves in the classical electromagnetic field, also exists in the form of particles.

By applying the renormalization procedure, calculations were finally made to explain the electron's anomalous magnetic moment (the deviation of the electron g-factor from 2) and vacuum polarization. These results agreed with experimental measurements to a remarkable degree, thus marking the end of a "war against infinities". but in 1951 he found a way around the problem of the infinities with a new method using external sources as currents coupled to gauge fields. Motivated by the former findings, Schwinger kept pursuing this approach in order to "quantumly" generalize the classical process of coupling external forces to the configuration space parameters known as Lagrange multipliers. He summarized his source theory in 1966 then expanded the theory's applications to quantum electrodynamics in his three volume-set titled: Particles, Sources, and Fields. Developments in pion physics, in which the new viewpoint was most successfully applied, convinced him of the great advantages of mathematical simplicity and conceptual clarity that its use bestowed.

In source theory there are no divergences, and no renormalization. It may be regarded as the calculational tool of field theory, but it is more general. Using source theory, Schwinger was able to calculate the anomalous magnetic moment of the electron, which he had done in 1947, but this time with no 'distracting remarks' about infinite quantities. The neglect of source theory by the physics community was a major disappointment for Schwinger:

{\partial(\partial\phi/\partial t)}\right] + \sum_{i=1}^3 \frac{\partial}{\partial x^i} \left[\frac{\partial\mathcal{L{\partial(\partial\phi/\partial x^i)}\right] - \frac{\partial\mathcal{L{\partial\phi} = 0,</math>

we obtain the equations of motion for the field, which describe the way it varies in time and space:

<math display="block">\left(\frac{\partial^2}{\partial t^2} - \nabla^2 + m^2\right)\phi = 0.</math>

This is known as the Klein–Gordon equation.

The Klein–Gordon equation is a wave equation, so its solutions can be expressed as a sum of normal modes (obtained via Fourier transform) as follows:

<math display="block">\phi(\mathbf{x}, t) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{\mathbf{p\left(a_{\mathbf{p e^{-i\omega_{\mathbf{pt + i\mathbf{p}\cdot\mathbf{x + a_{\mathbf{p^* e^{i\omega_{\mathbf{pt - i\mathbf{p}\cdot\mathbf{x\right),</math>

where is a complex number (normalized by convention), denotes complex conjugation, and is the frequency of the normal mode:

<math display="block">\omega_{\mathbf{p = \sqrt{|\mathbf{p}|^2 + m^2}.</math>

Thus each normal mode corresponding to a single can be seen as a classical harmonic oscillator with frequency .

Canonical quantization

The quantization procedure for the above classical field to a quantum operator field is analogous to the promotion of a classical harmonic oscillator to a quantum harmonic oscillator.

The displacement of a classical harmonic oscillator is described by

<math display="block">x(t) = \frac{1}{\sqrt{2\omega a e^{-i\omega t} + \frac{1}{\sqrt{2\omega a^* e^{i\omega t},</math>

where is a complex number (normalized by convention), and is the oscillator's frequency. Note that is the displacement of a particle in simple harmonic motion from the equilibrium position, not to be confused with the spatial label of a quantum field.

For a quantum harmonic oscillator, is promoted to a linear operator <math>\hat x(t)</math>:

<math display="block">\hat x(t) = \frac{1}{\sqrt{2\omega \hat a e^{-i\omega t} + \frac{1}{\sqrt{2\omega \hat a^\dagger e^{i\omega t}.</math>

Complex numbers and are replaced by the annihilation operator <math>\hat a</math> and the creation operator <math>\hat a^\dagger</math>, respectively, where denotes Hermitian conjugation. The commutation relation between the two is

<math display="block">\left[\hat a, \hat a^\dagger\right] = 1.</math>

The Hamiltonian of the simple harmonic oscillator can be written as

<math display="block">\hat H = \hbar\omega \hat{a}^\dagger \hat{a} +\frac{1}{2}\hbar\omega.</math>

The vacuum state <math>|0\rang</math>, which is the lowest energy state, is defined by

and has energy <math>\frac12\hbar\omega.</math>

One can easily check that <math>[\hat H, \hat{a}^\dagger]=\hbar\omega\hat{a}^\dagger,</math> which implies that <math>\hat{a}^\dagger</math> increases the energy of the simple harmonic oscillator by <math>\hbar\omega</math>. For example, the state <math>\hat{a}^\dagger|0\rang</math> is an eigenstate of energy <math>3\hbar\omega/2</math>.

Any energy eigenstate state of a single harmonic oscillator can be obtained from <math>|0\rang</math> by successively applying the creation operator <math>\hat a^\dagger</math>: and any state of the system can be expressed as a linear combination of the states

<math display="block">|n\rang \propto \left(\hat a^\dagger\right)^n|0\rang.</math>

A similar procedure can be applied to the real scalar field , by promoting it to a quantum field operator <math>\hat\phi</math>, while the annihilation operator <math>\hat a_{\mathbf{p</math>, the creation operator <math>\hat a_{\mathbf{p^\dagger</math> and the angular frequency <math>\omega_\mathbf {p}</math>are now for a particular :

<math display="block">\hat \phi(\mathbf{x}, t) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{\mathbf{p\left(\hat a_{\mathbf{p e^{-i\omega_{\mathbf{pt + i\mathbf{p}\cdot\mathbf{x + \hat a_{\mathbf{p^\dagger e^{i\omega_{\mathbf{pt - i\mathbf{p}\cdot\mathbf{x\right).</math>

Their commutation relations are:

<math display="block">\left[\hat a_{\mathbf p}, \hat a_{\mathbf q}^\dagger\right] = (2\pi)^3\delta(\mathbf{p} - \mathbf{q}),\quad \left[\hat a_{\mathbf p}, \hat a_{\mathbf q}\right] = \left[\hat a_{\mathbf p}^\dagger, \hat a_{\mathbf q}^\dagger\right] = 0,</math>

where is the Dirac delta function. The vacuum state <math>|0\rang</math> is defined by

<math display="block">\hat a_{\mathbf p}|0\rang = 0,\quad \text{for all }\mathbf p.</math>

Any quantum state of the field can be obtained from <math>|0\rang</math> by successively applying creation operators <math>\hat a_{\mathbf{p^\dagger</math> (or by a linear combination of such states), e.g.

<math display="block">\left(\hat a_{\mathbf{p}_3}^\dagger\right)^3 \hat a_{\mathbf{p}_2}^\dagger \left(\hat a_{\mathbf{p}_1}^\dagger\right)^2 |0\rang.</math>

While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles. The latter space is known as a Fock space, which can account for the fact that particle numbers are not fixed in relativistic quantum systems. The process of quantizing an arbitrary number of particles instead of a single particle is often also called second quantization.

The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields, Dirac fields, vector fields (e.g. the electromagnetic field), and even strings. However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory). In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes. To perform calculations on any realistic interacting theory, perturbation theory would be necessary.

The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a quartic interaction term could be introduced to the Lagrangian of the real scalar field:

<math display="block">\mathcal{L} = \frac 12 (\partial_\mu\phi)\left(\partial^\mu\phi\right) - \frac 12 m^2\phi^2 - \frac{\lambda}{4!}\phi^4,</math>

where is a spacetime index, <math>\partial_0 = \partial/\partial t,\ \partial_1 = \partial/\partial x^1</math>, etc. The summation over the index has been omitted following the Einstein notation. If the parameter is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory.

Path integrals

The path integral formulation of QFT is concerned with the direct computation of the scattering amplitude of a certain interaction process, rather than the establishment of operators and state spaces. To calculate the probability amplitude for a system to evolve from some initial state <math>|\phi_I\rang</math> at time to some final state <math>|\phi_F\rang</math> at , the total time is divided into small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let be the Hamiltonian (i.e. generator of time evolution), then

<math display="block">\lang \phi_F|e^{-iHT}|\phi_I\rang = \int d\phi_1\int d\phi_2\cdots\int d\phi_{N-1}\,\lang \phi_F|e^{-iHT/N}|\phi_{N-1}\rang\cdots\lang \phi_2|e^{-iHT/N}|\phi_1\rang\lang \phi_1|e^{-iHT/N}|\phi_I\rang.</math>

Taking the limit , the above product of integrals becomes the Feynman path integral:

<math display="block">\lang \phi_F|e^{-iHT}|\phi_I\rang = \int \mathcal{D}\phi(t)\,\exp\left\{i\int_0^T dt\,L\right\},</math>

where is the Lagrangian involving and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian via Legendre transformation. The initial and final conditions of the path integral are respectively

In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand.

Two-point correlation function

In calculations, one often encounters expression like<math display="block">\lang 0|T\{\phi(x)\phi(y)\}|0\rang

\quad \text{or} \quad

\lang \Omega |T\{\phi(x)\phi(y)\}|\Omega \rang</math>in the free or interacting theory, respectively. Here, <math>x</math> and <math>y</math> are position four-vectors, <math>T</math> is the time ordering operator that shuffles its operands so the time-components <math>x^0</math> and <math>y^0</math> increase from right to left, and <math>|\Omega\rang</math> is the ground state (vacuum state) of the interacting theory, different from the free ground state <math>| 0 \rang</math>. This expression represents the probability amplitude for the field to propagate from to , and goes by multiple names, like the two-point propagator, two-point correlation function, two-point Green's function or two-point function for short.

The free two-point function, also known as the Feynman propagator, can be found for the real scalar field by either canonical quantization or path integrals to be

<math display="block">\lang 0|T\{\phi(x)\phi(y)\} |0\rang \equiv D_F(x-y) = \lim_{\epsilon\to 0} \int\frac{d^4p}{(2\pi)^4} \frac{i}{p_\mu p^\mu - m^2 + i\epsilon} e^{-ip_\mu (x^\mu - y^\mu)}.</math>

In an interacting theory, where the Lagrangian or Hamiltonian contains terms <math>L_I(t)</math> or <math>H_I(t)</math> that describe interactions, the two-point function is more difficult to define. However, through both the canonical quantization formulation and the path integral formulation, it is possible to express it through an infinite perturbation series of the free two-point function.

In canonical quantization, the two-point correlation function can be written as:

<math display="block">\lang\Omega|T\{\phi(x)\phi(y)\}|\Omega\rang = \lim_{T\to\infty(1-i\epsilon)} \frac{\left\lang 0\left|T\left\{\phi_I(x)\phi_I(y)\exp\left[-i\int_{-T}^T dt\, H_I(t)\right]\right\}\right|0\right\rang}{\left\lang 0\left|T\left\{\exp\left[-i\int_{-T}^T dt\, H_I(t)\right]\right\}\right|0\right\rang},</math>

where is an infinitesimal number and is the field operator under the free theory. Here, the exponential should be understood as its power series expansion. For example, in <math>\phi^4</math>-theory, the interacting term of the Hamiltonian is <math display="inline">H_I(t) = \int d^3 x\,\frac{\lambda}{4!}\phi_I(x)^4</math>, and the expansion of the two-point correlator in terms of <math>\lambda</math> becomes<math display="block">\lang\Omega|T\{\phi(x)\phi(y)\}|\Omega\rang =

\frac{

\displaystyle \sum_{n=0}^\infty \frac{(-i \lambda)^n}{(4 !)^n n !} \int d^4 z_1 \cdots \int d^4 z_n \lang 0|T\{\phi_I(x)\phi_I(y)\phi_I(z_1)^4\cdots\phi_I(z_n)^4\}|0\rang}{

\displaystyle \sum_{n=0}^\infty \frac{(-i \lambda)^n}{(4 !)^n n !} \int d^4 z_1 \cdots \int d^4 z_n \lang 0|T\{ \phi_I(z_1)^4\cdots\phi_I(z_n)^4\}|0\rang

}.</math>This perturbation expansion expresses the interacting two-point function in terms of quantities <math>\lang 0 | \cdots | 0 \rang</math> that are evaluated in the free theory.

In the path integral formulation, the two-point correlation function can be written

<math display="block">\lang\Omega|T\{\phi(x)\phi(y)\}|\Omega\rang = \lim_{T\to\infty(1-i\epsilon)} \frac{\int\mathcal{D}\phi\,\phi(x)\phi(y)\exp\left[i\int_{-T}^T d^4z\,\mathcal{L}\right]}{\int\mathcal{D}\phi\,\exp\left[i\int_{-T}^T d^4z\,\mathcal{L}\right]},</math>

where <math>\mathcal{L}</math> is the Lagrangian density. As in the previous paragraph, the exponential can be expanded as a series in , reducing the interacting two-point function to quantities in the free theory.

Wick's theorem further reduce any -point correlation function in the free theory to a sum of products of two-point correlation functions. For example,

<math display="block">\begin{align}

\lang 0|T\{\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\}|0\rang &= \lang 0|T\{\phi(x_1)\phi(x_2)\}|0\rang \lang 0|T\{\phi(x_3)\phi(x_4)\}|0\rang\\

&+ \lang 0|T\{\phi(x_1)\phi(x_3)\}|0\rang \lang 0|T\{\phi(x_2)\phi(x_4)\}|0\rang\\

&+ \lang 0|T\{\phi(x_1)\phi(x_4)\}|0\rang \lang 0|T\{\phi(x_2)\phi(x_3)\}|0\rang.

\end{align}</math>

Since interacting correlation functions can be expressed in terms of free correlation functions, only the latter need to be evaluated in order to calculate all physical quantities in the (perturbative) interacting theory. This makes the Feynman propagator one of the most important quantities in quantum field theory.

Feynman diagram

Correlation functions in the interacting theory can be written as a perturbation series. Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a Feynman diagram. For example, the term in the two-point correlation function in the theory is

<math display="block">\frac{-i\lambda}{4!}\int d^4z\,\lang 0|T\{\phi(x)\phi(y)\phi(z)\phi(z)\phi(z)\phi(z)\}|0\rang.</math>

After applying Wick's theorem, one of the terms is

<math display="block">12\cdot \frac{-i\lambda}{4!}\int d^4z\, D_F(x-z)D_F(y-z)D_F(z-z).</math>

This term can instead be obtained from the Feynman diagram

:200px.

The diagram consists of

external vertices connected with one edge and represented by dots (here labeled <math>x</math> and <math>y</math>).
internal vertices connected with four edges and represented by dots (here labeled <math>z</math>).
edges connecting the vertices and represented by lines.

Every vertex corresponds to a single <math>\phi</math> field factor at the corresponding point in spacetime, while the edges correspond to the propagators between the spacetime points. The term in the perturbation series corresponding to the diagram is obtained by writing down the expression that follows from the so-called Feynman rules:

For every internal vertex <math>z_i</math>, write down a factor <math display="inline">-i \lambda \int d^4 z_i</math>.
For every edge that connects two vertices <math>z_i</math> and <math>z_j</math>, write down a factor <math>D_F(z_i-z_j)</math>.
Divide by the symmetry factor of the diagram.

With the symmetry factor <math>2</math>, following these rules yields exactly the expression above. By Fourier transforming the propagator, the Feynman rules can be reformulated from position space into momentum space.

In order to compute the -point correlation function to the -th order, list all valid Feynman diagrams with external points and or fewer vertices, and then use Feynman rules to obtain the expression for each term. To be precise,

<math display="block">\lang\Omega|T\{\phi(x_1)\cdots\phi(x_n)\}|\Omega\rang</math>

is equal to the sum of (expressions corresponding to) all connected diagrams with external points. (Connected diagrams are those in which every vertex is connected to an external point through lines. Components that are totally disconnected from external lines are sometimes called "vacuum bubbles".) In the interaction theory discussed above, every vertex must have four legs.

In realistic applications, the scattering amplitude of a certain interaction or the decay rate of a particle can be computed from the S-matrix, which itself can be found using the Feynman diagram method.

Feynman diagrams devoid of "loops" are called tree-level diagrams, which describe the lowest-order interaction processes; those containing loops are referred to as -loop diagrams, which describe higher-order contributions, or radiative corrections, to the interaction. Lines whose end points are vertices can be thought of as the propagation of virtual particles.

Renormalization

Feynman rules can be used to directly evaluate tree-level diagrams. However, naïve computation of loop diagrams such as the one shown above will result in divergent momentum integrals, which seems to imply that almost all terms in the perturbative expansion are infinite. The renormalisation procedure is a systematic process for removing such infinities.

Parameters appearing in the Lagrangian, such as the mass and the coupling constant , have no physical meaning — , , and the field strength are not experimentally measurable quantities and are referred to here as the bare mass, bare coupling constant, and bare field, respectively. The physical mass and coupling constant are measured in some interaction process and are generally different from the bare quantities. While computing physical quantities from this interaction process, one may limit the domain of divergent momentum integrals to be below some momentum cut-off , obtain expressions for the physical quantities, and then take the limit . This is an example of regularization, a class of methods to treat divergences in QFT, with being the regulator.

The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant. A different approach, called renormalized perturbation theory, is to use physically meaningful quantities from the very beginning. In the case of theory, the field strength is first redefined:

where is the bare field, is the renormalized field, and is a constant to be determined. The Lagrangian density becomes:

<math display="block">\mathcal{L} = \frac 12 (\partial_\mu\phi_r)(\partial^\mu\phi_r) - \frac 12 m_r^2\phi_r^2 - \frac{\lambda_r}{4!}\phi_r^4 + \frac 12 \delta_Z (\partial_\mu\phi_r)(\partial^\mu\phi_r) - \frac 12 \delta_m\phi_r^2 - \frac{\delta_\lambda}{4!}\phi_r^4,</math>

where and are the experimentally measurable, renormalized, mass and coupling constant, respectively, and

<math display="block">\delta_Z = Z-1,\quad \delta_m = m^2Z - m_r^2,\quad \delta_\lambda = \lambda Z^2 - \lambda_r</math>

are constants to be determined. The first three terms are the Lagrangian density written in terms of the renormalized quantities, while the latter three terms are referred to as "counterterms". As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as follows. First select a regularization scheme (such as the cut-off regularization introduced above or dimensional regularization); call the regulator . Compute Feynman diagrams, in which divergent terms will depend on . Then, define , , and such that Feynman diagrams for the counterterms will exactly cancel the divergent terms in the normal Feynman diagrams when the limit is taken. In this way, meaningful finite quantities are obtained.

It is only possible to eliminate all infinities to obtain a finite result in renormalizable theories, whereas in non-renormalizable theories infinities cannot be removed by the redefinition of a small number of parameters. The Standard Model of elementary particles is a renormalizable QFT, while quantum gravity is non-renormalizable.

Renormalization group

The renormalization group, developed by Kenneth Wilson, is a mathematical apparatus used to study the changes in physical parameters (coefficients in the Lagrangian) as the system is viewed at different scales. The way in which each parameter changes with scale is described by its β function. Correlation functions, which underlie quantitative physical predictions, change with scale according to the Callan–Symanzik equation.

As an example, the coupling constant in QED, namely the elementary charge , has the following β function:

<math display="block">\beta(e) \equiv \frac{1}{\Lambda}\frac{de}{d\Lambda} = \frac{e^3}{12\pi^2} + O\mathord\left(e^5\right),</math>

where is the energy scale under which the measurement of is performed. This differential equation implies that the observed elementary charge increases as the scale increases. The renormalized coupling constant, which changes with the energy scale, is also called the running coupling constant.

The coupling constant in quantum chromodynamics, a non-Abelian gauge theory based on the symmetry group , has the following β function:

<math display="block">\beta(g) \equiv \frac{1}{\Lambda}\frac{dg}{d\Lambda} = \frac{g^3}{16\pi^2}\left(-11 + \frac 23 N_f\right) + O\mathord\left(g^5\right),</math>

where is the number of quark flavours. In the case where (the Standard Model has ), the coupling constant decreases as the energy scale increases. Hence, while the strong interaction is strong at low energies, it becomes very weak in high-energy interactions, a phenomenon known as asymptotic freedom.

Conformal field theories (CFTs) are special QFTs that admit conformal symmetry. They are insensitive to changes in the scale, as all their coupling constants have vanishing β function. (The converse is not true, however — the vanishing of all β functions does not imply conformal symmetry of the theory.) Examples include string theory and supersymmetric Yang–Mills theory.

According to Wilson's picture, every QFT is fundamentally accompanied by its energy cut-off , i.e. that the theory is no longer valid at energies higher than , and all degrees of freedom above the scale are to be omitted. For example, the cut-off could be the inverse of the atomic spacing in a condensed matter system, and in elementary particle physics it could be associated with the fundamental "graininess" of spacetime caused by quantum fluctuations in gravity. The cut-off scale of theories of particle interactions lies far beyond current experiments. Even if the theory were very complicated at that scale, as long as its couplings are sufficiently weak, it must be described at low energies by a renormalizable effective field theory. The difference between renormalizable and non-renormalizable theories is that the former are insensitive to details at high energies, whereas the latter do depend on them. According to this view, non-renormalizable theories are to be seen as low-energy effective theories of a more fundamental theory. The failure to remove the cut-off from calculations in such a theory merely indicates that new physical phenomena appear at scales above , where a new theory is necessary.

Other theories

The quantization and renormalization procedures outlined in the preceding sections are performed for the free theory and theory of the real scalar field. A similar process can be done for other types of fields, including the complex scalar field, the vector field, and the Dirac field, as well as other types of interaction terms, including the electromagnetic interaction and the Yukawa interaction.

As an example, quantum electrodynamics contains a Dirac field representing the electron field and a vector field representing the electromagnetic field (photon field). (Despite its name, the quantum electromagnetic "field" actually corresponds to the classical electromagnetic four-potential, rather than the classical electric and magnetic fields.) The full QED Lagrangian density is:

<math display="block">\mathcal{L} = \bar\psi\left(i\gamma^\mu\partial_\mu - m\right)\psi - \frac 14 F_{\mu\nu}F^{\mu\nu} - e\bar\psi\gamma^\mu\psi A_\mu,</math>

where are Dirac matrices, <math>\bar\psi = \psi^\dagger\gamma^0</math>, and <math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu</math> is the electromagnetic field strength. The parameters in this theory are the (bare) electron mass and the (bare) elementary charge . The first and second terms in the Lagrangian density correspond to the free Dirac field and free vector fields, respectively. The last term describes the interaction between the electron and photon fields, which is treated as a perturbation from the free theories.

thumb

Shown above is an example of a tree-level Feynman diagram in QED. It describes an electron and a positron annihilating, creating an off-shell photon, and then decaying into a new pair of electron and positron. Time runs from left to right. Arrows pointing forward in time represent the propagation of electrons, while those pointing backward in time represent the propagation of positrons. A wavy line represents the propagation of a photon. Each vertex in QED Feynman diagrams must have an incoming and an outgoing fermion (positron/electron) leg as well as a photon leg.

Gauge symmetry

If the following transformation to the fields is performed at every spacetime point (a local transformation), then the QED Lagrangian remains unchanged, or invariant:

<math display="block">\psi(x) \to e^{i\alpha(x)}\psi(x),\quad A_\mu(x) \to A_\mu(x) + ie^{-1} e^{-i\alpha(x)}\partial_\mu e^{i\alpha(x)},</math>

where is any function of spacetime coordinates. If a theory's Lagrangian (or more precisely the action) is invariant under a certain local transformation, then the transformation is referred to as a gauge symmetry of the theory. Gauge symmetries form a group at every spacetime point. In the case of QED, the successive application of two different local symmetry transformations <math>e^{i\alpha(x)}</math> and <math>e^{i\alpha'(x)}</math> is yet another symmetry transformation <math>e^{i[\alpha(x)+\alpha'(x)]}</math>. For any , <math>e^{i\alpha(x)}</math> is an element of the group, thus QED is said to have gauge symmetry. The photon field may be referred to as the gauge boson.

is an Abelian group, meaning that the result is the same regardless of the order in which its elements are applied. QFTs can also be built on non-Abelian groups, giving rise to non-Abelian gauge theories (also known as Yang–Mills theories). Quantum chromodynamics, which describes the strong interaction, is a non-Abelian gauge theory with an gauge symmetry. It contains three Dirac fields representing quark fields as well as eight vector fields representing gluon fields, which are the gauge bosons. The QCD Lagrangian density is:

<math display="block">\mathcal{L} = i\bar\psi^i \gamma^\mu (D_\mu)^{ij} \psi^j - \frac 14 F_{\mu\nu}^aF^{a,\mu\nu} - m\bar\psi^i \psi^i,</math>

where is the gauge covariant derivative:

<math display="block">D_\mu = \partial_\mu - igA_\mu^a t^a,</math>

where is the coupling constant, are the eight generators of in the fundamental representation ( matrices),

<math display="block">F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc}A_\mu^b A_\nu^c,</math>

and are the structure constants of . Repeated indices are implicitly summed over following Einstein notation. This Lagrangian is invariant under the transformation:

<math display="block">\psi^i(x) \to U^{ij}(x)\psi^j(x),\quad A_\mu^a(x) t^a \to U(x)\left[A_\mu^a(x) t^a + ig^{-1} \partial_\mu\right]U^\dagger(x),</math>

where is an element of at every spacetime point :

<math display="block">U(x) = e^{i\alpha(x)^a t^a}.</math>

The preceding discussion of symmetries is on the level of the Lagrangian. In other words, these are "classical" symmetries. After quantization, some theories will no longer exhibit their classical symmetries, a phenomenon called anomaly. For instance, in the path integral formulation, despite the invariance of the Lagrangian density <math>\mathcal{L}[\phi,\partial_\mu\phi]</math> under a certain local transformation of the fields, the measure <math display="inline">\int\mathcal D\phi</math> of the path integral may change. For a theory describing nature to be consistent, it must not contain any anomaly in its gauge symmetry. The Standard Model of elementary particles is a gauge theory based on the group , in which all anomalies exactly cancel.

The theoretical foundation of general relativity, the equivalence principle, can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the Lorentz group.

Noether's theorem states that every continuous symmetry, i.e. the parameter in the symmetry transformation being continuous rather than discrete, leads to a corresponding conservation law. For example, the symmetry of QED implies charge conservation.

Gauge-transformations do not relate distinct quantum states. Rather, it relates two equivalent mathematical descriptions of the same quantum state. As an example, the photon field , being a four-vector, has four apparent degrees of freedom, but the actual state of a photon is described by its two degrees of freedom corresponding to the polarization. The remaining two degrees of freedom are said to be "redundant" — apparently different ways of writing can be related to each other by a gauge transformation and in fact describe the same state of the photon field. In this sense, gauge invariance is not a "real" symmetry, but a reflection of the "redundancy" of the chosen mathematical description.

To account for the gauge redundancy in the path integral formulation, one must perform the so-called Faddeev–Popov gauge fixing procedure. In non-Abelian gauge theories, such a procedure introduces new fields called "ghosts". Particles corresponding to the ghost fields are called ghost particles, which cannot be detected externally. A more rigorous generalization of the Faddeev–Popov procedure is given by BRST quantization.

Spontaneous symmetry-breaking

Spontaneous symmetry breaking is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it.

To illustrate the mechanism, consider a linear sigma model containing real scalar fields, described by the Lagrangian density:

<math display="block">\mathcal{L} = \frac 12 \left(\partial_\mu\phi^i\right)\left(\partial^\mu\phi^i\right) + \frac 12 \mu^2 \phi^i\phi^i - \frac{\lambda}{4} \left(\phi^i\phi^i\right)^2,</math>

where and are real parameters. The theory admits an global symmetry:

<math display="block">\phi^i \to R^{ij}\phi^j,\quad R\in\mathrm{O}(N).</math>

The lowest energy state (ground state or vacuum state) of the classical theory is any uniform field satisfying

<math display="block">\phi_0^i \phi_0^i = \frac{\mu^2}{\lambda}.</math>

Without loss of generality, let the ground state be in the -th direction:

<math display="block">\phi_0^i = \left(0,\cdots,0,\frac{\mu}{\sqrt{\lambda\right).</math>

The original fields can be rewritten as:

<math display="block">\phi^i(x) = \left(\pi^1(x),\cdots,\pi^{N-1}(x),\frac{\mu}{\sqrt{\lambda + \sigma(x)\right),</math>

and the original Lagrangian density as:

<math display="block">\mathcal{L} = \frac 12 \left(\partial_\mu\pi^k\right)\left(\partial^\mu\pi^k\right) + \frac 12 \left(\partial_\mu\sigma\right)\left(\partial^\mu\sigma\right) - \frac 12 \left(2\mu^2\right)\sigma^2 - \sqrt{\lambda}\mu\sigma^3 - \sqrt{\lambda}\mu\pi^k\pi^k\sigma - \frac{\lambda}{2} \pi^k\pi^k\sigma^2 - \frac{\lambda}{4}\left(\pi^k\pi^k\right)^2,</math>

where . The original global symmetry is no longer manifest, leaving only the subgroup . The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken.

Goldstone's theorem states that under spontaneous symmetry breaking, every broken continuous global symmetry leads to a massless field called the Goldstone boson. In the above example, has continuous symmetries (the dimension of its Lie algebra), while has . The number of broken symmetries is their difference, , which corresponds to the massless fields .

On the other hand, when a gauge (as opposed to global) symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the amplitude for emission or absorption of a longitudinally polarized massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson.

In the QFT of ferromagnetism, spontaneous symmetry breaking can explain the alignment of magnetic dipoles at low temperatures. In the Standard Model of elementary particles, the W and Z bosons, which would otherwise be massless as a result of gauge symmetry, acquire mass through spontaneous symmetry breaking of the Higgs boson, a process called the Higgs mechanism.

Supersymmetry

All experimentally known symmetries in nature relate bosons to bosons and fermions to fermions. Theorists have hypothesized the existence of a type of symmetry, called supersymmetry, that relates bosons and fermions.

The Standard Model obeys Poincaré symmetry, whose generators are the spacetime translations and the Lorentz transformations . In addition to these generators, supersymmetry in (3+1)-dimensions includes additional generators , called supercharges, which themselves transform as Weyl fermions. The symmetry group generated by all these generators is known as the super-Poincaré group. In general there can be more than one set of supersymmetry generators, , which generate the corresponding supersymmetry, supersymmetry, and so on. Supersymmetry can also be constructed in other dimensions, most notably in (1+1) dimensions for its application in superstring theory.

The Lagrangian of a supersymmetric theory must be invariant under the action of the super-Poincaré group. Examples of such theories include: Minimal Supersymmetric Standard Model (MSSM), supersymmetric Yang–Mills theory, and superstring theory. In a supersymmetric theory, every fermion has a bosonic superpartner and vice versa.

If supersymmetry is promoted to a local symmetry, then the resultant gauge theory is an extension of general relativity called supergravity.

Supersymmetry is a potential solution to many current problems in physics. For example, the hierarchy problem of the Standard Model—why the mass of the Higgs boson is not radiatively corrected (under renormalization) to a very high scale such as the grand unified scale or the Planck scale—can be resolved by relating the Higgs field and its super-partner, the Higgsino. Radiative corrections due to Higgs boson loops in Feynman diagrams are cancelled by corresponding Higgsino loops. Supersymmetry also offers answers to the grand unification of all gauge coupling constants in the Standard Model as well as the nature of dark matter.

Nevertheless, experiments have yet to provide evidence for the existence of supersymmetric particles. If supersymmetry were a true symmetry of nature, then it must be a broken symmetry, and the energy of symmetry breaking must be higher than those achievable by present-day experiments.

Other spacetimes

The theory, QED, QCD, as well as the whole Standard Model all assume a (3+1)-dimensional Minkowski space (3 spatial and 1 time dimensions) as the background on which the quantum fields are defined. However, QFT a priori imposes no restriction on the number of dimensions nor the geometry of spacetime.

In condensed matter physics, QFT is used to describe (2+1)-dimensional electron gases. In high-energy physics, string theory is a type of (1+1)-dimensional QFT, The Lagrangian of a QFT, hence its calculational results and physical predictions, depends on the geometry of the spacetime background.

Topological quantum field theory

The correlation functions and physical predictions of a QFT depend on the spacetime metric . For a special class of QFTs called topological quantum field theories (TQFTs), all correlation functions are independent of continuous changes in the spacetime metric. QFTs in curved spacetime generally change according to the geometry (local structure) of the spacetime background, while TQFTs are invariant under spacetime diffeomorphisms but are sensitive to the topology (global structure) of spacetime. This means that all calculational results of TQFTs are topological invariants of the underlying spacetime. Chern–Simons theory is an example of TQFT and has been used to construct models of quantum gravity. Applications of TQFT include the fractional quantum Hall effect and topological quantum computers. The world line trajectory of fractionalized particles (known as anyons) can form a link configuration in the spacetime, which relates the braiding statistics of anyons in physics to the

link invariants in mathematics. Topological quantum field theories (TQFTs) applicable to the frontier research of topological quantum matters include Chern-Simons-Witten gauge theories in 2+1 spacetime dimensions, other new exotic TQFTs in 3+1 spacetime dimensions and beyond.

Perturbative and non-perturbative methods

Using perturbation theory, the total effect of a small interaction term can be approximated order by order by a series expansion in the number of virtual particles participating in the interaction. Every term in the expansion may be understood as one possible way for (physical) particles to interact with each other via virtual particles, expressed visually using a Feynman diagram. The electromagnetic force between two electrons in QED is represented (to first order in perturbation theory) by the propagation of a virtual photon. In a similar manner, the W and Z bosons carry the weak interaction, while gluons carry the strong interaction. The interpretation of an interaction as a sum of intermediate states involving the exchange of various virtual particles only makes sense in the framework of perturbation theory. In contrast, non-perturbative methods in QFT treat the interacting Lagrangian as a whole without any series expansion. Instead of particles that carry interactions, these methods have spawned such concepts as 't Hooft–Polyakov monopole, domain wall, flux tube, and instanton. Examples of QFTs that are completely solvable non-perturbatively include minimal models of conformal field theory and the Thirring model.

Mathematical rigor

In spite of its overwhelming success in particle physics and condensed matter physics, QFT itself lacks a formal mathematical foundation. For example, according to Haag's theorem, there does not exist a well-defined interaction picture for QFT, which implies that perturbation theory of QFT, which underlies the entire Feynman diagram method, is fundamentally ill-defined.

However, perturbative quantum field theory, which only requires that quantities be computable as a formal power series without any convergence requirements, can be given a rigorous mathematical treatment. In particular, Kevin Costello's monograph Renormalization and Effective Field Theory provides a rigorous formulation of perturbative renormalization that combines both the effective-field theory approaches of Kadanoff, Wilson, and Polchinski, together with the Batalin-Vilkovisky approach to quantizing gauge theories. Furthermore, perturbative path-integral methods, typically understood as formal computational methods inspired from finite-dimensional integration theory, can be given a sound mathematical interpretation from their finite-dimensional analogues.

Since the 1950s, theoretical physicists and mathematicians have attempted to organize all QFTs into a set of axioms, in order to establish the existence of concrete models of relativistic QFT in a mathematically rigorous way and to study their properties. This line of study is called constructive quantum field theory, a subfield of mathematical physics, which has led to such results as CPT theorem, spin–statistics theorem, and Goldstone's theorem, the three-dimensional scalar field theories with a quartic interaction, etc.

Compared to ordinary QFT, topological quantum field theory and conformal field theory are better supported mathematically — both can be classified in the framework of representations of cobordisms.

Algebraic quantum field theory is another approach to the axiomatization of QFT, in which the fundamental objects are local operators and the algebraic relations between them. Axiomatic systems following this approach include Wightman axioms and Haag–Kastler axioms. One way to construct theories satisfying Wightman axioms is to use Osterwalder–Schrader axioms, which give the necessary and sufficient conditions for a real time theory to be obtained from an imaginary time theory by analytic continuation (Wick rotation).

Yang–Mills existence and mass gap, one of the Millennium Prize Problems, concerns the well-defined existence of Yang–Mills theories as set out by the above axioms. The full problem statement is as follows.

References

Bibliography

External links

Stanford Encyclopedia of Philosophy: "Quantum Field Theory", by Meinard Kuhlmann.
Siegel, Warren, 2005. Fields. .
Quantum Field Theory by P. J. Mulders