Quantum field theory

The quantum field theory ( QFT ) is a field of theoretical physics, are combined (for example, classical electrodynamics ) and quantum mechanics to the formation of an extended theory in the principles of classical field theories. It goes beyond quantum mechanics by uniformly describes particles and fields. Not only so-called observables (ie, observable quantities such as energy or momentum) are quantized, but also the interacting ( particle ) fields themselves; Fields and observables are thus treated analogously. The quantization of the fields are also called second quantization. This takes into account explicitly the formation and annihilation of elementary particles ( pair production, annihilation).

The methods of quantum field theory are used primarily in elementary particle physics and statistical mechanics apply. A distinction is made between relativistic quantum field theories that take into account the special theory of relativity and are often found in elementary particle physics applications, and non-relativistic quantum field theories that are relevant, for example in solid state physics.

The objects and methods of QFT are physically motivated, although many branches of mathematics are used. The Axiomatic quantum field theory attempts, is finding its basics and concepts in a rigorous mathematical framework.

  • 3.1 Standard Model
  • 3.2 Theory
  • 3.3 Quantum Electrodynamics
  • 3.4 Weak interaction
  • 3.5 quantum
  • 4.1 Spontaneous symmetry breaking
  • 4.2 Axiomatic quantum field theory


The quantum field theory is a further development of quantum physics and quantum mechanics beyond. The pre-existing quantum theories were their construction according to theories for systems with few particles. In order to describe systems with many particles not a new theory is in principle necessary, but the description of 1023 particles in a solid body with the methods of quantum mechanics without approximations due to the high computational cost, technically impossible.

A fundamental problem of quantum mechanics is to describe their inability systems with varying number of particles. The first attempts to quantization of the electromagnetic field were aimed to describe the emission of photons by an atom. There is also according to the relativistic Klein-Gordon equation and the Dirac equation, the above mentioned anti-particle solutions. With sufficient energy, it is then possible to create particle-antiparticle pairs, which makes a system with constant number of particles impossible.

To solve these problems, treating the object, which has been interpreted in the quantum mechanical wave function as a particle, as a quantum box. That is, it is treated similar to an observable quantum mechanics. This not only solves the aforementioned problems, but also eliminates inconsistencies of classical electrodynamics, as arise, for example, in the Abraham - Lorentz equation. In addition, we obtain justifications for the Pauli principle and the general spin- statistics theorem.


The quantum field theories were originally developed as a relativistic scattering theories. In bound systems, the particle energies are generally much smaller than the mass energy mc2. Therefore, in such cases usually accurate enough to work in the non-relativistic quantum mechanics and perturbation theory. In collisions between small particles, however, much higher energies can occur, so that relativistic effects must be considered.

The following section explains the steps to develop a relativistic scattering theory are necessary. First, the Lagrangian is to set up, then the fields are quantized. Finally, a scattering theory is described and solved a problem that arises through the renormalization of the quantized fields.


The first step to a quantum field theory is to find Lagrangians for quantum fields. This Lagrangians must provide as Euler-Lagrange equation which is generally well-known differential equation for the field. These are the Dirac equation for the photon and the Maxwell equations for a scalar field, the Klein-Gordon equation for a spinor field.

In the following, always 4- ( space-time ) -vector notation will be used. Here, the usual short notations are used, namely the short notation for differentials and the Einstein summation convention, which states that summing over one upstairs and one below index ( 0-3 ). In the unit system used is valid.

It denotes the Dirac matrices. is called the adjoint spinor. are the components of Feldstärketensors. In this case here, the Maxwell's equations are used in the formulation without the source of covariant terms ( charge and current density).

The Lagrangians listed above describe free fields, which do not interact. They yield the equations of motion for free fields. For interactions of the fields among themselves Lagrangians additional terms must be added. It is important to note the following points:

Terms are allowed, for example, where m and n are natural numbers (including zero) and k is the coupling constant. Interactions with the photon are usually realized by the covariant derivative of () in the Lagrangian for the free field. In this case, the electric charge e of the electron is here at the same time, the coupling constant of the electromagnetic field.

Field quantization

So far, no statement about the properties of the fields was made. In strong fields with a large number of boson excitations these can be treated semi- classically, in general, but you have to first develop a mechanism to describe the effects of the quantum nature of the fields. The development of such a mechanism is referred to as field quantization and it is the first step to make the behavior of the fields calculated. There are two different formalisms that involve different approach.

  • The older canonical formalism is based on the formalism of quantum mechanics. It is well suited to the fundamental characteristics of the fields, the spin statistics theorem to show how. Its disadvantage, however, is that many aspects appear quite arbitrary in this formalism. Furthermore, the calculation of interaction amplitudes and the field quantization in non- Abelian gauge theories is quite complicated.
  • The recent path-integral formalism is based on the principle of least action, that is, it is integrated over all field configurations, but not canceling contributions come at weak coupling only of paths near the minima of the action. The advantage of this formalism is that represents the calculation of interaction amplitudes as a relatively simple and the symmetries of the fields clearly expressed. The serious lack of mathematical perspective of this formalism is that the convergence of the path integral and thus the functioning of the methods of formalism is not rigorously proved mathematically. It is therefore partly rejected particularly in mathematical physics as a heuristic and " imprecise " or " nichtkonstruktiv ", although he also serves as the starting point of the lattice gauge theories, which are one of the main tools of the numerical treatment of quantum field theories.

The following are the basics of field quantization of free fields are explained in the two formalisms.

Canonical formalism

For the field quantization in the canonical formalism one uses the Hamiltonian formalism of classical mechanics. It assigns it to each field (or ) to a canonically conjugated field analogous to the canonical momentum. The field and its canonically conjugated field are then in the sense of quantum mechanics conjugate operators, so-called field operators, and satisfy an uncertainty relation, like position and momentum in quantum mechanics. The uncertainty principle can be either through a Kommutatorrelation ( for bosons after the spin- statistics theorem ) or a Antikommutatorrelation be realized ( for fermions ) analogous to the commutator of position and momentum. This relation has a positive Hamiltonian arise because this characterizes the energy of the system and negative energies are to be avoided. The Hamiltonian is obtained by forming the Hamiltonian function and then replaces the fields by operators.

Scalar fields

For scalar fields is obtained to be canonical conjugate field to field and as canonically conjugated. The required Kommutatorrelation is

It is common in quantum field theories can be expected in momentum space. This purpose we consider the Fourier representation of the field operator, which reads for the scalar field

In this case, the pulse and the step function which is 0 for negative argument and otherwise 1. Here and operators are, this also applies to, and to. Your commutators follow from the commutator of the field operators. The operator can be interpreted as an operator that produces a particle with the pulse, while an anti-particle produced by pulse. Correspondingly, and are interpreted as operators that annihilate a particle or antiparticle with momentum. The use of the commutation relations leads as desired to a positive definite Hamiltonian. Any number of scalar fields in the same condition ( Bose -Einstein statistics ).

Spinor fields

If one proceeds analogously for a spinor field, you get to as canonical conjugate field to field and as canonically conjugated. Thus, the required (anti) commutation relations arise to

Here, and Spinorindizes. Is then considered again analogous Fourier representation of the field operator, and calculates the Hamiltonian. A positive Hamiltonian, however, is obtained when spinor field only if you use anticommutators. These are written with curly braces, which was already anticipated in the above formulas. Because of this anticommutators the two applications of the same generation operator gives to a state of the zero state. This means that can never be two Spin-1/2-Teilchen in the same state ( Pauli principle). Therefore spinor fields obey Fermi- Dirac statistics.

Gauge fields

For gauge fields are the required commutation relations

Wherein the components of the Minkowski metric respectively. However, is obtained from the Lagrangian, which can not meet the required Kommutatorrelation. The quantization of gauge fields is possible only when defining a gauge condition. Establishing a suitable gauge condition, which allows access via commutation relations of fields, while maintaining Lorentz invariance of the Lagrangian is complicated. It usually uses a modification of the Lorenz gauge in order to make sense to define a canonically conjugated field can. The formalism is called after its developers Suraj N. Gupta and Konrad Bleuler as Gupta - Bleuler formalism.

Path integral

In the path integral formalism, the fields are not treated as operators, but as ordinary functions. The integral path is essentially a transition amplitude of a vacuum state at the time in a vacuum state at the time is, with all the possible intermediate configurations field (paths) is integrated by a phase factor that is determined by the effect. It has for the scalar field in the form

To even get though interactions with a transition from the vacuum to the vacuum fields can be created and destroyed. This is achieved not by using creation and annihilation operators, but by source fields in the path integral formalism. So it is added to the Lagrangian of a source term of the form. The source field J (x ) is to be different only in a finite interval on the time axis of zero. This means that the interacting fields exactly exist within this time interval. The full path integral for a free scalar field thus has the form

This can be due to the integration over an analogue of the Gaussian error integral bring in a form that depends in a certain way, only from the source field J (x ), namely:

It is given by so to speak, as the inverse of the Klein-Gordon operator. This object is called the time-ordered Green function or Feynman propagator. This is referred to as a path integral therefore generating functional of the propagator, since the derivations is correct to and effectively a multiplication by the propagator.

The behavior of the open field in the presence of sources is determined only by the propagator and the Source field. This result is as expected, because the behavior of a field that does not interact, is apparently determined only by the properties of the production and destruction and its free movement. The former stuck in the Source field and the movement behavior is determined by the Klein-Gordon operator, whose information content is here given by its inverse.

In the quantization of the path integral Spinorfeldes formalism, the problem occurs that the fields are treated as normal one hand zahlenwertige functions anticommute on the other side, however. Normal numbers commute, however. This difficulty can be solved by the Fermionfelder as elements of a Grassmann algebra, so-called Grassmann numbers, conceives. Mathematically, this means only that they are treated as anticommuting numbers. Through the Grassmann algebra, this approach is theoretically secured. The path integral with source fields and then has the form

This suggests, as in the scalar field, derive a form that only by in a certain way and depends. It can be again an analogue of the Gaussian integral apply that does not correspond to the usual formalism, but in some way toward "inverse" is. First, it is certainly necessary to develop an integral term for Grassmann numbers. Then the path integral can be put in the following form:

It is the inverse of the Dirac - operator, which is also referred to as Dirac propagator. Analogous to the scalar field also arises here a form which is expected to be determined only by the source fields and the dynamics of the fields.

The path integral for a gauge field is of the form

However, the operator has no inverse. The recognized by the fact that he gives when applied to vectors of type zero. At least one of its eigenvalues ​​is therefore zero, similar to a matrix ensures that the operator is not invertible.

Therefore, can not here the same procedure apply as in the scalar field and the spinor field. You have to add an additional term to the Lagrangian, so that one obtains an operator, for which there is an inverse. This is equivalent to define a calibration. Therefore, we designated the new term as eichfixierenden Term He is generally of the form. The corresponding thereto gauge condition is.

This, however, means that the Lagrangian depends on the choice of the calibration term f. This problem can be remedied by the introduction of so-called Faddejew - Popov ghosts. These ghosts are anticommuting scalar fields and thus contradict the spin- statistics theorem. Therefore, they can not act as free fields, but only as a so-called virtual particles. By choosing the so-called axial gauge, the incidence of these fields can be avoided, which would appear obvious as mathematical artifacts their interpretation. Their occurrence in other calibrations, however, from deeper theoretical reasons ( unitarity of the S- matrix ) is absolutely necessary for the consistency of the theory.

The full Lagrangian with eichfixierendem Term and mind fields depends on the gauge condition. For the Lorenz gauge it reads in non -Abelian gauge theories

It is the mind field and the anti - ghost field.

For abelian gauge theories such as electromagnetism, the last term takes regardless of the calibration to the mold. Therefore, this part of the path integral can be easily integrated and does not contribute to the dynamics.

The path integral also provides a connection with the distribution functions of statistical mechanics. To this end, the imaginary time coordinate in Minkowski space is analytically continued to the Euclidean space and instead of complex phase factors in the path integral is obtained real similar to the Boltzmann factors of statistical mechanics. In this form, this formulation is also the starting point of numerical simulations of field configurations (usually randomly selected in the Monte - Carlo method with a weighting of these Boltzmann factors ) in lattice calculations. They provide the most precise so far methods eg for the calculation of hadron masses in quantum chromodynamics.

Scattering processes

As already stated above, the aim of the preceding method is the description of a relativistic scattering theory. Although the methods of quantum field theories now also be used in other contexts, the scattering theory is still one of its main applications, so the basics are explained in the same at this point.

The central object of the scattering theory is the so-called scattering matrix S or matrix, the elements of which describe the probability of transition from an initial state to an initial state. The elements of the S matrix is referred to as the scattering amplitude. At the level of the fields, the S- matrix is thus determined by the equation

The S- matrix can essentially as a sum of vacuum expectation values ​​of time-ordered field operator products (including n- point functions, correlators or Green's functions referred to ) write. A proof of this so-called LSZ - decomposition is one of the first great successes of the axiomatic quantum field theory. In the example of a quantum field theory in which there is only one scalar field, the decomposition has the form

Here K is the Klein-Gordon operator, and T is the time ordering operator, the ascending orders the fields according to the value of time. If there happen to fields other than the scalar field, respectively, the corresponding Hamiltonians must be used. For a spinor field must be used, for example, the Dirac operator instead of the Klein-Gordon operator.

Thus, the calculation of the S- matrix, it is sufficient to calculate the time-ordered n-point functions.

Feynman rules and perturbation theory

As a useful tool to simplify the calculations of the n-point functions, the Feynman diagrams have been established. This shorthand notation was developed in 1950 by Richard Feynman and exploits that can be the terms that occur in the calculation of n- point functions, broken down into a small number of elementary building blocks. This term blocks pixels are then assigned. These rules according to which this assignment is done is referred to as Feynman rules. The Feynman diagrams allow thus to represent complicated terms in the form of small images.

In this case, there exists for every term in the Lagrangian a corresponding pixel. The mass term is treated together with the derivative term as a term that describes the free field. These terms are usually associated with different lines for different fields. Contrast, the interaction terms correspond to nodes, called vertices, where for each field that is in the interaction term, an appropriate line ends. Lines, which are connected at one end only with the chart are interpreted as real particles while lines connecting two vertices are interpreted as virtual particles. It can also be a time direction in the diagram set so that it can be interpreted as a type illustration of the scattering process. However, one must consider the complete computation of a given scattering amplitude all diagrams with the corresponding initial and final particles. If the Lagrangian quantum field theory contains interaction terms, these are infinitely many diagrams in general.

If the coupling constant is less than one, the terms with higher powers of the coupling constant become smaller. Since, according to the Feynman rules, each vertex stands for the multiplication with the corresponding coupling constant, the contributions of diagrams with many vertices are very small. The simplest diagrams thus make the greatest contribution to the scattering amplitude, while the diagrams with increasing complexity always provide smaller contributions simultaneously. In this manner, the principles of the perturbation can be applied while achieving good results for the scattering amplitudes by using only the low-order plots of the coupling constants can be calculated.


The Feynman diagrams with closed inner lines, the so-called loop diagrams (eg interaction of an electron with a "virtual" photons from the vacuum, the interaction of a photon with a virtual particle-antiparticle pairs created from the vacuum ), are usually divergent, since about all energy / momentum ( frequency / wavenumber ) is integrated. This has the consequence that can not be more complicated Feynman diagrams first calculate. This problem can be often described by a so-called renormalization fix, after an incorrect re-translation from English sometimes called " renormalization ".

There are basically two different ways of looking at this procedure. The first traditional view assigns the contributions of divergent loop diagrams in such a way that they correspond to a few parameters in the Lagrangian such as masses and coupling constants. One then proceeds counter- terms ( counter terms) in the Lagrangian, which cancel as an infinite "bare " values ​​of these parameters, these divergences. This is possible in quantum electrodynamics, as well as in quantum chromodynamics and other such gauge theories with other theories such as gravitation is not. There are many counter terms would be infinitely necessary, the theory is " not renormalizable ".

A second more recent view of the environment of the renormalization group describes the physics, depending on the energy sector through various "effective" field theories. For example, the coupling constant in quantum chromodynamics is energy-dependent, for small energies it tends to infinity ( confinement ) for high energies to zero ( asymptotic freedom ). While in QED the " bare " charges are effectively shielded by the vacuum polarization (pair creation and destruction ), in the case of Yang-Mills theories like QCD is more complicated because of the self-interaction of the charged gauge bosons.

It is believed that all coupling constants of physical theories converge at sufficiently high energies, and there the physics is then described by a grand unified theory of the fundamental forces. The behavior of the coupling constants and the possibility of phase transitions of energy is described by the theory of the renormalization. From such theoretical extrapolations, there have been in the 1990s first evidence for the existence of supersymmetric theories, for best meet the coupling constants at one point.

However, the technical procedure is independent of the view. It is first made ​​by an additional parameter is introduced in the calculation a RänderungRegularisierung. This parameter must be last again to zero or run indefinitely (depending on choice) to the original terms again to get. As long as the regularization parameter is assumed to be finite, the terms remain finite. One then forms the terms so that the infinities occur only in terms that are pure functions of the regularization. These terms are then omitted. Now set the regulation parameter zero or infinity, the result now remains finite.

This approach seems arbitrary at first glance, but the " omission " must be rules. This ensures that the renormalized coupling constants corresponding to the measured constants at low energies.


A special field of relativistic quantum mechanics relates solutions of the relativistic Klein-Gordon equation and the Dirac equation with negative energy. This would allow the particles to descend to infinite negative energy, which is not observed in the real world. In quantum mechanics to solve this problem by arbitrarily interpreted the solutions as entities with positive energy that move backwards in time; Thus, it transmits the wave function of the negative sign of the energy E to the time t, which is obvious because of the relation (h is Planck's constant and the energy difference associated with the frequency interval ).

Paul Dirac interpreted this backward moving solutions as antiparticles.

Concrete quantum field theories

Standard Model

By combining the electroweak model with the quantum result is a unified quantum field theory, the so-called Standard Model of elementary particle physics. It contains all the known particles and can explain most of the known processes.

At the same time it is known that the Standard Model can not be the final theory. On the one hand, gravity is not included, on the other hand there are a number of observations ( neutrino oscillations, dark matter ), after which an extension of the standard model seems necessary. In addition, the standard model contains many arbitrary parameters and explains, for example, not very different mass spectrum of elementary families.

Explained in the following quantum field theories are all included in the standard model.


The Lagrangian of the theory is

This quantum field theory is of great theoretical significance, since it is the simplest conceivable quantum field theory with an interaction and, in contrast to more realistic models, some exact mathematical statements can be made about their properties here. It describes a self- interacting real or complex scalar field.

In statistical physics, it plays a role as the simplest continuum model for the (very general ) Landau theory of second-order and critical phenomena Phase transitions. From the statistical interpretation, one gets at the same time a new and constructive approach to Renormierungsproblem by showing that the renormalization of the masses, charges and vertex functions by eliminating short-wave wave phenomena of the so-called state sum (English: "Partition Function" ) can be achieved can. Also the Higgs field of the standard model has a self- interaction, which is, however, supplemented by interactions with the other fields of the Standard Model. In these cases, the coupling constant is m2 negative, which would correspond to an imaginary ground. These fields are therefore referred to as tachyonische fields. However, this term refers to the Higgs field and not on the Higgs particle, called the Higgs boson, which is not a tachyon, but an ordinary particles with real mass. The Higgs particle is not described by the Higgs field but only by a certain portion of that field.

Quantum electrodynamics

The Lagrangian of quantum electrodynamics ( QED) is

The QED is the first physically successful quantum field theory. Describes the interaction of a Spinorfeldes with charge -e, which describes the electron with a calibration field, which describes the photon. This gives their equations of motion of electrodynamics by quantization of Maxwell's equations. The quantum electrodynamics explains with high accuracy the electromagnetic interaction between charged particles ( eg, electrons, muons, quarks ) by the exchange of virtual photons as well as the properties of electromagnetic radiation.

This can be about the chemical elements, their properties and bonds and the periodic table of elements to understand. The solid-state physics with the economically important semiconductor physics are derived ultimately from the QED. Concrete calculations are, however, carried out usually in the simplified but adequate formalism of quantum mechanics.

Weak interaction

The weak interaction, whose best-known effect of beta decay, a physically closed formulation assumes for unification with the QED in the electroweak standard model. The interaction is mediated by photons here, W and Z bosons.

Quantum chromodynamics

Another example of a QFT is the quantum chromodynamics ( QCD), which describes the strong interactions. In it, a part of which occur in the atomic nucleus interactions between protons and neutrons is reduced to the sub-nuclear interaction between quarks and gluons.

It is interesting in QCD that the gluons, which mediate the interaction, even interact with each other. ( That would be the example of QED about as if two penetrating rays of light would affect directly. ) Is a consequence of the gluonic self-interaction, that the elementary quarks can not be observed separately, but always in the form of quark- antiquark states or states three quarks (or antiquarks ) occur ( confinement). On the other hand, it follows that the coupling constant does not increase at high energies, but reduced. This behavior is called asymptotic freedom.

Further aspects

Spontaneous symmetry breaking

As already mentioned above, the theory is suitable for the description of systems with spontaneous symmetry breaking or critical points. The mass term is to be understood as part of the potential. For a real mass of this potential has then only a minimum while at imaginary mass the potential describes a w -shaped parabola of the fourth degree. If the field has more than one real component, is obtained more minima. For a complex field (with two real components ) is obtained, for example, the figure of rotation of the w- shaped parabola with a Minimakreis. This shape is also referred to as a Mexican Hat potential as the potential is reminiscent of the shape of a sombrero.

Each minimum now corresponds to a state of lowest energy, all of which are accepted by the field with equal probability. In each of these conditions, however, the field has a lower degree of symmetry, because the symmetry of the minima is lost with each other by selecting a minimum. This characteristic of the classical field theory carries over to the quantum field theory, so that there is the possibility of describing quantum systems with broken symmetry. Examples of such systems are the Ising model from thermodynamics that explains the spontaneous magnetization of a ferromagnet, and the Higgs mechanism to explain the masses of the gauge bosons in the weak interaction. Due to the resulting mass terms of the gauge bosons, the gauge symmetry is reduced viz.

Axiomatic quantum field theory

The Axiomatic quantum field theory attempts, starting from a set of possible less than mathematically or physically inevitable respected axioms, to obtain a consistent description of quantum field theory.

The axiomatic quantum field theory was, inter alia, from the Wightman axioms, created in 1956, founded. Another approach is that of Hague and Araki 1962 formulated algebraic quantum field theory, which is characterized by the Haag- Kastler axioms. The Osterwalder - Schrader axioms provide a third axiomatic approach to quantum field theory dar.

Some concrete results could be achieved with this approach, for example, the derivation of the spin-statistics theorem and the CPT theorem alone from the axioms, ie independent of a specific quantum field theory. An early success was developed in 1955 by Lehmann, Symanzik and Zimmermann LSZ reduction formula for the S- matrix. In addition, there is a reasonable by Bogoliubov, Medvedev and Polianov functional analytical approach to S- matrix theory exists (also called BMP - theory ).

Other applications in the field of classical statistics and quantum statistics are already well advanced. They range from the general derivation of the existence of thermodynamic quantities, Gibbs set of state variables such as pressure, internal energy and entropy to the proof of the existence of phase transitions and the exact discussion of important many-body systems:

  • Of the Bardeen -Cooper - Schrieffer model of superconductivity
  • The Heisenberg ferromagnet

Relation to other theories

Attempts to unite these quantum field theories with general relativity ( gravity ) to quantum gravity, have so far been unsuccessful. According to many researchers the quantization of gravity requires new, quantum field theory beyond concepts, since the space-time background is dynamically self. Examples of current research are the string theory, M-theory and loop quantum gravity. Next supersymmetry, the twistor theory and the finite quantum field theory provide important conceptual ideas that are currently being discussed in the professional world.

Also found in the solid state theory applications of the ( non-relativistic ) quantum field theory, mainly in the many-body theory.