# Quantum field theory

The quantum field theory ( QFT ) is an area of theoretical physics , in the principles of classic field theory (for example, the classical electrodynamics ) and the quantum mechanics are combined to form an expanded theory. It goes beyond quantum mechanics in that it describes particles and fields in a uniform manner. Not only so-called observables (i.e. observable quantities such as energy or momentum ) are quantized , but also the interacting (particle) fields themselves; Fields and observables are treated in the same way. The quantization of the fields is also known as the second quantization . This explicitly takes into account the creation and annihilation of elementary particles ( pair creation , annihilation ).

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

The objects and methods of QFT are physically motivated, even if many sub-areas of mathematics are used. The axiomatic quantum field theory tries, is finding its principles and concepts in a mathematically rigorous framework.

## motivation

The quantum field theory is a further development of quantum physics beyond quantum mechanics . The pre-existing quantum theories were structured according to theories for systems with few particles. In principle, no new theory is necessary to describe systems with many particles, but the description of e.g. 10 23 particles in a solid is technically impossible with the methods of quantum mechanics without approximations due to the high computational effort.

A fundamental problem in quantum mechanics is its inability to describe systems with varying numbers of particles. The first attempts to quantize the electromagnetic field aimed to describe the emission of photons by an atom. In addition, according to the relativistic Klein-Gordon equation and the Dirac equation, there are the above-mentioned antiparticle solutions. With sufficient energy it is then possible to generate particle-antiparticle pairs, which makes a system with a constant number of particles impossible.

To solve these problems, the object, which in quantum mechanics was interpreted as the wave function of a particle, is treated as a quantum field . This means that it is treated similarly to an observable in quantum mechanics. This not only solves the problems mentioned above, but also eliminates inconsistencies in classical electrodynamics, such as radiation feedback . In addition, one receives reasons for the Pauli principle and the more general spin statistics theorem .

## Basics

The quantum field theories were originally developed as relativistic scattering theories . In bound systems, the particle energies are generally significantly smaller than the mass energies mc 2 . Therefore, in such cases it is usually sufficiently precise to work with perturbation theory in non-relativistic quantum mechanics . In the case of collisions between small particles, however, much higher energies can occur, so that relativistic effects must be taken into account.

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

### Lagrangian

The first step to a quantum field theory is to find Lagrangian for the quantum fields. These Lagrangian densities must supply the generally known differential equation for the field as the Euler-Lagrange equation . These are the Klein-Gordon equation for a scalar field , the Dirac equation for a spinor field, and the Maxwell equation for the photon .

In the following, the 4-way (space-time) vector notation is used. The usual abbreviations are used, namely the abbreviation for differentials and Einstein's summation convention , which states that summing is carried out using an index above and below (from 0 to 3). As used unit system applies: . ${\ displaystyle \ textstyle \ partial _ {\ mu} = {\ frac {\ partial} {\ partial x ^ {\ mu}}}}$${\ displaystyle c = \ hbar = \ varepsilon _ {0} = 1}$

Lagrangian poems of various fields
field Field equation Lagrangian
Scalar (spin = 0) ${\ displaystyle \ phi \}$ ${\ displaystyle 0 = (\ square + m ^ {2}) \ phi}$ ${\ displaystyle {\ mathcal {L}} = (\ partial _ {\ mu} \ phi ^ {\ dagger}) (\ partial ^ {\, \ mu} \ phi) -m ^ {2} \ phi ^ { \ dagger} \ phi}$
Spinor (spin = 1/2) ${\ displaystyle \ psi \}$ ${\ displaystyle 0 = (i \ gamma ^ {\ mu} \ partial _ {\ mu} -m) \ psi}$ ${\ displaystyle {\ mathcal {L}} = {\ tfrac {i} {2}} \ left ({\ overline {\ psi}} \ gamma ^ {\ mu} (\ partial _ {\ mu} \ psi) - (\ partial _ {\ mu} {\ overline {\ psi}}) \ gamma ^ {\ mu} \ psi \ right) -m {\ overline {\ psi}} \ psi}$
Photon (spin 1) ${\ displaystyle A ^ {\ mu} \}$ ${\ displaystyle 0 = \ partial _ {\ mu} F ^ {\ mu \ nu} = \ square A ^ {\ nu} - \ partial ^ {\ nu} (\ partial _ {\ mu} A ^ {\ mu })}$ ${\ displaystyle {\ mathcal {L}} = - {\ tfrac {1} {4}} F _ {\ mu \ nu} F ^ {\ mu \ nu} = - {\ tfrac {1} {4}} ( \ partial _ {\ mu} A _ {\ nu} - \ partial _ {\ nu} A _ {\ mu}) (\ partial ^ {\ mu} A ^ {\ nu} - \ partial ^ {\ nu} A ^ {\ mu})}$

This denotes the Dirac matrices . is the so-called adjoint spinor. are the components of the field strength tensor . The Maxwell equations were used here in a covariant formulation without the source terms (charge and current density). ${\ displaystyle \ gamma ^ {\ mu}}$${\ displaystyle {\ overline {\ psi}} = \ psi ^ {\ dagger} \ gamma ^ {0}}$${\ displaystyle F _ {\ mu \ nu} = \ partial _ {\ mu} A _ {\ nu} - \ partial _ {\ nu} A _ {\ mu}}$

The Lagrangians listed above describe free fields that do not interact. They give the equations of motion for free fields. For interactions between the fields, additional terms must be added to the Lagrangian. Pay attention to the following points:

1. The added terms must all be scalar . This means that they are invariant under Poincaré transformations .
2. The added terms must have the dimension (length) −4 , since the Lagrangian is integrated in the scalar action over space-time. If necessary, this can be achieved using a constant factor with a suitable dimension. Such factors are called coupling constants .
3. In the case of interactions of calibration fields such as the photon with other fields, the Lagrangian must be calibration covariant. This means that the form of the Lagrangian under gauge transformations must not change.

Allowed terms are, for example, where m and n are natural numbers (including zero) and k is the coupling constant. Interactions with the photon are mostly realized by the covariant derivative ( ) in the Lagrangian for the free field. The electric charge e of the electron is also the coupling constant of the electromagnetic field. ${\ displaystyle k ({\ overline {\ psi}} \ psi) ^ {n} (\ phi ^ {\ dagger} \ phi) ^ {m} \ ,,}$${\ displaystyle \ partial _ {\ mu} \ rightarrow \ partial _ {\ mu} + ieA _ {\ mu}}$

### Field quantization

So far, no statement has been made about the properties of the fields. In the case of strong fields with a large number of boson excitations, these can be treated semi-classically, but in general a mechanism must first be developed to describe the effects of the quantum nature of the fields. The development of such a mechanism is called field quantization and it is the first step towards making the behavior of the fields predictable. There are two different formalisms that involve different procedures.

• The older canonical formalism is based on the formalism of quantum mechanics. He interprets the single-particle wave equations occurring there as the descriptions of amplitudes of a classical field theory, which in turn require a quantization according to the canonical commutation rules of quantum mechanics. The formalism is therefore suitable for showing fundamental properties of the fields, such as the spin statistics theorem . Its disadvantage, however, is that many aspects appear quite arbitrary in this formalism. In addition, the calculation of interaction amplitudes and field quantization in non-Abelian gauge theories are quite complicated.
• The newer path integral formalism is based on the principle of the smallest effect , that is, it is integrated across all field configurations, but contributions that do not cancel each other come only from paths close to the minima of the effect with weak coupling. The advantage of this formalism is that the calculation of interaction amplitudes is comparatively simple and the symmetries of the fields are clearly expressed. The serious shortcoming of this formalism from a mathematical point of view is that the convergence of the path integral and thus the functioning of the methods of the formalism has not been strictly mathematically proven. It is therefore partly rejected as heuristic and “imprecise” or “non-constructive”, especially in mathematical physics , although it also serves as the starting point for the lattice range theories , which are one of the main tools in the numerical treatment of quantum field theories.

The basics of field quantization for free fields in both formalisms are explained below.

#### Canonical formalism

For the field quantization in the canonical formalism, the Hamilton formalism of classical mechanics is used. Each field ( or ) is assigned a canonically conjugate field analogous to the canonical momentum. The field and its canonically conjugated field are then conjugate operators in the sense of quantum mechanics, so-called field operators , and fulfill an uncertainty relation , such as position and momentum in quantum mechanics. The uncertainty relation can be realized either by a commutator relation (for bosons according to the spin statistics theorem ) or an anti- commutator relation (for fermions) analogous to the commutator of position and momentum. The Hamilton operator , which characterizes the energy of the system, is obtained by forming the Hamilton function and replacing the fields with the field operators. As a rule, it is positively definite or at least must not have any unlimited negative eigenvalues, since such a system would fall into ever deeper energy eigenstates with any amount of energy given to the environment. ${\ displaystyle \ phi}$${\ displaystyle \ psi}$ ${\ displaystyle \ pi}$

##### Scalar fields

For scalar fields, the canonically conjugated field is zu and the canonically conjugated field is zu . The required commutator relation is ${\ displaystyle \ pi = \ partial _ {0} \ phi ^ {\ dagger}}$${\ displaystyle \, \ phi}$${\ displaystyle \ pi ^ {\ dagger} = \ partial _ {0} \ phi}$${\ displaystyle \ phi ^ {\ dagger} \}$

${\ displaystyle [\ phi ({\ vec {x}}, t), \ pi ({\ vec {y}}, t)] = [\ phi ^ {\ dagger} ({\ vec {x}}, t), \ pi ^ {\ dagger} ({\ vec {y}}, t)] = i \ delta ^ {(3)} ({\ vec {x}} - {\ vec {y}}). }$

It is common in quantum field theories to calculate in momentum space . To do this, consider the Fourier representation of the field operator, which reads for the scalar field

${\ displaystyle \ phi (x) = \ int {\ frac {\ mathrm {d} ^ {4} k} {(2 \ pi) ^ {4}}} 2 \ pi \ delta (k ^ {2} - m ^ {2}) \ theta (k_ {0}) \ left [a (k) e ^ {- ikx} + b ^ {\ dagger} (k) e ^ {ikx} \ right].}$

Here, the pulse and the step function , which is for negative argument and 0 otherwise. 1 As and operators, this also applies to , , and to. Their commutators follow from the commutator of the field operators. The operator can be interpreted as an operator that produces a particle with momentum while producing an antiparticle with momentum . Correspondingly, and can be interpreted as operators that destroy a particle or antiparticle with momentum . The use of the commutator relations leads, as desired, to a positive definite Hamilton operator. Any number of scalar fields can be in the same state ( Bose-Einstein statistics ). ${\ displaystyle \, k}$${\ displaystyle \, \ theta (k_ {0})}$${\ displaystyle \, \ phi (x)}$${\ displaystyle \, \ phi ^ {\ dagger} (x)}$${\ displaystyle \, a (k)}$${\ displaystyle a ^ {\ dagger} (k)}$${\ displaystyle \, b (k)}$${\ displaystyle b ^ {\ dagger} (k)}$${\ displaystyle a ^ {\ dagger} (k)}$${\ displaystyle \, k}$${\ displaystyle b ^ {\ dagger} (k)}$${\ displaystyle \, k}$${\ displaystyle \, a (k)}$${\ displaystyle \, b (k)}$${\ displaystyle \, k}$

##### Spinor fields

If you proceed in the same way for a spinor field , you get a canonically conjugated field zu and a canonically conjugated field zu . This results in the required (anti) commutator relations ${\ displaystyle \ pi = i \ psi ^ {\ dagger} \}$${\ displaystyle \ psi \}$${\ displaystyle {\ overline {\ pi}} = i \ gamma ^ {0} \ psi}$${\ displaystyle {\ overline {\ psi}} \}$

${\ displaystyle \ {\ psi _ {j} ({\ vec {x}}, t), \ pi _ {k} ({\ vec {y}}, t) \} = \ {{\ overline {\ psi}} _ {j} ({\ vec {x}}, t), {\ overline {\ pi}} _ {k} ({\ vec {y}}, t) \} = i \ delta _ { jk} \ delta ^ {(3)} ({\ vec {x}} - {\ vec {y}}).}$

Where and are spinor indices. The Fourier representation of the field operator is then again considered analogously and the Hamilton operator is calculated. A positive Hamilton operator can only be obtained for the spinor field if anti-commutators are used. These are written with curly brackets, which has already been anticipated in the formulas above. Because of these anti-commutators, applying the same generation operator twice to a state results in the zero state. This means that two spin 1/2 particles can never be in the same state (Pauli principle). Spinor fields therefore obey the Fermi-Dirac statistics . ${\ displaystyle j}$${\ displaystyle k}$

##### Calibration fields

For calibration fields, the required commutator relations are

${\ displaystyle [A _ {\ mu} ({\ vec {x}}, t), \ pi _ {\ nu} ({\ vec {y}}, t)] = ig _ {\ mu \ nu} \ delta ^ {(3)} ({\ vec {x}} - {\ vec {y}}),}$

where denotes the components of the Minkowski metric . However, one obtains from the Lagrangian what the required commutator relation cannot fulfill. The quantization of calibration fields is therefore only possible if a calibration condition is specified. The definition of a suitable calibration condition, which enables access via commutator relations of fields and at the same time preserves the Lorentz invariance of the Lagrangian, is complicated. ${\ displaystyle g _ {\ mu \ nu}}$${\ displaystyle \, \ pi _ {0} = 0}$

A modification of the Lorenz calibration is usually used in order to be able to meaningfully define a canonically conjugated field. The formalism is called the Gupta-Bleuler formalism after its developers Suraj N. Gupta and Konrad Bleuler .

An alternative is a physical calibration such as e.g. B. the temporal plus another calibration condition. Here two of the four polarizations of the calibration field are eliminated as physical degrees of freedom directly by the choice of the calibration and the subsequent implementation of Gauss's law as a condition for the physical states. The main advantage is the reduction of the Hilbert space to exclusively physical, transversal degrees of freedom. The disadvantage of this is the loss of a manifestly covariant formulation. ${\ displaystyle A_ {0} (x) = 0}$${\ displaystyle G (x) \, | {\ text {phys.}} \ rangle = 0}$

#### Path integral

In the path integral formalism, the fields are not treated as operators, but as simple functions. The path integral essentially represents a transition amplitude from a vacuum state at the point in time to a vacuum state at the point in time , integrating over all possible field configurations ( paths ) in between , with a phase factor that is determined by the effect. It has the form for the scalar field ${\ displaystyle t = - \ infty}$${\ displaystyle t = \ infty}$

${\ displaystyle Z \ propto \ int {\ mathcal {D}} \ phi \, \ exp {\ left \ {i \ int \ mathrm {d} ^ {4} x {\ mathcal {L}} (\ phi) \ right \}}}$.

However, in order to have interactions at all during a transition from vacuum to vacuum, fields must be able to be generated and destroyed. In path integral formalism, this is not achieved with the help of creation and annihilation operators, but with source fields. So a source term of the form is added to the Lagrangian . The source field J (x) should only differ from zero in a finite interval on the time axis. This means that the interacting fields exist exactly within this time interval. The full path integral for a free scalar field thus has the form ${\ displaystyle J ^ {\ dagger} (x) \ phi (x) + \ phi ^ {\ dagger} (x) J (x) \}$

${\ displaystyle Z [J] \ propto \ int {\ mathcal {D}} \ phi \, \ exp {\ left \ {i \ int \ mathrm {d} ^ {4} x \ left [(\ partial _ { \ mu} \ phi ^ {\ dagger}) (\ partial ^ {\, \ mu} \ phi) -m ^ {2} \ phi ^ {\ dagger} \ phi + J ^ {\ dagger} \ phi + \ phi ^ {\ dagger} J \ right] \ right \}}}$.

Because of the integration using an analogue of the Gaussian error integral, this can be brought into a form that only depends in a certain way on the source field J (x) , namely: ${\ displaystyle \, \ phi}$

${\ displaystyle Z [J] \ propto \ exp {\ left \ {- i \ int J ^ {\ dagger} (x) \ Delta _ {F} (xy) J (y) \, \ mathrm {d} ^ {4} x \, \ mathrm {d} ^ {4} y \ right \}}}$.

Thereby is given by as the inverse of the Klein-Gordon operator ( is the D'Alembert operator ). This object is called the time-ordered Green function or Feynman propagator . The path integral is therefore also called the generating functional of the propagator , since the derivatives according to and effectively correspond to a multiplication with the propagator. ${\ displaystyle \ Delta _ {F}}$${\ displaystyle (\ square + m ^ {2}) \ Delta _ {F} (x) = - \ delta ^ {(4)} (x)}$${\ displaystyle \ square}$${\ displaystyle J ^ {\ dagger}}$${\ displaystyle \, J}$

The behavior of the free field in the presence of sources is only determined by the propagator and the source field. This result corresponds to the expectation, because the behavior of a field that does not interact is evidently only determined by its properties during creation and annihilation and its free movement. The former are in the source field and the movement behavior is determined by the Klein-Gordon operator, the information content of which is given here by its inverse.

When quantizing the spinor field in the path integral formalism, the problem arises that on the one hand the fields are treated like normal numerical functions, but on the other hand they anti-commutate. However, normal numbers commute. This difficulty can be solved by considering the fermion fields as elements of a Graßmann algebra , so-called Graßmann numbers . Mathematically, this only means that they are treated like anticommuting numbers. This procedure is theoretically secured by the Graßmann algebra. The path integral with source fields and then has the form ${\ displaystyle {\ overline {\ eta}} \}$${\ displaystyle \ eta \}$

${\ displaystyle Z [\ eta, {\ overline {\ eta}}] \ propto \ int {\ mathcal {D}} {\ overline {\ psi}} {\ mathcal {D}} \ psi \, \ exp { \ left \ {i \ int \ mathrm {d} ^ {4} x \ left [\, {\ overline {\ psi}} (i \ gamma ^ {\, \ mu} \ partial _ {\ mu} -m ) \ psi + {\ overline {\ eta}} \ psi + {\ overline {\ psi}} \ eta \ right] \ right \}}}$.

As with the scalar field, a form can be derived from this, which in a certain way only depends on and . An analogue of the Gaussian integral can be used again , which does not correspond to the usual formalism, but is in a certain way "inverse" to it. In any case, it is first necessary to develop an integral term for Graßmann numbers. Then the path integral can be expressed in the following form: ${\ displaystyle {\ overline {\ eta}} \}$${\ displaystyle \ eta \}$

${\ displaystyle Z [\ eta, {\ overline {\ eta}}] \ propto \ exp {\ left \ {- i \ int {\ overline {\ eta}} (x) S (xy) \ eta (y) \, \ mathrm {d} ^ {4} x \, \ mathrm {d} ^ {4} y \ right \}}}$.

This is the inverse of the Dirac operator, which is also known as the Dirac propagator. Analogous to the scalar field, there is also a form here which, as expected, is only determined by the source fields and the dynamics of the fields. ${\ displaystyle S = (i \ gamma ^ {\, \ mu} \ partial _ {\ mu} + m) \ Delta _ {F}}$

The path integral for a calibration field is of the form

${\ displaystyle Z \ propto \ int {\ mathcal {D}} A _ {\ mu} \, \ exp {\ left \ {i \ int \ mathrm {d} ^ {4} x \ left [- {\ frac { 1} {2}} A ^ {\ mu} (g _ {\ mu \ nu} \ square - \ partial _ {\ mu} \ partial _ {\ nu}) A ^ {\ nu} \ right] \ right \ }}}$.

However, the operator does not have an inverse. This can be seen from the fact that it results in zero when applied to vectors of type . At least one of its eigenvalues ​​is zero, which, analogous to a matrix , ensures that the operator cannot be inverted. ${\ displaystyle (g _ {\ mu \ nu} \ square - \ partial _ {\ mu} \ partial _ {\ nu}) \}$${\ displaystyle \ partial _ {\ mu} v}$

Therefore the same procedure cannot be used here as with the scalar field and the spinor field. You have to add an additional term to the Lagrangian so that you get an operator to which there is an inverse. This is equivalent to setting a calibration. Therefore the new term is called the calibration-fixing term . It is general in shape . The corresponding calibration condition is . ${\ displaystyle {\ mathcal {L}} _ {gf} = {\ tfrac {1} {2 \ alpha}} f ^ {2} (A _ {\ mu})}$${\ displaystyle f (A _ {\ mu}) {\ stackrel {!} {=}} 0 \}$

However, this means that the Lagrangian depends on the choice of the gauge term f . This problem can be solved by introducing so-called Faddeev Popov ghosts . These ghosts are anti-commutating scalar fields and thus contradict the spin statistics theorem. Therefore, they cannot appear as free fields, but only as so-called virtual particles . By choosing the so-called axial calibration , the occurrence of these fields can be avoided, which makes their interpretation as mathematical artifacts seem obvious. Their appearance in other calibrations is, however, for deeper theoretical reasons (unitarity of the S matrix ) absolutely necessary for the consistency of the theory.

The complete Lagrangian with a calibration-fixing term and ghost fields depends on the calibration condition. For the Lorenz calibration it reads in non-Abelian calibration theories

${\ displaystyle {\ mathcal {L}} (A, {\ overline {\ eta}}, \ eta) = - {\ frac {1} {4}} F _ {\ mu \ nu} ^ {a} F ^ {\ mu \ nu \, a} - {\ frac {1} {2 \ alpha}} (\ partial _ {\ mu} A ^ {\ mu \, a}) ^ {2} - {\ bar {\ eta}} ^ {a} \ partial ^ {\ mu} (\ partial _ {\ mu} \ delta ^ {ac} -igf ^ {abc} A _ {\ mu} ^ {b}) \ eta ^ {c} }$

There is the spirit field and the anti-spirit field. ${\ displaystyle \ eta \}$${\ displaystyle {\ bar {\ eta}}}$

For Abelian gauge theories like electromagnetism, the last term takes the form regardless of the gauge . Therefore, this part of the path integral can be easily integrated and does not add to the dynamics. ${\ displaystyle {\ bar {\ eta}} \ square \ eta}$

The path integral also provides a connection with the distribution functions of statistical mechanics. For this purpose, the imaginary time coordinate in the Minkowski space is analytically continued into the Euclidean space and instead of complex phase factors in the path integral one obtains real ones similar to the Boltzmann factors of statistical mechanics. In this form, this formulation is also the starting point for numerical simulations of the field configurations (mostly selected at random using the Monte Carlo method with a weighting using these Boltzmann factors ) in grid calculations. They provide the most accurate methods to date, e.g. B. for the calculation of hadron masses in quantum chromodynamics.

### Scattering processes

As already stated above, the aim of the preceding procedure is the description of a relativistic scattering theory. Although the methods of quantum field theories are also used in other contexts today, scattering theory is still one of its main areas of application. Therefore, the basics of the same are explained here.

The central object of scattering theory is the so-called S-matrix or scattering matrix , the elements of which describe the transition probability from an initial state to an initial state. The elements of the S-matrix are called scattering amplitudes. At the field level, the S-matrix is ​​thus determined by the equation ${\ displaystyle | \ alpha _ {\ mathrm {in}} \ rangle}$${\ displaystyle | \ beta _ {\ mathrm {out}} \ rangle}$

${\ displaystyle \ phi _ {\ mathrm {out}} (x) = S ^ {\ dagger} \ phi _ {\ mathrm {in}} (x) S \}$.

The S-matrix can essentially be written as the sum of vacuum expectation values of time-ordered field operator products (also called n-point functions, correlators or Green's functions ). A proof of this so-called LSZ decomposition is one of the first great successes of 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

${\ displaystyle S = \ sum _ {n \ geq 0} {\ frac {1} {n!}} \ left (\ prod _ {i = 0} ^ {n} \ phi (x_ {i}) K ( x_ {i}) \ right) \ langle 0 | T \ left (\ phi (x_ {1}) \, ... \, \ phi (x_ {n}) \ right) | 0 \ rangle}$

Here K is the Klein-Gordon operator and T is the time order operator, which orders the fields in ascending order according to the value of time . If fields other than the scalar field occur, the respective Hamilton operators must be used. For a spinor field z. B. the Dirac operator can be used instead of the Klein-Gordon operator. ${\ displaystyle x_ {i} ^ {0}}$

To calculate the S matrix, it is sufficient to be able to calculate the time-ordered n-point functions . ${\ displaystyle \ langle 0 | T \ left (\ phi (x_ {1}) \, ... \, \ phi (x_ {n}) \ right) | 0 \ rangle}$

### Feynman rules and perturbation theory

The Feynman diagrams have proven to be a useful tool for simplifying the calculations of the n-point functions . This abbreviation was developed by Richard Feynman in 1950 and takes advantage of the fact that the terms that occur when calculating the n-point functions can be broken down into a small number of elementary components. Image elements are then assigned to these term modules. These rules, according to which this assignment is made, are called Feynman rules . The Feynman diagrams make it possible to represent complex terms in the form of small images.

There is a corresponding picture element for every term in the Lagrangian. The mass term is treated together with the derivative term as a term that describes the free field. Different lines are usually assigned to these terms for different fields. The interaction terms , on the other hand, correspond to nodes, so-called vertices , at which a corresponding line ends for each field in the interaction term. Lines that are only connected to the diagram at one end are interpreted as real particles, while lines that connect two vertices are interpreted as virtual particles . A time direction can also be set in the diagram so that it can be interpreted as a kind of illustration of the scattering process. However, in order to fully calculate a certain scattering amplitude, one must consider all diagrams with the corresponding start and end particles. If the Lagrangian of quantum field theory contains interaction terms, these are generally an infinite number of diagrams.

If the coupling constant is less than one, the terms with higher powers of the coupling constant become smaller and smaller. Since, according to Feynman's 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 therefore make the largest contribution to the scattering amplitude, while the diagrams, with increasing complexity, make ever smaller contributions. In this way, the principles of perturbation theory can be applied with good results for the scattering amplitudes by only calculating the low-order diagrams in the coupling constant.

### Renormalization

The Feynman diagrams with closed inner lines, the so-called loop diagrams (e.g. interaction of an electron with "virtual" photons from the vacuum, interaction of a photon with virtually generated particle-antiparticle pairs from the vacuum), are mostly divergent because over all energies / impulses (frequency / wave number) are integrated. As a result, more complicated Feynman diagrams cannot initially be calculated. However, this problem can often be remedied by a so-called renormalization process, sometimes also referred to as "renormalization" after an incorrect back translation from English.

There are basically two different ways of looking at this procedure. The first traditional view arranges the contributions of the diverging loop diagrams in such a way that they correspond to a few parameters in the Lagrangian such as masses and coupling constants. Then one introduces counter terms in the Lagrangian function, which, as infinite “naked” values ​​of these parameters, cancel out these divergences. This is possible in quantum electrodynamics, as well as in quantum chromodynamics and other such gauge theories, but not with other theories such as gravitation. There an infinite number of opposing terms would be necessary, the theory is “not renormalizable”.

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

It is assumed that all coupling constants of physical theories converge at sufficiently high energies, and there the physics is then described by a large, unified theory of basic forces. The behavior of coupling constants and the possibility of phase transitions with energy is described by the theory of the renormalization group. Such theoretical extrapolations gave the first indications in the 1990s of the existence of supersymmetric theories for which the coupling constants best meet at one point.

However, the technical approach is independent of the point of view. First, a regularization is carried out by introducing an additional parameter into the calculation. This parameter has to run towards zero or infinity again (depending on your choice) in order to get the original terms again. However, as long as the regularization parameter is assumed to be finite, the terms remain finite. The terms are then transformed in such a way that the infinities only occur in terms that are pure functions of the regularization parameter. These terms are then omitted. Then you set the regulation parameter zero or infinite, whereby the result now remains finite.

At first glance, this approach seems arbitrary, but the “omission” has to follow certain rules. This ensures that the renormalized coupling constants correspond to the measured constants at low energies.

### Antiparticle

A special area of ​​relativistic quantum mechanics concerns solutions of the relativistic Klein-Gordon equation and the Dirac equation with negative energy. This would allow particles to descend to infinite negative energy, which is not observed in reality. In quantum mechanics this problem is solved by arbitrarily interpreting the corresponding solutions as entities with positive energy moving backwards in time; the negative sign of the energy E is transferred to the time t in the wave function , which is obvious because of the relationship (  h  is Planck's constant and the frequency interval assigned to the energy difference ). ${\ displaystyle \ Delta E = h / \ Delta t}$${\ displaystyle h \ Delta f \, \, (= h / \ Delta t)}$${\ displaystyle \ Delta E}$

Paul Dirac interpreted these backward moving solutions as antiparticles .

## Concrete quantum field theories

### Standard model

Combining the electroweak model with quantum chromodynamics creates a unified quantum field theory, the so-called standard model of elementary particle physics. It contains all known particles and can explain most of the known processes.

At the same time it is known that the standard model cannot 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 ) according to which an expansion of the standard model seems necessary. In addition, the standard model contains many arbitrary parameters and explains e.g. B. the very different mass spectrum of the elementary particle families not.

The quantum field theories explained below are all included in the standard model.

### ϕ 4 theory

The Lagrangian of theory is ${\ displaystyle \ phi ^ {4}}$

${\ displaystyle {\ mathcal {L}} = (\ partial _ {\ mu} \ phi ^ {\ dagger}) (\ partial ^ {\ mu} \ phi) -m ^ {2} \ phi ^ {\ dagger } \ phi - {\ frac {\ lambda} {4}} (\ phi ^ {\ dagger} \ phi) ^ {2}}$

This quantum field theory is of great theoretical importance because it is the simplest conceivable quantum field theory with an interaction and, in contrast to more realistic models, some exact mathematical statements about its properties can be made 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 the second order phase transitions and critical phenomena. From the statistical interpretation one gets at the same time a new and constructive approach to the renormalization problem by showing that the renormalization of the masses, charges and vertex functions are achieved by eliminating short-wave wave phenomena from the so-called partition function can. The Higgs field of the standard model also has a self-interaction, which is, however, supplemented by interactions with the other fields of the standard model. In these cases the coupling constant m 2 is negative, which would correspond to an imaginary mass. These fields are therefore called Tachyonic fields. However, this designation refers to the Higgs field and not to the Higgs particle , the so-called Higgs boson, which is not a tachyon, but an ordinary particle with real mass. The Higgs particle is also not described by the Higgs field, but only by a certain proportion of this field. ${\ displaystyle {\ mathcal {Z}}}$${\ displaystyle \ phi ^ {4}}$

### Quantum electrodynamics

The Lagrangian of Quantum Electrodynamics (QED) is

${\ displaystyle {\ mathcal {L}} = i {\ overline {\ psi}} \ gamma ^ {\ mu} (\ partial _ {\ mu} -ieA _ {\ mu}) \ psi -m {\ overline { \ psi}} \ psi - {\ frac {1} {4}} F _ {\ mu \ nu} F ^ {\ mu \ nu}}$

QED is the first physically successful quantum field theory. It describes the interaction of a spinor field with charge -e , which describes the electron, with a calibration field, which describes the photon. Their equations of motion are obtained from electrodynamics by quantizing Maxwell's equations . Quantum electrodynamics explains with great accuracy the electromagnetic interaction between charged particles (e.g. electrons , muons , quarks ) by means of the exchange of virtual photons and the properties of electromagnetic radiation .

This makes it possible to understand the chemical elements , their properties and bonds and the periodic table of the elements. The solid state physics with the economically important semiconductor physics are derived ultimately from the QED. However, concrete calculations are usually carried out in the simplified but sufficient formalism of quantum mechanics .

### Weak interaction

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

### Quantum chromodynamics

Another example of a QFT is quantum chromodynamics (QCD), which describes the strong interaction . In it, some of the interactions between protons and neutrons occurring in the atomic nucleus are reduced to the subnuclear interaction between quarks and gluons .

What is interesting in QCD is that the gluons that mediate the interaction interact with each other themselves. (Using the example of QED, that would be as if two penetrating light rays were directly influencing each other.) A consequence of this gluonic self-interaction is that the elementary quarks cannot be observed individually, but always in the form of quark-antiquark states or states three quarks (or antiquarks) occur ( confinement ). On the other hand, it follows from this that the coupling constant does not increase at high energies, but decreases. This behavior is known as asymptotic freedom .

## Further aspects

### Spontaneous breaking of symmetry

As mentioned above, the theory is suitable for describing systems with spontaneous symmetry breaking or critical points. The mass term is understood as part of the potential. For a real mass this potential has only a minimum, while for an imaginary mass the potential describes a w-shaped parabola of the fourth degree. If the field has more than one real component, one gets even more minima. In the case of a complex field (with two real components), for example, the rotational figure of the w-shaped parabola with a mini circle is obtained. This shape is also known as the Mexican Hat Potential , as the potential is reminiscent of the shape of a sombrero . ${\ displaystyle \ phi ^ {4}}$

Every minimum now corresponds to a state of lowest energy, which the field accepts with equal probability. In each of these states, however, the field has a lesser degree of symmetry, since the symmetry of the minima with one another is lost by selecting a minimum. This property of classical field theory is transferred to quantum field theory, so that it is possible to describe quantum systems with broken symmetry. Examples of such systems are the Ising model from thermodynamics, which explains the spontaneous magnetization of a ferromagnet, and the Higgs mechanism , which explains the masses of the gauge bosons in the weak interaction. The calibration symmetry is namely reduced by the obtained mass terms of the calibration bosons.

### Axiomatic quantum field theory

Axiomatic quantum field theory tries to achieve a consistent description of quantum field theory based on a set of fewer axioms that are considered mathematically or physically inevitable.

The axiomatic quantum field theory was u. a. based on the Wightman axioms , created in 1956. Another approach is the algebraic quantum field theory formulated by Haag and Araki in 1962, which is characterized by the Haag-Kastler axioms . The Osterwalder-Schrader axioms represent a third axiomatic approach to quantum field theory.

A number of concrete results could be achieved with this approach, for example the derivation of the spin statistics theorem and the CPT theorem from the axioms alone, i.e. H. independent of a special quantum field theory. The LSZ reduction formula for the S matrix developed in 1955 by Lehmann , Symanzik and Zimmermann was an early success . In addition, there is a functional analytical approach to the S-matrix theory (also called BMP theory) established by Bogoliubov , Medvedev and Polianov.

Further applications in the field of classical statistics and quantum statistics are very advanced. They range from the general derivation of the existence of thermodynamic quantities, Gibbs ' theorem , state quantities such as pressure, internal energy and entropy to the proof of the existence of phase transitions and the exact treatment of important many-body systems:

## Relation to other theories

Attempts to combine these quantum field theories with the general theory of relativity (gravitation) to form quantum gravity have so far been unsuccessful. According to many researchers, the quantization of gravity requires new concepts that go beyond quantum field theory, since here the space-time background itself becomes dynamic. Examples from current research are string theory , M-theory and loop quantum gravity . Furthermore, the supersymmetry , the twistor theory , the finite quantum field theory and the topological quantum field theory provide important conceptual ideas that are currently being discussed in the professional world.

Applications of (non-relativistic) quantum field theory can also be found in solid-state theory, mainly in many-body theory .

## literature

General introductions to the topic (in alphabetical order of the (first) authors)

German:

• Christoph Berger: Elementary Particle Physics . 2nd edition, Springer, 2006
• Freeman Dyson : Quantum Field Theory. Springer Spectrum, 2014, ISBN 978-3-642-37677-1
• Walter Greiner u. a .: Theoretical physics . Verlag Harri Deutsch, volumes field quantization 1993, quantum electrodynamics 1994, calibration theory of weak interaction , 1994, quantum chromodynamics
• Gernot Münster: From quantum field theory to the standard model . de Gruyter, 2019, ISBN 978-3-11-063853-0

English:

More specific and related topics and