Quantum field theory

Quantum Field Theory (QFT) is a branch of physics that studies the behavior of quantum systems with an infinitely large number of degrees of freedom — quantum (or quantized) fields ; It is the theoretical basis for the description of microparticles, their interactions and transformations. It is precisely on quantum field theory that all high-energy physics , elementary particle physics, and condensed matter physics are based . The quantum field theory in the form of the Standard Model (with the addition of neutrino masses) is now the only experimentally confirmed theory capable of describing and predicting the behavior of elementary particles at high energies (that is, at energies significantly exceeding their rest energy ).

The KTP mathematical apparatus is the Hilbert state space ( Fock space ) of a quantum field and the operators acting in it. In contrast to quantum mechanics , “ particles ” as some indestructible elementary objects are absent in the QFT. Instead, the main objects here are the Fock space vectors , which describe all kinds of excitations of the quantum field. An analogue of a quantum-mechanical wave function in a QFT is a field operator (more precisely, a “field” is an operator-valued generalized function from which only after convolution with the main function is obtained an operator acting in the Hilbert state space) that can act on the vacuum vector of the Fock space (see vacuum ) and generate single-particle excitations of the quantum field. Physical observables here also correspond to operators composed of field operators.

In constructing quantum field theory, the key point was understanding the essence of the renormalization phenomenon.

Origin History

The basic equation of quantum mechanics - the Schrödinger equation - is relativistically non-invariant, which can be seen from the asymmetric occurrence of time and spatial coordinates in the equation. The non-relativistic Schrödinger equation corresponds to the classical relation between kinetic energy and particle momentum $E=p^{2}/2m$ ${\ displaystyle E = p ^ {2} / 2m}$ $E=p^2/2m$ . The relativistic ratio of energy and momentum has the form $E^{2}=p^{2}c^{2}+m^{2}c^{4}$ ${\ displaystyle E ^ {2} = p ^ {2} c ^ {2} + m ^ {2} c ^ {4}}$ $E^2=p^2c^2+m^2c^4$ . Assuming that the momentum operator in the relativistic case is the same as in the nonrelativistic region, and using this formula to construct the relativistic Hamiltonian by analogy, in 1926 a relativistically invariant equation was proposed for a free (spinless or zero-spin) particle ( the Klein equation - Gordon - Fock ). However, the problem with this equation is that it is difficult to interpret the wave function here as the probability amplitude, if only because - as can be shown - the probability density will not be a positive definite value.

A slightly different approach was implemented in 1928 by Dirac . Dirac tried to obtain a first-order differential equation in which the equality of the temporal coordinate and spatial coordinates was ensured. Since the momentum operator is proportional to the first derivative with respect to coordinates, the Dirac Hamiltonian must be linear with respect to the momentum operator. Taking into account the same relativistic relation between energy and momentum, restrictions are imposed on the square of this operator. Accordingly, linear “coefficients” must also satisfy a certain restriction, namely, their squares must be equal to unity and they must be mutually anticommutative . Thus, it certainly cannot be numerical coefficients. However, they can be matrices, with dimensions of at least 4, and the “wave function” - a four- component object, called the bispinor . As a result, the Dirac equation was obtained, in which the so-called Dirac 4-matrices and the four-component “wave function”. Formally, the Dirac equation is written in a form analogous to the Schrödinger equation with the Dirac Hamiltonian. However, this equation, as well as the Klein-Gordon equation, has solutions with negative energies. This circumstance was the reason for the prediction of antiparticles , which was later confirmed experimentally (discovery of the positron ). The presence of antiparticles is a consequence of the relativistic relationship between energy and momentum.

Thus, the transition to relativistically invariant equations leads to non-standard wave functions and many-particle interpretations. At the same time, by the end of the 1920s, a formalism was developed for a quantum description of many-particle systems (including systems with a variable number of particles), based on the creation and annihilation operators of particles. Quantum field theory is also based on these operators (expressed through them).

The Klein-Gordon and Dirac relativistic equations are considered in quantum field theory as equations for operator field functions. Correspondingly, the “new” Hilbert space of states of a system of quantum fields that are affected by the indicated field operators is introduced. Therefore, sometimes the procedure for quantizing fields is called "secondary quantization."

Classical Field Theory Formalism

Lagrange formalism

In Lagrangian mechanics , the Lagrange function $L$ ${\ displaystyle L}$ is a function of time and dynamic variables of the system and is written as a sum over all material points of the system. In the case of a continuous system, which is the field, the sum is replaced by the spatial integral of the density of the Lagrange function - the Lagrangian density ${\mathcal {L}}$ ${\ displaystyle {\ mathcal {L}}}$

L(t)=L(x^{0})=\int {\mathcal {L}}(x^{0},\mathbf {x} )d^{3}\mathbf {x} ,

{\ displaystyle L (t) = L (x ^ {0}) = \ int {\ mathcal {L}} (x ^ {0}, \ mathbf {x}) d ^ {3} \ mathbf {x}, }

where the spatial components of the 4-vector of coordinates are highlighted in bold, and the zero component is time.

Act $S$ ${\ displaystyle S}$ by definition, there is a time integral from the Lagrangian

S=\int dtL(t)=\int dx^{0}d^{3}\mathbf {x} {\mathcal {L}}(x^{0},\mathbf {x} )=\int d^{4}x{\mathcal {L}}(x),

{\ displaystyle S = \ int dtL (t) = \ int dx ^ {0} d ^ {3} \ mathbf {x} {\ mathcal {L}} (x ^ {0}, \ mathbf {x}) = \ int d ^ {4} x {\ mathcal {L}} (x),}

that is, the action in field theory is the four-dimensional integral of the Lagrangian density over four-dimensional space-time. Therefore, in field theory, the Lagrangian is usually called the Lagrangian density.

The field is described by the field function. $\psi (x)$ ${\ displaystyle \ psi (x)}$ , which can be a real or complex scalar (pseudoscalar), vector, spinor or other function. In field theory, it is assumed that the Lagrangian depends only on dynamic variables - on the field function and its derivatives, that is, there is no explicit dependence on the coordinates (an explicit dependence on the coordinates violates relativistic invariance). The locality of the theory requires that the Lagrangian contain a finite number of derivatives and not contain, for example, integral dependencies. Moreover, in order to obtain differential equations of no higher than second order (in order to correspond to classical mechanics) it is assumed that the Lagrangian depends only on the field function and its first derivatives

{\mathcal {L}}(x)={\mathcal {L}}(\psi (x),\partial _{\nu }{\psi }(x)).

{\ displaystyle {\ mathcal {L}} (x) = {\ mathcal {L}} (\ psi (x), \ partial _ {\ nu} {\ psi} (x)).}

The principle of least action (Hamilton principle) means that a real change in the state of the system occurs in such a way that the action is stationary (the variation of the action is zero). This principle allows to obtain field equations of motion - Euler-Lagrange equations , ^[1] :

{\frac {\partial }{\partial x^{\nu }}}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }\psi )}}\right)={\frac {\partial {\mathcal {L}}}{\partial \psi }}.

{\ displaystyle {\ frac {\ partial} {\ partial x ^ {\ nu}}} \ left ({\ frac {\ partial {\ mathcal {L}}} {\ partial (\ partial _ {\ nu} \ psi)}} \ right) = {\ frac {\ partial {\ mathcal {L}}} {\ partial \ psi}}.}

Since the physical properties of the system are determined by the action in which the Lagrangian is an integrand, this Lagrangian corresponds to a single action, but not vice versa. Namely, Lagrangians differing from each other in the complete 4-divergence of some 4-vector ${\mathcal {L}}'(x)={\mathcal {L}}(x)+\partial _{\nu }f^{\nu }(x)$ ${\ displaystyle {\ mathcal {L}} '(x) = {\ mathcal {L}} (x) + \ partial _ {\ nu} f ^ {\ nu} (x)}$ are physically equivalent.

Lagrangian of the field system

The Lagrangian of a system of non-interacting (free) fields is simply the sum of the Lagrangians of individual fields. The equations of motion for a system of free fields is a set of equations of motion of individual fields.

The interaction of the fields is taken into account in the Lagrangian by the addition of additional nonlinear terms. Thus, the total Lagrangian of the system of interacting fields is the sum of the free Lagrangians ${\mathcal {L}}_{0}$ ${\ displaystyle {\ mathcal {L}} _ {0}}$ and the Lagrangian of interaction ${\mathcal {L_{I}}}$ ${\ displaystyle {\ mathcal {L_ {I}}}}$ :

{\mathcal {L}}={\mathcal {L}}_{0}+{\mathcal {L_{I}}}.

{\ displaystyle {\ mathcal {L}} = {\ mathcal {L}} _ {0} + {\ mathcal {L_ {I}}}.}

The introduction of the interaction Lagrangian leads to heterogeneity and nonlinearity of the equations of motion. Interaction Lagrangians are usually polynomial functions of the participating fields (degrees not lower than the third), multiplied by some numerical constant - the so-called coupling constant . The interaction Lagrangian can be proportional to the third or fourth degree of the field function itself, to the product of various field functions (the total degree should not be lower than the third).

Hamiltonian formalism

From the Lagrangian formalism, we can pass to the Hamiltonian one by analogy with the Lagrangian and Hamiltonian mechanics. The field function here acts as a generalized (canonical) coordinate . Accordingly, it is also necessary to determine the generalized (canonical) momentum conjugate to this coordinate according to the standard formula:

\pi (t,\mathbf {x} )={\frac {{\partial }{\mathcal {L}}(\psi ,\partial _{\nu }{\psi })}{{\partial }{\dot {\psi }}(t,\mathbf {x} )}}.

{\ displaystyle \ pi (t, \ mathbf {x}) = {\ frac {{\ partial} {\ mathcal {L}} (\ psi, \ partial _ {\ nu} {\ psi})} {{\ partial} {\ dot {\ psi}} (t, \ mathbf {x})}}.}

Then the field Hamiltonian (density of the Hamiltonian) is equal by definition

{\mathcal {H}}=\pi {\dot {\psi }}-{\mathcal {L}}.

{\ displaystyle {\ mathcal {H}} = \ pi {\ dot {\ psi}} - {\ mathcal {L}}.}

The equations of motion in the Hamiltonian approach have the form:

{\dot {\psi }}={\frac {\partial {\mathcal {H}}}{\partial \pi }},{\dot {\pi }}=-{\frac {\partial {\mathcal {H}}}{\partial \psi }}.

{\ displaystyle {\ dot {\ psi}} = {\ frac {\ partial {\ mathcal {H}}} {\ partial \ pi}}, {\ dot {\ pi}} = - {\ frac {\ partial {\ mathcal {H}}} {\ partial \ psi}}.}

Dynamics of any quantities $F(\psi ,\pi )$ ${\ displaystyle F (\ psi, \ pi)}$ in the framework of the Hamiltonian formalism obey the following equation:

{\dot {F}}=\{F,{\mathcal {H}}\},

{\ displaystyle {\ dot {F}} = \ {F, {\ mathcal {H}} \},}

where curly brackets denote the Poisson bracket . Moreover, for the functions themselves $\psi$ ${\ displaystyle \ psi}$ and $\pi$ ${\ displaystyle \ pi}$ the following is true:

\{\psi (\mathbf {x} ,t),\pi (\mathbf {y} ,t)\}=1,\{\psi (\mathbf {x} ,t),\psi (\mathbf {y} ,t)\}=\{\pi (\mathbf {x} ,t),\pi (\mathbf {y} ,t)\}=0.

{\ displaystyle \ {\ psi (\ mathbf {x}, t), \ pi (\ mathbf {y}, t) \} = 1, \ {\ psi (\ mathbf {x}, t), \ psi ( \ mathbf {y}, t) \} = \ {\ pi (\ mathbf {x}, t), \ pi (\ mathbf {y}, t) \} = 0.}

Relations with the participation of Poisson brackets are usually the basis for the quantization of fields when field functions are replaced by the corresponding operators, and the Poisson brackets are replaced by a commutator of operators.

Symmetries in quantum field theory

Definition and types of symmetries

Symmetries in quantum field theory are called coordinate transformations and (or) field functions with respect to which the equations of motion are invariant, and hence the action is invariant. The transformations themselves form a group . Symmetries are called global if the corresponding transformations are independent of 4-coordinates. Otherwise, they speak of local symmetries. Symmetries can be discrete or continuous . In the latter case, the group of transformations is continuous ( topological ), that is, the group has a topology with respect to which group operations are continuous. In quantum field theory, however, a narrower class of groups is usually used — Lie groups , in which not only the topology, but also the structure of the differentiable manifold is introduced. Elements of such groups can be represented as differentiable (holomorphic or analytic) functions of a finite number of parameters. Transformation groups are usually considered in some representation - operator (matrix) parameter functions correspond to group elements.

Discrete symmetries. CPT Theorem

The most important types of conversion are:

$C$ ${\ displaystyle C}$ - Charge pairing - replacing field functions with paired ones.

$P$ ${\ displaystyle P}$ - Parity - reversing the signs of spatial components.

$T$ ${\ displaystyle T}$ - Time reversal - change the sign of the time component.

It is proved that in the local quantum field theory $C P T$ ${\ displaystyle CPT}$ -symmetry, that is, invariance with respect to the simultaneous application of these three transformations.

Continuous Symmetries. Noether's theorem

According to Noether's theorem, the invariance of the action functional with respect to $s$ ${\ displaystyle s}$ -parametric group of transformations leads to $s$ ${\ displaystyle s}$ dynamic field invariants, i.e., conservation laws. Namely, let the coordinate transformation be carried out using functions $F^{\mu }(x,\omega )$ ${\ displaystyle F ^ {\ mu} (x, \ omega)}$ , and the field function using the function $U(x,\omega )$ ${\ displaystyle U (x, \ omega)}$ where ${\omega }$ ${\ displaystyle {\ omega}}$ - aggregate $s$ ${\ displaystyle s}$ parameters. We denote $u_{k}$ ${\ displaystyle u_ {k}}$ derivative function value $U$ ${\ displaystyle U}$ by $k$ ${\ displaystyle k}$ -th parameter at a zero value of parameters, and through $f_{k}^{\mu }$ ${\ displaystyle f_ {k} ^ {\ mu}}$ - values of derived functions $F^{\mu }(x,\omega )$ ${\ displaystyle F ^ {\ mu} (x, \ omega)}$ by $k$ ${\ displaystyle k}$ -th parameter at a zero value of parameters. The indicated values are essentially generators of the corresponding transformation groups.

Then the Noether currents defined as $J_{k}^{\mu }={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}(\partial _{\nu }\psi f_{k}^{\nu }-u_{k})-f_{k}^{\mu }{\mathcal {L}}$ ${\ displaystyle J_ {k} ^ {\ mu} = {\ frac {\ partial {\ mathcal {L}}} {\ partial (\ partial _ {\ mu} \ psi)}} (\ partial _ {\ nu } \ psi f_ {k} ^ {\ nu} -u_ {k}) - f_ {k} ^ {\ mu} {\ mathcal {L}}}$ have the property $\partial _{\mu }J_{k}^{\mu }=0$ ${\ displaystyle \ partial _ {\ mu} J_ {k} ^ {\ mu} = 0}$ . The quantities conserved in time (“Noether charges”) are the spatial integrals over the zero component of the currents $C_{k}=\int d^{3}xJ_{k}^{0}$ ${\ displaystyle C_ {k} = \ int d ^ {3} xJ_ {k} ^ {0}}$

The fundamental symmetry inherent in all quantum field theories is relativistic invariance - invariance with respect to the inhomogeneous Lorentz group ( Poincare group ), i.e., with respect to space-time translations and Lorentz rotations. Another global symmetry for complex fields is the global gauge symmetry - symmetry with respect to the one-parameter group $U(1)$ ${\ displaystyle U (1)}$ - groups of multiplications by $e^{i\alpha }$ ${\ displaystyle e ^ {i \ alpha}}$ . It is connected with the requirement that the Lagrangian be real and the physical quantities observed, which leads to dependence on complex fields only through quadratic forms, which are products of mutually complex conjugate functions and their derivatives. Therefore, multiplying by a unitary phase factor $e^{i\alpha }$ ${\ displaystyle e ^ {i \ alpha}}$ does not lead to any changes.

The table below gives general expressions for Noether currents and charges for basic global symmetries and the corresponding conservation laws.

Symmetry	Noether currents	Noether charges and conservation laws
Space-time broadcasts	Energy-momentum tensor: $T_{\nu }^{\mu }={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\partial _{\nu }\psi -{\delta }_{\nu }^{\mu }{\mathcal {L}}$ ${\ displaystyle T _ {\ nu} ^ {\ mu} = {\ frac {\ partial {\ mathcal {L}}} {\ partial (\ partial _ {\ mu} \ psi)}} \ partial _ {\ nu } \ psi - {\ delta} _ {\ nu} ^ {\ mu} {\ mathcal {L}}}$ . In particular ${\mathcal {H}}=T_{0}^{0}={\frac {\partial {\mathcal {L}}}{\partial (\partial _{0}\psi )}}\partial _{0}\psi -{\mathcal {L}}$ ${\ displaystyle {\ mathcal {H}} = T_ {0} ^ {0} = {\ frac {\ partial {\ mathcal {L}}} {\ partial (\ partial _ {0} \ psi)}} \ partial _ {0} \ psi - {\ mathcal {L}}}$ - Hamiltonian (density) of the field.	The law of conservation of 4-momentum: $P^{\nu }=\int d^{3}xT^{0\nu }$ ${\ displaystyle P ^ {\ nu} = \ int d ^ {3} xT ^ {0 \ nu}}$ , in particular energy (Hamiltonian) $H=P^{0}$ ${\ displaystyle H = P ^ {0}}$
Lorentz rotation	The tensor of (total) momentum $M^{\tau (\rho \sigma )}=M_{0}^{\tau (\rho \sigma )}+S^{\tau (\rho \sigma )}$ ${\ displaystyle M ^ {\ tau (\ rho \ sigma)} = M_ {0} ^ {\ tau (\ rho \ sigma)} + S ^ {\ tau (\ rho \ sigma)}}$ where $M_{0}^{\tau (\rho \sigma )}=x^{\sigma }T^{\rho \tau }-x^{\rho }T^{\sigma \tau }$ ${\ displaystyle M_ {0} ^ {\ tau (\ rho \ sigma)} = x ^ {\ sigma} T ^ {\ rho \ tau} -x ^ {\ rho} T ^ {\ sigma \ tau}}$ - orbital moment tensor, $S^{\tau (\rho \sigma )}=-{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\tau }\psi _{i})}}A_{i}^{j(\rho \sigma )}\psi _{j}$ ${\ displaystyle S ^ {\ tau (\ rho \ sigma)} = - {\ frac {\ partial {\ mathcal {L}}} {\ partial (\ partial _ {\ tau} \ psi _ {i})} } A_ {i} ^ {j (\ rho \ sigma)} \ psi _ {j}}$ Is the spin moment tensor (spin), where $A_{i}^{j(\rho \sigma )}$ ${\ displaystyle A_ {i} ^ {j (\ rho \ sigma)}}$ - parameters of the transformation of field functions during Lorentz rotations. For scalar fields $S^{\tau (\rho \sigma )}=0$ ${\ displaystyle S ^ {\ tau (\ rho \ sigma)} = 0}$	The law of conservation of the full moment $M^{\rho \sigma }=M_{0}^{\rho \sigma }+S^{\rho \sigma }$ ${\ displaystyle M ^ {\ rho \ sigma} = M_ {0} ^ {\ rho \ sigma} + S ^ {\ rho \ sigma}}$ - spatial integral of $M^{0(\rho \sigma )}$ ${\ displaystyle M ^ {0 (\ rho \ sigma)}}$
Global Gauge Symmetry $U(1)$ ${\ displaystyle U (1)}$ - multiplication of the complex field function by $e^{i\alpha }$ ${\ displaystyle e ^ {i \ alpha}}$	4-vector of charged current: $J^{\mu }=i\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi ^{})}}\psi ^{}-{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\psi \right)$ ${\ displaystyle J ^ {\ mu} = i \ left ({\ frac {\ partial {\ mathcal {L}}} {\ partial (\ partial _ {\ mu} \ psi ^ {})}} \ psi ^ {} - {\ frac {\ partial {\ mathcal {L}}} {\ partial (\ partial _ {\ mu} \ psi)}} \ psi \ right)}$ . Для вещественных полей равны нулю.	Закон сохранения заряда ( электрический заряд , барионный заряд , странность , очарование и т. д.): $Q=\int d^{3}\mathbf {x} J^{0}$ $Q=\int d^{3}\mathbf {x} J^{0}$ . Для вещественных полей равен нулю.

Основные характеристики базовых полей

Ниже в таблице приведены описание и основные характеристики простейших полей, являющихся базовыми при построении реальных квантово-полевых теорий — скалярные, векторные и спинорные поля.

Characteristic	Скалярное поле	Векторное поле	Спинорное поле
Полевая функция	$\phi (x)={\frac {1}{\sqrt {2}}}(\phi _{1}(x)+i\phi _{2}(x))$ $\phi (x)={\frac {1}{\sqrt {2}}}(\phi _{1}(x)+i\phi _{2}(x))$ — в общем случае комплексная функция. $\phi ^{}(x)$ $\phi ^{}(x)$ — комплексно-сопряжённая функция. If a $\phi ^{}(x)=\phi (x)$ $\phi ^{}(x)=\phi (x)$ (то есть $\phi _{2}(x)=0$ $\phi _{2}(x)=0$ ), то имеем вещественное скалярное поле $\phi _{1}(x)$ $\phi _{1}(x)$ (переобозначив её просто как $\phi (x)$ $\phi (x)$ )	$A^{\mu }(x)$ $A^{\mu }(x)$ — векторная функция (4-вектор), в общем случае с комплексными компонентами (заряженное векторное поле). Вещественное (нейтральное) векторное поле получается из условия равенства $A_{\mu }^{}=A_{\mu }$ $A_{\mu }^{}=A_{\mu }$ (комплексное поле приравнивается тогда к вещественному, делённому на ${\sqrt {2}}$ ${\sqrt {2}}$ )	$\psi (x)$ $\psi (x)$ -четырёхкомпонентная функция (биспинор)-столбец, ${\bar {\psi }}=\psi ^{}\gamma ^{0}$ ${\bar {\psi }}=\psi ^{}\gamma ^{0}$ — дираковски сопряжённая четырёхкомпонентная функция-строка, $\gamma ^{\mu }$ $\gamma ^{\mu }$ — матрицы Дирака
Характер описываемых частиц	Частица со спином 0. Для вещественного поля — нейтральная, для комплексного — заряженная.	Частицы со спином 1 (проекции $0,\pm 1$ $0,\pm 1$ ), заряженные или нейтральные	Заряженные частицы со спином 1/2 ( $\pm 1/2$ $\pm 1/2$ )
Лагранжиан $({\mathcal {L}})$ $({\mathcal {L}})$	$\partial _{\mu }\phi ^{}\partial ^{\mu }\phi -m^{2}\phi ^{}\phi ={\mathcal {L}}(\phi _{1})+{\mathcal {L}}(\phi _{2})$ $\partial _{\mu }\phi ^{}\partial ^{\mu }\phi -m^{2}\phi ^{}\phi ={\mathcal {L}}(\phi _{1})+{\mathcal {L}}(\phi _{2})$ where ${\mathcal {L}}(\phi )={\frac {1}{2}}(\partial _{\mu }\phi \partial ^{\mu }\phi -m^{2}\phi ^{2})$ ${\mathcal {L}}(\phi )={\frac {1}{2}}(\partial _{\mu }\phi \partial ^{\mu }\phi -m^{2}\phi ^{2})$ — лагранжиан для вещественного поля $\phi$ ${\ displaystyle \ phi}$	$-{\frac {1}{2}}F_{\mu \nu }^{}F^{\mu \nu }+m^{2}A_{\mu }^{}A^{\mu }$ $-{\frac {1}{2}}F_{\mu \nu }^{}F^{\mu \nu }+m^{2}A_{\mu }^{}A^{\mu }$ where $F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }$ $F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }$ Для вещественного поля $-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+{\frac {1}{2}}m^{2}A_{\mu }A^{\mu }$ $-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+{\frac {1}{2}}m^{2}A_{\mu }A^{\mu }$	${\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi$ ${\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi$
Уравнения движения Эйлера-Лагранжа	$(\partial _{\mu }\partial ^{\mu }+m^{2})\phi =0$ $(\partial _{\mu }\partial ^{\mu }+m^{2})\phi =0$ ( уравнение Клейна-Гордона — верно и для сопряжённой функции)	$-(\partial _{\nu }\partial ^{\nu }+m^{2})A^{\mu }+\partial ^{\mu }\partial _{\nu }A^{\nu }=0$ $-(\partial _{\nu }\partial ^{\nu }+m^{2})A^{\mu }+\partial ^{\mu }\partial _{\nu }A^{\nu }=0$ ( Уравнение Прока ) Дифференцирование по $x^{\mu }$ $x^{\mu }$ приводит (если $m\neq 0$ $m\neq 0$ ) к $\partial _{\nu }A^{\nu }=0$ $\partial _{\nu }A^{\nu }=0$ . С этим условием (Лоренца) $(\partial _{\nu }\partial ^{\nu }+m^{2})A^{\mu }=0$ $(\partial _{\nu }\partial ^{\nu }+m^{2})A^{\mu }=0$	$(i\gamma ^{\mu }\partial _{\mu }-m)\psi =0$ $(i\gamma ^{\mu }\partial _{\mu }-m)\psi =0$ — уравнение Дирака
Тензор энергии-импульса $(T^{\mu \nu })$ $(T^{\mu \nu })$ , гамильтониан $({\mathcal {H}}=T^{00})$ $({\mathcal {H}}=T^{00})$ , 4-импульс $(P^{\nu })$ $(P^{\nu })$	$\partial ^{\mu }\phi ^{}\partial ^{\nu }\phi -g^{\mu \nu }{\mathcal {L}}$ $\partial ^{\mu }\phi ^{}\partial ^{\nu }\phi -g^{\mu \nu }{\mathcal {L}}$ , ${\mathcal {H}}=\pi ^{}\pi +\nabla \phi ^{}\nabla \phi +m^{2}\phi ^{}\phi$ ${\mathcal {H}}=\pi ^{}\pi +\nabla \phi ^{}\nabla \phi +m^{2}\phi ^{}\phi$ where $\pi ={\dot {\phi }}$ $\pi ={\dot {\phi }}$ , для вещественного поля — ${\frac {1}{2}}(\pi ^{2}+(\nabla \phi )^{2}+m^{2}\phi ^{2})$ ${\frac {1}{2}}(\pi ^{2}+(\nabla \phi )^{2}+m^{2}\phi ^{2})$	$F_{}^{\mu \sigma }\partial ^{\nu }A_{\sigma }+\partial ^{\nu }A_{\sigma }^{}F^{\mu \sigma }-g^{\mu \nu }{\mathcal {L}}$ $F_{}^{\mu \sigma }\partial ^{\nu }A_{\sigma }+\partial ^{\nu }A_{\sigma }^{}F^{\mu \sigma }-g^{\mu \nu }{\mathcal {L}}$	${\frac {i}{2}}({\bar {\psi }}\gamma ^{\mu }\partial ^{\nu }\psi -\partial ^{\nu }{\bar {\psi }}\gamma ^{\mu }\psi )$ ${\frac {i}{2}}({\bar {\psi }}\gamma ^{\mu }\partial ^{\nu }\psi -\partial ^{\nu }{\bar {\psi }}\gamma ^{\mu }\psi )$ , ${\mathcal {H}}={\frac {i}{2}}(\psi ^{}{\dot {\psi }}-{\dot {\psi }}^{}\psi )$ ${\mathcal {H}}={\frac {i}{2}}(\psi ^{}{\dot {\psi }}-{\dot {\psi }}^{}\psi )$
4-вектор тока $(J^{\mu })$ $(J^{\mu })$ и заряд $(Q)$ ${\displaystyle (Q)}$	$J^{\mu }=i(\phi ^{}\partial ^{\mu }\phi -\phi \partial ^{\mu }\phi ^{})$ $J^{\mu }=i(\phi ^{}\partial ^{\mu }\phi -\phi \partial ^{\mu }\phi ^{})$ , $Q=\int d^{3}\mathbf {x} (\phi ^{}{\dot {\phi }}-\phi {\dot {\phi }}^{})$ $Q=\int d^{3}\mathbf {x} (\phi ^{}{\dot {\phi }}-\phi {\dot {\phi }}^{})$ для вещественного поля равны нулю	$i(A_{\nu }^{}F^{\mu \nu }-F_{}^{\mu \nu }A_{\nu })$ $i(A_{\nu }^{}F^{\mu \nu }-F_{}^{\mu \nu }A_{\nu })$	$J^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi$ $J^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi$ , $Q=\int d^{3}\mathbf {x} (\psi ^{}\psi )$ $Q=\int d^{3}\mathbf {x} (\psi ^{}\psi )$
Спин-тензор $(S^{\tau (\mu \nu )})$ $(S^{\tau (\mu \nu )})$	0	$A_{}^{\mu }F^{\nu \tau }-F_{}^{\mu \tau }A^{\nu }+F_{}^{\nu \tau }A^{\mu }-A_{}^{\nu }F^{\mu \tau }$ $A_{}^{\mu }F^{\nu \tau }-F_{}^{\mu \tau }A^{\nu }+F_{}^{\nu \tau }A^{\mu }-A_{}^{\nu }F^{\mu \tau }$	${\frac {1}{4}}{\bar {\psi }}(\gamma ^{\tau }\sigma ^{\mu \nu }+\sigma ^{\mu \nu }\gamma ^{\tau })\psi$ ${\frac {1}{4}}{\bar {\psi }}(\gamma ^{\tau }\sigma ^{\mu \nu }+\sigma ^{\mu \nu }\gamma ^{\tau })\psi$ where $\sigma ^{\mu \nu }={\frac {1}{2i}}[\gamma ^{\mu },\gamma ^{\nu }]$ $\sigma ^{\mu \nu }={\frac {1}{2i}}[\gamma ^{\mu },\gamma ^{\nu }]$

Локальные симметрии и калибровочные поля

Локальные преобразования можно определить как умножение полевой функции на некоторую функцию, зависящую от 4-координат. Например, локальные преобразования группы $U(1)$ ${\displaystyle U(1)}$ — фазовое преобразование, зависящее от конкретной пространственно-временной точки, то есть умножение на $e^{i\alpha (x)}$ $e^{i\alpha (x)}$ . Как отмечалось выше, все комплексные поля симметричны относительно аналогичных глобальных преобразований. Однако они часто неинвариантны относительно локальных преобразований. В частности, описанные выше скалярные и спинорные поля неинвариантны относительно локальных калибровочных преобразований. Причина этого — неинвариантность относительно такого преобразования обычной производной. Если ввести дополнительное поле $A_{\mu }$ $A_{\mu }$ и заменить производную в лагранжиане на т. н. калибровочно-ковариантную производную

D_{\mu }=\partial _{\mu }-ieA_{\mu },

D_{\mu }=\partial _{\mu }-ieA_{\mu },

то полученный лагранжиан будет инвариантен относительно локальных калибровочных преобразований. Однако полученный таким образом лагранжиан будет по сути содержать взаимодействие двух полей — исходного и калибровочного $A_{\mu }$ $A_{\mu }$ . По общему правилу в таком случае необходимо ввести в общий лагранжиан также слагаемое, отвечающее за лагранжиан свободного калибровочного поля. Этот лагранжиан тоже должен быть калибровочно инвариантен и выбирается как лагранжиан свободного безмассового векторного поля $-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }$ $-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }$ . В итоге, например, для спинорного поля получаем лагранжиан квантовой электродинамики (КЭД):

{\mathcal {L}}_{QED}={\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }={\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi +e{\bar {\psi }}\gamma ^{\mu }A_{\mu }\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu },

{\mathcal {L}}_{QED}={\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }={\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi +e{\bar {\psi }}\gamma ^{\mu }A_{\mu }\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu },

то есть данный лагранжиан включает лагранжиан свободного спинорного поля Дирака, калибровочного (электромагнитного) поля и лагранжиан взаимодействия этих полей. Аналогичным образом можно написать калибровочно инвариантный лагранжиан комплексного скалярного поля — лагранжиан скалярной КЭД .

{\mathcal {L}}_{SQED}=(D_{\mu }\phi )^{*}D^{\mu }\phi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }.

{\mathcal {L}}_{SQED}=(D_{\mu }\phi )^{*}D^{\mu }\phi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }.

Таким образом, требование локальной калибровочной инвариантности лагранжиана относительно фазового преобразования (группа $U(1)$ ${\displaystyle U(1)}$ ) приводит к появлению калибровочного поля, в данном случае — электромагнитного поля, с которым взаимодействует «основное» поле.

Указанный подход можно обобщить на случай других локальных групп симметрии. В общем случае это приводит к появлению так называемых калибровочных полей Янга-Миллса . Ковариантная производная в этом случае имеет вид:

D_{\mu }=I\partial _{\mu }-igT^{a}A_{\mu }^{a},

{\ displaystyle D _ {\ mu} = I \ partial _ {\ mu} -igT ^ {a} A _ {\ mu} ^ {a},}

Where $T^{a}$ ${\ displaystyle T ^ {a}}$ - transformation generators of the corresponding group (in the case of U (1), there was one generator equal to unity).

Using such a covariant derivative, for example, the Lagrangian of quantum chromodynamics (QCD) is constructed, corresponding to the group $SU(3)$ ${\ displaystyle SU (3)}$ :

{\mathcal {L}}_{\mathrm {QCD} }={\bar {q}}\left(i\gamma ^{\mu }D_{\mu }-mI\right)q-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }

{\ displaystyle {\ mathcal {L}} _ {\ mathrm {QCD}} = {\ bar {q}} \ left (i \ gamma ^ {\ mu} D _ {\ mu} -mI \ right) q- { \ frac {1} {4}} G _ {\ mu \ nu} ^ {a} G_ {a} ^ {\ mu \ nu}}

where

G_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf_{bc}^{a}A_{\mu }^{b}A_{\nu }^{c}\,,

{\ displaystyle G _ {\ mu \ nu} ^ {a} = \ partial _ {\ mu} A _ {\ nu} ^ {a} - \ partial _ {\ nu} A _ {\ mu} ^ {a} + gf_ {bc} ^ {a} A _ {\ mu} ^ {b} A _ {\ nu} ^ {c} \ ,,}

Where $f_{bc}^{a}$ ${\ displaystyle f_ {bc} ^ {a}}$ - structural constants of the group involved in the switch generators (Gell-Mann matrices): $[T^{a},T^{b}]=if_{c}^{ab}T^{c}$ ${\ displaystyle [T ^ {a}, T ^ {b}] = if_ {c} ^ {ab} T ^ {c}}$

Impulse presentation

To solve the equations of motion, we can go to the so-called momentum representation using the Fourier transform :

\psi (x)={\frac {1}{(2\pi )^{2}}}\int d^{4}pf(p)e^{ipx},

{\ displaystyle \ psi (x) = {\ frac {1} {(2 \ pi) ^ {2}}} \ int d ^ {4} pf (p) e ^ {ipx},}

taking into account the properties of the Fourier image $f(p)$ ${\ displaystyle f (p)}$ , in particular the Fourier transform of derivatives $\partial _{\mu }\psi (x)$ ${\ displaystyle \ partial _ {\ mu} \ psi (x)}$ is equal to $ip_{\mu }f(p)$ ${\ displaystyle ip _ {\ mu} f (p)}$ .

Finding a solution to the equations of motion can be shown by the example of the Klein-Gordon equation.

Solution of the equation and impulse representation of the Klein-Gordon field

Passing to the momentum representation, the Klein-Gordon equation for the Fourier transform of the field function will have the form:

(p^{2}-m^{2})f(p)=0.

{\ displaystyle (p ^ {2} -m ^ {2}) f (p) = 0.}

Consequently, $f(p)={\sqrt {2\pi }}\delta (p^{2}-m^{2}){\tilde {f}}(p)$ ${\ displaystyle f (p) = {\ sqrt {2 \ pi}} \ delta (p ^ {2} -m ^ {2}) {\ tilde {f}} (p)}$ (factor ${\sqrt {2\pi }}$ ${\ displaystyle {\ sqrt {2 \ pi}}}$ - for convenience), where ${\tilde {f}}(p)$ ${\ displaystyle {\ tilde {f}} (p)}$ - arbitrary function $p$ ${\ displaystyle p}$ defined on the "mass surface" $p^{2}-m^{2}=0$ ${\ displaystyle p ^ {2} -m ^ {2} = 0}$ or highlighting the temporary component $p_{0}=\pm {\sqrt {\mathbf {p} ^{2}+m^{2}}}$ ${\ displaystyle p_ {0} = \ pm {\ sqrt {\ mathbf {p} ^ {2} + m ^ {2}}}}$ (the bold part is the spatial part of the 4-vector of the pulse, that is, the usual pulse). Then the impulse representation has the form:

\phi (x)={\frac {1}{(2\pi )^{3/2}}}\int d^{4}p\delta (p^{2}-m^{2})e^{ipx}{\tilde {f}}(p).

{\ displaystyle \ phi (x) = {\ frac {1} {(2 \ pi) ^ {3/2}}} \ int d ^ {4} p \ delta (p ^ {2} -m ^ {2 }) e ^ {ipx} {\ tilde {f}} (p).}

The presence of a delta function under the integral sign means that, in essence, integration is carried out not over the entire 4-dimensional momentum space, but only over two floors of the three-dimensional hyperboloid determined by the mass surface equation. Two signs in front of the square root define two independent solutions with which the field function is divided into two components (each separately relativistic is invariant)

\phi (x)=\phi ^{+}(x)+\phi ^{-}(x).

{\ displaystyle \ phi (x) = \ phi ^ {+} (x) + \ phi ^ {-} (x).}

Then the impulse representation of two independent solutions has the form

\phi {\pm }(x)={\frac {1}{(2\pi )^{3/2}}}\int d^{4}p\delta (p^{2}-m^{2}){\tilde {f}}^{\pm }(p)e^{\pm ipx}.

{\ displaystyle \ phi {\ pm} (x) = {\ frac {1} {(2 \ pi) ^ {3/2}}} int d ^ {4} p \ delta (p ^ {2} - m ^ {2}) {\ tilde {f}} ^ {\ pm} (p) e ^ {\ pm ipx}.}

Integrating over the time component $p_{0}$ ${\ displaystyle p_ {0}}$ we get

\phi ^{\pm }(x)={\frac {1}{(2\pi )^{3/2}}}\int {\frac {d^{3}\mathbf {p} }{\sqrt {2p_{0}}}}e^{\pm ipx}a^{\pm }(\mathbf {p} )

{\ displaystyle \ phi ^ {\ pm} (x) = {\ frac {1} {(2 \ pi) ^ {3/2}}} \ int {\ frac {d ^ {3} \ mathbf {p} } {\ sqrt {2p_ {0}}}} e ^ {\ pm ipx} a ^ {\ pm} (\ mathbf {p})}

where

a^{\pm }={\tilde {f}}^{\pm }/{\sqrt {2p_{0}}}.

{\ displaystyle a ^ {\ pm} = {\ tilde {f}} ^ {\ pm} / {\ sqrt {2p_ {0}}}.}

Using the impulse representation of field functions, one can also obtain other characteristics of the field in the impulse representation. We show this by the example of a 4-momentum for the same real Klein-Gordon scalar field.

Conclusion of the impulse representation for the 4-momentum of the Klein-Gordon field

To obtain an impulse representation of the field characteristics, it is necessary to express these field characteristics in terms of functions $\phi ^{\pm }(x)$ ${\ displaystyle \ phi ^ {\ pm} (x)}$ , and then use the impulse representations of the latest functions. For example? the Hamiltonian of the field is $H=\int d^{3}\mathbf {x} {\mathcal {H}}=1/2\int d^{3}\mathbf {x} [(\partial _{\nu }\phi (x))^{2}+m^{2}\phi ^{2}(x)]$ ${\ displaystyle H = \ int d ^ {3} \ mathbf {x} {\ mathcal {H}} = 1/2 \ int d ^ {3} \ mathbf {x} [(\ partial _ {\ nu} \ phi (x)) ^ {2} + m ^ {2} \ phi ^ {2} (x)]}$ . If we substitute here the expansion of the field function into two terms, we obtain in square brackets various pairwise products of positive and negative frequency field functions and their derivatives. However, it can be shown that works with the same sign actually give a zero contribution. To do this, you need to use the momentum representation and the fact that the product of two integrals is a double integral over all possible combinations of arguments:

\int d^{3}\mathbf {x} [(\partial _{\nu }\phi ^{\pm }(x))^{2}+m^{2}(\phi ^{\pm }(x))^{2}]={\frac {1}{(2\pi )^{3}}}\int \int {\frac {d^{3}\mathbf {p} d^{3}\mathbf {p'} }{2{\sqrt {p_{0}p'_{0}}}}}a^{\pm }(\mathbf {p} )a^{\pm }(\mathbf {p'} )e^{\pm i(p_{0}+p'_{0})x_{0}}(m^{2}-p_{\nu }p'_{\nu })\int d^{3}\mathbf {x} e^{\mp i(\mathbf {p} +\mathbf {p'} )}.

{\ displaystyle \ int d ^ {3} \ mathbf {x} [(\ partial _ {\ nu} \ phi ^ {\ pm} (x)) ^ {2} + m ^ {2} (\ phi ^ { \ pm} (x)) ^ {2}] = {\ frac {1} {(2 \ pi) ^ {3}}} \ int \ int {\ frac {d ^ {3} \ mathbf {p} d ^ {3} \ mathbf {p '}} {2 {\ sqrt {p_ {0} p' _ {0}}}} a ^ {\ pm} (\ mathbf {p}) a ^ {\ pm} (\ mathbf {p '}) e ^ {\ pm i (p_ {0} + p' _ {0}) x_ {0}} (m ^ {2} -p _ {\ nu} p '_ {\ nu }) \ int d ^ {3} \ mathbf {x} e ^ {\ mp i (\ mathbf {p} + \ mathbf {p '})}.}

The last integral in this expression is, as is known, the delta function $\delta (\mathbf {p} +\mathbf {p'} )$ ${\ displaystyle \ delta (\ mathbf {p} + \ mathbf {p '})}$ therefore, the whole expression can be non-zero only if this delta function is not zero, which is possible only under the condition $\mathbf {p'} =-\mathbf {p}$ ${\ displaystyle \ mathbf {p '} = - \ mathbf {p}}$ (where also $p_{0}=p'_{0}$ ${\ displaystyle p_ {0} = p '_ {0}}$ ) But in this case, the expression in parentheses $m^{2}-p_{\nu }p'_{\nu }=m^{2}-(p_{0}^{2}+\mathbf {p} _{n}\mathbf {p'} _{n})=m^{2}-p_{0}^{2}+\mathbf {p} ^{2}$ ${\ displaystyle m ^ {2} -p _ {\ nu} p '_ {\ nu} = m ^ {2} - (p_ {0} ^ {2} + \ mathbf {p} _ {n} \ mathbf { p '} _ {n}) = m ^ {2} -p_ {0} ^ {2} + \ mathbf {p} ^ {2}}$ , which is zero. Therefore, the entire original expression is also zero. Thus, the initial integral for the Hamiltonian should be expressed only in terms of products of functions with different signs. Using a similar approach, we get that

\int d^{3}\mathbf {x} [\partial _{\nu }\phi ^{+}(x)\partial _{\nu }\phi ^{-}(x)+m^{2}\phi ^{+}(x)\phi ^{-}(x)]={\frac {1}{(2\pi )^{3}}}\int \int {\frac {d^{3}\mathbf {p} d^{3}\mathbf {p'} }{2{\sqrt {p_{0}p'_{0}}}}}a^{+}(\mathbf {p} )a^{-}(\mathbf {p'} )e^{i(p_{0}-p'_{0})x_{0}}(m^{2}+p_{\nu }p'_{\nu })\int d^{3}\mathbf {x} e^{-i(\mathbf {p} -\mathbf {p'} )}.

{\ displaystyle \ int d ^ {3} \ mathbf {x} [\ partial _ {\ nu} \ phi ^ {+} (x) \ partial _ {\ nu} \ phi ^ {-} (x) + m ^ {2} \ phi ^ {+} (x) \ phi ^ {-} (x)] = {\ frac {1} {(2 \ pi) ^ {3}}} \ int \ int {\ frac { d ^ {3} \ mathbf {p} d ^ {3} \ mathbf {p '}} {2 {\ sqrt {p_ {0} p' _ {0}}}} a ^ {+} (\ mathbf {p}) a ^ {-} (\ mathbf {p '}) e ^ {i (p_ {0} -p' _ {0}) x_ {0}} (m ^ {2} + p _ {\ nu } p '_ {\ nu}) \ int d ^ {3} \ mathbf {x} e ^ {- i (\ mathbf {p} - \ mathbf {p'})}.}

In this case, the last integral gives a delta function $\delta (\mathbf {p} -\mathbf {p'} )$ ${\ displaystyle \ delta (\ mathbf {p} - \ mathbf {p '})}$ therefore, there must be equality $\mathbf {p} =\mathbf {p'}$ ${\ displaystyle \ mathbf {p} = \ mathbf {p '}}$ to provide a nonzero contribution to the integral. Then $m^{2}+p_{\nu }p'_{\nu }=m^{2}+p_{0}^{2}+\mathbf {p} ^{2}=2p_{0}^{2}$ ${\ displaystyle m ^ {2} + p _ {\ nu} p '_ {\ nu} = m ^ {2} + p_ {0} ^ {2} + \ mathbf {p} ^ {2} = 2p_ {0 } ^ {2}}$ . From here we finally get

\int d^{3}\mathbf {x} [\partial _{\nu }\phi ^{+}(x)\partial _{\nu }\phi ^{-}(x)+m^{2}\phi ^{+}(x)\phi ^{-}(x)]={\frac {1}{(2\pi )^{3}}}\int d^{3}\mathbf {p} p_{0}a^{+}(\mathbf {p} )a^{-}(\mathbf {p} ).

{\ displaystyle \ int d ^ {3} \ mathbf {x} [\ partial _ {\ nu} \ phi ^ {+} (x) \ partial _ {\ nu} \ phi ^ {-} (x) + m ^ {2} \ phi ^ {+} (x) \ phi ^ {-} (x)] = {\ frac {1} {(2 \ pi) ^ {3}}} \ int d ^ {3} \ mathbf {p} p_ {0} a ^ {+} (\ mathbf {p}) a ^ {-} (\ mathbf {p}).}

Similarly to the Hamiltonian, one can obtain a similar expression for other components of the 4-vector of momentum. As a result, we obtain the general expression for the 4-momentum:

P^{\mu }={\frac {1}{2}}\int d^{3}\mathbf {p} p^{\mu }(a^{+}(\mathbf {p} )a^{-}(\mathbf {p} )+a^{-}(\mathbf {p} )a^{+}(\mathbf {p} ))=\int d^{3}\mathbf {p} p^{\mu }a^{+}(\mathbf {p} )a(\mathbf {p} ).

{\ displaystyle P ^ {\ mu} = {\ frac {1} {2}} \ int d ^ {3} \ mathbf {p} p ^ {\ mu} (a ^ {+} (\ mathbf {p} ) a ^ {-} (\ mathbf {p}) + a ^ {-} (\ mathbf {p}) a ^ {+} (\ mathbf {p})) = \ int d ^ {3} \ mathbf { p} p ^ {\ mu} a ^ {+} (\ mathbf {p}) a (\ mathbf {p}).}

The first expression is necessary for quantization - when the order of multiplication plays a role due to the non-commutativity of the operators in the general case.

Characteristic	Scalar field	Vector field	Spinor field
Impulse representation of a field function: $f(\mathbf {p} )$ ${\ displaystyle f (\ mathbf {p})}$ in expression ${\frac {1}{(2\pi )^{3/2}}}\int {\frac {d^{3}\mathbf {p} }{\sqrt {2p_{0}}}}f(\mathbf {p} )$ ${\ displaystyle {\ frac {1} {(2 \ pi) ^ {3/2}}} \ int {\ frac {d ^ {3} \ mathbf {p}} {\ sqrt {2p_ {0}}} } f (\ mathbf {p})}$ . $a$ ${\ displaystyle a}$ responsible for the particle $b$ ${\ displaystyle b}$ - for the antiparticle.	$e^{-ipx}a(\mathbf {p} )+e^{ipx}b^{+}(\mathbf {p} )$ ${\ displaystyle e ^ {- ipx} a (\ mathbf {p}) + e ^ {ipx} b ^ {+} (\ mathbf {p})}$ , for the real field $b=a$ ${\ displaystyle b = a}$ .	$\sum _{n=1}^{3}e_{\mu }^{n}(\mathbf {p} )(e^{-ipx}a_{n}(\mathbf {p} )+e^{ipx}b_{n}^{+}(\mathbf {p} ))$ ${\ displaystyle \ sum _ {n = 1} ^ {3} e _ {\ mu} ^ {n} (\ mathbf {p}) (e ^ {- ipx} a_ {n} (\ mathbf {p}) + e ^ {ipx} b_ {n} ^ {+} (\ mathbf {p}))}$	$\sum _{r=1}^{2}(e^{-ipx}a_{r}(\mathbf {p} )u_{r}(\mathbf {p} )+e^{ipx}b_{r}^{+}(\mathbf {p} )v_{r}(\mathbf {p} )$ ${\ displaystyle \ sum _ {r = 1} ^ {2} (e ^ {- ipx} a_ {r} (\ mathbf {p}) u_ {r} (\ mathbf {p}) + e ^ {ipx} b_ {r} ^ {+} (\ mathbf {p}) v_ {r} (\ mathbf {p})}$
Density $n(\mathbf {p} )$ ${\ displaystyle n (\ mathbf {p})}$ particles with momentum $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ . Total number of particles $N=\int d^{3}\mathbf {p} n(\mathbf {p} )$ ${\ displaystyle N = \ int d ^ {3} \ mathbf {p} n (\ mathbf {p})}$ . 4-pulse field $P^{\nu }=\int d^{3}\mathbf {p} p^{\nu }n(\mathbf {p} )$ ${\ displaystyle P ^ {\ nu} = \ int d ^ {3} \ mathbf {p} p ^ {\ nu} n (\ mathbf {p})}$	$(a^{+}(\mathbf {p} )a(\mathbf {p} )+b^{+}(\mathbf {p} )b(\mathbf {p} ))$ ${\ displaystyle (a ^ {+} (\ mathbf {p}) a (\ mathbf {p}) + b ^ {+} (\ mathbf {p}) b (\ mathbf {p}))}$	$\sum _{n=1}^{3}(a_{n}^{+}(\mathbf {p} )a_{n}(\mathbf {p} )+b_{n}^{+}(\mathbf {p} )b_{n}(\mathbf {p} ))$ ${\ displaystyle \ sum _ {n = 1} ^ {3} (a_ {n} ^ {+} (\ mathbf {p}) a_ {n} (\ mathbf {p}) + b_ {n} ^ {+ } (\ mathbf {p}) b_ {n} (\ mathbf {p}))}$	$\sum _{r=1}^{2}(a_{r}^{+}(\mathbf {p} )a_{r}(\mathbf {p} )-b_{r}^{+}(\mathbf {p} )b_{r}(\mathbf {p} ))$ ${\ displaystyle \ sum _ {r = 1} ^ {2} (a_ {r} ^ {+} (\ mathbf {p}) a_ {r} (\ mathbf {p}) -b_ {r} ^ {+ } (\ mathbf {p}) b_ {r} (\ mathbf {p}))}$
Charge $(Q)$ ${\ displaystyle (Q)}$	$(a^{+}(\mathbf {p} )a(\mathbf {p} )-b^{+}(\mathbf {p} )b(\mathbf {p} ))$ ${\ displaystyle (a ^ {+} (\ mathbf {p}) a (\ mathbf {p}) -b ^ {+} (\ mathbf {p}) b (\ mathbf {p}))}$ , for the real field is zero	$\sum _{n=1}^{3}(a_{n}^{+}(\mathbf {p} )a_{n}(\mathbf {p} )-b_{n}^{+}(\mathbf {p} )b_{n}(\mathbf {p} ))$ ${\ displaystyle \ sum _ {n = 1} ^ {3} (a_ {n} ^ {+} (\ mathbf {p}) a_ {n} (\ mathbf {p}) -b_ {n} ^ {+ } (\ mathbf {p}) b_ {n} (\ mathbf {p}))}$	$\sum _{r=1}^{2}(a_{r}^{+}(\mathbf {p} )a_{r}(\mathbf {p} )-b_{r}^{+}(\mathbf {p} )b_{r}(\mathbf {p} ))$ ${\ displaystyle \ sum _ {r = 1} ^ {2} (a_ {r} ^ {+} (\ mathbf {p}) a_ {r} (\ mathbf {p}) -b_ {r} ^ {+ } (\ mathbf {p}) b_ {r} (\ mathbf {p}))}$
Projection of the spin on the direction of the momentum	0	$b_{1}^{+}a_{1}-b_{1}a_{1}^{+}+b_{2}a_{2}^{+}-b_{2}^{+}a_{2}$ ${\ displaystyle b_ {1} ^ {+} a_ {1} -b_ {1} a_ {1} ^ {+} + b_ {2} a_ {2} ^ {+} - b_ {2} ^ {+} a_ {2}}$ , indices 1 and 2 correspond to particles with spin projections $\pm 1$ ${\ displaystyle \ pm 1}$ , and the third index - to particles with a zero projection of the spin

Field quantization

Quantization means the transition from fields (field functions) to the corresponding operators (operator-valued functions) acting on the vector (amplitude) of the state Φ . By analogy with ordinary quantum mechanics, the state vector completely characterizes the physical state of the system of quantized wave fields. A state vector is a vector in some linear space called the Fock space .

The basic postulate of quantizing wave fields is that the operators of dynamic variables are expressed in terms of field operators in the same way as the classical expression of these quantities in terms of field functions (taking into account the order of multiplication, since the multiplication of operators is generally non-commutative, in contrast to the product of ordinary functions ) The Poisson bracket (see the Hamiltonian formalism) is replaced by the commutator of the corresponding operators. In particular, the classical Hamiltonian formalism transforms into a quantum one as follows:

[\psi (\mathbf {x} ,t),\psi (\mathbf {x'} ,t)]=[\pi (\mathbf {x} ,t),\pi (\mathbf {x'} ,t)]=0

{\ displaystyle [\ psi (\ mathbf {x}, t), \ psi (\ mathbf {x '}, t)] = [\ pi (\ mathbf {x}, t), \ pi (\ mathbf {x '}, t)] = 0}

,

[\pi (\mathbf {x} ,t),\psi (\mathbf {x'} ,t)]=-i\delta ^{(3)}(\mathbf {x-x'} ).

{\ displaystyle [\ pi (\ mathbf {x}, t), \ psi (\ mathbf {x '}, t)] = - i \ delta ^ {(3)} (\ mathbf {x-x'}) .}

These are the so-called Bose-Einstein commutation relations based on the usual commutator - the difference between the “direct” and “inverse” product of operators

[A,B]=AB-BA.

{\ displaystyle [A, B] = AB-BA.}

Fermi-Dirac switching relations are based on an anticommutator — the sum of the “direct” and “inverse” product of operators:

[A,B]_{+}=AB+BA.

{\ displaystyle [A, B] _ {+} = AB + BA.}

The quanta of the first fields obey the Bose-Einstein statistics and are called bosons , and the quanta of the second fields obey the Fermi-Dirac statistics and are called fermions . Bose-Einstein quantization of fields turns out to be consistent for particles with a whole spin, and for particles with a half-integer spin, Fermi – Dirac quantization is consistent. Thus, fermions are particles with a half-integer spin, and bosons are particles with a whole.

From the commutation relations for the field function (generalized coordinate) and the corresponding generalized momentum, we can obtain commutation relations for the creation and annihilation operators of quanta

[a_{\mathbf {p} },a_{\mathbf {p'} }^{+}]=\delta (\mathbf {p} -\mathbf {p'} ),[a_{\mathbf {p} },a_{\mathbf {p'} }]=[a_{\mathbf {p} }^{+},a_{\mathbf {p'} }^{+}]=0.

{\ displaystyle [a _ {\ mathbf {p}}, a _ {\ mathbf {p '}} ^ {+}] = \ delta (\ mathbf {p} - \ mathbf {p'}), [a _ {\ mathbf {p}}, a _ {\ mathbf {p '}}] = [a _ {\ mathbf {p}} ^ {+}, a _ {\ mathbf {p'}} ^ {+}] = 0.}

Field as a set of harmonic oscillators

The field can be represented as an infinite number of harmonic field oscillators. This can be shown by the example of the Klein-Gordon field. The three-dimensional (in three spatial coordinates) Fourier transform of the field function satisfies the following equation (Fourier transform of the Klein-Gordon equation)

\partial _{t}^{2}\phi (\mathbf {p} ,t)+(\mathbf {p} ^{2}+m^{2})\phi (\mathbf {p} ,t)=0,

{\ displaystyle \ partial _ {t} ^ {2} \ phi (\ mathbf {p}, t) + (\ mathbf {p} ^ {2} + m ^ {2}) \ phi (\ mathbf {p} , t) = 0,}

what is the differential equation for a harmonic oscillator with a frequency $\omega ={\sqrt {\mathbf {p} ^{2}+m^{2}}}$ ${\ displaystyle \ omega = {\ sqrt {\ mathbf {p} ^ {2} + m ^ {2}}}}$ each fixed fashion $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ Fourier decompositions. For each such quantum harmonic oscillator , as is known from quantum mechanics, stationary states $\phi _{n}$ ${\ displaystyle \ phi _ {n}}$ can be connected with each other by raising and lowering operators as follows

{\hat {a}}^{+}{\phi }_{n}={\sqrt {n+1}}{\phi }_{n+1}

{\ displaystyle {\ hat {a}} ^ {+} {\ phi} _ {n} = {\ sqrt {n + 1}} {\ phi} _ {n + 1}}

,

{\hat {a}}{\phi }_{n}={\sqrt {n}}{\phi }_{n-1},

{\ displaystyle {\ hat {a}} {\ phi} _ {n} = {\ sqrt {n}} {\ phi} _ {n-1},}

and the Hamiltonian is $H={\hbar \omega }{({\hat {n}}+1/2)}$ ${\ displaystyle H = {\ hbar \ omega} {({\ hat {n}} + 1/2)}}$ where ${\hat {n}}={a}^{+}a$ ${\ displaystyle {\ hat {n}} = {a} ^ {+} a}$ . Accordingly, the oscillator energy is quantized $E_{n}={\hbar }{\omega }(n+1/2)$ ${\ displaystyle E_ {n} = {\ hbar} {\ omega} (n + 1/2)}$ where $n$ ${\ displaystyle n}$ - quantum number-eigenvalues of the operator ${\hat {n}}={a}^{+}a$ ${\ displaystyle {\ hat {n}} = {a} ^ {+} a}$ .

Thus, the application of an increasing or decreasing operator changes the quantum number $n$ ${\ displaystyle n}$ per unit and leads to the same change in the energy of the oscillator ( equidistance spectrum ), which can be interpreted as the birth of a new or the destruction of a quantum field with energy ${\hbar }{\omega }$ ${\ displaystyle {\ hbar} {\ omega}}$ . It is this interpretation that allows the use of the above operators, such as the creation and annihilation operators . Any state with an index $n$ ${\ displaystyle n}$ can be represented as an action $n$ ${\ displaystyle n}$ birth operators to the "zero" state:

{\phi }_{n}={\frac {({\hat {a}}^{+})^{n}}{\sqrt {n!}}}\phi _{0}.

{\ displaystyle {\ phi} _ {n} = {\ frac {({\ hat {a}} ^ {+}) ^ {n}} {\ sqrt {n!}}} \ phi _ {0}. }

When $N$ ${\ displaystyle N}$ Oscillators the Hamiltonian of the system is equal to the sum of the Hamiltonians of the individual oscillators. For each such oscillator, you can define your own creation operators ${\hat {a}}_{k}^{+},k=1,...,N$ ${\ displaystyle {\ hat {a}} _ {k} ^ {+}, k = 1, ..., N}$ . Therefore, an arbitrary quantum state of such a system can be described using occupation numbers $n_{k}$ ${\ displaystyle n_ {k}}$ - the number of operators of a given variety k acting on a vacuum:

\phi (n_{1},...,n_{N})=\prod _{(k)}{\frac {({\hat {a_{k}}}^{+})^{n_{k}}}{\sqrt {n_{k}!}}}\phi _{0}.

{\ displaystyle \ phi (n_ {1}, ..., n_ {N}) = \ prod _ {(k)} {\ frac {({\ hat {a_ {k}}} ^ {+}) ^ {n_ {k}}} {\ sqrt {n_ {k}!}}} \ phi _ {0}.}

This representation is called the representation of fill numbers . The essence of this representation is that instead of setting the state vector as a function of coordinates (coordinate representation) or as a function of impulses (impulse representation), the state of the system is characterized by the number of the excited state - the fill number.

Fock space. Vacuum. Fock View

In quantum field theory, the Hamiltonian, originally expressed as a function $\psi$ ${\ displaystyle \ psi}$ and $\pi$ ${\ displaystyle \ pi}$ , is also ultimately expressed through the corresponding creation and annihilation operators of field quanta. The main principle is preserved - any operators (including the Hamiltonian) are expressed through these creation and annihilation operators as well as the corresponding functions before quantization. The only difference is that the order in which operators are written matters, since operators, in contrast to ordinary functions, are generally non-commutative.

All creation and annihilation operators and their combinations, the operators of the fields themselves and their derivatives - they all act in the infinite- dimensional Fock space . In the Fock space, vacuum is first determined (vacuum state) $\Phi _{0}$ ${\ displaystyle \ Phi _ {0}}$ or $|0\rangle$ ${\ displaystyle | 0 \ rangle}$ , by analogy with the zero state of a quantum oscillator. Vacuum is defined as

a(\mathbf {p} )|0\rangle =\langle 0|a^{+}(\mathbf {p} )=0,\langle 0|0\rangle =1.

{\ displaystyle a (\ mathbf {p}) | 0 \ rangle = \ langle 0 | a ^ {+} (\ mathbf {p}) = 0, \ langle 0 | 0 \ rangle = 1.}

Arbitrary states are defined as excitations of a vacuum of the following form:

|f\rangle =\int d^{3}\mathbf {p_{1}} d^{3}\mathbf {p_{2}} ...d^{3}\mathbf {p_{k}} f(\mathbf {p_{1}} ,\mathbf {p_{2}} ,...,\mathbf {p_{k}} )a^{+}(\mathbf {p_{1}} )a^{+}(\mathbf {p_{2}} )...a^{+}(\mathbf {p_{k}} )|0\rangle .

{\ displaystyle | f \ rangle = \ int d ^ {3} \ mathbf {p_ {1}} d ^ {3} \ mathbf {p_ {2}} ... d ^ {3} \ mathbf {p_ {k }} f (\ mathbf {p_ {1}}, \ mathbf {p_ {2}}, ..., \ mathbf {p_ {k}}) a ^ {+} (\ mathbf {p_ {1}}) a ^ {+} (\ mathbf {p_ {2}}) ... a ^ {+} (\ mathbf {p_ {k}}) | 0 \ rangle.}

This is the Fock representation for the k-particle state. The functions f are ordinary quantum-mechanical wave functions. Usually they are assumed to be quadratically integrable so that the norms of state vectors are finite quantities. However, states with an infinite norm also make sense. For example, the state $a^{+}(\mathbf {p} )|0\rangle$ ${\ displaystyle a ^ {+} (\ mathbf {p}) | 0 \ rangle}$ has an infinite rate $(\delta (0))$ ${\ displaystyle (\ delta (0))}$ However, this state corresponds to a single-particle state with a definite momentum, and if we consider the spatial density of such particles, then it turns out to be finite.

Normal and chronological work. Wick's Theorem

It follows from the definition of vacuum that the vacuum average of the product of any number of creation and annihilation operators, in which all the creation operators are to the left of all the annihilation operators, is zero . The corresponding order of writing birth and annihilation operators is called normal form or normal ordering . To emphasize that the operators are normally ordered, the corresponding products are enclosed in brackets from colons, for example, $:\phi (x)\phi (y):$ ${\ displaystyle: \ phi (x) \ phi (y):}$ or you can indicate under the sign of some conditional operator ${\mathcal {N}}\{\phi (x)\phi (y)\}$ ${\ displaystyle {\ mathcal {N}} \ {\ phi (x) \ phi (y) \}}$

The normal form, obviously, is connected with the normal form through the commutator of operators, namely, the “ordinary” form is equal to the normal form plus the (anti) commutator of the corresponding operators (“incorrectly” ordered). For example,

\phi (x)\phi (y)=\phi ^{+}(x)\phi ^{+}(y)+\phi ^{+}(x)\phi ^{-}(y)+\phi ^{-}(x)\phi ^{+}(y)+\phi ^{-}(x)\phi ^{-}(y).

{\ displaystyle \ phi (x) \ phi (y) = \ phi ^ {+} (x) \ phi ^ {+} (y) + \ phi ^ {+} (x) \ phi ^ {-} (y ) + \ phi ^ {-} (x) \ phi ^ {+} (y) + \ phi ^ {-} (x) \ phi ^ {-} (y).}

In this record, only one term is not written in normal form, respectively, you can write

\phi (x)\phi (y)=(\phi ^{+}(x)\phi ^{+}(y)+\phi ^{+}(x)\phi ^{-}(y)+\phi ^{+}(y)\phi ^{-}(x)+\phi ^{-}(x)\phi ^{-}(y))+(\phi ^{-}(x)\phi ^{+}(y)-\phi ^{+}(y)\phi ^{-}(x))={\mathcal {N}}\{\phi (x)\phi (y)\}+[\phi ^{-}(x),\phi ^{+}(y)].

{\ displaystyle \ phi (x) \ phi (y) = (\ phi ^ {+} (x) \ phi ^ {+} (y) + \ phi ^ {+} (x) \ phi ^ {-} ( y) + \ phi ^ {+} (y) \ phi ^ {-} (x) + \ phi ^ {-} (x) \ phi ^ {-} (y)) + (\ phi ^ {-} ( x) \ phi ^ {+} (y) - \ phi ^ {+} (y) \ phi ^ {-} (x)) = {\ mathcal {N}} \ {\ phi (x) \ phi (y ) \} + [\ phi ^ {-} (x), \ phi ^ {+} (y)].}

Thus, the vacuum average from the original product of the operators will essentially be determined only by the last commutator.

A chronological work is defined as ordered by time variable (the zero component of 4-coordinates):

Tf_{1}(x_{1})f_{2}(x_{2})...f_{n}(x_{n})=(-1)^{\sigma }f_{i_{1}}(x_{i_{1}})f_{i_{2}}(x_{i_{2}})...f_{i_{n}}(x_{i_{n}})

{\ displaystyle Tf_ {1} (x_ {1}) f_ {2} (x_ {2}) ... f_ {n} (x_ {n}) = (- 1) ^ {\ sigma} f_ {i_ { 1}} (x_ {i_ {1}}) f_ {i_ {2}} (x_ {i_ {2}}) ... f_ {i_ {n}} (x_ {i_ {n}})}

where

x_{i_{1}}^{0}>x_{i_{1}}^{0}>...>x_{i_{n}}^{0},

{\ displaystyle x_ {i_ {1}} ^ {0}> x_ {i_ {1}} ^ {0}> ...> x_ {i_ {n}} ^ {0},}

Where ${\sigma }$ ${\ displaystyle {\ sigma}}$ - the number of permutations of fermion fields among themselves during T-ordering (permutation of bosonic fields does not affect the sign).

Consider the simplest case of the product of a pair of field functions at different spatio-temporal points $\phi (x)\phi (y)$ ${\ displaystyle \ phi (x) \ phi (y)}$ . As indicated above, this product of operators can be expressed in terms of normal form plus a commutator. Under the sign of chronological ordering, here you need to make a modification - instead of the switch, you need to use the so-called convolution ${\overline {\phi (x)\phi (y)}}$ ${\ displaystyle {\ overline {\ phi (x) \ phi (y)}}}$ equal to the switch $[\phi ^{-}(x),\phi ^{+}(y)]$ ${\ displaystyle [\ phi ^ {-} (x), \ phi ^ {+} (y)]}$ , if a $x^{0}>y^{0}$ ${\ displaystyle x ^ {0}> y ^ {0}}$ and switch $[\phi ^{-}(y),\phi ^{+}(x)]$ ${\ displaystyle [\ phi ^ {-} (y), \ phi ^ {+} (x)]}$ , if a $y^{0}>x^{0}$ ${\ displaystyle y ^ {0}> x ^ {0}}$ . Thus, the chronological product of two field functions is equal to their product in normal form plus convolution:

{\mathcal {T}}\{\phi (x)\phi (y)\}={\mathcal {N}}\{\phi (x)\phi (y)\}+{\overline {\phi (x)\phi (y)}}.

{\ displaystyle {\ mathcal {T}} \ {\ phi (x) \ phi (y) \} = {\ mathcal {N}} \ {\ phi (x) \ phi (y) \} + {\ overline {\ phi (x) \ phi (y)}}.}

Wick's theorem generalizes this representation to the case of an arbitrary number of factors:

{\mathcal {T}}\{f_{1}f_{2}...f_{n})\}=\sum (-1)^{\sigma }{\overline {f_{i_{1}}f_{i_{2}}}}...{\overline {f_{i_{k-1}}f_{i_{k}}}}{\mathcal {N}}\{f_{i_{k+1}}...f_{i_{n}}\},

{\ displaystyle {\ mathcal {T}} \ {f_ {1} f_ {2} ... f_ {n}) \} = \ sum (-1) ^ {\ sigma} {\ overline {f_ {i_ { 1}} f_ {i_ {2}}}} ... {\ overline {f_ {i_ {k-1}} f_ {i_ {k}}}} {\ mathcal {N}} \ {f_ {i_ { k + 1}} ... f_ {i_ {n}} \},}

where the sum is taken over all possible pairwise convolutions of functions ( $k$ ${\ displaystyle k}$ - even numbers from 0 to $n$ ${\ displaystyle n}$ )

Basic switching relationships

We define an explicit expression for the vacuum average of the product of the field operators of the Klein-Gordon scalar field, taking into account the foregoing

\langle 0|\phi (x)\phi (y)|0\rangle =\langle 0|[\phi ^{-}(x)\phi ^{+}(y)]|0\rangle =[\phi ^{-}(x)\phi ^{+}(y)]={\frac {1}{(2\pi )^{3}}}\int \int {\frac {d^{3}\mathbf {p} d^{3}\mathbf {p'} }{2{\sqrt {p_{0}p'_{0}}}}}e^{-ip(x-y)}[a(\mathbf {p} )a^{+}(\mathbf {p} )]=

{\ displaystyle \ langle 0 | \ phi (x) \ phi (y) | 0 \ rangle = \ langle 0 | [\ phi ^ {-} (x) \ phi ^ {+} (y)] | 0 \ rangle = [\ phi ^ {-} (x) \ phi ^ {+} (y)] = {\ frac {1} {(2 \ pi) ^ {3}}} \ int \ int {\ frac {d ^ {3} \ mathbf {p} d ^ {3} \ mathbf {p '}} {2 {\ sqrt {p_ {0} p' _ {0}}}} e ^ {- ip (xy)} [ a (\ mathbf {p}) a ^ {+} (\ mathbf {p})] =}

$={\frac {1}{(2\pi )^{3}}}\int \int {\frac {d^{3}\mathbf {p} d^{3}\mathbf {p'} }{2{\sqrt {p_{0}p'_{0}}}}}e^{-ip(x-y)}\delta (\mathbf {p} -\mathbf {p'} )={\frac {1}{(2\pi )^{3}}}\int {\frac {d^{3}\mathbf {p} }{2p_{0}}}e^{-ip(x-y)}.$ ${\ displaystyle = {\ frac {1} {(2 \ pi) ^ {3}}} \ int \ int {\ frac {d ^ {3} \ mathbf {p} d ^ {3} \ mathbf {p ' }} {2 {\ sqrt {p_ {0} p '_ {0}}}}} e ^ {- ip (xy)} \ delta (\ mathbf {p} - \ mathbf {p'}) = {\ frac {1} {(2 \ pi) ^ {3}}} \ int {\ frac {d ^ {3} \ mathbf {p}} {2p_ {0}}} e ^ {- ip (xy)}. }$

Denote this function as $D^{-}(x-y)$ ${\ displaystyle D ^ {-} (xy)}$ . This is the amplitude of particle propagation from the point $y$ ${\ displaystyle y}$ exactly $x$ ${\ displaystyle x}$ . It can be shown that this function is Lorentz-invariant. Obviously, the field function commutator is expressed through this function as follows:

[\phi (x)\phi (y)]=D^{-}(x-y)-D^{-}(y-x)=D(x-y)={\frac {1}{(2\pi )^{3}}}\int {\frac {d^{3}\mathbf {p} }{2p_{0}}}(e^{-ip(x-y)}-e^{-ip(y-x)}).

{\ displaystyle [\ phi (x) \ phi (y)] = D ^ {-} (xy) -D ^ {-} (yx) = D (xy) = {\ frac {1} {(2 \ pi ) ^ {3}}} \ int {\ frac {d ^ {3} \ mathbf {p}} {2p_ {0}}} (e ^ {- ip (xy)} - e ^ {- ip (yx) }).}

For any space-like interval $(x-y)^{2}<0$ ${\ displaystyle (xy) ^ {2} <0}$ you can choose a reporting system so that $x-y$ ${\ displaystyle xy}$ changed sign, and due to Lorentz invariance, this means that the corresponding commutator is equal to zero. This means that at points separated by a space-like interval, measurements are possible and they do not affect each other. That is, no measurement can affect another measurement outside the light cone. This means observing the causality principle in quantum field theory. For complex fields, the causality principle requires a particle-antiparticle pair with the same masses and opposite “charges”.

Field operator switches with birth and annihilation operators are easier to derive. Let us present these switching relations without derivation.

For scalar field $[a^{\pm }(\mathbf {p} ),\phi (x)]=\pm {\frac {e^{\pm ipx}}{(2\pi )^{3/2}{\sqrt {2p_{0}}}}}$ ${\ displaystyle [a ^ {\ pm} (\ mathbf {p}), \ phi (x)] = \ pm {\ frac {e ^ {\ pm ipx}} {(2 \ pi) ^ {3/2 } {\ sqrt {2p_ {0}}}}}}$

For spinor field $[a_{r}(\mathbf {p} ),{\bar {\psi }}(x)]={\frac {e^{ipx}{\bar {u}}_{r}(\mathbf {p} )}{(2\pi )^{3/2}{\sqrt {2p_{0}}}}}$ ${\ displaystyle [a_ {r} (\ mathbf {p}), {\ bar {\ psi}} (x)] = {\ frac {e ^ {ipx} {\ bar {u}} _ {r} ( \ mathbf {p})} {(2 \ pi) ^ {3/2} {\ sqrt {2p_ {0}}}}}}$

For electromagnetic field $[a_{\lambda }(\mathbf {p} ),A_{\mu }(x)]=-{\frac {e^{ipx}e_{\mu }^{\lambda }(\mathbf {p} )}{(2\pi )^{3/2}{\sqrt {2p_{0}}}}}$ ${\ displaystyle [a _ {\ lambda} (\ mathbf {p}), A _ {\ mu} (x)] = - {\ frac {e ^ {ipx} e _ {\ mu} ^ {\ lambda} (\ mathbf {p})} {(2 \ pi) ^ {3/2} {\ sqrt {2p_ {0}}}}}}$

Propagators

Рассмотрим вакуумное среднее от хронологического произведения двух полевых операторов скалярного поля:

D^{c}(x-y)=i\langle 0|T\phi (x)\phi (y)|0\rangle =i(\theta (x_{0}-y_{0})D^{-}(x-y)+\theta (y_{0}-x_{0})D^{-}(y-x))={\frac {i}{(2\pi )^{3}}}\int {\frac {d^{3}\mathbf {p} }{2p_{0}}}(\theta (x_{0}-y_{0})e^{-ip(x-y)}+\theta (y_{0}-x_{0})e^{-ip(y-x)}).

D^{c}(xy)=i\langle 0|T\phi (x)\phi (y)|0\rangle =i(\theta (x_{0}-y_{0})D^{-}(xy)+\theta (y_{0}-x_{0})D^{-}(yx))={\frac {i}{(2\pi )^{3}}}\int {\frac {d^{3}\mathbf {p} }{2p_{0}}}(\theta (x_{0}-y_{0})e^{-ip(xy)}+\theta (y_{0}-x_{0})e^{-ip(yx)}).

Очевидно, функция $D^{c}(x)$ $D^{c}(x)$ является четной. Непосредственно можно убедиться, что данная функция является функцией Грина для оператора Клейна-Гордона, то есть

(\partial ^{2}+m^{2})D^{c}(x)=\delta (x).

(\partial ^{2}+m^{2})D^{c}(x)=\delta (x).

Следовательно, 4-мерный фурье-образ этой функции должен быть пропорционален $(m^{2}-p^{2})^{-1}$ $(m^{2}-p^{2})^{-1}$ . Однако, в силу неопределённости в точках на массовой поверхности $m^{2}-p^{2}=0$ $m^{2}-p^{2}=0$ импульсное представление данной функции записывают следующим образом:

D^{c}(x)={\frac {1}{(2\pi )^{4}}}\int {\frac {d^{4}pe^{-ipx}}{m^{2}-p^{2}-i\epsilon }},

D^{c}(x)={\frac {1}{(2\pi )^{4}}}\int {\frac {d^{4}pe^{-ipx}}{m^{2}-p^{2}-i\epsilon }},

Where $\epsilon$ $\epsilon$ — бесконечно малая величина, которая задаёт обходы полюсов $p_{0}=\pm {\sqrt {\mathbf {p} ^{2}+m^{2}}}$ $p_{0}=\pm {\sqrt {\mathbf {p} ^{2}+m^{2}}}$ при интегрировании по $p_{0}$ $p_{0}$ .

Пропагаторы базовых полей (ненулевыми являются только свертки одинаковых полей противоположных зарядов)

Field	Value	Formula
Вещественное или комплексное скалярное поле	$\langle 0\|T\phi (x)\phi ^{}(y)\|0\rangle$ $\langle 0\|T\phi (x)\phi ^{}(y)\|0\rangle$	$-iD^{c}(x-y)={\frac {i}{(2\pi )^{4}}}\int {\frac {d^{4}pe^{-ip(x-y)}}{p^{2}-m^{2}+i\epsilon }}$ $-iD^{c}(xy)={\frac {i}{(2\pi )^{4}}}\int {\frac {d^{4}pe^{-ip(xy)}}{p^{2}-m^{2}+i\epsilon }}$
Спинорное поле	$\langle 0\|T\psi (x){\bar {\psi }}(y)\|0\rangle$ $\langle 0\|T\psi (x){\bar {\psi }}(y)\|0\rangle$	$({\hat {\partial }}_{x}-im)D^{c}(x-y)={\frac {i}{(2\pi )^{4}}}\int {\frac {d^{4}pe^{-ip(x-y)}({\hat {p}}+m)}{p^{2}-m^{2}+i\epsilon }}$ $({\hat {\partial }}_{x}-im)D^{c}(xy)={\frac {i}{(2\pi )^{4}}}\int {\frac {d^{4}pe^{-ip(xy)}({\hat {p}}+m)}{p^{2}-m^{2}+i\epsilon }}$
Массивное векторное поле	$\langle 0\|TU_{\mu }(x)U_{\nu }^{}(y)\|0\rangle$ $\langle 0\|TU_{\mu }(x)U_{\nu }^{}(y)\|0\rangle$	${\frac {-i}{(2\pi )^{4}}}\int {\frac {d^{4}pe^{-ip(x-y)}(g_{\mu \nu }-p_{\mu }p_{\nu }/m^{2})}{p^{2}-m^{2}+i\epsilon }}$ ${\frac {-i}{(2\pi )^{4}}}\int {\frac {d^{4}pe^{-ip(xy)}(g_{\mu \nu }-p_{\mu }p_{\nu }/m^{2})}{p^{2}-m^{2}+i\epsilon }}$
Вещественное безмассовое векторное (электромагнитное) поле	$\langle 0\|TA_{\mu }(x)A_{\nu }(y)\|0\rangle$ $\langle 0\|TA_{\mu }(x)A_{\nu }(y)\|0\rangle$	${\frac {-i}{(2\pi )^{4}}}\int {\frac {d^{4}pe^{-ip(x-y)}g_{\mu \nu }}{p^{2}+i\epsilon }}$ ${\frac {-i}{(2\pi )^{4}}}\int {\frac {d^{4}pe^{-ip(xy)}g_{\mu \nu }}{p^{2}+i\epsilon }}$

S-матрица

Пусть задано начальное состояние полей $|in\rangle$ $|in\rangle$ in the “distant” past and the final state in the “distant” future $|out\rangle$ ${\ displaystyle | out \ rangle}$ . It is assumed that in the “distant” past and future there is no interaction, but it is “turned on” in some finite spatio-temporal region. Operator $S$ ${\ displaystyle S}$ that transfers the initial state to the final state is called the scattering operator:

|out\rangle =S|in\rangle .

{\ displaystyle | out \ rangle = S | in \ rangle.}

Accordingly, the amplitude ${\mathcal {M}}$ ${\ displaystyle {\ mathcal {M}}}$ transition from the initial state to the final state is equal to:

{\mathcal {M}}=\langle out|S|in\rangle .

{\ displaystyle {\ mathcal {M}} = \ langle out | S | in \ rangle.}

The scattering operator can be expressed in terms of matrix elements in some basis. The corresponding infinite-dimensional matrix is called the scattering matrix or $S$ ${\ displaystyle S}$ -matrix. The squares of the moduli of the matrix elements determine the probabilities of transitions between the basis vectors of the initial and final states.

Based on the general requirements of relativistic covariance, causality, unitarity, as well as the correspondence principle, it can be shown that $S$ ${\ displaystyle S}$ -matrix (operator) is expressed through the interaction Lagrangian as follows (sometimes this formula is also obtained using perturbation theory):

S=Te^{i\int d^{4}x{\mathcal {L}}_{I}(\phi (x))}=\sum _{n}{\frac {i^{n}}{n!}}T(\int d^{4}x{\mathcal {L}}_{I}(x))^{n}=\sum _{n}{\frac {i^{n}}{n!}}\int T\prod _{j=1}^{n}d^{4}x_{j}{\mathcal {L}}_{I}(x_{j}),

{\ displaystyle S = Te ^ {i \ int d ^ {4} x {\ mathcal {L}} _ {I} (\ phi (x))} = \ sum _ {n} {\ frac {i ^ { n}} {n!}} T (\ int d ^ {4} x {\ mathcal {L}} _ {I} (x)) ^ {n} = \ sum _ {n} {\ frac {i ^ {n}} {n!}} \ int T \ prod _ {j = 1} ^ {n} d ^ {4} x_ {j} {\ mathcal {L}} _ {I} (x_ {j}) ,}

$T e$ ${\ displaystyle Te}$ - chronological exhibitor, $T$ ${\ displaystyle T}$ -exponent, understood as expansion into the above infinite series in $T$ ${\ displaystyle T}$ -products (chronological works) $T\prod _{j=1}^{n}$ ${\ displaystyle T \ prod _ {j = 1} ^ {n}}$ .

Let the initial state have the form $|in\rangle =a^{+}(\mathbf {p_{1}} )...a^{+}(\mathbf {p_{s}} )|0\rangle$ ${\ displaystyle | in \ rangle = a ^ {+} (\ mathbf {p_ {1}}) ... a ^ {+} (\ mathbf {p_ {s}}) | 0 \ rangle}$ , and the final state $|out\rangle =a^{+}(\mathbf {p'_{1}} )...a^{+}(\mathbf {p'_{r}} )|0\rangle$ ${\ displaystyle | out \ rangle = a ^ {+} (\ mathbf {p '_ {1}}) ... a ^ {+} (\ mathbf {p' _ {r}}) | 0 \ rangle}$ . Then the contribution $n$ ${\ displaystyle n}$ th order of perturbation theory will be equal to the vacuum mean of the following form (coupling constant $g$ ${\ displaystyle g}$ derived from the interaction Lagrangian):

{\frac {i^{n}g^{n}}{n!}}\langle 0|a(\mathbf {p'_{1}} )...a(\mathbf {p'_{r}} )\int T\prod _{j=1}^{n}d^{4}x_{j}{\mathcal {L}}_{I}(x_{j})a^{+}(\mathbf {p_{1}} )...a^{+}(\mathbf {p_{s}} )|0\rangle .

{\ displaystyle {\ frac {i ^ {n} g ^ {n}} {n!}} \ langle 0 | a (\ mathbf {p '_ ​​{1}}) ... a (\ mathbf {p' _ {r}}) \ int T \ prod _ {j = 1} ^ {n} d ^ {4} x_ {j} {\ mathcal {L}} _ {I} (x_ {j}) a ^ { +} (\ mathbf {p_ {1}}) ... a ^ {+} (\ mathbf {p_ {s}}) | 0 \ rangle.}

Taking into account Wick's theorem, such averages will be expanded into terms in which all convolutions in these terms are deduced from the vacuum mean and the remaining field operators in normal form will only participate in (anti) commutators with operators of the initial and final state, generating standard contributions from such switches. A nonzero contribution can be made only by those terms in which the number and type of fields under the normal product sign will correspond to the type and total number of particles in the initial and final states. These nonzero contributions are also deduced from the sign of the vacuum average (because they are not operators, too) and in these terms there remain factors with vacuum plates without operators $\langle 0|0\rangle$ ${\ displaystyle \ langle 0 | 0 \ rangle}$ , which is equal to unity by definition. Thus, in the final expressions there are no operators and vacuum envelopes left, convolutions and expressions for commutators of field operators with operators of initial and final states remain. Convolutions are replaced by their impulse representations - propagators, and integration over spatio-temporal coordinates eliminates all exponentials, replacing them with delta functions from the sums of 4 pulses. Impulse integrals also destroy most of these delta functions. What kind of finite expressions are obtained can be formalized using the rules and the corresponding Feynman diagrams.

Example

Feynman Rules and Charts

Functional Integral

Spontaneous symmetry breaking. Higgs Mechanism

Divergences. Regularization, renormalization, renormalization group

Axiomatic quantum field theory

Bogolyubov's approach

Whiteman approach

Haag Ruel's Approach

Nonlocal quantum field theory

Based on the assumption of nonlocality of interactions. The interactions of the quantum fields under consideration do not occur at a point, but in a region of space. This assumption avoids ultraviolet divergences .