Editing Quantum field theory (section)

==Principles==
For simplicity, [[natural units]] are used in the following sections, in which the [[reduced Planck constant]] {{math|''ħ''}} and the [[speed of light]] {{math|''c''}} are both set to one.

===Classical fields===
{{See also|Classical field theory}}

A classical [[field (physics)|field]] is a [[function (mathematics)|function]] of spatial and time coordinates.<ref name="tong1">{{harvnb|Tong|2015|loc=Chapter 1}}</ref> Examples include the [[gravitational field]] in [[Newtonian gravity]] {{math|'''g'''('''x''', ''t'')}} and the [[electric field]] {{math|'''E'''('''x''', ''t'')}} and [[magnetic field]] {{math|'''B'''('''x''', ''t'')}} in [[classical electromagnetism]]. A classical field can be thought of as a numerical quantity assigned to every point in space that changes in time. Hence, it has infinitely many [[degrees of freedom (mechanics)|degrees of freedom]].<ref name="tong1" /><ref>In fact, its number of degrees of freedom is uncountable, because the vector space dimension of the space of continuous (differentiable, real analytic) functions on even a finite dimensional Euclidean space is uncountable.  On the other hand, subspaces (of these function spaces) that one typically considers, such as Hilbert spaces (e.g. the space of square integrable real valued functions) or separable Banach spaces (e.g. the space of continuous real-valued functions on a compact interval, with the uniform convergence norm), have denumerable (i. e. countably infinite) dimension in the category of Banach spaces (though still their Euclidean vector space dimension is uncountable), so in these restricted contexts, the number of degrees of freedom (interpreted now as the vector space dimension of a dense subspace rather than the vector space dimension of the function space of interest itself) is denumerable.</ref>

Many phenomena exhibiting quantum mechanical properties cannot be explained by classical fields alone. Phenomena such as the [[photoelectric effect]] are best explained by discrete particles ([[photon]]s), rather than a spatially continuous field. The goal of quantum field theory is to describe various quantum mechanical phenomena using a modified concept of fields.

[[Canonical quantization]] and [[path integral formulation|path integral]]s are two common formulations of QFT.<ref name="zee">{{cite book |last=Zee |first=A. |date=2010 |title=Quantum Field Theory in a Nutshell |url=https://archive.org/details/isbn_9780691140346 |url-access=registration |publisher=Princeton University Press |isbn=978-0-691-01019-9 |author-link=Anthony Zee }}</ref>{{rp|61}} To motivate the fundamentals of QFT, an overview of classical field theory follows.

The simplest classical field is a real [[scalar field]] — a [[real number]] at every point in space that changes in time. It is denoted as {{math|''ϕ''('''x''', ''t'')}}, where {{math|'''x'''}} is the position vector, and {{math|''t''}} is the time. Suppose the [[Lagrangian (field theory)|Lagrangian]] of the field, <math>L</math>, is
:<math>L = \int d^3x\,\mathcal{L} = \int d^3x\,\left[\frac 12 \dot\phi^2 - \frac 12 (\nabla\phi)^2 - \frac 12 m^2\phi^2\right],</math>
where <math>\mathcal{L}</math> is the Lagrangian density, <math>\dot\phi</math> is the time-derivative of the field, {{math|∇}} is the gradient operator, and {{math|''m''}} is a real parameter (the "mass" of the field). Applying the [[Euler–Lagrange equation]] on the Lagrangian:{{r|peskin|page1=16}}
:<math>\frac{\partial}{\partial t} \left[\frac{\partial\mathcal{L}}{\partial(\partial\phi/\partial t)}\right] + \sum_{i=1}^3 \frac{\partial}{\partial x^i} \left[\frac{\partial\mathcal{L}}{\partial(\partial\phi/\partial x^i)}\right] - \frac{\partial\mathcal{L}}{\partial\phi} = 0,</math>
we obtain the [[equations of motion]] for the field, which describe the way it varies in time and space:
:<math>\left(\frac{\partial^2}{\partial t^2} - \nabla^2 + m^2\right)\phi = 0.</math>
This is known as the [[Klein–Gordon equation]].{{r|peskin|page1=17}}

The Klein–Gordon equation is a [[wave equation]], so its solutions can be expressed as a sum of [[normal mode]]s (obtained via [[Fourier transform]]) as follows:
:<math>\phi(\mathbf{x}, t) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{\mathbf{p}}}}\left(a_{\mathbf{p}} e^{-i\omega_{\mathbf{p}}t + i\mathbf{p}\cdot\mathbf{x}} + a_{\mathbf{p}}^* e^{i\omega_{\mathbf{p}}t - i\mathbf{p}\cdot\mathbf{x}}\right),</math>
where {{math|''a''}} is a [[complex number]] (normalized by convention), {{math|*}} denotes [[complex conjugation]], and {{math|''ω''<sub>'''p'''</sub>}} is the frequency of the normal mode:
:<math>\omega_{\mathbf{p}} = \sqrt{|\mathbf{p}|^2 + m^2}.</math>
Thus each normal mode corresponding to a single {{math|'''p'''}} can be seen as a classical [[harmonic oscillator]] with frequency {{math|''ω''<sub>'''p'''</sub>}}.{{r|peskin|page1=21,26}}

===Canonical quantization===
{{Main|Canonical quantization}}

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

The displacement of a classical harmonic oscillator is described by
:<math>x(t) = \frac{1}{\sqrt{2\omega}} a e^{-i\omega t} + \frac{1}{\sqrt{2\omega}} a^* e^{i\omega t},</math>
where {{math|''a''}} is a complex number (normalized by convention), and {{math|''ω''}} is the oscillator's frequency. Note that {{math|''x''}} is the displacement of a particle in simple harmonic motion from the equilibrium position,  not to be confused with the spatial label {{math|'''x'''}} of a quantum field.

For a quantum harmonic oscillator, {{math|''x''(''t'')}} is promoted to a [[linear operator]] <math>\hat x(t)</math>:
:<math>\hat x(t) = \frac{1}{\sqrt{2\omega}} \hat a e^{-i\omega t} + \frac{1}{\sqrt{2\omega}} \hat a^\dagger e^{i\omega t}.</math>
Complex numbers {{math|''a''}} and {{math|''a''<sup>*</sup>}} are replaced by the [[annihilation operator]] <math>\hat a</math> and the [[creation operator]] <math>\hat a^\dagger</math>, respectively, where {{math|†}} denotes [[Hermitian conjugation]]. The [[commutation relation]] between the two is
:<math>\left[\hat a, \hat a^\dagger\right] = 1.</math>
The [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the simple harmonic oscillator can be written as
:<math>\hat H = \hbar\omega \hat{a}^\dagger \hat{a} +\frac{1}{2}\hbar\omega.</math>
The [[vacuum state]] <math>|0\rang</math>, which is the lowest energy state, is defined by
:<math>\hat a|0\rang = 0</math>
and has energy <math>\frac12\hbar\omega.</math>
One can easily check that <math>[\hat H, \hat{a}^\dagger]=\hbar\omega\hat{a}^\dagger,</math> which implies that <math>\hat{a}^\dagger</math> increases the energy of the simple harmonic oscillator by <math>\hbar\omega</math>. For example, the state <math>\hat{a}^\dagger|0\rang</math> is an eigenstate of energy <math>3\hbar\omega/2</math>.
Any energy eigenstate state of a single harmonic oscillator can be obtained from <math>|0\rang</math> by successively applying the creation operator <math>\hat a^\dagger</math>:{{r|peskin|page1=20}} and any state of the system can be expressed as a linear combination of the states
:<math>|n\rang \propto \left(\hat a^\dagger\right)^n|0\rang.</math>

A similar procedure can be applied to the real scalar field {{math|''ϕ''}}, by promoting it to a quantum field operator <math>\hat\phi</math>, while the annihilation operator <math>\hat a_{\mathbf{p}}</math>, the creation operator <math>\hat a_{\mathbf{p}}^\dagger</math> and the angular frequency <math>\omega_\mathbf {p}</math>are now for a particular {{math|'''p'''}}:
:<math>\hat \phi(\mathbf{x}, t) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{\mathbf{p}}}}\left(\hat a_{\mathbf{p}} e^{-i\omega_{\mathbf{p}}t + i\mathbf{p}\cdot\mathbf{x}} + \hat a_{\mathbf{p}}^\dagger e^{i\omega_{\mathbf{p}}t - i\mathbf{p}\cdot\mathbf{x}}\right).</math>
Their commutation relations are:{{r|peskin|page1=21}}
:<math>\left[\hat a_{\mathbf p}, \hat a_{\mathbf q}^\dagger\right] = (2\pi)^3\delta(\mathbf{p} - \mathbf{q}),\quad \left[\hat a_{\mathbf p}, \hat a_{\mathbf q}\right] = \left[\hat a_{\mathbf p}^\dagger, \hat a_{\mathbf q}^\dagger\right] = 0,</math>
where {{math|''δ''}} is the [[Dirac delta function]]. The vacuum state <math>|0\rang</math> is defined by
:<math>\hat a_{\mathbf p}|0\rang = 0,\quad \text{for all }\mathbf p.</math>
Any quantum state of the field can be obtained from <math>|0\rang</math> by successively applying creation operators <math>\hat a_{\mathbf{p}}^\dagger</math> (or by a linear combination of such states), e.g. {{r|peskin|page1=22}}
:<math>\left(\hat a_{\mathbf{p}_3}^\dagger\right)^3 \hat a_{\mathbf{p}_2}^\dagger \left(\hat a_{\mathbf{p}_1}^\dagger\right)^2 |0\rang.</math>

While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles. The latter space is known as a [[Fock space]], which can account for the fact that particle numbers are not fixed in relativistic quantum systems.<ref>{{cite journal |last1=Fock |first1=V. |author-link=Vladimir Fock |date=1932-03-10 |title=Konfigurationsraum und zweite Quantelung |journal=Zeitschrift für Physik |volume=75 |issue=9–10 |pages=622–647 |doi=10.1007/BF01344458 |language=de |bibcode=1932ZPhy...75..622F |s2cid=186238995 }}</ref> The process of quantizing an arbitrary number of particles instead of a single particle is often also called [[second quantization]].{{r|peskin|page1=19}}

The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields, [[Dirac field]]s,{{r|peskin|page1=52}} [[vector field]]s (''e.g.'' the electromagnetic field), and even [[string theory|strings]].<ref>{{cite book |last1=Becker |first1=Katrin |last2=Becker |first2=Melanie|author-link2=Melanie Becker |last3=Schwarz |first3=John H. |date=2007 |title=String Theory and M-Theory |url=https://archive.org/details/stringtheorymthe00beck_649 |url-access=limited |publisher=Cambridge University Press |page=[https://archive.org/details/stringtheorymthe00beck_649/page/n53 36] |isbn=978-0-521-86069-7 |author-link3=John Henry Schwarz }}</ref> However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory). In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes. To perform calculations on any realistic interacting theory, [[perturbation theory (quantum mechanics)|perturbation theory]] would be necessary.

The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a [[quartic interaction]] term could be introduced to the Lagrangian of the real scalar field:{{r|peskin|page1=77}}
:<math>\mathcal{L} = \frac 12 (\partial_\mu\phi)\left(\partial^\mu\phi\right) - \frac 12 m^2\phi^2 - \frac{\lambda}{4!}\phi^4,</math>
where {{math|''μ''}} is a spacetime index, <math>\partial_0 = \partial/\partial t,\ \partial_1 = \partial/\partial x^1</math>, etc. The summation over the index {{math|''μ''}} has been omitted following the [[Einstein notation]]. If the parameter {{math|''λ''}} is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory.

===Path integrals===
{{Main|Path integral formulation}}

The [[path integral formulation]] of QFT is concerned with the direct computation of the [[scattering amplitude]] of a certain interaction process, rather than the establishment of operators and state spaces. To calculate the [[probability amplitude]] for a system to evolve from some initial state <math>|\phi_I\rang</math> at time {{math|''t'' {{=}} 0}} to some final state <math>|\phi_F\rang</math> at {{math|''t'' {{=}} ''T''}}, the total time {{math|''T''}} is divided into {{math|''N''}} small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let {{math|''H''}} be the [[Hamiltonian (quantum mechanics)|Hamiltonian]] (''i.e.'' [[time evolution operator|generator of time evolution]]), then{{r|zee|page1=10}}
:<math>\lang \phi_F|e^{-iHT}|\phi_I\rang = \int d\phi_1\int d\phi_2\cdots\int d\phi_{N-1}\,\lang \phi_F|e^{-iHT/N}|\phi_{N-1}\rang\cdots\lang \phi_2|e^{-iHT/N}|\phi_1\rang\lang \phi_1|e^{-iHT/N}|\phi_I\rang.</math>
Taking the limit {{math|''N'' → ∞}}, the above product of integrals becomes the Feynman path integral:{{r|peskin|zee|page1=282|page2=12}}
:<math>\lang \phi_F|e^{-iHT}|\phi_I\rang = \int \mathcal{D}\phi(t)\,\exp\left\{i\int_0^T dt\,L\right\},</math>
where {{math|''L''}} is the Lagrangian involving {{math|''ϕ''}} and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian {{math|''H''}} via [[Legendre transformation]]. The initial and final conditions of the path integral are respectively
:<math>\phi(0) = \phi_I,\quad \phi(T) = \phi_F.</math>
In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand.

===Two-point correlation function===
{{Main|Correlation function (quantum field theory)}}

In calculations, one often encounters expression like<math display="block">\lang 0|T\{\phi(x)\phi(y)\}|0\rang
\quad \text{or} \quad
\lang \Omega |T\{\phi(x)\phi(y)\}|\Omega \rang</math>in the free or interacting theory, respectively. Here, <math>x</math> and <math>y</math> are position [[four-vector]]s, <math>T</math> is the [[time ordering]] operator that shuffles its operands so the time-components <math>x^0</math> and <math>y^0</math> increase from right to left, and <math>|\Omega\rang</math> is the ground state (vacuum state) of the interacting theory, different from the free ground state <math>| 0 \rang</math>. This expression represents the probability amplitude for the field to propagate from {{math|''y''}} to {{math|''x''}}, and goes by multiple names, like the two-point [[propagator]], two-point [[correlation function (quantum field theory)|correlation function]], two-point [[Green's function]] or two-point function for short.{{r|peskin|page1=82}}

The free two-point function, also known as the [[Feynman propagator]], can be found for the real scalar field by either canonical quantization or path integrals to be{{r|peskin|zee|page1=31,288|page2=23}}
:<math>\lang 0|T\{\phi(x)\phi(y)\} |0\rang \equiv D_F(x-y) = \lim_{\epsilon\to 0} \int\frac{d^4p}{(2\pi)^4} \frac{i}{p_\mu p^\mu - m^2 + i\epsilon} e^{-ip_\mu (x^\mu - y^\mu)}.</math>
In an interacting theory, where the Lagrangian or Hamiltonian contains terms <math>L_I(t)</math> or <math>H_I(t)</math> that describe interactions, the two-point function is more difficult to define. However, through both the canonical quantization formulation and the path integral formulation, it is possible to express it through an infinite perturbation series of the ''free'' two-point function.

In canonical quantization, the two-point correlation function can be written as:{{r|peskin|page1=87}}
:<math>\lang\Omega|T\{\phi(x)\phi(y)\}|\Omega\rang = \lim_{T\to\infty(1-i\epsilon)} \frac{\left\lang 0\left|T\left\{\phi_I(x)\phi_I(y)\exp\left[-i\int_{-T}^T dt\, H_I(t)\right]\right\}\right|0\right\rang}{\left\lang 0\left|T\left\{\exp\left[-i\int_{-T}^T dt\, H_I(t)\right]\right\}\right|0\right\rang},</math>
where {{math|''ε''}} is an [[infinitesimal]] number and {{math|''ϕ<sub>I</sub>''}} is the field operator under the free theory. Here, the [[Exponential function|exponential]] should be understood as its [[power series]] expansion. For example, in <math>\phi^4</math>-theory, the interacting term of the Hamiltonian is <math display="inline">H_I(t) = \int d^3 x\,\frac{\lambda}{4!}\phi_I(x)^4</math>,{{r|peskin|page1=84}} and the expansion of the two-point correlator in terms of <math>\lambda</math> becomes<math display="block">\lang\Omega|T\{\phi(x)\phi(y)\}|\Omega\rang = 
\frac{
    \displaystyle \sum_{n=0}^\infty \frac{(-i \lambda)^n}{(4 !)^n n !} \int d^4 z_1 \cdots \int d^4 z_n \lang 0|T\{\phi_I(x)\phi_I(y)\phi_I(z_1)^4\cdots\phi_I(z_n)^4\}|0\rang}{
    \displaystyle \sum_{n=0}^\infty \frac{(-i \lambda)^n}{(4 !)^n n !} \int d^4 z_1 \cdots \int d^4 z_n \lang 0|T\{                  \phi_I(z_1)^4\cdots\phi_I(z_n)^4\}|0\rang
}.</math>This perturbation expansion expresses the interacting two-point function in terms of quantities <math>\lang 0 | \cdots | 0 \rang</math> that are evaluated in the ''free'' theory.

In the path integral formulation, the two-point correlation function can be written{{r|peskin|page1=284}}
:<math>\lang\Omega|T\{\phi(x)\phi(y)\}|\Omega\rang = \lim_{T\to\infty(1-i\epsilon)} \frac{\int\mathcal{D}\phi\,\phi(x)\phi(y)\exp\left[i\int_{-T}^T d^4z\,\mathcal{L}\right]}{\int\mathcal{D}\phi\,\exp\left[i\int_{-T}^T d^4z\,\mathcal{L}\right]},</math>
where <math>\mathcal{L}</math> is the Lagrangian density. As in the previous paragraph, the exponential can be expanded as a series in {{math|''λ''}}, reducing the interacting two-point function to quantities in the free theory.

[[Wick's theorem]] further reduce any {{math|''n''}}-point correlation function in the free theory to a sum of products of two-point correlation functions. For example,
:<math>\begin{align}
\lang 0|T\{\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\}|0\rang &= \lang 0|T\{\phi(x_1)\phi(x_2)\}|0\rang \lang 0|T\{\phi(x_3)\phi(x_4)\}|0\rang\\
&+ \lang 0|T\{\phi(x_1)\phi(x_3)\}|0\rang \lang 0|T\{\phi(x_2)\phi(x_4)\}|0\rang\\
&+ \lang 0|T\{\phi(x_1)\phi(x_4)\}|0\rang \lang 0|T\{\phi(x_2)\phi(x_3)\}|0\rang.
\end{align}</math>
Since interacting correlation functions can be expressed in terms of free correlation functions, only the latter need to be evaluated in order to calculate all physical quantities in the (perturbative) interacting theory.{{r|peskin|page1=90}} This makes the Feynman propagator one of the most important quantities in quantum field theory.

===Feynman diagram===
{{Main|Feynman diagram}}

Correlation functions in the interacting theory can be written as a perturbation series. Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a [[Feynman diagram]]. For example, the {{math|''λ''<sup>1</sup>}} term in the two-point correlation function in the {{math|''ϕ''<sup>4</sup>}} theory is
:<math>\frac{-i\lambda}{4!}\int d^4z\,\lang 0|T\{\phi(x)\phi(y)\phi(z)\phi(z)\phi(z)\phi(z)\}|0\rang.</math>
After applying Wick's theorem, one of the terms is
:<math>12\cdot \frac{-i\lambda}{4!}\int d^4z\, D_F(x-z)D_F(y-z)D_F(z-z).</math>
This term can instead be obtained from the Feynman diagram

:[[File:Phi-4 one-loop.svg|200px]].

The diagram consists of

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

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

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

With the symmetry factor <math>2</math>, following these rules yields exactly the expression above. By Fourier transforming the propagator, the Feynman rules can be reformulated from position space into momentum space.{{r|peskin|page1=91–94}}

In order to compute the {{math|''n''}}-point correlation function to the {{math|''k''}}-th order, list all valid Feynman diagrams with {{math|''n''}} external points and {{math|''k''}} or fewer vertices, and then use Feynman rules to obtain the expression for each term. To be precise,
:<math>\lang\Omega|T\{\phi(x_1)\cdots\phi(x_n)\}|\Omega\rang</math>
is equal to the sum of (expressions corresponding to) all connected diagrams with {{math|''n''}} external points. (Connected diagrams are those in which every vertex is connected to an external point through lines. Components that are totally disconnected from external lines are sometimes called "vacuum bubbles".) In the {{math|''ϕ''<sup>4</sup>}} interaction theory discussed above, every vertex must have four legs.{{r|peskin|page1=98}}

In realistic applications, the scattering amplitude of a certain interaction or the [[decay rate]] of a particle can be computed from the [[S-matrix]], which itself can be found using the Feynman diagram method.{{r|peskin|page1=102–115}}

Feynman diagrams devoid of "loops" are called tree-level diagrams, which describe the lowest-order interaction processes; those containing {{math|''n''}} loops are referred to as {{math|''n''}}-loop diagrams, which describe higher-order contributions, or radiative corrections, to the interaction.{{r|zee|page1=44}} Lines whose end points are vertices can be thought of as the propagation of [[virtual particle]]s.{{r|peskin|page1=31}}

===Renormalization===
{{Main|Renormalization}}

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

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

The approach illustrated above is called bare perturbation theory, as calculations involve only the bare quantities such as mass and coupling constant. A different approach, called renormalized perturbation theory, is to use physically meaningful quantities from the very beginning. In the case of {{math|''ϕ''<sup>4</sup>}} theory, the field strength is first redefined:
:<math>\phi = Z^{1/2}\phi_r,</math>
where {{math|''ϕ''}} is the bare field, {{math|''ϕ<sub>r</sub>''}} is the renormalized field, and {{math|''Z''}} is a constant to be determined. The Lagrangian density becomes:
:<math>\mathcal{L} = \frac 12 (\partial_\mu\phi_r)(\partial^\mu\phi_r) - \frac 12 m_r^2\phi_r^2 - \frac{\lambda_r}{4!}\phi_r^4 + \frac 12 \delta_Z (\partial_\mu\phi_r)(\partial^\mu\phi_r) - \frac 12 \delta_m\phi_r^2 - \frac{\delta_\lambda}{4!}\phi_r^4,</math>
where {{math|''m<sub>r</sub>''}} and {{math|''λ<sub>r</sub>''}} are the experimentally measurable, renormalized, mass and coupling constant, respectively, and
:<math>\delta_Z = Z-1,\quad \delta_m = m^2Z - m_r^2,\quad \delta_\lambda = \lambda Z^2 - \lambda_r</math>
are constants to be determined. The first three terms are the {{math|''ϕ''<sup>4</sup>}} Lagrangian density written in terms of the renormalized quantities, while the latter three terms are referred to as "counterterms". As the Lagrangian now contains more terms, so the Feynman diagrams should include additional elements, each with their own Feynman rules. The procedure is outlined as follows. First select a regularization scheme (such as the cut-off regularization introduced above or [[dimensional regularization]]); call the regulator {{math|Λ}}. Compute Feynman diagrams, in which divergent terms will depend on {{math|Λ}}. Then, define {{math|''δ<sub>Z</sub>''}}, {{math|''δ<sub>m</sub>''}}, and {{math|''δ<sub>λ</sub>''}} such that Feynman diagrams for the counterterms will exactly cancel the divergent terms in the normal Feynman diagrams when the limit {{math|Λ → ∞}} is taken. In this way, meaningful finite quantities are obtained.{{r|peskin|page1=323–326}}

<!--"Is it true?" The renormalization procedure will lead to the same quantitative result and physical prediction irrespective of the regularization scheme chosen. -->It is only possible to eliminate all infinities to obtain a finite result in renormalizable theories, whereas in non-renormalizable theories infinities cannot be removed by the redefinition of a small number of parameters. The [[Standard Model]] of elementary particles is a renormalizable QFT,{{r|peskin|page1=719–727}} while [[quantum gravity]] is non-renormalizable.{{r|peskin|zee|page1=798|page2=421}}

====Renormalization group====
{{Main|Renormalization group}}

The [[renormalization group]], developed by [[Kenneth G. Wilson|Kenneth Wilson]], is a mathematical apparatus used to study the changes in physical parameters (coefficients in the Lagrangian) as the system is viewed at different scales.{{r|peskin|page1=393}} The way in which each parameter changes with scale is described by its [[beta function (physics)|''β'' function]].{{r|peskin|page1=417}} Correlation functions, which underlie quantitative physical predictions, change with scale according to the [[Callan–Symanzik equation]].{{r|peskin|page1=410–411}}

As an example, the coupling constant in QED, namely the [[elementary charge]] {{math|''e''}}, has the following ''β'' function:
:<math>\beta(e) \equiv \frac{1}{\Lambda}\frac{de}{d\Lambda} = \frac{e^3}{12\pi^2} + O\mathord\left(e^5\right),</math>
where {{math|Λ}} is the energy scale under which the measurement of {{math|''e''}} is performed. This [[differential equation]] implies that the observed elementary charge increases as the scale increases.<ref>{{cite arXiv |last=Fujita |first=Takehisa |eprint=hep-th/0606101 |title=Physics of Renormalization Group Equation in QED |date=2008-02-01 }}</ref> The renormalized coupling constant, which changes with the energy scale, is also called the running coupling constant.{{r|peskin|page1=420}}

The coupling constant {{math|''g''}} in [[quantum chromodynamics]], a non-Abelian gauge theory based on the symmetry group {{math|[[special unitary group|SU(3)]]}}, has the following ''β'' function:
:<math>\beta(g) \equiv \frac{1}{\Lambda}\frac{dg}{d\Lambda} = \frac{g^3}{16\pi^2}\left(-11 + \frac 23 N_f\right) + O\mathord\left(g^5\right),</math>
where {{math|''N<sub>f</sub>''}} is the number of [[quark]] [[flavour (particle physics)|flavours]]. In the case where {{math|''N<sub>f</sub>'' ≤ 16}} (the Standard Model has {{math|''N<sub>f</sub>'' {{=}} 6}}), the coupling constant {{math|''g''}} decreases as the energy scale increases. Hence, while the strong interaction is strong at low energies, it becomes very weak in high-energy interactions, a phenomenon known as [[asymptotic freedom]].{{r|peskin|page1=531}}

[[Conformal field theories]] (CFTs) are special QFTs that admit [[conformal symmetry]]. They are insensitive to changes in the scale, as all their coupling constants have vanishing ''β'' function. (The converse is not true, however — the vanishing of all ''β'' functions does not imply conformal symmetry of the theory.)<ref>{{Cite journal |last1=Aharony |first1=Ofer |last2=Gur-Ari |first2=Guy |last3=Klinghoffer |first3=Nizan |arxiv=1501.06664 |title=The Holographic Dictionary for Beta Functions of Multi-trace Coupling Constants |journal=Journal of High Energy Physics |volume=2015 |issue=5 |pages=31 |date=2015-05-19 |bibcode=2015JHEP...05..031A |doi=10.1007/JHEP05(2015)031 |s2cid=115167208 }}</ref> Examples include [[string theory]]<ref name="polchinski1">{{cite book |last=Polchinski |first=Joseph |date=2005 |title=String Theory |volume=1 |publisher=Cambridge University Press |isbn=978-0-521-67227-6 |author-link=Joseph Polchinski }}</ref> and [[N = 4 supersymmetric Yang–Mills theory|{{math|''N'' {{=}} 4}} supersymmetric Yang–Mills theory]].<ref>{{cite arXiv |last=Kovacs |first=Stefano |eprint=hep-th/9908171 |title={{math|''N'' {{=}} 4}} supersymmetric Yang–Mills theory and the AdS/SCFT correspondence |date=1999-08-26 }}</ref>

According to Wilson's picture, every QFT is fundamentally accompanied by its energy cut-off {{math|Λ}}, ''i.e.'' that the theory is no longer valid at energies higher than {{math|Λ}}, and all degrees of freedom above the scale {{math|Λ}} are to be omitted. For example, the cut-off could be the inverse of the atomic spacing in a condensed matter system, and in elementary particle physics it could be associated with the fundamental "graininess" of spacetime caused by quantum fluctuations in gravity. The cut-off scale of theories of particle interactions lies far beyond current experiments. Even if the theory were very complicated at that scale, as long as its couplings are sufficiently weak, it must be described at low energies by a renormalizable [[effective field theory]].{{r|peskin|page1=402–403}} The difference between renormalizable and non-renormalizable theories is that the former are insensitive to details at high energies, whereas the latter do depend on them.{{r|shifman|page1=2}} According to this view, non-renormalizable theories are to be seen as low-energy effective theories of a more fundamental theory. The failure to remove the cut-off {{math|Λ}} from calculations in such a theory merely indicates that new physical phenomena appear at scales above {{math|Λ}}, where a new theory is necessary.{{r|zee|page1=156}}

===Other theories===
The quantization and renormalization procedures outlined in the preceding sections are performed for the free theory and [[quartic interaction|{{math|''ϕ''<sup>4</sup>}} theory]] of the real scalar field. A similar process can be done for other types of fields, including the [[complex numbers|complex]] scalar field, the [[vector field]], and the [[Dirac field]], as well as other types of interaction terms, including the electromagnetic interaction and the [[Yukawa interaction]].

As an example, [[quantum electrodynamics]] contains a Dirac field {{math|''ψ''}} representing the [[electron]] field and a vector field {{math|''A<sup>μ</sup>''}} representing the electromagnetic field ([[photon]] field). (Despite its name, the quantum electromagnetic "field" actually corresponds to the classical [[electromagnetic four-potential]], rather than the classical electric and magnetic fields.) The full QED Lagrangian density is:
:<math>\mathcal{L} = \bar\psi\left(i\gamma^\mu\partial_\mu - m\right)\psi - \frac 14 F_{\mu\nu}F^{\mu\nu} - e\bar\psi\gamma^\mu\psi A_\mu,</math>
where {{math|''γ<sup>μ</sup>''}} are [[Dirac matrices]], <math>\bar\psi = \psi^\dagger\gamma^0</math>, and <math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu</math> is the [[electromagnetic field strength]]. The parameters in this theory are the (bare) electron mass {{math|''m''}} and the (bare) [[elementary charge]] {{math|''e''}}. The first and second terms in the Lagrangian density correspond to the free Dirac field and free vector fields, respectively. The last term describes the interaction between the electron and photon fields, which is treated as a perturbation from the free theories.{{r|peskin|page1=78}}
[[File:Electron-positron-annihilation.svg|thumb]]

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

====Gauge symmetry====
{{Main|Gauge theory}}

If the following transformation to the fields is performed at every spacetime point {{math|''x''}} (a local transformation), then the QED Lagrangian remains unchanged, or invariant:
:<math>\psi(x) \to e^{i\alpha(x)}\psi(x),\quad A_\mu(x) \to A_\mu(x) + ie^{-1} e^{-i\alpha(x)}\partial_\mu e^{i\alpha(x)},</math>
where {{math|''α''(''x'')}} is any function of spacetime coordinates. If a theory's Lagrangian (or more precisely the [[action (physics)|action]]) is invariant under a certain local transformation, then the transformation is referred to as a [[gauge symmetry]] of the theory.{{r|peskin|page1=482–483}} Gauge symmetries form a [[group (mathematics)|group]] at every spacetime point. In the case of QED, the successive application of two different local symmetry transformations <math>e^{i\alpha(x)}</math> and <math>e^{i\alpha'(x)}</math> is yet another symmetry transformation <math>e^{i[\alpha(x)+\alpha'(x)]}</math>. For any {{math|''α''(''x'')}}, <math>e^{i\alpha(x)}</math> is an element of the {{math|[[U(1)]]}} group, thus QED is said to have {{math|U(1)}} gauge symmetry.{{r|peskin|page1=496}} The photon field {{math|''A<sub>μ</sub>''}} may be referred to as the {{math|U(1)}} [[gauge boson]].

{{math|U(1)}} is an [[Abelian group]], meaning that the result is the same regardless of the order in which its elements are applied. QFTs can also be built on [[non-Abelian group]]s, giving rise to [[Yang–Mills theory|non-Abelian gauge theories]] (also known as Yang–Mills theories).{{r|peskin|page1=489}} [[Quantum chromodynamics]], which describes the strong interaction, is a non-Abelian gauge theory with an {{math|[[special unitary group|SU(3)]]}} gauge symmetry. It contains three Dirac fields {{math|''ψ<sup>i</sup>'', ''i'' {{=}} 1,2,3}} representing [[quark]] fields as well as eight vector fields {{math|''A<sup>a,μ</sup>'', ''a'' {{=}} 1,...,8}} representing [[gluon]] fields, which are the {{math|SU(3)}} gauge bosons.{{r|peskin|page1=547}} The QCD Lagrangian density is:{{r|peskin|page1=490–491}}
:<math>\mathcal{L} = i\bar\psi^i \gamma^\mu (D_\mu)^{ij} \psi^j - \frac 14 F_{\mu\nu}^aF^{a,\mu\nu} - m\bar\psi^i \psi^i,</math>
where {{math|''D<sub>μ</sub>''}} is the gauge [[covariant derivative]]:
:<math>D_\mu = \partial_\mu - igA_\mu^a t^a,</math>
where {{math|''g''}} is the coupling constant, {{math|''t<sup>a</sup>''}} are the eight [[Lie algebra|generators]] of {{math|SU(3)}} in the [[fundamental representation]] ({{math|3×3}} matrices),
:<math>F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc}A_\mu^b A_\nu^c,</math>
and {{math|''f<sup>abc</sup>''}} are the [[structure constants]] of {{math|SU(3)}}. Repeated indices {{math|''i'',''j'',''a''}} are implicitly summed over following Einstein notation. This Lagrangian is invariant under the transformation:
:<math>\psi^i(x) \to U^{ij}(x)\psi^j(x),\quad A_\mu^a(x) t^a \to U(x)\left[A_\mu^a(x) t^a + ig^{-1} \partial_\mu\right]U^\dagger(x),</math>
where {{math|''U''(''x'')}} is an element of {{math|SU(3)}} at every spacetime point {{math|''x''}}:
:<math>U(x) = e^{i\alpha(x)^a t^a}.</math>

The preceding discussion of symmetries is on the level of the Lagrangian. In other words, these are "classical" symmetries. After quantization, some theories will no longer exhibit their classical symmetries, a phenomenon called [[anomaly (physics)|anomaly]]. For instance, in the path integral formulation, despite the invariance of the Lagrangian density <math>\mathcal{L}[\phi,\partial_\mu\phi]</math> under a certain local transformation of the fields, the [[measure (mathematics)|measure]] <math display="inline">\int\mathcal D\phi</math> of the path integral may change.{{r|zee|page1=243}} For a theory describing nature to be consistent, it must not contain any anomaly in its gauge symmetry. The Standard Model of elementary particles is a gauge theory based on the group {{math|SU(3) × SU(2) × U(1)}}, in which all anomalies exactly cancel.{{r|peskin|page1=705–707}}

The theoretical foundation of [[general relativity]], the [[equivalence principle]], can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the [[Lorentz group]].<ref>Veltman, M. J. G. (1976). ''Methods in Field Theory, Proceedings of the Les Houches Summer School, Les Houches, France, 1975''.</ref>

[[Noether's theorem]] states that every continuous symmetry, ''i.e.'' the parameter in the symmetry transformation being continuous rather than discrete, leads to a corresponding [[conservation law]].{{r|peskin|zee|page1=17–18|page2=73}} For example, the {{math|U(1)}} symmetry of QED implies [[charge conservation]].<ref>{{cite journal |last1=Brading |first1=Katherine A.|author1-link=Katherine Brading |date=March 2002 |title=Which symmetry? Noether, Weyl, and conservation of electric charge |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |volume=33 |issue=1 |pages=3–22 |doi=10.1016/S1355-2198(01)00033-8 |bibcode=2002SHPMP..33....3B |citeseerx=10.1.1.569.106 }}</ref>

Gauge-transformations do not relate distinct quantum states. Rather, it relates two equivalent mathematical descriptions of the same quantum state. As an example, the photon field {{math|''A<sup>μ</sup>''}}, being a [[four-vector]], has four apparent degrees of freedom, but the actual state of a photon is described by its two degrees of freedom corresponding to the [[photon polarization|polarization]]. The remaining two degrees of freedom are said to be "redundant" — apparently different ways of writing {{math|''A<sup>μ</sup>''}} can be related to each other by a gauge transformation and in fact describe the same state of the photon field. In this sense, gauge invariance is not a "real" symmetry, but a reflection of the "redundancy" of the chosen mathematical description.{{r|zee|page1=168}}

To account for the gauge redundancy in the path integral formulation, one must perform the so-called [[Faddeev–Popov ghost|Faddeev–Popov]] [[gauge fixing]] procedure. In non-Abelian gauge theories, such a procedure introduces new fields called "ghosts". Particles corresponding to the ghost fields are called ghost particles, which cannot be detected externally.{{r|peskin|page1=512–515}} A more rigorous generalization of the Faddeev–Popov procedure is given by [[BRST quantization]].{{r|peskin|page1=517}}

====Spontaneous symmetry-breaking====
{{Main|Spontaneous symmetry breaking}}

[[Spontaneous symmetry breaking]] is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it.{{r|peskin|page1=347}}

To illustrate the mechanism, consider a linear [[sigma model]] containing {{math|''N''}} real scalar fields, described by the Lagrangian density:
:<math>\mathcal{L} = \frac 12 \left(\partial_\mu\phi^i\right)\left(\partial^\mu\phi^i\right) + \frac 12 \mu^2 \phi^i\phi^i - \frac{\lambda}{4} \left(\phi^i\phi^i\right)^2,</math>
where {{math|''μ''}} and {{math|''λ''}} are real parameters. The theory admits an {{math|[[orthogonal group|O(''N'')]]}} global symmetry:
:<math>\phi^i \to R^{ij}\phi^j,\quad R\in\mathrm{O}(N).</math>
The lowest energy state (ground state or vacuum state) of the classical theory is any uniform field {{math|''ϕ''<sub>0</sub>}} satisfying
:<math>\phi_0^i \phi_0^i = \frac{\mu^2}{\lambda}.</math>
Without loss of generality, let the ground state be in the {{math|''N''}}-th direction:
:<math>\phi_0^i = \left(0,\cdots,0,\frac{\mu}{\sqrt{\lambda}}\right).</math>
The original {{math|''N''}} fields can be rewritten as:
:<math>\phi^i(x) = \left(\pi^1(x),\cdots,\pi^{N-1}(x),\frac{\mu}{\sqrt{\lambda}} + \sigma(x)\right),</math>
and the original Lagrangian density as:
:<math>\mathcal{L} = \frac 12 \left(\partial_\mu\pi^k\right)\left(\partial^\mu\pi^k\right) + \frac 12 \left(\partial_\mu\sigma\right)\left(\partial^\mu\sigma\right) - \frac 12 \left(2\mu^2\right)\sigma^2 - \sqrt{\lambda}\mu\sigma^3 - \sqrt{\lambda}\mu\pi^k\pi^k\sigma - \frac{\lambda}{2} \pi^k\pi^k\sigma^2 - \frac{\lambda}{4}\left(\pi^k\pi^k\right)^2,</math>
where {{math|''k'' {{=}} 1, ..., ''N'' − 1}}. The original {{math|O(''N'')}} global symmetry is no longer manifest, leaving only the [[subgroup]] {{math|O(''N'' − 1)}}. The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken.{{r|peskin|page1=349–350}}

[[Goldstone's theorem]] states that under spontaneous symmetry breaking, every broken continuous global symmetry leads to a massless field called the Goldstone boson. In the above example, {{math|O(''N'')}} has {{math|''N''(''N'' − 1)/2}} continuous symmetries (the dimension of its [[Lie algebra]]), while {{math|O(''N'' − 1)}} has {{math|(''N'' − 1)(''N'' − 2)/2}}. The number of broken symmetries is their difference, {{math|''N'' − 1}}, which corresponds to the {{math|''N'' − 1}} massless fields {{math|''π<sup>k</sup>''}}.{{r|peskin|page1=351}}

On the other hand, when a gauge (as opposed to global) symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the amplitude for emission or absorption of a longitudinally polarized massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson.{{r|peskin|page1=743–744}}

In the QFT of [[ferromagnetism]], spontaneous symmetry breaking can explain the alignment of [[magnetic dipole]]s at low temperatures.{{r|zee|page1=199}} In the Standard Model of elementary particles, the [[W and Z bosons]], which would otherwise be massless as a result of gauge symmetry, acquire mass through spontaneous symmetry breaking of the [[Higgs boson]], a process called the [[Higgs mechanism]].{{r|peskin|page1=690}}

====Supersymmetry====
{{Main|Supersymmetry}}

All experimentally known symmetries in nature relate [[boson]]s to bosons and [[fermion]]s to fermions. Theorists have hypothesized the existence of a type of symmetry, called [[supersymmetry]], that relates bosons and fermions.{{r|peskin|zee|page1=795|page2=443}}

The Standard Model obeys [[Poincaré group|Poincaré symmetry]], whose generators are the spacetime [[translation (geometry)|translations]] {{math|''P<sup>μ</sup>''}} and the [[Lorentz transformations]] {{math|''J<sub>μν</sub>''}}.<ref name="WeinbergQFT">{{cite book |last=Weinberg |first=Steven |date=1995 |title=The Quantum Theory of Fields |publisher=Cambridge University Press |isbn=978-0-521-55001-7 |author-link=Steven Weinberg |url-access=registration |url=https://archive.org/details/quantumtheoryoff00stev }}</ref>{{rp|58–60}} In addition to these generators, supersymmetry in (3+1)-dimensions includes additional generators {{math|''Q<sub>α</sub>''}}, called [[supercharge]]s, which themselves transform as [[Weyl fermion]]s.{{r|peskin|zee|page1=795|page2=444}} The symmetry group generated by all these generators is known as the [[super-Poincaré group]]. In general there can be more than one set of supersymmetry generators, {{math|''Q<sub>α</sub><sup>I</sup>'', ''I'' {{=}} 1, ..., ''N''}}, which generate the corresponding {{math|''N'' {{=}} 1}} supersymmetry, {{math|''N'' {{=}} 2}} supersymmetry, and so on.{{r|peskin|zee|page1=795|page2=450}} Supersymmetry can also be constructed in other dimensions,<ref>{{cite arXiv |last1=de Wit |first1=Bernard |last2=Louis |first2=Jan |eprint=hep-th/9801132 |title=Supersymmetry and Dualities in various dimensions |date=1998-02-18 }}</ref> most notably in (1+1) dimensions for its application in [[superstring theory]].<ref>{{cite book |last=Polchinski |first=Joseph |date=2005 |title=String Theory |volume=2 |publisher=Cambridge University Press |isbn=978-0-521-67228-3 |author-link=Joseph Polchinski }}</ref>

The Lagrangian of a supersymmetric theory must be invariant under the action of the super-Poincaré group.{{r|zee|page1=448}} Examples of such theories include: [[Minimal Supersymmetric Standard Model]] (MSSM), [[N {{=}} 4 supersymmetric Yang–Mills theory|{{math|''N'' {{=}} 4}} supersymmetric Yang–Mills theory]],{{r|zee|page1=450}} and superstring theory. In a supersymmetric theory, every fermion has a bosonic [[superpartner]] and vice versa.{{r|zee|page1=444}}

If supersymmetry is promoted to a local symmetry, then the resultant gauge theory is an extension of general relativity called [[supergravity]].<ref name="NathArnowitt">{{cite journal | last1 = Nath | first1 = P. | last2 = Arnowitt | first2 = R. | year = 1975 | title = Generalized Super-Gauge Symmetry as a New Framework for Unified Gauge Theories | journal = Physics Letters B | volume = 56 | issue = 2| page = 177 | doi=10.1016/0370-2693(75)90297-x| bibcode = 1975PhLB...56..177N }}</ref>

Supersymmetry is a potential solution to many current problems in physics. For example, the [[hierarchy problem]] of the Standard Model—why the mass of the Higgs boson is not radiatively corrected (under renormalization) to a very high scale such as the [[Grand Unified Theory|grand unified scale]] or the [[Planck mass|Planck scale]]—can be resolved by relating the [[Higgs field]] and its super-partner, the [[Higgsino]]. Radiative corrections due to Higgs boson loops in Feynman diagrams are cancelled by corresponding Higgsino loops. Supersymmetry also offers answers to the grand unification of all gauge coupling constants in the Standard Model as well as the nature of [[dark matter]].{{r|peskin|page1=796–797}}<ref>{{Cite journal |last=Munoz |first=Carlos |arxiv=1701.05259 |title=Models of Supersymmetry for Dark Matter |journal=EPJ Web of Conferences |volume=136 |pages=01002 |date=2017-01-18 |bibcode=2017EPJWC.13601002M |doi=10.1051/epjconf/201713601002 |s2cid=55199323 }}</ref>

Nevertheless, experiments have yet to provide evidence for the existence of supersymmetric particles. If supersymmetry were a true symmetry of nature, then it must be a broken symmetry, and the energy of symmetry breaking must be higher than those achievable by present-day experiments.{{r|peskin|zee|page1=797|page2=443}}

====Other spacetimes====
The {{math|''ϕ''<sup>4</sup>}} theory, QED, QCD, as well as the whole Standard Model all assume a (3+1)-dimensional [[Minkowski space]] (3 spatial and 1 time dimensions) as the background on which the quantum fields are defined. However, QFT ''a priori'' imposes no restriction on the number of dimensions nor the geometry of spacetime.

In [[condensed matter physics]], QFT is used to describe [[two-dimensional electron gas|(2+1)-dimensional electron gases]].<ref>{{cite book |last1=Morandi |first1=G. |last2=Sodano |first2=P. |last3=Tagliacozzo |first3=A. |last4=Tognetti |first4=V. |date=2000 |title=Field Theories for Low-Dimensional Condensed Matter Systems |url=https://www.springer.com/us/book/9783540671770 |publisher=Springer |isbn=978-3-662-04273-1 }}</ref> In [[high-energy physics]], [[string theory]] is a type of (1+1)-dimensional QFT,{{r|zee|page1=452}}<ref name="polchinski1" /> while [[Kaluza–Klein theory]] uses gravity in [[extra dimensions]] to produce gauge theories in lower dimensions.{{r|zee|page1=428–429}}

In Minkowski space, the flat [[metric tensor (general relativity)|metric]] {{math|''η<sub>μν</sub>''}} is used to [[raising and lowering indices|raise and lower]] spacetime indices in the Lagrangian, ''e.g.''
:<math>A_\mu A^\mu = \eta_{\mu\nu} A^\mu A^\nu,\quad \partial_\mu\phi \partial^\mu\phi = \eta^{\mu\nu}\partial_\mu\phi \partial_\nu\phi,</math>
where {{math|''η<sup>μν</sup>''}} is the inverse of {{math|''η<sub>μν</sub>''}} satisfying {{math|''η<sup>μρ</sup>η<sub>ρν</sub>'' {{=}} ''δ<sup>μ</sup><sub>ν</sub>''}}. 
For [[quantum field theory in curved spacetime|QFTs in curved spacetime]] on the other hand, a general metric (such as the [[Schwarzschild metric]] describing a [[black hole]]) is used:
:<math>A_\mu A^\mu = g_{\mu\nu} A^\mu A^\nu,\quad \partial_\mu\phi \partial^\mu\phi = g^{\mu\nu}\partial_\mu\phi \partial_\nu\phi,</math>
where {{math|''g<sup>μν</sup>''}} is the inverse of {{math|''g<sub>μν</sub>''}}. 
For a real scalar field, the Lagrangian density in a general spacetime background is
:<math>\mathcal{L} = \sqrt{|g|}\left(\frac 12 g^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi - \frac 12 m^2\phi^2\right),</math>
where {{math|''g'' {{=}} det(''g<sub>μν</sub>'')}}, and {{math|∇<sub>''μ''</sub>}} denotes the [[covariant derivative]].<ref>{{cite book |last1=Parker |first1=Leonard E. |last2=Toms |first2=David J. |date=2009 |title=Quantum Field Theory in Curved Spacetime |url=https://archive.org/details/quantumfieldtheo00park |url-access=limited |publisher=Cambridge University Press |page=[https://archive.org/details/quantumfieldtheo00park/page/n58 43] |isbn=978-0-521-87787-9 }}</ref> The Lagrangian of a QFT, hence its calculational results and physical predictions, depends on the geometry of the spacetime background.

====Topological quantum field theory====
{{Main|Topological quantum field theory}}

The correlation functions and physical predictions of a QFT depend on the spacetime metric {{math|''g<sub>μν</sub>''}}. For a special class of QFTs called [[topological quantum field theories]] (TQFTs), all correlation functions are independent of continuous changes in the spacetime metric.<ref>{{cite arXiv |last1=Ivancevic |first1=Vladimir G. |last2=Ivancevic |first2=Tijana T. |eprint=0810.0344v5 |title=Undergraduate Lecture Notes in Topological Quantum Field Theory |class=math-th |date=2008-12-11 }}</ref>{{rp|36}} QFTs in curved spacetime generally change according to the ''geometry'' (local structure) of the spacetime background, while TQFTs are invariant under spacetime [[diffeomorphism]]s but are sensitive to the ''[[topology]]'' (global structure) of spacetime. This means that all calculational results of TQFTs are [[topological invariant]]s of the underlying spacetime. [[Chern–Simons theory]] is an example of TQFT and has been used to construct models of quantum gravity.<ref>{{cite book |last=Carlip |first=Steven |author-link=Steve Carlip |date=1998 |title=Quantum Gravity in 2+1 Dimensions |url=https://www.cambridge.org/core/books/quantum-gravity-in-21-dimensions/D2F727B6822014270F423D82501E674A |publisher=Cambridge University Press |pages=27–29 |isbn=9780511564192 |doi=10.1017/CBO9780511564192 |arxiv=2312.12596 }}</ref> Applications of TQFT include the [[fractional quantum Hall effect]] and [[topological quantum computer]]s.<ref>{{cite journal |last1=Carqueville |first1=Nils |last2=Runkel |first2=Ingo |arxiv=1705.05734 |title=Introductory lectures on topological quantum field theory |journal=Banach Center Publications |year=2018 |volume=114 |pages=9–47 |doi=10.4064/bc114-1 |s2cid=119166976 }}</ref>{{rp|1–5}} The world line trajectory of fractionalized particles (known as [[anyons]]) can form a link configuration in the spacetime,<ref>{{Cite journal |author-link=Edward Witten |first=Edward |last=Witten |title=Quantum Field Theory and the Jones Polynomial |journal=[[Communications in Mathematical Physics]] |volume=121 |issue=3 |pages=351–399 |year=1989 |mr=0990772 |bibcode = 1989CMaPh.121..351W |doi = 10.1007/BF01217730 |s2cid=14951363 |url=http://projecteuclid.org/euclid.cmp/1104178138 }}</ref> which relates the braiding statistics of anyons in physics to the
link invariants in mathematics. Topological quantum field theories (TQFTs) applicable to the frontier research of topological quantum matters include Chern-Simons-Witten gauge theories in 2+1 spacetime dimensions, other new exotic TQFTs in 3+1 spacetime dimensions and beyond.<ref>{{Cite journal |first1=Pavel|last1=Putrov |first2=Juven |last2=Wang | first3=Shing-Tung | last3=Yau    |title=Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions |journal=[[Annals of Physics]] |volume=384 |issue=C |pages=254–287 |year=2017|doi =10.1016/j.aop.2017.06.019|arxiv=1612.09298 |bibcode=2017AnPhy.384..254P |s2cid=119578849 }}</ref>

===Perturbative and non-perturbative methods===
Using [[perturbation theory (quantum mechanics)|perturbation theory]], the total effect of a small interaction term can be approximated order by order by a series expansion in the number of [[virtual particle]]s participating in the interaction. Every term in the expansion may be understood as one possible way for (physical) particles to interact with each other via virtual particles, expressed visually using a [[Feynman diagram]]. The [[electromagnetic force]] between two electrons in QED is represented (to first order in perturbation theory) by the propagation of a virtual photon. In a similar manner, the [[W and Z bosons]] carry the weak interaction, while [[gluon]]s carry the strong interaction. The interpretation of an interaction as a sum of intermediate states involving the exchange of various virtual particles only makes sense in the framework of perturbation theory. In contrast, non-perturbative methods in QFT treat the interacting Lagrangian as a whole without any series expansion. Instead of particles that carry interactions, these methods have spawned such concepts as [['t Hooft–Polyakov monopole]], [[domain wall]], [[flux tube]], and [[instanton]].<ref name="shifman">{{cite book |last=Shifman |first=M. |author-link=Mikhail Shifman |date=2012 |title=Advanced Topics in Quantum Field Theory |publisher=Cambridge University Press |isbn=978-0-521-19084-8 }}</ref> Examples of QFTs that are completely solvable non-perturbatively include [[Minimal model (physics)|minimal models]] of [[conformal field theory]]<ref>{{cite book |last1=Di Francesco |first1=Philippe |last2=Mathieu |first2=Pierre |last3=Sénéchal |first3=David |date=1997 |title=Conformal Field Theory |publisher=Springer |isbn=978-1-4612-7475-9 |url=https://books.google.com/books?id=5u7jBwAAQBAJ }}</ref> and the [[Thirring model]].<ref>{{Cite journal |last=Thirring |first=W. |author-link=Walter Thirring |year=1958 |title=A Soluble Relativistic Field Theory? |journal=[[Annals of Physics]] |volume=3 |issue=1|pages=91–112 |bibcode=1958AnPhy...3...91T |doi=10.1016/0003-4916(58)90015-0}}</ref>