Editing Quantum field theory (section)

===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}}