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Scalar electrodynamics
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<h2 id="matter-content-and-lagrangian">Matter content and Lagrangian</h2> <h3>Matter content</h3> <p>The model consists of a complex scalar field ϕ ( x ) {\displaystyle \phi (x)} minimally coupled to a gauge field A μ ( x ) {\displaystyle A_{\mu }(x)} . </p><p>This article discusses the theory on flat spacetime R 1 , 3 {\displaystyle \mathbb {R} ^{1,3}} (<a href="/facts/Minkowski_space/vDkfkQJn">Minkowski space</a>) so these fields can be treated (naïvely) as functions ϕ : R 1 , 3 → C {\displaystyle \phi :\mathbb {R} ^{1,3}\rightarrow \mathbb {C} } , and A μ : R 1 , 3 → ( R 1 , 3 ) ∗ {\displaystyle A_{\mu }:\mathbb {R} ^{1,3}\rightarrow (\mathbb {R} ^{1,3})^{*}} . The theory can also be defined for curved spacetime but these definitions must be replaced with a more subtle one. The gauge field is also known as a <a href="/facts/Principal_connection/UchSqJ0C">principal connection</a>, specifically a principal U ( 1 ) {\displaystyle {\text{U}}(1)} connection. </p> <h3>Lagrangian</h3> <p>The dynamics is given by the Lagrangian density </p><p> L = ( D μ ϕ ) ∗ D μ ϕ − V ( ϕ ∗ ϕ ) − 1 4 F μ ν F μ ν = ( ∂ μ ϕ ) ∗ ( ∂ μ ϕ ) − i e ( ( ∂ μ ϕ ) ∗ ϕ − ϕ ∗ ( ∂ μ ϕ ) ) A μ + e 2 A μ A μ ϕ ∗ ϕ − V ( ϕ ∗ ϕ ) − 1 4 F μ ν F μ ν {\displaystyle {\begin{array}{lcl}{\mathcal {L}}&=&(D_{\mu }\phi )^{*}D^{\mu }\phi -V(\phi ^{*}\phi )-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\\&=&(\partial _{\mu }\phi )^{*}(\partial ^{\mu }\phi )-ie((\partial _{\mu }\phi )^{*}\phi -\phi ^{*}(\partial _{\mu }\phi ))A^{\mu }+e^{2}A_{\mu }A^{\mu }\phi ^{*}\phi -V(\phi ^{*}\phi )-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\end{array}}} </p><p>where </p> <ul><li> F μ ν = ( ∂ μ A ν − ∂ ν A μ ) {\displaystyle F_{\mu \nu }=(\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu })} is the electromagnetic field strength, or <a href="/facts/Curvature_form/LWp7DMDM">curvature</a> of the connection.</li> <li> D μ ϕ = ( ∂ μ ϕ − i e A μ ϕ ) {\displaystyle D_{\mu }\phi =(\partial _{\mu }\phi -ieA_{\mu }\phi )} is the covariant derivative of the field ϕ {\displaystyle \phi } </li> <li> e = − | e | < 0 {\displaystyle e=-|e|<0} is the electric charge</li> <li> V ( ϕ ∗ ϕ ) {\displaystyle V(\phi ^{*}\phi )} is the potential for the complex scalar field.</li></ul> <h3>Gauge-invariance</h3> <p>This model is invariant under gauge transformations parameterized by λ ( x ) {\displaystyle \lambda (x)} . This is a real-valued function λ : R 1 , 3 → R . {\displaystyle \lambda :\mathbb {R} ^{1,3}\rightarrow \mathbb {R} .} </p><p> ϕ ′ ( x ) = e i e λ ( x ) ϕ ( x ) and A μ ′ ( x ) = A μ ( x ) + ∂ μ λ ( x ) . {\displaystyle \phi '(x)=e^{ie\lambda (x)}\phi (x)\quad {\textrm {and}}\quad A_{\mu }'(x)=A_{\mu }(x)+\partial _{\mu }\lambda (x).} </p> <h4>Differential-geometric view</h4> <p>From the geometric viewpoint, λ {\displaystyle \lambda } is an infinitesimal change of trivialization, which generates the finite change of trivialization e i e λ : R 1 , 3 → U ( 1 ) . {\displaystyle e^{ie\lambda }:\mathbb {R} ^{1,3}\rightarrow {\text{U}}(1).} In physics, it is customary to work under an implicit choice of trivialization, hence a gauge transformation really can be viewed as a change of trivialization. </p> <h2 id="higgs-mechanism">Higgs mechanism</h2> <p>If the potential is such that its minimum occurs at non-zero value of | ϕ | {\displaystyle |\phi |} , this model exhibits the <a href="/facts/Higgs_mechanism/Vs5Cbxyy">Higgs mechanism</a>. This can be seen by studying fluctuations about the lowest energy configuration: one sees that the gauge field behaves as a massive field with its mass proportional to e {\displaystyle e} times the minimum value of | ϕ | {\displaystyle |\phi |} . As shown in 1973 by Nielsen and Olesen, this model, in 2 + 1 {\displaystyle 2+1} dimensions, admits time-independent finite energy configurations corresponding to vortices carrying magnetic flux. The magnetic flux carried by these vortices are quantized (in units of 2 π e {\displaystyle {\tfrac {2\pi }{e}}} ) and appears as a topological charge associated with the topological current </p><p> J t o p μ = ϵ μ ν ρ F ν ρ . {\displaystyle J_{top}^{\mu }=\epsilon ^{\mu \nu \rho }F_{\nu \rho }\ .} </p><p>These vortices are similar to the vortices appearing in type-II superconductors. This analogy was used by Nielsen and Olesen in obtaining their solutions. </p> <h3>Example</h3> <p>A simple choice of potential for demonstrating the Higgs mechanism is </p> V ( | ϕ | 2 ) = λ ( | ϕ | 2 − Φ 2 ) 2 . {\displaystyle V(|\phi |^{2})=\lambda (|\phi |^{2}-\Phi ^{2})^{2}.} <p>The potential is minimized at | ϕ | = Φ {\displaystyle |\phi |=\Phi } , which is chosen to be greater than zero. This produces a circle of minima, with values Φ e i θ {\displaystyle \Phi e^{i\theta }} , for θ {\displaystyle \theta } a real number. </p> <h2 id="scalar-chromodynamics">Scalar chromodynamics</h2> <p class="note">See also: <a href="/facts/Scalar_chromodynamics/nHxBM8xq">scalar chromodynamics</a></p> <p>This theory can be generalized from a theory with U ( 1 ) {\displaystyle U(1)} gauge symmetry containing a scalar field ϕ {\displaystyle \phi } valued in C {\displaystyle \mathbb {C} } coupled to a gauge field A μ {\displaystyle A_{\mu }} to a theory with gauge symmetry under the gauge group G {\displaystyle G} , a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a>. </p><p>The scalar field ϕ {\displaystyle \phi } is valued in a representation space of the gauge group G {\displaystyle G} , making it a <a href="/facts/Vector_space/M1pxMLx2">vector</a>; the label of "scalar" field refers only to the transformation of ϕ {\displaystyle \phi } under the action of the <a href="/facts/Lorentz_group/t1APc2Ye">Lorentz group</a>, so it is still referred to as a scalar field, in the sense of a <a href="/facts/Lorentz_scalar/USPNCG1l">Lorentz scalar</a>. The gauge-field is a g {\displaystyle {\mathfrak {g}}} -valued 1-form, where g {\displaystyle {\mathfrak {g}}} is the <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> of G. </p> <ul><li>H. B. Nielsen and P. Olesen (1973). "Vortex-line models for dual strings". <i>Nuclear Physics B</i>. 61: 45–61. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1973NuPhB..61...45N">1973NuPhB..61...45N</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0550-3213%2873%2990350-7">10.1016/0550-3213(73)90350-7</a>.</li></ul> <ul><li>Peskin, M and Schroeder, D.; <i>An Introduction to Quantum Field Theory</i> (Westview Press, 1995) <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-201-50397-2</li></ul>